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+A Stochastic ADMM Algorithm for Large-Scale Ptychography with Weighted
+Difference of Anisotropic and Isotropic Total Variation∗
+Kevin Bui† and Zichao (Wendy) Di ‡
+Abstract. Ptychography is an imaging technique that has various scientific applications, ranging from biology to
+optics. The method scans the object of interest in a series of overlapping positions, thereby generating
+a set of multiple Fourier magnitude measurements that are potentially corrupted by noise. From
+these measurements, an image of the object can be reconstructed depending on how the related
+inverse problem is formulated and solved. In this paper, we propose a class of variational models
+that incorporate the weighted anisotropic–isotropic total variation (AITV), an effective regularizer
+for image recovery. This class of models is applicable to measurements corrupted by either Gaussian
+or Poisson noise. In order to have the models applicable for large number of ptychographic scans,
+we design an efficient stochastic alternating direction method of multipliers algorithm and establish
+its convergence.
+Numerical experiments demonstrate that from a large set of highly corrupted
+Fourier measurements, the proposed stochastic algorithm with AITV regularization can reconstruct
+complex-valued images with satisfactory quality, especially for the phase components.
+AMS subject classifications. 65F22, 65K10, 68U10, 90C06, 90C15, 90C26
+Key words. phase retrieval, total variation, ptychography, ADMM, Poisson/Gaussian noise, nonconvex opti-
+mization, stochastic optimization
+1. Introduction. Ptychography is a popular imaging technique that combines both co-
+herent diffractive imaging and scanning transmission microscopy. It has been used in various
+industrial and scientific applications, including biology [34, 45, 58], crystallography [14], and
+optics [44, 48]. To perform a ptychographic experiment (see Figure 1), a coherent beam is
+scanned across the object of interest, where each scan may have overlapping positions with
+another. The scanning procedure provides a set of phaseless measurements that can be used
+to reconstruct an image of the object of interest.
+We describe the 2D ptychography in the discrete setting. Let z ∈ Cn2 be the object of
+interest with n × n pixels and ω ∈ Cm2 be the localized 2D probe with m × m pixels, where
+m < n. Both the object z and the probe ω are expressed as vectors in lexiographical order.
+We denote the set of N masks by {Sj}N
+j=1, where each Sj ∈ Rm2×n2 is a binary matrix that
+represents a (m×m)-size window over the image z. The set of phaseless measurements {dj}N
+j=1
+is obtained by dj = |F(Pjz)|2 = |F(ω◦Sjz)|2, where Pj := ω◦Sj is the jth probe, F ∈ Cm2×m2
+is the 2D discrete Fourier operator, the operation ◦ is elementwise multiplication, and the
+operation | · | is the elementwise absolute value of a vector. We aim to solve the following
+ptychographic phase retrieval problems. When the probe is unknown, the blind ptychographic
+∗Submitted to the editors DATE.
+Funding: This material is based upon work supported by the U.S. Department of Energy, Office of Science,
+under contract number DE-AC02-06CH11357.
+†Department of Mathematics, University of California at Irvine, Irvine, CA 92697 USA (kevinb3@uci.edu).
+‡Mathematics and Computer Science Division, Argonne National Laboratory, Lemont, IL (wendydi@anl.gov).
+1
+arXiv:2301.02386v1  [math.NA]  6 Jan 2023
+
+2
+KEVIN BUI AND ZICHAO (WENDY) DI
+Probe
+Beam
+Object
+Measurements
+Figure 1: Schematic of a ptychography experiment.
+phase retrieval problem is expressed by
+BP-PR:
+To find ω ∈ Cm2 and z ∈ Cn2 such that |F(ω ◦ Sjz)|2 = dj, j = 1, . . . , N.
+(1.1)
+When the probe ω is known, (1.1) reduces to the non-blind case where we only find z ∈ Cn2.
+Multiple algorithms have been developed to solve the non-blind and blind phase retrieval
+problems. One of the most popular methods is the ptychographical iterative engine (PIE) [42],
+where later refinements led to ePIE [33] and rPIE [32]. The PIE methods are based on gradient
+descent applied to each measurement sequentially. Other gradient-based methods for phase
+retrieval include Wirtinger flow [6] and its variants [13, 56, 57], which use careful initialization
+by a spectral method and adaptive step sizes. PIE is also one class of projection algorithms
+for phase retrieval.
+Other projection-based algorithms are hybrid projection-reflection [1],
+Douglas-Rachford splitting [39, 46], and the relaxed averaged alternating reflections [31]. The
+phase retrieval problem can be formulated as a semidefinite optimization problem. For exam-
+ple, PhaseLift [7] solves the phase retrieval problem as a trace (nuclear) norm minimization
+problem. A nonconvex variant called PhaseLiftOff subtracts the trace norm by the Frobenius
+norm in the objective function [55]. PhaseCut proposes a different semidefinite formulation
+of the phase retrieval problem by explicitly separating the amplitude and phase variables and
+optimize only the values of the phase variables [47]. The phase retrieval problem can alter-
+natively be written as a saddle point problem [51], solved by alternating direction method
+of multipliers (ADMM) [4]. A globally convergent ADMM algorithm has recently been de-
+veloped to solve the BP-PR problem [8]. Another globally convergent algorithm is proximal
+alternating linearized minimization (PALM) [2], which has also been adapted to solve the
+BP-PR problem [12, 21]. For a detailed survey of numerical algorithms for phase retrieval,
+please refer to [17].
+For large-scale ptychography, when a huge number of scans are collected, many of the
+aforementioned algorithms may be inapplicable or may need to be adapted because of the
+
+STOCHASTIC ADMM FOR PTYCHOGRAPHY
+3
+demanding memory footprint and computational cost. Various parallel algorithms have been
+developed. For example, an asynchronous parallel version of ePIE has been implemented on
+GPUs, where each partition of a measurement set is asynchronously processed to obtain a sub-
+image and the sub-images are later fused together to form the entire image [35]. A parallel
+version of relaxed averaged alternating reflections has recently been developed for GPU impl-
+mentation [15]. Unfortunately, some of these parallel algorithms require a GPU, which many
+computers do not have. However, there are efficient algorithms for large-scale ptychography
+without the need for a GPU, although having one could speed up the processing time. A
+multigrid optimization framework has been proposed to accelerate large-scale gradient-based
+methods for phase retrieval [52]. An overlapping domain decomposition method combined
+with ADMM leads to a highly parallel algorithm with good load balance [9]. To the best
+of our knowledge, there does not yet exist a stochastic optimization algorithm for large-scale
+ptychography that iteratively processes a batch of measurements. Such an algorithm would be
+useful for practitioners who do not have access to multiple cores to perform parallel computing.
+To improve the image reconstruction quality in phase retrieval, total variation (TV) [43]
+has been incorporated for the cases when the measurements are corrupted with Gaussian noise
+[11] or with Poisson noise [10]. Both cases consider the isotropic TV approximation:
+∥∇z∥2,1 =
+n2
+�
+i=1
+�
+|(∇xz)i|2 + |(∇yz)i|2,
+(1.2)
+where ∇x and ∇y are the horizontal and vertical difference operators, respectively, and (∇xz)i
+and (∇yz)i are the ith entries of the vectors ∇xz and ∇yz, respectively. However, it has been
+known that isotropic TV tends to blur oblique edges.
+An alternative approximation that
+preserves sharper edges is the anisotropic TV [16]:
+∥∇z∥1 =
+n2
+�
+i=1
+(|(∇xz)i| + |(∇yz)i|) .
+(1.3)
+Overall, TV is meant to approximate the ℓ0 “norm” of the image gradient, i.e., ∥∇z∥0, be-
+cause TV is based on the ℓ1 norm, a convex relaxation of ℓ0. A nonconvex alternative to ℓ1 is
+ℓ1 − αℓ2, 0 < α ≤ 1, which performs well in recovering sparse solutions in various compressed
+sensing problems [27, 28, 29, 54]. The superior performance of ℓ1 − αℓ2 in sparse recovery has
+motivated the development of the weighted difference of anisotropic and isotropic total varia-
+tion (AITV) [30], which applies ℓ1 −αℓ2 on each gradient vector of an image. Mathematically,
+AITV is formulated by
+∥∇z∥1 − α∥∇z∥2,1 =
+n2
+�
+i=1
+�
+|(∇xz)i| + |(∇yz)i| − α
+�
+|(∇xz)i|2 + |(∇yz)i|2
+�
+.
+(1.4)
+AITV has demonstrated better performance than TV in image denoising, image deconvolution,
+image segmentation, and MRI reconstruction [5, 30, 38], especially in preserving sharper edges.
+In this work, we consider the large-scale ptychography problem where the measurements
+are corrupted by either Gaussian or Poisson noise.
+To improve the image reconstruction
+
+4
+KEVIN BUI AND ZICHAO (WENDY) DI
+quality, the AITV regularization is incorporated. The overall problem is formulated as a gen-
+eral variational problem, where we develop an ADMM algorithm to solve it. The ADMM
+algorithm has subproblems that can be approximately solved by stochastic gradient descent
+(SGD) [40]. Although SGD is a generic and popular algorithm for unconstrained optimization
+problems whose objective functions have a finite-sum structure, it may not be directly appli-
+cable to the subproblem being solved in the case of ptychography. Hence, we show how to
+appropriately apply SGD in order to develop our specialized stochastic ADMM algorithm. To
+further modify the algorithm, we incorporate adaptive step size based on the PIE algorithms
+[32, 33, 42]. Instead of using all measurements per iteration, this stochastic ADMM algorithm
+can iteratively process a batch of measurements to accurately perform image reconstruction.
+The paper is organized as follows. In Section 2, we review notations and definitions that
+will be used throughout the paper.
+Next in Section 3 we describe the AITV-regularized
+variational models to solve the image ptychography problem. Within this section, we design
+the stochastic ADMM algorithms to solve these models. Convergence analysis of the stochatsic
+ADMM algorithm follows in Section 4. In Section 5, we illustrate the performance of our
+proposed stochastic ADMM algorithms and compare them with other competing algorithms.
+Lastly, in Section 6, we conclude the paper with summary and future works.
+2. Preliminaries. In this section, we describe basic notations used throughout the paper.
+Let z ∈ Cn2. The ith entry of z is denoted by (z)i. The vector 1 is a vector whose entries
+are all ones. The vector 0 is also defined similarly. The real transpose and conjugate transpose
+of z are denoted by z⊤ and z∗, respectively. The same superscript notations are used for the
+real transpose and conjugate transpose of matrices. The sign of a complex value z′ ∈ C is
+given by
+sgn(z′) =
+�
+�
+�
+z′
+|z′|
+if z′ ̸= 0,
+c ∈ {c′ ∈ C : |c′| = 1}
+if z′ = 0.
+The sign of a vector z ∈ Cn2 is denoted by sgn(z) and is defined elementwise by sgn(z)i =
+sgn(zi), i = 1, . . . , n2. The standard basis vectors of Cn2 are denoted by {ei}n2
+i=1, where ei is
+a vector whose ith component is 1 while all other components are zeros. The diagonal matrix
+of a vector z ∈ Cn2 is denoted by Dz = Diag(z) = z1⊤ ◦ In2×n2.
+For p = (px, py) ∈ Cn2 × Cn2, its ith entry is pi =
+�(px)i
+(py)i
+�
+∈ C2. We define the following
+norms on Cn2 × Cn2:
+∥p∥1 =
+n2
+�
+i=1
+|(px)i| + |(py)i|,
+∥p∥2 =
+�
+�
+�
+�
+n2
+�
+i=1
+|(px)i|2 + |(py)i|2,
+∥p∥2,1 =
+n2
+�
+i=1
+�
+|(px)i|2 + |(py)i|2 =
+n2
+�
+i=1
+∥pi∥2.
+The discrete gradient operator ∇ : Cn2 → Cn2 × Cn2 when specifically applied to the image
+
+STOCHASTIC ADMM FOR PTYCHOGRAPHY
+5
+z is given by ∇z = (∇xz, ∇yz), where ∇x and ∇y are the forward horizontal and vertical
+difference operators.
+We also define the proximal operator of a function f : Cn2 → R ∪ {+∞} as
+proxf(·)(z′) = arg min
+z
+f(z) + 1
+2∥z − z′∥2
+2, ∀z′ ∈ Cn2.
+3. Proposed Model. Throughout the paper, we assume that among the mask set {Sj}N
+j=1,
+there exists j′ such that ∥Sj′ei∥1 = 1 for each i = 1, . . . , n2. This assumption ensures that
+each pixel of an image z ∈ Cn2 is sampled at least once.
+Suppose that the measurements {dj}N
+j=1 are corrupted by independent and identically
+distributed (iid) noise, i.e., dj
+iid
+∼ Noise(|F(ω ◦ Sjz)|2) for j = 1, . . . , N. We assume that the
+noise is either Gaussian or Poisson, both of which are common in phase retrieval. Given an
+unknown probe ω, the blind variational model [8] is
+min
+ω∈Cm2,z∈Cn2
+N
+�
+j=1
+B(|F(ω ◦ Sjz)|2, dj),
+(3.1)
+where
+B(g, f) =
+�
+�
+�
+�
+�
+�
+�
+�
+�
+1
+2∥√g −
+�
+f∥2
+2,
+amplitude Gaussian metric (AGM) [51],
+1
+2⟨g − f ◦ log(g), 1⟩,
+intensity Poisson metric (IPM) [10].
+(3.2)
+Note that √· is elementwise square root. When the probe ω is known, (3.1) simplifies to the
+non-blind case as a special case where we only need to find z ∈ Cn2. Hence, throughout the
+rest of this section, we will focus on the blind case. To improve image recovery, we propose a
+class of AITV-regularized variants of (3.1).
+3.1. AITV model. For image ptychography, we propose the following AITV-regularized
+model:
+min
+ω∈Cm2,z∈Cn2
+N
+�
+j=1
+B(|F(ω ◦ Sjz)|2, dj) + λ (∥∇z∥1 − α∥∇z∥2,1) , λ > 0, α ∈ [0, 1].
+(3.3)
+To develop an ADMM algorithm of (3.3), we introduce auxiliary variables u = (u1, . . . , uN) ∈
+Cm2×N and v = (vx, vy) ∈ Cn2 ×Cn2 so that we obtain an equivalent constrained optimization
+problem
+min
+u,v,z
+N
+�
+j=1
+B(|uj|2, dj) + λ (∥v∥1 − α∥v∥2,1) s.t.
+uj = F(ω ◦ Sjz), j = 1, . . . , N, and v = ∇z.
+(3.4)
+
+6
+KEVIN BUI AND ZICHAO (WENDY) DI
+The augmented Lagrangian of (3.4) is
+L(u, ω, v, z, Λ, y) =
+N
+�
+j=1
+�
+B(|uj|2, dj) + R (⟨Λj, uj − F(ω ◦ Sjz)⟩) + β1
+2 ∥uj − F(ω ◦ Sjz)∥2
+2
+�
++ λ (∥v∥1 − α∥v∥2,1) + R (⟨y, v − ∇z⟩) + β2
+2 ∥v − ∇z∥2
+2,
+(3.5)
+where R(·) denotes the real component of a complex number; ⟨·, ·⟩ denotes the complex inner
+product between two vectors; Λ = (Λ1, . . . , ΛN) ∈ Cm2×N and y = (yx, yy) ∈ Cn2×n2 are
+Lagrange multipliers; and β1, β2 > 0 are penalty parameters. The ADMM algorithm iterates
+as follows:
+ut+1 ∈ arg min
+u
+L(u, ωt, vt, zt, Λt, yt),
+(3.6a)
+ωt+1 ∈ arg min
+ω
+L(ut+1, ω, vt, zt, Λt, yt),
+(3.6b)
+vt+1 ∈ arg min
+v
+L(ut+1, ωt+1, v, zt, Λt, yt),
+(3.6c)
+zt+1 ∈ arg min
+z
+L(ut+1, ωt+1, vt+1, z, Λt, yt),
+(3.6d)
+Λt+1
+j
+= Λt
+j + β1
+�
+ut+1
+j
+− F(ωt+1 ◦ Sjzt+1)
+�
+,
+j = 1, . . . , N,
+(3.6e)
+yt+1 = yt + β2
+�
+vt+1 − ∇zt+1�
+.
+(3.6f)
+Next we explain how to solve each subproblem.
+3.1.1. u-subproblem. In (3.6a), we solve uj independently of each other. For each j =
+1, . . . , N, we have
+ut+1
+j
+∈ arg min
+uj
+B(|uj|2, dj) + R
+�
+⟨Λt
+j, uj − F(P t
+j zt)⟩
+�
++ β1
+2 ∥uj − F(P t
+j zt)∥2
+2
+= arg min
+uj
+1
+β1
+B(|uj|2, dj) + 1
+2
+����uj − F(P t
+j zt) + 1
+β1
+Λt
+j
+����
+2
+2
+= prox 1
+β1 B(|·|2,dj)
+�
+F(P t
+j zt) − 1
+β1
+Λt
+j
+�
+,
+(3.7)
+where P t
+j = ωt ◦ Sj. The proximal operator for each fidelity term in (3.2) has a closed-form
+solution provided in [10, 11], so we have
+ut+1
+j
+=
+�
+�
+�
+�
+�
+�
+�
+√
+dj+β1
+���F(P t
+j zt)− 1
+β1 Λt
+j
+���
+1+β1
+◦ sgn
+�
+F(P t
+j zt) − 1
+β1 Λt
+j
+�
+,
+AGM,
+β1|F(P t
+j zt)− 1
+β1 Λt
+j|+
+��
+β1|F(P t
+j zt)− 1
+β1 Λt
+j|
+�2
++4(1+β1)dj
+2(1+β1)
+◦ sgn
+�
+F(P t
+j zt) − 1
+β1 Λt
+j
+�
+,
+IPM.
+(3.8)
+
+STOCHASTIC ADMM FOR PTYCHOGRAPHY
+7
+3.1.2. ω-subproblem. The ω-subproblem (3.6b) can be rewritten as
+ωt+1 ∈ arg min
+ω
+N
+�
+j=1
+�
+�β1
+2
+�����F−1
+�
+ut+1
+j
++
+Λt
+j
+β1
+�
+− ω ◦ Sjzt
+�����
+2
+2
+�
+� ,
+(3.9)
+which shows that updating ω requires access to all N probes. Hence, we develop an alternative
+update scheme that uses only b ≤ N probes. Instead of solving (3.6b) exactly, we linearize it
+as done in [26, 37] to obtain
+ωt+1 ∈ arg min
+ω
+⟨∇ωL(ut+1, ωt, zt, Λt, yt), ω − ωt⟩ +
+1
+2δtω
+∥ω − ωt∥2
+2
+(3.10)
+for some constant δt
+ω > 0 at iteration t. Then (3.10) is equivalent to performing gradient
+descent with step size δt
+ω:
+ωt+1 = ωt − δt
+ω∇ωL(ut+1, ωt, vt, zt, Λt, yt).
+(3.11)
+Next we approximate ∇Lω by its stochastic estimator ˜∇ωL. thereby updating ωt+1 by SGD
+with step size δt
+ω > 0:
+ωt+1 = ωt − δt
+ω ˜∇ωL(ut+1, ωt, vt, zt, Λt, yt).
+(3.12)
+We derive some candidates for ˜∇ωL. Let Gt
+j(ω) = β1
+2
+���F−1 �
+ut+1
+j
++
+Λt
+j
+β1
+�
+− ω ◦ Sjzt���
+2
+2, which
+means that ∇Gt
+j(ω) = −β1(Sjzt)∗ ◦
+�
+F−1 �
+ut+1
+j
++
+Λt
+j
+β1
+�
+− ω ◦ Sjzt�
+. (3.11) can be rewritten as
+a gradient descent step with Nδt
+ω:
+ωt+1 = ωt − Nδt
+ω
+�
+� 1
+N
+N
+�
+j=1
+∇Gt
+j(ωt)
+�
+� .
+(3.13)
+The SGD estimator [3, 40] of 1
+N
+�N
+j=1 ∇Gt
+j(ωt) is 1
+b
+�
+j∈nt ∇Gt
+j(ωt), where nt ⊂ {1, . . . , N} is
+a sub-batch of N masks such that |nt| = b. Then the SGD variant (after scaling δt
+ω) of (3.11)
+is
+ωt+1 = ωt − δt
+ω
+�
+�1
+b
+�
+j∈nt
+∇Gt
+j(ωt)
+�
+� .
+(3.14)
+This implies that one candidate stochastic estimator for ∇ωL is
+˜∇SGD
+ω
+L(ut+1, ωt, vt, zt, Λt, yt) = 1
+b
+�
+j∈nt
+∇Gt
+j(ωt)
+= −β1
+b
+�
+j∈nt
+(Sjzt)∗ ◦
+�
+F−1
+�
+ut+1
+j
++
+Λt
+j
+β1
+�
+− ωt ◦ Sjzt
+�
+.
+(3.15)
+
+8
+KEVIN BUI AND ZICHAO (WENDY) DI
+We can further modify (3.15) by incorporating spatially varying step sizes inspired by the PIE
+algorithms [32, 33, 42]. Let
+Φt
+j =
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+1
+∥Sjzt∥2∞
+,
+ePIE [33],
+∥Sjzt∥11
+∥Sjzt∥∞ (|Sjzt|2 + γω∥Sjzt∥2∞1),
+PIE [42],
+1
+(1 − γω)|Sjzt|2 + γω∥Sjzt∥2∞1,
+rPIE [32],
+(3.16)
+where γω ∈ [0, 1] for PIE and rPIE and division is elementwise. Incorporating Φt
+j into (3.15),
+we have another class of stochastic estimators
+˜∇PIE
+ω
+L(ut+1, ωt, vt, zt, Λt, yt) = −β1
+b
+�
+j∈nt
+Φt
+j ◦ (Sjzt)∗ ◦
+�
+F−1
+�
+ut+1
+j
++
+Λt
+j
+β1
+�
+− ωt ◦ Sjzt
+�
+.
+(3.17)
+3.1.3. v-subproblem. Expanding (3.6c) gives
+vt+1 ∈ arg min
+v
+λ
+β2
+(∥v∥1 − α∥v∥2,1) + 1
+2
+����v − ∇zt + yt
+β2
+����
+2
+2
+= arg min
+v
+n2
+�
+i=1
+λ
+β2
+(∥vi∥1 − α∥vi∥2) + 1
+2
+����vi − (∇zt)i + (yt)i
+β2
+����
+2
+2
+,
+(3.18)
+which means that the solution vt+1 can be solved elementwise. As a result, the subproblem
+simplifies to
+(vt+1)i = prox λ
+β2 (∥·∥1−α∥·∥2)
+�
+(∇zt)i − (yt)i
+β2
+�
+.
+(3.19)
+A closed-form solution for the proximal operator of ℓ1 − αℓ2 is provided in [27] but only for
+real-valued vectors. We generalize it to the complex case in Lemma 3.1, whose proof is delayed
+to Appendix A.
+Lemma 3.1. Given x′ ∈ Cn, λ > 0. and α ≥ 0, we have the following cases:
+1. When ∥x′∥∞ > λ, we have
+x∗ = (∥ξ∥2 + αλ)
+ξ
+∥ξ∥2
+, where ξ = sgn(x′) ◦ max(|x′| − λ, 0).
+2. When (1 − α)λ < ∥x′∥∞ ≤ λ, we have x∗ as a 1-sparse vector such that one chooses
+an index i ∈ arg maxj(|(x′)j|) and have
+(x∗)j =
+�
+(|(x′)j| + (α − 1)λ) sgn((x′)j)
+if j = i,
+0
+if j ̸= i.
+3. When ∥x′∥∞ ≤ (1 − α)λ, we have x∗ = 0.
+Then x∗ is an optimal solution to
+proxλ(∥·∥1−α∥·∥2)(x′) = arg min
+x
+∥x∥1 − α∥x∥2 + 1
+2λ∥x − x′∥2
+2.
+(3.20)
+
+STOCHASTIC ADMM FOR PTYCHOGRAPHY
+9
+3.1.4. z-subproblem. (3.6d) can be rewritten as
+zt+1 ∈ arg min
+z
+N
+�
+j=1
+�
+�β1
+2
+�����ut+1
+j
+− F(P t+1
+j
+z) +
+Λt
+j
+β1
+�����
+2
+2
+�
+� + β2
+2
+����vt+1 − ∇z + yt
+β2
+����
+2
+2
+,
+(3.21)
+which implies that zt+1 must satisfy the first-order optimality condition
+�
+�β1
+N
+�
+j=1
+(P t+1
+j
+)∗P t+1
+j
+− β2∆
+�
+� zt+1 =
+N
+�
+j=1
+β1(P t+1
+j
+)∗F−1
+�
+ut+1
+j
++
+Λt
+j
+β1
+�
++ β2∇⊤
+�
+vt+1 + yt
+β2
+�
+,
+(3.22)
+where the Laplacian ∆ = −∇⊤∇. Since the coefficient matrix of zt+1 is invertible, solving
+(3.22) can be performed exactly, but it could be computationally expensive if the matrix
+system is extremely large because of the image size of z. Since the coefficient matrix tends
+to be sparse, conjugate gradient [22] can be used to solve (3.22) like in [10, 11], but it needs
+access to all N probes and requires at most n2 iterations to attain an exact solution, assuming
+exact arithmetic. Moreover, it is sensitive to roundoff error [19].
+Alternatively, like in Section 3.1.2 we linearize (3.21) to obtain the gradient descent step
+with step size δt
+z > 0:
+zt+1 = zt − δt
+z∇zL(ut+1, ωt+1, vt+1, zt, Λt, yt).
+(3.23)
+We approximate ∇zL by its stochastic estimator ˜∇zL that only has access to b ≤ N probes.
+Replacing ∇zL with ˜∇zL in (3.23) gives
+zt+1 = zt − δt
+z ˜∇zL(ut+1, ωt+1, vt+1, zt, Λt, yt).
+(3.24)
+To design candidates for ˜∇zL, we will use the following lemma:
+Lemma 3.2. Let S ∈ Rm2×n2. If ei ∈ ker(S) for some index i, then for any x ∈ Cm2, we
+have (S⊤x)i = 0.
+Proof. We have (S⊤x)i = ⟨S⊤x, ei⟩ = ⟨x, Sei⟩ = ⟨x, 0⟩ = 0.
+For brevity, we denote the vectors
+At
+j = −β1
+�
+(P t+1
+j
+)∗F−1
+�
+ut+1
+j
++
+Λt
+j
+β1
+�
+− (P t+1
+j
+)∗P t+1
+j
+zt
+�
+,
+Bt = −β2
+�
+∇⊤
+�
+vt+1 + yt
+β2
+�
++ ∆zt
+�
+.
+(3.25)
+At each element i = 1, . . . , n2, (3.23) becomes
+(zt+1)i = (zt)i − δt
+z(∇zL(ut+1, ωt+1, vt+1, zt, Λt, yt))i = (zt)i − δt
+z
+�
+�
+N
+�
+j=1
+(At
+j)i + (Bt)i
+�
+� .
+(3.26)
+
+10
+KEVIN BUI AND ZICHAO (WENDY) DI
+By Lemma 3.2, since (P t+1
+j
+)∗ = (ωt+1 ◦ Sj)∗ = S⊤
+j D(ωt+1)∗, we have (At
+j)i = 0 if ei ∈ ker(Sj),
+which means that element i is not scanned by the mask matrix Sj. For each i = 1, . . . , n2,
+we define Ni = {j : ei ̸∈ ker(Sj)} to be the set of indices corresponding to the mask matrices
+that scan element i. As a result, (3.26) reduces to and can be rewritten as
+(zt+1)i = (zt)i − δt
+z
+�
+� �
+j∈Ni
+(At
+j)i + (Bt)i
+�
+� = (zt)i − |Ni|δt
+z
+�
+� 1
+|Ni|
+�
+j∈Ni
+�
+(At
+j)i +
+1
+|Ni|(Bt)i
+��
+� .
+(3.27)
+Comparing (3.26) and (3.27), we observe that
+1
+|Ni|
+�
+j∈Ni
+�
+(At
+j)i +
+1
+|Ni|(Bt)i
+�
+∝ (∇zL(ut+1, ωt+1, vt+1, zt, Λt, yt))i.
+Thus, a candidate for the stochastic estimator ˜∇zL is the SGD estimator ˜∇SGD
+z
+L given by
+�
+˜∇SGD
+z
+L(ut+1, ωt+1, vt+1, zt, Λt, yt)
+�
+i =
+1
+|nt
+i|
+�
+j∈nt
+i
+�
+(At
+j)i +
+1
+|Ni|(Bt)i
+�
+,
+(3.28)
+where nt
+i ⊂ Ni is a mini-batch sampled from Ni at iteration t [3, 40].
+We can further modify (3.28) by incorporating spatially varying step sizes inspired by the
+PIE algorithms [32, 33, 42]. We define
+Ψi,j =
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+1
+∥ωt+1∥2∞
+ePIE [33],
+∥P t+1
+j
+ei∥1
+∥ωt+1∥∞
+�
+∥P t+1
+j
+ei∥2
+1 + γz∥ωt+1∥2∞
+�
+PIE [42],
+1
+(1 − γz)∥P t+1
+j
+ei∥2
+1 + γz∥ωt+1∥2∞
+rPIE [32],
+(3.29)
+with γz ∈ [0, 1] for PIE and rPIE. A class of PIE candidates for the stochastic estimator is
+�
+˜∇PIE
+z
+L(ut+1, ωt+1, vt+1, zt, Λt, yt)
+�
+i =
+1
+|nt
+i|
+�
+j∈nt
+i
+Ψi,j
+�
+(At
+j)i +
+1
+|Ni|(Bt)i
+�
+.
+(3.30)
+The overall stochastic ADMM algorithm that solves (3.3) is provided by Algorithm 3.1.
+Notice that the non-blind problem is just a special case with the probe ω fixed.
+4. Convergence Analysis. We discuss the convergence of Algorithm 3.1. Although global
+convergence for ADMM can be established using Kurdyka-�Lojasiewicz assumptions [49], the
+result does not apply for our models because our models contain the gradient operator, which
+does not satisfy the necessary surjectivity assumption. Hence, we will prove up to subse-
+quential convergence. The convergence analysis is based on the analyses done in [10, 11, 51],
+where under certain assumptions, they showed that the iterate subsequences of the ADMM
+
+STOCHASTIC ADMM FOR PTYCHOGRAPHY
+11
+Algorithm 3.1 Stochastic ADMM to solve (3.3)
+Input:
+set of masks {Sj}N
+j=1; model parameters λ > 0, α ∈ [0, 1]; penalty parameters β1, β2 > 0; sequence of step sizes
+{(δt
+ω, δt
+z)}∞
+t=1; batch size b ≤ N; PIE factors γz, γω ∈ [0, 1].
+1: Initialize ω0, z0, {u0
+j}N
+j=1 = {Λ0
+j}N
+j=1, y0 = ∇z0.
+2: for t = 0 to T − 1 do
+3:
+Uniformly sample without replacement the subset nt ⊂ {1, . . . , N} of batch size b.
+4:
+Compute nt
+i from nt, i.e., nt
+i =
+�
+j∈nt
+∥Sjei∥1.
+5:
+Update ut+1
+j
+according to (3.8) for each j ∈ nt.
+6:
+if ω is unknown then
+7:
+Update ωt+1 = ωt − δt
+ω ˜∇ωL(ut+1, ωt, vt, zt, Λt, yt). See (3.15) and (3.17) for a candidate ˜∇ωL.
+8:
+else
+9:
+ωt+1 = ωt.
+10:
+end if
+11:
+Compute
+(vt+1)i = prox λ
+β2 (∥·∥1−α∥·∥2)
+�
+(∇zt)i − (yt)i
+β2
+�
+for all i = 1, . . . , n2; see Lemma 3.1.
+12:
+Update (zt+1)i = (zt)i − δt
+z
+�
+˜∇zL(ut+1, ωt+1, vt+1, zt, Λt, yt)
+�
+i for all i such that nt
+i ̸= 0. See (3.28) and (3.30)
+for a candidate ˜∇zL.
+13:
+Compute
+Λt+1
+j
+= Λt
+j + β1
+�
+ut+1
+j
+− F(ωt+1 ◦ Sjzt+1)
+�
+, ∀j ∈ nt,
+yt+1 = yt + β2
+�
+vt+1 − ∇zt+1�
+.
+14: end for
+Output: ω∗ = ωT , z∗ = zT
+algorithms converge to Karush-Kuhn-Tucker (KKT) points. To simplify notation, let Z =
+(u, ω, v, z) and Ω = (Λ, y). We also write L(ω), for example, to represent the Lagrangian with
+respect to ω with all other variables fixed at their most recent values. A KKT point (Z⋆, Ω⋆)
+of the Lagrangian (3.5) satisfies the KKT conditions given by
+0 ∈
+�
+�
+�
+�
+�
+∂|u⋆
+j|(|u⋆
+j| −
+�
+dj) + Λ⋆
+j,
+if AGM,
+∂|u⋆
+j|
+�
+|u⋆
+j| −
+�
+dj
+|u⋆
+j|
+�
++ Λ⋆
+j,
+if IPM,
+for j = 1, . . . , N,
+(4.1a)
+−y⋆
+λ ∈ ∂(∥v⋆∥1 − α∥v⋆∥2,1),
+(4.1b)
+u⋆
+j = F(ω⋆ ◦ Sjz⋆)
+for j = 1, . . . , N,
+(4.1c)
+v⋆ = ∇z⋆,
+(4.1d)
+∇ωL(Z⋆, Ω⋆) = 0,
+(4.1e)
+∇zL(Z⋆, Ω⋆) = 0.
+(4.1f)
+
+12
+KEVIN BUI AND ZICHAO (WENDY) DI
+Because we implement SGD to solve for the probe ω and the image z in the ADMM algorithm,
+we replace (4.1e) and (4.1f) with the following conditions, respectively:
+E
+�
+∥∇ωL(Z⋆, Ω⋆)∥2
+2
+�
+= 0
+(4.1e′)
+E
+�
+∥∇zL(Z⋆, Ω⋆)∥2
+2
+�
+= 0.
+(4.1f′)
+We say a point (Z⋆, Ω⋆) is a stochastic KKT point if it satisfies (4.1a)-(4.1d) and (4.1e′)-(4.1f′).
+Where Et denotes the expectation conditioned on the first t iterations of the stochastic
+ADMM algorithm, we impose the following assumption adapted from [3, Assumption 4.3]
+relating to the stochastic gradient estimators ˜∇ωL and ˜∇zL.
+Assumption 4.1. Let {(Zt, Ωt)}∞
+t=1 be a sequence of iterates generated by Algorithm 3.1.
+Suppose that at each iteration t, the stochastic gradient estimators
+˜∇ωL(ωt) := ˜∇ωL(ut+1, ωt, vt, zt, Λt, yt) and ˜∇zL(zt) := ˜∇ωL(ut+1, ωt+1, vt+1, zt, Λt, yt) sat-
+isfy the following:
+(a) There exist constants KU ≥ KL > 0 such that
+R
+�
+Et
+�
+⟨∇ωL(ωt), ˜∇ωL(ωt)⟩
+��
+≥ KLEt
+�
+∥∇ωL(ωt)∥2
+2
+�
+(4.3)
+���Et[ ˜∇ωL(ωt)]
+���
+2
+2 ≤ KUEt
+�
+∥∇ωL(ωt)∥2
+2
+�
+(4.4)
+R
+�
+Et
+�
+⟨∇zL(zt)), ˜∇zL(zt)⟩
+��
+≥ KLEt
+�
+∥∇zL(zt)∥2
+2
+�
+(4.5)
+���Et[ ˜∇zL(zt)]
+���
+2
+2 ≤ KUEt
+�
+∥∇zL(zt)∥2
+2
+�
+.
+(4.6)
+(b) There exists constant M, MV ≥ 0 such that
+Et
+�
+∥ ˜∇ωL(ωt)∥2
+2
+�
+−
+���Et[ ˜∇ωL(ωt)]
+���
+2
+2 ≤ M + MV Et
+���∇ωL(ωt)
+��2
+2
+�
+(4.7)
+Et
+�
+∥ ˜∇zL(zt)∥2
+2
+�
+−
+���Et[ ˜∇zL(zt)]
+���
+2
+2 ≤ M + MV Et
+���∇zL(zt)
+��2
+2
+�
+.
+(4.8)
+To prove the convergence of Algorithm 3.1, we require the following preliminary results.
+Lemma 4.2 provides useful inequalities while Proposition 4.3 bounds the iterates {(Zt, Ωt)}∞
+t=1
+and establishes some bounded property of the gradients {(∇ωL(ωt), ∇zL(zt))}∞
+t=1.
+Lemma 4.2. Let {(Zt, Ωt)}∞
+t=1 be a sequence of iterates generated by Algorithm 3.1 that
+satisfies Assumption 4.1. Suppose that {(ωt, zt)}∞
+t=1 is bounded. For each iteration t, we have
+Et[L(ωt+1)] − Et[L(ωt)] ≤ −
+�
+KL − δt
+ωLω(MV + KU)
+2
+�
+δt
+ωEt
+���∇ωL(ωt)
+��2
+2
+�
++ (δt
+ω)2LωM
+2
+(4.9)
+Et[L(zt+1)] − Et[L(zt)] ≤ −
+�
+KL − δt
+zLz(MV + KU)
+2
+�
+δt
+zEt
+���∇zL(zt)
+��2
+2
+�
++ (δt
+z)2LzM
+2
+(4.10)
+for some constants Lω, Lz > 0.
+
+STOCHASTIC ADMM FOR PTYCHOGRAPHY
+13
+Proposition 4.3. Let {(Zt, Ωt)}∞
+t=1 be a sequence of iterates generated by Algorithm 3.1 that
+satisfies Assumption 4.1. Suppose {(ωt, zt)}∞
+t=1 is bounded, �∞
+t=1 ∥Ωt+1 − Ωt∥2
+2 < ∞, and
+∞
+�
+t=1
+δt
+ω = ∞,
+∞
+�
+t=1
+(δt
+ω)2 < ∞,
+∞
+�
+t=1
+δt
+z = ∞,
+∞
+�
+t=1
+(δt
+z)2 < ∞.
+(4.11)
+Then {(Zt, Ωt)}∞
+t=1 is bounded and
+∞
+�
+t=1
+E
+�
+δt
+ω∥∇ωL(ωt)∥2
+2 + δt
+z∥∇zL(zt)∥2
+2
+�
+< ∞.
+(4.12)
+The convergence of Algorithm 3.1 is finally established below (see proofs in Appendix B).
+Theorem 4.4. Let {(Zt, Ωt)}∞
+t=1 be generated by Algorithm 3.1. Under the same assump-
+tion as Proposition 4.3, there exists a subsequence of {(Zt, Ωt)}∞
+t=1 whose accumulation point
+(Z⋆, Ω⋆) is a stochastic KKT point almost surely (a.s.) of (3.5).
+We note that the requirement �∞
+t=1 ∥Ωt+1 − Ωt∥2
+2 < ∞ is rather strong, but similar
+assumption was made in other nonconvex ADMM algorithms [24, 25, 53] that do not satisfy
+the necessary assumptions for global convergence [49].
+5. Numerical Results. In this section, we evaluate the performance of Algorithm 3.1 on
+two complex images presented in Figure 2. The probe size used for both images is 256 × 256,
+and the probe is scanned across an image from left to right and top to bottom, giving us
+N = 100 measurements. The measurements {dj}N
+j=1 are either corrupted by Gaussian noise
+or Poisson noise. More specifically, when the measurements are corrupted by Gaussian noise,
+we have
+dj = (|F(Pjz)| + ϵ)2,
+(5.1)
+where ϵ is an i.i.d. Gaussian random vector. When the measurements are corrupted by Poisson
+noise, we have
+dj = Poisson(|F(Pjzζ)|),
+(5.2)
+where zζ = ζz for some constant ζ > 0. Note that Poisson noise is stronger when ζ is smaller.
+For numerical evaluation, we compute the SSIMs [50] of the magnitudes and phases be-
+tween the reconstructed image z∗∗ and the ground-truth image zg, where z∗∗
+i
+= ζ∗z∗
+i+t∗ is
+adjusted for scaling by ζ∗ and translation by t∗ and (ζ∗, t∗) = arg min
+ζ∈C,t∈Z
+n2
+�
+i=1
+|ζz∗
+i+t − zg
+i |2. We
+compare the proposed stochastic ADMM algorithms with its deterministic, full-batch coun-
+terparts (i.e., (3.22) and (3.9) are solved exactly) and its isotropic TV counterparts based on
+
+14
+KEVIN BUI AND ZICHAO (WENDY) DI
+0.5
+0.55
+0.6
+0.65
+0.7
+0.75
+0.8
+0.85
+0.9
+0.95
+(a)
+0.05
+0.1
+0.15
+0.2
+0.25
+0.3
+0.35
+(b)
+5
+10
+15
+20
+25
+30
+(c)
+0.1
+0.2
+0.3
+0.4
+0.5
+0.6
+0.7
+0.8
+0.9
+(d)
+0
+0.05
+0.1
+0.15
+0.2
+0.25
+0.3
+0.35
+(e)
+1
+2
+3
+4
+5
+6
+7
+8
+9
+10
+(f)
+Figure 2: Two complex sample images and their probes examined in the experiments. Left
+column: sample magnitude; middle column: sample phase with inserted proportionally sized
+probe magnitude; left column: the magnitude differences between the ground-truth probe and
+the initial probe ω0.
+Table 1: Parameter settings for each method. Note that b refers to the batch size.
+Total
+Epochs
+β1 = β2
+δt
+z
+Ψi,j
+δt
+ω
+Φi,j
+AGM
+600
+0.25
+�
+�
+�
+�
+�
+2
+√
+b
+1 ≤ t ≤ 300
+1
+5
+√
+b
+300 < t ≤ 450
+1
+50
+√
+b
+450 < t ≤ 600
+rPIE
+(γz = 0.1)
+�
+�
+�
+�
+�
+√
+b × 10−3
+1 ≤ t ≤ 300
+√
+b × 10−4
+300 < t ≤ 450
+√
+b × 10−5
+450 ≤ t ≤ 600
+rPIE
+(γω = 0.025)
+IPM
+300
+0.25
+�
+�
+�
+�
+�
+15
+√
+b
+1 ≤ t ≤ 150
+3
+2
+√
+b
+150 < t ≤ 225
+3
+20
+√
+b
+225 < t ≤ 300
+ePIE
+�
+�
+�
+�
+�
+2
+√
+b × 10−3
+1 ≤ t ≤ 300
+2
+√
+b × 10−4
+300 < t ≤ 450
+2
+√
+b × 10−5
+450 ≤ t ≤ 600
+ePIE
+[10, 11]. The results are also compared with Douglas-Rachford splitting [46], rPIE [33], and
+PHeBIE [21]. We follow the implementation of Douglas-Rachford from [8].
+We initialize z0 =
+1
+√
+2(1+i1) when using AGM for Gaussian-corrupted measurements and
+z0 =
+ζ
+√
+2(1 + i1) when using IPM for Poisson-corrupted measurements. When performing the
+blind experiments using Algorithm 3.1, ω0 is initialized as the perturbation of the ground-
+truth probe. The magnitude differences between the initial probe and the ground-truth probe
+are shown in Figure 2. The selected parameters, except for λ, are summarized in Table 1.
+The initial step sizes for δt
+z and δt
+ω are determined empirically, and motivated by (4.11), we
+decrease them by a factor of 10 at the 1/2 and 3/4 of the total epochs. Decreasing the step
+size in this way is a popular technique, especially in the deep learning community [18, 20].
+Inspired from [18], the step sizes are multiplied by a factor of
+√
+b so that they are scaled
+accordingly to the batch size b. For AITV regularization, we examine α ∈ {0.2, 0.4, 0.6, 0.8}
+
+-
+-
+-
+-
+-
+-
+-STOCHASTIC ADMM FOR PTYCHOGRAPHY
+15
+Table 2: SSIM results of the algorithms applied to the Gaussian corrupted measurements
+with SNR = 40. The stochastic algorithms (e.g., AITV and isoTV, b ∈ {5, 10, 20, 50}) are ran
+three times to obtain the average SSIM values. Bold indicates best value; underline indicates
+second best value.
+Non-blind
+Blind
+Chip
+Cameraman/Baboon
+Chip
+Cameraman/Baboon
+mag.
+SSIM
+phase
+SSIM
+mag.
+SSIM
+phase
+SSIM
+mag.
+SSIM
+phase
+SSIM
+mag.
+SSIM
+phase
+SSIM
+DR
+0.8130
+0.8089
+0.8701
+0.5191
+0.8008
+0.7642
+0.8009
+0.3207
+rPIE
+0.8886
+0.9073
+0.8930
+0.6055
+0.9070
+0.9120
+0.8890
+0.6145
+PHeBIE
+0.8004
+0.8019
+0.8725
+0.5718
+0.8612
+0.8438
+0.8846
+0.5756
+isoTV (b = 5)
+0.9501
+0.9027
+0.9393
+0.7578
+0.9426
+0.8919
+0.9324
+0.7547
+isoTV (b = 10)
+0.9498
+0.9004
+0.9387
+0.7475
+0.9429
+0.8891
+0.9326
+0.7477
+isoTV (b = 20)
+0.9514
+0.8981
+0.9385
+0.7302
+0.9447
+0.8850
+0.9298
+0.7289
+isoTV (b = 50)
+0.9355
+0.9193
+0.9294
+0.7050
+0.9322
+0.9047
+0.9153
+0.7025
+isoTV (full batch)
+0.9578
+0.9145
+0.9769
+0.7338
+0.9527
+0.8698
+0.9589
+0.5774
+AITV (b = 5)
+0.9585
+0.9556
+0.9438
+0.7720
+0.9490
+0.9477
+0.9373
+0.7775
+AITV (b = 10)
+0.9620
+0.9579
+0.9515
+0.7747
+0.9534
+0.9481
+0.9450
+0.7772
+AITV (b = 20)
+0.9629
+0.9583
+0.9538
+0.7707
+0.9547
+0.9470
+0.9468
+0.7690
+AITV (b = 50)
+0.9585
+0.9550
+0.9490
+0.7358
+0.9514
+0.9432
+0.9391
+0.7342
+AITV (full batch)
+0.9674
+0.9513
+0.9814
+0.7463
+0.9676
+0.9296
+0.9725
+0.5956
+and determine that α = 0.8 yields the best results across all of our numerical examples. The
+batch sizes we examine are b ∈ {5, 10, 20, 50} for Gaussian noise and b ∈ {5, 10, 20, 25} for
+Poisson noise. For each parameter setting and image, we run three trials to obtain the mean
+SSIM values.
+The code for the experiments is available at https://github.com/kbui1993/Stochastic
+ADMM Ptycho.
+5.1. Gaussian noise. The SNR of the noisy measurements [10] is given by
+SNR
+�
+{
+�
+dj}N
+j=1, {|F(Pjz)|}N
+j=1
+�
+= −10 log10
+�
+�
+�
+�
+�
+�
+�
+N
+�
+j=1
+∥
+�
+dj − |F(Pjz)|∥2
+2
+N
+�
+j=1
+∥F(Pjz)∥2
+2
+�
+�
+�
+�
+�
+�
+�
+,
+so determined by the SNR value, the noise level ϵ in (5.1) can be calculated by
+ϵ =
+�
+�
+�
+�
+�
+�
+10−SNR/10
+N
+�
+j=1
+∥F(Pjz)∥2
+2
+Nm2
+.
+For both the non-blind and blind case, we examine the case when SNR = 40 for the
+noisy measurements. We set λ = 10.0. The numerical results are recorded in Table 2. For
+
+16
+KEVIN BUI AND ZICHAO (WENDY) DI
+(a) isoTV (b = 50)
+(b) AITV (b = 20)
+(c) AITV (full)
+(d) DR
+(e) rPIE
+(f) PHeBIE
+(g) isoTV (b = 50)
+(h) AITV (b = 20)
+(i) AITV (full)
+(j) DR
+(k) rPIE
+(l) PHeBIE
+(m) isoTV (b = 5)
+(n) AITV (b = 10)
+(o) AITV (full)
+(p) DR
+(q) rPIE
+(r) PHeBIE
+(s) isoTV (b = 5)
+(t) AITV (b = 10)
+(u) AITV (full)
+(v) DR
+(w) rPIE
+(x) PHeBIE
+Figure 3: Reconstructions of the non-blind case for the Gaussian noise. Top two rows: recon-
+structions of Figures 2a-2b; bottom two rows: reconstructions of Figs. 2d-2e.
+both the non-blind and blind cases, DR, rPIE, and PHeBIE yield magnitude images with
+the worst SSIM values, and AITV outperforms its corresponding isotropic TV counterpart by
+having better SSIM values for both the magnitude and phase images. The stochastic AITV,
+particularly b = 10 or b = 20, has slightly lower magnitude SSIM values by at most 0.04 than
+the best results obtained from the deterministic, full-batch AITV. In fact, stochastic AITV
+attains the second best magnitude SSIM values in three out of the four cases considered.
+On the other hand, stochastic AITV with either b = 10 or b = 20 has the best phase SSIM
+values, outperforming its deterministic version by up to 0.19.
+For the blind case of the
+cameraman/baboon image, the deterministic AITV and isotropic TV have worse phase SSIM
+values than their stochastic counterparts. In general, the stochastic algorithm does best in
+recovering phase images with superior SSIM values while recovering the magnitude images
+with comparable SSIM values as the deterministic algorithm.
+The reconstructed images for the non-blind experiments are presented in Figure 3. DR,
+rPIE, PHeBIE, and the stochastic algorithms have artifacts in all four corners of the magnitude
+images because the corners are scanned significantly less than in the middle of the image.
+
+-STOCHASTIC ADMM FOR PTYCHOGRAPHY
+17
+(a) isoTV (b = 5)
+(b) AITV (b = 20)
+(c) AITV (full)
+(d) DR
+(e) rPIE
+(f) PHeBIE
+(g) isoTV (b = 5)
+(h) AITV (b = 20)
+(i) AITV (full)
+(j) DR
+(k) rPIE
+(l) PHeBIE
+(m) isoTV (b = 10) (n) AITV (b = 10)
+(o) AITV (full)
+(p) DR
+(q) rPIE
+(r) PHeBIE
+(s) isoTV (b = 10)
+(t) AITV (b = 10)
+(u) AITV (full)
+(v) DR
+(w) rPIE
+(x) PHeBIE
+Figure 4: Reconstructions of the blind case for Gaussian noise.
+100
+101
+102
+103
+epoch
+105
+106
+107
+108
+Figure 5: Amplitude Gaussian metric plotted across 600 epochs for the blind algorithms on
+the complex image given by Figure 2d-2e.
+
+-18
+KEVIN BUI AND ZICHAO (WENDY) DI
+20
+30
+40
+50
+60
+70
+0
+0.1
+0.2
+0.3
+0.4
+0.5
+0.6
+0.7
+0.8
+0.9
+1
+Phase SSIM
+20
+30
+40
+50
+60
+70
+0.5
+0.55
+0.6
+0.65
+0.7
+0.75
+0.8
+0.85
+0.9
+0.95
+1
+Magnitude SSIM
+Figure 6: Magnitude and phase SSIMs over different Gaussian noise level for the complex
+image given by Figure 2d-2e for the blind case.
+However, the deterministic AITV has no artifacts because (3.22) is solved exactly for the
+image solution z. As a result, it has higher magnitude SSIM values than their stochastic
+counterparts. Nevertheless, the stochastic algorithms yield better phase images with less noise
+artifacts than any other algorithms. For example, the phase images of Figure 2e reconstructed
+from stochastic isoTV and AITV have the least amount of cameraman remnants.
+Figure 4 shows the results of the blind algorithms. The phase images reconstructed by
+the deterministic AITV, DR, ePIE and PHeBIE are significantly worse than the stochastic
+algorithms.
+For example in Figure 2b, the contrasts of the reconstructed images by the
+deterministic AITV, DR, and PHeBIE are inconsistent as they become darker from left to
+right while the contrasts are more consistent with the stochastic algorithms. For Figure 2e,
+the stochastic algorithms perform the best in recovering the phase image while deterministic
+AITV is unable to recover the left half of the image and DR, rPIE, and PHeBIE have strong
+remnants of the cameraman present. Like in the non-blind case, stochastic AITV reconstructs
+the phase image the best.
+In Figure 5, we examine the convergence of the blind algorithms applied to the Camera-
+man/Baboon image by recording their AGM values for each epoch. We omit the convergence
+curves for isoTV since their curves are similar to their AITV counterparts. Overall, the curves
+for our proposed stochastic algorithms are decreasing, validating the numerical convergence
+of Algorithm 3.1 with AGM fidelity. However, their curves are slightly above the determinis-
+tic ADMM algorithm and rPIE. The reason why rPIE outperforms the AITV algorithms is
+because it seeks to only minimize AGM while the AITV algorithms minimize a larger objec-
+tive function given by (3.5). Overall, after several hundred epochs, our proposed stochastic
+algorithms can give comparable AGM values as the deterministic AITV and rPIE algorithms.
+
+STOCHASTIC ADMM FOR PTYCHOGRAPHY
+19
+Table 3: SSIM results of the algorithms applied to the Poisson corrupted measurements with
+η = 0.01. The stochastic algorithms (e.g., AITV and isoTV, b ∈ {5, 10, 20, 25}) are ran three
+times to obtain the average SSIM values. Bold indicates best value; underline indicates second
+best value.
+Non-blind
+Blind
+Chip
+Cameraman/Baboon
+Chip
+Cameraman/Baboon
+mag.
+SSIM
+phase
+SSIM
+mag.
+SSIM
+phase
+SSIM
+mag.
+SSIM
+phase
+SSIM
+mag.
+SSIM
+phase
+SSIM
+DR
+0.8523
+0.8455
+0.8704
+0.5043
+0.8431
+0.7387
+0.7630
+0.2529
+PHeBIE
+0.9404
+0.9398
+0.9271
+0.6791
+0.9280
+0.9082
+0.8678
+0.5470
+isoTV (b = 5)
+0.9460
+0.9151
+0.9301
+0.7193
+0.9394
+0.9001
+0.9238
+0.7105
+isoTV (b = 10)
+0.9365
+0.9212
+0.9250
+0.7085
+0.9342
+0.9064
+0.9188
+0.6979
+isoTV (b = 20)
+0.9335
+0.9397
+0.9224
+0.7008
+0.9355
+0.9248
+0.9153
+0.6905
+isoTV (b = 25)
+0.9353
+0.9465
+0.9220
+0.6992
+0.9376
+0.9315
+0.9150
+0.6907
+isoTV (full batch)
+0.9767
+0.9590
+0.9773
+0.7093
+0.9655
+0.9192
+0.9588
+0.4920
+AITV (b = 5)
+0.9526
+0.9680
+0.9319
+0.7477
+0.9409
+0.9530
+0.9242
+0.7418
+AITV (b = 10)
+0.9590
+0.9685
+0.9375
+0.7366
+0.9472
+0.9533
+0.9321
+0.7318
+AITV (b = 20)
+0.9598
+0.9682
+0.9383
+0.7171
+0.9494
+0.9526
+0.9322
+0.7114
+AITV (b = 25)
+0.9585
+0.9676
+0.9373
+0.7112
+0.9492
+0.9525
+0.9307
+0.7055
+AITV (full batch)
+0.9803
+0.9644
+0.9782
+0.7084
+0.9741
+0.9354
+0.9671
+0.4975
+Lastly, we analyze the robustness of the blind algorithms applied to Figures 2d-2e cor-
+rupted by different levels of Gaussian noise, from SNR 25 to 65. The fidelity parameter λ
+varies for different noise level of the image: λ = 100 for SNR = 25; λ = 50 for SNR = 30,
+35; λ = 10 for SNR =40, 45; λ = 5 for SNR = 50, 55, 60; and λ = 3 for SNR = 65. The
+SSIMs for the magnitude and phase images across different SNRs are plotted in Figure 6.
+For SNR ≥ 40, the deterministic algorithms have the best magnitude SSIMs than the other
+algorithms while their stochastic counterparts have slightly lower SSIMs. When SNR < 40,
+the stochastic algorithms perform the best. In fact, stochastic AITV has magnitude SSIM at
+least 0.90 across different noise levels. For the phase image, the stochastic algorithms have
+the highest SSIMs up to SNR = 55. For SNR ≥ 60, the rPIE algorithm has the best phase
+SSIM while stochastic AITV has the second best. Overall, stochastic AITV is the most stable
+across different levels of Gaussian noise.
+5.2. Poisson noise. For both the non-blind and blind case, we examine the measurements
+corrupted with Poisson noise with η = 0.01 according to (5.2). We set λ = 0.15. The numeri-
+cal results are recorded in Table 3. Note that rPIE results are excluded because the algorithm
+is tailored towards measurements corrupted with Gaussian noise [32]. Across all cases, de-
+terministic AITV attains the highest magnitude SSIM values and stochastic AITV attains
+the highest phase SSIM values while DR performs the worst in reconstructing images from
+Poisson-corrupted measurements. We observe general improvement in SSIM values for both
+magnitude and phase images by using AITV over isoTV. Although the stochastic algorithms
+have lower SSIM values than their deterministic counterparts for the magnitude images, the
+difference is at most 0.047 for AITV and at most 0.056 for isoTV. Moreover, the SSIM val-
+ues of the magnitude images from the stochastic algorithms are at least 0.91. Similar to the
+
+20
+KEVIN BUI AND ZICHAO (WENDY) DI
+Gaussian noise case, stochastic AITV reconstructs the phase image well while recovering the
+magnitude image with satisfactory quality.
+We examine the robustness of the blind algorithms on Figures 2d-2e with different level of
+Poisson noise. The noise levels we examine are η ∈ {0.005k}9
+k=1. We set the fidelity parameter
+to be λ = 15 × η. The SSIMs for the magnitude and phase images across different Poisson
+noise levels are plotted in Figure 7. We observe that the deterministic algorithms yield the
+best magnitude SSIMs while the stochastic algorithms yield the best phase SSIMs. DR yields
+the worst results for both magnitude and phase components. Although stochastic AITV yields
+the third best SSIMs for the magnitude image, its SSIMs are at least 0.90. Moreover, it has
+the best phase SSIMs, significantly more than its deterministic counterpart by at about 0.20.
+In summary, stochastic AITV is a robust method across different levels of Poisson noise.
+6. Conclusion. In this work, we propose AITV-regularized variational models for image
+ptychography, where the measurements are corrupted by either Gaussian or Poisson noise.
+To adapt the algorithm for large number of measurements, we design a stochastic ADMM
+algorithm that incorporates adaptive step sizes based on the PIE algorithms. Overall, us-
+ing both AITV regularization and stochastic ADMM, we are able to reconstruct an image
+of satisfactory quality from heavily corrupted measurements as demonstrated in our numer-
+ical experiments. In fact, the phase component of the image is best recovered through our
+proposed algorithms. Lastly, we prove theoretical convergence for the proposed stochastic
+ADMM algorithm under certain conditions and demonstrate numerical convergence in our
+experiments.
+Future directions include the design of a globally convergent algorithm for the AITV-
+regularized ptychography model, and incorporation of variance-reduced stochastic gradient
+estimators, such as SVRG [23] and SARAH [36], to accelerate convergence and improve re-
+construction quality.
+Appendix A. Proof of Lemma 3.1.
+Proof. If x′ = 0, then it is trivial, so for the rest of the proof, we assume that x′ ̸= 0.
+Suppose x∗ is the optimal solution to (3.20). We show that sgn(x∗) = sgn(x′). If x∗ = 0,
+then we can choose ci ∈ {c′ ∈ C : |c′| = 1} such that sgn(x∗)i = ci = sgn(x′)i for each
+i = 1, . . . , n2, giving the desired result. Suppose that x∗ ̸= 0. Because ∥ · ∥1 − α∥ · ∥2,1 is
+rotation invariant, we only need to examine and expand the quadratic term in (3.20). We see
+that
+∥x∗ − x′∥2
+2 =
+n2
+�
+i=1
+|(x∗)i − (x′)i|2 =
+n2
+�
+i=1
+�
+|(x∗)i|2 + |(x′)i|2 − 2|(x∗)i||(x′)i| cos θi
+�
+,
+where θi is the angle between the components (x∗)i and (x′)i. This term is minimized when
+θi = 0 for all i. This means that sgn(x∗)i = sgn(x′)i for all i, or otherwise x∗ would not be an
+optimal solution to (3.20). Hence, sgn(x∗) = sgn(x′).
+After establishing that sgn(x∗) = sgn(x′), we simplify (3.20) to an optimization problem
+with respect to |x| given by
+|x∗| =
+arg min
+ρ∈Rn, (ρ)i≥0
+∥ρ∥1 − α∥ρ∥2 + 1
+2λ∥ρ − |x′|∥2
+2.
+
+STOCHASTIC ADMM FOR PTYCHOGRAPHY
+21
+0
+0.01
+0.02
+0.03
+0.04
+0.05
+0
+0.1
+0.2
+0.3
+0.4
+0.5
+0.6
+0.7
+0.8
+0.9
+1
+Phase SSIM
+0
+0.01
+0.02
+0.03
+0.04
+0.05
+0.5
+0.55
+0.6
+0.65
+0.7
+0.75
+0.8
+0.85
+0.9
+0.95
+1
+Magnitude SSIM
+Figure 7: Magnitude and phase SSIMs over different Poisson noise level for the complex image
+given by Figure 2d-2e for the blind case.
+Therefore, by applying [27, Lemma 1] to the optimization problem followed by multiplying
+the solution |x∗| by sgn(x∗), we obtain the desired results.
+Appendix B. Proofs of Section 4.
+Before proving our main results, we present prelimi-
+nary tools necessary for the convergence analysis.
+Definition B.1 ([41]). Let h : Rn2 → (−∞, +∞] be a proper and lower semicontinuouous
+function and dom h := {x ∈ Rn2 : h(x) < ∞}.
+(a) The Fr´echet subdifferential of h at the point x ∈ dom h is the set
+ˆ∂h(x) =
+�
+v ∈ Rn2 : lim inf
+y̸=x,y→x
+h(y) − h(x) − ⟨v, y − x⟩
+∥y − x∥
+≥ 0
+�
+.
+(b) The limiting subdifferential of h at the point x ∈ dom h is the set
+∂h(x) =
+�
+v ∈ Rn2 : ∃{(xt, yt)}∞
+t=1 s.t. xt → x, h(xt) → h(x), ˆ∂h(xt) ∋ yt → y
+�
+.
+We note that the limiting subdifferential is closed [41]:
+(xt, yt) → (x, y), h(xt) → h(x), yt ∈ ∂h(xt) =⇒ y ∈ ∂h(x).
+After establishing the definitions of subdifferentials in the real case, we extend them to the
+complex case. If z = z1 + z2i, where z1, z2 ∈ Rn2, then the limiting subdifferential of a proper
+and lower semicontinuous function f : Cn2 → (−∞, +∞] is defined by
+∂f(z) := ∂z1f(z) + ∂z2f(z)i.
+(B.1)
+If the function f is continuously differentiable at the point z, then with slight abuse of notation,
+we denote its gradient by ∇f(z), and ∂f(z) = {∇f(z)} [41].
+
+22
+KEVIN BUI AND ZICHAO (WENDY) DI
+B.1. Proof of Lemma 4.2.
+Proof. If {(ωt, zt)}∞
+t=1 is bounded, then there exists a constant C such that ∥ωt∥∞, ∥zt∥∞ ≤
+C for all t ∈ N. We establish that L(ω) has a Lipschitz continuous gradient with respect to
+ω. For any ω1, ω2 ∈ Cm2 at iteration t, we have
+∥∇ωL(ω2) − ∇ωL(ω1)∥2 ≤
+������
+N
+�
+j=1
+β1(Sjzt)∗ ◦ (ω2 − ω1) ◦ (Sjzt)
+������
+2
+≤
+�
+�
+N
+�
+j=1
+β1
+��(Sjzt)∗ ◦ (Sjzt)
+��
+∞
+�
+� ∥ω2 − ω1∥2 ≤ β1NC2∥ω2 − ω1∥2.
+Hence, we observe that L(ω) has a Lipschitz continuous gradient with Lipschitz constant
+Lω := β1NC2. By the descent property [8, Definition 1], at iteration t we have
+L(ωt+1) − L(ωt) ≤ R(⟨∇ωL(ωt), ωt+1 − ωt⟩) + Lω
+2 ∥ωt+1 − ωt∥2
+2
+= −δt
+ωR(⟨∇ωL(ωt), ˜∇ωL(ωt)⟩) + Lω(δt
+ω)2
+2
+∥ ˜∇ωL(ωt)∥2
+2,
+where the last equality is due to (3.12). Taking the expectation with respect to the first t
+iterations, we obtain
+Et[L(ωt+1)] − Et[L(ωt)] = −δt
+ωR
+�
+Et
+�
+⟨∇ωL(ωt), ˜∇ωL(ωt)⟩
+��
++ Lω(δt
+ω)2
+2
+Et
+�
+∥ ˜∇ωL(ωt)∥2
+2
+�
+≤ −
+�
+KL − δt
+ωLω(MV + KU)
+2
+�
+δt
+ωEt
+���∇ωL(ωt)
+��2
+2
+�
++ (δt
+ω)2LωM
+2
+,
+where the last inequality is due to combining (4.3), (4.4). and (4.7). Similarly, we can estimate
+(4.10) because we can compute that L(z) has a Lipschitz continuous gradient with Lipschitz
+constant Lz := β1NC2 + β2∥∆∥ and follow the same steps as above.
+B.2. Proof of Proposition 4.3.
+Proof. Because �∞
+t=1 ∥Ωt+1 −Ωt∥2
+2 < ∞, we have lim
+t→∞ Λt+1
+j
+−Λt
+j = 0 for each j = 1, . . . , N
+and lim
+t→∞ yt+1 − yt = 0, which implies from (3.6e)-(3.6f) that
+lim
+t→∞ ut
+j − F(ωt ◦ Sjzt) = 0
+∀j = 1, . . . , N,
+(B.2)
+lim
+t→∞ vt − ∇zt = 0.
+(B.3)
+It follows that {(ut, vt)}∞
+t=1 is bounded since {(ωt, zt)}∞
+t=1 is bounded. By (3.8), when B(·, ·)
+is AGM, we have
+∥ut+1
+j
+∥2 =
+������
+�
+dj + β1
+���F(ωt ◦ Sjzt) − 1
+β1 Λt
+j
+���
+1 + β1
+������
+2
+≥
+β1
+1 + β1
+� 1
+β1
+��Λt
+j
+��
+2 − ∥F(ωt ◦ Sjzt)∥2
+�
+,
+
+STOCHASTIC ADMM FOR PTYCHOGRAPHY
+23
+or equivalently,
+(1 + β1)∥ut+1
+j
+∥2 + β1∥F(ωt ◦ Sjzt)∥2 ≥ ∥Λt
+j∥2.
+(B.4)
+Similarly, when B(·, ·) is IPM, we have the same inequality as (B.4). As a result, {Λt}∞
+t=1 is
+bounded. Finally, we show that {yt}∞
+t=1 is bounded. By Lemma 3.1, we have two cases. When
+����(∇zt)i − (yt)i
+β2
+����
+∞
+≤ λ
+β2
+,
+we have
+λ
+β2
+≥
+����(∇zt)i − (yt)i
+β2
+����
+∞
+≥ 1
+β2
+��(yt)i
+��
+∞ −
+��(∇zt)i
+��
+∞ ,
+or λ + β2∥(∇zt)i∥∞ ≥ ∥(yt)i∥∞. Otherwise, we have
+∥(vt+1)i∥∞ ≥
+����
+����(∇zt)i − (yt)i
+β2
+���� − λ
+β2
+����
+∞
+≥
+����(∇zt)i − (yt)i
+β2
+����
+∞
+− λ
+β2
+≥ 1
+β2
+��(yt)i
+��
+∞ −
+��(∇zt)i
+��
+∞ − λ
+β2
+,
+or β2
+�
+∥(vt+1)i∥∞ +
+��(∇zt)i
+��
+∞
+�
++ λ ≥
+��(yt)i
+��
+∞. Altogether, {yt}∞
+t=1 is bounded since
+{(vt, zt)}∞
+t=1 is bounded. Therefore, we establish that {(Zt, Ωt)}∞
+t=1 is bounded.
+We see that
+L(Z, Ω) =
+N
+�
+j=1
+�
+B(|uj|2, dj) + β1
+2
+����uj − F(ω ◦ Sjz) + Λj
+β1
+����
+2
+2
+−
+1
+2β1
+∥Λj∥2
+2
+�
++ λ(∥v∥1 − α∥v∥2,1) + β2
+2
+����v − ∇z + y
+β2
+����
+2
+2
+−
+1
+2β2
+∥y∥2
+2
+≥
+N
+�
+j=1
+�
+B(|uj|2, dj) −
+1
+2β1
+∥Λj∥2
+2
+�
+−
+1
+2β2
+∥y∥2
+2.
+Because B(·, ·) is bounded below according to (3.2) and {Ωt}∞
+t=1 is bounded, {L(Zt, Ωt)}∞
+t=1
+is bounded below by some constant Linf. By (3.6a) and (3.6c), we have L(ut+1) ≤ L(ut) and
+L(vt+1) ≤ L(vt), respectively, so taking expectation with respect to the first t iterations, we
+obtain
+Et[L(ωt)] = Et[L(ut+1)] ≤ Et[L(ut)],
+(B.5)
+Et[L(zt)] = Et[L(vt+1)] ≤ Et[L(vt)] = Et[L(ωt+1)].
+(B.6)
+In addition, we have
+L(Λt+1) − L(Λt) =
+N
+�
+j=1
+R(⟨Λt+1
+j
+− Λt
+j, ut+1
+j
+− F(ωt+1 ◦ Sjzt+1)⟩)
+
+24
+KEVIN BUI AND ZICHAO (WENDY) DI
+= 1
+β1
+N
+�
+j=1
+���Λt+1
+j
+− Λt
+j
+���
+2
+2 = 1
+β1
+∥Λt+1 − Λt∥2
+2,
+where the second to last equality is due to (3.6e). Taking expectation with respect to the first
+t iterations gives
+Et[L(Λt+1)] − Et[L(Λt)] = 1
+β1
+Et
+���Λt+1 − Λt��2
+2
+�
+.
+(B.7)
+Similarly, we obtain
+Et[L(yt+1)] − Et[L(yt)] = 1
+β2
+Et[∥yt+1 − yt∥2
+2].
+(B.8)
+Summing up (4.9)-(4.10), (B.5)-(B.8) and taking total expectation, we have
+E[L(Zt+1, Ωt+1)] − E[L(Zt, Ωt)] ≤ 1
+β1
+Et
+���Λt+1 − Λt��2
+2
+�
++ 1
+β2
+E[∥yt+1 − yt∥2
+2]
+−
+�
+KL − δt
+ωLω(MV + KU)
+2
+�
+δt
+ωE
+���∇ωL(ωt)
+��2
+2
+�
++ (δt
+ω)2LωM
+2
+−
+�
+KL − δt
+zLz(MV + KU)
+2
+�
+δt
+zE
+���∇zL(zt)
+��2
+2
+�
++ (δt
+z)2LzM
+2
+.
+(B.9)
+By (4.11), lim
+t→∞ δt
+ω = 0 and lim
+t→∞ δt
+z = 0, which means that we can assume without generality
+that δt
+ωLω, δt
+zLz <
+KL
+MV +KU for all t ∈ N. Hence, summing up t = 1, . . . , T, we obtain
+Linf − E[L(Z0, Ω0)] ≤ E[L(ZT+1, ΩT+1)] − E[L(Z0, Ω0)]
+≤
+T
+�
+t=1
+�
+C∥Ωt+1 − Ωt∥2
+2 − KLδt
+ω
+2
+E
+���∇ωL(ωt)
+��2
+2
+�
+− KLδt
+z
+2
+E
+���∇zL(zt)
+��2
+2
+�
++ (δt
+ω)2LωM
+2
++ (δt
+z)2LzM
+2
+�
+,
+where C = max{ 1
+β1 , 1
+β2 }. Rearranging the inequality and letting T → ∞ give us
+0 ≤KL
+2
+∞
+�
+t=1
+�
+δt
+ωE
+���∇ωL(ωt)
+��2
+2
+�
++ δt
+zE
+���∇zL(zt)
+��2
+2
+��
+≤
+E[L(Z0, Ω0)] − Linf +
+∞
+�
+t=1
+�
+C∥Ωt+1 − Ωt∥2
+2 + (δt
+ω)2LωM
+2
++ (δt
+z)2LzM
+2
+�
+.
+By the assumption, the right-hand side is bounded, so therefore it implies (4.12).
+
+STOCHASTIC ADMM FOR PTYCHOGRAPHY
+25
+B.3. Proof of Theorem 4.4.
+Proof. If {(wt, zt)}∞
+t=1 is bounded, then ∥wt∥2, ∥zt∥2 ≤ C for all t ∈ N for some constant
+C > 0. At a given iteration t, we have ∇ωL(ωt+1) = 0 by the first-order optimality condition
+of (3.6b) and ∇ωL(ω) to be Lipschitz. It follows that
+∥∇ωL(ωt)∥2 = ∥∇ωL(ωt+1) − ∇ωL(ωt)∥2 ≤ Lω∥ωt+1 − ωt∥2 ≤ 2CLω.
+Similarly, we have ∥∇zL(zt)∥2 ≤ 2CLz. As a result, by squaring the inequalities and tak-
+ing their expectation,
+��
+E
+���∇ωL(ωt)
+��2
+2
+�
+, E
+���∇zL(zt)
+��2
+2
+���∞
+t=1 is bounded. The step size
+condition (4.11) implies that lim
+t→∞ δt
+ω = 0 and lim
+t→∞ δt
+z = 0. Since the SGD steps for ω and z
+are
+ωt+1 = ωt − δt
+ω ˜∇L(ωt), zt+1 = zt − δt
+z ˜∇L(zt),
+it follows that by taking expectation with respect to the first t iterations and using Assumption
+4.1,
+Et[
+��ωt+1 − ωt��2
+2] = (δt
+ω)2Et[∥ ˜∇ωL(ωt)∥2
+2] ≤ (δt
+ω)2 �
+M + (MV + KU)Et
+���∇ωL(ωt)
+��2
+2
+��
+,
+Et[
+��zt+1 − zt��2
+2] = (δt
+z)2Et[∥ ˜∇zL(zt)∥2
+2] ≤ (δt
+z)2 �
+M + (MV + KU)Et
+���∇zL(zt)
+��2
+2
+��
+.
+Because
+��
+E
+���∇ωL(ωt)
+��2
+2
+�
+, E
+���∇zL(zt)
+��2
+2
+���∞
+t=1 is bounded, we apply total expectation and
+take the limit to obtain
+lim
+t→∞ E[
+��ωt+1 − ωt��2
+2] ≤ lim
+t→∞(δt
+ω)2 �
+M + (MV + KU)E
+���∇ωL(ωt)
+��2
+2
+��
+= 0,
+(B.10)
+lim
+t→∞ E[
+��zt+1 − zt��2
+2] ≤ lim
+t→∞(δt
+z)2 �
+M + (MV + KU)E
+���∇zL(zt)
+��2
+2
+��
+= 0.
+(B.11)
+Earlier, in the proof of Proposition 4.3, we obtain
+lim
+t→∞ ut
+j − F(ωt ◦ Sjzt) = 0,
+∀j = 1, . . . , N,
+(B.12)
+lim
+t→∞ vt − ∇zt = 0.
+(B.13)
+By Proposition 4.3, we have {(Zt, Ωt)}∞
+t=1 to be bounded and (4.12) to be true.
+Because {Zt}∞
+t=1 is bounded, there exists a convergent subsequence {(Ztk, Ωtk)}∞
+k=1 and
+a point (Z⋆, Ω⋆) such that lim
+k→∞(Ztk, Ωtk) = (Z⋆, Ω⋆). In addition, (4.12) and (B.10)-(B.13)
+hold for the subsequence and its further subsequences. (B.10)-(B.11) imply that there is a
+further subsequence {(Ztkℓ, Ωtkℓ)}∞
+ℓ=1 such that
+lim
+ℓ→∞ ωtkℓ+1 − ωtkℓ = 0 a.s.,
+(B.14)
+lim
+ℓ→∞ ztkℓ+1 − ztkℓ = 0 a.s.
+(B.15)
+
+26
+KEVIN BUI AND ZICHAO (WENDY) DI
+From (4.12), we obtain lim
+ℓ→∞ E
+�
+δ
+tkℓ
+ω ∥∇ωL(ωtkℓ)∥2
+2 + δ
+tkℓ
+z ∥∇zL(ztkℓ)∥2
+2
+�
+= 0, so we obtain
+lim inf
+ℓ→∞ E[∥∇ωL(ωtkℓ)∥2
+2] = 0
+(B.16)
+lim inf
+ℓ→∞ E[∥∇zL(ztkℓ)∥2
+2] = 0.
+(B.17)
+Now we show that (Z⋆, Ω⋆) is a stochastic KKT point a.s. From (B.12)-(B.13), we have
+u⋆
+j = lim
+ℓ→∞ u
+tkℓ
+j
+= lim
+ℓ→∞ F(ωtkℓ ◦ Sjztkℓ) = F(ω⋆ ◦ Sjz⋆),
+(B.18)
+v⋆ = lim
+ℓ→∞ vtkℓ = lim
+ℓ→∞ ∇ztkℓ = ∇z⋆.
+(B.19)
+By (B.12) and (B.14)-(B.15), we have
+lim
+ℓ→∞ utkℓ+1 = lim
+ℓ→∞ F
+�
+ωtkℓ+1 ◦ Sjztkℓ+1�
+= F
+�
+lim
+ℓ→∞ ωtkℓ+1 ◦ Sj
+�
+lim
+ℓ→∞ ztkℓ+1
+��
+= F
+�
+lim
+ℓ→∞ ωtkℓ ◦ Sj
+�
+lim
+ℓ→∞ ztkℓ
+��
+= lim
+ℓ→∞ F
+�
+ωtkℓ ◦ Sjztkℓ�
+= lim
+ℓ→∞ utkℓ a.s.
+As a result, we have
+lim
+ℓ→∞ utkℓ+1 − utkℓ = 0 a.s.
+(B.20)
+Similarly, by (B.13) and (B.15), we have
+lim
+ℓ→∞ vtkℓ+1 − vtkℓ = 0 a.s.
+(B.21)
+For the sake of brevity, we will omit “a.s.” henceforth.
+Next we prove (4.1a). Suppose that B(·, ·) is AGM. At iteration tkℓ, the first-order opti-
+mality condition of (3.6a) is
+0 ∈ ∂
+���u
+tkℓ+1
+j
+���
+�
+|u
+tkℓ+1
+j
+| −
+�
+dj
+�
++ Λ
+tkℓ
+j
++ β1
+�
+u
+tkℓ+1
+j
+− F(ωtkℓ ◦ Sjzkℓ)
+�
+,
+(B.22)
+where
+�
+∂
+���u
+tkℓ+1
+j
+���
+�
+i =
+�
+�
+�
+�
+�
+�
+�
+(u
+tkℓ+1
+j
+)i
+|(u
+tkℓ+1
+j
+)i|
+,
+if (u
+tkℓ+1
+j
+)i ̸= 0
+{u ∈ C : |u| ≤ 1}
+if (u
+tkℓ+1
+j
+)i = 0.
+If (u⋆
+j)i ̸= 0 for some i ∈ {1, . . . , n2}, then there exists a neighborhood Br((u⋆
+j)i) = {u ∈ C :
+|u − (u⋆
+j)i| ≤ r} such that all points in Br((u⋆
+j)i) are nonzero. By (B.20) and the fact that
+lim
+ℓ→∞ utkℓ = u⋆, we have (u
+tkℓ+1
+j
+)i ̸= 0 for all tkℓ sufficiently large. As a result, (B.22) becomes
+0 =
+(u
+tkℓ+1
+j
+)i
+|(u
+tkℓ+1
+j
+)i|
+�
+|(u
+tkℓ+1
+j
+)i| − (
+�
+dj)i
+�
++ (Λ
+tkℓ
+j )i + β1
+�
+(u
+tkℓ+1
+j
+)i − (F(ωtkℓ ◦ Sjzkℓ))i
+�
+,
+(B.23)
+
+STOCHASTIC ADMM FOR PTYCHOGRAPHY
+27
+By (B.12) and (B.20), we take the limit to obtain
+0 =
+(u⋆
+j)i
+|(u⋆
+j)i|
+�
+|(u⋆
+j)i| − (
+�
+dj)i
+�
++ (Λ⋆
+j)i.
+(B.24)
+On the other hand, when (u⋆
+j)i = 0 for some index i, we have (u
+tkℓ
+j )i ∈ Br((u⋆
+j)i) \ {(u⋆
+j)i} to
+be nonzero. Taking the absolute value of (B.23) gives us
+���(Λ
+tkℓ
+j )i + β1
+�
+(u
+tkℓ+1
+j
+)i − (F(ωtkℓ ◦ Sjzkℓ))i
+���� =
+���|(u
+tkℓ+1
+j
+)i| − (
+�
+dj)i
+��� .
+Again, by (B.12) and (B.20), taking the limit yields
+��(Λ⋆
+j)i
+�� =
+���(
+�
+dj)i
+��� ,
+which implies that there exists u′ ∈ {u ∈ C : |u| ≤ 1} such that −u′(
+�
+dj)i + (Λ⋆
+j)i = 0.
+This result and (B.24) give us (4.1a) for the AGM case.
+As for the IPM case, because
+lim
+x→0+ x − d log x = +∞, it follows that (u⋆
+j)i ̸= 0 for all i, so we only need to worry about the
+nonzero case. Hence, verifying (4.1a) for IPM requires computation similar to the nonzero
+case for AGM, so it is omitted for brevity.
+At iteration tkℓ, the first-order optimality condition of (3.6c) is
+−ytkℓ
+λ
+− β2
+λ
+�
+vtkℓ − ∇ztkℓ�
+∈ ∂
+�
+∥vtkℓ+1∥1 − α∥vtkℓ+1∥2,1
+�
+.
+(B.25)
+By (B.13) and (B.21), we have
+lim
+ℓ→∞ −ytkℓ
+λ
+− β2
+λ
+�
+vtkℓ − ∇ztkℓ�
+= −y⋆
+λ ,
+lim
+ℓ→∞ vtkℓ+1 = lim
+ℓ→∞ vtkℓ = v⋆.
+By continuity, lim
+ℓ→∞ ∥vtkℓ+1∥1 − α∥vtkℓ+1∥2,1 = ∥v⋆∥1 − α∥v⋆∥2,1. Altogether, by closedness of
+limiting subdifferential, we establish (4.1b).
+Lastly, we prove (4.1e′)-(4.1f′). At iteration tkℓ, we have
+∇ωL(ωtkℓ) = −
+N
+�
+j=1
+�
+�
+�β1(Sjztkℓ)∗ ◦
+�
+�F−1
+�
+�u
+tkℓ+1
+j
++
+Λ
+tkℓ
+j
+β1
+�
+� − ωtkℓ ◦ Sjztkℓ
+�
+�
+�
+�
+� .
+Taking the limit, we see that
+∇ωL(Z⋆, Ω⋆) = −
+N
+�
+j=1
+�
+β1(Sjz⋆)∗ ◦
+�
+F−1
+�
+u⋆
+j +
+Λ⋆
+j
+β1
+�
+− ω⋆ ◦ Sjz⋆
+��
+= lim
+ℓ→∞ ∇ωL(ωtkℓ).
+Applying Fatou’s Lemma to (B.16), we have E
+�
+∥∇ωL(Z⋆, Ω⋆)∥2
+2
+�
+≤ lim inf
+ℓ→∞ E[∥∇ωL(ωtkℓ)∥2
+2] =
+0. Proving (4.1f′) is similar to proving (4.1e′), so we omit the proof. Altogether, (Z⋆, Ω⋆) is a
+stochastic KKT point a.s.
+
+28
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+Under consideration for publication in J. Plasma Phys.
+1
+Energetic bounds on gyrokinetic instabilities.
+Part III. Generalized free energy.
+G. G. Plunk1†, and P. Helander1
+1Max-Planck-Institut für Plasmaphysik, 17491 Greifswald, Germany
+(Received xx; revised xx; accepted xx)
+Free energy, widely used as a measure of turbulence intensity in weakly collisional
+plasmas, has been recently found to be a suitable basis to describe both linear and
+nonlinear growth in a wide class gyrokinetic systems. The simplicity afforded by this
+approach is accompanied by some drawbacks, notably the lack of any explicit treatment
+of wave-particle effects, which makes the theory unable to describe things like stability
+thresholds or dependence on the geometry of the background magnetic field. As a step
+toward overcoming these limitations, we propose an extension of the theory based on
+a generalization of free energy. With this it is demonstrated that resonance effects
+are recovered, and the bounds on growth are significantly reduced. The simplicity
+and efficient computation of the associated “optimal” growth rates makes the theory
+potentially applicable to stellarator optimization.
+1. Introduction
+This is the third paper in a series (Helander and Plunk 2022; Plunk and Helander 2022),
+in which we develop a linear and nonlinear stability theory based on gyrokinetic energy
+balance. The last two papers used Helmholtz free energy, and introduced the concept of
+optimal mode growth for fully electromagnetic gyrokinetic. The present paper proposes
+a generalized energetic measure of fluctuations, allowing the inclusion of additional
+instability mechanisms. We do this first for a simple case, namely the electrostatic limit
+(low plasma β) with only one kinetic species (ions), with the electrons being treated
+adiabatically. These simplifications limit the application to ion-temperature-gradient
+(ITG) driven turbulence, though the central result of the paper is capable of treating
+completely general details of the magnetic geometry.
+Free energy is a useful concept for understanding nonlinear and linear aspects of
+plasma turbulence. At the level of linear instabilities it is common to speak of a source
+of free energy that drives modes. Indeed, without a source of free energy, provided
+by background plasma gradients (density, temperature, flows), there can be no linear
+instabilities (nor can there be subcritical turbulence Landreman et al. (2015); Plunk and
+Helander (2022)). However, there is usually another ingredient that arises in the detailed
+analysis of normal linear instabilities, namely the wave-particle resonance. In gyrokinetic
+theory, this involves parallel motion (along the magnetic field) and magnetic drift, and
+the resonance is physically linked to the work that the electrostatic field performs on
+gyrocenter motion. However, the terms needed to capture this do not contribute to free
+energy balance, and the influence of resonance therefore cannot be accounted for by the
+optimal modes that we introduced in our previous works.
+In this work we propose a new measure of gyrokinetic fluctuations, a generalization
+of the concept of free energy, that incorporates the resonance mechanism, and, via the
+† Email address for correspondence: gplunk@ipp.mpg.de
+arXiv:2301.00988v1  [physics.plasm-ph]  3 Jan 2023
+
+2
+G. G. Plunk and P. Helander
+magnetic drift, the full details of the background magnetic geometry. We demonstrate
+the existence of a class of quadratic measures closely related to Helmholtz free energy
+that behave as positive-definite norms for fluctuations in the distribution function. The
+corresponding energy balance equation is then used to derive a theory of optimal modes
+that most efficiently extract this energy from its source. The growth rate of these optimal
+modes provides a rigorous upper bound on the growth rate of linear instabilities, and this
+bound is shown to be lower than that obtained previously from Helmholtz free energy.
+By studying some simple limits, we show that we recover some expected behavior of both
+the slab and toroidal branches of the ITG mode.
+2. Definitions and gyrokinetic energy balance
+The ion gyrokinetic equation in the electrostatic limit is written
+∂gk
+∂t + v∥
+∂gk
+∂l + i˜ωdgk + 1
+B2
+�
+k′
+B · (k × k′)δφk′gk−k′ = eiF0
+Ti
+� ∂
+∂t + iωT
+∗
+�
+δφk,
+(2.1)
+where g is the gyro-center dependent part of the perturbed ion distribution function, i.e.
+fi = (1 − eiδφ(r)/Ti) Fi0 + g(R, Ei, µi, t). Its phase space variables are the energy Ea =
+mav2/2 + eaΦ(ψ) and the magnetic moment µa = mav2
+⊥/(2B), and the perpendicular
+wavenumber is k = k⊥ = kψ∇ψ + kα∇α with kψ and kα independent of the arc length
+l along the magnetic field, and ψ and α defined via B = Bb = ∇ψ × ∇α. We neglect
+collisions here†, and used the simplified notation gk = gi,k, and ω∗ = ω∗i, etc because the
+adiabatic approximation ge,k = 0 is assumed throughout‡. We will also assume kρi ∼ 1,
+implying kρe ≪ 1. The gyrokinetic free energy balance equation obtained in this limit
+reads
+d
+dt
+�
+k
+H = 2
+�
+k
+D,
+(2.2)
+where the drive term D is
+D(k, t) = Im ei
+��
+gkωT
+∗ δφ
+∗
+kd3v
+�
+,
+(2.3)
+and the free energy, expressed in terms of the gyrocenter distribution function
+H(k, t) =
+�
+Ti
+� |gk|2
+Fi0
+d3v −
+�
+a
+nae2
+a
+Ta
+|δφk|2
+�
+,
+(2.4)
+where the space average is defined as (see also Helander and Plunk (2022) for general-
+izations)
+⟨· · ·⟩ = lim
+L→∞
+� L
+−L
+(· · · )dl
+B
+� � L
+−L
+dl
+B .
+(2.5)
+The diamagnetic frequencies are
+† We do not retain collisions, since we will not be able to fix the sign of its contribution in
+our later analysis.
+‡ Here we do not include the customary correction for the zonal component (Dorland and
+Hammett 1993), but it does not affect the subsequent analysis, as the growth of this component
+is always zero because it has no free energy source (D = 0 for kα = 0).
+
+Energetic bounds. Part III
+3
+ω∗a = kαTa
+ea
+d ln na
+dψ
+,
+ωT
+∗a = ω∗a
+�
+1 + ηa
+�mav2
+2Ta
+− 3
+2
+��
+,
+and the magnetic drift frequency is
+˜ωd = k · vd,
+where the magnetic drift velocity is vd = ˆb × ((v2
+⊥/2)∇ ln B + v2
+∥κ)/Ωi, κ = ˆb · ∇ˆb, and
+Ωa = eaB/ma is gyrofrequency. Assuming ∇ ln B ≈ κ (low plasma β), we can separate
+the drift frequency into velocity-dependent and space-dependent factors following Plunk
+et al. (2014)†:
+˜ωd = ωd(l)
+�
+v2
+⊥
+2v2
+th
++
+v2
+∥
+v2
+th
+�
+.
+(2.6)
+The gyro-averaged electrostatic potential is denoted
+δφk = J0
+�k⊥v⊥
+Ωi
+�
+δφk,
+and the quasi-neutrality condition is
+�
+a
+nae2
+a
+Ta
+δφk = ei
+�
+gkJ0d3v, ,
+(2.7)
+where Jn = Jn(k⊥v⊥/Ωi). Following our previous convention, we define the free energy
+as twice that which appears in some other publications. Henceforth, we suppress the
+k-subscripts.
+2.1. Electrostatic energy and positive-definiteness of free energy
+It is useful to decompose the free energy into a part associated with a perturbed
+distribution function and a part associated with fluctuations in the electrostatic field,
+i.e.
+H = G + E,
+(2.8)
+where
+G = −TiSi =
+�
+Ti
+� |δF|2
+Fi0
+d3v
+�
+(2.9)
+E =
+�
+(τ + 1 − Γ0) nie2
+i
+Ti
+|δφ|2
+�
+.
+(2.10)
+Recall the conventional definitions Γn(b) = exp(−b)In(b) and b = k2
+⊥ρ2
+i = k2
+⊥Ti/(miΩ2
+i ),
+and τ = (eTi)/(eiTe). Note that δF = g−(eiδφ/Ti)F0 is the gyro-averaged perturbed dis-
+† Actually, there is spatial dependence in both v⊥ and v∥, since these are not the proper
+gyrokinetic phase-space variables, but a separation like this is useful to make contact with
+known limits from gyrokinetic theory of the ITG mode.
+
+4
+G. G. Plunk and P. Helander
+tribution function, and these two contributions to H can be identified as the gyrokinetic
+perturbed entropy and the gyrokinetic field energy.
+Although the general electromagnetic free energy admits a similar form as Eqn. 2.8
+(see for instance Helander and Plunk (2022)), we note that the electrostatic limit is
+distinguished by the fact that the field contribution E is itself a nonlinear invariant of
+the gyrokinetic system (Schekochihin et al. 2009), and its conservation may be viewed
+as an additional constraint on the nonlinear dynamics, with consequences e.g. for the
+cascade and production of large-scale E × B flows (PLUNK et al. 2010).
+For what follows, we need the electrostatic energy balance equation. This is obtained
+by multiplying the ion gyrokinetic equation by eiδφ
+∗ integrate over velocity, average over
+the parallel coordinate l, and sum over perpendicular wavenumber k, yielding (Helander
+et al. 2013)
+d
+dt
+�
+k
+E = 2
+�
+k
+K,
+(2.11)
+where the drive term K is
+K = −Re ei
+��
+δφ
+∗ �
+v∥
+∂
+∂l + iωd
+�
+gd3v
+�
+.
+(2.12)
+This is composed to two contributions, one coming from the parallel streaming term,
+and the other coming from the magnetic drift term. The first contribution has a simple
+physical interpretation, as the rate of energy exchanged between particles and the parallel
+electric field (i.e. the volume average of the parallel current multiplied by the parallel
+electric field), while the second term describes an analogous process in the perpendicular
+direction associated with the drift motion of gyrocenters.
+Eqn. 2.8 is a physically transparent form that makes it clear that the free energy H is
+a positive-definite norm for the distribution function g†, i.e.
+H ⩾ 0, and H = 0 iff g = 0
+(2.13)
+over all of phase space, ℓ and v. To see this, note that the quantities G and E are both
+positive, i.e. G ⩾ 0, obviously, and E ⩾ 0 because Γ0 ⩽ 1. Therefore if H = 0 then both
+E = 0 and G = 0. The first implies δφ = 0 everywhere, while the second implies δF = 0
+over all of phase space; δφ = 0 and δF = 0 obviously implies g = 0.
+We note that positive-definiteness is a desirable property of an energetic measure
+that can be useful for setting bounds on the growth rate of fluctuations; if a non-
+zero fluctuation (g ̸= 0) has zero measure M then the rate of growth d ln M/dt can
+be unbounded.
+Although we mainly consider a plasma with a single kinetic ion species and adiabatic
+electrons, the concepts and the formalism carry over to the more general case of a plasma
+with an arbitrary number of kinetic species, as shown in Appendix A. An important
+limitation, however, is that magnetic fluctuations and collisions are neglected.
+2.2. Generalized Free Energy
+The positive definiteness of H suggests a family of related quadratic energetic measures
+that are also positive definite. In particular it is clear that something of the form
+† By extension, using Eqn. 2.7, H can be shown to also be a positive-definite norm for the total
+deviation of the distribution function δf = g − (eiφ/Ti)F0 from the zeroth-order Maxwellian.
+
+Energetic bounds. Part III
+5
+˜H = H − ∆E,
+(2.14)
+will be positive-definite, by the same arguments of the previous section, for particular
+values of the parameter ∆. For instance the choice ∆ < 1 allows trivial generalization of
+the arguments, but we will see that the value can be extended beyond this.
+To find a range of permissible values of ∆, we will consider a diagonalization of ˜H,
+meaning that we will define a distribution function ˜g, which allows the energy to be
+expressed using to the Euclidean norm,
+˜H = ||˜g||2 = (˜g, ˜g),
+(2.15)
+where we have introduced the inner product
+(˜g1, ˜g2) =
+�
+Ti
+� ˜g∗
+1˜g2
+F0
+d3v
+�
+.
+(2.16)
+To find the relationship between ˜g and g, we introduce the Ansatz ˜g = g −νJ0F0eiδφ/Ti,
+substitute this into Eqn. 2.15, using also Eqn. 2.10, and solve for the free parameter ν,
+yielding
+ν = 1
+Γ0
+�
+1 + τ −
+�
+(1 + τ − Γ0)(1 + τ − ∆Γ0)
+�
+,
+(2.17)
+where we have taken the negative root for convenience. Observe that in order for ν to be
+real, we must have
+∆ ⩽ (1 + τ)/Γ0.
+(2.18)
+The parameter ∆ can of course be negative, in which case its magnitude is unbounded.
+Noting that Γ0 generally depends on k, we may also assume the more restrictive ∆ ⩽
+(1 + τ) to ensure that ˜H remains a nonlinear invariant.
+We pause to note that the choice ∆ = 0 yields a novel form of the conventional
+(Helmholtz) free energy, immediately suggesting what can be considered as the phase-
+space density of free energy, namely the quantity Ti|˜g|2/F0, for which there has not yet
+been an expression available.†
+It is useful now to write quasi-neutrality in terms of ˜g,
+ei
+Ti
+δφ = α
+ni
+�
+˜gJ0d3v,
+(2.19)
+where
+α =
+1
+�
+(1 + τ − Γ0)(1 + τ − ∆Γ0)
+.
+(2.20)
+Finally, we can show that ˜H is positive-definite. First, positivity follows from Eqn. 2.15,
+and it is obvious from Eqn. 2.7 that if g = 0 then δφ = 0 so that E and H both vanish,
+implying ˜H = 0. On the other hand, if we assume that ˜H = 0, then Eqn. 2.15 implies that
+˜g = 0, and Eqn. 2.19 implies that δφ = 0, from which we conclude g = 0. In summary,
+˜H ⩾ 0 and ˜H = 0 iff g = 0.
+† The idea for a phase-space density of free energy (i.e. a quantity that can be directly
+integrated over phase space to yield the total free energy) was suggested by Teaca.
+
+6
+G. G. Plunk and P. Helander
+3. Modes of optimal growth
+A key point in introducing the generalization of free energy ˜H is that this quantity
+introduces wave-particle effects (parallel resonance and drift resonance) that enter the
+electrostatic energy balance equation, Eqn. 2.11. Note that the case ∆ = 0 (i.e. the
+“conventional” Helmholtz free energy) is included as a limit ∆ = 0 and so the most
+stringent bound on growth obtained from the generalized free energy will be at least as
+good as the known bound obtained from the Helmholtz free energy.
+Note that, as long as the parameter ∆ is independent of k, the quantity ˜H is conserved
+by the nonlinearity, i.e. under summation over k. This is because it is a linear combination
+of two nonlinear invariants. One simply combines Eqns. 2.2 and 2.11 to obtain
+d
+dt
+�
+k
+˜H = 2
+�
+k
+(D − ∆K),
+for ∆ independent of k,
+(3.1)
+i.e. the change of this measure is due to the drive terms of electrostatic and free energy,
+and is otherwise conserved by the turbulent interactions. It is potentially useful to also
+consider ∆ that does depend on k, for the purpose of obtaining bounds on linear growth,
+but the nonlinear implications will be less clear in that case.
+In direct analogy to how modes of optimal free energy growth were defined, we
+introduce a rate Λ
+Λ = (D − ∆K)/ ˜H
+(3.2)
+to be optimized over the space of ion distribution functions g. We note the bound on
+conventional gyrokinetic instability growth rates,
+γL ⩽ max
+g
+Λ.
+(3.3)
+Having already found a diagonal form of the generalized free energy, Eqn. 2.15, we
+need not use a variational approach to find the states of extremal Λ. We simply identify
+the Hermitian linear operators associated with the input of free energy and electrostatic
+energy, i.e.
+D = (˜g, D˜g),
+(3.4)
+K = (˜g, K˜g)
+(3.5)
+Using Eqn. 2.19 and Eqns. 2.3 and 2.12, and some straightforward algebra (see Appendix
+B), we obtain
+D˜g = iα
+2ni
+J0F0ηω∗
+�
+d3v′J′
+0˜g′
+�� v
+vth
+�2
+−
+� v′
+vth
+�2�
+,
+(3.6)
+where primes denote evaluation at v′ and vth =
+�
+2Ti/mi. For convenience, the operator
+K can be split into its parallel and perpendicular components as K = K∥ + Kd, for which
+we obtain
+Kd˜g = iα
+2ni
+ωd(ℓ)F0J0
+�
+d3v′J′
+0˜g′
+�
+�
+�
+v⊥
+√
+2vth
+�2
++
+� v∥
+vth
+�2
+−
+�
+v′
+⊥
+√
+2vth
+�2
+−
+�
+v′
+∥
+vth
+�2�
+� ,
+(3.7)
+
+Energetic bounds. Part III
+7
+and
+K∥˜g = α
+2ni
+F0
+�
+J0
+�
+−B ∂
+∂l
+� 1
+B
+�
+d3v′v′
+∥J′
+0˜g′
+�
++
+�
+d3v′v′
+∥
+∂J′
+0
+∂l ˜g′
+�
++v∥
+∂
+∂l
+�
+J0
+�
+d3v′J′
+0˜g′
+��
+.
+(3.8)
+In deriving Eqn. 3.8, it is important to note that the parallel derivative is taken at fixed
+magnetic moment and particle energy, and that the velocity-space volume element d3v
+is proportional to B/v∥ in these variables. More details are given in Appendix A. The
+kinetic eigenvalue problem can be stated now as
+Λ˜g = (D − ∆K) ˜g,
+(3.9)
+where solutions (Λ, g(l, v)) realize modes of optimal growth of ˜H. The analysis of this
+eigenproblem is greatly simplified by adopting a moment form.
+3.1. Moment form of eigenproblem
+As found in the preceding papers, there are natural moments that appear in the
+energy input terms that can be identified to reduce the dimensionality of the problem
+substantially. Upon inspecting the energy balance equations one finds the following key
+dimensionless integrals:
+κ1 =
+�
+d3vJ0˜g/ni,
+(3.10)
+κ2 =
+�
+d3v
+� v2
+v2
+th
+�
+J0˜g/ni,
+(3.11)
+κ3 =
+�
+d3v
+�
+v2
+⊥
+2v2
+th
++
+v2
+∥
+v2
+th
+�
+J0˜g/ni,
+(3.12)
+κ4 =
+�
+d3v
+� v∥
+vth
+�
+J0˜g/ni,
+(3.13)
+κ5 =
+�
+d3v
+� v∥
+vth
+� ∂J0
+∂l ˜g/ni,
+(3.14)
+where κ1 is a density-like moment, κ2 and κ3 are pressure-like, κ4 is parallel ion flow,
+while κ5 is more abstract.
+It is easy to recognize these integrals on the right hand side of Eqns. 3.6, 3.7 and
+3.8, and straightforward to rewrite those equations in moment form. The dimensional
+reduction is achieved by taking moments of the these equations to obtain a coupled set
+of five fluid equations. These, which are given in Appendix C, can be combined, leading,
+after lengthy algebra, to a relatively simple second order ordinary differential equation,
+the main result of this paper:
+�4Λ2
+α2 + (∆ωdG3 − ηω∗G1)2 − G0
+�
+(ηω∗)2G2 − 2∆ωdηω∗G4 + ∆2ω2
+dG5
+��
+κ1
+= ∆2v2
+thG0B
+�
+− ∂
+∂l
+�G0,2
+B
+∂κ1
+∂l
+�
++ G′′
+0,2
+B κ1 − ∂
+∂l
+�G′
+0,2
+B
+�
+κ1
+�
+,
+(3.15)
+
+8
+G. G. Plunk and P. Helander
+The functions Gm,n, G′
+m,n, and G′′
+m,n, which depend on arc length via b(l) and B(l), are
+defined in terms of integrals involving Bessel functions, and are evaluated in Appendix
+D. The other b-dependent factors (G0-G5) can be expressed in terms of Gm,n, and are
+evaluated in terms of more elementary Bessel functions in Appendix D.1.
+In Eqn. 3.15 we see the eigenvalue Λ entering quadratically, reflecting the fact that
+there will be two real roots, one positive and one negative, owing to Hermiticity and
+time-reversal symmetriy of the full eigenproblem, Eqn. 3.9. Note that the terms arising
+from the parallel drive of electrostatic energy are placed on the right hand side. In the
+following section, we will consider some simple limits of this equation, and leave its more
+general solution for a future publication.
+4. Simple limits
+In this section we will consider some simple limits applied to Eqn. 3.15, and draw some
+comparison to linear theory of the main instability targeted by limit of this paper, the
+ion temperature gradient (ITG) mode (see for instance Plunk et al. (2014)). To start,
+we note that taking ∆ = 0, so that ˜H becomes the conventional Helmholtz free energy,
+yields
+Λ2 =
+(ηω∗)2
+4(1 + τ − Γ0)(1 + τ)
+�
+G0G2 − G2
+1
+�
+,
+(4.1)
+which matches Eqn. 6.20 of Helander and Plunk (2022).
+In considering other simplifications, we first should note that the adiabatic electron ap-
+proximation already neglects a trapped particle population, which is not really consistent
+unless we take the magnetic field strength to be independent of arc length
+∂B
+∂l = 0.
+(4.2)
+We have avoided making this approximation explicitly, since the present paper lays the
+foundation for extensions, in which it will be useful to include variation in B. Making
+the approximation now leads to minor simplifications of Eqn. 3.15, where all the explicit
+factors of B drop out of the right-hand side. A more significant simplification is achieved
+by assuming unsheared and uniform magnetic geometry, in particular
+∂b
+∂l = 0,
+(4.3)
+∂ωd
+∂l
+= 0,
+(4.4)
+In this limit, all of the coefficients of Eqn. 3.15 are constants, and a simple dispersion
+relation is the obtained by taking ∂κ1/∂l = ik∥κ1. We find
+4Λ2
+α2 + (∆ωdG3 − ηω∗G1)2 − G0
+�
+(ηω∗)2G2 − 2∆ωdηω∗G4 + ∆2ω2
+dG5
+�
+= ∆2k2
+∥v2
+thG2
+0/2.
+(4.5)
+were we have used G0,2 = G0/2. As noted in Section 2.2, the quantity ∆ is a free
+parameter, over which we can optimize Λ to improve the bounds on the growth rate of
+fluctuations.
+
+Energetic bounds. Part III
+9
+4.1. Slab ITG mode
+Setting ωd = 0 leaves only the slab branch of the ITG mode, driven by the temperature
+gradient, and involving ion parallel resonance. Eqn. 4.5 reduces to
+4Λ2
+(ηω∗)2α2 = G0G2 − G2
+1 + ∆2κ−2
+∥ G2
+0/2,
+(4.6)
+where κ∥ = ηω∗/(k∥vth). Because G0G2−G2
+1 ⩾ 0, the two contributions on the right hand
+side are both positive but the solution for which Λ is minimal is actually not obtained
+for ∆ = 0, due to the implicit dependence of α on ∆ given by Eqn. 2.20.
+To obtain the value of ∆ which yields an optimal bound, we can look for extrema of
+Λ2/(ηω∗)2, i.e.
+d
+d∆
+�
+G0G2 − G2
+1 + ∆2κ−2
+∥ G2
+0/2
+(1 + τ − Γ0)(1 + τ − ∆Γ0)
+�
+= 0.
+(4.7)
+This results in a quadratic equation for ∆ that is still rather complicated so we will
+consider the limit b → 0; see Appendix D.2 for the relevant limits of Gm,n, etc. Applying
+the limit to Eqn. 4.6 yields
+Λ2
+(ηω∗)2 =
+3 + ∆2/κ2
+∥
+8τ(1 + τ − ∆)
+(4.8)
+This solution diverges as ∆ approaches 1+τ; recall that this is the upper limit allowed
+by Eqn. 2.18. It also grows in an unbounded fashion as ∆ → −∞. There is an optimal
+value giving minimal |Λ|, obtained by solving Eqn. 4.7 in this limit. This solution, denoted
+as ∆min, is
+∆min = 1 + τ −
+�
+(1 + τ)2 + 3κ2
+∥
+(4.9)
+where the negative root has been selected to be consistent with Eqn. 2.18. Substituting
+this solution into Eqn. 4.8 gives
+Λ2
+min =
+(ηω∗)2
+4¯κ2
+∥τ(1 + τ)
+��
+1 + 3¯κ2
+∥ − 1
+�
+,
+(4.10)
+where we define ¯κ∥ = κ∥/(1 + τ). This reaches its maximum value in the limit ¯κ∥ → 0,
+and is a decreasing function of |¯κ∥|, i.e.
+Λ2
+min =
+�
+3
+8τ(τ+1)(ηω∗)2,
+for ¯κ∥ → 0,
+√
+3
+4τ |ηω∗k∥vth|,
+for |¯κ∥| ≫ 1,
+(4.11)
+Physically, the first result implies that when drive (ηω∗) is much smaller than the parallel
+transit frequency (k∥vth), the best bound is equal to that obtained by free energy (∆ = 0).
+In this case, the bound is consistent from expectations of the growth rate of a resonant
+slab ITG mode, i.e. γL ∼ ηω∗.
+In the opposite limit (¯κ∥ ≫ 1), however, when the drive large, i.e. in the so-called
+non-resonant or “fluid” limit, we obtain a much lower bound, essentially the geometric
+mean of the drive and the parallel transit frequency k∥vth. We note that this bound is
+not as low as what is obtained from the non-resonant solution of the dispersion relation
+(without density gradient), i.e. γL ∼ ηω1/3
+∗
+(k∥vth)2/3 (Plunk et al. 2014), but nevertheless
+captures the expected weakening (relative to the resonant result) qualitatively.
+
+10
+G. G. Plunk and P. Helander
+It is interesting to observe that this latter limit corresponds to ∆min → −∞, making
+˜H in some sense dominated by the electrostatic component.
+4.2. Toroidal ITG mode
+Now taking k∥vth to be small, we can neglect the right-hand side of Eqn. 4.5, leaving
+4Λ2
+α2 = G0
+�
+(ηω∗)2G2 − 2∆ωdηω∗G4 + ∆2ω2
+dG5
+�
+− (∆ωdG3 − ηω∗G1)2 .
+(4.12)
+To derive the optimal choice of ∆, we again take the b → 0 limit and obtain from
+Eqn. 4.12
+Λ2
+(ηω∗)2 = 3∆2 − 8∆κd + 6κ2
+d
+16κ2
+dτ(τ + 1 − ∆) ,
+(4.13)
+where we define κd = ηω∗/ωd. Note the similar qualitative behavior with ∆ as Eqn. 4.6,
+namely its divergence at the ∆ → 1 + τ, and unbounded growth as ∆ → −∞. The key
+difference here arises in the linear term in the drive parameter κ; this is expected from
+the theory of the toroidal ITG mode since the sign of the drift frequency (associated with
+so-called ‘good’ and ’bad’ magnetic curvature) is important for the resonance.
+We now find the value of ∆ that minimizes Λ:
+∆min = (1 + τ)
+�
+1 −
+�
+2¯κ2
+d − 8¯κd/3 + 1
+�
+,
+(4.14)
+where we define the parameter ¯κd = κd/(1 + τ). Substituting this into Eqn. 4.13 yields
+Λ2
+min = (ηω∗)2
+�
+8¯κd (ζ (¯κd) − 1) + 3 (ζ (¯κd) − 1)2 + 6¯κ2
+d
+16τ(τ + 1)¯κ2
+dζ (¯κd)
+�
+,
+(4.15)
+with ζ =
+�
+2¯κ2
+d − 8¯κd/3 + 1. This expression for Λmin is naturally separated into a
+factor that depends only on the instability parameter ¯κd, from which we can derive
+the asymptotic behavior. To show the behavior of this factor we plot the quantity
+τ(1 + τ)Λ2
+min/(ηω∗)2 in Fig. 1. The overall behavior of Λmin is captured by the following
+limits
+Λ2
+min =
+�
+�
+�
+�
+�
+�
+�
+|ηω∗ωd|
+�
+3
+√
+2+4σ
+8τ
+�
+,
+for |¯κd| ≫ 1,
+(ηω∗)2
+24τ(1+τ),
+for ¯κd → 0,
+3(ηω∗)2
+8τ(1+τ),
+for ¯κd → 4/3
+(4.16)
+where we denote σ = ±1 as the sign of κd. At large drive (|¯κd| ≫ 1; |ηω∗| ≫ |ωd|(1 + τ))
+we recover the expected non-resonant (“fluid”) behavior of the toroidal ITG mode with no
+density gradient, namely γL ∼ √ηω∗ωd. Note that this growth rate is much smaller than
+the bound found by merely considering the Helmholtz free energy (Helander and Plunk
+2022). Although we do not see the complete stabilization (Λ = 0) at negative values of
+¯κd (opposite sign of ωd and ηω∗) expected from theory, there is a strong asymmetry,
+with |Λ| having its larger values at positive ¯κd and being comparatively much smaller for
+negative ¯κd.
+The value ¯κd = 4/3 (i.e. ηω∗ = 4(1 + τ)ωd/3) achieves the maximal value of Λmin
+at fixed ηω∗, and therefore is evocative of the resonance condition for the toroidal ITG
+modes ηω∗ ∼ ωd (Biglari et al. 1989). This value of ¯κd is obtained by solving for ∆min = 0,
+
+Energetic bounds. Part III
+11
+-1
+1
+2
+4/3
+3
+4
+0.1
+0.2
+0.3
+Figure 1. Bound of the growth rate of the toroidal ITG mode, obtained from the optimal
+growth of generalized free energy, plotted versus the instability parameter
+¯κd = ηω∗/[ωd(1 + τ)].
+explaining why it produces the worst bound, i.e. that given by optimal growth of Helmoltz
+free energy.
+It is noteworthy that for the limit ¯κd → 0 (ωd ≫ ηω∗/(1 + τ)) our method yields a
+value of |Λ| that is a factor of 1/3 reduced as compared to the resonant case, again at
+least qualitatively reproducing the expected stabilization of the toroidal ITG mode in
+this limit.
+5. Conclusion
+We have demonstrated that the use of a generalized form of free energy ˜H introduces
+some of the physics of wave-particle resonance that is missing in the theory of optimal
+mode growth of Helmholtz free energy (Helander and Plunk 2021, 2022; Plunk and
+Helander 2022). The growth rates of optimal modes of generalized free energy provide
+a rigorous upper bound on the growth of conventional gyrokinetic instabilities (“normal
+modes”), which is always below the Helmholtz bound, as it must be, given that the
+Helmholtz free energy is a special case of the generalized measure. Moreover, optimal
+modes of generalized free energy depend on the magnetic-field geometry to a greater
+extent than those associated with Helmholtz free energy. The difference in growth rates
+can be very large. For instance, in the important case of a strongly driven toroidal ITG
+mode, the Helmholtz bound is larger by a factor of order ηω∗/ωd ≫ 1.
+A single ordinary differential equation has been derived for optimal modes, allowing
+general magnetic geometry. We found solutions of this equation in some simple limits to
+demonstrate that it indeed recovers, at least qualitatively, some of the physical effects
+expected from the theory of linear ITG modes, including sensitivity to the ratio of the
+frequencies associated with drive and resonance, and transition of the instability when
+this ratio is near one. Density gradient dependence of ITG mode is absent from both
+electrostatic and free energy input terms, assuming adiabatic electrons, so its effect is
+not accounted for by the theory presented here.
+The results of this work have possible implications for “turbulence optimization”,
+i.e. the endeavor to shape the equilibrium magnetic geometry for low turbulence in
+
+12
+G. G. Plunk and P. Helander
+stellarators. The general result, Eqn. 3.15, allows in principle for the inclusion of the
+complete geometric information contained in gyrokinetics, that is needed to run gyroki-
+netic simulations. However, the solution of this equation should be far simpler and more
+efficient, due to the reduction of velocity space to a single moment. The results found for
+the toroidal branch of the ITG hint at a possible optimization strategy. Consider fixed
+plasma conditions, i.e. a given temperature gradient (ηω∗) and temperature ratio (τ): At
+high drive (ηω∗/(1+τ) > 4ωd/3), minimization of the optimal growth rate |Λ| is achieved
+by minimization of the magnetic drift ωd (i.e. magnetic curvature), corresponding to
+minimization of the strongly-driven (non-resonant) toroidal ITG mode. On the other
+hand, at low drive (ηω∗/(1 + τ) < 4ωd/3) the increase of ωd is favored, corresponding to
+a weakening of the marginally unstable ITG mode, i.e. an increase of the threshold of
+instability. The latter case corresponds to “critical gradient” optimization, an idea which
+has recently been developed (Roberg-Clark et al. 2022a,b).
+More general solutions of the optimal mode equation, and the application to optimiza-
+tion will be pursued in future works. Other special limits can also be explored including,
+adiabatic ion limits, appropriate for studying trapped electron and universal instabilities.
+Electromagnetic generalizations are also possible: although it is not clear how to construct
+an electromagnetic form of generalized free energy ˜H that is a nonlinear invariant, it is
+certainly possible to consider related measures that focus on linear bounds.
+Funding. This work has been carried out within the framework of the EUROfusion
+Consortium, funded by the European Union via the Euratom Research and Training
+Programme (Grant Agreement No 101052200 — EUROfusion). Views and opinions
+expressed are however those of the author(s) only and do not necessarily reflect those of
+the European Union or the European Commission. Neither the European Union nor the
+European Commission can be held responsible for them. This work was partly supported
+by a grant from the Simons Foundation (560651, PH).
+Declaration of Interests. The authors report no conflict of interest.
+Author ORCID. G. G. Plunk, https://orcid.org/0000-0002-4012-4038; P. Helander,
+https://orcid.org/0000-0002-0460-590X.
+Appendix A. Several kinetic species
+For a plasma with an arbitrary number of particle species, we multiply each gyrokinetic
+equation
+∂ga,k
+∂t
++ v∥
+∂ga,k
+∂l
++ iωdaga,k + 1
+B2
+�
+k′
+B · (k × k′)δφk′ga,k−k′
+(A 1)
+= eaFa0
+Ta
+� ∂
+∂t + iωT
+∗a
+�
+δφk ,
+(A 2)
+by eaδφ
+∗
+k, integrate over velocity space, take the real part and the average ⟨· · · ⟩ over the
+flux tube, and sum over all species a and wave vectors k. In other words, we apply the
+operator
+Re
+�
+a,k
+ea
+��
+δφ
+∗
+k (· · · ) d3v
+�
+.
+(A 3)
+Since the expression
+Re (k × k′)δφ
+∗
+k′δφkga,k−k′ = 1
+2(k × k′)
+�
+δφ
+∗
+k′δφkga,k−k′ + δφk′δφ
+∗
+kg∗
+a,k−k′
+�
+(A 4)
+
+Energetic bounds. Part III
+13
+= 1
+2(k × k′)
+�
+δφ−k′δφkga,k−k′ + δφk′δφ−kga,−k+k′�
+(A 5)
+changes sign under an exchange of k and k′, the nonlinear terms cancel upon summation
+over k and k′, and we obtain
+Re
+�
+a,k
+ea
+��
+δφ
+∗
+k
+�∂ga,k
+∂t
++ v∥
+∂ga,k
+∂l
++ iωdaga,k − eaFa0
+Ta
+∂δφk
+∂t
+�
+d3v
+�
+= 0.
+(A 6)
+The quasineutrality equation (2.7) can be used to write the first term as
+Re
+�
+a,k
+ea
+�
+δφ
+∗
+k
+∂ga,k
+∂t d3v = d
+dt
+�
+k
+nae2
+a
+2Ta
+|δφk|2 .
+(A 7)
+We thus arrive at the electrostatic energy balance equation
+d
+dt
+�
+k
+nae2
+a
+2Ta
+�
+[1 − Γ0(bak)] |δφk|2�
+(A 8)
+= −Re
+�
+a,k
+�
+ea
+�
+δφ
+∗
+k
+�
+v∥
+∂ga,k
+∂l
++ iωdaga,k
+�
+d3v
+�
+,
+(A 9)
+which is the generalization of Eqn. 2.11 to several species. The right-hand side can be
+interpreted as minus the work done by the electric field on the various particle species.
+The first term in this expression contains
+�
+J0av∥
+∂ga,k
+∂l
+d3v =
+�
+σ
+2πσB
+m2a
+� ∞
+0
+dEa
+� Ea/B
+0
+J0a
+∂ga,k
+∂l
+dµa
+(A 10)
+= B ∂
+∂l
+� 1
+B
+�
+J0av∥ga,kd3v
+�
+−
+� ∂J0a
+∂l v∥ga,kd3v,
+(A 11)
+which we used in Eqn. 3.8, with σ = v∥/|v∥|, Ea = mav2/2 and µa = mav2
+⊥/(2B).
+Appendix B. Derivation of operators D and K
+We first write forms of D and K explicit in ˜g, noting that the contribution to D
+proportional to the density gradient is zero by use of quasi-neutrality with the adiabatic
+electron approximation (the factor ηω∗ ∝ dTi/dψ appears in what follows, but never
+ω∗ ∝ dni/dψ alone). For the same reason, the terms proportional to ν, involved in the
+transformation from g to ˜g, do not contribute, so expressing energy input in terms of ˜g
+merely has the consequence of introducing the overall factor α. The expressions are
+D = −Ti
+iα
+ni
+��
+˜g(v)˜g∗(v′) ηω∗
+� v2
+v2
+th
+�
+J0J′
+0d3vd3v′
+�
++ c.c.,
+(B 1)
+K∥ = −Ti
+α
+2ni
+��
+˜g∗(v′)
+�
+v∥
+∂˜g(v)
+∂l
+�
+J0J′
+0d3vd3v′
+�
++ c.c.,
+(B 2)
+Kd = −Ti
+iα
+2ni
+��
+˜g∗(v′)ωd˜g(v)J0J′
+0d3vd3v′
+�
++ c.c.,
+(B 3)
+
+14
+G. G. Plunk and P. Helander
+where we separate K = K∥ + Kd and ‘c.c.’ denotes complex conjutage. These can be
+re-expressed in terms of linear operators by writing them in the form
+D = (˜g, D˜g) =
+�
+Ti
+�
+d3v ˜g∗
+F0
+D˜g
+�
+,
+(B 4)
+K∥ = (˜g, K∥˜g) =
+�
+Ti
+�
+d3v ˜g∗
+F0
+K∥˜g
+�
+,
+(B 5)
+Kd = (˜g, Kd˜g) =
+�
+Ti
+�
+d3v ˜g∗
+F0
+Kd˜g
+�
+.
+(B 6)
+Identifying D and Kd is simply a matter of exchanging labels of dummy variables of
+integration (v for v′, etc.). Manipulating the expression for K∥ to reveal K∥ is more
+involved. We also need to integrate by parts in l and will need to use the fact that ∂/∂l
+is performed at fixed phase space variables Ei and µ = miv2
+⊥/(2B). The velocity space
+volume element contains an important factor of 1/v∥, which generally depends on l and
+does not itself commute with ∂/∂l:
+d3v = 2πv⊥dv⊥v∥ =
+�
+σ
+2πBdEidµi
+m2
+i |v∥|
+(B 7)
+where σ denotes the sign of v∥.
+Appendix C. Moment form of eigenproblem
+The three terms of Eqn. 3.9 can be rewritten in terms of the moments of ˜g (Eqn. 3.10):
+D˜g = iα
+2 ηω∗J0F0
+� v2
+v2
+th
+κ1 − κ2
+�
+,
+(C 1)
+K∥˜g = α
+2 F0
+�
+vthJ0
+�
+−B ∂
+∂l
+�κ4
+B
+�
++ κ5
+�
++ v∥
+∂
+∂l (J0κ1)
+�
+,
+(C 2)
+Kd˜g = iα
+2 ωdJ0F0
+��
+v2
+⊥
+2v2
+th
++
+v2
+∥
+v2
+th
+�
+κ1 − κ3
+�
+,
+(C 3)
+(C 4)
+Then, taking moments of Eqn. 3.9 yields the following five equations
+2Λ
+α κ1 = iηω∗ (G1κ1 − G0κ2) − i∆ωd (G3κ1 − G0κ3) − ∆G0vth
+�
+κ5 − B ∂
+∂l
+�κ4
+B
+��
+,(C 5)
+2Λ
+α κ2 = iηω∗ (G2κ1 − G1κ2) − i∆ωd (G2κ1 − G1κ3) − ∆G1vth
+�
+κ5 − B ∂
+∂l
+�κ4
+B
+��
+,(C 6)
+2Λ
+α κ3 = iηω∗ (G4κ1 − G3κ2) − i∆ωd (G5κ1 − G3κ3) − ∆G3vth
+�
+κ5 − B ∂
+∂l
+�κ4
+B
+��
+,(C 7)
+2Λ
+α κ4 = −∆vth
+�
+G′
+0,2κ1 + G0,2
+∂κ1
+∂l
+�
+,(C 8)
+2Λ
+α κ5 = −∆vth
+�
+G′′
+0,2κ1 + G′
+0,2
+∂κ1
+∂l
+�
+.
+(C 9)
+See the next section where the integrals Gm,n, etc., are defined and evaluated. Note that
+the final two equations can be immediately used to eliminate κ4 and κ5, leaving a system
+
+Energetic bounds. Part III
+15
+of three equations. The second and third equations are used together to find forms for κ2
+and κ3 in terms of κ1, and these forms are substituted into the first equation to obtain
+the final form, in terms of κ1 only, given by Eqn. 3.15.
+Appendix D. Bessel-type integrals
+The following definitions, mostly copied from Plunk and Helander (2022), are needed to
+perform the various integrals that appear in the moment equations for our eiegenproblem.
+First we need a general form of Weber’s integral,
+In(p, a1, a2) =
+� ∞
+0
+exp(−pt2)Jn(a1t)Jn(a2t)tdt
+= 1
+2p exp
+�−a2
+1 − a2
+2
+4p
+�
+In
+�a1a2
+2p
+�
+(D 1)
+where In is the modified Bessel function of order n. The integrals we need to evaluate
+can be conveniently found in terms of In. We define
+G⊥m(b) = 2
+� ∞
+0
+xm+1
+⊥
+exp(−x2
+⊥)J2
+0(
+√
+2bx2
+⊥)dx⊥,
+(D 2a)
+G(1)
+⊥m(b) = 2
+� ∞
+0
+xm+2
+⊥
+exp(−x2
+⊥)J0(
+√
+2bx2
+⊥)J1(
+√
+2bx2
+⊥)dx⊥,
+(D 2b)
+G(2)
+⊥m(b) = 2
+� ∞
+0
+xm+3
+⊥
+exp(−x2
+⊥)J2
+1(
+√
+2bx2
+⊥)dx⊥,
+(D 2c)
+where m is assumed to be even. Now we note that these integrals can be evaluated in
+terms of Weber’s integral:
+G⊥m(b) = 2
+��
+− d
+dp
+�m/2
+I0(p,
+√
+2b,
+√
+2b)
+�
+p=1
+,
+(D 3a)
+G(1)
+⊥m(b) = 2
+��
+− d
+dp
+�m/2 �
+− d
+dλ
+�
+I0(p, λ,
+√
+2b)
+�
+p=1,λ=
+√
+2b
+,
+(D 3b)
+G(2)
+⊥m(b) = 2
+��
+− d
+dp
+�m/2 �
+− d
+dλ1
+� �
+− d
+dλ2
+�
+I0(p, λ1, λ2)
+�
+p=1,λ1=λ2=
+√
+2b
+. (D 3c)
+The above relations allows us to evaluate the functions
+Gm,n(b) = G⊥m(b)G∥n,
+(D 4a)
+G(1)
+m,n(b) = G(1)
+⊥m(b)G∥n,
+(D 4b)
+G(2)
+m,n(b) = G(2)
+⊥m(b)G∥n.
+(D 4c)
+where
+G∥n =
+1
+√π
+� ∞
+−∞
+exp(−x2
+∥)xn
+∥dx∥ = 1 + (−1)n
+2√π
+ΓE
+�1 + n
+2
+�
+,
+(D 5)
+
+16
+G. G. Plunk and P. Helander
+and ΓE is the Euler gamma function. Finally, we can evaluate the integrals G′
+m,n, and
+G′
+m,n. We define
+G′
+⊥m(b) = 2
+� ∞
+0
+xm+1
+⊥
+exp(−x2
+⊥)J0
+∂J0
+∂l dx⊥,
+(D 6)
+G′′
+⊥m(b) = 2
+� ∞
+0
+xm+1
+⊥
+exp(−x2
+⊥)
+�∂J0
+∂l
+�2
+dx⊥,
+(D 7)
+Relating x⊥ to the proper gyrokinetic phase space variable µ, that is x2
+⊥ = µB/Ti allows
+the derivatives to be evaluated
+∂J0
+∂l = −
+�
+2/B ∂
+∂l(B
+√
+b)x2
+⊥J1
+(D 8)
+so that we can write
+G′
+⊥m(b) = −
+�
+2
+B
+∂
+∂l
+�
+B
+√
+b
+�
+G(1)
+⊥m(b),
+(D 9)
+G′′
+⊥m(b) = 2
+B
+� ∂
+∂l
+�
+B
+√
+b
+��2
+G(2)
+⊥m(b).
+(D 10)
+These expressions allow us to evaluate
+G′
+m,n = G′
+⊥mG∥n,
+(D 11)
+G′′
+m,n = G′′
+⊥mG∥n.
+(D 12)
+D.1. Explicit expressions for some Bessel integrals
+The b(l)-dependent factors in Eqn. 3.15 can be written as
+G0 = G0,0,
+(D 13)
+G1 = G2,0 + G0,2,
+(D 14)
+G2 = G4,0 + 2G2,2 + G0,4,
+(D 15)
+G3 = 1
+2G2,0 + G0,2,
+(D 16)
+G4 = 1
+2G4,0 + 3
+2G2,2 + G0,4,
+(D 17)
+G5 = 1
+4G4,0 + G2,2 + G0,4,
+(D 18)
+which can be evaluated using the identities of the previous section in terms of the familiar
+Γn(b) of gyrokinetic theory (suppressing its argument for succinctness):
+
+Energetic bounds. Part III
+17
+G0 = Γ0,
+(D 19)
+G1 =
+�3
+2 − b
+�
+Γ0 + bΓ1,
+(D 20)
+G2 = 1
+4
+��
+6b2 − 20b + 15
+�
+Γ0 + 2b ((10 − 4b)Γ1 + bΓ2)
+�
+,
+(D 21)
+G3 = 1
+2 (bΓ1 − (b − 2)Γ0) ,
+(D 22)
+G4 = 1
+4
+��
+3b2 − 11b + 10
+�
+Γ0 + b ((11 − 4b)Γ1 + bΓ2)
+�
+,
+(D 23)
+G5 = 1
+8
+��
+3b2 − 12b + 14
+�
+Γ0 + b (bΓ2 − 4(b − 3)Γ1)
+�
+.
+(D 24)
+where we recall
+Γn(b) = exp(−b)In(b)
+(D 25)
+For completeness, we evaluate the few remaining factors that enter Eqn. 3.15.
+G0,2 = Γ0
+2 ,
+(D 26)
+G(1)
+0,2 =
+√
+b (Γ0 − Γ1)
+2
+√
+2
+,
+(D 27)
+G(2)
+0,2 = 1
+8 (3bΓ0 + (2 − 4b)Γ1 + bΓ2) ,
+(D 28)
+and using Eqn. D 11
+G′
+0,2 = −
+�
+2
+B
+∂
+∂l
+�
+B
+√
+b
+�
+G(1)
+0,2,
+(D 29)
+G′′
+0,2 = 2
+B
+� ∂
+∂l
+�
+B
+√
+b
+��2
+G(2)
+0,2.
+(D 30)
+D.2. Limit b → 0
+In the limit b → 0 we obtain
+G0 = 1,
+(D 31)
+G1 = 3
+2,
+(D 32)
+G2 = 15
+4 ,
+(D 33)
+G3 = 1,
+(D 34)
+G4 = 5
+2,
+(D 35)
+G5 = 7
+4.
+(D 36)
+and
+
+18
+G. G. Plunk and P. Helander
+G0,2 = 1
+2,
+(D 37)
+G(1)
+0,2 = 0,
+(D 38)
+G(2)
+0,2 = 0,
+(D 39)
+REFERENCES
+P. Helander and G.G. Plunk.
+Energetic bounds on gyrokinetic instabilities. Part 1.
+Fundamentals. Journal of Plasma Physics, 88(2):905880207, 2022. .
+G.G. Plunk and P. Helander. Energetic bounds on gyrokinetic instabilities. Part 2. Modes of
+optimal growth. Journal of Plasma Physics, 88(3):905880313, 2022. .
+Matt Landreman, Gabriel G. Plunk, and William Dorland. Generalized universal instability:
+transient linear amplification and subcritical turbulence. Journal of Plasma Physics, 81
+(5):905810501, 2015. .
+W. Dorland and G. W. Hammett. Gyrofluid turbulence models with kinetic effects. Physics
+of Fluids B: Plasma Physics, 5(3):812–835, 1993. . URL https://doi.org/10.1063/1.
+860934.
+G. G. Plunk, P. Helander, P. Xanthopoulos, and J. W. Connor. Collisionless microinstabilities
+in stellarators. III. the ion-temperature-gradient mode. Physics of Plasmas, 21(3):032112,
+2014. . URL https://doi.org/10.1063/1.4868412.
+A. A. Schekochihin, S. C. Cowley, W. Dorland, G. W. Hammett, G. G. Howes, E. Quataert, and
+T. Tatsuno. Astrophysical gyrokinetics: Kinetic and fluid turbulent cascades in magnetized
+weakly collisional plasmas. The Astrophysical Journal Supplement Series, 182(1):310, may
+2009. . URL https://dx.doi.org/10.1088/0067-0049/182/1/310.
+G. G. PLUNK, S. C. COWLEY, A. A. SCHEKOCHIHIN, and T. TATSUNO. Two-dimensional
+gyrokinetic turbulence. Journal of Fluid Mechanics, 664:407–435, 2010. .
+P. Helander, J. H. E. Proll, and G. G. Plunk. Collisionless microinstabilities in stellarators. i.
+analytical theory of trapped-particle modes. Physics of Plasmas, 20(12):122505, 2013. .
+URL https://doi.org/10.1063/1.4846818.
+Bogdan Teaca. private communication.
+H. Biglari, P. H. Diamond, and M. N. Rosenbluth. Toroidal ion-pressure-gradient-driven drift
+instabilities and transport revisited. Physics of Fluids B: Plasma Physics, 1(1):109–118,
+1989. . URL http://link.aip.org/link/?PFB/1/109/1.
+P. Helander and G. G. Plunk. Upper bounds on gyrokinetic instabilities in magnetized plasmas.
+Phys. Rev. Lett., 127:155001, Oct 2021. . URL https://link.aps.org/doi/10.1103/
+PhysRevLett.127.155001.
+G. T. Roberg-Clark, G. G. Plunk, and P. Xanthopoulos. Coarse-grained gyrokinetics for the
+critical ion temperature gradient in stellarators. Phys. Rev. Research, 4:L032028, Aug
+2022a. . URL https://link.aps.org/doi/10.1103/PhysRevResearch.4.L032028.
+G. T. Roberg-Clark, P. Xanthopoulos, and G. G. Plunk. Reduction of electrostatic turbulence
+in a quasi-helically symmetric stellarator via critical gradient optimization, 2022b. URL
+https://arxiv.org/abs/2210.16030.
+
diff --git a/4dAzT4oBgHgl3EQfEPoC/content/tmp_files/load_file.txt b/4dAzT4oBgHgl3EQfEPoC/content/tmp_files/load_file.txt
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@@ -0,0 +1,642 @@
+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfEPoC/content/2301.00988v1.pdf,len=641
+page_content='Under consideration for publication in J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfEPoC/content/2301.00988v1.pdf'}
+page_content=' Plasma Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfEPoC/content/2301.00988v1.pdf'}
+page_content='  1  Energetic bounds on gyrokinetic instabilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfEPoC/content/2301.00988v1.pdf'}
+page_content='  Part III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfEPoC/content/2301.00988v1.pdf'}
+page_content=' Generalized free energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfEPoC/content/2301.00988v1.pdf'}
+page_content='  G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfEPoC/content/2301.00988v1.pdf'}
+page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfEPoC/content/2301.00988v1.pdf'}
+page_content=' Plunk1†, and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfEPoC/content/2301.00988v1.pdf'}
+page_content=' Helander1  1Max-Planck-Institut für Plasmaphysik, 17491 Greifswald, Germany  (Received xx;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfEPoC/content/2301.00988v1.pdf'}
+page_content=' revised xx;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfEPoC/content/2301.00988v1.pdf'}
+page_content=' accepted xx)  Free energy, widely used as a measure of turbulence intensity in weakly collisional  plasmas, has been recently found to be a suitable basis to describe both linear and  nonlinear growth in a wide class gyrokinetic systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfEPoC/content/2301.00988v1.pdf'}
+page_content=' The simplicity afforded by this  approach is accompanied by some drawbacks, notably the lack of any explicit treatment  of wave-particle effects, which makes the theory unable to describe things like stability  thresholds or dependence on the geometry of the background magnetic field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfEPoC/content/2301.00988v1.pdf'}
+page_content=' As a step  toward overcoming these limitations, we propose an extension of the theory based on  a generalization of free energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfEPoC/content/2301.00988v1.pdf'}
+page_content=' With this it is demonstrated that resonance effects  are recovered, and the bounds on growth are significantly reduced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfEPoC/content/2301.00988v1.pdf'}
+page_content=' The simplicity  and efficient computation of the associated “optimal” growth rates makes the theory  potentially applicable to stellarator optimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfEPoC/content/2301.00988v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfEPoC/content/2301.00988v1.pdf'}
+page_content=' Introduction  This is the third paper in a series (Helander and Plunk 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfEPoC/content/2301.00988v1.pdf'}
+page_content=' Plunk and Helander 2022),  in which we develop a linear and nonlinear stability theory based on gyrokinetic energy  balance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfEPoC/content/2301.00988v1.pdf'}
+page_content=' The last two papers used Helmholtz free energy, and introduced the concept of  optimal mode growth for fully electromagnetic gyrokinetic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfEPoC/content/2301.00988v1.pdf'}
+page_content=' The present paper proposes  a generalized energetic measure of fluctuations, allowing the inclusion of additional  instability mechanisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfEPoC/content/2301.00988v1.pdf'}
+page_content=' We do this first for a simple case, namely the electrostatic limit  (low plasma β) with only one kinetic species (ions), with the electrons being treated  adiabatically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfEPoC/content/2301.00988v1.pdf'}
+page_content=' These simplifications limit the application to ion-temperature-gradient  (ITG) driven turbulence, though the central result of the paper is capable of treating  completely general details of the magnetic geometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfEPoC/content/2301.00988v1.pdf'}
+page_content='  Free energy is a useful concept for understanding nonlinear and linear aspects of  plasma turbulence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfEPoC/content/2301.00988v1.pdf'}
+page_content=' At the level of linear instabilities it is common to speak of a source  of free energy that drives modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfEPoC/content/2301.00988v1.pdf'}
+page_content=' Indeed, without a source of free energy, provided  by background plasma gradients (density, temperature, flows), there can be no linear  instabilities (nor can there be subcritical turbulence Landreman et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfEPoC/content/2301.00988v1.pdf'}
+page_content=' (2015);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfEPoC/content/2301.00988v1.pdf'}
+page_content=' Plunk and  Helander (2022)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfEPoC/content/2301.00988v1.pdf'}
+page_content=' However, there is usually another ingredient that arises in the detailed  analysis of normal linear instabilities, namely the wave-particle resonance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfEPoC/content/2301.00988v1.pdf'}
+page_content=' In gyrokinetic  theory, this involves parallel motion (along the magnetic field) and magnetic drift, and  the resonance is physically linked to the work that the electrostatic field performs on  gyrocenter motion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfEPoC/content/2301.00988v1.pdf'}
+page_content=' However, the terms needed to capture this do not contribute to free  energy balance, and the influence of resonance therefore cannot be accounted for by the  optimal modes that we introduced in our previous works.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfEPoC/content/2301.00988v1.pdf'}
+page_content='  In this work we propose a new measure of gyrokinetic fluctuations, a generalization  of the concept of free energy, that incorporates the resonance mechanism, and, via the  † Email address for correspondence: gplunk@ipp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfEPoC/content/2301.00988v1.pdf'}
+page_content='mpg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfEPoC/content/2301.00988v1.pdf'}
+page_content='de  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfEPoC/content/2301.00988v1.pdf'}
+page_content='00988v1  [physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfEPoC/content/2301.00988v1.pdf'}
+page_content='plasm-ph]  3 Jan 2023  2  G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfEPoC/content/2301.00988v1.pdf'}
+page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfEPoC/content/2301.00988v1.pdf'}
+page_content=' Plunk and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfEPoC/content/2301.00988v1.pdf'}
+page_content=' Helander  magnetic drift, the full details of the background magnetic geometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfEPoC/content/2301.00988v1.pdf'}
+page_content=' We demonstrate  the existence of a class of quadratic measures closely related to Helmholtz free energy  that behave as positive-definite norms for fluctuations in the distribution function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfEPoC/content/2301.00988v1.pdf'}
+page_content=' The  corresponding energy balance equation is then used to derive a theory of optimal modes  that most efficiently extract this energy from its source.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfEPoC/content/2301.00988v1.pdf'}
+page_content=' The growth rate of these optimal  modes provides a rigorous upper bound on the growth rate of linear instabilities, and this  bound is shown to be lower than that obtained previously from Helmholtz free energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfEPoC/content/2301.00988v1.pdf'}
+page_content='  By studying some simple limits, we show that we recover some expected behavior of both  the slab and toroidal branches of the ITG mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfEPoC/content/2301.00988v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfEPoC/content/2301.00988v1.pdf'}
+page_content=' Definitions and gyrokinetic energy balance  The ion gyrokinetic equation in the electrostatic limit is written  ∂gk  ∂t + v∥  ∂gk  ∂l + i˜ωdgk + 1  B2  �  k′  B · (k × k′)δφk′gk−k′ = eiF0  Ti  � ∂  ∂t + iωT  ∗  �  δφk,  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfEPoC/content/2301.00988v1.pdf'}
+page_content='1)  where g is the gyro-center dependent part of the perturbed ion distribution function, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfEPoC/content/2301.00988v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfEPoC/content/2301.00988v1.pdf'}
+page_content='  fi = (1 − eiδφ(r)/Ti) Fi0 + g(R, Ei, µi, t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfEPoC/content/2301.00988v1.pdf'}
+page_content=' Its phase space variables are the energy Ea =  mav2/2 + eaΦ(ψ) and the magnetic moment µa = mav2  ⊥/(2B), and the perpendicular  wavenumber is k = k⊥ = kψ∇ψ + kα∇α with kψ and kα independent of the arc length  l along the magnetic field, and ψ and α defined via B = Bb = ∇ψ × ∇α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfEPoC/content/2301.00988v1.pdf'}
+page_content=' We neglect  collisions here†, and used the simplified notation gk = gi,k, and ω∗ = ω∗i, etc because the  adiabatic approximation ge,k = 0 is assumed throughout‡.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfEPoC/content/2301.00988v1.pdf'}
+page_content=' We will also assume kρi ∼ 1,  implying kρe ≪ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfEPoC/content/2301.00988v1.pdf'}
+page_content=' The gyrokinetic free energy balance equation obtained in this limit  reads  d  dt  �  k  H = 2  �  k  D,  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfEPoC/content/2301.00988v1.pdf'}
+page_content='2)  where the drive term D is  D(k, t) = Im ei  ��  gkωT  ∗ δφ  ∗  kd3v  �  ,  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfEPoC/content/2301.00988v1.pdf'}
+page_content='3)  and the free energy, expressed in terms of the gyrocenter distribution function  H(k, t) =  �  Ti  � |gk|2  Fi0  d3v −  �  a  nae2  a  Ta  |δφk|2  �  ,  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfEPoC/content/2301.00988v1.pdf'}
+page_content='4)  where the space average is defined as (see also Helander and Plunk (2022) for general-  izations)  ⟨· · ·⟩ = lim  L→∞  � L  −L  (· · · )dl  B  � � L  −L  dl  B .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfEPoC/content/2301.00988v1.pdf'}
+page_content='  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfEPoC/content/2301.00988v1.pdf'}
+page_content='5)  The diamagnetic frequencies are  † We do not retain collisions, since we will not be able to fix the sign of its contribution in  our later analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfEPoC/content/2301.00988v1.pdf'}
+page_content='  ‡ Here we do not include the customary correction for the zonal component (Dorland and  Hammett 1993), but it does not affect the subsequent analysis, as the growth of this component  is always zero because it has no free energy source (D = 0 for kα = 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfEPoC/content/2301.00988v1.pdf'}
+page_content='  Energetic bounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfEPoC/content/2301.00988v1.pdf'}
+page_content=' Part III  3  ω∗a = kαTa  ea  d ln na  dψ  ,  ωT  ∗a = ω∗a  �  1 + ηa  �mav2  2Ta  − 3  2  ��  ,  and the magnetic drift frequency is  ˜ωd = k · vd,  where the magnetic drift velocity is vd = ˆb × ((v2  ⊥/2)∇ ln B + v2  ∥κ)/Ωi, κ = ˆb · ∇ˆb, and  Ωa = eaB/ma is gyrofrequency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfEPoC/content/2301.00988v1.pdf'}
+page_content=' Assuming ∇ ln B ≈ κ (low plasma β), we can separate  the drift frequency into velocity-dependent and space-dependent factors following Plunk  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfEPoC/content/2301.00988v1.pdf'}
+page_content=' (2014)†:  ˜ωd = ωd(l)  �  v2  ⊥  2v2  th  +  v2  ∥  v2  th  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfEPoC/content/2301.00988v1.pdf'}
+page_content='  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfEPoC/content/2301.00988v1.pdf'}
+page_content='6)  The gyro-averaged electrostatic potential is denoted  δφk = J0  �k⊥v⊥  Ωi  �  δφk,  and the quasi-neutrality condition is  �  a  nae2  a  Ta  δφk = ei  �  gkJ0d3v, ,  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfEPoC/content/2301.00988v1.pdf'}
+page_content='7)  where Jn = Jn(k⊥v⊥/Ωi).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfEPoC/content/2301.00988v1.pdf'}
+page_content=' Following our previous convention, we define the free energy  as twice that which appears in some other publications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfEPoC/content/2301.00988v1.pdf'}
+page_content=' Henceforth, we suppress the  k-subscripts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfEPoC/content/2301.00988v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfEPoC/content/2301.00988v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfEPoC/content/2301.00988v1.pdf'}
+page_content=' Electrostatic energy and positive-definiteness of free energy  It is useful to decompose the free energy into a part associated with a perturbed  distribution function and a part associated with fluctuations in the electrostatic field,  i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfEPoC/content/2301.00988v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfEPoC/content/2301.00988v1.pdf'}
+page_content='  H = G + E,  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfEPoC/content/2301.00988v1.pdf'}
+page_content='8)  where  G = −TiSi =  �  Ti  � |δF|2  Fi0  d3v  �  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfEPoC/content/2301.00988v1.pdf'}
+page_content='9)  E =  �  (τ + 1 − Γ0) nie2  i  Ti  |δφ|2  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfEPoC/content/2301.00988v1.pdf'}
+page_content='  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfEPoC/content/2301.00988v1.pdf'}
+page_content='10)  Recall the conventional definitions Γn(b) = exp(−b)In(b) and b = k2  ⊥ρ2  i = k2  ⊥Ti/(miΩ2  i ),  and τ = (eTi)/(eiTe).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfEPoC/content/2301.00988v1.pdf'}
+page_content=' Note that δF = g−(eiδφ/Ti)F0 is the gyro-averaged perturbed dis-  † Actually, there is spatial dependence in both v⊥ and v∥, since these are not the proper  gyrokinetic phase-space variables, but a separation like this is useful to make contact with  known limits from gyrokinetic theory of the ITG mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfEPoC/content/2301.00988v1.pdf'}
+page_content='  4  G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfEPoC/content/2301.00988v1.pdf'}
+page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfEPoC/content/2301.00988v1.pdf'}
+page_content=' Plunk and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfEPoC/content/2301.00988v1.pdf'}
+page_content=' Helander  tribution function, and these two contributions to H can be identified as the gyrokinetic  perturbed entropy and the gyrokinetic field energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfEPoC/content/2301.00988v1.pdf'}
+page_content='  Although the general electromagnetic free energy admits a similar form as Eqn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfEPoC/content/2301.00988v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfEPoC/content/2301.00988v1.pdf'}
+page_content='8  (see for instance Helander and Plunk (2022)), we note that the electrostatic limit is  distinguished by the fact that the field contribution E is itself a nonlinear invariant of  the gyrokinetic system (Schekochihin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfEPoC/content/2301.00988v1.pdf'}
+page_content=' 2009), and its conservation may be viewed  as an additional constraint on the nonlinear dynamics, with consequences e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfEPoC/content/2301.00988v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfEPoC/content/2301.00988v1.pdf'}
+page_content=' for the  cascade and production of large-scale E × B flows (PLUNK et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfEPoC/content/2301.00988v1.pdf'}
+page_content=' 2010).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfEPoC/content/2301.00988v1.pdf'}
+page_content='  For what follows, we need the electrostatic energy balance equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfEPoC/content/2301.00988v1.pdf'}
+page_content=' This is obtained  by multiplying the ion gyrokinetic equation by eiδφ  ∗ integrate over velocity, average over  the parallel coordinate l, and sum over perpendicular wavenumber k, yielding (Helander  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfEPoC/content/2301.00988v1.pdf'}
+page_content=' 2013)  d  dt  �  k  E = 2  �  k  K,  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfEPoC/content/2301.00988v1.pdf'}
+page_content='11)  where the drive term K is  K = −Re ei  ��  δφ  ∗ �  v∥  ∂  ∂l + iωd  �  gd3v  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfEPoC/content/2301.00988v1.pdf'}
+page_content='  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfEPoC/content/2301.00988v1.pdf'}
+page_content='12)  This is composed to two contributions, one coming from the parallel streaming term,  and the other coming from the magnetic drift term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfEPoC/content/2301.00988v1.pdf'}
+page_content=' The first contribution has a simple  physical interpretation, as the rate of energy exchanged between particles and the parallel  electric field (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfEPoC/content/2301.00988v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfEPoC/content/2301.00988v1.pdf'}
+page_content=' the volume average of the parallel current multiplied by the parallel  electric field), while the second term describes an analogous process in the perpendicular  direction associated with the drift motion of gyrocenters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfEPoC/content/2301.00988v1.pdf'}
+page_content='  Eqn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfEPoC/content/2301.00988v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfEPoC/content/2301.00988v1.pdf'}
+page_content='8 is a physically transparent form that makes it clear that the free energy H is  a positive-definite norm for the distribution function g†, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfEPoC/content/2301.00988v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfEPoC/content/2301.00988v1.pdf'}
+page_content='  H ⩾ 0, and H = 0 iff g = 0  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfEPoC/content/2301.00988v1.pdf'}
+page_content='13)  over all of phase space, ℓ and v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfEPoC/content/2301.00988v1.pdf'}
+page_content=' To see this, note that the quantities G and E are both  positive, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfEPoC/content/2301.00988v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfEPoC/content/2301.00988v1.pdf'}
+page_content=' G ⩾ 0, obviously, and E ⩾ 0 because Γ0 ⩽ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfEPoC/content/2301.00988v1.pdf'}
+page_content=' Therefore if H = 0 then both  E = 0 and G = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfEPoC/content/2301.00988v1.pdf'}
+page_content=' The first implies δφ = 0 everywhere, while the second implies δF = 0  over all of phase space;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfEPoC/content/2301.00988v1.pdf'}
+page_content=' δφ = 0 and δF = 0 obviously implies g = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfEPoC/content/2301.00988v1.pdf'}
+page_content='  We note that positive-definiteness is a desirable property of an energetic measure  that can be useful for setting bounds on the growth rate of fluctuations;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfEPoC/content/2301.00988v1.pdf'}
+page_content=' if a non-  zero fluctuation (g ̸= 0) has zero measure M then the rate of growth d ln M/dt can  be unbounded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfEPoC/content/2301.00988v1.pdf'}
+page_content='  Although we mainly consider a plasma with a single kinetic ion species and adiabatic  electrons, the concepts and the formalism carry over to the more general case of a plasma  with an arbitrary number of kinetic species, as shown in Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfEPoC/content/2301.00988v1.pdf'}
+page_content=' An important  limitation, however, is that magnetic fluctuations and collisions are neglected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfEPoC/content/2301.00988v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfEPoC/content/2301.00988v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfEPoC/content/2301.00988v1.pdf'}
+page_content=' Generalized Free Energy  The positive definiteness of H suggests a family of related quadratic energetic measures  that are also positive definite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfEPoC/content/2301.00988v1.pdf'}
+page_content=' In particular it is clear that something of the form  † By extension, using Eqn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfEPoC/content/2301.00988v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfEPoC/content/2301.00988v1.pdf'}
+page_content='7, H can be shown to also be a positive-definite norm for the total  deviation of the distribution function δf = g − (eiφ/Ti)F0 from the zeroth-order Maxwellian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfEPoC/content/2301.00988v1.pdf'}
+page_content='  Energetic bounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfEPoC/content/2301.00988v1.pdf'}
+page_content=' Part III  5  ˜H = H − ∆E,  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfEPoC/content/2301.00988v1.pdf'}
+page_content='14)  will be positive-definite, by the same arguments of the previous section, for particular  values of the parameter ∆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfEPoC/content/2301.00988v1.pdf'}
+page_content=' For instance the choice ∆ < 1 allows trivial generalization of  the arguments, but we will see that the value can be extended beyond this.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfEPoC/content/2301.00988v1.pdf'}
+page_content='  To find a range of permissible values of ∆, we will consider a diagonalization of ˜H,  meaning that we will define a distribution function ˜g, which allows the energy to be  expressed using to the Euclidean norm,  ˜H = ||˜g||2 = (˜g, ˜g),  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfEPoC/content/2301.00988v1.pdf'}
+page_content='15)  where we have introduced the inner product  (˜g1, ˜g2) =  �  Ti  � ˜g∗  1˜g2  F0  d3v  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfEPoC/content/2301.00988v1.pdf'}
+page_content='  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfEPoC/content/2301.00988v1.pdf'}
+page_content='16)  To find the relationship between ˜g and g, we introduce the Ansatz ˜g = g −νJ0F0eiδφ/Ti,  substitute this into Eqn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfEPoC/content/2301.00988v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfEPoC/content/2301.00988v1.pdf'}
+page_content='15, using also Eqn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfEPoC/content/2301.00988v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfEPoC/content/2301.00988v1.pdf'}
+page_content='10, and solve for the free parameter ν,  yielding  ν = 1  Γ0  �  1 + τ −  �  (1 + τ − Γ0)(1 + τ − ∆Γ0)  �  ,  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfEPoC/content/2301.00988v1.pdf'}
+page_content='17)  where we have taken the negative root for convenience.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfEPoC/content/2301.00988v1.pdf'}
+page_content=' Observe that in order for ν to be  real, we must have  ∆ ⩽ (1 + τ)/Γ0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfEPoC/content/2301.00988v1.pdf'}
+page_content='  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfEPoC/content/2301.00988v1.pdf'}
+page_content='18)  The parameter ∆ can of course be negative, in which case its magnitude is unbounded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfEPoC/content/2301.00988v1.pdf'}
+page_content='  Noting that Γ0 generally depends on k, we may also assume the more restrictive ∆ ⩽  (1 + τ) to ensure that ˜H remains a nonlinear invariant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfEPoC/content/2301.00988v1.pdf'}
+page_content='  We pause to note that the choice ∆ = 0 yields a novel form of the conventional  (Helmholtz) free energy, immediately suggesting what can be considered as the phase-  space density of free energy, namely the quantity Ti|˜g|2/F0, for which there has not yet  been an expression available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfEPoC/content/2301.00988v1.pdf'}
+page_content='†  It is useful now to write quasi-neutrality in terms of ˜g,  ei  Ti  δφ = α  ni  �  ˜gJ0d3v,  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfEPoC/content/2301.00988v1.pdf'}
+page_content='19)  where  α =  1  �  (1 + τ − Γ0)(1 + τ − ∆Γ0)  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfEPoC/content/2301.00988v1.pdf'}
+page_content='  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfEPoC/content/2301.00988v1.pdf'}
+page_content='20)  Finally, we can show that ˜H is positive-definite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfEPoC/content/2301.00988v1.pdf'}
+page_content=' First, positivity follows from Eqn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfEPoC/content/2301.00988v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfEPoC/content/2301.00988v1.pdf'}
+page_content='15,  and it is obvious from Eqn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfEPoC/content/2301.00988v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfEPoC/content/2301.00988v1.pdf'}
+page_content='7 that if g = 0 then δφ = 0 so that E and H both vanish,  implying ˜H = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfEPoC/content/2301.00988v1.pdf'}
+page_content=' On the other hand, if we assume that ˜H = 0, then Eqn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfEPoC/content/2301.00988v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfEPoC/content/2301.00988v1.pdf'}
+page_content='15 implies that  ˜g = 0, and Eqn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfEPoC/content/2301.00988v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfEPoC/content/2301.00988v1.pdf'}
+page_content='19 implies that δφ = 0, from which we conclude g = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfEPoC/content/2301.00988v1.pdf'}
+page_content=' In summary,  ˜H ⩾ 0 and ˜H = 0 iff g = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfEPoC/content/2301.00988v1.pdf'}
+page_content='  † The idea for a phase-space density of free energy (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfEPoC/content/2301.00988v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfEPoC/content/2301.00988v1.pdf'}
+page_content=' a quantity that can be directly  integrated over phase space to yield the total free energy) was suggested by Teaca.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfEPoC/content/2301.00988v1.pdf'}
+page_content='  6  G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfEPoC/content/2301.00988v1.pdf'}
+page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfEPoC/content/2301.00988v1.pdf'}
+page_content=' Plunk and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfEPoC/content/2301.00988v1.pdf'}
+page_content=' Helander  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfEPoC/content/2301.00988v1.pdf'}
+page_content=' Modes of optimal growth  A key point in introducing the generalization of free energy ˜H is that this quantity  introduces wave-particle effects (parallel resonance and drift resonance) that enter the  electrostatic energy balance equation, Eqn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfEPoC/content/2301.00988v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfEPoC/content/2301.00988v1.pdf'}
+page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfEPoC/content/2301.00988v1.pdf'}
+page_content=' Note that the case ∆ = 0 (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfEPoC/content/2301.00988v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfEPoC/content/2301.00988v1.pdf'}
+page_content=' the  “conventional” Helmholtz free energy) is included as a limit ∆ = 0 and so the most  stringent bound on growth obtained from the generalized free energy will be at least as  good as the known bound obtained from the Helmholtz free energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfEPoC/content/2301.00988v1.pdf'}
+page_content='  Note that, as long as the parameter ∆ is independent of k, the quantity ˜H is conserved  by the nonlinearity, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfEPoC/content/2301.00988v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfEPoC/content/2301.00988v1.pdf'}
+page_content=' under summation over k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfEPoC/content/2301.00988v1.pdf'}
+page_content=' This is because it is a linear combination  of two nonlinear invariants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfEPoC/content/2301.00988v1.pdf'}
+page_content=' One simply combines Eqns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfEPoC/content/2301.00988v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfEPoC/content/2301.00988v1.pdf'}
+page_content='2 and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfEPoC/content/2301.00988v1.pdf'}
+page_content='11 to obtain  d  dt  �  k  ˜H = 2  �  k  (D − ∆K),  for ∆ independent of k,  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfEPoC/content/2301.00988v1.pdf'}
+page_content='1)  i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfEPoC/content/2301.00988v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfEPoC/content/2301.00988v1.pdf'}
+page_content=' the change of this measure is due to the drive terms of electrostatic and free energy,  and is otherwise conserved by the turbulent interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfEPoC/content/2301.00988v1.pdf'}
+page_content=' It is potentially useful to also  consider ∆ that does depend on k, for the purpose of obtaining bounds on linear growth,  but the nonlinear implications will be less clear in that case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfEPoC/content/2301.00988v1.pdf'}
+page_content='  In direct analogy to how modes of optimal free energy growth were defined, we  introduce a rate Λ  Λ = (D − ∆K)/ ˜H  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfEPoC/content/2301.00988v1.pdf'}
+page_content='2)  to be optimized over the space of ion distribution functions g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfEPoC/content/2301.00988v1.pdf'}
+page_content=' We note the bound on  conventional gyrokinetic instability growth rates,  γL ⩽ max  g  Λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfEPoC/content/2301.00988v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfEPoC/content/2301.00988v1.pdf'}
+page_content='3)  Having already found a diagonal form of the generalized free energy, Eqn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfEPoC/content/2301.00988v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfEPoC/content/2301.00988v1.pdf'}
+page_content='15, we  need not use a variational approach to find the states of extremal Λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfEPoC/content/2301.00988v1.pdf'}
+page_content=' We simply identify  the Hermitian linear operators associated with the input of free energy and electrostatic  energy, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfEPoC/content/2301.00988v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfEPoC/content/2301.00988v1.pdf'}
+page_content='  D = (˜g, D˜g),  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfEPoC/content/2301.00988v1.pdf'}
+page_content='4)  K = (˜g, K˜g)  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfEPoC/content/2301.00988v1.pdf'}
+page_content='5)  Using Eqn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfEPoC/content/2301.00988v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfEPoC/content/2301.00988v1.pdf'}
+page_content='19 and Eqns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfEPoC/content/2301.00988v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfEPoC/content/2301.00988v1.pdf'}
+page_content='3 and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfEPoC/content/2301.00988v1.pdf'}
+page_content='12, and some straightforward algebra (see Appendix  B), we obtain  D˜g = iα  2ni  J0F0ηω∗  �  d3v′J′  0˜g′  �� v  vth  �2  −  � v′  vth  �2�  ,  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfEPoC/content/2301.00988v1.pdf'}
+page_content='6)  where primes denote evaluation at v′ and vth =  �  2Ti/mi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfEPoC/content/2301.00988v1.pdf'}
+page_content=' For convenience, the operator  K can be split into its parallel and perpendicular components as K = K∥ + Kd, for which  we obtain  Kd˜g = iα  2ni  ωd(ℓ)F0J0  �  d3v′J′  0˜g′  �  �  �  v⊥  √  2vth  �2  +  � v∥  vth  �2  −  �  v′  ⊥  √  2vth  �2  −  �  v′  ∥  vth  �2�  � ,  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfEPoC/content/2301.00988v1.pdf'}
+page_content='7)  Energetic bounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfEPoC/content/2301.00988v1.pdf'}
+page_content=' Part III  7  and  K∥˜g = α  2ni  F0  �  J0  �  −B ∂  ∂l  � 1  B  �  d3v′v′  ∥J′  0˜g′  �  +  �  d3v′v′  ∥  ∂J′  0  ∂l ˜g′  �  +v∥  ∂  ∂l  �  J0  �  d3v′J′  0˜g′  ��  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfEPoC/content/2301.00988v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfEPoC/content/2301.00988v1.pdf'}
+page_content='8)  In deriving Eqn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfEPoC/content/2301.00988v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfEPoC/content/2301.00988v1.pdf'}
+page_content='8, it is important to note that the parallel derivative is taken at fixed  magnetic moment and particle energy, and that the velocity-space volume element d3v  is proportional to B/v∥ in these variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfEPoC/content/2301.00988v1.pdf'}
+page_content=' More details are given in Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfEPoC/content/2301.00988v1.pdf'}
+page_content=' The  kinetic eigenvalue problem can be stated now as  Λ˜g = (D − ∆K) ˜g,  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfEPoC/content/2301.00988v1.pdf'}
+page_content='9)  where solutions (Λ, g(l, v)) realize modes of optimal growth of ˜H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfEPoC/content/2301.00988v1.pdf'}
+page_content=' The analysis of this  eigenproblem is greatly simplified by adopting a moment form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfEPoC/content/2301.00988v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfEPoC/content/2301.00988v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfEPoC/content/2301.00988v1.pdf'}
+page_content=' Moment form of eigenproblem  As found in the preceding papers, there are natural moments that appear in the  energy input terms that can be identified to reduce the dimensionality of the problem  substantially.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfEPoC/content/2301.00988v1.pdf'}
+page_content=' Upon inspecting the energy balance equations one finds the following key  dimensionless integrals:  κ1 =  �  d3vJ0˜g/ni,  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfEPoC/content/2301.00988v1.pdf'}
+page_content='10)  κ2 =  �  d3v  � v2  v2  th  �  J0˜g/ni,  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfEPoC/content/2301.00988v1.pdf'}
+page_content='11)  κ3 =  �  d3v  �  v2  ⊥  2v2  th  +  v2  ∥  v2  th  �  J0˜g/ni,  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfEPoC/content/2301.00988v1.pdf'}
+page_content='12)  κ4 =  �  d3v  � v∥  vth  �  J0˜g/ni,  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfEPoC/content/2301.00988v1.pdf'}
+page_content='13)  κ5 =  �  d3v  � v∥  vth  � ∂J0  ∂l ˜g/ni,  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfEPoC/content/2301.00988v1.pdf'}
+page_content='14)  where κ1 is a density-like moment, κ2 and κ3 are pressure-like, κ4 is parallel ion flow,  while κ5 is more abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfEPoC/content/2301.00988v1.pdf'}
+page_content='  It is easy to recognize these integrals on the right hand side of Eqns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfEPoC/content/2301.00988v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfEPoC/content/2301.00988v1.pdf'}
+page_content='6, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfEPoC/content/2301.00988v1.pdf'}
+page_content='7 and  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfEPoC/content/2301.00988v1.pdf'}
+page_content='8, and straightforward to rewrite those equations in moment form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfEPoC/content/2301.00988v1.pdf'}
+page_content=' The dimensional  reduction is achieved by taking moments of the these equations to obtain a coupled set  of five fluid equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfEPoC/content/2301.00988v1.pdf'}
+page_content=' These, which are given in Appendix C, can be combined, leading,  after lengthy algebra, to a relatively simple second order ordinary differential equation,  the main result of this paper:  �4Λ2  α2 + (∆ωdG3 − ηω∗G1)2 − G0  �  (ηω∗)2G2 − 2∆ωdηω∗G4 + ∆2ω2  dG5  ��  κ1  = ∆2v2  thG0B  �  − ∂  ∂l  �G0,2  B  ∂κ1  ∂l  �  + G′′  0,2  B κ1 − ∂  ∂l  �G′  0,2  B  �  κ1  �  ,  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfEPoC/content/2301.00988v1.pdf'}
+page_content='15)  8  G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfEPoC/content/2301.00988v1.pdf'}
+page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfEPoC/content/2301.00988v1.pdf'}
+page_content=' Plunk and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfEPoC/content/2301.00988v1.pdf'}
+page_content=' Helander  The functions Gm,n, G′  m,n, and G′′  m,n, which depend on arc length via b(l) and B(l), are  defined in terms of integrals involving Bessel functions, and are evaluated in Appendix  D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfEPoC/content/2301.00988v1.pdf'}
+page_content=' The other b-dependent factors (G0-G5) can be expressed in terms of Gm,n, and are  evaluated in terms of more elementary Bessel functions in Appendix D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfEPoC/content/2301.00988v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfEPoC/content/2301.00988v1.pdf'}
+page_content='  In Eqn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfEPoC/content/2301.00988v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfEPoC/content/2301.00988v1.pdf'}
+page_content='15 we see the eigenvalue Λ entering quadratically, reflecting the fact that  there will be two real roots, one positive and one negative, owing to Hermiticity and  time-reversal symmetriy of the full eigenproblem, Eqn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfEPoC/content/2301.00988v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfEPoC/content/2301.00988v1.pdf'}
+page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfEPoC/content/2301.00988v1.pdf'}
+page_content=' Note that the terms arising  from the parallel drive of electrostatic energy are placed on the right hand side.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfEPoC/content/2301.00988v1.pdf'}
+page_content=' In the  following section, we will consider some simple limits of this equation, and leave its more  general solution for a future publication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfEPoC/content/2301.00988v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfEPoC/content/2301.00988v1.pdf'}
+page_content=' Simple limits  In this section we will consider some simple limits applied to Eqn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfEPoC/content/2301.00988v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfEPoC/content/2301.00988v1.pdf'}
+page_content='15, and draw some  comparison to linear theory of the main instability targeted by limit of this paper, the  ion temperature gradient (ITG) mode (see for instance Plunk et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfEPoC/content/2301.00988v1.pdf'}
+page_content=' (2014)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfEPoC/content/2301.00988v1.pdf'}
+page_content=' To start,  we note that taking ∆ = 0, so that ˜H becomes the conventional Helmholtz free energy,  yields  Λ2 =  (ηω∗)2  4(1 + τ − Γ0)(1 + τ)  �  G0G2 − G2  1  �  ,  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfEPoC/content/2301.00988v1.pdf'}
+page_content='1)  which matches Eqn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfEPoC/content/2301.00988v1.pdf'}
+page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfEPoC/content/2301.00988v1.pdf'}
+page_content='20 of Helander and Plunk (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfEPoC/content/2301.00988v1.pdf'}
+page_content='  In considering other simplifications, we first should note that the adiabatic electron ap-  proximation already neglects a trapped particle population, which is not really consistent  unless we take the magnetic field strength to be independent of arc length  ∂B  ∂l = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfEPoC/content/2301.00988v1.pdf'}
+page_content='  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfEPoC/content/2301.00988v1.pdf'}
+page_content='2)  We have avoided making this approximation explicitly, since the present paper lays the  foundation for extensions, in which it will be useful to include variation in B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfEPoC/content/2301.00988v1.pdf'}
+page_content=' Making  the approximation now leads to minor simplifications of Eqn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfEPoC/content/2301.00988v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfEPoC/content/2301.00988v1.pdf'}
+page_content='15, where all the explicit  factors of B drop out of the right-hand side.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfEPoC/content/2301.00988v1.pdf'}
+page_content=' A more significant simplification is achieved  by assuming unsheared and uniform magnetic geometry, in particular  ∂b  ∂l = 0,  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfEPoC/content/2301.00988v1.pdf'}
+page_content='3)  ∂ωd  ∂l  = 0,  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfEPoC/content/2301.00988v1.pdf'}
+page_content='4)  In this limit, all of the coefficients of Eqn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfEPoC/content/2301.00988v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfEPoC/content/2301.00988v1.pdf'}
+page_content='15 are constants, and a simple dispersion  relation is the obtained by taking ∂κ1/∂l = ik∥κ1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfEPoC/content/2301.00988v1.pdf'}
+page_content=' We find  4Λ2  α2 + (∆ωdG3 − ηω∗G1)2 − G0  �  (ηω∗)2G2 − 2∆ωdηω∗G4 + ∆2ω2  dG5  �  = ∆2k2  ∥v2  thG2  0/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfEPoC/content/2301.00988v1.pdf'}
+page_content='  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfEPoC/content/2301.00988v1.pdf'}
+page_content='5)  were we have used G0,2 = G0/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfEPoC/content/2301.00988v1.pdf'}
+page_content=' As noted in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfEPoC/content/2301.00988v1.pdf'}
+page_content='2, the quantity ∆ is a free  parameter, over which we can optimize Λ to improve the bounds on the growth rate of  fluctuations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfEPoC/content/2301.00988v1.pdf'}
+page_content='  Energetic bounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfEPoC/content/2301.00988v1.pdf'}
+page_content=' Part III  9  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfEPoC/content/2301.00988v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfEPoC/content/2301.00988v1.pdf'}
+page_content=' Slab ITG mode  Setting ωd = 0 leaves only the slab branch of the ITG mode, driven by the temperature  gradient, and involving ion parallel resonance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfEPoC/content/2301.00988v1.pdf'}
+page_content=' Eqn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfEPoC/content/2301.00988v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfEPoC/content/2301.00988v1.pdf'}
+page_content='5 reduces to  4Λ2  (ηω∗)2α2 = G0G2 − G2  1 + ∆2κ−2  ∥ G2  0/2,  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfEPoC/content/2301.00988v1.pdf'}
+page_content='6)  where κ∥ = ηω∗/(k∥vth).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfEPoC/content/2301.00988v1.pdf'}
+page_content=' Because G0G2−G2  1 ⩾ 0, the two contributions on the right hand  side are both positive but the solution for which Λ is minimal is actually not obtained  for ∆ = 0, due to the implicit dependence of α on ∆ given by Eqn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfEPoC/content/2301.00988v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfEPoC/content/2301.00988v1.pdf'}
+page_content='20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfEPoC/content/2301.00988v1.pdf'}
+page_content='  To obtain the value of ∆ which yields an optimal bound, we can look for extrema of  Λ2/(ηω∗)2, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfEPoC/content/2301.00988v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfEPoC/content/2301.00988v1.pdf'}
+page_content='  d  d∆  �  G0G2 − G2  1 + ∆2κ−2  ∥ G2  0/2  (1 + τ − Γ0)(1 + τ − ∆Γ0)  �  = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfEPoC/content/2301.00988v1.pdf'}
+page_content='  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfEPoC/content/2301.00988v1.pdf'}
+page_content='7)  This results in a quadratic equation for ∆ that is still rather complicated so we will  consider the limit b → 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfEPoC/content/2301.00988v1.pdf'}
+page_content=' see Appendix D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfEPoC/content/2301.00988v1.pdf'}
+page_content='2 for the relevant limits of Gm,n, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfEPoC/content/2301.00988v1.pdf'}
+page_content=' Applying  the limit to Eqn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfEPoC/content/2301.00988v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfEPoC/content/2301.00988v1.pdf'}
+page_content='6 yields  Λ2  (ηω∗)2 =  3 + ∆2/κ2  ∥  8τ(1 + τ − ∆)  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfEPoC/content/2301.00988v1.pdf'}
+page_content='8)  This solution diverges as ∆ approaches 1+τ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfEPoC/content/2301.00988v1.pdf'}
+page_content=' recall that this is the upper limit allowed  by Eqn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfEPoC/content/2301.00988v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfEPoC/content/2301.00988v1.pdf'}
+page_content='18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfEPoC/content/2301.00988v1.pdf'}
+page_content=' It also grows in an unbounded fashion as ∆ → −∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfEPoC/content/2301.00988v1.pdf'}
+page_content=' There is an optimal  value giving minimal |Λ|, obtained by solving Eqn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfEPoC/content/2301.00988v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfEPoC/content/2301.00988v1.pdf'}
+page_content='7 in this limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfEPoC/content/2301.00988v1.pdf'}
+page_content=' This solution, denoted  as ∆min, is  ∆min = 1 + τ −  �  (1 + τ)2 + 3κ2  ∥  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfEPoC/content/2301.00988v1.pdf'}
+page_content='9)  where the negative root has been selected to be consistent with Eqn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfEPoC/content/2301.00988v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfEPoC/content/2301.00988v1.pdf'}
+page_content='18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfEPoC/content/2301.00988v1.pdf'}
+page_content=' Substituting  this solution into Eqn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfEPoC/content/2301.00988v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfEPoC/content/2301.00988v1.pdf'}
+page_content='8 gives  Λ2  min =  (ηω∗)2  4¯κ2  ∥τ(1 + τ)  ��  1 + 3¯κ2  ∥ − 1  �  ,  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfEPoC/content/2301.00988v1.pdf'}
+page_content='10)  where we define ¯κ∥ = κ∥/(1 + τ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfEPoC/content/2301.00988v1.pdf'}
+page_content=' This reaches its maximum value in the limit ¯κ∥ → 0,  and is a decreasing function of |¯κ∥|, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfEPoC/content/2301.00988v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfEPoC/content/2301.00988v1.pdf'}
+page_content='  Λ2  min =  �  3  8τ(τ+1)(ηω∗)2,  for ¯κ∥ → 0,  √  3  4τ |ηω∗k∥vth|,  for |¯κ∥| ≫ 1,  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfEPoC/content/2301.00988v1.pdf'}
+page_content='11)  Physically, the first result implies that when drive (ηω∗) is much smaller than the parallel  transit frequency (k∥vth), the best bound is equal to that obtained by free energy (∆ = 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfEPoC/content/2301.00988v1.pdf'}
+page_content='  In this case, the bound is consistent from expectations of the growth rate of a resonant  slab ITG mode, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfEPoC/content/2301.00988v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfEPoC/content/2301.00988v1.pdf'}
+page_content=' γL ∼ ηω∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfEPoC/content/2301.00988v1.pdf'}
+page_content='  In the opposite limit (¯κ∥ ≫ 1), however, when the drive large, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfEPoC/content/2301.00988v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfEPoC/content/2301.00988v1.pdf'}
+page_content=' in the so-called  non-resonant or “fluid” limit, we obtain a much lower bound, essentially the geometric  mean of the drive and the parallel transit frequency k∥vth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfEPoC/content/2301.00988v1.pdf'}
+page_content=' We note that this bound is  not as low as what is obtained from the non-resonant solution of the dispersion relation  (without density gradient), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfEPoC/content/2301.00988v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfEPoC/content/2301.00988v1.pdf'}
+page_content=' γL ∼ ηω1/3  ∗  (k∥vth)2/3 (Plunk et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfEPoC/content/2301.00988v1.pdf'}
+page_content=' 2014), but nevertheless  captures the expected weakening (relative to the resonant result) qualitatively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfEPoC/content/2301.00988v1.pdf'}
+page_content='  10  G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfEPoC/content/2301.00988v1.pdf'}
+page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfEPoC/content/2301.00988v1.pdf'}
+page_content=' Plunk and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfEPoC/content/2301.00988v1.pdf'}
+page_content=' Helander  It is interesting to observe that this latter limit corresponds to ∆min → −∞, making  ˜H in some sense dominated by the electrostatic component.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfEPoC/content/2301.00988v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfEPoC/content/2301.00988v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfEPoC/content/2301.00988v1.pdf'}
+page_content=' Toroidal ITG mode  Now taking k∥vth to be small, we can neglect the right-hand side of Eqn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfEPoC/content/2301.00988v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfEPoC/content/2301.00988v1.pdf'}
+page_content='5, leaving  4Λ2  α2 = G0  �  (ηω∗)2G2 − 2∆ωdηω∗G4 + ∆2ω2  dG5  �  − (∆ωdG3 − ηω∗G1)2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfEPoC/content/2301.00988v1.pdf'}
+page_content='  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfEPoC/content/2301.00988v1.pdf'}
+page_content='12)  To derive the optimal choice of ∆, we again take the b → 0 limit and obtain from  Eqn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfEPoC/content/2301.00988v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfEPoC/content/2301.00988v1.pdf'}
+page_content='12  Λ2  (ηω∗)2 = 3∆2 − 8∆κd + 6κ2  d  16κ2  dτ(τ + 1 − ∆) ,  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfEPoC/content/2301.00988v1.pdf'}
+page_content='13)  where we define κd = ηω∗/ωd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfEPoC/content/2301.00988v1.pdf'}
+page_content=' Note the similar qualitative behavior with ∆ as Eqn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfEPoC/content/2301.00988v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfEPoC/content/2301.00988v1.pdf'}
+page_content='6,  namely its divergence at the ∆ → 1 + τ, and unbounded growth as ∆ → −∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfEPoC/content/2301.00988v1.pdf'}
+page_content=' The key  difference here arises in the linear term in the drive parameter κ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfEPoC/content/2301.00988v1.pdf'}
+page_content=' this is expected from  the theory of the toroidal ITG mode since the sign of the drift frequency (associated with  so-called ‘good’ and ’bad’ magnetic curvature) is important for the resonance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfEPoC/content/2301.00988v1.pdf'}
+page_content='  We now find the value of ∆ that minimizes Λ:  ∆min = (1 + τ)  �  1 −  �  2¯κ2  d − 8¯κd/3 + 1  �  ,  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfEPoC/content/2301.00988v1.pdf'}
+page_content='14)  where we define the parameter ¯κd = κd/(1 + τ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfEPoC/content/2301.00988v1.pdf'}
+page_content=' Substituting this into Eqn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfEPoC/content/2301.00988v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfEPoC/content/2301.00988v1.pdf'}
+page_content='13 yields  Λ2  min = (ηω∗)2  �  8¯κd (ζ (¯κd) − 1) + 3 (ζ (¯κd) − 1)2 + 6¯κ2  d  16τ(τ + 1)¯κ2  dζ (¯κd)  �  ,  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfEPoC/content/2301.00988v1.pdf'}
+page_content='15)  with ζ =  �  2¯κ2  d − 8¯κd/3 + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfEPoC/content/2301.00988v1.pdf'}
+page_content=' This expression for Λmin is naturally separated into a  factor that depends only on the instability parameter ¯κd, from which we can derive  the asymptotic behavior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfEPoC/content/2301.00988v1.pdf'}
+page_content=' To show the behavior of this factor we plot the quantity  τ(1 + τ)Λ2  min/(ηω∗)2 in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfEPoC/content/2301.00988v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfEPoC/content/2301.00988v1.pdf'}
+page_content=' The overall behavior of Λmin is captured by the following  limits  Λ2  min =  �  �  �  �  �  �  �  |ηω∗ωd|  �  3  √  2+4σ  8τ  �  ,  for |¯κd| ≫ 1,  (ηω∗)2  24τ(1+τ),  for ¯κd → 0,  3(ηω∗)2  8τ(1+τ),  for ¯κd → 4/3  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfEPoC/content/2301.00988v1.pdf'}
+page_content='16)  where we denote σ = ±1 as the sign of κd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfEPoC/content/2301.00988v1.pdf'}
+page_content=' At large drive (|¯κd| ≫ 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfEPoC/content/2301.00988v1.pdf'}
+page_content=' |ηω∗| ≫ |ωd|(1 + τ))  we recover the expected non-resonant (“fluid”) behavior of the toroidal ITG mode with no  density gradient, namely γL ∼ √ηω∗ωd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfEPoC/content/2301.00988v1.pdf'}
+page_content=' Note that this growth rate is much smaller than  the bound found by merely considering the Helmholtz free energy (Helander and Plunk  2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfEPoC/content/2301.00988v1.pdf'}
+page_content=' Although we do not see the complete stabilization (Λ = 0) at negative values of  ¯κd (opposite sign of ωd and ηω∗) expected from theory, there is a strong asymmetry,  with |Λ| having its larger values at positive ¯κd and being comparatively much smaller for  negative ¯κd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfEPoC/content/2301.00988v1.pdf'}
+page_content='  The value ¯κd = 4/3 (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfEPoC/content/2301.00988v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfEPoC/content/2301.00988v1.pdf'}
+page_content=' ηω∗ = 4(1 + τ)ωd/3) achieves the maximal value of Λmin  at fixed ηω∗, and therefore is evocative of the resonance condition for the toroidal ITG  modes ηω∗ ∼ ωd (Biglari et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfEPoC/content/2301.00988v1.pdf'}
+page_content=' 1989).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfEPoC/content/2301.00988v1.pdf'}
+page_content=' This value of ¯κd is obtained by solving for ∆min = 0,  Energetic bounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfEPoC/content/2301.00988v1.pdf'}
+page_content=' Part III  11  1  1  2  4/3  3  4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfEPoC/content/2301.00988v1.pdf'}
+page_content='1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfEPoC/content/2301.00988v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfEPoC/content/2301.00988v1.pdf'}
+page_content='3  Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfEPoC/content/2301.00988v1.pdf'}
+page_content=' Bound of the growth rate of the toroidal ITG mode, obtained from the optimal  growth of generalized free energy, plotted versus the instability parameter  ¯κd = ηω∗/[ωd(1 + τ)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfEPoC/content/2301.00988v1.pdf'}
+page_content='  explaining why it produces the worst bound, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfEPoC/content/2301.00988v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfEPoC/content/2301.00988v1.pdf'}
+page_content=' that given by optimal growth of Helmoltz  free energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfEPoC/content/2301.00988v1.pdf'}
+page_content='  It is noteworthy that for the limit ¯κd → 0 (ωd ≫ ηω∗/(1 + τ)) our method yields a  value of |Λ| that is a factor of 1/3 reduced as compared to the resonant case, again at  least qualitatively reproducing the expected stabilization of the toroidal ITG mode in  this limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfEPoC/content/2301.00988v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfEPoC/content/2301.00988v1.pdf'}
+page_content=' Conclusion  We have demonstrated that the use of a generalized form of free energy ˜H introduces  some of the physics of wave-particle resonance that is missing in the theory of optimal  mode growth of Helmholtz free energy (Helander and Plunk 2021, 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfEPoC/content/2301.00988v1.pdf'}
+page_content=' Plunk and  Helander 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfEPoC/content/2301.00988v1.pdf'}
+page_content=' The growth rates of optimal modes of generalized free energy provide  a rigorous upper bound on the growth of conventional gyrokinetic instabilities (“normal  modes”), which is always below the Helmholtz bound, as it must be, given that the  Helmholtz free energy is a special case of the generalized measure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfEPoC/content/2301.00988v1.pdf'}
+page_content=' Moreover, optimal  modes of generalized free energy depend on the magnetic-field geometry to a greater  extent than those associated with Helmholtz free energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfEPoC/content/2301.00988v1.pdf'}
+page_content=' The difference in growth rates  can be very large.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfEPoC/content/2301.00988v1.pdf'}
+page_content=' For instance, in the important case of a strongly driven toroidal ITG  mode, the Helmholtz bound is larger by a factor of order ηω∗/ωd ≫ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfEPoC/content/2301.00988v1.pdf'}
+page_content='  A single ordinary differential equation has been derived for optimal modes, allowing  general magnetic geometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfEPoC/content/2301.00988v1.pdf'}
+page_content=' We found solutions of this equation in some simple limits to  demonstrate that it indeed recovers, at least qualitatively, some of the physical effects  expected from the theory of linear ITG modes, including sensitivity to the ratio of the  frequencies associated with drive and resonance, and transition of the instability when  this ratio is near one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfEPoC/content/2301.00988v1.pdf'}
+page_content=' Density gradient dependence of ITG mode is absent from both  electrostatic and free energy input terms, assuming adiabatic electrons, so its effect is  not accounted for by the theory presented here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfEPoC/content/2301.00988v1.pdf'}
+page_content='  The results of this work have possible implications for “turbulence optimization”,  i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfEPoC/content/2301.00988v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfEPoC/content/2301.00988v1.pdf'}
+page_content=' the endeavor to shape the equilibrium magnetic geometry for low turbulence in  12  G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfEPoC/content/2301.00988v1.pdf'}
+page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfEPoC/content/2301.00988v1.pdf'}
+page_content=' Plunk and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfEPoC/content/2301.00988v1.pdf'}
+page_content=' Helander  stellarators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfEPoC/content/2301.00988v1.pdf'}
+page_content=' The general result, Eqn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfEPoC/content/2301.00988v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfEPoC/content/2301.00988v1.pdf'}
+page_content='15, allows in principle for the inclusion of the  complete geometric information contained in gyrokinetics, that is needed to run gyroki-  netic simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfEPoC/content/2301.00988v1.pdf'}
+page_content=' However, the solution of this equation should be far simpler and more  efficient, due to the reduction of velocity space to a single moment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfEPoC/content/2301.00988v1.pdf'}
+page_content=' The results found for  the toroidal branch of the ITG hint at a possible optimization strategy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfEPoC/content/2301.00988v1.pdf'}
+page_content=' Consider fixed  plasma conditions, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfEPoC/content/2301.00988v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfEPoC/content/2301.00988v1.pdf'}
+page_content=' a given temperature gradient (ηω∗) and temperature ratio (τ): At  high drive (ηω∗/(1+τ) > 4ωd/3), minimization of the optimal growth rate |Λ| is achieved  by minimization of the magnetic drift ωd (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfEPoC/content/2301.00988v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfEPoC/content/2301.00988v1.pdf'}
+page_content=' magnetic curvature), corresponding to  minimization of the strongly-driven (non-resonant) toroidal ITG mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfEPoC/content/2301.00988v1.pdf'}
+page_content=' On the other  hand, at low drive (ηω∗/(1 + τ) < 4ωd/3) the increase of ωd is favored, corresponding to  a weakening of the marginally unstable ITG mode, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfEPoC/content/2301.00988v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfEPoC/content/2301.00988v1.pdf'}
+page_content=' an increase of the threshold of  instability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfEPoC/content/2301.00988v1.pdf'}
+page_content=' The latter case corresponds to “critical gradient” optimization, an idea which  has recently been developed (Roberg-Clark et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfEPoC/content/2301.00988v1.pdf'}
+page_content=' 2022a,b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfEPoC/content/2301.00988v1.pdf'}
+page_content='  More general solutions of the optimal mode equation, and the application to optimiza-  tion will be pursued in future works.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfEPoC/content/2301.00988v1.pdf'}
+page_content=' Other special limits can also be explored including,  adiabatic ion limits, appropriate for studying trapped electron and universal instabilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfEPoC/content/2301.00988v1.pdf'}
+page_content='  Electromagnetic generalizations are also possible: although it is not clear how to construct  an electromagnetic form of generalized free energy ˜H that is a nonlinear invariant, it is  certainly possible to consider related measures that focus on linear bounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfEPoC/content/2301.00988v1.pdf'}
+page_content='  Funding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfEPoC/content/2301.00988v1.pdf'}
+page_content=' This work has been carried out within the framework of the EUROfusion  Consortium, funded by the European Union via the Euratom Research and Training  Programme (Grant Agreement No 101052200 — EUROfusion).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfEPoC/content/2301.00988v1.pdf'}
+page_content=' Views and opinions  expressed are however those of the author(s) only and do not necessarily reflect those of  the European Union or the European Commission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfEPoC/content/2301.00988v1.pdf'}
+page_content=' Neither the European Union nor the  European Commission can be held responsible for them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfEPoC/content/2301.00988v1.pdf'}
+page_content=' This work was partly supported  by a grant from the Simons Foundation (560651, PH).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfEPoC/content/2301.00988v1.pdf'}
+page_content='  Declaration of Interests.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfEPoC/content/2301.00988v1.pdf'}
+page_content=' The authors report no conflict of interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfEPoC/content/2301.00988v1.pdf'}
+page_content='  Author ORCID.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfEPoC/content/2301.00988v1.pdf'}
+page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfEPoC/content/2301.00988v1.pdf'}
+page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfEPoC/content/2301.00988v1.pdf'}
+page_content=' Plunk, https://orcid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfEPoC/content/2301.00988v1.pdf'}
+page_content='org/0000-0002-4012-4038;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfEPoC/content/2301.00988v1.pdf'}
+page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfEPoC/content/2301.00988v1.pdf'}
+page_content=' Helander,  https://orcid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfEPoC/content/2301.00988v1.pdf'}
+page_content='org/0000-0002-0460-590X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfEPoC/content/2301.00988v1.pdf'}
+page_content='  Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfEPoC/content/2301.00988v1.pdf'}
+page_content=' Several kinetic species  For a plasma with an arbitrary number of particle species, we multiply each gyrokinetic  equation  ∂ga,k  ∂t  + v∥  ∂ga,k  ∂l  + iωdaga,k + 1  B2  �  k′  B · (k × k′)δφk′ga,k−k′  (A 1)  = eaFa0  Ta  � ∂  ∂t + iωT  ∗a  �  δφk ,  (A 2)  by eaδφ  ∗  k, integrate over velocity space, take the real part and the average ⟨· · · ⟩ over the  flux tube, and sum over all species a and wave vectors k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfEPoC/content/2301.00988v1.pdf'}
+page_content=' In other words, we apply the  operator  Re  �  a,k  ea  ��  δφ  ∗  k (· · · ) d3v  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfEPoC/content/2301.00988v1.pdf'}
+page_content='  (A 3)  Since the expression  Re (k × k′)δφ  ∗  k′δφkga,k−k′ = 1  2(k × k′)  �  δφ  ∗  k′δφkga,k−k′ + δφk′δφ  ∗  kg∗  a,k−k′  �  (A 4)  Energetic bounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfEPoC/content/2301.00988v1.pdf'}
+page_content=' Part III  13  = 1  2(k × k′)  �  δφ−k′δφkga,k−k′ + δφk′δφ−kga,−k+k′�  (A 5)  changes sign under an exchange of k and k′, the nonlinear terms cancel upon summation  over k and k′, and we obtain  Re  �  a,k  ea  ��  δφ  ∗  k  �∂ga,k  ∂t  + v∥  ∂ga,k  ∂l  + iωdaga,k − eaFa0  Ta  ∂δφk  ∂t  �  d3v  �  = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfEPoC/content/2301.00988v1.pdf'}
+page_content='  (A 6)  The quasineutrality equation (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfEPoC/content/2301.00988v1.pdf'}
+page_content='7) can be used to write the first term as  Re  �  a,k  ea  �  δφ  ∗  k  ∂ga,k  ∂t d3v = d  dt  �  k  nae2  a  2Ta  |δφk|2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfEPoC/content/2301.00988v1.pdf'}
+page_content='  (A 7)  We thus arrive at the electrostatic energy balance equation  d  dt  �  k  nae2  a  2Ta  �  [1 − Γ0(bak)] |δφk|2�  (A 8)  = −Re  �  a,k  �  ea  �  δφ  ∗  k  �  v∥  ∂ga,k  ∂l  + iωdaga,k  �  d3v  �  ,  (A 9)  which is the generalization of Eqn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfEPoC/content/2301.00988v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfEPoC/content/2301.00988v1.pdf'}
+page_content='11 to several species.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfEPoC/content/2301.00988v1.pdf'}
+page_content=' The right-hand side can be  interpreted as minus the work done by the electric field on the various particle species.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfEPoC/content/2301.00988v1.pdf'}
+page_content='  The first term in this expression contains  �  J0av∥  ∂ga,k  ∂l  d3v =  �  σ  2πσB  m2a  � ∞  0  dEa  � Ea/B  0  J0a  ∂ga,k  ∂l  dµa  (A 10)  = B ∂  ∂l  � 1  B  �  J0av∥ga,kd3v  �  −  � ∂J0a  ∂l v∥ga,kd3v,  (A 11)  which we used in Eqn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfEPoC/content/2301.00988v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfEPoC/content/2301.00988v1.pdf'}
+page_content='8, with σ = v∥/|v∥|, Ea = mav2/2 and µa = mav2  ⊥/(2B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfEPoC/content/2301.00988v1.pdf'}
+page_content='  Appendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfEPoC/content/2301.00988v1.pdf'}
+page_content=' Derivation of operators D and K  We first write forms of D and K explicit in ˜g, noting that the contribution to D  proportional to the density gradient is zero by use of quasi-neutrality with the adiabatic  electron approximation (the factor ηω∗ ∝ dTi/dψ appears in what follows, but never  ω∗ ∝ dni/dψ alone).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfEPoC/content/2301.00988v1.pdf'}
+page_content=' For the same reason, the terms proportional to ν, involved in the  transformation from g to ˜g, do not contribute, so expressing energy input in terms of ˜g  merely has the consequence of introducing the overall factor α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfEPoC/content/2301.00988v1.pdf'}
+page_content=' The expressions are  D = −Ti  iα  ni  ��  ˜g(v)˜g∗(v′) ηω∗  � v2  v2  th  �  J0J′  0d3vd3v′  �  + c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfEPoC/content/2301.00988v1.pdf'}
+page_content='c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfEPoC/content/2301.00988v1.pdf'}
+page_content=',  (B 1)  K∥ = −Ti  α  2ni  ��  ˜g∗(v′)  �  v∥  ∂˜g(v)  ∂l  �  J0J′  0d3vd3v′  �  + c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfEPoC/content/2301.00988v1.pdf'}
+page_content='c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfEPoC/content/2301.00988v1.pdf'}
+page_content=',  (B 2)  Kd = −Ti  iα  2ni  ��  ˜g∗(v′)ωd˜g(v)J0J′  0d3vd3v′  �  + c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfEPoC/content/2301.00988v1.pdf'}
+page_content='c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfEPoC/content/2301.00988v1.pdf'}
+page_content=',  (B 3)  14  G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfEPoC/content/2301.00988v1.pdf'}
+page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfEPoC/content/2301.00988v1.pdf'}
+page_content=' Plunk and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfEPoC/content/2301.00988v1.pdf'}
+page_content=' Helander  where we separate K = K∥ + Kd and ‘c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfEPoC/content/2301.00988v1.pdf'}
+page_content='c.’ denotes complex conjutage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfEPoC/content/2301.00988v1.pdf'}
+page_content=' These can be  re-expressed in terms of linear operators by writing them in the form  D = (˜g, D˜g) =  �  Ti  �  d3v ˜g∗  F0  D˜g  �  ,  (B 4)  K∥ = (˜g, K∥˜g) =  �  Ti  �  d3v ˜g∗  F0  K∥˜g  �  ,  (B 5)  Kd = (˜g, Kd˜g) =  �  Ti  �  d3v ˜g∗  F0  Kd˜g  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfEPoC/content/2301.00988v1.pdf'}
+page_content='  (B 6)  Identifying D and Kd is simply a matter of exchanging labels of dummy variables of  integration (v for v′, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfEPoC/content/2301.00988v1.pdf'}
+page_content=').' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfEPoC/content/2301.00988v1.pdf'}
+page_content=' Manipulating the expression for K∥ to reveal K∥ is more  involved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfEPoC/content/2301.00988v1.pdf'}
+page_content=' We also need to integrate by parts in l and will need to use the fact that ∂/∂l  is performed at fixed phase space variables Ei and µ = miv2  ⊥/(2B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfEPoC/content/2301.00988v1.pdf'}
+page_content=' The velocity space  volume element contains an important factor of 1/v∥, which generally depends on l and  does not itself commute with ∂/∂l:  d3v = 2πv⊥dv⊥v∥ =  �  σ  2πBdEidµi  m2  i |v∥|  (B 7)  where σ denotes the sign of v∥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfEPoC/content/2301.00988v1.pdf'}
+page_content='  Appendix C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfEPoC/content/2301.00988v1.pdf'}
+page_content=' Moment form of eigenproblem  The three terms of Eqn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfEPoC/content/2301.00988v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfEPoC/content/2301.00988v1.pdf'}
+page_content='9 can be rewritten in terms of the moments of ˜g (Eqn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfEPoC/content/2301.00988v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfEPoC/content/2301.00988v1.pdf'}
+page_content='10):  D˜g = iα  2 ηω∗J0F0  � v2  v2  th  κ1 − κ2  �  ,  (C 1)  K∥˜g = α  2 F0  �  vthJ0  �  −B ∂  ∂l  �κ4  B  �  + κ5  �  + v∥  ∂  ∂l (J0κ1)  �  ,  (C 2)  Kd˜g = iα  2 ωdJ0F0  ��  v2  ⊥  2v2  th  +  v2  ∥  v2  th  �  κ1 − κ3  �  ,  (C 3)  (C 4)  Then, taking moments of Eqn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfEPoC/content/2301.00988v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfEPoC/content/2301.00988v1.pdf'}
+page_content='9 yields the following five equations  2Λ  α κ1 = iηω∗ (G1κ1 − G0κ2) − i∆ωd (G3κ1 − G0κ3) − ∆G0vth  �  κ5 − B ∂  ∂l  �κ4  B  ��  ,(C 5)  2Λ  α κ2 = iηω∗ (G2κ1 − G1κ2) − i∆ωd (G2κ1 − G1κ3) − ∆G1vth  �  κ5 − B ∂  ∂l  �κ4  B  ��  ,(C 6)  2Λ  α κ3 = iηω∗ (G4κ1 − G3κ2) − i∆ωd (G5κ1 − G3κ3) − ∆G3vth  �  κ5 − B ∂  ∂l  �κ4  B  ��  ,(C 7)  2Λ  α κ4 = −∆vth  �  G′  0,2κ1 + G0,2  ∂κ1  ∂l  �  ,(C 8)  2Λ  α κ5 = −∆vth  �  G′′  0,2κ1 + G′  0,2  ∂κ1  ∂l  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfEPoC/content/2301.00988v1.pdf'}
+page_content='  (C 9)  See the next section where the integrals Gm,n, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfEPoC/content/2301.00988v1.pdf'}
+page_content=', are defined and evaluated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfEPoC/content/2301.00988v1.pdf'}
+page_content=' Note that  the final two equations can be immediately used to eliminate κ4 and κ5, leaving a system  Energetic bounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfEPoC/content/2301.00988v1.pdf'}
+page_content=' Part III  15  of three equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfEPoC/content/2301.00988v1.pdf'}
+page_content=' The second and third equations are used together to find forms for κ2  and κ3 in terms of κ1, and these forms are substituted into the first equation to obtain  the final form, in terms of κ1 only, given by Eqn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfEPoC/content/2301.00988v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfEPoC/content/2301.00988v1.pdf'}
+page_content='15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfEPoC/content/2301.00988v1.pdf'}
+page_content='  Appendix D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfEPoC/content/2301.00988v1.pdf'}
+page_content=' Bessel-type integrals  The following definitions, mostly copied from Plunk and Helander (2022), are needed to  perform the various integrals that appear in the moment equations for our eiegenproblem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfEPoC/content/2301.00988v1.pdf'}
+page_content='  First we need a general form of Weber’s integral,  In(p, a1, a2) =  � ∞  0  exp(−pt2)Jn(a1t)Jn(a2t)tdt  = 1  2p exp  �−a2  1 − a2  2  4p  �  In  �a1a2  2p  �  (D 1)  where In is the modified Bessel function of order n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfEPoC/content/2301.00988v1.pdf'}
+page_content=' The integrals we need to evaluate  can be conveniently found in terms of In.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfEPoC/content/2301.00988v1.pdf'}
+page_content=' We define  G⊥m(b) = 2  � ∞  0  xm+1  ⊥  exp(−x2  ⊥)J2  0(  √  2bx2  ⊥)dx⊥,  (D 2a)  G(1)  ⊥m(b) = 2  � ∞  0  xm+2  ⊥  exp(−x2  ⊥)J0(  √  2bx2  ⊥)J1(  √  2bx2  ⊥)dx⊥,  (D 2b)  G(2)  ⊥m(b) = 2  � ∞  0  xm+3  ⊥  exp(−x2  ⊥)J2  1(  √  2bx2  ⊥)dx⊥,  (D 2c)  where m is assumed to be even.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfEPoC/content/2301.00988v1.pdf'}
+page_content=' Now we note that these integrals can be evaluated in  terms of Weber’s integral:  G⊥m(b) = 2  ��  − d  dp  �m/2  I0(p,  √  2b,  √  2b)  �  p=1  ,  (D 3a)  G(1)  ⊥m(b) = 2  ��  − d  dp  �m/2 �  − d  dλ  �  I0(p, λ,  √  2b)  �  p=1,λ=  √  2b  ,  (D 3b)  G(2)  ⊥m(b) = 2  ��  − d  dp  �m/2 �  − d  dλ1  � �  − d  dλ2  �  I0(p, λ1, λ2)  �  p=1,λ1=λ2=  √  2b  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfEPoC/content/2301.00988v1.pdf'}
+page_content=' (D 3c)  The above relations allows us to evaluate the functions  Gm,n(b) = G⊥m(b)G∥n,  (D 4a)  G(1)  m,n(b) = G(1)  ⊥m(b)G∥n,  (D 4b)  G(2)  m,n(b) = G(2)  ⊥m(b)G∥n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfEPoC/content/2301.00988v1.pdf'}
+page_content='  (D 4c)  where  G∥n =  1  √π  � ∞  −∞  exp(−x2  ∥)xn  ∥dx∥ = 1 + (−1)n  2√π  ΓE  �1 + n  2  �  ,  (D 5)  16  G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfEPoC/content/2301.00988v1.pdf'}
+page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfEPoC/content/2301.00988v1.pdf'}
+page_content=' Plunk and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfEPoC/content/2301.00988v1.pdf'}
+page_content=' Helander  and ΓE is the Euler gamma function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfEPoC/content/2301.00988v1.pdf'}
+page_content=' Finally, we can evaluate the integrals G′  m,n, and  G′  m,n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfEPoC/content/2301.00988v1.pdf'}
+page_content=' We define  G′  ⊥m(b) = 2  � ∞  0  xm+1  ⊥  exp(−x2  ⊥)J0  ∂J0  ∂l dx⊥,  (D 6)  G′′  ⊥m(b) = 2  � ∞  0  xm+1  ⊥  exp(−x2  ⊥)  �∂J0  ∂l  �2  dx⊥,  (D 7)  Relating x⊥ to the proper gyrokinetic phase space variable µ, that is x2  ⊥ = µB/Ti allows  the derivatives to be evaluated  ∂J0  ∂l = −  �  2/B ∂  ∂l(B  √  b)x2  ⊥J1  (D 8)  so that we can write  G′  ⊥m(b) = −  �  2  B  ∂  ∂l  �  B  √  b  �  G(1)  ⊥m(b),  (D 9)  G′′  ⊥m(b) = 2  B  � ∂  ∂l  �  B  √  b  ��2  G(2)  ⊥m(b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfEPoC/content/2301.00988v1.pdf'}
+page_content='  (D 10)  These expressions allow us to evaluate  G′  m,n = G′  ⊥mG∥n,  (D 11)  G′′  m,n = G′′  ⊥mG∥n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfEPoC/content/2301.00988v1.pdf'}
+page_content='  (D 12)  D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfEPoC/content/2301.00988v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfEPoC/content/2301.00988v1.pdf'}
+page_content=' Explicit expressions for some Bessel integrals  The b(l)-dependent factors in Eqn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfEPoC/content/2301.00988v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfEPoC/content/2301.00988v1.pdf'}
+page_content='15 can be written as  G0 = G0,0,  (D 13)  G1 = G2,0 + G0,2,  (D 14)  G2 = G4,0 + 2G2,2 + G0,4,  (D 15)  G3 = 1  2G2,0 + G0,2,  (D 16)  G4 = 1  2G4,0 + 3  2G2,2 + G0,4,  (D 17)  G5 = 1  4G4,0 + G2,2 + G0,4,  (D 18)  which can be evaluated using the identities of the previous section in terms of the familiar  Γn(b) of gyrokinetic theory (suppressing its argument for succinctness):  Energetic bounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfEPoC/content/2301.00988v1.pdf'}
+page_content=' Part III  17  G0 = Γ0,  (D 19)  G1 =  �3  2 − b  �  Γ0 + bΓ1,  (D 20)  G2 = 1  4  ��  6b2 − 20b + 15  �  Γ0 + 2b ((10 − 4b)Γ1 + bΓ2)  �  ,  (D 21)  G3 = 1  2 (bΓ1 − (b − 2)Γ0) ,  (D 22)  G4 = 1  4  ��  3b2 − 11b + 10  �  Γ0 + b ((11 − 4b)Γ1 + bΓ2)  �  ,  (D 23)  G5 = 1  8  ��  3b2 − 12b + 14  �  Γ0 + b (bΓ2 − 4(b − 3)Γ1)  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfEPoC/content/2301.00988v1.pdf'}
+page_content='  (D 24)  where we recall  Γn(b) = exp(−b)In(b)  (D 25)  For completeness, we evaluate the few remaining factors that enter Eqn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfEPoC/content/2301.00988v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfEPoC/content/2301.00988v1.pdf'}
+page_content='15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfEPoC/content/2301.00988v1.pdf'}
+page_content='  G0,2 = Γ0  2 ,  (D 26)  G(1)  0,2 =  √  b (Γ0 − Γ1)  2  √  2  ,  (D 27)  G(2)  0,2 = 1  8 (3bΓ0 + (2 − 4b)Γ1 + bΓ2) ,  (D 28)  and using Eqn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfEPoC/content/2301.00988v1.pdf'}
+page_content=' D 11  G′  0,2 = −  �  2  B  ∂  ∂l  �  B  √  b  �  G(1)  0,2,  (D 29)  G′′  0,2 = 2  B  � ∂  ∂l  �  B  √  b  ��2  G(2)  0,2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfEPoC/content/2301.00988v1.pdf'}
+page_content='  (D 30)  D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfEPoC/content/2301.00988v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfEPoC/content/2301.00988v1.pdf'}
+page_content=' Limit b → 0  In the limit b → 0 we obtain  G0 = 1,  (D 31)  G1 = 3  2,  (D 32)  G2 = 15  4 ,  (D 33)  G3 = 1,  (D 34)  G4 = 5  2,  (D 35)  G5 = 7  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfEPoC/content/2301.00988v1.pdf'}
+page_content='  (D 36)  and  18  G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfEPoC/content/2301.00988v1.pdf'}
+page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfEPoC/content/2301.00988v1.pdf'}
+page_content=' Plunk and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfEPoC/content/2301.00988v1.pdf'}
+page_content=' Helander  G0,2 = 1  2,  (D 37)  G(1)  0,2 = 0,  (D 38)  G(2)  0,2 = 0,  (D 39)  REFERENCES  P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfEPoC/content/2301.00988v1.pdf'}
+page_content=' Helander and G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfEPoC/content/2301.00988v1.pdf'}
+page_content='G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfEPoC/content/2301.00988v1.pdf'}
+page_content=' Plunk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfEPoC/content/2301.00988v1.pdf'}
+page_content='  Energetic bounds on gyrokinetic instabilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfEPoC/content/2301.00988v1.pdf'}
+page_content=' Part 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfEPoC/content/2301.00988v1.pdf'}
+page_content='  Fundamentals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfEPoC/content/2301.00988v1.pdf'}
+page_content=' Journal of Plasma Physics, 88(2):905880207, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfEPoC/content/2301.00988v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfEPoC/content/2301.00988v1.pdf'}
+page_content='  G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfEPoC/content/2301.00988v1.pdf'}
+page_content='G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfEPoC/content/2301.00988v1.pdf'}
+page_content=' Plunk and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfEPoC/content/2301.00988v1.pdf'}
+page_content=' Helander.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfEPoC/content/2301.00988v1.pdf'}
+page_content=' Energetic bounds on gyrokinetic instabilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfEPoC/content/2301.00988v1.pdf'}
+page_content=' Part 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfEPoC/content/2301.00988v1.pdf'}
+page_content=' Modes of  optimal growth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfEPoC/content/2301.00988v1.pdf'}
+page_content=' Journal of Plasma Physics, 88(3):905880313, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfEPoC/content/2301.00988v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfEPoC/content/2301.00988v1.pdf'}
+page_content='  Matt Landreman, Gabriel G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfEPoC/content/2301.00988v1.pdf'}
+page_content=' Plunk, and William Dorland.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfEPoC/content/2301.00988v1.pdf'}
+page_content=' Generalized universal instability:  transient linear amplification and subcritical turbulence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfEPoC/content/2301.00988v1.pdf'}
+page_content=' Journal of Plasma Physics, 81  (5):905810501, 2015.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfEPoC/content/2301.00988v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfEPoC/content/2301.00988v1.pdf'}
+page_content='  W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfEPoC/content/2301.00988v1.pdf'}
+page_content=' Dorland and G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfEPoC/content/2301.00988v1.pdf'}
+page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfEPoC/content/2301.00988v1.pdf'}
+page_content=' Hammett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfEPoC/content/2301.00988v1.pdf'}
+page_content=' Gyrofluid turbulence models with kinetic effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfEPoC/content/2301.00988v1.pdf'}
+page_content=' Physics  of Fluids B: Plasma Physics, 5(3):812–835, 1993.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfEPoC/content/2301.00988v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfEPoC/content/2301.00988v1.pdf'}
+page_content=' URL https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfEPoC/content/2301.00988v1.pdf'}
+page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfEPoC/content/2301.00988v1.pdf'}
+page_content='1063/1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfEPoC/content/2301.00988v1.pdf'}
+page_content='  860934.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfEPoC/content/2301.00988v1.pdf'}
+page_content='  G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfEPoC/content/2301.00988v1.pdf'}
+page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfEPoC/content/2301.00988v1.pdf'}
+page_content=' Plunk, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfEPoC/content/2301.00988v1.pdf'}
+page_content=' Helander, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfEPoC/content/2301.00988v1.pdf'}
+page_content=' Xanthopoulos, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfEPoC/content/2301.00988v1.pdf'}
+page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfEPoC/content/2301.00988v1.pdf'}
+page_content=' Connor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfEPoC/content/2301.00988v1.pdf'}
+page_content=' Collisionless microinstabilities  in stellarators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfEPoC/content/2301.00988v1.pdf'}
+page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfEPoC/content/2301.00988v1.pdf'}
+page_content=' the ion-temperature-gradient mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfEPoC/content/2301.00988v1.pdf'}
+page_content=' Physics of Plasmas, 21(3):032112,  2014.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfEPoC/content/2301.00988v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfEPoC/content/2301.00988v1.pdf'}
+page_content=' URL https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfEPoC/content/2301.00988v1.pdf'}
+page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfEPoC/content/2301.00988v1.pdf'}
+page_content='1063/1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfEPoC/content/2301.00988v1.pdf'}
+page_content='4868412.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfEPoC/content/2301.00988v1.pdf'}
+page_content='  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfEPoC/content/2301.00988v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfEPoC/content/2301.00988v1.pdf'}
+page_content=' Schekochihin, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfEPoC/content/2301.00988v1.pdf'}
+page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfEPoC/content/2301.00988v1.pdf'}
+page_content=' Cowley, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfEPoC/content/2301.00988v1.pdf'}
+page_content=' Dorland, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfEPoC/content/2301.00988v1.pdf'}
+page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfEPoC/content/2301.00988v1.pdf'}
+page_content=' Hammett, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfEPoC/content/2301.00988v1.pdf'}
+page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfEPoC/content/2301.00988v1.pdf'}
+page_content=' Howes, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfEPoC/content/2301.00988v1.pdf'}
+page_content=' Quataert, and  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfEPoC/content/2301.00988v1.pdf'}
+page_content=' Tatsuno.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfEPoC/content/2301.00988v1.pdf'}
+page_content=' Astrophysical gyrokinetics: Kinetic and fluid turbulent cascades in magnetized  weakly collisional plasmas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfEPoC/content/2301.00988v1.pdf'}
+page_content=' The Astrophysical Journal Supplement Series, 182(1):310, may  2009.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfEPoC/content/2301.00988v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfEPoC/content/2301.00988v1.pdf'}
+page_content=' URL https://dx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfEPoC/content/2301.00988v1.pdf'}
+page_content='doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfEPoC/content/2301.00988v1.pdf'}
+page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfEPoC/content/2301.00988v1.pdf'}
+page_content='1088/0067-0049/182/1/310.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfEPoC/content/2301.00988v1.pdf'}
+page_content='  G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfEPoC/content/2301.00988v1.pdf'}
+page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfEPoC/content/2301.00988v1.pdf'}
+page_content=' PLUNK, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfEPoC/content/2301.00988v1.pdf'}
+page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfEPoC/content/2301.00988v1.pdf'}
+page_content=' COWLEY, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfEPoC/content/2301.00988v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfEPoC/content/2301.00988v1.pdf'}
+page_content=' SCHEKOCHIHIN, and T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfEPoC/content/2301.00988v1.pdf'}
+page_content=' TATSUNO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfEPoC/content/2301.00988v1.pdf'}
+page_content=' Two-dimensional  gyrokinetic turbulence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfEPoC/content/2301.00988v1.pdf'}
+page_content=' Journal of Fluid Mechanics, 664:407–435, 2010.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfEPoC/content/2301.00988v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfEPoC/content/2301.00988v1.pdf'}
+page_content='  P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfEPoC/content/2301.00988v1.pdf'}
+page_content=' Helander, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfEPoC/content/2301.00988v1.pdf'}
+page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfEPoC/content/2301.00988v1.pdf'}
+page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfEPoC/content/2301.00988v1.pdf'}
+page_content=' Proll, and G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfEPoC/content/2301.00988v1.pdf'}
+page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfEPoC/content/2301.00988v1.pdf'}
+page_content=' Plunk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfEPoC/content/2301.00988v1.pdf'}
+page_content=' Collisionless microinstabilities in stellarators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfEPoC/content/2301.00988v1.pdf'}
+page_content=' i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfEPoC/content/2301.00988v1.pdf'}
+page_content='  analytical theory of trapped-particle modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfEPoC/content/2301.00988v1.pdf'}
+page_content=' Physics of Plasmas, 20(12):122505, 2013.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfEPoC/content/2301.00988v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfEPoC/content/2301.00988v1.pdf'}
+page_content='  URL https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfEPoC/content/2301.00988v1.pdf'}
+page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfEPoC/content/2301.00988v1.pdf'}
+page_content='1063/1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfEPoC/content/2301.00988v1.pdf'}
+page_content='4846818.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfEPoC/content/2301.00988v1.pdf'}
+page_content='  Bogdan Teaca.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfEPoC/content/2301.00988v1.pdf'}
+page_content=' private communication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfEPoC/content/2301.00988v1.pdf'}
+page_content='  H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfEPoC/content/2301.00988v1.pdf'}
+page_content=' Biglari, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfEPoC/content/2301.00988v1.pdf'}
+page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfEPoC/content/2301.00988v1.pdf'}
+page_content=' Diamond, and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfEPoC/content/2301.00988v1.pdf'}
+page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfEPoC/content/2301.00988v1.pdf'}
+page_content=' Rosenbluth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfEPoC/content/2301.00988v1.pdf'}
+page_content=' Toroidal ion-pressure-gradient-driven drift  instabilities and transport revisited.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfEPoC/content/2301.00988v1.pdf'}
+page_content=' Physics of Fluids B: Plasma Physics, 1(1):109–118,  1989.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfEPoC/content/2301.00988v1.pdf'}
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+page_content='PFB/1/109/1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfEPoC/content/2301.00988v1.pdf'}
+page_content='  P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfEPoC/content/2301.00988v1.pdf'}
+page_content=' Helander and G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfEPoC/content/2301.00988v1.pdf'}
+page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfEPoC/content/2301.00988v1.pdf'}
+page_content=' Plunk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfEPoC/content/2301.00988v1.pdf'}
+page_content=' Upper bounds on gyrokinetic instabilities in magnetized plasmas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfEPoC/content/2301.00988v1.pdf'}
+page_content='  Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfEPoC/content/2301.00988v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfEPoC/content/2301.00988v1.pdf'}
+page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfEPoC/content/2301.00988v1.pdf'}
+page_content=', 127:155001, Oct 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfEPoC/content/2301.00988v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfEPoC/content/2301.00988v1.pdf'}
+page_content=' URL https://link.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfEPoC/content/2301.00988v1.pdf'}
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+page_content='org/doi/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfEPoC/content/2301.00988v1.pdf'}
+page_content='1103/  PhysRevLett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfEPoC/content/2301.00988v1.pdf'}
+page_content='127.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfEPoC/content/2301.00988v1.pdf'}
+page_content='155001.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfEPoC/content/2301.00988v1.pdf'}
+page_content='  G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfEPoC/content/2301.00988v1.pdf'}
+page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfEPoC/content/2301.00988v1.pdf'}
+page_content=' Roberg-Clark, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfEPoC/content/2301.00988v1.pdf'}
+page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfEPoC/content/2301.00988v1.pdf'}
+page_content=' Plunk, and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfEPoC/content/2301.00988v1.pdf'}
+page_content=' Xanthopoulos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfEPoC/content/2301.00988v1.pdf'}
+page_content=' Coarse-grained gyrokinetics for the  critical ion temperature gradient in stellarators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfEPoC/content/2301.00988v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfEPoC/content/2301.00988v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfEPoC/content/2301.00988v1.pdf'}
+page_content=' Research, 4:L032028, Aug  2022a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfEPoC/content/2301.00988v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfEPoC/content/2301.00988v1.pdf'}
+page_content=' URL https://link.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfEPoC/content/2301.00988v1.pdf'}
+page_content='aps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfEPoC/content/2301.00988v1.pdf'}
+page_content='org/doi/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfEPoC/content/2301.00988v1.pdf'}
+page_content='1103/PhysRevResearch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfEPoC/content/2301.00988v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfEPoC/content/2301.00988v1.pdf'}
+page_content='L032028.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfEPoC/content/2301.00988v1.pdf'}
+page_content='  G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfEPoC/content/2301.00988v1.pdf'}
+page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfEPoC/content/2301.00988v1.pdf'}
+page_content=' Roberg-Clark, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfEPoC/content/2301.00988v1.pdf'}
+page_content=' Xanthopoulos, and G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfEPoC/content/2301.00988v1.pdf'}
+page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfEPoC/content/2301.00988v1.pdf'}
+page_content=' Plunk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfEPoC/content/2301.00988v1.pdf'}
+page_content=' Reduction of electrostatic turbulence  in a quasi-helically symmetric stellarator via critical gradient optimization, 2022b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfEPoC/content/2301.00988v1.pdf'}
+page_content=' URL  https://arxiv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfEPoC/content/2301.00988v1.pdf'}
+page_content='org/abs/2210.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfEPoC/content/2301.00988v1.pdf'}
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+arXiv:2301.12147v1  [math.AP]  28 Jan 2023
+LOGISTIC ELLIPTIC EQUATION WITH A NONLINEAR BOUNDARY
+CONDITION ARISING FROM COASTAL FISHERY HARVESTING II
+KENICHIRO UMEZU
+Abstract. We study the positive solutions of the logistic elliptic equation with a nonlinear
+Neumann boundary condition that models coastal fishery harvesting ([18]). An essential role is
+played by the smallest eigenvalue of the Dirichlet eigenvalue problem, with respect to which a
+noncritical case is studied in [32]. In this paper, we extend our analysis to the critical case and
+further study the noncritical case for a more precise description of the positive solution set. Our
+approach relies on the energy method, sub- and supersolutions, and implicit function analysis.
+1. Introduction
+This paper is devoted to the study of the positive solutions for the following logistic elliptic
+equation with a nonlinear boundary condition arising from coastal fishery harvesting ([18]):
+
+
+
+
+
+−∆u = u − up
+in Ω,
+u ≥ 0
+in Ω,
+∂u
+∂ν = −λuq
+on ∂Ω.
+(1.1)
+Here, Ω ⊂ RN, N ≥ 1, is a bounded domain with smooth boundary ∂Ω, ∆ = �N
+i=1
+∂2
+∂x2
+i is the
+usual Laplacian in RN, 0 < q < 1 < p, λ ≥ 0 is a parameter, and ν is the unit outer normal to
+∂Ω. Unless stated otherwise, throughout this paper we assume the subcritical condition
+p < N + 2
+N − 2
+for N > 2.
+(1.2)
+In the case of p = 2, the unknown function u ≥ 0 ecologically represents the biomass of fish that
+inhabit a lake Ω, obeying the logistic law ([8]), and the nonlinear boundary condition means
+fishery harvesting with the harvesting effort λ on the lake coast ∂Ω, obeying the Cobb–Douglas
+production function ([18, Subsection 2.1]).
+A nonnegative function u ∈ H1(Ω) is called a nonnegative (weak) solution of (1.1) if u satisfies
+�
+Ω
+�
+∇u∇ϕ − uϕ + upϕ
+�
++ λ
+�
+∂Ω
+uqϕ = 0,
+ϕ ∈ H1(Ω)
+(1.3)
+(we may regard (λ, u) as a nonnegative solution of (1.1)). It is seen that problem (1.1) has
+a solution (λ, 0) for every λ > 0, called a trivial solution.
+The sets {(λ, 0) : λ ≥ 0} and
+{(λ, 0) : λ > 0} are said to be the trivial lines. We know ([30]) that a nonnegative solution u of
+(1.1) belongs to the space W 1,r(Ω) for r > N (consequently, Cθ(Ω) for θ ∈ (0, 1)). Moreover, a
+nontrivial nonnegative solution u of (1.1) satisfies that u ∈ C2+θ(Ω) for θ ∈ (0, 1), and u > 0
+in Ω ([17], [27]), which is called a positive solution. Indeed, if u > 0 in Ω, then u ∈ C2+θ(Ω)
+by the bootstrap argument using elliptic regularity, and u satisfies (1.1) pointwisely in Ω in the
+classical sense. However, we do not know if u > 0 on the entirety of ∂Ω for a positive solution u
+2020 Mathematics Subject Classification. 35J65, 35B32, 35J25, 92D40.
+Key words and phrases. logistic elliptic equation, concave–convex nonlinearity, positive solution, uniqueness,
+stability, sub- and supersolutions, energy method, boundary harvesting.
+1
+
+of (1.1). As a matter of fact, Hopf’s boundary point lemma ([27]) does not work because of the
+lack of the one-sided Lipschitz condition [26, (4.1.19)] for mapping 0 ≤ u �→ (−uq) for u close to
+0.
+For a positive solution (λ, u) of (1.1) satisfying u > 0 in Ω, we call γ1 = γ1(λ, u) ∈ R the
+smallest eigenvalue of the linearized eigenvalue problem at (λ, u)
+�
+−∆ϕ = ϕ − pup−1ϕ + γϕ
+in Ω,
+∂ϕ
+∂ν = −λquq−1ϕ + γϕ
+on ∂Ω.
+(1.4)
+It is well known that γ1 is simple with a positive eigenfunction ϕ1 ∈ C2+θ(Ω) satisfying ϕ1 > 0
+in Ω. Indeed, γ1 is characterized by the variational formula
+γ1 = inf
+��
+Ω
+�
+|∇ϕ|2 − ϕ2 + pup−1ϕ2
+�
++ λ
+�
+∂Ω
+quq−1ϕ2 : ϕ ∈ H1(Ω),
+�
+Ω
+ϕ2 +
+�
+∂Ω
+ϕ2 = 1
+�
+.
+A positive solution u > 0 in Ω of (1.1) is said to be asymptotically stable, weakly stable, and
+unstable if γ1 > 0, γ1 ≥ 0, and γ1 < 0, respectively.
+Problem (1.1) possesses a sublinear nonlinearity at infinity and also a concave–convex nature.
+Thus, the global uniqueness of a positive solution of (1.1) for every λ > 0 would not be so easy
+to deduce. For nonlinear elliptic problems with a concave–convex nature, we refer to [4, 31, 1,
+5, 11, 12, 19]. The sublinear nonlinearity (−uq) that appears in (1.1) induces the absorption
+effect on ∂Ω. Sublinear boundary conditions of the uq type were explored in [16, 14, 15, 28, 29].
+The case of an incoming flux on ∂Ω was studied in [16, 15]. The mixed case of absorption and
+an incoming flux on ∂Ω was studied in [14]. The absorption case was also studied in [28, 29],
+where a similar type of logistic elliptic equation with an indefinite weight has been analyzed for
+the existence and multiplicity of nontrivial nonnegative solutions.
+An important role is played by the smallest eigenvalue βΩ > 0 of the Dirichlet eigenvalue
+problem
+�
+−∆φ = βφ
+in Ω,
+φ = 0
+on ∂Ω.
+It is well known that βΩ is simple with a positive eigenfunction φΩ ∈ H1
+0(Ω) (implying φΩ ∈
+C2+θ(Ω) by elliptic regularity). Indeed, φΩ > 0 in Ω, and
+c1 ≤ −∂φΩ
+∂ν ≤ c2
+on ∂Ω
+(1.5)
+for some 0 < c1 < c2. Moreover, βΩ is characterized by the variational formula
+βΩ = inf
+��
+Ω
+|∇φ|2 : φ ∈ H1
+0(Ω),
+�
+Ω
+φ2 = 1
+�
+.
+If βΩ < 1, then uD ∈ H1
+0(Ω) ∩ C2+θ(Ω) denotes the unique positive solution of the Dirichlet
+logistic problem ([7])
+�
+−∆u = u − up
+in Ω,
+u = 0
+on ∂Ω.
+(1.6)
+The existence, nonexistence, and multiplicity of positive solutions for (1.1) in the case where
+βΩ ̸= 1 were studied in the author’s previous work [32, Theorems 1.1, 1.2, 1.4, 1.5], which
+Theorem 0 summarizes.
+Theorem 0.
+(I) A positive solution u of (1.1) satisfies that u < 1 in Ω and u > 0 on Γ ⊂ ∂Ω with the
+condition |Γ| > 0.
+2
+
+(II) There exists λ∗ > 0 such that problem (1.1) has a positive solution curve
+C0 = {(λ, uλ) : 0 ≤ λ ≤ λ∗},
+(1.7)
+emanating from (λ, u) = (0, 1), that satisfies the following three conditions:
+• λ �→ uλ ∈ C2+θ(Ω) is C∞,
+• uλ > 0 in Ω,
+• uλ is asymptotically stable.
+Moreover, the positive solutions of (1.1) near (λ, u) = (0, 1) in R × C2+θ(Ω) form C0.
+Let λ be the positive value defined as
+λ = sup{λ > 0 : (1.1) has a positive solution for λ}.
+(1.8)
+Then, the following assertions hold.
+(III) Assume that βΩ < 1. Then, we have the following (as in Figure 1).
+(i) λ = ∞, and more precisely, problem (1.1) possesses a positive solution u for every
+λ > 0 such that u > 0 in Ω.
+(ii) (λ, uλ) ∈ C0 is a unique positive solution of (1.1) for 0 < λ ≤ λ∗ (by making λ∗ in
+(1.7) smaller if necessary).
+(iii) un → uD in H1(Ω) for a positive solution (λn, un) of (1.1) with λn → ∞.
+(iv) The positive solution set {(λ, u)} does not meet the trivial line {(λ, 0) : λ ≥ 0} in
+the topology of H1(Ω) (nor C(Ω)).
+(IV) Assume that βΩ > 1. Then, we have the following (as in Figure 2).
+(i) λ < ∞.
+(ii) There exists a bounded subcontinuum (closed connected subset) �C0 = {(λ, u)} of
+nonnegative solutions of (1.1) in [0, ∞) × C(Ω) joining (λ, u) = (0, 1) and (0, 0)
+such that �C0 \ {(0, 0)} includes C0 and consists of positive solutions of (1.1). Par-
+ticularly, problem (1.1) has at least two positive solutions for λ > 0 small.
+(iii) The positive solution set {(λ, u)} does not meet the trivial line {(λ, 0) : λ > 0} in
+the topology of H1(Ω) (nor C(Ω)).
+(iv) γ1(λn, un) < 0 for a positive solution (λn, un) of (1.1) such that (λn, un) → (0, 0)
+in R × H1(Ω), provided that un > 0 in Ω, i.e., un is unstable.
+Remark 0.
+(i) Assertions (I) and (II) hold for every case of βΩ > 0.
+(ii) Assertions (II) and (III-i) hold for any p > 1.
+(iii) Problem (1.1) with λ = 0 has exactly two nonnegative solutions (λ, u) = (0, 0), (0, 1).
+Thus, Theorem 0(I) is used to show easily that in every case of βΩ > 0, the positive
+solution set {(λ, u)} of (1.1) meets at most (0, 0) and (0, 1) on {(0, u) : u ≥ 0}, i.e., if
+(λn, un) is a positive solution of (1.1) such that (λn, un) → (0, u) in H1(Ω) (equivalently
+C(Ω) by elliptic regularity), then either u = 0 or 1.
+In this paper, we extend our consideration to the case where βΩ = 1 and further study the
+positive solution set in the case where βΩ < 1. Our first main result concerns the case where
+βΩ < 1. On the basis of Theorem 0(III), we present the uniqueness and stability of a positive
+solution of (1.1) for λ > 0 large and also the strong positivity of the positive solutions for every
+λ > 0.
+Theorem 1.1. Assume that βΩ < 1. Then, the following assertions hold (see Figure 1):
+3
+
+(i) There exists λ∗ ≥ λ∗ such that the positive solution of (1.1) ensured by Theorem 0(III-i)
+is unique for every λ > λ∗ (say uλ); more precisely, the positive solutions of (1.1) for
+λ > λ∗ form a C∞ curve C∞ = {(λ, uλ) : λ∗ < λ} (i.e., λ �→ uλ ∈ C2+θ(Ω) is C∞),
+which satisfies the following conditions:
+(a) uλ is asymptotically stable,
+(b) uλ −→ uD in H1(Ω) as λ → ∞,
+(c) uλ is decreasing, i.e., uλ1 > uλ2 in Ω if λ1 < λ2. Furthermore, if 0 < λ1 < λ2 with
+the condition that λ1 ≤ λ∗ < λ2, then u > uλ2 in Ω for a positive solution u of
+(1.1) for λ = λ1.
+(ii) u > 0 in Ω for a positive solution u of (1.1) for every λ > 0 (strong positivity).
+Figure 1. Possible positive solution sets in the case where βΩ < 1.
+Remark 1.2.
+(i) Assertions (i-a) and (i-b) hold for any p > 1 (see Remark 3.3).
+(ii) For (λ, uλ) ∈ C0 with 0 ≤ λ ≤ λ∗ in (1.7), we present similar results as those in assertions
+(i-c). Indeed, λ �→ uλ is decreasing for 0 < λ ≤ λ∗; if 0 < λ1 < λ2 with the condition
+that λ1 ≤ λ∗ < λ2, then uλ1 > u in Ω for a positive solution u of (1.1) with λ = λ2.
+(iii) It is an open question to get the global uniqueness for a positive solution of (1.1) for
+all λ > 0, i.e., λ∗ = λ∗. In this case, [0, ∞) ∋ λ �→ uλ is C∞ and decreasing.
+(iv) For uniqueness and stability analysis of positive solutions for large parameters in non-
+linear elliptic problems, we refer to [9, 34, 33, 24, 10, 25, 13, 20, 21, 22].
+Our second main result is the counterpart of Theorem 0(III-i) and (III-iii) for the case where
+βΩ = 1 and pq > 1.
+Theorem 1.3. Assume that βΩ = 1 and pq > 1. Then, problem (1.1) possesses a positive
+solution uλ for every λ > 0 such that uλ > 0 in Ω, which satisfies that
+uλ −→ 0
+and
+uλ
+∥uλ∥ −→ φΩ
+in H1(Ω)
+as λ → ∞.
+(1.9)
+Remark 1.4.
+(i) The existence assertion holds for any p > 1; thus, so does assertion (1.9) (see Remark
+3.3).
+(ii) Similarly as in Theorem 0(III-iii), assertion (1.9) is valid if we assume a positive solution
+(λ, uλ) (which may take zero value somewhere on ∂Ω) of (1.1) with λ → ∞.
+Our third main result is the counterpart of Theorem 0(III-iv), (IV-i), and (IV-iii) for the case
+where βΩ = 1.
+4
+
+u
+u
+A
+1
+1
+Co
+Co
+....
+入
+入
+*Y
+\*
+0
+入*
+\*
+0Theorem 1.5. Assume that βΩ = 1. Then, the following three assertions hold.
+(i) If pq ≤ 1, then λ < ∞ where λ > 0 is defined by (1.8).
+(ii) If pq ̸= 1, then the positive solution set {(λ, u)} of (1.1) does not meet the trivial line
+{(λ, 0) : λ > 0} in the topology of H1(Ω) (nor C(Ω)).
+(iii) If pq ≥ 1, then it does not meet {(0, 0)} in the topology of H1(Ω) (nor C(Ω)).
+Theorem 1.5 provides a guess (Remark 1.6) for the global extension of the C∞ positive solution
+curve C0 given by (1.7) in the case where βΩ = 1 and pq ≤ 1.
+Remark 1.6. Assume that βΩ = 1 and pq ≤ 1.
+Let �C0 = {(λ, u)} ⊂ [0, ∞) × C(Ω) be
+the component (maximal, closed, and connected subset) of nonnegative solutions of (1.1) that
+includes C0. From Theorems 0(I) and Theorem 1.5(i), �C0 \ {(0, 1)} ⊂ {(λ, u) ∈ [0, ∞) × C(Ω) :
+λ ≤ λ, u < 1 in Ω}. If we suppose that Γ0 :=
+�
+�C0 \ {(0, 1)}
+�
+∩{(λ, 0), (0, u) : λ ≥ 0, u ≥ 0} ̸= ∅,
+then Theorem 1.5(ii),(iii) show that Γ0 = {(0, 0)} and Γ0 ⊂ {(λ, 0) : λ ≥ Λ0} for some Λ0 > 0
+when pq < 1 and pq = 1, respectively. The existence of �C0 is still an open question. Suggested
+positive solution sets are illustrated in Figures 2 and 3.
+Figure 2. Suggested positive solution set in the case where βΩ = 1 and pq < 1.
+Figure 3. Suggested positive solution set in the case where βΩ = 1 and pq = 1,
+and λc ∈ Γ0.
+We conclude the Introduction by mentioning the stability of the trivial solution u = 0. A
+linearized stability analysis does not work for u = 0 because u �→ uq is not differentiable at
+u = 0.
+Instead, by the construction of suitable sub- and supersolutions of (1.1), we try to
+employ the Lyapunov stability criterion [26, Chapter 5] on the basis of the monotone iteration
+method, which is developed in Section 5.
+Notation: ∥ · ∥ denotes the usual norm of H1(Ω).
+un ⇀ u∞ means that un weakly
+converges to u∞ in H1(Ω). H1
+0(Ω) = {u ∈ H1(Ω) : u = 0 on ∂Ω}.
+�
+Ω fdx for f ∈ L1(Ω)
+5
+
+u
+1
+入
+入cu
+1and
+�
+∂Ω gdσ for g ∈ L1(∂Ω) are simply written as
+�
+Ω f and
+�
+∂Ω g, respectively. | · |
+represents both the Lebesgue measure in Ω and the surface measure on ∂Ω.
+The remainder of this paper is organized as follows. Sections 2 and 3 are devoted to the
+preparation for the proofs of Theorems 1.1, 1.3 and 1.5. In Section 2, we develop the energy
+method for the energy functional associated with (1.1).
+In Section 3, we use the sub- and
+supersolution method to prove existence and positivity results for positive solutions of (1.1).
+We give proofs for Theorems 1.1 and 1.3 in Section 3. In Section 4, we prove Theorem 1.5.
+Section 5 is devoted to a stability analysis of the trivial solution u = 0, which is based on
+Lemma 3.1 and Theorem 5.1.
+2. Energy method
+Let
+E(u) =
+�
+Ω
+(|∇u|2 − u2),
+u ∈ H1(Ω);
+then, the next lemma is used several times in the following arguments.
+Lemma 2.1. Let {un} ⊂ H1(Ω) satisfy E(un) ≤ 0, un ⇀ u∞, and un → u∞ in L2(Ω). Then,
+u∞ ̸= 0 if ∥un∥ ≥ C for some C > 0.
+Proof. By the weak lower semicontinuity, E(u∞) ≤ limn E(un) ≤ limn E(un) ≤ 0. If u∞ = 0,
+then ∥un∥ → 0, as desired.
+□
+We start by proving the following two propositions, which provide the asymptotic profile of
+a positive solution of (1.1) as λ → ∞. It is understood that uD = 0 if βΩ = 1.
+Proposition 2.2. Assume that βΩ ≤ 1. Let (λn, un) be a positive solution of (1.1) with λn → ∞.
+Then, un → uD in H1(Ω).
+Proof. We first assume that βΩ < 1. Because un < 1 in Ω, we substitute u = ϕ = un into (1.3)
+to deduce that
+�
+Ω
+|∇un|2 =
+�
+Ω
+�
+u2
+n − up+1
+n
+�
+− λn
+�
+∂Ω
+uq+1
+n
+(2.1)
+≤
+�
+Ω
+u2
+n ≤ |Ω|;
+thus, ∥un∥ is bounded. Immediately, up to a subsequence, un ⇀ u∞ ≥ 0, un → u∞ in L2(Ω)
+and L2(∂Ω), and un → u∞ a.e. in Ω for some u∞ ∈ H1(Ω). We then infer that
+�
+∂Ω
+uq+1
+n
+= 1
+λn
+�
+−
+�
+Ω
+|∇un|2 +
+�
+Ω
+�
+u2
+n − up+1
+n
+��
+≤ 1
+λn
+�
+Ω
+u2
+n −→ 0,
+(2.2)
+which implies that
+�
+∂Ω uq+1
+∞
+= 0; thus, u∞ ∈ H1
+0(Ω). From (1.3) with (λ, u) = (λn, un), it follows
+that
+�
+Ω
+�
+∇un∇ϕ − unϕ + up
+nϕ
+�
+= 0,
+ϕ ∈ H1
+0(Ω).
+Taking the limit, u∞ is a nonnegative solution of (1.6), where we have used the Lebesgue
+dominated convergence theorem to deduce that
+�
+Ω up
+nϕ →
+�
+Ω up
+∞ϕ.
+Then, we verify that u∞ ̸= 0. Since E(un) ≤ 0, the weak lower semicontinuity means that
+E(u∞) ≤ lim
+n→∞
+E(un) ≤ lim
+n→∞ E(un) ≤ 0.
+6
+
+If u∞ = 0, then it follows that ∥un∥ → 0. Here, we may assume that un → 0 a.e. in Ω. Say that
+wn =
+un
+∥un∥; then, up to a subsequence, wn ⇀ w∞ ≥ 0, wn → w∞ in L2(Ω) and L2(∂Ω), and
+wn → w∞ a.e. in Ω. Since ∥wn∥ = 1, we deduce that w∞ ̸= 0 using Lemma 2.1. However, we
+observe from (2.2) that
+�
+∂Ω
+wq+1
+n
+≤ 1
+λn
+�
+Ω
+w2
+n∥un∥1−q −→ 0.
+This implies that w∞ ∈ H1
+0(Ω). From (1.3) with (λ, u) = (λn, un), we see that
+�
+Ω
+�
+∇wn∇ϕ − wnϕ + wnϕ up−1
+n
+�
+= 0,
+ϕ ∈ H1
+0(Ω).
+Taking the limit,
+�
+Ω(∇w∞ϕ − w∞ϕ) = 0, where we have used the Lebesgue dominated conver-
+gence theorem to obtain that
+�
+Ω
+��wnϕ up−1
+n
+�� ≤
+��
+Ω
+w2
+n
+� 1
+2 ��
+Ω
+ϕ2u2(p−1)
+n
+� 1
+2
+≤ C
+��
+Ω
+ϕ2u2(p−1)
+n
+� 1
+2
+−→ 0.
+This implies that w∞ is a nontrivial nonnegative solution of the problem
+�
+−∆w = w
+in Ω,
+w = 0
+on ∂Ω.
+Thus, we deduce that βΩ = 1, which contradicts the assumption. The assertion that u∞ ≥ 0
+and u∞ ̸= 0 means that u∞ is the unique positive solution uD of (1.6) by the strong maximum
+principle.
+It remains to show that un → u∞ in H1(Ω).
+Observing that E(u∞) +
+�
+Ω up+1
+∞
+= 0 and
+E(un) ≤ −
+�
+Ω up+1
+n
+, we deduce that
+E(u∞) ≤ lim
+n→∞
+E(un) ≤ lim
+n→∞ E(un) ≤ − lim
+n→∞
+�
+Ω
+up+1
+n
+= −
+�
+Ω
+up+1
+∞
+= E(u∞),
+where we have used the Lebesgue dominated convergence theorem again. Thus, E(un) → E(u∞),
+i.e., ∥un∥ → ∥u∞∥. Since un ⇀ u∞, the desired assertion follows.
+Next, we assume that βΩ = 1. Then, u∞ is a nonnegative solution of (1.6), and indeed u∞ = 0
+because βΩ = 1 ([7]). Thus, ∥un∥ → 0, as desired.
+□
+In the case where βΩ = 1, we observe that E(u) ≥ 0 for u ∈ H1
+0(Ω). Indeed, we note that
+�
+u ∈ H1(Ω) : E(u) = 0
+�
+= ⟨φΩ⟩ := {sφΩ : s ∈ R}.
+We then investigate the asymptotic profile of a positive solution (λn, un) of (1.1) with ∥un∥ → 0.
+Proposition 2.3. Assume that βΩ = 1. Let (λn, un) be a positive solution of (1.1) such that
+λn ≥ λ for some λ > 0 and ∥un∥ → 0. Then, we obtain that
+un
+∥un∥ → φΩ in H1(Ω).
+Proof. Say that wn =
+un
+∥un∥ and, up to a subsequence, wn ⇀ w∞ ≥ 0, and wn → w∞ in L2(Ω)
+and L2(∂Ω) for some w∞ ∈ H1(Ω). From (2.1) it follows that
+λ
+�
+∂Ω
+uq+1
+n
+≤
+�
+Ω
+u2
+n.
+We use the condition ∥un∥ → 0 to infer that
+�
+∂Ω
+wq+1
+n
+≤ ∥un∥1−q
+λ
+�
+Ω
+w2
+n −→ 0;
+thus,
+�
+∂Ω wq+1
+∞
+= 0, i.e., w∞ ∈ H1
+0(Ω).
+7
+
+Since βΩ = 1 and E(wn) ≤ 0, the weak lower semicontinuity means that
+0 ≤ E(w∞) ≤ lim
+n→∞
+E(wn) ≤ lim
+n→∞ E(wn) ≤ 0,
+implying that E(wn) → E(w∞) = 0, i.e., ∥wn∥ → ∥w∞∥. Since wn ⇀ w∞, we deduce that
+wn → w∞ in H1(Ω) and w∞ = φΩ with ∥φΩ∥ = 1. Finally, because φΩ is unique, the desired
+conclusion follows.
+□
+Remark 2.4. If we construct a positive solution (λn, un) of (1.1) without using (1.2), then
+Propositions 2.2 and 2.3 remain valid for all p > 1.
+For further analysis of the asymptotic behavior of a positive solution (λn, un) of (1.1) with
+the condition that λn ≥ λ and ∥un∥ → 0, the orthogonal decomposition H1(Ω) = ⟨φΩ⟩⊕V using
+⟨φΩ⟩ is useful, where V denotes the orthogonal complement of ⟨φΩ⟩ that is given explicitly as
+V =
+�
+v ∈ H1(Ω) :
+�
+Ω
+�
+∇v∇φΩ + vφΩ
+�
+= 0
+�
+.
+Note that ⟨φΩ⟩ and V are both closed subspaces of H1(Ω) and ∥u∥ is equivalent to |s| + ∥v∥ for
+u = sφΩ + v ∈ H1(Ω) = ⟨φΩ⟩ ⊕ V .
+Using the orthogonal decomposition,
+un = snφΩ + vn ∈ ⟨φΩ⟩ ⊕ V
+(2.3)
+for a positive solution (λn, un) of (1.1) such that λn ≥ λ for some λ > 0 and ∥un∥ → 0 (under
+the assumption of Proposition 2.3). Since
+un
+∥un∥ → φΩ in H1(Ω), it follows that
+sn
+∥un∥ → 1,
+(2.4)
+∥vn∥
+∥un∥ → 0,
+(2.5)
+∥vn∥
+sn
+→ 0.
+(2.6)
+Because of (2.4), we may assume that sn > 0. Note that vn ≥ 0 on ∂Ω because φΩ = 0 on ∂Ω.
+We then deduce the following result, which plays a crucial role in the proof of Theorem 1.5.
+Lemma 2.5. Assume that βΩ = 1. Let {vn} be as introduced by (2.3). Then, there exists c > 0
+such that
+E(vn) + c
+�
+∂Ω
+vq+1
+n
+≤ 0
+for sufficiently large n,
+(2.7)
+provided that one of the following conditions is satisfied.
+(a) pq < 1,
+(b) pq = 1 and λn → ∞,
+(c) pq > 1 and λn is bounded above.
+Proof. Substituting un = snφΩ + vn into (2.1), we deduce that
+2sn
+��
+Ω
+∇φΩ∇vn − φΩvn
+�
++ E(vn) +
+�
+Ω
+(snφΩ + vn)p+1 + λn
+�
+∂Ω
+vq+1
+n
+= 0.
+(2.8)
+Using the divergence theorem,
+�
+Ω
+φΩvn =
+�
+Ω
+−∆φΩvn =
+�
+Ω
+∇φΩ∇vn +
+�
+∂Ω
+�
+−∂φΩ
+∂ν
+�
+vn;
+(2.9)
+8
+
+thus, (2.8) implies that
+−2sn
+�
+∂Ω
+�
+−∂φΩ
+∂ν
+�
+vn + E(vn) +
+�
+Ω
+(snφΩ + vn)p+1 + λn
+�
+∂Ω
+vq+1
+n
+= 0.
+It follows that
+E(vn) + λn
+2
+�
+∂Ω
+vq+1
+n
++ In ≤ 0
+(2.10)
+with
+In = λn
+2
+�
+∂Ω
+vq+1
+n
+− 2sn
+�
+∂Ω
+�
+−∂φΩ
+∂ν
+�
+vn.
+(2.11)
+Once we verify that
+In ≥ 0
+for sufficiently large n,
+(2.12)
+we obtain (2.7) and complete the proof. To verify (2.12), we use the test function ϕ = φΩ to
+deduce that
+�
+Ω
+�
+∇un∇φΩ − unφΩ + up
+nφΩ
+�
+= 0.
+(2.13)
+Substituting un = snφΩ + vn into (2.13) and combining (2.9) with (2.13) provide
+�
+∂Ω
+�
+−∂φΩ
+∂ν
+� vn
+sp
+n
+=
+�
+Ω
+�
+φΩ + vn
+sn
+�p
+φΩ.
+(2.14)
+We then consider either case (a) or (b). From (2.6), we deduce that
+�
+Ω
+�
+φΩ + vn
+sn
+�p
+φΩ −→
+�
+Ω
+φp+1
+Ω
+> 0.
+Taking into account (1.5), we may derive from (2.14) that
+csp
+n ≤
+�
+∂Ω
+vn
+for some c > 0. By H¨older’s inequality, it follows that
+csp
+n ≤
+�
+∂Ω
+vn ≤ |∂Ω|
+q
+q+1
+��
+∂Ω
+vq+1
+n
+�
+1
+q+1
+.
+(2.15)
+Combining (2.11) with (2.15) and using H¨older’s inequality, there exist c, ˜c > 0 such that
+In ≥ λn
+2
+�
+∂Ω
+vq+1
+n
+− c sn
+��
+∂Ω
+vq+1
+n
+�
+1
+q+1
+=
+�
+λn
+2
+��
+∂Ω
+vq+1
+n
+�
+q
+q+1
+− c sn
+� ��
+∂Ω
+vq+1
+n
+�
+1
+q+1
+≥ {˜c λn spq
+n − c sn}
+��
+∂Ω
+vq+1
+n
+�
+1
+q+1
+= spq
+n
+�
+˜cλn − c s1−pq
+n
+� ��
+∂Ω
+vq+1
+n
+�
+1
+q+1
+.
+Since sn → 0 from (2.4), assertion (2.12) follows.
+9
+
+We next consider case (c) and verify (2.12). By combining (2.11) with (2.14), it follows from
+(2.11) that
+In = sp+1
+n
+�λn
+2
+�
+∂Ω
+vq+1
+n
+sp+1
+n
+− 2
+�
+Ω
+�
+φΩ + vn
+sn
+�p
+φΩ
+�
+.
+(2.16)
+Furthermore, we use the test function ϕ = 1 in (1.3) to infer that
+−
+�
+Ω
+un +
+�
+Ω
+up
+n + λn
+�
+∂Ω
+uq
+n = 0.
+Substituting un = snφΩ + vn,
+−
+�
+Ω
+�
+φΩ + vn
+sn
+�
++ sp−1
+n
+�
+Ω
+�
+φΩ + vn
+sn
+�p
++
+�
+∂Ω
+λnvq
+n
+sn
+= 0,
+which implies that
+�
+∂Ω
+λnvq
+n
+sn
+−→
+�
+Ω
+φΩ > 0,
+where we have used condition (2.6). Thus, we may deduce that
+c sn
+λn
+≤
+�
+∂Ω
+vq
+n
+for some c > 0. Using H¨older’s inequality, we deduce that
+c
+� sn
+λn
+� q+1
+q
+≤
+�
+∂Ω
+vq+1
+n
+(2.17)
+for some c > 0. Combining (2.16) with (2.17),
+In ≥ sp+1
+n
+�
+c s
+1
+q −p
+n
+λ
+− 1
+q
+n
+− 2
+�
+Ω
+�
+φΩ + vn
+sn
+�p
+φΩ
+�
+for some c > 0. We observe that
+s
+1
+q −p
+n
+→ ∞,
+λ
+− 1
+q
+n
+≥ c
+by some constant c > 0,
+�
+Ω
+�
+φΩ + vn
+sn
+�p
+φΩ →
+�
+Ω
+φp+1
+Ω
+> 0.
+Thus, assertion (2.12) follows.
+□
+We conclude this section with the establishment of the uniqueness and stability results for a
+positive solution of (1.1) in the case where βΩ < 1.
+Proposition 2.6. Assume that βΩ < 1. Then, there exists Λ > 0 such that if λ > Λ, then the
+following two assertions hold:
+(i) Problem (1.1) has at most one positive solution.
+(ii) A positive solution u of (1.1) satisfying u > 0 in Ω is asymptotically stable.
+Proof. We recall that the unique positive solution uD of (1.6) is asymptotically stable, i.e.,
+γ1,D = inf
+��
+Ω
+�
+|∇ϕ|2 − ϕ2 + pup−1
+D
+ϕ2�
+: ϕ ∈ H1
+0(Ω),
+�
+Ω
+ϕ2 = 1
+�
+> 0.
+(2.18)
+10
+
+(i) Assume to the contrary that problem (1.1) has two distinct positive solutions (λn, un) and
+(λn, vn) with λn → ∞. Note that un, vn < 1 in Ω. We may assume that un, vn → uD in H1(Ω)
+and a.e. in Ω. The difference wn = un − vn (may change sign) allows that
+�
+Ω
+�
+∇wn∇ϕ − wnϕ
+�
++
+�
+Ω
+�
+(vn + wn)p − vp
+n
+�
+ϕ + λn
+�
+Γn
+(vn + wn)q − vq
+n
+wn
+wnϕ = 0
+(2.19)
+for ϕ ∈ H1(Ω), where Γn = {x ∈ ∂Ω : wn(x) ̸= 0}. Note that ∥wn∥ → 0, and wn → 0 a.e. in Ω.
+Substituting ϕ = wn into (2.19), the mean value theorem shows the existence of θn ∈ (0, 1)
+such that
+E(wn) +
+�
+Ω
+p(vn + θnwn)p−1w2
+n + λn
+�
+Γn
+(vn + wn)q − vq
+n
+wn
+w2
+n = 0;
+thus, ψn =
+wn
+∥wn∥ implies that
+E(ψn) +
+�
+Ω
+p(vn + θnwn)p−1ψ2
+n + λn
+�
+Γn
+(vn + wn)q − vq
+n
+wn
+ψ2
+n = 0.
+(2.20)
+From ∥ψn∥ = 1, we infer that up to a subsequence, ψn ⇀ ψ∞, ψn → ψ∞ in L2(Ω) and L2(∂Ω)
+for some ψ∞ ∈ H1(Ω). Then, we claim that ψ∞ ∈ H1
+0(Ω) and ψ∞ ̸= 0. From (2.20), we deduce
+that
+λn
+�
+Γn
+(vn + wn)q − vq
+n
+wn
+ψ2
+n ≤
+�
+Ω
+ψ2
+n ≤ 1.
+We observe that
+(vn + wn)q − vq
+n
+wn
+≥ q
+on Γn
+because un, vn < 1 in Ω. It follows that λnq
+�
+∂Ω ψ2
+n ≤ 1. Passing to the limit, we deduce that
+ψ∞ ∈ H1
+0(Ω). Indeed, ψ∞ ̸= 0 by Lemma 2.1.
+Then, we assert in (2.20) that
+�
+Ω
+p(vn + θnwn)p−1ψ2
+n −→
+�
+Ω
+pup−1
+D
+ψ2
+∞.
+Indeed, we use
+�
+Ω
+p(vn + θnwn)p−1ψ2
+n =
+�
+Ω
+p(vn + θnwn)p−1ψ2
+∞ +
+�
+Ω
+p(vn + θnwn)p−1(ψ2
+n − ψ2
+∞).
+Since vn → uD and wn → 0 a.e. in Ω and un, vn < 1 in Ω, the Lebesgue dominated convergence
+theorem shows that
+�
+Ω
+p(vn + θnwn)p−1ψ2
+∞ −→
+�
+Ω
+pup−1
+D
+ψ2
+∞.
+Using the fact
+�
+Ω |ψ2
+n − ψ2
+∞| → 0 yields
+�
+Ω
+p(vn + θnwn)p−1(ψ2
+n − ψ2
+∞) −→ 0,
+as desired.
+Then, the weak lower semicontinuity allows us to deduce from (2.20) that
+�
+Ω
+�
+|∇ψ∞|2 − ψ2
+∞ + pup−1
+D
+ψ2
+∞
+�
+≤ lim
+n→∞
+�
+E(ψn) +
+�
+Ω
+p(vn + θnwn)p−1ψ2
+n
+�
+≤ 0,
+which contradicts ψ∞ ∈ H1
+0(Ω) and ψ∞ ̸= 0 in view of (2.18).
+11
+
+(ii) On the basis of (1.4), we claim that γ1 > 0 for sufficiently large λ > 0.
+Assume by
+contradiction that a positive solution (λn, un) of (1.1) with the condition that λn → ∞ and
+un > 0 in Ω satisfies γn := γ1,n(λn, un) ≤ 0. This means that
+�
+Ω
+�
+|∇ϕn|2 − ϕ2
+n + pup−1
+n
+ϕ2
+n
+�
++ λnq
+�
+∂Ω
+uq−1
+n
+ϕ2
+n = γn ≤ 0,
+(2.21)
+where ϕn := ϕ1,n, normalized as
+�
+Ω
+ϕ2
+n +
+�
+∂Ω
+ϕ2
+n = 1.
+(2.22)
+Because
+�
+Ω |∇ϕn|2 ≤
+�
+Ω ϕ2
+n ≤ 1 from (2.21) and (2.22), ∥ϕn∥ is bounded, which implies that up
+to a subsequence, ϕn ⇀ ϕ∞, ϕn → ϕ∞ in L2(Ω) and L2(∂Ω), and ϕn → ϕ∞ a.e. in Ω for some
+ϕ∞ ∈ H1(Ω). Since uq−1
+n
+≥ 1 from Theorem 0(I), assertion (2.21) gives us
+λnq
+�
+∂Ω
+ϕ2
+n ≤
+�
+Ω
+ϕ2
+n ≤ 1.
+Passing to the limit, ϕ∞ = 0 on ∂Ω; thus, ϕ∞ ∈ H1
+0(Ω). From (2.22),
+�
+Ω ϕ2
+∞ = 1 is also deduced.
+By the weak lower semicontinuity, we derive from (2.21) that
+�
+Ω
+�
+|∇ϕ∞|2 − ϕ2
+∞ + pup−1
+D
+ϕ2
+∞
+�
+≤ lim
+n→∞
+�
+Ω
+�
+|∇ϕn|2 − ϕ2
+n + pup−1
+n
+ϕ2
+n
+�
+≤ 0
+(2.23)
+Indeed, on the basis of the facts that un → uD in H1(Ω) and un < 1 in Ω (see Theorem 0(I)
+and Proposition 2.2), the Lebesgue dominated convergence theorem shows that
+�
+Ω
+up−1
+n
+ϕ2
+n =
+�
+Ω
+up−1
+n
+ϕ2
+∞ +
+�
+Ω
+up−1
+n
+(ϕ2
+n − ϕ2
+∞) −→
+�
+Ω
+up−1
+D
+ϕ2
+∞,
+where we have used that un → uD a.e. in Ω. Assertion (2.23) contradicts ϕ∞ ∈ H1
+0(Ω) and
+�
+Ω ϕ2
+∞ = 1 in view of (2.18).
+□
+Remark 2.7. If we construct a positive solution u of (1.1) such that u > 0 in Ω without using
+(1.2), then assertion (ii) of Proposition 2.6 remains valid for all p > 1.
+3. Sub- and supersolutions
+Consider the case where βΩ < 1 or where βΩ = 1 and pq > 1. Then, we first construct small
+positive subsolutions of (1.1) and use them to establish an a priori lower bound for positive
+solutions (λ, u) of (1.1) satisfying u > 0 in Ω. For a fixed τ > 0 and with a parameter ε > 0, we
+set
+φε(x) = ε(φΩ(x) + ετ),
+x ∈ Ω,
+which implies that φε ∈ C2+θ(Ω) and φε > 0 in Ω.
+We then use φε to formulate the following a priori lower bound for positive solutions u > 0
+in Ω of (1.1).
+Lemma 3.1. Assume that βΩ < 1 or that βΩ = 1 and pq > 1. Let τ > 1−q
+q
+when βΩ < 1, and
+let 1−q
+q
+< τ < p − 1 when βΩ = 1 and pq > 1. Then, for Λ > 0 there exists ε = ε(τ, Λ) > 0 such
+that φε is a subsolution of (1.1), provided that λ ∈ [0, Λ] and ε ∈ (0, ε]. Furthermore, u ≥ φε
+in Ω for a positive solution u > 0 in Ω of (1.1) with λ ∈ [0, Λ]. Here, ε does not depend on
+λ ∈ [0, Λ].
+12
+
+Proof. We only consider the case where βΩ = 1 and pq > 1. The case where βΩ < 1 is proved
+similarly. First, we verify the former assertion. We take 0 < ε ≤ 1 and then use the condition
+p − τ − 1 > 0 to deduce that
+−∆φε − φε + φp
+ε ≤ ε1+τ
+�
+−1 + εp−τ−1
+�
+1 + max
+Ω
+φΩ
+�p�
+≤ 0
+in Ω
+if ε > 0 is small. For Λ > 0 we use (1.5) and the condition τ > 1−q
+q
+to deduce that
+∂φε
+∂ν + λφq
+ε ≤ −c1ε + λε(1+τ)q ≤ ε(−c1 + Λεq+τq−1) ≤ 0
+on ∂Ω
+if 0 < ε ≤ (c1/Λ)1/(q+τq−1). The desired assertion follows.
+Next, we argue by contradiction to verify the latter assertion. Assume by contradiction that
+u ̸≥ φε in Ω for some positive solution u > 0 in Ω with λ ∈ [0, Λ]. Because ε �→ φε is increasing
+and φε → 0 uniformly in Ω, we can take ε1 ∈ (0, ε) such that
+�
+u ≥ φε1
+in Ω,
+u(x1) = φε1(x1)
+for some x1 ∈ Ω.
+(3.1)
+Take c > 0 such that u, φε1 ≥ c in Ω; then, choose K > 0 sufficiently large so that fK(t) =
+Kt+t−tp is increasing for t ∈
+�
+0, maxΩ u
+�
+and M > 0 sufficiently large so that M −Λqcq−1 > 0.
+We use the subsolution φε1 (not a positive solution of (1.1)) to deduce that
+(−∆ + K)(u − φε1) ≥ fK(u) − fK(φε1) ≥ 0 (and ̸≡ 0)
+in Ω,
+(3.2)
+and for x ∈ ∂Ω satisfying u > φε1,
+� ∂
+∂ν + M
+�
+(u − φε1) ≥ −λuq + λφq
+ε1 + M(u − φε1)
+=
+�
+M − λuq − φq
+ε1
+u − φε1
+�
+(u − φε1)
+≥ (M − Λqcq−1)(u − φε1) > 0.
+(3.3)
+Thus, the strong maximum principle and boundary point lemma are applicable to infer that
+u − φε1 > 0 in Ω, which contradicts (3.1).
+□
+On the basis of Lemma 3.1, we construct minimal and maximal positive solutions of (1.1) as
+follows.
+Proposition 3.2. Assume that βΩ < 1 or that βΩ = 1 and pq > 1. Then, problem (1.1) has
+a minimal positive solution uλ ∈ C2+θ(Ω) and a maximal positive solution uλ ∈ C2+θ(Ω) for
+each λ > 0 such that 0 < uλ ≤ uλ in Ω, meaning that any positive solution u of (1.1) with the
+condition that u > 0 in Ω satisfies uλ ≤ u ≤ uλ in Ω. Moreover, both uλ and uλ are weakly
+stable, i.e., γ1(λ, uλ), γ1(λ, uλ) ≥ 0.
+Proof. It is clear that (λ, 1) is a supersolution of (1.1). Choose ε0 > 0 such that φε0 ≤ 1 in Ω;
+then, Lemma 3.1 states that (λ, φε0) is a subsolution of (1.1). By Theorem 0(I) and Lemma 3.1,
+a positive solution u > 0 in Ω of (1.1) implies that φε0 ≤ u ≤ 1 in Ω. Thus, this proposition is
+a direct consequence of applying [3, (2.1)Theorem] and [2, Proposition 7.8].
+□
+Remark 3.3. In view of the construction, Lemma 3.1 and Proposition 3.2 remain valid for
+any p > 1; therefore, Propositions 2.2, 2.3, 2.6(ii) hold for any p > 1 with the positive solution
+(λn, un) of (1.1) with λn → ∞ that is constructed by Proposition 3.2 (see Remarks 2.4 and 2.7).
+13
+
+In the case where βΩ < 1, Propositions 2.6 and 3.2 ensure the existence of a unique positive
+solution u(λ) of (1.1) for λ > Λ, which is asymptotically stable. Using the implicit function
+theorem provides us with the following result.
+Corollary 3.4. Assume that βΩ < 1. Then, {(λ, u(λ)) : λ > Λ} is a C∞ curve, i.e., λ �→
+u(λ) ∈ C2+θ(Ω) is C∞. Moreover, it is decreasing, i.e., u(λ1) > u(λ2) in Ω for λ2 > λ1 > Λ.
+Proof. We verify the first assertion.
+Let (λ0, u(λ0)) be the unique positive solution of (1.1)
+for λ0 > Λ. Since γ1(λ0, u(λ0)) > 0, the implicit function theorem applies at (λ0, u0); then,
+we deduce, thanks to the uniqueness, that {(λ, u(λ)) : λ1 < λ ≤ λ2} is a C∞ curve for λ1 <
+λ0 < λ2 such that u(λ) is asymptotically stable. The implicit function theorem applies again at
+(λ2, u(λ2)); then, the curve is continued until λ = λ3 > λ2. Repeating the same procedure, the
+curve is continued to λ = ∞ thanks to the a priori upper and lower bounds (Theorem 0(I) and
+Lemma 3.1), as desired.
+We next verify the second assertion. If λ1 < λ2, then u(λ1) is a supersolution of (1.1) for
+λ = λ2. By Lemma 3.1, it is possible to construct a subsolution φε of (1.1) for λ = λ2 such
+that 0 < φε ≤ u(λ1) in Ω. The sub- and supersolution method applies, and problem (1.1) has
+a positive solution u for λ = λ2 such that φε ≤ u ≤ u(λ1) in Ω, where u ̸≡ u(λ1). Thanks
+to Proposition 2.6(i), we obtain u = u(λ2). The desired assertion follows by using the strong
+maximum principle and boundary point lemma (as developed in (3.2) and (3.3)).
+□
+We conclude this section by employing the weak sub- and supersolution method [23] to show
+global strong positivity for a positive solution of (1.1) in the case where βΩ < 1.
+Proposition 3.5. Assume that βΩ < 1. Then, a positive solution (λ, u) of (1.1) satisfies that
+u > 0 in Ω.
+Proof. Assume by contradiction that problem (1.1) possesses a positive solution (λ0, u0) for
+λ0 > 0 such that u0 = 0 somewhere on ∂Ω. Let λ = λ1 > max(λ0, Λ), for which problem (1.1)
+has at most one positive solution by Proposition 2.6(i). Then, u0 is a weak supersolution of
+(1.1) for λ = λ1. Indeed,
+0 =
+�
+Ω
+�
+∇u0∇ϕ − u0ϕ + up
+0ϕ
+�
++ λ0
+�
+∂Ω
+uq
+0ϕ
+≤
+�
+Ω
+�
+∇u0∇ϕ − u0ϕ + up
+0ϕ
+�
++ λ1
+�
+∂Ω
+uq
+0ϕ,
+ϕ ∈ H1(Ω) and ϕ ≥ 0.
+We next construct a weak subsolution of (1.1) for λ = λ1 that is smaller than or equal to
+u0. Note that u0 ∈ Cθ(Ω) and u0 > 0 in Ω. From βΩ < 1, it follows by the continuity and
+monotonicity of βΩ with respect to Ω that we can choose a subdomain Ω1 ⋐ Ω with smooth
+boundary ∂Ω1 such that βΩ1 < 1. Then, we deduce that
+u0 ≥ c
+in Ω1
+(3.4)
+for some c > 0. We also deduce that if ε > 0 is sufficiently small, then
+−∆(εφΩ1) ≤ εφΩ1 − (εφΩ1)p
+in Ω1,
+where φΩ1 is a positive eigenfunction associated with βΩ1. Consequently, the divergence theorem
+is applied to
+�
+Ω −∆(εφΩ1)ϕ for ϕ ∈ H1(Ω) and ϕ ≥ 0 to deduce that
+�
+Ω1
+�
+∇(εφΩ1)∇ϕ − (εφΩ1)ϕ + (εφΩ1)pϕ
+�
+≤ 0.
+(3.5)
+14
+
+Define
+Φε =
+�
+εφΩ1
+in Ω1,
+0
+in Ω \ Ω1,
+and Φε ∈ H1(Ω) ∩ C(Ω). By virtue of (3.5), the linking technique [6, Lemma I.1] yields
+�
+Ω
+�
+∇Φε∇ϕ − Φεϕ + Φp
+ε ϕ
+�
++ λ1
+�
+∂Ω
+Φq
+ε ϕ =
+�
+Ω1
+�
+∇Φε∇ϕ − Φεϕ + Φp
+ε ϕ
+�
+≤ 0.
+Thanks to (3.4), we can take ε > 0 such that Φε ≤ u0 in Ω, as desired.
+The weak sub- and supersolution method [23, Subsection 2.2] is now applicable to deduce
+the existence of a positive solution (λ1, u1) of (1.1) such that Φε ≤ u1 ≤ u0 in Ω. Particularly,
+u1 = 0 somewhere on ∂Ω because so is u0. However, this contradicts Proposition 3.2 in view of
+the uniqueness.
+□
+Proof of Theorem 1.1. The uniqueness assertion follows from Proposition 2.6(i). Assertions (i-
+a) and (i-b) follow from Propositions 2.6(ii) and 2.2, respectively. Assertion (i-c) is verified by
+Corollary 3.4 and an analogous argument as in the proof of Corollary 3.4. Assertion (ii) follows
+from Proposition 3.5.
+□
+Proof of Theorem 1.3. The existence part follows from Proposition 3.2. Assertion (1.9) follows
+from Propositions 2.2 and 2.3.
+□
+4. Proof of Theorem 1.5
+This section is devoted to the proof of Theorem 1.5.
+(i) We prove assertion (i). Assume by contradiction that problem (1.1) has a positive solution
+(λn, un) with λn → ∞. Then, Proposition 2.2 shows that ∥un∥ → 0; thus, Proposition 2.3 shows
+that
+un
+∥un∥ → φΩ in H1(Ω). We apply Lemma 2.5(a) and (b); then, for un = snφΩ+vn ∈ ⟨φΩ⟩⊕V
+as in (2.3), we have (2.7) with (2.4)–(2.6).
+Observe from (2.5) that ∥vn∥ → 0. Say that ψn =
+vn
+∥vn∥; then, up to a subsequence, ψn ⇀
+ψ∞ ≥ 0, and ψn → ψ∞ in L2(Ω) and L2(∂Ω) for some ψ∞ ∈ H1(Ω). From (2.7), it follows that
+c
+�
+∂Ω
+ψq+1
+n
+≤ −E(ψn)∥vn∥1−q −→ 0,
+so that
+�
+∂Ω ψq+1
+∞
+= 0, i.e., ψ∞ ∈ H1
+0(Ω). Lastly, we use the condition E(ψn) ≤ 0 derived from
+(2.7) to follow the argument in the last paragraph of the proof of Proposition 2.3; then, we arrive
+at the contradiction ψ∞ = φΩ ∈ ⟨φΩ⟩ ∩ V = {0}.
+(ii) We verify assertion (ii). We remark that the convergences (λn, un) → (λ∞, 0) with λ∞ ≥ 0
+in R × H1(Ω) and R × C(Ω) are equivalent for a positive solution (λn, un) of (1.1) with λn > 0.
+This is verified by the bootstrap argument [32, Lemma 3.3]. In fact, the proof of assertion (ii)
+is similar to that for assertion (i). Assume by contradiction that problem (1.1) has a positive
+solution (λn, un) with the condition that λn → λ∞ > 0 and ∥un∥ → 0. Lemma 2.5(a) and (c)
+apply; then, we arrive at a contradiction.
+(iii) To verify assertion (iii), we prove the following three auxiliary lemmas. Say that Un =
+λ
+−
+1
+1−q
+n
+un.
+Lemma 4.1. There exists C > 0 such that ∥Un∥ ≤ C for a positive solution (λn, un) of (1.1)
+with λn > 0 satisfying that (λn, un) → (0, 0) in R × H1(Ω).
+15
+
+Proof. Assume by contradiction that ∥Un∥ → ∞. Say that wn =
+Un
+∥Un∥; then, up to a subse-
+quence, wn ⇀ w∞ ≥ 0, and wn → w∞ in L2(Ω) and L2(∂Ω) for some w∞ ∈ H1(Ω). Since
+E(wn) ≤ 0, Lemma 2.1 provides w∞ ̸= 0.
+Recall that (λ, U) = (λn, Un) satisfies
+�
+Ω
+�
+∇U∇ϕ − Uϕ + λ
+p−1
+1−q U pϕ
+�
++
+�
+∂Ω
+U qϕ = 0,
+ϕ ∈ H1(Ω).
+(4.1)
+Using the test function ϕ = 1 in (4.1), we deduce that
+�
+Ω
+Un = λ
+p−1
+1−q
+n
+�
+Ω
+U p
+n +
+�
+∂Ω
+U q
+n =
+�
+Ω
+up−1
+n
+Un +
+�
+∂Ω
+U q
+n,
+implying
+�
+Ω
+wn =
+�
+Ω
+up−1
+n
+wn +
+�
+∂Ω
+wq
+n∥Un∥q−1.
+(4.2)
+We may assume that un → 0 a.e. in Ω, and since un < 1 in Ω, we deduce that
+�
+Ω
+up−1
+n
+wn =
+�
+Ω
+up−1
+n
+w∞ +
+�
+Ω
+up−1
+n
+(wn − w∞) −→ 0,
+by applying the Lebesgue dominated convergence theorem and using the condition wn → w∞
+in L2(Ω).
+Then, passing to the limit in (4.2) yields
+�
+Ω w∞ = 0, i.e., w∞ = 0, which is a
+contradiction.
+□
+Lemma 4.2. Assume that βΩ = 1. Then, there is no positive solution U of (4.1) for λ = 0.
+Proof. If it exists, then from (4.1) with λ = 0 and ϕ = 1, it follows that U > 0 on Γ ⊂ ∂Ω with
+|Γ| > 0, implying
+�
+∂Ω
+∂φΩ
+∂ν U < 0. We use the test function ϕ = φΩ to deduce that
+�
+Ω
+�
+∇U∇φΩ − UφΩ
+�
+= 0.
+However, the divergence theorem leads us to the contradiction
+�
+Ω
+φΩU =
+�
+Ω
+−∆φΩU =
+�
+Ω
+∇φΩ∇U −
+�
+∂Ω
+∂φΩ
+∂ν U >
+�
+Ω
+∇φΩ∇U.
+□
+Lemma 4.3. Assume that βΩ = 1 and pq ≥ 1. Then, there exists C > 0 such that ∥Un∥ ≥ C
+for a positive solution (λn, Un) of (4.1) with λn → 0+.
+Proof. Assume by contradiction that (λn, Un) → (0, 0) in R × H1(Ω) for a positive solution
+(λn, Un) of (4.1). Say that wn =
+Un
+∥Un∥; then, up to a subsequence, wn ⇀ w∞ ≥ 0, wn → w∞ in
+Lp+1(Ω) and L2(∂Ω) for some w∞ ∈ H1(Ω). From (4.1) with (λ, U) = (λn, Un) and ϕ = Un, it
+follows that
+�
+Ω
+�
+|∇Un|2 − U 2
+n + λ
+p−1
+1−q
+n
+U p+1
+n
+�
++
+�
+∂Ω
+U q+1
+n
+= 0.
+(4.3)
+We then deduce that
+�
+∂Ω wq+1
+n
+≤
+�
+Ω w2
+n∥Un∥1−q → 0; thus,
+�
+∂Ω wq+1
+∞
+= 0, i.e., w∞ ∈ H1
+0(Ω). We
+also deduce from (4.3) that E(wn) =
+�
+Ω(|∇wn|2 − w2
+n) ≤ 0. Thus, we derive that wn → φΩ in
+H1(Ω) using a similar argument as in the last paragraph of the proof of Proposition 2.3.
+For a contradiction, we use the same strategy developed in the proof of assertion (i). To this
+end, we consider the orthogonal decomposition Un = snφΩ + vn ∈ ⟨φΩ⟩ ⊕ V as in (2.3); then,
+16
+
+we obtain (2.4) to (2.6) with un replaced by Un. As in the proof of Lemma 2.5, we deduce the
+following counterpart of (2.10) and (2.11) for (4.3):
+E(vn) + 1
+2
+�
+∂Ω
+vq+1
+n
++ Jn ≤ 0,
+with
+Jn = 1
+2
+�
+∂Ω
+vq+1
+n
+− 2sn
+�
+∂Ω
+�
+−∂φΩ
+∂ν
+�
+vn.
+(4.4)
+In the same spirit of Lemma 2.5 ((2.12)), we establish
+E(vn) + 1
+2
+�
+∂Ω
+vq+1
+n
+≤ 0
+for sufficiently large n,
+(4.5)
+by verifying that
+Jn ≥ 0
+for sufficiently large n.
+(4.6)
+Analogously to (2.14), we obtain
+�
+∂Ω
+�
+−∂φΩ
+∂ν
+�
+vn = λ
+p−1
+1−q
+n
+sp
+n
+�
+Ω
+�
+φΩ + vn
+sn
+�p
+φΩ.
+Using this assertion, we deduce from (4.4) that
+Jn = sp+1
+�1
+2
+�
+∂Ω
+vq+1
+n
+sp+1 − 2λ
+p−1
+1−q
+n
+�
+Ω
+�
+φΩ + vn
+sn
+�p
+φΩ
+�
+.
+(4.7)
+Furthermore, we use the test function ϕ = 1 in (4.1) to obtain
+−
+�
+Ω
+Un + λ
+p−1
+1−q
+n
+�
+Ω
+U p
+n +
+�
+∂Ω
+U q
+n = 0.
+Substituting Un = snφΩ + vn,
+−
+�
+Ω
+�
+φΩ + vn
+sn
+�
++ λ
+p−1
+1−q
+n
+sp−1
+n
+�
+Ω
+�
+φΩ + vn
+sn
+�p
++
+�
+∂Ω
+vq
+n
+sn
+= 0,
+from which we use (2.6) with Un to infer that
+�
+∂Ω
+vq
+n
+sn
+−→
+�
+Ω
+φΩ > 0.
+Then, we may deduce that
+csn ≤
+�
+∂Ω
+vq
+n
+for some c > 0. By H¨older’s inequality, we deduce that
+cs
+q+1
+q
+≤
+�
+∂Ω
+vq+1
+for some c > 0. We use this inequality to derive from (4.7) that
+Jn ≥ sp+1
+�
+cs
+1
+q −p
+n
+− 2λ
+p−1
+1−q
+n
+�
+Ω
+�
+φΩ + vn
+sn
+�p
+φΩ
+�
+for some c > 0; thus, (4.6) follows. Assertion (4.5) has been now established.
+We end the proof of this lemma. Observe from (2.5) with Un that ∥vn∥ → 0. Then, we
+develop the same argument as in the second paragraph of the proof of assertion (i) to arrive at
+the same contradiction.
+□
+17
+
+Employing the above lemmas, we then verify assertion (iii). Assume by contradiction that
+(λn, un) → (0, 0) in R × H1(Ω) for a positive solution (λn, un) of (1.1) with λn > 0. Then,
+(λn, Un) with Un = λ
+−
+1
+1−q
+n
+un admits a positive solution of (4.1). Since Un is bounded in H1(Ω)
+by Lemma 4.1, we deduce that up to a subsequence, Un ⇀ U∞ ≥ 0, and Un → U∞ in Lp+1(Ω)
+and L2(∂Ω) for some U∞ ∈ H1(Ω). Thanks to Lemma 4.3, we apply Lemma 2.1 to obtain
+U∞ ̸= 0.
+Furthermore, substituting (λ, U) = (λn, Un) into (4.1) and then taking the limit, we deduce
+that
+�
+Ω
+�
+∇U∞∇ϕ − U∞ϕ
+�
++
+�
+∂Ω
+U q
+∞ϕ = 0.
+This implies that U∞ is a nonnegative solution of (4.1) for λ = 0. Finally, Lemma 4.2 provides
+U∞ = 0, which is a contradiction.
+The proof of Theorem 1.5 is complete.
+□
+Remark 4.4. Assertions (ii) and (iii) of Theorem 1.5 are also derived from Lemma 3.1 when
+βΩ = 1 and pq > 1.
+5. Stability analysis of the trivial solution
+In the last section, we consider the stability of the trivial solution u = 0. It is worthwhile
+to mention that a linearized stability analysis does not work for u = 0 because u �→ uq is not
+differentiable at u = 0.
+The corresponding initial-boundary value problem is formulated as
+follows:
+
+
+
+
+
+∂u
+∂t (t, x) = ∆u + u − up
+in (0, ∞) × Ω,
+∂u
+∂ν = −λuq
+on (0, ∞) × ∂Ω,
+u(0, x) = u0(x) ≥ 0
+in Ω.
+(5.1)
+We use the method of monotone iterations to determine the Lyapunov stability of the trivial
+solution u = 0 (see [26, Definition 5.1.1]).
+When βΩ < 1 or when βΩ = 1 and pq > 1, we observe from Lemma 3.1 that u = 0 is unstable
+in the following sense: for u0 ∈ C2(Ω) sufficiently small such that u0 > 0 in Ω, the positive
+solution u(t, x) of (5.1) corresponding to the initial value u0 moves away from 0 as t → ∞.
+When βΩ > 1, for ε, δ, τ > 0, we set
+ψδ,ε,τ(x) = δ(φΩ(x) + ε)τ,
+x ∈ Ω.
+Let Ωρ := {x ∈ Ω : dist(x, ∂Ω) < ρ} for ρ > 0 be a tubular neighborhood of ∂Ω. Then, by (1.5),
+for ρ0 > 0 small, we can choose a constant c3 = c3(ρ0) > 0 such that |∇φΩ|2 ≥ c3 in Ωρ for
+0 < ρ ≤ ρ0. If 0 < ρ ≤ ρ0, then there exists c4 = c4(ρ) > 0 such that φΩ ≥ c4 in Eρ := Ω \ Ωρ.
+The following result would then provide useful information about the stability of the trivial
+solution u = 0.
+Theorem 5.1. Assume that βΩ > 1. Then, for
+1
+βΩ < τ < 1 and ε > 0 small, there exists δ1 > 0
+such that ψδ,ε,τ is a supersolution of (1.1) whenever 0 < δ ≤ δ1.
+Proof. We write ψδ,ε,τ simply as φδ,ε. By direct computations, we obtain
+∇ψδ,ε = δτ(φΩ + ε)τ−1∇φΩ,
+(5.2)
+∆ψδ,ε = δτ(τ − 1)(φΩ + ε)τ−2|∇φΩ|2 + δτ(φΩ + ε)τ−1∆φΩ.
+(5.3)
+18
+
+We see from (5.3) that for x ∈ Ωρ,
+∆ψδ,ε + ψδ,ε − ψp
+δ,ε ≤ δτ(τ − 1)(φΩ + ε)τ−2|∇φΩ|2 + δ(φΩ + ε)τ
+= δ(φΩ + ε)τ−2
+
+
+−τ(1 − τ)c3 +
+�
+ε + max
+Ωρ
+φΩ
+�2
+
+ .
+We then find 0 < ρ1 ≤ ρ0 and ε1 > 0 such that
+�
+ε + max
+Ωρ1
+φΩ
+�2
+≤ τ(1 − τ)c3
+for 0 < ε ≤ ε1,
+and then,
+−∆ψδ,ε + ψδ,ε − ψp
+δ,ε ≤ 0
+in Ωρ1.
+Let us fix c4 = c4(ρ1), and let 0 < ε ≤ ε1. We also see from (5.3) that for x ∈ Eρ1,
+∆ψδ,ε + ψδ,ε − ψp
+δ,ε ≤ δτ(φΩ + ε)τ−1(−βΩ)φΩ + δ(φΩ + ε)τ
+≤ δ(φΩ + ε)τ−1 {(1 − τβΩ)c4 + ε} .
+We then determine 0 < ε2 ≤ ε1 such that (1 − τβΩ)c4 + ε2 ≤ 0, and then,
+−∆ψδ,ε2 + ψδ,ε2 − ψp
+δ,ε2 ≤ 0
+in Eρ1.
+Finally, using (1.5), we see from (5.2) that
+∂ψδ,ε2
+∂ν
++ λψq
+δ,ε2 ≥ δq(−δ1−qτετ−1
+2
+c2 + λετq
+2 ) ≥ 0
+on ∂Ω,
+if 0 < δ ≤ δ1 for some δ1 > 0.
+In summary, ψδ,ε2, 0 < δ ≤ δ1, is as desired.
+□
+From Theorem 5.1, it might be claimed that u = 0 is asymptotically stable for the case where
+βΩ > 1, meaning that for u0 in the order interval [0, ψδ1,ε2,τ], the positive solution u(t, x) of (5.1)
+associated with u0 tends to 0 as t → ∞. If this occurs, then Theorem 0(II) means that problem
+(5.1) is bistable with two nonnegative stable equilibria for 0 < λ ≤ λ∗ (one is uλ, and the other
+is u = 0), which presents ecologically a conditional persistence strategy for the harvesting effort
+λ. However, the difficulty arises from the fact that the monotone iteration scheme does not
+work for (5.1) in the order interval [0, ψδ1,ε2,τ] because u �→ (−uq) does not satisfy the one-sided
+Lipschitz condition [26, (4.1.19)] for u close to 0. Rigorous verification of the claim is an open
+question.
+References
+[1] S. Alama, Semilinear elliptic equations with sublinear indefinite nonlinearities, Adv. Differential
+Equations 4 (1999), 813–842.
+[2] H. Amann, Fixed point equations and nonlinear eigenvalue problems in ordered Banach spaces, SIAM
+Rev. 18 (1976), 620–709.
+[3] H. Amann, Nonlinear elliptic equations with nonlinear boundary conditions, New developments in
+differential equations (Proc. 2nd Scheveningen Conf., Scheveningen, 1975), pp. 43–63, North-Holland
+Math. Studies, Vol. 21, North-Holland, Amsterdam, 1976.
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+Edinburgh Sect. A 131 (2001), 733–765.
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+autonomous problems with concave–convex nonlinearities, Nonlinear Anal. 93 (2013), 226–235.
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+Soc. Edinburgh Sect. A 136 (2006), 779–784.
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+Math. J. 55 (2013), 399–409.
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+Glasg. Math. J. 58 (2016), 461–469.
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+20
+
+[32] K. Umezu, Logistic elliptic equation with a nonlinear boundary condition arising from coastal fishery
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+Department of Mathematics, Faculty of Education, Ibaraki University, Mito 310-8512, Japan
+Email address: kenichiro.umezu.math@vc.ibaraki.ac.jp
+21
+
diff --git a/7tFLT4oBgHgl3EQfsi8k/content/tmp_files/load_file.txt b/7tFLT4oBgHgl3EQfsi8k/content/tmp_files/load_file.txt
new file mode 100644
index 0000000000000000000000000000000000000000..5055849d1899740deeb1e8013f414e93d99cbb82
--- /dev/null
+++ b/7tFLT4oBgHgl3EQfsi8k/content/tmp_files/load_file.txt
@@ -0,0 +1,1071 @@
+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf,len=1070
+page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content='12147v1  [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content='AP]  28 Jan 2023  LOGISTIC ELLIPTIC EQUATION WITH A NONLINEAR BOUNDARY  CONDITION ARISING FROM COASTAL FISHERY HARVESTING II  KENICHIRO UMEZU  Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content=' We study the positive solutions of the logistic elliptic equation with a nonlinear  Neumann boundary condition that models coastal fishery harvesting ([18]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content=' An essential role is  played by the smallest eigenvalue of the Dirichlet eigenvalue problem, with respect to which a  noncritical case is studied in [32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content=' In this paper, we extend our analysis to the critical case and  further study the noncritical case for a more precise description of the positive solution set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content=' Our  approach relies on the energy method, sub- and supersolutions, and implicit function analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content=' Introduction  This paper is devoted to the study of the positive solutions for the following logistic elliptic  equation with a nonlinear boundary condition arising from coastal fishery harvesting ([18]):  \uf8f1  \uf8f4  \uf8f2  \uf8f4  \uf8f3  −∆u = u − up  in Ω,  u ≥ 0  in Ω,  ∂u  ∂ν = −λuq  on ∂Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content='  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content='1)  Here, Ω ⊂ RN, N ≥ 1, is a bounded domain with smooth boundary ∂Ω, ∆ = �N  i=1  ∂2  ∂x2  i is the  usual Laplacian in RN, 0 < q < 1 < p, λ ≥ 0 is a parameter, and ν is the unit outer normal to  ∂Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content=' Unless stated otherwise, throughout this paper we assume the subcritical condition  p < N + 2  N − 2  for N > 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content='  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content='2)  In the case of p = 2, the unknown function u ≥ 0 ecologically represents the biomass of fish that  inhabit a lake Ω, obeying the logistic law ([8]), and the nonlinear boundary condition means  fishery harvesting with the harvesting effort λ on the lake coast ∂Ω, obeying the Cobb–Douglas  production function ([18, Subsection 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content='1]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content='  A nonnegative function u ∈ H1(Ω) is called a nonnegative (weak) solution of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content='1) if u satisfies  �  Ω  �  ∇u∇ϕ − uϕ + upϕ  �  + λ  �  ∂Ω  uqϕ = 0,  ϕ ∈ H1(Ω)  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content='3)  (we may regard (λ, u) as a nonnegative solution of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content='1)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content=' It is seen that problem (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content='1) has  a solution (λ, 0) for every λ > 0, called a trivial solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content='  The sets {(λ, 0) : λ ≥ 0} and  {(λ, 0) : λ > 0} are said to be the trivial lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content=' We know ([30]) that a nonnegative solution u of  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content='1) belongs to the space W 1,r(Ω) for r > N (consequently, Cθ(Ω) for θ ∈ (0, 1)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content=' Moreover, a  nontrivial nonnegative solution u of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content='1) satisfies that u ∈ C2+θ(Ω) for θ ∈ (0, 1), and u > 0  in Ω ([17], [27]), which is called a positive solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content=' Indeed, if u > 0 in Ω, then u ∈ C2+θ(Ω)  by the bootstrap argument using elliptic regularity, and u satisfies (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content='1) pointwisely in Ω in the  classical sense.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content=' However, we do not know if u > 0 on the entirety of ∂Ω for a positive solution u  2020 Mathematics Subject Classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content=' 35J65, 35B32, 35J25, 92D40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content='  Key words and phrases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content=' logistic elliptic equation, concave–convex nonlinearity, positive solution, uniqueness,  stability, sub- and supersolutions, energy method, boundary harvesting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content='  1  of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content=' As a matter of fact, Hopf’s boundary point lemma ([27]) does not work because of the  lack of the one-sided Lipschitz condition [26, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content='19)] for mapping 0 ≤ u �→ (−uq) for u close to  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content='  For a positive solution (λ, u) of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content='1) satisfying u > 0 in Ω, we call γ1 = γ1(λ, u) ∈ R the  smallest eigenvalue of the linearized eigenvalue problem at (λ, u)  �  −∆ϕ = ϕ − pup−1ϕ + γϕ  in Ω,  ∂ϕ  ∂ν = −λquq−1ϕ + γϕ  on ∂Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content='  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content='4)  It is well known that γ1 is simple with a positive eigenfunction ϕ1 ∈ C2+θ(Ω) satisfying ϕ1 > 0  in Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content=' Indeed, γ1 is characterized by the variational formula  γ1 = inf  ��  Ω  �  |∇ϕ|2 − ϕ2 + pup−1ϕ2  �  + λ  �  ∂Ω  quq−1ϕ2 : ϕ ∈ H1(Ω),  �  Ω  ϕ2 +  �  ∂Ω  ϕ2 = 1  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content='  A positive solution u > 0 in Ω of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content='1) is said to be asymptotically stable, weakly stable, and  unstable if γ1 > 0, γ1 ≥ 0, and γ1 < 0, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content='  Problem (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content='1) possesses a sublinear nonlinearity at infinity and also a concave–convex nature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content='  Thus, the global uniqueness of a positive solution of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content='1) for every λ > 0 would not be so easy  to deduce.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content=' For nonlinear elliptic problems with a concave–convex nature, we refer to [4, 31, 1,  5, 11, 12, 19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content=' The sublinear nonlinearity (−uq) that appears in (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content='1) induces the absorption  effect on ∂Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content=' Sublinear boundary conditions of the uq type were explored in [16, 14, 15, 28, 29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content='  The case of an incoming flux on ∂Ω was studied in [16, 15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content=' The mixed case of absorption and  an incoming flux on ∂Ω was studied in [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content=' The absorption case was also studied in [28, 29],  where a similar type of logistic elliptic equation with an indefinite weight has been analyzed for  the existence and multiplicity of nontrivial nonnegative solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content='  An important role is played by the smallest eigenvalue βΩ > 0 of the Dirichlet eigenvalue  problem  �  −∆φ = βφ  in Ω,  φ = 0  on ∂Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content='  It is well known that βΩ is simple with a positive eigenfunction φΩ ∈ H1  0(Ω) (implying φΩ ∈  C2+θ(Ω) by elliptic regularity).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content=' Indeed, φΩ > 0 in Ω, and  c1 ≤ −∂φΩ  ∂ν ≤ c2  on ∂Ω  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content='5)  for some 0 < c1 < c2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content=' Moreover, βΩ is characterized by the variational formula  βΩ = inf  ��  Ω  |∇φ|2 : φ ∈ H1  0(Ω),  �  Ω  φ2 = 1  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content='  If βΩ < 1, then uD ∈ H1  0(Ω) ∩ C2+θ(Ω) denotes the unique positive solution of the Dirichlet  logistic problem ([7])  �  −∆u = u − up  in Ω,  u = 0  on ∂Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content='  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content='6)  The existence, nonexistence, and multiplicity of positive solutions for (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content='1) in the case where  βΩ ̸= 1 were studied in the author’s previous work [32, Theorems 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content='1, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content='2, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content='4, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content='5], which  Theorem 0 summarizes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content='  Theorem 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content='  (I) A positive solution u of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content='1) satisfies that u < 1 in Ω and u > 0 on Γ ⊂ ∂Ω with the  condition |Γ| > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content='  2  (II) There exists λ∗ > 0 such that problem (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content='1) has a positive solution curve  C0 = {(λ, uλ) : 0 ≤ λ ≤ λ∗},  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content='7)  emanating from (λ, u) = (0, 1), that satisfies the following three conditions:  λ �→ uλ ∈ C2+θ(Ω) is C∞,  uλ > 0 in Ω,  uλ is asymptotically stable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content='  Moreover, the positive solutions of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content='1) near (λ, u) = (0, 1) in R × C2+θ(Ω) form C0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content='  Let λ be the positive value defined as  λ = sup{λ > 0 : (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content='1) has a positive solution for λ}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content='  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content='8)  Then, the following assertions hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content='  (III) Assume that βΩ < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content=' Then, we have the following (as in Figure 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content='  (i) λ = ∞, and more precisely, problem (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content='1) possesses a positive solution u for every  λ > 0 such that u > 0 in Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content='  (ii) (λ, uλ) ∈ C0 is a unique positive solution of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content='1) for 0 < λ ≤ λ∗ (by making λ∗ in  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content='7) smaller if necessary).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content='  (iii) un → uD in H1(Ω) for a positive solution (λn, un) of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content='1) with λn → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content='  (iv) The positive solution set {(λ, u)} does not meet the trivial line {(λ, 0) : λ ≥ 0} in  the topology of H1(Ω) (nor C(Ω)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content='  (IV) Assume that βΩ > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content=' Then, we have the following (as in Figure 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content='  (i) λ < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content='  (ii) There exists a bounded subcontinuum (closed connected subset) �C0 = {(λ, u)} of  nonnegative solutions of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content='1) in [0, ∞) × C(Ω) joining (λ, u) = (0, 1) and (0, 0)  such that �C0 \\ {(0, 0)} includes C0 and consists of positive solutions of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content=' Par-  ticularly, problem (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content='1) has at least two positive solutions for λ > 0 small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content='  (iii) The positive solution set {(λ, u)} does not meet the trivial line {(λ, 0) : λ > 0} in  the topology of H1(Ω) (nor C(Ω)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content='  (iv) γ1(λn, un) < 0 for a positive solution (λn, un) of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content='1) such that (λn, un) → (0, 0)  in R × H1(Ω), provided that un > 0 in Ω, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content=', un is unstable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content='  Remark 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content='  (i) Assertions (I) and (II) hold for every case of βΩ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content='  (ii) Assertions (II) and (III-i) hold for any p > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content='  (iii) Problem (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content='1) with λ = 0 has exactly two nonnegative solutions (λ, u) = (0, 0), (0, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content='  Thus, Theorem 0(I) is used to show easily that in every case of βΩ > 0, the positive  solution set {(λ, u)} of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content='1) meets at most (0, 0) and (0, 1) on {(0, u) : u ≥ 0}, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content=', if  (λn, un) is a positive solution of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content='1) such that (λn, un) → (0, u) in H1(Ω) (equivalently  C(Ω) by elliptic regularity), then either u = 0 or 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content='  In this paper, we extend our consideration to the case where βΩ = 1 and further study the  positive solution set in the case where βΩ < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content=' Our first main result concerns the case where  βΩ < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content=' On the basis of Theorem 0(III), we present the uniqueness and stability of a positive  solution of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content='1) for λ > 0 large and also the strong positivity of the positive solutions for every  λ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content='  Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content=' Assume that βΩ < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content=' Then, the following assertions hold (see Figure 1):  3  (i) There exists λ∗ ≥ λ∗ such that the positive solution of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content='1) ensured by Theorem 0(III-i)  is unique for every λ > λ∗ (say uλ);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content=' more precisely, the positive solutions of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content='1) for  λ > λ∗ form a C∞ curve C∞ = {(λ, uλ) : λ∗ < λ} (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content=', λ �→ uλ ∈ C2+θ(Ω) is C∞),  which satisfies the following conditions:  (a) uλ is asymptotically stable,  (b) uλ −→ uD in H1(Ω) as λ → ∞,  (c) uλ is decreasing, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content=', uλ1 > uλ2 in Ω if λ1 < λ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content=' Furthermore, if 0 < λ1 < λ2 with  the condition that λ1 ≤ λ∗ < λ2, then u > uλ2 in Ω for a positive solution u of  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content='1) for λ = λ1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content='  (ii) u > 0 in Ω for a positive solution u of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content='1) for every λ > 0 (strong positivity).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content='  Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content=' Possible positive solution sets in the case where βΩ < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content='  Remark 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content='  (i) Assertions (i-a) and (i-b) hold for any p > 1 (see Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content='  (ii) For (λ, uλ) ∈ C0 with 0 ≤ λ ≤ λ∗ in (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content='7), we present similar results as those in assertions  (i-c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content=' Indeed, λ �→ uλ is decreasing for 0 < λ ≤ λ∗;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content=' if 0 < λ1 < λ2 with the condition  that λ1 ≤ λ∗ < λ2, then uλ1 > u in Ω for a positive solution u of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content='1) with λ = λ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content='  (iii) It is an open question to get the global uniqueness for a positive solution of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content='1) for  all λ > 0, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content=', λ∗ = λ∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content=' In this case, [0, ∞) ∋ λ �→ uλ is C∞ and decreasing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content='  (iv) For uniqueness and stability analysis of positive solutions for large parameters in non-  linear elliptic problems, we refer to [9, 34, 33, 24, 10, 25, 13, 20, 21, 22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content='  Our second main result is the counterpart of Theorem 0(III-i) and (III-iii) for the case where  βΩ = 1 and pq > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content='  Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content=' Assume that βΩ = 1 and pq > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content=' Then, problem (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content='1) possesses a positive  solution uλ for every λ > 0 such that uλ > 0 in Ω, which satisfies that  uλ −→ 0  and  uλ  ∥uλ∥ −→ φΩ  in H1(Ω)  as λ → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content='  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content='9)  Remark 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content='  (i) The existence assertion holds for any p > 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content=' thus, so does assertion (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content='9) (see Remark  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content='  (ii) Similarly as in Theorem 0(III-iii), assertion (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content='9) is valid if we assume a positive solution  (λ, uλ) (which may take zero value somewhere on ∂Ω) of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content='1) with λ → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content='  Our third main result is the counterpart of Theorem 0(III-iv), (IV-i), and (IV-iii) for the case  where βΩ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content='  4  u  u  A  1  1  Co  Co  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content='.  入  入  Y  \\*  0  入*  \\*  0Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content=' Assume that βΩ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content=' Then, the following three assertions hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content='  (i) If pq ≤ 1, then λ < ∞ where λ > 0 is defined by (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content='8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content='  (ii) If pq ̸= 1, then the positive solution set {(λ, u)} of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content='1) does not meet the trivial line  {(λ, 0) : λ > 0} in the topology of H1(Ω) (nor C(Ω)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content='  (iii) If pq ≥ 1, then it does not meet {(0, 0)} in the topology of H1(Ω) (nor C(Ω)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content='  Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content='5 provides a guess (Remark 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content='6) for the global extension of the C∞ positive solution  curve C0 given by (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content='7) in the case where βΩ = 1 and pq ≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content='  Remark 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content=' Assume that βΩ = 1 and pq ≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content='  Let �C0 = {(λ, u)} ⊂ [0, ∞) × C(Ω) be  the component (maximal, closed, and connected subset) of nonnegative solutions of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content='1) that  includes C0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content=' From Theorems 0(I) and Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content='5(i), �C0 \\ {(0, 1)} ⊂ {(λ, u) ∈ [0, ∞) × C(Ω) :  λ ≤ λ, u < 1 in Ω}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content=' If we suppose that Γ0 :=  �  �C0 \\ {(0, 1)}  �  ∩{(λ, 0), (0, u) : λ ≥ 0, u ≥ 0} ̸= ∅,  then Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content='5(ii),(iii) show that Γ0 = {(0, 0)} and Γ0 ⊂ {(λ, 0) : λ ≥ Λ0} for some Λ0 > 0  when pq < 1 and pq = 1, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content=' The existence of �C0 is still an open question.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content=' Suggested  positive solution sets are illustrated in Figures 2 and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content='  Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content=' Suggested positive solution set in the case where βΩ = 1 and pq < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content='  Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content=' Suggested positive solution set in the case where βΩ = 1 and pq = 1,  and λc ∈ Γ0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content='  We conclude the Introduction by mentioning the stability of the trivial solution u = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content=' A  linearized stability analysis does not work for u = 0 because u �→ uq is not differentiable at  u = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content='  Instead, by the construction of suitable sub- and supersolutions of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content='1), we try to  employ the Lyapunov stability criterion [26, Chapter 5] on the basis of the monotone iteration  method, which is developed in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content='  Notation: ∥ · ∥ denotes the usual norm of H1(Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content='  un ⇀ u∞ means that un weakly  converges to u∞ in H1(Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content=' H1  0(Ω) = {u ∈ H1(Ω) : u = 0 on ∂Ω}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content='  �  Ω fdx for f ∈ L1(Ω)  5  u  1  入  入cu  1and  �  ∂Ω gdσ for g ∈ L1(∂Ω) are simply written as  �  Ω f and  �  ∂Ω g, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content=' | · |  represents both the Lebesgue measure in Ω and the surface measure on ∂Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content='  The remainder of this paper is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content=' Sections 2 and 3 are devoted to the  preparation for the proofs of Theorems 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content='1, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content='3 and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content=' In Section 2, we develop the energy  method for the energy functional associated with (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content='  In Section 3, we use the sub- and  supersolution method to prove existence and positivity results for positive solutions of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content='  We give proofs for Theorems 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content='1 and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content='3 in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content=' In Section 4, we prove Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content='  Section 5 is devoted to a stability analysis of the trivial solution u = 0, which is based on  Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content='1 and Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content=' Energy method  Let  E(u) =  �  Ω  (|∇u|2 − u2),  u ∈ H1(Ω);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content='  then, the next lemma is used several times in the following arguments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content='  Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content=' Let {un} ⊂ H1(Ω) satisfy E(un) ≤ 0, un ⇀ u∞, and un → u∞ in L2(Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content=' Then,  u∞ ̸= 0 if ∥un∥ ≥ C for some C > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content=' By the weak lower semicontinuity, E(u∞) ≤ limn E(un) ≤ limn E(un) ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content=' If u∞ = 0,  then ∥un∥ → 0, as desired.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content='  □  We start by proving the following two propositions, which provide the asymptotic profile of  a positive solution of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content='1) as λ → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content=' It is understood that uD = 0 if βΩ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content='  Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content=' Assume that βΩ ≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content=' Let (λn, un) be a positive solution of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content='1) with λn → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content='  Then, un → uD in H1(Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content=' We first assume that βΩ < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content=' Because un < 1 in Ω, we substitute u = ϕ = un into (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content='3)  to deduce that  �  Ω  |∇un|2 =  �  Ω  �  u2  n − up+1  n  �  − λn  �  ∂Ω  uq+1  n  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content='1)  ≤  �  Ω  u2  n ≤ |Ω|;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content='  thus, ∥un∥ is bounded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content=' Immediately, up to a subsequence, un ⇀ u∞ ≥ 0, un → u∞ in L2(Ω)  and L2(∂Ω), and un → u∞ a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content=' in Ω for some u∞ ∈ H1(Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content=' We then infer that  �  ∂Ω  uq+1  n  = 1  λn  �  −  �  Ω  |∇un|2 +  �  Ω  �  u2  n − up+1  n  ��  ≤ 1  λn  �  Ω  u2  n −→ 0,  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content='2)  which implies that  �  ∂Ω uq+1  ∞  = 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content=' thus, u∞ ∈ H1  0(Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content=' From (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content='3) with (λ, u) = (λn, un), it follows  that  �  Ω  �  ∇un∇ϕ − unϕ + up  nϕ  �  = 0,  ϕ ∈ H1  0(Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content='  Taking the limit, u∞ is a nonnegative solution of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content='6), where we have used the Lebesgue  dominated convergence theorem to deduce that  �  Ω up  nϕ →  �  Ω up  ∞ϕ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content='  Then, we verify that u∞ ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content=' Since E(un) ≤ 0, the weak lower semicontinuity means that  E(u∞) ≤ lim  n→∞  E(un) ≤ lim  n→∞ E(un) ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content='  6  If u∞ = 0, then it follows that ∥un∥ → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content=' Here, we may assume that un → 0 a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content=' in Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content=' Say that  wn =  un  ∥un∥;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content=' then, up to a subsequence, wn ⇀ w∞ ≥ 0, wn → w∞ in L2(Ω) and L2(∂Ω), and  wn → w∞ a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content=' in Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content=' Since ∥wn∥ = 1, we deduce that w∞ ̸= 0 using Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content=' However, we  observe from (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content='2) that  �  ∂Ω  wq+1  n  ≤ 1  λn  �  Ω  w2  n∥un∥1−q −→ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content='  This implies that w∞ ∈ H1  0(Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content=' From (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content='3) with (λ, u) = (λn, un), we see that  �  Ω  �  ∇wn∇ϕ − wnϕ + wnϕ up−1  n  �  = 0,  ϕ ∈ H1  0(Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content='  Taking the limit,  �  Ω(∇w∞ϕ − w∞ϕ) = 0, where we have used the Lebesgue dominated conver-  gence theorem to obtain that  �  Ω  ��wnϕ up−1  n  �� ≤  ��  Ω  w2  n  � 1  2 ��  Ω  ϕ2u2(p−1)  n  � 1  2  ≤ C  ��  Ω  ϕ2u2(p−1)  n  � 1  2  −→ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content='  This implies that w∞ is a nontrivial nonnegative solution of the problem  �  −∆w = w  in Ω,  w = 0  on ∂Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content='  Thus, we deduce that βΩ = 1, which contradicts the assumption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content=' The assertion that u∞ ≥ 0  and u∞ ̸= 0 means that u∞ is the unique positive solution uD of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content='6) by the strong maximum  principle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content='  It remains to show that un → u∞ in H1(Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content='  Observing that E(u∞) +  �  Ω up+1  ∞  = 0 and  E(un) ≤ −  �  Ω up+1  n  , we deduce that  E(u∞) ≤ lim  n→∞  E(un) ≤ lim  n→∞ E(un) ≤ − lim  n→∞  �  Ω  up+1  n  = −  �  Ω  up+1  ∞  = E(u∞),  where we have used the Lebesgue dominated convergence theorem again.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content=' Thus, E(un) → E(u∞),  i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content=', ∥un∥ → ∥u∞∥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content=' Since un ⇀ u∞, the desired assertion follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content='  Next, we assume that βΩ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content=' Then, u∞ is a nonnegative solution of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content='6), and indeed u∞ = 0  because βΩ = 1 ([7]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content=' Thus, ∥un∥ → 0, as desired.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content='  □  In the case where βΩ = 1, we observe that E(u) ≥ 0 for u ∈ H1  0(Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content=' Indeed, we note that  �  u ∈ H1(Ω) : E(u) = 0  �  = ⟨φΩ⟩ := {sφΩ : s ∈ R}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content='  We then investigate the asymptotic profile of a positive solution (λn, un) of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content='1) with ∥un∥ → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content='  Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content=' Assume that βΩ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content=' Let (λn, un) be a positive solution of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content='1) such that  λn ≥ λ for some λ > 0 and ∥un∥ → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content=' Then, we obtain that  un  ∥un∥ → φΩ in H1(Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content=' Say that wn =  un  ∥un∥ and, up to a subsequence, wn ⇀ w∞ ≥ 0, and wn → w∞ in L2(Ω)  and L2(∂Ω) for some w∞ ∈ H1(Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content=' From (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content='1) it follows that  λ  �  ∂Ω  uq+1  n  ≤  �  Ω  u2  n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content='  We use the condition ∥un∥ → 0 to infer that  �  ∂Ω  wq+1  n  ≤ ∥un∥1−q  λ  �  Ω  w2  n −→ 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content='  thus,  �  ∂Ω wq+1  ∞  = 0, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content=', w∞ ∈ H1  0(Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content='  7  Since βΩ = 1 and E(wn) ≤ 0, the weak lower semicontinuity means that  0 ≤ E(w∞) ≤ lim  n→∞  E(wn) ≤ lim  n→∞ E(wn) ≤ 0,  implying that E(wn) → E(w∞) = 0, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content=', ∥wn∥ → ∥w∞∥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content=' Since wn ⇀ w∞, we deduce that  wn → w∞ in H1(Ω) and w∞ = φΩ with ∥φΩ∥ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content=' Finally, because φΩ is unique, the desired  conclusion follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content='  □  Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content=' If we construct a positive solution (λn, un) of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content='1) without using (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content='2), then  Propositions 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content='2 and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content='3 remain valid for all p > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content='  For further analysis of the asymptotic behavior of a positive solution (λn, un) of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content='1) with  the condition that λn ≥ λ and ∥un∥ → 0, the orthogonal decomposition H1(Ω) = ⟨φΩ⟩⊕V using  ⟨φΩ⟩ is useful, where V denotes the orthogonal complement of ⟨φΩ⟩ that is given explicitly as  V =  �  v ∈ H1(Ω) :  �  Ω  �  ∇v∇φΩ + vφΩ  �  = 0  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content='  Note that ⟨φΩ⟩ and V are both closed subspaces of H1(Ω) and ∥u∥ is equivalent to |s| + ∥v∥ for  u = sφΩ + v ∈ H1(Ω) = ⟨φΩ⟩ ⊕ V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content='  Using the orthogonal decomposition,  un = snφΩ + vn ∈ ⟨φΩ⟩ ⊕ V  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content='3)  for a positive solution (λn, un) of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content='1) such that λn ≥ λ for some λ > 0 and ∥un∥ → 0 (under  the assumption of Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content=' Since  un  ∥un∥ → φΩ in H1(Ω), it follows that  sn  ∥un∥ → 1,  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content='4)  ∥vn∥  ∥un∥ → 0,  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content='5)  ∥vn∥  sn  → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content='  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content='6)  Because of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content='4), we may assume that sn > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content=' Note that vn ≥ 0 on ∂Ω because φΩ = 0 on ∂Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content='  We then deduce the following result, which plays a crucial role in the proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content='  Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content=' Assume that βΩ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content=' Let {vn} be as introduced by (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content=' Then, there exists c > 0  such that  E(vn) + c  �  ∂Ω  vq+1  n  ≤ 0  for sufficiently large n,  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content='7)  provided that one of the following conditions is satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content='  (a) pq < 1,  (b) pq = 1 and λn → ∞,  (c) pq > 1 and λn is bounded above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content=' Substituting un = snφΩ + vn into (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content='1), we deduce that  2sn  ��  Ω  ∇φΩ∇vn − φΩvn  �  + E(vn) +  �  Ω  (snφΩ + vn)p+1 + λn  �  ∂Ω  vq+1  n  = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content='  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content='8)  Using the divergence theorem,  �  Ω  φΩvn =  �  Ω  −∆φΩvn =  �  Ω  ∇φΩ∇vn +  �  ∂Ω  �  −∂φΩ  ∂ν  �  vn;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content='  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content='9)  8  thus, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content='8) implies that  −2sn  �  ∂Ω  �  −∂φΩ  ∂ν  �  vn + E(vn) +  �  Ω  (snφΩ + vn)p+1 + λn  �  ∂Ω  vq+1  n  = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content='  It follows that  E(vn) + λn  2  �  ∂Ω  vq+1  n  + In ≤ 0  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content='10)  with  In = λn  2  �  ∂Ω  vq+1  n  − 2sn  �  ∂Ω  �  −∂φΩ  ∂ν  �  vn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content='  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content='11)  Once we verify that  In ≥ 0  for sufficiently large n,  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content='12)  we obtain (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content='7) and complete the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content=' To verify (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content='12), we use the test function ϕ = φΩ to  deduce that  �  Ω  �  ∇un∇φΩ − unφΩ + up  nφΩ  �  = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content='  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content='13)  Substituting un = snφΩ + vn into (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content='13) and combining (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content='9) with (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content='13) provide  �  ∂Ω  �  −∂φΩ  ∂ν  � vn  sp  n  =  �  Ω  �  φΩ + vn  sn  �p  φΩ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content='  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content='14)  We then consider either case (a) or (b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content=' From (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content='6), we deduce that  �  Ω  �  φΩ + vn  sn  �p  φΩ −→  �  Ω  φp+1  Ω  > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content='  Taking into account (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content='5), we may derive from (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content='14) that  csp  n ≤  �  ∂Ω  vn  for some c > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content=' By H¨older’s inequality, it follows that  csp  n ≤  �  ∂Ω  vn ≤ |∂Ω|  q  q+1  ��  ∂Ω  vq+1  n  �  1  q+1  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content='  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content='15)  Combining (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content='11) with (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content='15) and using H¨older’s inequality, there exist c, ˜c > 0 such that  In ≥ λn  2  �  ∂Ω  vq+1  n  − c sn  ��  ∂Ω  vq+1  n  �  1  q+1  =  �  λn  2  ��  ∂Ω  vq+1  n  �  q  q+1  − c sn  � ��  ∂Ω  vq+1  n  �  1  q+1  ≥ {˜c λn spq  n − c sn}  ��  ∂Ω  vq+1  n  �  1  q+1  = spq  n  �  ˜cλn − c s1−pq  n  � ��  ∂Ω  vq+1  n  �  1  q+1  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content='  Since sn → 0 from (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content='4), assertion (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content='12) follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content='  9  We next consider case (c) and verify (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content='12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content=' By combining (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content='11) with (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content='14), it follows from  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content='11) that  In = sp+1  n  �λn  2  �  ∂Ω  vq+1  n  sp+1  n  − 2  �  Ω  �  φΩ + vn  sn  �p  φΩ  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content='  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content='16)  Furthermore, we use the test function ϕ = 1 in (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content='3) to infer that  −  �  Ω  un +  �  Ω  up  n + λn  �  ∂Ω  uq  n = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content='  Substituting un = snφΩ + vn,  −  �  Ω  �  φΩ + vn  sn  �  + sp−1  n  �  Ω  �  φΩ + vn  sn  �p  +  �  ∂Ω  λnvq  n  sn  = 0,  which implies that  �  ∂Ω  λnvq  n  sn  −→  �  Ω  φΩ > 0,  where we have used condition (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content='6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content=' Thus, we may deduce that  c sn  λn  ≤  �  ∂Ω  vq  n  for some c > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content=' Using H¨older’s inequality, we deduce that  c  � sn  λn  � q+1  q  ≤  �  ∂Ω  vq+1  n  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content='17)  for some c > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content=' Combining (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content='16) with (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content='17),  In ≥ sp+1  n  �  c s  1  q −p  n  λ  − 1  q  n  − 2  �  Ω  �  φΩ + vn  sn  �p  φΩ  �  for some c > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content=' We observe that  s  1  q −p  n  → ∞,  λ  − 1  q  n  ≥ c  by some constant c > 0,  �  Ω  �  φΩ + vn  sn  �p  φΩ →  �  Ω  φp+1  Ω  > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content='  Thus, assertion (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content='12) follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content='  □  We conclude this section with the establishment of the uniqueness and stability results for a  positive solution of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content='1) in the case where βΩ < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content='  Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content=' Assume that βΩ < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content=' Then, there exists Λ > 0 such that if λ > Λ, then the  following two assertions hold:  (i) Problem (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content='1) has at most one positive solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content='  (ii) A positive solution u of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content='1) satisfying u > 0 in Ω is asymptotically stable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content=' We recall that the unique positive solution uD of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content='6) is asymptotically stable, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content=',  γ1,D = inf  ��  Ω  �  |∇ϕ|2 − ϕ2 + pup−1  D  ϕ2�  : ϕ ∈ H1  0(Ω),  �  Ω  ϕ2 = 1  �  > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content='  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content='18)  10  (i) Assume to the contrary that problem (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content='1) has two distinct positive solutions (λn, un) and  (λn, vn) with λn → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content=' Note that un, vn < 1 in Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content=' We may assume that un, vn → uD in H1(Ω)  and a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content=' in Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content=' The difference wn = un − vn (may change sign) allows that  �  Ω  �  ∇wn∇ϕ − wnϕ  �  +  �  Ω  �  (vn + wn)p − vp  n  �  ϕ + λn  �  Γn  (vn + wn)q − vq  n  wn  wnϕ = 0  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content='19)  for ϕ ∈ H1(Ω), where Γn = {x ∈ ∂Ω : wn(x) ̸= 0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content=' Note that ∥wn∥ → 0, and wn → 0 a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content=' in Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content='  Substituting ϕ = wn into (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content='19), the mean value theorem shows the existence of θn ∈ (0, 1)  such that  E(wn) +  �  Ω  p(vn + θnwn)p−1w2  n + λn  �  Γn  (vn + wn)q − vq  n  wn  w2  n = 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content='  thus, ψn =  wn  ∥wn∥ implies that  E(ψn) +  �  Ω  p(vn + θnwn)p−1ψ2  n + λn  �  Γn  (vn + wn)q − vq  n  wn  ψ2  n = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content='  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content='20)  From ∥ψn∥ = 1, we infer that up to a subsequence, ψn ⇀ ψ∞, ψn → ψ∞ in L2(Ω) and L2(∂Ω)  for some ψ∞ ∈ H1(Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content=' Then, we claim that ψ∞ ∈ H1  0(Ω) and ψ∞ ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content=' From (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content='20), we deduce  that  λn  �  Γn  (vn + wn)q − vq  n  wn  ψ2  n ≤  �  Ω  ψ2  n ≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content='  We observe that  (vn + wn)q − vq  n  wn  ≥ q  on Γn  because un, vn < 1 in Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content=' It follows that λnq  �  ∂Ω ψ2  n ≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content=' Passing to the limit, we deduce that  ψ∞ ∈ H1  0(Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content=' Indeed, ψ∞ ̸= 0 by Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content='  Then, we assert in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content='20) that  �  Ω  p(vn + θnwn)p−1ψ2  n −→  �  Ω  pup−1  D  ψ2  ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content='  Indeed, we use  �  Ω  p(vn + θnwn)p−1ψ2  n =  �  Ω  p(vn + θnwn)p−1ψ2  ∞ +  �  Ω  p(vn + θnwn)p−1(ψ2  n − ψ2  ∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content='  Since vn → uD and wn → 0 a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content=' in Ω and un, vn < 1 in Ω, the Lebesgue dominated convergence  theorem shows that  �  Ω  p(vn + θnwn)p−1ψ2  ∞ −→  �  Ω  pup−1  D  ψ2  ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content='  Using the fact  �  Ω |ψ2  n − ψ2  ∞| → 0 yields  �  Ω  p(vn + θnwn)p−1(ψ2  n − ψ2  ∞) −→ 0,  as desired.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content='  Then, the weak lower semicontinuity allows us to deduce from (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content='20) that  �  Ω  �  |∇ψ∞|2 − ψ2  ∞ + pup−1  D  ψ2  ∞  �  ≤ lim  n→∞  �  E(ψn) +  �  Ω  p(vn + θnwn)p−1ψ2  n  �  ≤ 0,  which contradicts ψ∞ ∈ H1  0(Ω) and ψ∞ ̸= 0 in view of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content='18).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content='  11  (ii) On the basis of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content='4), we claim that γ1 > 0 for sufficiently large λ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content='  Assume by  contradiction that a positive solution (λn, un) of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content='1) with the condition that λn → ∞ and  un > 0 in Ω satisfies γn := γ1,n(λn, un) ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content=' This means that  �  Ω  �  |∇ϕn|2 − ϕ2  n + pup−1  n  ϕ2  n  �  + λnq  �  ∂Ω  uq−1  n  ϕ2  n = γn ≤ 0,  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content='21)  where ϕn := ϕ1,n, normalized as  �  Ω  ϕ2  n +  �  ∂Ω  ϕ2  n = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content='  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content='22)  Because  �  Ω |∇ϕn|2 ≤  �  Ω ϕ2  n ≤ 1 from (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content='21) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content='22), ∥ϕn∥ is bounded, which implies that up  to a subsequence, ϕn ⇀ ϕ∞, ϕn → ϕ∞ in L2(Ω) and L2(∂Ω), and ϕn → ϕ∞ a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content=' in Ω for some  ϕ∞ ∈ H1(Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content=' Since uq−1  n  ≥ 1 from Theorem 0(I), assertion (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content='21) gives us  λnq  �  ∂Ω  ϕ2  n ≤  �  Ω  ϕ2  n ≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content='  Passing to the limit, ϕ∞ = 0 on ∂Ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content=' thus, ϕ∞ ∈ H1  0(Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content=' From (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content='22),  �  Ω ϕ2  ∞ = 1 is also deduced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content='  By the weak lower semicontinuity, we derive from (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content='21) that  �  Ω  �  |∇ϕ∞|2 − ϕ2  ∞ + pup−1  D  ϕ2  ∞  �  ≤ lim  n→∞  �  Ω  �  |∇ϕn|2 − ϕ2  n + pup−1  n  ϕ2  n  �  ≤ 0  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content='23)  Indeed, on the basis of the facts that un → uD in H1(Ω) and un < 1 in Ω (see Theorem 0(I)  and Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content='2), the Lebesgue dominated convergence theorem shows that  �  Ω  up−1  n  ϕ2  n =  �  Ω  up−1  n  ϕ2  ∞ +  �  Ω  up−1  n  (ϕ2  n − ϕ2  ∞) −→  �  Ω  up−1  D  ϕ2  ∞,  where we have used that un → uD a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content=' in Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content=' Assertion (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content='23) contradicts ϕ∞ ∈ H1  0(Ω) and  �  Ω ϕ2  ∞ = 1 in view of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content='18).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content='  □  Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content=' If we construct a positive solution u of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content='1) such that u > 0 in Ω without using  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content='2), then assertion (ii) of Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content='6 remains valid for all p > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content=' Sub- and supersolutions  Consider the case where βΩ < 1 or where βΩ = 1 and pq > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content=' Then, we first construct small  positive subsolutions of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content='1) and use them to establish an a priori lower bound for positive  solutions (λ, u) of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content='1) satisfying u > 0 in Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content=' For a fixed τ > 0 and with a parameter ε > 0, we  set  φε(x) = ε(φΩ(x) + ετ),  x ∈ Ω,  which implies that φε ∈ C2+θ(Ω) and φε > 0 in Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content='  We then use φε to formulate the following a priori lower bound for positive solutions u > 0  in Ω of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content='  Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content=' Assume that βΩ < 1 or that βΩ = 1 and pq > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content=' Let τ > 1−q  q  when βΩ < 1, and  let 1−q  q  < τ < p − 1 when βΩ = 1 and pq > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content=' Then, for Λ > 0 there exists ε = ε(τ, Λ) > 0 such  that φε is a subsolution of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content='1), provided that λ ∈ [0, Λ] and ε ∈ (0, ε].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content=' Furthermore, u ≥ φε  in Ω for a positive solution u > 0 in Ω of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content='1) with λ ∈ [0, Λ].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content=' Here, ε does not depend on  λ ∈ [0, Λ].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content='  12  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content=' We only consider the case where βΩ = 1 and pq > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content=' The case where βΩ < 1 is proved  similarly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content=' First, we verify the former assertion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content=' We take 0 < ε ≤ 1 and then use the condition  p − τ − 1 > 0 to deduce that  −∆φε − φε + φp  ε ≤ ε1+τ  �  −1 + εp−τ−1  �  1 + max  Ω  φΩ  �p�  ≤ 0  in Ω  if ε > 0 is small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content=' For Λ > 0 we use (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content='5) and the condition τ > 1−q  q  to deduce that  ∂φε  ∂ν + λφq  ε ≤ −c1ε + λε(1+τ)q ≤ ε(−c1 + Λεq+τq−1) ≤ 0  on ∂Ω  if 0 < ε ≤ (c1/Λ)1/(q+τq−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content=' The desired assertion follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content='  Next, we argue by contradiction to verify the latter assertion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content=' Assume by contradiction that  u ̸≥ φε in Ω for some positive solution u > 0 in Ω with λ ∈ [0, Λ].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content=' Because ε �→ φε is increasing  and φε → 0 uniformly in Ω, we can take ε1 ∈ (0, ε) such that  �  u ≥ φε1  in Ω,  u(x1) = φε1(x1)  for some x1 ∈ Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content='1)  Take c > 0 such that u, φε1 ≥ c in Ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content=' then, choose K > 0 sufficiently large so that fK(t) =  Kt+t−tp is increasing for t ∈  �  0, maxΩ u  �  and M > 0 sufficiently large so that M −Λqcq−1 > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content='  We use the subsolution φε1 (not a positive solution of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content='1)) to deduce that  (−∆ + K)(u − φε1) ≥ fK(u) − fK(φε1) ≥ 0 (and ̸≡ 0)  in Ω,  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content='2)  and for x ∈ ∂Ω satisfying u > φε1,  � ∂  ∂ν + M  �  (u − φε1) ≥ −λuq + λφq  ε1 + M(u − φε1)  =  �  M − λuq − φq  ε1  u − φε1  �  (u − φε1)  ≥ (M − Λqcq−1)(u − φε1) > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content='3)  Thus, the strong maximum principle and boundary point lemma are applicable to infer that  u − φε1 > 0 in Ω, which contradicts (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content='  □  On the basis of Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content='1, we construct minimal and maximal positive solutions of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content='1) as  follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content='  Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content=' Assume that βΩ < 1 or that βΩ = 1 and pq > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content=' Then, problem (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content='1) has  a minimal positive solution uλ ∈ C2+θ(Ω) and a maximal positive solution uλ ∈ C2+θ(Ω) for  each λ > 0 such that 0 < uλ ≤ uλ in Ω, meaning that any positive solution u of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content='1) with the  condition that u > 0 in Ω satisfies uλ ≤ u ≤ uλ in Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content=' Moreover, both uλ and uλ are weakly  stable, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content=', γ1(λ, uλ), γ1(λ, uλ) ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content=' It is clear that (λ, 1) is a supersolution of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content=' Choose ε0 > 0 such that φε0 ≤ 1 in Ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content='  then, Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content='1 states that (λ, φε0) is a subsolution of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content=' By Theorem 0(I) and Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content='1,  a positive solution u > 0 in Ω of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content='1) implies that φε0 ≤ u ≤ 1 in Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content=' Thus, this proposition is  a direct consequence of applying [3, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content='1)Theorem] and [2, Proposition 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content='8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content='  □  Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content=' In view of the construction, Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content='1 and Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content='2 remain valid for  any p > 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content=' therefore, Propositions 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content='2, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content='3, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content='6(ii) hold for any p > 1 with the positive solution  (λn, un) of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content='1) with λn → ∞ that is constructed by Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content='2 (see Remarks 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content='4 and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content='7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content='  13  In the case where βΩ < 1, Propositions 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content='6 and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content='2 ensure the existence of a unique positive  solution u(λ) of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content='1) for λ > Λ, which is asymptotically stable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content=' Using the implicit function  theorem provides us with the following result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content='  Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content=' Assume that βΩ < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content=' Then, {(λ, u(λ)) : λ > Λ} is a C∞ curve, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content=', λ �→  u(λ) ∈ C2+θ(Ω) is C∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content=' Moreover, it is decreasing, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content=', u(λ1) > u(λ2) in Ω for λ2 > λ1 > Λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content=' We verify the first assertion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content='  Let (λ0, u(λ0)) be the unique positive solution of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content='1)  for λ0 > Λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content=' Since γ1(λ0, u(λ0)) > 0, the implicit function theorem applies at (λ0, u0);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content=' then,  we deduce, thanks to the uniqueness, that {(λ, u(λ)) : λ1 < λ ≤ λ2} is a C∞ curve for λ1 <  λ0 < λ2 such that u(λ) is asymptotically stable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content=' The implicit function theorem applies again at  (λ2, u(λ2));' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content=' then, the curve is continued until λ = λ3 > λ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content=' Repeating the same procedure, the  curve is continued to λ = ∞ thanks to the a priori upper and lower bounds (Theorem 0(I) and  Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content='1), as desired.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content='  We next verify the second assertion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content=' If λ1 < λ2, then u(λ1) is a supersolution of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content='1) for  λ = λ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content=' By Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content='1, it is possible to construct a subsolution φε of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content='1) for λ = λ2 such  that 0 < φε ≤ u(λ1) in Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content=' The sub- and supersolution method applies, and problem (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content='1) has  a positive solution u for λ = λ2 such that φε ≤ u ≤ u(λ1) in Ω, where u ̸≡ u(λ1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content=' Thanks  to Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content='6(i), we obtain u = u(λ2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content=' The desired assertion follows by using the strong  maximum principle and boundary point lemma (as developed in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content='2) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content='3)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content='  □  We conclude this section by employing the weak sub- and supersolution method [23] to show  global strong positivity for a positive solution of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content='1) in the case where βΩ < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content='  Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content=' Assume that βΩ < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content=' Then, a positive solution (λ, u) of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content='1) satisfies that  u > 0 in Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content=' Assume by contradiction that problem (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content='1) possesses a positive solution (λ0, u0) for  λ0 > 0 such that u0 = 0 somewhere on ∂Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content=' Let λ = λ1 > max(λ0, Λ), for which problem (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content='1)  has at most one positive solution by Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content='6(i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content=' Then, u0 is a weak supersolution of  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content='1) for λ = λ1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content=' Indeed,  0 =  �  Ω  �  ∇u0∇ϕ − u0ϕ + up  0ϕ  �  + λ0  �  ∂Ω  uq  0ϕ  ≤  �  Ω  �  ∇u0∇ϕ − u0ϕ + up  0ϕ  �  + λ1  �  ∂Ω  uq  0ϕ,  ϕ ∈ H1(Ω) and ϕ ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content='  We next construct a weak subsolution of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content='1) for λ = λ1 that is smaller than or equal to  u0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content=' Note that u0 ∈ Cθ(Ω) and u0 > 0 in Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content=' From βΩ < 1, it follows by the continuity and  monotonicity of βΩ with respect to Ω that we can choose a subdomain Ω1 ⋐ Ω with smooth  boundary ∂Ω1 such that βΩ1 < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content=' Then, we deduce that  u0 ≥ c  in Ω1  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content='4)  for some c > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content=' We also deduce that if ε > 0 is sufficiently small, then  −∆(εφΩ1) ≤ εφΩ1 − (εφΩ1)p  in Ω1,  where φΩ1 is a positive eigenfunction associated with βΩ1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content=' Consequently, the divergence theorem  is applied to  �  Ω −∆(εφΩ1)ϕ for ϕ ∈ H1(Ω) and ϕ ≥ 0 to deduce that  �  Ω1  �  ∇(εφΩ1)∇ϕ − (εφΩ1)ϕ + (εφΩ1)pϕ  �  ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content='5)  14  Define  Φε =  �  εφΩ1  in Ω1,  0  in Ω \\ Ω1,  and Φε ∈ H1(Ω) ∩ C(Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content=' By virtue of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content='5), the linking technique [6, Lemma I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content='1] yields  �  Ω  �  ∇Φε∇ϕ − Φεϕ + Φp  ε ϕ  �  + λ1  �  ∂Ω  Φq  ε ϕ =  �  Ω1  �  ∇Φε∇ϕ − Φεϕ + Φp  ε ϕ  �  ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content='  Thanks to (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content='4), we can take ε > 0 such that Φε ≤ u0 in Ω, as desired.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content='  The weak sub- and supersolution method [23, Subsection 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content='2] is now applicable to deduce  the existence of a positive solution (λ1, u1) of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content='1) such that Φε ≤ u1 ≤ u0 in Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content=' Particularly,  u1 = 0 somewhere on ∂Ω because so is u0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content=' However, this contradicts Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content='2 in view of  the uniqueness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content='  □  Proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content=' The uniqueness assertion follows from Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content='6(i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content=' Assertions (i-  a) and (i-b) follow from Propositions 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content='6(ii) and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content='2, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content=' Assertion (i-c) is verified by  Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content='4 and an analogous argument as in the proof of Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content=' Assertion (ii) follows  from Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content='  □  Proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content=' The existence part follows from Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content=' Assertion (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content='9) follows  from Propositions 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content='2 and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content='  □  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content=' Proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content='5  This section is devoted to the proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content='  (i) We prove assertion (i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content=' Assume by contradiction that problem (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content='1) has a positive solution  (λn, un) with λn → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content=' Then, Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content='2 shows that ∥un∥ → 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content=' thus, Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content='3 shows  that  un  ∥un∥ → φΩ in H1(Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content=' We apply Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content='5(a) and (b);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content=' then, for un = snφΩ+vn ∈ ⟨φΩ⟩⊕V  as in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content='3), we have (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content='7) with (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content='4)–(2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content='6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content='  Observe from (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content='5) that ∥vn∥ → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content=' Say that ψn =  vn  ∥vn∥;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content=' then, up to a subsequence, ψn ⇀  ψ∞ ≥ 0, and ψn → ψ∞ in L2(Ω) and L2(∂Ω) for some ψ∞ ∈ H1(Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content=' From (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content='7), it follows that  c  �  ∂Ω  ψq+1  n  ≤ −E(ψn)∥vn∥1−q −→ 0,  so that  �  ∂Ω ψq+1  ∞  = 0, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content=', ψ∞ ∈ H1  0(Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content=' Lastly, we use the condition E(ψn) ≤ 0 derived from  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content='7) to follow the argument in the last paragraph of the proof of Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content='3;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content=' then, we arrive  at the contradiction ψ∞ = φΩ ∈ ⟨φΩ⟩ ∩ V = {0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content='  (ii) We verify assertion (ii).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content=' We remark that the convergences (λn, un) → (λ∞, 0) with λ∞ ≥ 0  in R × H1(Ω) and R × C(Ω) are equivalent for a positive solution (λn, un) of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content='1) with λn > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content='  This is verified by the bootstrap argument [32, Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content='3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content=' In fact, the proof of assertion (ii)  is similar to that for assertion (i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content=' Assume by contradiction that problem (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content='1) has a positive  solution (λn, un) with the condition that λn → λ∞ > 0 and ∥un∥ → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content='5(a) and (c)  apply;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content=' then, we arrive at a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content='  (iii) To verify assertion (iii), we prove the following three auxiliary lemmas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content=' Say that Un =  λ  −  1  1−q  n  un.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content='  Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content=' There exists C > 0 such that ∥Un∥ ≤ C for a positive solution (λn, un) of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content='1)  with λn > 0 satisfying that (λn, un) → (0, 0) in R × H1(Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content='  15  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content=' Assume by contradiction that ∥Un∥ → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content=' Say that wn =  Un  ∥Un∥;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content=' then, up to a subse-  quence, wn ⇀ w∞ ≥ 0, and wn → w∞ in L2(Ω) and L2(∂Ω) for some w∞ ∈ H1(Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content=' Since  E(wn) ≤ 0, Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content='1 provides w∞ ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content='  Recall that (λ, U) = (λn, Un) satisfies  �  Ω  �  ∇U∇ϕ − Uϕ + λ  p−1  1−q U pϕ  �  +  �  ∂Ω  U qϕ = 0,  ϕ ∈ H1(Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content='  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content='1)  Using the test function ϕ = 1 in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content='1), we deduce that  �  Ω  Un = λ  p−1  1−q  n  �  Ω  U p  n +  �  ∂Ω  U q  n =  �  Ω  up−1  n  Un +  �  ∂Ω  U q  n,  implying  �  Ω  wn =  �  Ω  up−1  n  wn +  �  ∂Ω  wq  n∥Un∥q−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content='  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content='2)  We may assume that un → 0 a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content=' in Ω, and since un < 1 in Ω, we deduce that  �  Ω  up−1  n  wn =  �  Ω  up−1  n  w∞ +  �  Ω  up−1  n  (wn − w∞) −→ 0,  by applying the Lebesgue dominated convergence theorem and using the condition wn → w∞  in L2(Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content='  Then, passing to the limit in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content='2) yields  �  Ω w∞ = 0, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content=', w∞ = 0, which is a  contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content='  □  Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content=' Assume that βΩ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content=' Then, there is no positive solution U of (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content='1) for λ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content=' If it exists, then from (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content='1) with λ = 0 and ϕ = 1, it follows that U > 0 on Γ ⊂ ∂Ω with  |Γ| > 0, implying  �  ∂Ω  ∂φΩ  ∂ν U < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content=' We use the test function ϕ = φΩ to deduce that  �  Ω  �  ∇U∇φΩ − UφΩ  �  = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content='  However, the divergence theorem leads us to the contradiction  �  Ω  φΩU =  �  Ω  −∆φΩU =  �  Ω  ∇φΩ∇U −  �  ∂Ω  ∂φΩ  ∂ν U >  �  Ω  ∇φΩ∇U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content='  □  Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content=' Assume that βΩ = 1 and pq ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content=' Then, there exists C > 0 such that ∥Un∥ ≥ C  for a positive solution (λn, Un) of (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content='1) with λn → 0+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content=' Assume by contradiction that (λn, Un) → (0, 0) in R × H1(Ω) for a positive solution  (λn, Un) of (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content=' Say that wn =  Un  ∥Un∥;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content=' then, up to a subsequence, wn ⇀ w∞ ≥ 0, wn → w∞ in  Lp+1(Ω) and L2(∂Ω) for some w∞ ∈ H1(Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content=' From (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content='1) with (λ, U) = (λn, Un) and ϕ = Un, it  follows that  �  Ω  �  |∇Un|2 − U 2  n + λ  p−1  1−q  n  U p+1  n  �  +  �  ∂Ω  U q+1  n  = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content='  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content='3)  We then deduce that  �  ∂Ω wq+1  n  ≤  �  Ω w2  n∥Un∥1−q → 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content=' thus,  �  ∂Ω wq+1  ∞  = 0, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content=', w∞ ∈ H1  0(Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content=' We  also deduce from (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content='3) that E(wn) =  �  Ω(|∇wn|2 − w2  n) ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content=' Thus, we derive that wn → φΩ in  H1(Ω) using a similar argument as in the last paragraph of the proof of Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content='  For a contradiction, we use the same strategy developed in the proof of assertion (i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content=' To this  end, we consider the orthogonal decomposition Un = snφΩ + vn ∈ ⟨φΩ⟩ ⊕ V as in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content='3);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content=' then,  16  we obtain (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content='4) to (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content='6) with un replaced by Un.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content=' As in the proof of Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content='5, we deduce the  following counterpart of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content='10) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content='11) for (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content='3):  E(vn) + 1  2  �  ∂Ω  vq+1  n  + Jn ≤ 0,  with  Jn = 1  2  �  ∂Ω  vq+1  n  − 2sn  �  ∂Ω  �  −∂φΩ  ∂ν  �  vn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content='  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content='4)  In the same spirit of Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content='5 ((2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content='12)), we establish  E(vn) + 1  2  �  ∂Ω  vq+1  n  ≤ 0  for sufficiently large n,  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content='5)  by verifying that  Jn ≥ 0  for sufficiently large n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content='  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content='6)  Analogously to (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content='14), we obtain  �  ∂Ω  �  −∂φΩ  ∂ν  �  vn = λ  p−1  1−q  n  sp  n  �  Ω  �  φΩ + vn  sn  �p  φΩ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content='  Using this assertion, we deduce from (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content='4) that  Jn = sp+1  �1  2  �  ∂Ω  vq+1  n  sp+1 − 2λ  p−1  1−q  n  �  Ω  �  φΩ + vn  sn  �p  φΩ  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content='  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content='7)  Furthermore, we use the test function ϕ = 1 in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content='1) to obtain  −  �  Ω  Un + λ  p−1  1−q  n  �  Ω  U p  n +  �  ∂Ω  U q  n = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content='  Substituting Un = snφΩ + vn,  −  �  Ω  �  φΩ + vn  sn  �  + λ  p−1  1−q  n  sp−1  n  �  Ω  �  φΩ + vn  sn  �p  +  �  ∂Ω  vq  n  sn  = 0,  from which we use (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content='6) with Un to infer that  �  ∂Ω  vq  n  sn  −→  �  Ω  φΩ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content='  Then, we may deduce that  csn ≤  �  ∂Ω  vq  n  for some c > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content=' By H¨older’s inequality, we deduce that  cs  q+1  q  ≤  �  ∂Ω  vq+1  for some c > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content=' We use this inequality to derive from (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content='7) that  Jn ≥ sp+1  �  cs  1  q −p  n  − 2λ  p−1  1−q  n  �  Ω  �  φΩ + vn  sn  �p  φΩ  �  for some c > 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content=' thus, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content='6) follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content=' Assertion (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content='5) has been now established.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content='  We end the proof of this lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content=' Observe from (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content='5) with Un that ∥vn∥ → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content=' Then, we  develop the same argument as in the second paragraph of the proof of assertion (i) to arrive at  the same contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content='  □  17  Employing the above lemmas, we then verify assertion (iii).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content=' Assume by contradiction that  (λn, un) → (0, 0) in R × H1(Ω) for a positive solution (λn, un) of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content='1) with λn > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content=' Then,  (λn, Un) with Un = λ  −  1  1−q  n  un admits a positive solution of (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content=' Since Un is bounded in H1(Ω)  by Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content='1, we deduce that up to a subsequence, Un ⇀ U∞ ≥ 0, and Un → U∞ in Lp+1(Ω)  and L2(∂Ω) for some U∞ ∈ H1(Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content=' Thanks to Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content='3, we apply Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content='1 to obtain  U∞ ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content='  Furthermore, substituting (λ, U) = (λn, Un) into (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content='1) and then taking the limit, we deduce  that  �  Ω  �  ∇U∞∇ϕ − U∞ϕ  �  +  �  ∂Ω  U q  ∞ϕ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content='  This implies that U∞ is a nonnegative solution of (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content='1) for λ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content=' Finally, Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content='2 provides  U∞ = 0, which is a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content='  The proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content='5 is complete.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content='  □  Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content=' Assertions (ii) and (iii) of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content='5 are also derived from Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content='1 when  βΩ = 1 and pq > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content=' Stability analysis of the trivial solution  In the last section, we consider the stability of the trivial solution u = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content=' It is worthwhile  to mention that a linearized stability analysis does not work for u = 0 because u �→ uq is not  differentiable at u = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content='  The corresponding initial-boundary value problem is formulated as  follows:  \uf8f1  \uf8f4  \uf8f2  \uf8f4  \uf8f3  ∂u  ∂t (t, x) = ∆u + u − up  in (0, ∞) × Ω,  ∂u  ∂ν = −λuq  on (0, ∞) × ∂Ω,  u(0, x) = u0(x) ≥ 0  in Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content='  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content='1)  We use the method of monotone iterations to determine the Lyapunov stability of the trivial  solution u = 0 (see [26, Definition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content='1]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content='  When βΩ < 1 or when βΩ = 1 and pq > 1, we observe from Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content='1 that u = 0 is unstable  in the following sense: for u0 ∈ C2(Ω) sufficiently small such that u0 > 0 in Ω, the positive  solution u(t, x) of (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content='1) corresponding to the initial value u0 moves away from 0 as t → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content='  When βΩ > 1, for ε, δ, τ > 0, we set  ψδ,ε,τ(x) = δ(φΩ(x) + ε)τ,  x ∈ Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content='  Let Ωρ := {x ∈ Ω : dist(x, ∂Ω) < ρ} for ρ > 0 be a tubular neighborhood of ∂Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content=' Then, by (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content='5),  for ρ0 > 0 small, we can choose a constant c3 = c3(ρ0) > 0 such that |∇φΩ|2 ≥ c3 in Ωρ for  0 < ρ ≤ ρ0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content=' If 0 < ρ ≤ ρ0, then there exists c4 = c4(ρ) > 0 such that φΩ ≥ c4 in Eρ := Ω \\ Ωρ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content='  The following result would then provide useful information about the stability of the trivial  solution u = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content='  Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content=' Assume that βΩ > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content=' Then, for  1  βΩ < τ < 1 and ε > 0 small, there exists δ1 > 0  such that ψδ,ε,τ is a supersolution of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content='1) whenever 0 < δ ≤ δ1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content=' We write ψδ,ε,τ simply as φδ,ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content=' By direct computations, we obtain  ∇ψδ,ε = δτ(φΩ + ε)τ−1∇φΩ,  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content='2)  ∆ψδ,ε = δτ(τ − 1)(φΩ + ε)τ−2|∇φΩ|2 + δτ(φΩ + ε)τ−1∆φΩ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content='  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content='3)  18  We see from (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content='3) that for x ∈ Ωρ,  ∆ψδ,ε + ψδ,ε − ψp  δ,ε ≤ δτ(τ − 1)(φΩ + ε)τ−2|∇φΩ|2 + δ(φΩ + ε)τ  = δ(φΩ + ε)τ−2  \uf8f1  \uf8f2  \uf8f3−τ(1 − τ)c3 +  �  ε + max  Ωρ  φΩ  �2\uf8fc  \uf8fd  \uf8fe .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content='  We then find 0 < ρ1 ≤ ρ0 and ε1 > 0 such that  �  ε + max  Ωρ1  φΩ  �2  ≤ τ(1 − τ)c3  for 0 < ε ≤ ε1,  and then,  −∆ψδ,ε + ψδ,ε − ψp  δ,ε ≤ 0  in Ωρ1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content='  Let us fix c4 = c4(ρ1), and let 0 < ε ≤ ε1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content=' We also see from (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content='3) that for x ∈ Eρ1,  ∆ψδ,ε + ψδ,ε − ψp  δ,ε ≤ δτ(φΩ + ε)τ−1(−βΩ)φΩ + δ(φΩ + ε)τ  ≤ δ(φΩ + ε)τ−1 {(1 − τβΩ)c4 + ε} .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content='  We then determine 0 < ε2 ≤ ε1 such that (1 − τβΩ)c4 + ε2 ≤ 0, and then,  −∆ψδ,ε2 + ψδ,ε2 − ψp  δ,ε2 ≤ 0  in Eρ1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content='  Finally, using (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content='5), we see from (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content='2) that  ∂ψδ,ε2  ∂ν  + λψq  δ,ε2 ≥ δq(−δ1−qτετ−1  2  c2 + λετq  2 ) ≥ 0  on ∂Ω,  if 0 < δ ≤ δ1 for some δ1 > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content='  In summary, ψδ,ε2, 0 < δ ≤ δ1, is as desired.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content='  □  From Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content='1, it might be claimed that u = 0 is asymptotically stable for the case where  βΩ > 1, meaning that for u0 in the order interval [0, ψδ1,ε2,τ], the positive solution u(t, x) of (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content='1)  associated with u0 tends to 0 as t → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content=' If this occurs, then Theorem 0(II) means that problem  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content='1) is bistable with two nonnegative stable equilibria for 0 < λ ≤ λ∗ (one is uλ, and the other  is u = 0), which presents ecologically a conditional persistence strategy for the harvesting effort  λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content=' However, the difficulty arises from the fact that the monotone iteration scheme does not  work for (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content='1) in the order interval [0, ψδ1,ε2,τ] because u �→ (−uq) does not satisfy the one-sided  Lipschitz condition [26, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content='19)] for u close to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content=' Rigorous verification of the claim is an open  question.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
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+page_content='  Department of Mathematics, Faculty of Education, Ibaraki University, Mito 310-8512, Japan  Email address: kenichiro.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content='umezu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
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+page_content='ibaraki.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
+page_content='jp  21' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfsi8k/content/2301.12147v1.pdf'}
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+Sample efficient graph classification using binary Gaussian boson sampling
+Amanuel Anteneh∗
+Department of Computer Science, University of Virginia, Charlottesville, Virginia 22903, USA†
+Olivier Pfister‡
+Department of Physics, University of Virginia, Charlottesville, Virginia 22903, USA
+(Dated: January 4, 2023)
+We present a variation of a quantum algorithm for the machine learning task of classification with
+graph-structured data. The algorithm implements a feature extraction strategy that is based on
+Gaussian boson sampling (GBS) a near term model of quantum computing. However, unlike the
+currently proposed algorithms for this problem, our GBS setup only requires binary (light/no light)
+detectors, as opposed to photon number resolving detectors. These detectors are technologically
+simpler and can operate at room temperature, making our algorithm less complex and less costly
+to implement on the physical hardware. We also investigate the connection between graph theory
+and the matrix function called the Torontonian which characterizes the probabilities of binary GBS
+detection events.
+I.
+INTRODUCTION
+Graphs are one of the most versatile data structures
+used in computing, and developing machine learning
+methods for working with graph-structured data has
+been a growing sub-field of machine learning research.
+Graph classification, in particular, has useful applica-
+tions in fields such as bioinformatics, network science
+and computer vision as many of the objects studied in
+these fields can easily be represented as graphs. How-
+ever, using graph-structured data with machine learning
+models is not a straightforward task.
+This is because
+one of the most common ways of representing a graph
+for computational applications, i.e., as an adjacency ma-
+trix, cannot be easily used as an input to machine learn-
+ing classifiers which primarily take vector-valued data as
+their inputs. Therefore, a common way of working with
+graph-structured data is by defining a feature map φ that
+maps a graph G to a vector in a Hilbert space called a
+feature space. From there a function κ, called a kernel,
+is defined that measures the similarity of two graphs in
+the feature space.
+An example of a feature map from
+R2 → R3 is shown in Fig. 1.
+Kernel methods refer to machine learning algorithms
+that learn by comparing pairs of data points using this
+similarity measure. In our context we have a set of graphs
+G and we call a kernel κ a graph kernel if it is a function
+of the form κ : G × G → R [1, 2]. The most common
+example of a kernel function is the feature space’s inner
+product κ(x, x′) = ⟨φ(x), φ(x′)⟩. The goal of such meth-
+ods is to construct mappings to feature vectors whose en-
+tries (the features) relate to relevant information about
+the graphs. Using a Gaussian boson sampling (GBS) de-
+vice to construct graph kernels was an idea first proposed
+∗ asa2rc@virginia.edu
+† Current address:
+Booz Allen Hamilton, Arlington, Virginia
+22202, USA
+‡ olivier.pfister@gmail.com
+FIG. 1: In the original input space R2 the data points,
+which belong either to the class ‘red’ or ‘blue’, are not
+separable by a linear function (the decision boundary)
+but after mapping the points to feature vectors in a
+higher dimensional space R3 a linear function is able to
+separate the two classes. This linear decision boundary
+can be calculated by supervised machine learning
+models such as a support vector machine. In our case
+the input space is the set of all undirected graphs which
+we denote as G.
+by Schuld et al. in Ref. 3.
+Boson sampling was first proposed by Aaronson and
+Arkhipov [4] as a task—generating the photon-counting
+outcomes of the “quantum Galton board” constituted by
+an M ×M optical interferometer fed with single photons
+into some of its input ports—that is strongly believed
+to be intractable to classical computers. The reason for
+this intractability is that calculating the probability dis-
+tribution for generating random outcomes using Monte
+Carlo simulations requires calculating the permanent of
+an M × M matrix. Calculating the permanent of a gen-
+eral matrix is known to be #P-complete [5] which is a
+class of problems comparable to the class of NP-complete
+problems in their difficulty. Gaussian boson sampling [6]
+is a variant of boson sampling in which the single-photon
+inputs are replaced with single-mode squeezed states, as
+produced, for example, by two-photon-emitting optical
+arXiv:2301.01232v1  [quant-ph]  3 Jan 2023
+
+Linear Decision
+Boundary
+Input Space
+Feature Space
+d3
+d22
+parametric amplifiers [7]. The GBS probability distribu-
+tion is governed by the Hafnian of an M × M matrix.
+Calculating the Hafnian of a general square matrix can
+be reduced to the task of calculating permanents there-
+fore calculating the Hafnian is also #P-complete. In both
+cases, a quantum machine implementing boson sampling
+or GBS can easily sample from these hard-to-calculate
+probability distributions, just because they are “wired-
+in,” and this constitutes the “quantum advantage” that
+was recently demonstrated in optical experiments [8, 9].
+Note also that the initial “quantum supremacy” result
+obtained by Google on a superconducting qubit array [10]
+was a quantum (circuit) sampling result as well.
+Beyond these necessary initial steps of demonstrating
+that quantum hardware can indeed reach regions inacces-
+sible to classical hardware, a subsequent question is that
+of the utility of a sampling task. Whereas the usefulness
+of sampling in and of itself is far from established, we
+know that the histograms produced by statistically sig-
+nificant sampling constitute empirical probability distri-
+butions that tend toward the true, classically intractable
+probability distributions for sample numbers linear in
+the number of possible outcomes [11]. The problem is
+that this very number of possible outcomes grows expo-
+nentially with M in a M-qubit quantum circuit in gen-
+eral [12], and exponentially or super-exponentially with
+M in an M-optical-mode boson or Gaussian boson sam-
+pler, which dispels any notion of quantum advantage for
+calculating the corresponding quantum probability dis-
+tributions.
+One direction that has been explored out of this co-
+nundrum is the binning of GBS measurements results
+into outcome classes whose cardinality scales favorably
+(e.g. polynomially) with the problem size (the GBS mode
+number). The immediate downside of such an approach
+is loss of information it entails, which impacts usefulness.
+However, graph classification using feature vectors and
+coarse-graining might provide advantageous GBS appli-
+cations. This was first pointed out by Schuld et al. [3].
+In this paper, we show that a technologically simpler
+version of GBS, which we term binary GBS, can achieve
+comparable or better performance. The paper is struc-
+tured as follows. In Sec.II we give broad reminders about
+GBS and graph theory (with details in Appendix A) and
+the current GBS graph kernel from Ref. 3.
+We then
+present our graph kernel in Sec.III along with results
+from numerical experiments and analyses of its complex-
+ity, features and advantages.
+II.
+REMINDERS ABOUT GAUSSIAN BOSON
+SAMPLING (GBS) AND GRAPH THEORY
+A.
+Gaussian Boson Sampling
+As mentioned above, an M-mode GBS devise com-
+prises M single-mode-squeezing (SMS) inputs, an M ×M
+optical interferometer, and M photon-number-resolving
+FIG. 2: Example of a 3-mode Gaussian boson sampler.
+Mode i ∈ {1, 2, 3} starts in the vacuum state |0⟩, is then
+squeezed by ˆS(ri) and passes through the network of
+two beamsplitters (the interferometer) before the
+number of photons in each mode is measured by the
+detectors Di∈{1,2,3}.
+(PNR) detectors, see Fig.2 for an example. The latter
+have come of age in superconducting devices such as tran-
+sition edge sensors [13] and superconducting nanowire
+single-photon detectors [14].
+Both the former and the
+latter have recently been used to make PNR measure-
+ments of as many as 100 photons [15, 16].
+An M-mode Gaussian boson sampler prepares a Gaus-
+sian (Wigner function) quantum state by the M squeez-
+ers and the interferometer.
+The squeezers output
+squeezed light into the interferometer and the photons
+are then passed through the interferometer after which
+the M detectors detect what modes the photons end up
+in resulting in a detection event.
+We denote a detec-
+tion event as n = (n1, ..., nM), where ni is the photon
+count in the ith mode and the total number of photons
+is n = �M
+i=1 ni.
+We know consider binary detectors, such as single-
+photon avalanche photodiodes, which are single-photon
+sensitive but aren’t PNR and give the same signal how-
+ever many photons were absorbed. In this case, we have
+ni ∈ {0, 1} where ni = 0 indicated zero photons were
+detected in that mode and ni = 1 indicates that at least
+one photon was detected. When using binary detectors
+we no longer know the total photon number n so we use
+N to denote the number of detectors that detect photons
+leading to �M
+i=1 ni = N ≤ M.
+An M-mode Gaussian state is fully described by a co-
+variance matrix Σ ∈ R2M×2M and a displacement vector
+d ∈ R2M [17].
+B.
+Graph theory
+In this paper we define a graph G = (V, E) as a
+set of vertices V
+= {v1, v2, ...} and a set of edges
+E = {(v1, v1), (v1, v2), ...(vi, vj), ...} that connect vertices
+if the edge value is not zero. A graph can be unweighted,
+with all nonzero edge weights equal to 1, or weighted, for
+example with real edge weights in GBS. For undirected
+graphs, which is what we will exclusively work with in
+
+<ol
+S(r1)
+D
+B(p2, T2)
+<ol
+S(r2)
+D.
+B(p1, T1)
+<ol
+S(r3)
+D3
+this paper, (vi, vj) = (vj, vi), ∀i, j. The size of a graph is
+equal to the cardinality |V | of its vertex set. The degree
+of a vertex v is the number of edges that are connected to
+it. The maximum degree of a graph is the largest degree
+of a vertex in its vertex set.
+Graphs can be represented in a number of ways such as
+a diagram, Fig.3a, or a more computationally useful way
+as an adjacency matrix, Fig.3b. The adjacency matrix
+of an undirected graph G with |V | vertices is a |V | × |V |
+symmetric matrix A with entries aij where aij is the
+weight of the edge connecting vertices i and j.
+1
+2
+3
+4
+3
+6
+4
+9
+(a) Undirected weighted
+4-vertex graph with 4 edges
+�
+���
+0 4 3 0
+4 0 6 9
+3 6 0 0
+0 9 0 0
+�
+���
+(b) Adjacency matrix of graph
+FIG. 3: 4-vertex graph and its corresponding adjacency
+matrix
+C.
+Sample complexity of GBS
+1.
+The problem with using GBS beyond sampling
+The sample complexity of a machine learning algo-
+rithm refers to the number of samples or amount of data
+required to learn some target function. In the case of
+GBS applications it refers to the number of samples we
+need to generate from the GBS device to learn or approx-
+imate a probability distribution over some set of the pho-
+ton detection events. This complexity type is extremely
+important to examine for any applications of GBS as it
+could potential render certain applications of GBS in-
+tractable for larger problem sizes.
+For example it was shown that the GBS device uti-
+lizing PNR detectors can encode the graph isomorphism
+problem [18]. This is done by encoding two graphs into
+two GBS devices and sampling each S times. The S sam-
+ples could then be used, in principle, to reconstruct the
+probability distribution over all possible detection events
+n for a given M and n. However, this cannot be done effi-
+ciently enough to provide a quantum advantage. Indeed,
+we know from Refs. 11, 19 that reconstructing a proba-
+bility distribution D over a discrete finite set Ω of cardi-
+nality |Ω| from an empirical distribution ˆD constructed
+from samples from D we require
+S =
+�2(ln(2)|Ω| + ln( 1
+δ ))
+ϵ2
+�
+(1)
+samples to guarantee that
+p(||D − ˆD||1 ≥ ϵ) ≤ δ
+(2)
+where ||D − ˆD||1 denotes the L1 distance between D and
+ˆD. In other words we require
+O
+�|Ω| + ln( 1
+δ )
+ϵ2
+�
+(3)
+samples to ensure with probability at most δ that the
+sum of the absolute values of the errors on the empirical
+probability distribution is ϵ or greater.
+This means the number of samples we need to approx-
+imate a probability distribution scales linearly with the
+number of elements in it’s sample space i.e. the num-
+ber of outcomes. In the case where D is the probability
+distribution over the set of all possible PNR detection
+events the number of such events for a given number of
+modes M and maximum number of photons n is
+|Ω| =
+�n + M − 1
+M − 1
+�
+= (n + M − 1)!
+n!(M − 1)!
+(4)
+which in number theory is also known as the formula for
+the number of weak compositions of an integer n into M
+parts. As shown in appendices B and C under the as-
+sumption that the number of mode scales quadratically
+with the number of photons, M ∈ O(n2), this quantity
+grows super exponentially with M and in general scales
+as O((n + M − 1)M−1) meaning that as the size of the
+graphs increase, and therefore as the number of modes
+of the GBS device increase, we require an exponential
+number of samples to ensure the algorithm can give us
+the correct result within a certain probability. Therefore
+while the algorithm may in principle be able to decide
+graph isomorphism, it is sample inefficient to an expo-
+nential degree making it intractable to implement even
+with a fault tolerant quantum computer.
+2.
+Coarse graining of sample distributions
+However a method was suggested in [18] to coarse-grain
+the probability distribution by combining outcomes into
+groups called orbits. Coarse-graining in this sense means
+to construct a new probability distribution over the set
+of these groups, the cardinality of which is less than the
+original set of all possible detection events. An orbit On
+consists of a detection event n and all of its permutations.
+For example the orbit that contains the detection event
+n = (1, 2, 2) also contains the detection events (2, 1, 2)
+and (2, 2, 1). The number of orbits for a 4-mode GBS
+device is equal to the number of ways one can write n1 +
+n2 + n3 + n4 = n, where the order of the summands does
+not matter. This is called the number of partitions of
+the integer n into M parts and from the number theory
+literature [20] it is known to behave asymptotically as
+|Ω| ≈
+eπ
+�
+2(n−M)
+3
+4
+√
+3(n − M), M ≤ n ≤ 2M.
+(5)
+
+4
+If we assume the number of photons grows linearly with
+the number of modes, n ∈ Θ(M) → n = 2M, we have
+the following asymptotic bound on the number of orbits
+1
+4
+√
+3M eπ√
+2M
+3 ∈ O(eπ√
+2M
+3
+M
+).
+(6)
+This means the number of orbits, which is now the num-
+ber of outcomes |Ω| from Eq. 1, grows like M −1eπ√
+2M/3
+meaning we would have a sample complexity of
+O
+�
+M −1eπ√
+2M/3 + ln( 1
+δ )
+ϵ2
+�
+(7)
+which is subexponential but still intractable for large M.
+3.
+Sample complexity of previously proposed GBS graph
+kernels
+The first GBS based graph kernel proposed in [3]
+maps a graph G to feature vectors in a feature space
+φ : G → f = (f1, f2, ..., fD) ∈ RD. Where fi = p(Oi
+n)
+is the probability of detecting a detection event from the
+orbit Oi
+n. This kernel was shown to perform well against
+three of the four classical kernels we use as benchmarks
+in this paper. However a shortcoming of this method is
+that the sample complexity is O(M −1eπ√
+2M/3).
+The second GBS kernel is of the form φ : G →
+f = (f1, f2, ..., fD) ∈ RD, with fi = p(Mi
+n,∆s) where
+p(Mi
+n,∆s) is the probability of detecting a detection event
+that belongs to the “meta-orbit” Mi
+n,∆s. A meta-orbit
+Mn,∆s is uniquely defined by a total photon number n
+and ∆s which is defined as
+∆s = {n :
+�
+i
+ni = n ∧ ∀i : ni ≤ s }.
+(8)
+Therefore a meta-orbit consists of all detection events
+where total photon number is equal to n, where no detec-
+tor counts more than s photons. It is claimed that this
+strategy partitions the set of all PNR detection events
+into a polynomial number of subsets in n [21].
+III.
+THE ALGORITHM
+A.
+GBS with binary detectors and it relation to
+graph theory
+While the relationship between GBS with PNR de-
+tectors and graph theory has been thoroughly explored,
+there has been little exploration of how GBS with bi-
+nary detectors fits into the picture. In this section we
+shed some light on the relationship between the two.
+As stated before when using binary detectors the detec-
+tion outcomes are of the form nbin = (n1, ..., nM) where
+ni ∈ {0, 1} ∀i and ni = 1 indicates the ith detector de-
+tected one or more photons. The probability of detect-
+ing a detection outcome with binary detectors is charac-
+terized by a matrix function called the Torontonian, to
+which the same arguments for classical intractability as
+for the Hafnian can be extended [22]. The probability of
+a given binary detection event nbin is given by
+p(nbin) = Tor(Onbin)
+�
+det(Q)
+= Tor(X ˜Anbin)
+�
+det(Q)
+(9)
+where Q is defined in Eq. (A2), O = I − Q−1, and Tor()
+is the Torontonian of a 2N × 2N matrix A defined as
+Tor(A) =
+�
+Z∈P ([N])
+(−1)|Z|
+1
+�
+det(I − AZ)
+.
+(10)
+Where P([N]) is the power set, the set of all possible
+subsets, of the set [N] = {1, 2, ..., N}. The probability of
+a PNR detection event n can be written in terms of the
+matrix O as
+p(n) =
+1
+�
+det(Q)
+Haf( ˜An)
+n!
+=
+1
+�
+det(Q)
+Haf(XOn)
+n!
+(11)
+The probability of a binary GBS detection event is
+simply the sum of all probabilities of the corresponding
+PNR detection events.
+A useful example to illustrate
+this is a 4-mode Gaussian boson sampler programmed
+according to some adjacency matrix A of a graph G. Sup-
+pose we use binary detectors and measure the detection
+event nbin = (1, 0, 1, 0).
+The corresponding detection
+events when using PNR detectors would be of the form
+n = (n1, 0, n3, 0) where n1, n3 > 0. We will define N to
+be the set of all possible 4-mode PNR detection events
+with 0’s in the 2nd and 4th index, i.e. only the 2nd and
+4th detectors detect no photons. From this we have
+p((1, 0, 1, 0)) = Tor(X ˜A(1,0,1,0))
+�
+det(Q)
+=
+�
+n∈N
+p(n) =
+�
+n∈N
+Haf2(An)
+n!
+�
+det(Q)
+.
+(12)
+This means the Torontonian of X ˜A is proportional to an
+infinite sum of Hafnians as there are an infinite number
+of integer lists of the form (n1, 0, n3, 0) where n1, n3 > 0.
+In a real GBS experiment, however, the energy is finite
+and therefore the measured probabilities of these events
+would be equal to a finite version of this sum where all
+detection events with total photon number greater than
+some cutoff photon number vanish from the series.
+In terms of graph theory this means the probability of
+detecting nbin = (1, 0, 1, 0) is proportional to the sum of
+the squared Hafnians of all possible subgraphs of G of
+unbounded size with their 2nd and 4th vertices removed.
+But again in practice the maximum size of the subgraphs
+
+5
+will always be bounded by some maximum photon num-
+ber for a real GBS experiment. More generally we have
+p(nbin) = Tor(Onbin)
+�
+det(Q)
+= Tor(X ˜Anbin)
+�
+det(Q)
+=
+�
+n∈N
+Haf2(An)
+n!
+�
+det(Q)
+(13)
+where N is the set of all PNR events that correspond to
+the binary detection event nbin [23].
+B.
+Constructing the feature vectors
+Once the GBS device is programmed we generate S
+samples from the device. For our algorithm we use bi-
+nary detectors so each sample is a list of length M with
+entries either 0 or 1. Once we have these samples we use
+them to construct the feature vector of which we have
+two definitions based on two coarse-graining strategies.
+The first is based on what we call the µ coarse-graining
+strategy where we group together detection events that
+contain exactly i detector ‘clicks’ or ones.
+For exam-
+ple the detection events (1, 0, 0) and (0, 0, 1) would be
+grouped together since they both contain exactly 1 detec-
+tor click. These groups can also be thought of as ‘binary
+orbits’ since they contain a detection event and all its per-
+mutations. This strategy partitions the set of all binary
+detection events into a linear number of disjoint subsets
+in N. Using this strategy we can define the feature map
+as φ : G → f = (f0, f1, ..., fN) ∈ RN. Where N is the
+maximum number of detector clicks and fi =
+Si
+S with
+Si being the number of samples which contain exactly i
+ones. Equivalently this is the probability of detecting an
+event where exactly i detectors detect a photon.
+The second feature map is based on what we call the
+ν coarse-graining strategy. For a 5 mode boson sampler
+utilizing binary detectors with maximum click number
+5 there are |Ω| = 32 possible detection outcomes. This
+coarse-graining strategy groups together detection events
+whose first 5 modes are one of these 32 outcomes. For
+example the detection event nbin = (0, 1, 0, 0, 1, 0, 1) be-
+longs in the group associated with the detection event
+(0, 1, 0, 0, 1) since they are equal if one is only concerned
+with the first 5 modes.
+This strategy partitions the
+set of all detection events of 5 or more modes into a
+constant number of subsets, i.e. 32. The feature map
+based on this strategy is defined as φ : G → f =
+(f[0,0,0,0,0], f[1,0,0,0,0], ..., fn) ∈ R32. Where fn is the prob-
+ability of detecting an event where the first 5 modes cor-
+respond to one of the 32 possible detection outcomes. For
+example f[1,0,0,0,0] is the probability that the first detec-
+tor detects photons and the following 4 detectors detect
+vacuum.
+Once we construct the feature vector for each graph in
+the data set we input them to a machine learning classi-
+fier such as a support vector machine.
+C.
+Numerical experiments
+We used The Walrus python library to classically sam-
+ple from the GBS output distribution when running our
+experiments and the GraKel python library to fetch the
+data sets and simulate the classical graph kernels [24, 25].
+Classically sampling from a GBS output distribution is
+very time intensive even when using binary detectors so
+we choose to follow the choice made in [3] and discard
+graphs with greater than 25 and less than 6 vertices for
+each data set. Before sampling from the GBS device we
+have four parameters we can set: the maximum number
+of detector clicks allowed N, the average photon number
+¯n, the displacement on each mode of the GBS device d
+and lastly the number of samples generated by the GBS
+device S. We set N = 6, ¯n = 5 and d = 0 for our re-
+sults reported here leading to probability distribution of
+32 outcomes using the ν coarse-graining strategy and 7
+outcomes using the µ coarse-graining strategy. Using Eq.
+1 with δ = 0.01 and ϵ = 0.06 we require about S = 15000
+samples for the ν feature vectors and about S = 6000
+samples for the µ feature vectors.
+For the machine learning classifier we use a support
+vector machine with an RBF kernel κrbf. We obtain the
+accuracies in Table II by running a double 10-fold cross-
+validation 10 times. The inner fold performs a grid search
+through the discrete set of values [10−4, 10−3, ..., 102, 103]
+on the C hyper-parameter of the SVM which controls
+the penalty on misclassifications. We tested our graph
+kernel on the same data sets used in [3]. We also ignored
+vertex labels, vertex attributes and edge attributes and
+converted all adjacency matrices to be unweighted.
+Four classical graph kernels were used as a bench-
+mark for our algorithms classification accuracy.
+The
+subgraph matching kernel (SM) with time complexity
+O(kM k+1) where M is the number of vertices and k the
+size of the subgraphs being considered [26], the graphlet
+sampling kernel (GS) with worst case time complexity
+O(M k) which can be optimized to O(Mdk−1) for graphs
+of bounded degree with the restriction that k ∈ {3, 4, 5},
+where k is the graphlet size and d is the maximum de-
+gree of the graph [27], the random walk kernel (RW) with
+time complexity O(M 3) [28] and the shortest path kernel
+(SP) with time complexity O(M 4) [29]. For the graphlet
+sampling kernel we set maximum graphlet size to k = 5
+and draw 5174 samples, for the random walk kernel we
+use fast computation and a geometric kernel type with
+the decay factor set to λ = 10−3, for the subgraph match-
+ing kernel we set maximum subgraph size to k = 5 and
+for the shortest path kernel we used the Floyd–Warshall
+algorithm to calculate shortest paths. The accuracies of
+all four classical kernels, the original GBS graph kernels
+from [3] with n = 6 and our kernel are shown in Table II.
+
+6
+TABLE I: Graph data set statistics after prepossessing. A more detailed description of these data sets can be found
+in appendix B of [3].
+Data set
+# of graphs # of classes avg. # of vertices avg. # of edges
+AIDS
+1723
+2
+11.11
+11.29
+BZR MD
+257
+2
+20.10
+197.69
+COX2 MD
+118
+2
+23.90
+274.40
+ENZYMES
+204
+6
+18.56
+36.30
+ER MD
+357
+2
+19.27
+185.15
+FINGERPRINT
+1080
+3
+10.58
+9.10
+IMDB-BINARY
+806
+2
+15.98
+63.32
+MUTAG
+179
+2
+17.48
+19.23
+NCI1
+1853
+2
+19.77
+21.27
+PROTEINS
+515
+2
+15.77
+29.37
+PTC FM
+284
+2
+13.64
+13.99
+TABLE II: Average test accuracies of the support vector machine with different data sets and graph kernels. The
+error reported is the standard deviation between 10 repeats of double cross validation. GS, RW, SM and SP refer to
+the graphlet sampling, random walk, subgraph matching and shortest path kernels respectively. GBSbin
+ν
+and GBSbin
+µ
+denotes our GBS kernel with binary detectors that use the ν and µ coarse-graining strategies to construct the
+feature vectors respectively. GBSbin+
+ν
+denotes that the feature associated with detecting vacuum [0, 0, 0, 0, 0] in the
+first 5 modes was dropped from all feature vectors. GBSPNR and GBSPNR+ refer to the original GBS kernels with
+PNR detectors that use orbit and meta-orbit probabilities as features respectively with a displacement of d on each
+mode. *Runtime > 7 days
+Data set
+GBSbin+
+ν
+GBSbin
+ν
+GBSbin
+µ
+GS
+RW
+SM
+SP
+AIDS
+98.47(0.10)
+98.74(0.20)
+99.53(0.05)
+99.30(0.07)
+53.11(11.90)
+77.85(2.44)
+99.34(0.09)
+BZR MD
+60.14(1.28)
+61.73(0.89)
+58.79(1.17)
+51.42(3.51)
+64.54(0.36)
+time out*
+50.82(1.76)
+COX2 MD
+51.62(2.76)
+50.18(2.96)
+51.30(3.86)
+49.01(3.18)
+48.98(4.78)
+time out*
+48.11(4.30)
+ENZYMES
+48.10(1.18)
+41.75(2.35)
+19.83(1.43)
+34.59(2.54)
+19.50(2.29)
+37.38(1.60)
+22.15(1.88)
+ER MD
+67.74(0.94)
+69.19(0.33)
+68.84(0.50)
+48.88(4.53)
+70.32(0.02)
+time out*
+45.23(4.35)
+FINGERPRINT
+64.45(0.78)
+65.53(0.86)
+63.56(0.67)
+65.25(1.30)
+33.63(3.57)
+46.89(0.56)
+46.22(1.02)
+IMDB-BINARY
+60.69(0.84)
+61.35(0.98)
+67.34(0.38)
+68.49(0.63)
+67.78(0.38)
+time out*
+65.50(0.27)
+MUTAG
+84.63(0.91)
+85.94(0.98)
+81.37(0.90)
+80.80(0.91)
+83.22(0.04)
+83.24(1.27)
+82.74(1.65)
+NCI1
+63.45(0.57)
+56.99(1.69)
+59.09(1.02)
+50.34(3.22)
+50.96(3.58)
+time out*
+53.40(2.25)
+PROTEINS
+65.95(1.03)
+63.38(0.73)
+63.11(0.55)
+65.75(0.94)
+56.91(1.39)
+62.93(0.83)
+63.63(0.41)
+PTC FM
+52.63(3.95)
+57.47(2.72)
+59.17(1.58)
+60.74(1.48)
+50.95(3.68)
+56.36(2.66)
+55.38(4.04)
+Data set
+GBSPNR (d = 0)
+GBSPNR (d = 0.25)
+GBSPNR+ (d = 0)
+GBSPNR+ (d = 0.25)
+AIDS
+99.60(0.05)
+99.62(0.03)
+99.58(0.06)
+99.61(0.05)
+BZR MD
+62.73(0.71)
+62.13(1.44)
+62.01(1.43)
+63.16(2.11)
+COX2 MD
+44.98(1.80)
+50.11(0.97)
+57.84(4.04)
+57.89(2.62)
+ENZYMES
+22.29(1.60)
+28.01(1.83)
+25.72(2.60)
+40.42(2.02)
+ER MD
+70.36(0.78)
+70.41(0.47)
+71.01(1.26)
+71.05(0.83)
+FINGERPRINT
+65.42(0.49)
+65.85(0.36)
+66.19(00.84)
+66.26(4.29)
+IMDB-BINARY
+64.09(0.34)
+68.71(0.59)
+68.14(0.71)
+67.60(0.75)
+MUTAG
+86.41(0.33)
+85.58(0.59)
+85.64(0.78)
+84.46(0.44)
+NCI1
+63.61(0.00)
+62.79(0.00)
+63.59(0.17)
+63.11(0.93)
+PROTEINS
+66.88(0.22)
+66.14(0.48)
+65.73(0.69)
+66.16(0.76)
+PTC FM
+53.84(0.96)
+52.45(1.78)
+59.14(1.72)
+56.25(2.04)
+D.
+Complexity analysis
+In this section we discuss, in addition to the time and
+space complexity, the sample complexity of our algo-
+rithm.
+1.
+Sample Complexity
+Since the ni’s for binary detection events can be either
+0 or 1 we can think of the detection outcomes as binary
+strings of length M with at most M ones. The number
+of binary strings of length M with exactly i ones is
+�M
+i
+�
+.
+So the number of possible binary detection events, the
+number of binary strings of length M with at most M
+
+7
+ones, is given by
+|Ω| =
+M
+�
+i=0
+�M
+i
+�
+.
+(14)
+We can show this function grows like 2M using the bino-
+mial expansion
+2M = (1 + 1)M =
+M
+�
+i=0
+�M
+i
+�
+1M−i1i =
+M
+�
+i=0
+�M
+i
+�
+.
+(15)
+Therefore we could not simply use the probability of the
+individual detection events as features without coarse-
+graining even when using binary detectors as we would
+still need a prohibitively large number of samples to ap-
+proximate their probabilities to within a constant error.
+This was the reason for introducing the ν and µ coarse-
+graining strategies.
+Since the number of outcomes of the µ distribution
+scales linearly with N which is ≤ M the sample com-
+plexity of approximating the µ coarse-grained probability
+distribution is
+O
+�M + ln( 1
+δ )
+ϵ2
+�
+(16)
+which reduces to O(M) for constant ϵ and δ. The sample
+complexity of approximating the ν coarse-grained prob-
+ability distribution is
+O
+�32 + ln( 1
+δ )
+ϵ2
+�
+(17)
+which reduces to O(1) for constant ϵ and δ.
+2.
+Space Complexity
+The size of the ν feature vectors is constant with re-
+spect to the graph size so the space required is O(1) and
+for the µ feature vectors the size grows linearly with N
+which is ≤ M so the space required is O(M).
+How-
+ever storing the adjacency matrix of the graphs requires
+O(M 2) space complexity.
+3.
+Time Complexity
+The time complexity is determined by the most com-
+putationally time intensive step of the algorithm which
+is encoding the adjacency matrix into the GBS device.
+This is the case because the encoding process requires
+taking the Takagi decomposition of the matrix A which
+for a M × M matrix has time complexity O(M 3) as it
+is a special case of the singular value decomposition [30].
+However there do exist quantum algorithms for comput-
+ing the singular value decomposition of a matrix with
+complexity that is polylogarithmic in the size of the ma-
+trix [31]. In particular the quantum singular value esti-
+mation algorithm for a m × n matrix presented in [32]
+has complexity O(polylog(mn)/ϵ) where ϵ is an additive
+error.
+E.
+Feature analysis & comparison to classical
+kernels
+Fig. 4 shows the results of performing a principal com-
+ponent analysis on the feature vectors generated using
+the ν coarse-graining strategy for various datasets. The
+analysis shows that the feature associated with vacuum
+[0, 0, 0, 0, 0] contributes by far the most in the support of
+the first principal component. The analysis also suggests
+that in some cases the first 10 or so features contribute
+the most to the support of all of the first four principal
+components but in other cases, such as with FINGER-
+PRINT, most features contribute more or less equally.
+Our graph kernel has a time complexity that is equiva-
+lent to the random walk kernel and better than the short-
+est path kernel by a factor of M while outperforming both
+on most data sets. Furthermore the time complexity of
+our kernel is not exponential in the size of the subgraphs
+we are probing like the subgraph matching kernel. The
+graphlet sampling kernel does have a more favorable com-
+plexity of O(Mdk−1) for graphs with maximum degree
+d. However it’s important to note that many real world
+graphs are what are called ‘scale-free networks’ and from
+the network science literature [33] the maximum degree
+of these graphs grows polynomially with the graph size.
+Therefore it is possible that the the maximum degree
+of these graphs grows linearly with the graph size i.g.
+d ∈ O(M) which would lead to a complexity of O(M k)
+for the graphlet sampling kernel. What is also interesting
+is that GBS kernels seems to provide more distinguish-
+ing power than some classical kernels for graphs with no
+vertex and edge labels like those used in our simulations.
+Take for example the ENZYMES dataset for which the
+binary GBS kernel achieves a classification accuracy of
+≈ 48% while the shortest path kernel reaches about 23%.
+If we instead choose to not ignore vertex labels we found
+the shortest path kernel gives a classification accuracy of
+about 50%. Since the GBS features are related to Haf-
+nians this suggests that features related to the number
+of perfect matchings of a graph could be more useful for
+distinguishing graphs of different classes when one has no
+information about the attributes of the graph nodes.
+IV.
+CONCLUSION
+We proposed a variation of an algorithm for the ma-
+chine learning task of classification with graph-structured
+data that uses a Gaussian boson sampler utilizing only
+binary detectors. We show that our algorithm out per-
+forms four classical graph kernels for the task of graph
+
+8
+FIG. 4: Results of the principal component analysis (PCA) on the ν feature vector entries for the ENZYMES,
+MUTAG, IMDB BINARY and FINGERPRINT datasets. The heatmaps show the weight/coefficient associated with
+each feature with regard to the first four principal components.
+classification on many data sets. This is most evident
+with regard to the ENZYMES data set where the ν fea-
+ture map outperforms all methods. The feature corre-
+sponding to detecting vacuum in the first 5 modes plays
+a particularity important role as shown by the princi-
+pal component analysis as it is related to the Hafnian of
+all possible subgraphs of G with their first 5 vertices re-
+moved. We also show that it is sample efficient, a major
+issue for applications of GBS, and has a time complexity
+that is comparable with the classical strategies.
+The fact that a GBS kernel using only binary detec-
+tors produces such accuracies suggests that technologi-
+cally more feasible—binary detectors such as SPADs do
+not operate at cryogenic temperatures such as supercon-
+ducting PNR ones—GBS devices could have useful appli-
+cations for machine learning with graph-structured data.
+We believe that GBS with PNR detectors should also be
+explored more for this application with particular atten-
+tion given to coarse-graining strategies that both reduce
+the sample complexity as well as provide features that
+capture useful information about the graphs.
+A number of questions remain open for investigation
+such as how vertex and edge labels can be encoded into
+the GBS device. Also as stated earlier it is known that
+the existence of a polynomial-time classical algorithm for
+exact sampling from the output probability distribution
+of a boson sampling or Gaussian boson sampling device
+would imply the collapse of the polynomial hierarchy to
+the third level and thus the existence of such an algorithm
+is believed to be very unlikely [34]. This result can also
+be extended to GBS with binary detectors [22]. However
+it is not known, although some work has been done in
+this area [21], if such arguments exist for algorithms that
+sample from coarse-grained versions of these probability
+distributions such as those defined in [3] or our work. It is
+vital to know if such arguments exist as they would imply
+these quantum kernels are also likely hard to simulate
+classically.
+ACKNOWLEDGMENTS
+We thank Maria Schuld, Kamil Br´adler, Scott Aaron-
+son, Ignacio Cirac, Miller Eaton, Nicol´as Quesada, An-
+drew Blance, Shreyas Murthy and Sefonias Maereg for
+useful advice and discussions. We thank Research Com-
+puting at the University of Virginia for providing access
+to, and support with, the Rivanna computing cluster.
+
+ENZYMES
+FINGERPRNT
+Feautures
+1
+1 
+01 - [0, 0, 0, 0, 0]
+02 - [1, 0, 0, 0, 0]
+5 -
+5 -
+1.00
+03 - [0. 1, 0, 0, 0
+04 - [0, 0, 1, 0, 0]
+9 -
+- 6
+05 - [0, 0, 0, 1, 0]
+tures
+eautures
+0.75
+06 - 0. 0. 0, 0, 1
+13
+ 13
+07 - [1, 1, 0, 0, 0]
+08 - [1, 0, 1, 0, 0]
+17
+17
+eau
+09 - [1, 0, 0, 1, 0]
+0.50
+10 - [1, 0, 0, 0, 1]
+21
+21
+11 - [0, 1, 1, 0, 0]
+25 -
+25 
+12 - [0, 1, 0, 1, 0
+- 0.25
+13 - [0, 1, 0, 0, 1]
+29 -
+29
+14 - [0, 0, 1, 1, 0]
+15 - [0, 0, 1, 0, 1]
+16 - [0, 0, 0, 1, 1]
+PC1
+PC2
+PC3
+PC4
+PC1
+PC2
+PC3
+PC4
+- 0.00
+MUTAG
+17 - [1, 1, 1, 0, 0]
+IMDBBINARY
+18 - [1, 1, 0, 1, 0]
+1 
+19 - [1, 1, 0, 0, 1]
+5
+-0.25
+5 -
+20 - [1, 0, 1, 1, 0]
+21 - [1, 0, 1, 0, 1]
+22 - [1, 0, 0, 1, 1
+23 - [0, 1, 1, 1, 0]
+-0.50
+es
+13
+13
+24 - [0, 1, 1, 0, 1]
+eautur
+25 - [0, 1, 0, 1, 1]
+17
+17
+26 - [0, 0, 1, 1, 1]
+-0.75
+27 - [1, 1, 1, 1, 0
+21
+E 21
+28 - [1, 1, 1, 0, 1]
+29 - [1, 1, 0, 1, 1
+25 -
+25
+30 - [1, 0, 1, 1, 1]
+-1.00
+31 - [0, 1, 1, 1, 1]
+29 -
+29 -
+32 - [1, 1, 1, 1, 1]
+PC1
+PC2
+PC3
+PC4
+PC1
+PC2
+PC3
+PC49
+This work was supported by NSF grant PHY-2112867.
+Appendix A: Reminders about standard GBS
+1.
+GBS with PNR detectors
+There has been substantial work done already on the
+connection between graph theory and Gaussain boson
+sampling with PNR detectors [18, 35, 36].
+Here we
+present the important concepts. Any undirected graph
+G with no self-loops and |V | = M vertices can be en-
+coded into a M-mode GBS setup consisting of a set of
+M squeezers followed by an interferometer of beamsplit-
+ters according to its adjacency matrix A. Once the graph
+is encoded into the GBS device the probability of detect-
+ing a specific detection event n = (n1, ..., nM) is equal
+to
+p(n) =
+1
+�
+det(Q)
+Haf( ˜An)
+n!
+=
+1
+�
+det(Q)
+Haf2(An)
+n!
+(A1)
+with
+Q = (I2M − X ˜A)−1,
+X =
+�
+0 I
+I 0
+�
+,
+(A2)
+n! = n1!×...×nM!, ˜A = (A⊕A) and Haf() denoting the
+Hafnian of a 2M × 2M matrix. The Hafnian is a matrix
+function defined mathematically as
+Haf(A) =
+�
+π∈SM
+�
+(u,v)∈π
+Au,v,
+(A3)
+where SM is the partition of the set {1, 2, ..., 2M} into
+unordered disjoint pairs.
+For example if M = 2 then
+SM = ({(1, 2), (3, 4)}, {(1, 4), (2, 3)}, {(1, 3), (2, 4)}).
+If
+A is the adjacency matrix of a unweighted graph then
+the Hafnian is equal to the number of perfect matchings
+of the vertices of the graph.
+A perfect matching is a
+partition of the vertex set of a graph into pairs such that
+each vertex is connected to exactly one edge from the
+edge set. All perfect matchings of a complete 4-vertex
+graph are shown in Fig.5.
+An is the n×n submatrix of A induced according to the
+photon detection event n. An is obtained by repeating
+the ith row and column according to the measurement
+pattern n. If ni = 0 then the ith row and column are
+deleted from A but if ni > 0 then the ith row and col-
+umn are repeated ni times. For example the probability
+of detecting the event where each mode has exactly one
+photon n = (1, 1, ..., 1) would be proportional to the Haf-
+nian of the original matrix A since An = A. What this
+means in terms of the graph is that vertex i and all its
+edges are either deleted if ni = 0 or duplicated ni times
+if ni > 0. Therefore the probability of a detection event
+n is proportional to the squared Hafnian of the subgraph
+Gn corresponding to the induced adjacency matrix An.
+1
+2
+3
+4
+(a) Complete graph of 4
+vertices
+1
+2
+3
+4
+1
+2
+3
+4
+1
+2
+3
+4
+(b) The three perfect matchings of
+the complete 4-vertex graph
+FIG. 5: The complete graph of 4 vertices and its
+corresponding perfect matching
+FIG. 6: Table of different photon detection events n
+and the corresponding subgraphs Gn they induce and
+the value of the squared Hafnians of those subgraphs.
+The probability of the detection event where each
+detector detects one photon corresponds to the Hafnian
+of the graph encoded into the GBS. We can see in the
+third graph from the top when a detector detects 2
+photons the corresponding vertex is duplicated.
+Examples of different detection events and their corre-
+sponding induced subgraphs are shown in Fig.6.
+These induced subgraphs are of even size since the
+number of photons detected is always even due to the
+fact that the inputs are squeezed states. However when
+displacement is applied to the modes of the GBS the
+probability of detecting an odd number of photons is in
+general not zero anymore and the probability of individ-
+ual detection events is characterized by the loop Hafnian
+lHaf() as opposed to the Hafnian [37, 38]. We don’t apply
+displacement for the experiments done in this paper.
+
+Gn
+Haf(An)
+n
+2
+4
+(1,1,1,1)
+3
+4
+(1, 0, 0, 1)
+1
+4
+2
+(1,1,1,2)
+0
+4
+3
+410
+2.
+Encoding a graph into a GBS device
+To map a graph to a feature vector we must first pro-
+gram the GBS device, by setting the squeezing parame-
+ters and beamsplitter angles of the device, according to
+the adjacency matrix A of the graph. Any adjacency ma-
+trix A ∈ RM×M of an undirected graph of M vertices can
+be mapped to a symmetric, positive definite 2M ×2M co-
+variance matrix Σ of a pure Gaussian state of M modes
+via the following procedure. First a doubled adjacency
+matrix ˜A is constructed,
+˜A = c
+�
+A 0
+0 A
+�
+= c(A ⊕ A),
+(A4)
+where c is a rescaling constant chosen such that 0 < c <
+1/λmax where λmax is the maximum singular value of
+A [3]. We use ˜A as, unlike A, it is guaranteed to map
+to a covariance matrix of a pure Gaussian state which
+is easier to prepare than a mixed one [35].
+This also
+has the advantage of allowing us to utilize the identity
+Haf(A ⊕ A) = Haf2(A) to relate ˜A to A.
+To map ˜A
+to a covariance matrix Σ we use the following matrix
+equations
+Σ = Q−I2M/2, with Q = (I2M −X ˜A)−1,
+X =
+�
+0 I
+I 0
+�
+.
+(A5)
+To program the GBS device to sample from the probabil-
+ity distribution corresponding to the covariance matrix Σ
+of the pure Gaussian state we need the unitary matrix
+U that characterizes the interferometer of the device as
+well as the squeezing parameters r1, ..., rM of each the
+M squeezers. We can obtain these values by taking the
+Takagi decomposition of A which is of the form
+A = Udiag(λ1, ..., λM)U T .
+(A6)
+The squeezing parameters are determined by the sin-
+gular values λ1, ..., λM and c via the relationship ri =
+tanh−1(cλi).
+The singular values and c also uniquely
+determine the mean photon number ¯n of the device ac-
+cording to
+¯n =
+M
+�
+i=1
+(cλi)2
+1 − (cλi)2 =
+M
+�
+i=1
+sinh2(ri).
+(A7)
+The rescaling constant c can be used to adjust ¯n as multi-
+plying A by c scales it’s singular values without changing
+the structure of the graph other than scaling all edge
+weights by c. The matrix U can be decomposed to give
+the parameters of the beamsplitter gates of the interfer-
+ometer [39].
+The GBS device, if using PNR detectors, now samples
+from the probability distribution
+p(n) =
+1
+�
+det(Q)
+Haf( ˜An)
+n!
+=
+1
+�
+det(Q)
+Haf2(An)
+n!
+. (A8)
+Appendix B: Super Exponential Growth of GBS
+Detection Events for M ∈ O(n2)
+Lemma 1.
+(n+M−1)!
+n!(M−1)! ∈ ω(
+√
+M
+√
+M) for n = ⌊
+√
+M⌋
+Proof.
+(n + M − 1)!
+n!(M − 1)!
+n=⌊
+√
+M⌋
+−−−−−−→ (⌊
+√
+M⌋ + M − 1)!
+(⌊
+√
+M⌋)!(M − 1)!
+(⌊
+√
+M⌋ + M − 1)!
+(⌊
+√
+M⌋)!(M − 1)!
+= [�⌊
+√
+M⌋
+i=1
+(M − 1 + i)](M − 1)!
+[�⌊
+√
+M⌋
+i=1
+i](M − 1)!
+= [�⌊
+√
+M⌋
+i=1
+(M − 1 + i)]
+[�⌊
+√
+M⌋
+i=1
+i]
+=
+⌊
+√
+M⌋
+�
+i=1
+[M − 1
+i
++ 1]
+>
+⌊
+√
+M⌋
+�
+i=1
+[M − 1
+⌊
+√
+M⌋
++ 1]
+= (M − 1
+⌊
+√
+M⌋
++ 1)⌊
+√
+M⌋
+= (
+M
+⌊
+√
+M⌋
++ 1 −
+1
+⌊
+√
+M⌋
+)⌊
+√
+M⌋
+≥ ⌊
+√
+M⌋
+⌊
+√
+M⌋
+Therefore (n+M−1)!
+n!(M−1)! ∈ ω(
+√
+M
+√
+M) for n = ⌊
+√
+M⌋. We
+drop the floor function for simplicity as ⌊x⌋ ∈ Θ(x).
+Appendix C: Induction Proof for
+�n
+k
+�
+∈ Θ(nk)
+Lemma 2.
+�n
+k
+�
+∈ Θ(nk)
+Proof. Base Case: k = 2
+�n
+2
+�
+= n(n − 1)
+2!
+lim
+n→∞
+n(n−1)
+2!
+n2
+= 1
+2!
+0 < 1
+2! < ∞
+∴
+�n
+2
+�
+∈ Θ(n2)
+Assume result holds up to k = ℓ
+�n
+ℓ
+�
+= n(n − 1)(n − 2) · · · (n − ℓ − 1)
+ℓ!
+∈ Θ(nℓ)
+
+11
+Inductive Step: k = ℓ + 1
+� n
+ℓ + 1
+�
+= n(n − 1)(n − 2) · · · (n − (ℓ + 2))
+(ℓ + 1)!
+lim
+n→∞
+n(n−1)(n−2)···(n−ℓ−2)
+(ℓ+1)!
+nℓ+1
+= lim
+n→∞
+n(n−1)(n−2)···(n−ℓ−1)
+ℓ!
+nℓ
+(n−ℓ−2)
+ℓ+1
+n
+= lim
+n→∞
+n(n−1)(n−2)···(n−ℓ−1)
+ℓ!
+nℓ
+lim
+n→∞
+(n−ℓ−2)
+ℓ+1
+n
+≡ 1
+ℓ!
+1
+ℓ + 1
+=
+1
+(ℓ + 1)!
+0 <
+1
+(ℓ + 1)! < ∞
+∴
+� n
+ℓ + 1
+�
+∈ Θ(nℓ+1)
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diff --git a/89AzT4oBgHgl3EQfSfsp/content/tmp_files/load_file.txt b/89AzT4oBgHgl3EQfSfsp/content/tmp_files/load_file.txt
new file mode 100644
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@@ -0,0 +1,1058 @@
+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQfSfsp/content/2301.01232v1.pdf,len=1057
+page_content='Sample efficient graph classification using binary Gaussian boson sampling  Amanuel Anteneh∗  Department of Computer Science, University of Virginia, Charlottesville, Virginia 22903, USA†  Olivier Pfister‡  Department of Physics, University of Virginia, Charlottesville, Virginia 22903, USA  (Dated: January 4, 2023)  We present a variation of a quantum algorithm for the machine learning task of classification with  graph-structured data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQfSfsp/content/2301.01232v1.pdf'}
+page_content=' The algorithm implements a feature extraction strategy that is based on  Gaussian boson sampling (GBS) a near term model of quantum computing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQfSfsp/content/2301.01232v1.pdf'}
+page_content=' However, unlike the  currently proposed algorithms for this problem, our GBS setup only requires binary (light/no light)  detectors, as opposed to photon number resolving detectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQfSfsp/content/2301.01232v1.pdf'}
+page_content=' These detectors are technologically  simpler and can operate at room temperature, making our algorithm less complex and less costly  to implement on the physical hardware.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQfSfsp/content/2301.01232v1.pdf'}
+page_content=' We also investigate the connection between graph theory  and the matrix function called the Torontonian which characterizes the probabilities of binary GBS  detection events.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQfSfsp/content/2301.01232v1.pdf'}
+page_content='  I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQfSfsp/content/2301.01232v1.pdf'}
+page_content='  INTRODUCTION  Graphs are one of the most versatile data structures  used in computing, and developing machine learning  methods for working with graph-structured data has  been a growing sub-field of machine learning research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQfSfsp/content/2301.01232v1.pdf'}
+page_content='  Graph classification, in particular, has useful applica-  tions in fields such as bioinformatics, network science  and computer vision as many of the objects studied in  these fields can easily be represented as graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQfSfsp/content/2301.01232v1.pdf'}
+page_content=' How-  ever, using graph-structured data with machine learning  models is not a straightforward task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQfSfsp/content/2301.01232v1.pdf'}
+page_content='  This is because  one of the most common ways of representing a graph  for computational applications, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQfSfsp/content/2301.01232v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQfSfsp/content/2301.01232v1.pdf'}
+page_content=', as an adjacency ma-  trix, cannot be easily used as an input to machine learn-  ing classifiers which primarily take vector-valued data as  their inputs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQfSfsp/content/2301.01232v1.pdf'}
+page_content=' Therefore, a common way of working with  graph-structured data is by defining a feature map φ that  maps a graph G to a vector in a Hilbert space called a  feature space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQfSfsp/content/2301.01232v1.pdf'}
+page_content=' From there a function κ, called a kernel,  is defined that measures the similarity of two graphs in  the feature space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQfSfsp/content/2301.01232v1.pdf'}
+page_content='  An example of a feature map from  R2 → R3 is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQfSfsp/content/2301.01232v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQfSfsp/content/2301.01232v1.pdf'}
+page_content='  Kernel methods refer to machine learning algorithms  that learn by comparing pairs of data points using this  similarity measure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQfSfsp/content/2301.01232v1.pdf'}
+page_content=' In our context we have a set of graphs  G and we call a kernel κ a graph kernel if it is a function  of the form κ : G × G → R [1, 2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQfSfsp/content/2301.01232v1.pdf'}
+page_content=' The most common  example of a kernel function is the feature space’s inner  product κ(x, x′) = ⟨φ(x), φ(x′)⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQfSfsp/content/2301.01232v1.pdf'}
+page_content=' The goal of such meth-  ods is to construct mappings to feature vectors whose en-  tries (the features) relate to relevant information about  the graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQfSfsp/content/2301.01232v1.pdf'}
+page_content=' Using a Gaussian boson sampling (GBS) de-  vice to construct graph kernels was an idea first proposed  ∗ asa2rc@virginia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQfSfsp/content/2301.01232v1.pdf'}
+page_content='edu  † Current address:  Booz Allen Hamilton, Arlington, Virginia  22202, USA  ‡ olivier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQfSfsp/content/2301.01232v1.pdf'}
+page_content='pfister@gmail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQfSfsp/content/2301.01232v1.pdf'}
+page_content='com  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQfSfsp/content/2301.01232v1.pdf'}
+page_content=' 1: In the original input space R2 the data points,  which belong either to the class ‘red’ or ‘blue’, are not  separable by a linear function (the decision boundary)  but after mapping the points to feature vectors in a  higher dimensional space R3 a linear function is able to  separate the two classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQfSfsp/content/2301.01232v1.pdf'}
+page_content=' This linear decision boundary  can be calculated by supervised machine learning  models such as a support vector machine.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQfSfsp/content/2301.01232v1.pdf'}
+page_content=' In our case  the input space is the set of all undirected graphs which  we denote as G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQfSfsp/content/2301.01232v1.pdf'}
+page_content='  by Schuld et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQfSfsp/content/2301.01232v1.pdf'}
+page_content=' in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQfSfsp/content/2301.01232v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQfSfsp/content/2301.01232v1.pdf'}
+page_content='  Boson sampling was first proposed by Aaronson and  Arkhipov [4] as a task—generating the photon-counting  outcomes of the “quantum Galton board” constituted by  an M ×M optical interferometer fed with single photons  into some of its input ports—that is strongly believed  to be intractable to classical computers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQfSfsp/content/2301.01232v1.pdf'}
+page_content=' The reason for  this intractability is that calculating the probability dis-  tribution for generating random outcomes using Monte  Carlo simulations requires calculating the permanent of  an M × M matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQfSfsp/content/2301.01232v1.pdf'}
+page_content=' Calculating the permanent of a gen-  eral matrix is known to be #P-complete [5] which is a  class of problems comparable to the class of NP-complete  problems in their difficulty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQfSfsp/content/2301.01232v1.pdf'}
+page_content=' Gaussian boson sampling [6]  is a variant of boson sampling in which the single-photon  inputs are replaced with single-mode squeezed states, as  produced, for example, by two-photon-emitting optical  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQfSfsp/content/2301.01232v1.pdf'}
+page_content='01232v1  [quant-ph]  3 Jan 2023  Linear Decision  Boundary  Input Space  Feature Space  d3  d22  parametric amplifiers [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQfSfsp/content/2301.01232v1.pdf'}
+page_content=' The GBS probability distribu-  tion is governed by the Hafnian of an M × M matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQfSfsp/content/2301.01232v1.pdf'}
+page_content='  Calculating the Hafnian of a general square matrix can  be reduced to the task of calculating permanents there-  fore calculating the Hafnian is also #P-complete.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQfSfsp/content/2301.01232v1.pdf'}
+page_content=' In both  cases, a quantum machine implementing boson sampling  or GBS can easily sample from these hard-to-calculate  probability distributions, just because they are “wired-  in,” and this constitutes the “quantum advantage” that  was recently demonstrated in optical experiments [8, 9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQfSfsp/content/2301.01232v1.pdf'}
+page_content='  Note also that the initial “quantum supremacy” result  obtained by Google on a superconducting qubit array [10]  was a quantum (circuit) sampling result as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQfSfsp/content/2301.01232v1.pdf'}
+page_content='  Beyond these necessary initial steps of demonstrating  that quantum hardware can indeed reach regions inacces-  sible to classical hardware, a subsequent question is that  of the utility of a sampling task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQfSfsp/content/2301.01232v1.pdf'}
+page_content=' Whereas the usefulness  of sampling in and of itself is far from established, we  know that the histograms produced by statistically sig-  nificant sampling constitute empirical probability distri-  butions that tend toward the true, classically intractable  probability distributions for sample numbers linear in  the number of possible outcomes [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQfSfsp/content/2301.01232v1.pdf'}
+page_content=' The problem is  that this very number of possible outcomes grows expo-  nentially with M in a M-qubit quantum circuit in gen-  eral [12], and exponentially or super-exponentially with  M in an M-optical-mode boson or Gaussian boson sam-  pler, which dispels any notion of quantum advantage for  calculating the corresponding quantum probability dis-  tributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQfSfsp/content/2301.01232v1.pdf'}
+page_content='  One direction that has been explored out of this co-  nundrum is the binning of GBS measurements results  into outcome classes whose cardinality scales favorably  (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQfSfsp/content/2301.01232v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQfSfsp/content/2301.01232v1.pdf'}
+page_content=' polynomially) with the problem size (the GBS mode  number).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQfSfsp/content/2301.01232v1.pdf'}
+page_content=' The immediate downside of such an approach  is loss of information it entails, which impacts usefulness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQfSfsp/content/2301.01232v1.pdf'}
+page_content='  However, graph classification using feature vectors and  coarse-graining might provide advantageous GBS appli-  cations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQfSfsp/content/2301.01232v1.pdf'}
+page_content=' This was first pointed out by Schuld et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQfSfsp/content/2301.01232v1.pdf'}
+page_content=' [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQfSfsp/content/2301.01232v1.pdf'}
+page_content='  In this paper, we show that a technologically simpler  version of GBS, which we term binary GBS, can achieve  comparable or better performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQfSfsp/content/2301.01232v1.pdf'}
+page_content=' The paper is struc-  tured as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQfSfsp/content/2301.01232v1.pdf'}
+page_content=' In Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQfSfsp/content/2301.01232v1.pdf'}
+page_content='II we give broad reminders about  GBS and graph theory (with details in Appendix A) and  the current GBS graph kernel from Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQfSfsp/content/2301.01232v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQfSfsp/content/2301.01232v1.pdf'}
+page_content='  We then  present our graph kernel in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQfSfsp/content/2301.01232v1.pdf'}
+page_content='III along with results  from numerical experiments and analyses of its complex-  ity, features and advantages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQfSfsp/content/2301.01232v1.pdf'}
+page_content='  II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQfSfsp/content/2301.01232v1.pdf'}
+page_content='  REMINDERS ABOUT GAUSSIAN BOSON  SAMPLING (GBS) AND GRAPH THEORY  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQfSfsp/content/2301.01232v1.pdf'}
+page_content='  Gaussian Boson Sampling  As mentioned above, an M-mode GBS devise com-  prises M single-mode-squeezing (SMS) inputs, an M ×M  optical interferometer, and M photon-number-resolving  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQfSfsp/content/2301.01232v1.pdf'}
+page_content=' 2: Example of a 3-mode Gaussian boson sampler.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQfSfsp/content/2301.01232v1.pdf'}
+page_content='  Mode i ∈ {1, 2, 3} starts in the vacuum state |0⟩, is then  squeezed by ˆS(ri) and passes through the network of  two beamsplitters (the interferometer) before the  number of photons in each mode is measured by the  detectors Di∈{1,2,3}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQfSfsp/content/2301.01232v1.pdf'}
+page_content='  (PNR) detectors, see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQfSfsp/content/2301.01232v1.pdf'}
+page_content='2 for an example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQfSfsp/content/2301.01232v1.pdf'}
+page_content=' The latter  have come of age in superconducting devices such as tran-  sition edge sensors [13] and superconducting nanowire  single-photon detectors [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQfSfsp/content/2301.01232v1.pdf'}
+page_content='  Both the former and the  latter have recently been used to make PNR measure-  ments of as many as 100 photons [15, 16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQfSfsp/content/2301.01232v1.pdf'}
+page_content='  An M-mode Gaussian boson sampler prepares a Gaus-  sian (Wigner function) quantum state by the M squeez-  ers and the interferometer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQfSfsp/content/2301.01232v1.pdf'}
+page_content='  The squeezers output  squeezed light into the interferometer and the photons  are then passed through the interferometer after which  the M detectors detect what modes the photons end up  in resulting in a detection event.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQfSfsp/content/2301.01232v1.pdf'}
+page_content='  We denote a detec-  tion event as n = (n1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQfSfsp/content/2301.01232v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQfSfsp/content/2301.01232v1.pdf'}
+page_content=', nM), where ni is the photon  count in the ith mode and the total number of photons  is n = �M  i=1 ni.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQfSfsp/content/2301.01232v1.pdf'}
+page_content='  We know consider binary detectors, such as single-  photon avalanche photodiodes, which are single-photon  sensitive but aren’t PNR and give the same signal how-  ever many photons were absorbed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQfSfsp/content/2301.01232v1.pdf'}
+page_content=' In this case, we have  ni ∈ {0, 1} where ni = 0 indicated zero photons were  detected in that mode and ni = 1 indicates that at least  one photon was detected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQfSfsp/content/2301.01232v1.pdf'}
+page_content=' When using binary detectors  we no longer know the total photon number n so we use  N to denote the number of detectors that detect photons  leading to �M  i=1 ni = N ≤ M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQfSfsp/content/2301.01232v1.pdf'}
+page_content='  An M-mode Gaussian state is fully described by a co-  variance matrix Σ ∈ R2M×2M and a displacement vector  d ∈ R2M [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQfSfsp/content/2301.01232v1.pdf'}
+page_content='  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQfSfsp/content/2301.01232v1.pdf'}
+page_content='  Graph theory  In this paper we define a graph G = (V, E) as a  set of vertices V  = {v1, v2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQfSfsp/content/2301.01232v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQfSfsp/content/2301.01232v1.pdf'}
+page_content='} and a set of edges  E = {(v1, v1), (v1, v2), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQfSfsp/content/2301.01232v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQfSfsp/content/2301.01232v1.pdf'}
+page_content='(vi, vj), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQfSfsp/content/2301.01232v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQfSfsp/content/2301.01232v1.pdf'}
+page_content='} that connect vertices  if the edge value is not zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQfSfsp/content/2301.01232v1.pdf'}
+page_content=' A graph can be unweighted,  with all nonzero edge weights equal to 1, or weighted, for  example with real edge weights in GBS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQfSfsp/content/2301.01232v1.pdf'}
+page_content=' For undirected  graphs, which is what we will exclusively work with in  <ol  S(r1)  D  B(p2, T2)  <ol  S(r2)  D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQfSfsp/content/2301.01232v1.pdf'}
+page_content='  B(p1, T1)  <ol  S(r3)  D3  this paper, (vi, vj) = (vj, vi), ∀i, j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQfSfsp/content/2301.01232v1.pdf'}
+page_content=' The size of a graph is  equal to the cardinality |V | of its vertex set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQfSfsp/content/2301.01232v1.pdf'}
+page_content=' The degree  of a vertex v is the number of edges that are connected to  it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQfSfsp/content/2301.01232v1.pdf'}
+page_content=' The maximum degree of a graph is the largest degree  of a vertex in its vertex set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQfSfsp/content/2301.01232v1.pdf'}
+page_content='  Graphs can be represented in a number of ways such as  a diagram, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQfSfsp/content/2301.01232v1.pdf'}
+page_content='3a, or a more computationally useful way  as an adjacency matrix, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQfSfsp/content/2301.01232v1.pdf'}
+page_content='3b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQfSfsp/content/2301.01232v1.pdf'}
+page_content=' The adjacency matrix  of an undirected graph G with |V | vertices is a |V | × |V |  symmetric matrix A with entries aij where aij is the  weight of the edge connecting vertices i and j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQfSfsp/content/2301.01232v1.pdf'}
+page_content='  1  2  3  4  3  6  4  9  (a) Undirected weighted  4-vertex graph with 4 edges  �  ���  0 4 3 0  4 0 6 9  3 6 0 0  0 9 0 0  �  ���  (b) Adjacency matrix of graph  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQfSfsp/content/2301.01232v1.pdf'}
+page_content=' 3: 4-vertex graph and its corresponding adjacency  matrix  C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQfSfsp/content/2301.01232v1.pdf'}
+page_content='  Sample complexity of GBS  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQfSfsp/content/2301.01232v1.pdf'}
+page_content='  The problem with using GBS beyond sampling  The sample complexity of a machine learning algo-  rithm refers to the number of samples or amount of data  required to learn some target function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQfSfsp/content/2301.01232v1.pdf'}
+page_content=' In the case of  GBS applications it refers to the number of samples we  need to generate from the GBS device to learn or approx-  imate a probability distribution over some set of the pho-  ton detection events.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQfSfsp/content/2301.01232v1.pdf'}
+page_content=' This complexity type is extremely  important to examine for any applications of GBS as it  could potential render certain applications of GBS in-  tractable for larger problem sizes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQfSfsp/content/2301.01232v1.pdf'}
+page_content='  For example it was shown that the GBS device uti-  lizing PNR detectors can encode the graph isomorphism  problem [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQfSfsp/content/2301.01232v1.pdf'}
+page_content=' This is done by encoding two graphs into  two GBS devices and sampling each S times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQfSfsp/content/2301.01232v1.pdf'}
+page_content=' The S sam-  ples could then be used, in principle, to reconstruct the  probability distribution over all possible detection events  n for a given M and n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQfSfsp/content/2301.01232v1.pdf'}
+page_content=' However, this cannot be done effi-  ciently enough to provide a quantum advantage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQfSfsp/content/2301.01232v1.pdf'}
+page_content=' Indeed,  we know from Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQfSfsp/content/2301.01232v1.pdf'}
+page_content=' 11, 19 that reconstructing a proba-  bility distribution D over a discrete finite set Ω of cardi-  nality |Ω| from an empirical distribution ˆD constructed  from samples from D we require  S =  �2(ln(2)|Ω| + ln( 1  δ ))  ϵ2  �  (1)  samples to guarantee that  p(||D − ˆD||1 ≥ ϵ) ≤ δ  (2)  where ||D − ˆD||1 denotes the L1 distance between D and  ˆD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQfSfsp/content/2301.01232v1.pdf'}
+page_content=' In other words we require  O  �|Ω| + ln( 1  δ )  ϵ2  �  (3)  samples to ensure with probability at most δ that the  sum of the absolute values of the errors on the empirical  probability distribution is ϵ or greater.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQfSfsp/content/2301.01232v1.pdf'}
+page_content='  This means the number of samples we need to approx-  imate a probability distribution scales linearly with the  number of elements in it’s sample space i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQfSfsp/content/2301.01232v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQfSfsp/content/2301.01232v1.pdf'}
+page_content=' the num-  ber of outcomes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQfSfsp/content/2301.01232v1.pdf'}
+page_content=' In the case where D is the probability  distribution over the set of all possible PNR detection  events the number of such events for a given number of  modes M and maximum number of photons n is  |Ω| =  �n + M − 1  M − 1  �  = (n + M − 1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQfSfsp/content/2301.01232v1.pdf'}
+page_content='  n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQfSfsp/content/2301.01232v1.pdf'}
+page_content=' (M − 1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQfSfsp/content/2301.01232v1.pdf'}
+page_content='  (4)  which in number theory is also known as the formula for  the number of weak compositions of an integer n into M  parts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQfSfsp/content/2301.01232v1.pdf'}
+page_content=' As shown in appendices B and C under the as-  sumption that the number of mode scales quadratically  with the number of photons, M ∈ O(n2), this quantity  grows super exponentially with M and in general scales  as O((n + M − 1)M−1) meaning that as the size of the  graphs increase, and therefore as the number of modes  of the GBS device increase, we require an exponential  number of samples to ensure the algorithm can give us  the correct result within a certain probability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQfSfsp/content/2301.01232v1.pdf'}
+page_content=' Therefore  while the algorithm may in principle be able to decide  graph isomorphism, it is sample inefficient to an expo-  nential degree making it intractable to implement even  with a fault tolerant quantum computer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQfSfsp/content/2301.01232v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQfSfsp/content/2301.01232v1.pdf'}
+page_content='  Coarse graining of sample distributions  However a method was suggested in [18] to coarse-grain  the probability distribution by combining outcomes into  groups called orbits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQfSfsp/content/2301.01232v1.pdf'}
+page_content=' Coarse-graining in this sense means  to construct a new probability distribution over the set  of these groups, the cardinality of which is less than the  original set of all possible detection events.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQfSfsp/content/2301.01232v1.pdf'}
+page_content=' An orbit On  consists of a detection event n and all of its permutations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQfSfsp/content/2301.01232v1.pdf'}
+page_content='  For example the orbit that contains the detection event  n = (1, 2, 2) also contains the detection events (2, 1, 2)  and (2, 2, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQfSfsp/content/2301.01232v1.pdf'}
+page_content=' The number of orbits for a 4-mode GBS  device is equal to the number of ways one can write n1 +  n2 + n3 + n4 = n, where the order of the summands does  not matter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQfSfsp/content/2301.01232v1.pdf'}
+page_content=' This is called the number of partitions of  the integer n into M parts and from the number theory  literature [20] it is known to behave asymptotically as  |Ω| ≈  eπ  �  2(n−M)  3  4  √  3(n − M), M ≤ n ≤ 2M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQfSfsp/content/2301.01232v1.pdf'}
+page_content='  (5)  4  If we assume the number of photons grows linearly with  the number of modes, n ∈ Θ(M) → n = 2M, we have  the following asymptotic bound on the number of orbits  1  4  √  3M eπ√  2M  3 ∈ O(eπ√  2M  3  M  ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQfSfsp/content/2301.01232v1.pdf'}
+page_content='  (6)  This means the number of orbits, which is now the num-  ber of outcomes |Ω| from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQfSfsp/content/2301.01232v1.pdf'}
+page_content=' 1, grows like M −1eπ√  2M/3  meaning we would have a sample complexity of  O  �  M −1eπ√  2M/3 + ln( 1  δ )  ϵ2  �  (7)  which is subexponential but still intractable for large M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQfSfsp/content/2301.01232v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQfSfsp/content/2301.01232v1.pdf'}
+page_content='  Sample complexity of previously proposed GBS graph  kernels  The first GBS based graph kernel proposed in [3]  maps a graph G to feature vectors in a feature space  φ : G → f = (f1, f2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQfSfsp/content/2301.01232v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQfSfsp/content/2301.01232v1.pdf'}
+page_content=', fD) ∈ RD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQfSfsp/content/2301.01232v1.pdf'}
+page_content=' Where fi = p(Oi  n)  is the probability of detecting a detection event from the  orbit Oi  n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQfSfsp/content/2301.01232v1.pdf'}
+page_content=' This kernel was shown to perform well against  three of the four classical kernels we use as benchmarks  in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQfSfsp/content/2301.01232v1.pdf'}
+page_content=' However a shortcoming of this method is  that the sample complexity is O(M −1eπ√  2M/3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQfSfsp/content/2301.01232v1.pdf'}
+page_content='  The second GBS kernel is of the form φ : G →  f = (f1, f2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQfSfsp/content/2301.01232v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQfSfsp/content/2301.01232v1.pdf'}
+page_content=', fD) ∈ RD, with fi = p(Mi  n,∆s) where  p(Mi  n,∆s) is the probability of detecting a detection event  that belongs to the “meta-orbit” Mi  n,∆s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQfSfsp/content/2301.01232v1.pdf'}
+page_content=' A meta-orbit  Mn,∆s is uniquely defined by a total photon number n  and ∆s which is defined as  ∆s = {n :  �  i  ni = n ∧ ∀i : ni ≤ s }.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQfSfsp/content/2301.01232v1.pdf'}
+page_content='  (8)  Therefore a meta-orbit consists of all detection events  where total photon number is equal to n, where no detec-  tor counts more than s photons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQfSfsp/content/2301.01232v1.pdf'}
+page_content=' It is claimed that this  strategy partitions the set of all PNR detection events  into a polynomial number of subsets in n [21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQfSfsp/content/2301.01232v1.pdf'}
+page_content='  III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQfSfsp/content/2301.01232v1.pdf'}
+page_content='  THE ALGORITHM  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQfSfsp/content/2301.01232v1.pdf'}
+page_content='  GBS with binary detectors and it relation to  graph theory  While the relationship between GBS with PNR de-  tectors and graph theory has been thoroughly explored,  there has been little exploration of how GBS with bi-  nary detectors fits into the picture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQfSfsp/content/2301.01232v1.pdf'}
+page_content=' In this section we  shed some light on the relationship between the two.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQfSfsp/content/2301.01232v1.pdf'}
+page_content='  As stated before when using binary detectors the detec-  tion outcomes are of the form nbin = (n1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQfSfsp/content/2301.01232v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQfSfsp/content/2301.01232v1.pdf'}
+page_content=', nM) where  ni ∈ {0, 1} ∀i and ni = 1 indicates the ith detector de-  tected one or more photons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQfSfsp/content/2301.01232v1.pdf'}
+page_content=' The probability of detect-  ing a detection outcome with binary detectors is charac-  terized by a matrix function called the Torontonian, to  which the same arguments for classical intractability as  for the Hafnian can be extended [22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQfSfsp/content/2301.01232v1.pdf'}
+page_content=' The probability of  a given binary detection event nbin is given by  p(nbin) = Tor(Onbin)  �  det(Q)  = Tor(X ˜Anbin)  �  det(Q)  (9)  where Q is defined in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQfSfsp/content/2301.01232v1.pdf'}
+page_content=' (A2), O = I − Q−1, and Tor()  is the Torontonian of a 2N × 2N matrix A defined as  Tor(A) =  �  Z∈P ([N])  (−1)|Z|  1  �  det(I − AZ)  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQfSfsp/content/2301.01232v1.pdf'}
+page_content='  (10)  Where P([N]) is the power set, the set of all possible  subsets, of the set [N] = {1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQfSfsp/content/2301.01232v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQfSfsp/content/2301.01232v1.pdf'}
+page_content=', N}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQfSfsp/content/2301.01232v1.pdf'}
+page_content=' The probability of  a PNR detection event n can be written in terms of the  matrix O as  p(n) =  1  �  det(Q)  Haf( ˜An)  n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQfSfsp/content/2301.01232v1.pdf'}
+page_content='  =  1  �  det(Q)  Haf(XOn)  n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQfSfsp/content/2301.01232v1.pdf'}
+page_content='  (11)  The probability of a binary GBS detection event is  simply the sum of all probabilities of the corresponding  PNR detection events.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQfSfsp/content/2301.01232v1.pdf'}
+page_content='  A useful example to illustrate  this is a 4-mode Gaussian boson sampler programmed  according to some adjacency matrix A of a graph G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQfSfsp/content/2301.01232v1.pdf'}
+page_content=' Sup-  pose we use binary detectors and measure the detection  event nbin = (1, 0, 1, 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQfSfsp/content/2301.01232v1.pdf'}
+page_content='  The corresponding detection  events when using PNR detectors would be of the form  n = (n1, 0, n3, 0) where n1, n3 > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQfSfsp/content/2301.01232v1.pdf'}
+page_content=' We will define N to  be the set of all possible 4-mode PNR detection events  with 0’s in the 2nd and 4th index, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQfSfsp/content/2301.01232v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQfSfsp/content/2301.01232v1.pdf'}
+page_content=' only the 2nd and  4th detectors detect no photons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQfSfsp/content/2301.01232v1.pdf'}
+page_content=' From this we have  p((1, 0, 1, 0)) = Tor(X ˜A(1,0,1,0))  �  det(Q)  =  �  n∈N  p(n) =  �  n∈N  Haf2(An)  n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQfSfsp/content/2301.01232v1.pdf'}
+page_content='  �  det(Q)  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQfSfsp/content/2301.01232v1.pdf'}
+page_content='  (12)  This means the Torontonian of X ˜A is proportional to an  infinite sum of Hafnians as there are an infinite number  of integer lists of the form (n1, 0, n3, 0) where n1, n3 > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQfSfsp/content/2301.01232v1.pdf'}
+page_content='  In a real GBS experiment, however, the energy is finite  and therefore the measured probabilities of these events  would be equal to a finite version of this sum where all  detection events with total photon number greater than  some cutoff photon number vanish from the series.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQfSfsp/content/2301.01232v1.pdf'}
+page_content='  In terms of graph theory this means the probability of  detecting nbin = (1, 0, 1, 0) is proportional to the sum of  the squared Hafnians of all possible subgraphs of G of  unbounded size with their 2nd and 4th vertices removed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQfSfsp/content/2301.01232v1.pdf'}
+page_content='  But again in practice the maximum size of the subgraphs  5  will always be bounded by some maximum photon num-  ber for a real GBS experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQfSfsp/content/2301.01232v1.pdf'}
+page_content=' More generally we have  p(nbin) = Tor(Onbin)  �  det(Q)  = Tor(X ˜Anbin)  �  det(Q)  =  �  n∈N  Haf2(An)  n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQfSfsp/content/2301.01232v1.pdf'}
+page_content='  �  det(Q)  (13)  where N is the set of all PNR events that correspond to  the binary detection event nbin [23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQfSfsp/content/2301.01232v1.pdf'}
+page_content='  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQfSfsp/content/2301.01232v1.pdf'}
+page_content='  Constructing the feature vectors  Once the GBS device is programmed we generate S  samples from the device.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQfSfsp/content/2301.01232v1.pdf'}
+page_content=' For our algorithm we use bi-  nary detectors so each sample is a list of length M with  entries either 0 or 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQfSfsp/content/2301.01232v1.pdf'}
+page_content=' Once we have these samples we use  them to construct the feature vector of which we have  two definitions based on two coarse-graining strategies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQfSfsp/content/2301.01232v1.pdf'}
+page_content='  The first is based on what we call the µ coarse-graining  strategy where we group together detection events that  contain exactly i detector ‘clicks’ or ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQfSfsp/content/2301.01232v1.pdf'}
+page_content='  For exam-  ple the detection events (1, 0, 0) and (0, 0, 1) would be  grouped together since they both contain exactly 1 detec-  tor click.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQfSfsp/content/2301.01232v1.pdf'}
+page_content=' These groups can also be thought of as ‘binary  orbits’ since they contain a detection event and all its per-  mutations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQfSfsp/content/2301.01232v1.pdf'}
+page_content=' This strategy partitions the set of all binary  detection events into a linear number of disjoint subsets  in N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQfSfsp/content/2301.01232v1.pdf'}
+page_content=' Using this strategy we can define the feature map  as φ : G → f = (f0, f1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQfSfsp/content/2301.01232v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQfSfsp/content/2301.01232v1.pdf'}
+page_content=', fN) ∈ RN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQfSfsp/content/2301.01232v1.pdf'}
+page_content=' Where N is the  maximum number of detector clicks and fi =  Si  S with  Si being the number of samples which contain exactly i  ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQfSfsp/content/2301.01232v1.pdf'}
+page_content=' Equivalently this is the probability of detecting an  event where exactly i detectors detect a photon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQfSfsp/content/2301.01232v1.pdf'}
+page_content='  The second feature map is based on what we call the  ν coarse-graining strategy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQfSfsp/content/2301.01232v1.pdf'}
+page_content=' For a 5 mode boson sampler  utilizing binary detectors with maximum click number  5 there are |Ω| = 32 possible detection outcomes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQfSfsp/content/2301.01232v1.pdf'}
+page_content=' This  coarse-graining strategy groups together detection events  whose first 5 modes are one of these 32 outcomes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQfSfsp/content/2301.01232v1.pdf'}
+page_content=' For  example the detection event nbin = (0, 1, 0, 0, 1, 0, 1) be-  longs in the group associated with the detection event  (0, 1, 0, 0, 1) since they are equal if one is only concerned  with the first 5 modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQfSfsp/content/2301.01232v1.pdf'}
+page_content='  This strategy partitions the  set of all detection events of 5 or more modes into a  constant number of subsets, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQfSfsp/content/2301.01232v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQfSfsp/content/2301.01232v1.pdf'}
+page_content=' 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQfSfsp/content/2301.01232v1.pdf'}
+page_content=' The feature map  based on this strategy is defined as φ : G → f =  (f[0,0,0,0,0], f[1,0,0,0,0], .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQfSfsp/content/2301.01232v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQfSfsp/content/2301.01232v1.pdf'}
+page_content=', fn) ∈ R32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQfSfsp/content/2301.01232v1.pdf'}
+page_content=' Where fn is the prob-  ability of detecting an event where the first 5 modes cor-  respond to one of the 32 possible detection outcomes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQfSfsp/content/2301.01232v1.pdf'}
+page_content=' For  example f[1,0,0,0,0] is the probability that the first detec-  tor detects photons and the following 4 detectors detect  vacuum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQfSfsp/content/2301.01232v1.pdf'}
+page_content='  Once we construct the feature vector for each graph in  the data set we input them to a machine learning classi-  fier such as a support vector machine.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQfSfsp/content/2301.01232v1.pdf'}
+page_content='  C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQfSfsp/content/2301.01232v1.pdf'}
+page_content='  Numerical experiments  We used The Walrus python library to classically sam-  ple from the GBS output distribution when running our  experiments and the GraKel python library to fetch the  data sets and simulate the classical graph kernels [24, 25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQfSfsp/content/2301.01232v1.pdf'}
+page_content='  Classically sampling from a GBS output distribution is  very time intensive even when using binary detectors so  we choose to follow the choice made in [3] and discard  graphs with greater than 25 and less than 6 vertices for  each data set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQfSfsp/content/2301.01232v1.pdf'}
+page_content=' Before sampling from the GBS device we  have four parameters we can set: the maximum number  of detector clicks allowed N, the average photon number  ¯n, the displacement on each mode of the GBS device d  and lastly the number of samples generated by the GBS  device S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQfSfsp/content/2301.01232v1.pdf'}
+page_content=' We set N = 6, ¯n = 5 and d = 0 for our re-  sults reported here leading to probability distribution of  32 outcomes using the ν coarse-graining strategy and 7  outcomes using the µ coarse-graining strategy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQfSfsp/content/2301.01232v1.pdf'}
+page_content=' Using Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQfSfsp/content/2301.01232v1.pdf'}
+page_content='  1 with δ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQfSfsp/content/2301.01232v1.pdf'}
+page_content='01 and ϵ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQfSfsp/content/2301.01232v1.pdf'}
+page_content='06 we require about S = 15000  samples for the ν feature vectors and about S = 6000  samples for the µ feature vectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQfSfsp/content/2301.01232v1.pdf'}
+page_content='  For the machine learning classifier we use a support  vector machine with an RBF kernel κrbf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQfSfsp/content/2301.01232v1.pdf'}
+page_content=' We obtain the  accuracies in Table II by running a double 10-fold cross-  validation 10 times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQfSfsp/content/2301.01232v1.pdf'}
+page_content=' The inner fold performs a grid search  through the discrete set of values [10−4, 10−3, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQfSfsp/content/2301.01232v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQfSfsp/content/2301.01232v1.pdf'}
+page_content=', 102, 103]  on the C hyper-parameter of the SVM which controls  the penalty on misclassifications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQfSfsp/content/2301.01232v1.pdf'}
+page_content=' We tested our graph  kernel on the same data sets used in [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQfSfsp/content/2301.01232v1.pdf'}
+page_content=' We also ignored  vertex labels, vertex attributes and edge attributes and  converted all adjacency matrices to be unweighted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQfSfsp/content/2301.01232v1.pdf'}
+page_content='  Four classical graph kernels were used as a bench-  mark for our algorithms classification accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQfSfsp/content/2301.01232v1.pdf'}
+page_content='  The  subgraph matching kernel (SM) with time complexity  O(kM k+1) where M is the number of vertices and k the  size of the subgraphs being considered [26],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQfSfsp/content/2301.01232v1.pdf'}
+page_content=' the graphlet  sampling kernel (GS) with worst case time complexity  O(M k) which can be optimized to O(Mdk−1) for graphs  of bounded degree with the restriction that k ∈ {3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQfSfsp/content/2301.01232v1.pdf'}
+page_content=' 4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQfSfsp/content/2301.01232v1.pdf'}
+page_content=' 5},' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQfSfsp/content/2301.01232v1.pdf'}
+page_content='  where k is the graphlet size and d is the maximum de-  gree of the graph [27],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQfSfsp/content/2301.01232v1.pdf'}
+page_content=' the random walk kernel (RW) with  time complexity O(M 3) [28] and the shortest path kernel  (SP) with time complexity O(M 4) [29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQfSfsp/content/2301.01232v1.pdf'}
+page_content=' For the graphlet  sampling kernel we set maximum graphlet size to k = 5  and draw 5174 samples, for the random walk kernel we  use fast computation and a geometric kernel type with  the decay factor set to λ = 10−3, for the subgraph match-  ing kernel we set maximum subgraph size to k = 5 and  for the shortest path kernel we used the Floyd–Warshall  algorithm to calculate shortest paths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQfSfsp/content/2301.01232v1.pdf'}
+page_content=' The accuracies of  all four classical kernels, the original GBS graph kernels  from [3] with n = 6 and our kernel are shown in Table II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQfSfsp/content/2301.01232v1.pdf'}
+page_content='  6  TABLE I: Graph data set statistics after prepossessing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQfSfsp/content/2301.01232v1.pdf'}
+page_content=' A more detailed description of these data sets can be found  in appendix B of [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQfSfsp/content/2301.01232v1.pdf'}
+page_content='  Data set  # of graphs # of classes avg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQfSfsp/content/2301.01232v1.pdf'}
+page_content=' # of vertices avg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQfSfsp/content/2301.01232v1.pdf'}
+page_content=' # of edges  AIDS  1723  2  11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQfSfsp/content/2301.01232v1.pdf'}
+page_content='11  11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQfSfsp/content/2301.01232v1.pdf'}
+page_content='29  BZR MD  257  2  20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQfSfsp/content/2301.01232v1.pdf'}
+page_content='10  197.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQfSfsp/content/2301.01232v1.pdf'}
+page_content='69  COX2 MD  118  2  23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQfSfsp/content/2301.01232v1.pdf'}
+page_content='90  274.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQfSfsp/content/2301.01232v1.pdf'}
+page_content='40  ENZYMES  204  6  18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQfSfsp/content/2301.01232v1.pdf'}
+page_content='56  36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQfSfsp/content/2301.01232v1.pdf'}
+page_content='30  ER MD  357  2  19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQfSfsp/content/2301.01232v1.pdf'}
+page_content='27  185.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQfSfsp/content/2301.01232v1.pdf'}
+page_content='15  FINGERPRINT  1080  3  10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQfSfsp/content/2301.01232v1.pdf'}
+page_content='58  9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQfSfsp/content/2301.01232v1.pdf'}
+page_content='10  IMDB-BINARY  806  2  15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQfSfsp/content/2301.01232v1.pdf'}
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+page_content='32  MUTAG  179  2  17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQfSfsp/content/2301.01232v1.pdf'}
+page_content='48  19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQfSfsp/content/2301.01232v1.pdf'}
+page_content='23  NCI1  1853  2  19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQfSfsp/content/2301.01232v1.pdf'}
+page_content='77  21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQfSfsp/content/2301.01232v1.pdf'}
+page_content='27  PROTEINS  515  2  15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQfSfsp/content/2301.01232v1.pdf'}
+page_content='77  29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQfSfsp/content/2301.01232v1.pdf'}
+page_content='37  PTC FM  284  2  13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQfSfsp/content/2301.01232v1.pdf'}
+page_content='64  13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQfSfsp/content/2301.01232v1.pdf'}
+page_content='99  TABLE II: Average test accuracies of the support vector machine with different data sets and graph kernels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQfSfsp/content/2301.01232v1.pdf'}
+page_content=' The  error reported is the standard deviation between 10 repeats of double cross validation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQfSfsp/content/2301.01232v1.pdf'}
+page_content=' GS, RW, SM and SP refer to  the graphlet sampling, random walk, subgraph matching and shortest path kernels respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQfSfsp/content/2301.01232v1.pdf'}
+page_content=' GBSbin  ν  and GBSbin  µ  denotes our GBS kernel with binary detectors that use the ν and µ coarse-graining strategies to construct the  feature vectors respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQfSfsp/content/2301.01232v1.pdf'}
+page_content=' GBSbin+  ν  denotes that the feature associated with detecting vacuum [0, 0, 0, 0, 0] in the  first 5 modes was dropped from all feature vectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQfSfsp/content/2301.01232v1.pdf'}
+page_content=' GBSPNR and GBSPNR+ refer to the original GBS kernels with  PNR detectors that use orbit and meta-orbit probabilities as features respectively with a displacement of d on each  mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQfSfsp/content/2301.01232v1.pdf'}
+page_content=' *Runtime > 7 days  Data set  GBSbin+  ν  GBSbin  ν  GBSbin  µ  GS  RW  SM  SP  AIDS  98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQfSfsp/content/2301.01232v1.pdf'}
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+page_content='50)  48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQfSfsp/content/2301.01232v1.pdf'}
+page_content='88(4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQfSfsp/content/2301.01232v1.pdf'}
+page_content='53)  70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQfSfsp/content/2301.01232v1.pdf'}
+page_content='32(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQfSfsp/content/2301.01232v1.pdf'}
+page_content='02)  time out*  45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQfSfsp/content/2301.01232v1.pdf'}
+page_content='23(4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQfSfsp/content/2301.01232v1.pdf'}
+page_content='35)  FINGERPRINT  64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQfSfsp/content/2301.01232v1.pdf'}
+page_content='45(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQfSfsp/content/2301.01232v1.pdf'}
+page_content='78)  65.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQfSfsp/content/2301.01232v1.pdf'}
+page_content='53(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQfSfsp/content/2301.01232v1.pdf'}
+page_content='86)  63.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQfSfsp/content/2301.01232v1.pdf'}
+page_content='56(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQfSfsp/content/2301.01232v1.pdf'}
+page_content='67)  65.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQfSfsp/content/2301.01232v1.pdf'}
+page_content='25(1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQfSfsp/content/2301.01232v1.pdf'}
+page_content='30)  33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQfSfsp/content/2301.01232v1.pdf'}
+page_content='63(3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQfSfsp/content/2301.01232v1.pdf'}
+page_content='57)  46.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQfSfsp/content/2301.01232v1.pdf'}
+page_content='89(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQfSfsp/content/2301.01232v1.pdf'}
+page_content='56)  46.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQfSfsp/content/2301.01232v1.pdf'}
+page_content='22(1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQfSfsp/content/2301.01232v1.pdf'}
+page_content='02)  IMDB-BINARY  60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQfSfsp/content/2301.01232v1.pdf'}
+page_content='69(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQfSfsp/content/2301.01232v1.pdf'}
+page_content='84)  61.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQfSfsp/content/2301.01232v1.pdf'}
+page_content='35(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQfSfsp/content/2301.01232v1.pdf'}
+page_content='98)  67.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQfSfsp/content/2301.01232v1.pdf'}
+page_content='34(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQfSfsp/content/2301.01232v1.pdf'}
+page_content='38)  68.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQfSfsp/content/2301.01232v1.pdf'}
+page_content='49(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQfSfsp/content/2301.01232v1.pdf'}
+page_content='63)  67.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQfSfsp/content/2301.01232v1.pdf'}
+page_content='78(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQfSfsp/content/2301.01232v1.pdf'}
+page_content='38)  time out*  65.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQfSfsp/content/2301.01232v1.pdf'}
+page_content='50(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQfSfsp/content/2301.01232v1.pdf'}
+page_content='27)  MUTAG  84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQfSfsp/content/2301.01232v1.pdf'}
+page_content='63(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQfSfsp/content/2301.01232v1.pdf'}
+page_content='91)  85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQfSfsp/content/2301.01232v1.pdf'}
+page_content='94(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQfSfsp/content/2301.01232v1.pdf'}
+page_content='98)  81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQfSfsp/content/2301.01232v1.pdf'}
+page_content='37(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQfSfsp/content/2301.01232v1.pdf'}
+page_content='90)  80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQfSfsp/content/2301.01232v1.pdf'}
+page_content='80(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQfSfsp/content/2301.01232v1.pdf'}
+page_content='91)  83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQfSfsp/content/2301.01232v1.pdf'}
+page_content='22(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQfSfsp/content/2301.01232v1.pdf'}
+page_content='04)  83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQfSfsp/content/2301.01232v1.pdf'}
+page_content='24(1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQfSfsp/content/2301.01232v1.pdf'}
+page_content='27)  82.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQfSfsp/content/2301.01232v1.pdf'}
+page_content='74(1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQfSfsp/content/2301.01232v1.pdf'}
+page_content='65)  NCI1  63.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQfSfsp/content/2301.01232v1.pdf'}
+page_content='45(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQfSfsp/content/2301.01232v1.pdf'}
+page_content='57)  56.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQfSfsp/content/2301.01232v1.pdf'}
+page_content='99(1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQfSfsp/content/2301.01232v1.pdf'}
+page_content='69)  59.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQfSfsp/content/2301.01232v1.pdf'}
+page_content='09(1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQfSfsp/content/2301.01232v1.pdf'}
+page_content='02)  50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQfSfsp/content/2301.01232v1.pdf'}
+page_content='34(3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQfSfsp/content/2301.01232v1.pdf'}
+page_content='22)  50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQfSfsp/content/2301.01232v1.pdf'}
+page_content='96(3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQfSfsp/content/2301.01232v1.pdf'}
+page_content='58)  time out*  53.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQfSfsp/content/2301.01232v1.pdf'}
+page_content='40(2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQfSfsp/content/2301.01232v1.pdf'}
+page_content='25)  PROTEINS  65.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQfSfsp/content/2301.01232v1.pdf'}
+page_content='95(1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQfSfsp/content/2301.01232v1.pdf'}
+page_content='03)  63.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQfSfsp/content/2301.01232v1.pdf'}
+page_content='38(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQfSfsp/content/2301.01232v1.pdf'}
+page_content='73)  63.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQfSfsp/content/2301.01232v1.pdf'}
+page_content='11(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQfSfsp/content/2301.01232v1.pdf'}
+page_content='55)  65.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQfSfsp/content/2301.01232v1.pdf'}
+page_content='75(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQfSfsp/content/2301.01232v1.pdf'}
+page_content='94)  56.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQfSfsp/content/2301.01232v1.pdf'}
+page_content='91(1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQfSfsp/content/2301.01232v1.pdf'}
+page_content='39)  62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQfSfsp/content/2301.01232v1.pdf'}
+page_content='93(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQfSfsp/content/2301.01232v1.pdf'}
+page_content='83)  63.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQfSfsp/content/2301.01232v1.pdf'}
+page_content='63(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQfSfsp/content/2301.01232v1.pdf'}
+page_content='41)  PTC FM  52.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQfSfsp/content/2301.01232v1.pdf'}
+page_content='63(3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQfSfsp/content/2301.01232v1.pdf'}
+page_content='95)  57.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQfSfsp/content/2301.01232v1.pdf'}
+page_content='47(2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQfSfsp/content/2301.01232v1.pdf'}
+page_content='72)  59.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQfSfsp/content/2301.01232v1.pdf'}
+page_content='17(1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQfSfsp/content/2301.01232v1.pdf'}
+page_content='58)  60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQfSfsp/content/2301.01232v1.pdf'}
+page_content='74(1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQfSfsp/content/2301.01232v1.pdf'}
+page_content='48)  50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQfSfsp/content/2301.01232v1.pdf'}
+page_content='95(3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQfSfsp/content/2301.01232v1.pdf'}
+page_content='68)  56.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQfSfsp/content/2301.01232v1.pdf'}
+page_content='36(2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQfSfsp/content/2301.01232v1.pdf'}
+page_content='66)  55.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQfSfsp/content/2301.01232v1.pdf'}
+page_content='38(4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQfSfsp/content/2301.01232v1.pdf'}
+page_content='04)  Data set  GBSPNR (d = 0)  GBSPNR (d = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQfSfsp/content/2301.01232v1.pdf'}
+page_content='25)  GBSPNR+ (d = 0)  GBSPNR+ (d = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQfSfsp/content/2301.01232v1.pdf'}
+page_content='25)  AIDS  99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQfSfsp/content/2301.01232v1.pdf'}
+page_content='60(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQfSfsp/content/2301.01232v1.pdf'}
+page_content='05)  99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQfSfsp/content/2301.01232v1.pdf'}
+page_content='62(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQfSfsp/content/2301.01232v1.pdf'}
+page_content='03)  99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQfSfsp/content/2301.01232v1.pdf'}
+page_content='58(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQfSfsp/content/2301.01232v1.pdf'}
+page_content='06)  99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQfSfsp/content/2301.01232v1.pdf'}
+page_content='61(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQfSfsp/content/2301.01232v1.pdf'}
+page_content='05)  BZR MD  62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQfSfsp/content/2301.01232v1.pdf'}
+page_content='73(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQfSfsp/content/2301.01232v1.pdf'}
+page_content='71)  62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQfSfsp/content/2301.01232v1.pdf'}
+page_content='13(1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQfSfsp/content/2301.01232v1.pdf'}
+page_content='44)  62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQfSfsp/content/2301.01232v1.pdf'}
+page_content='01(1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQfSfsp/content/2301.01232v1.pdf'}
+page_content='43)  63.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQfSfsp/content/2301.01232v1.pdf'}
+page_content='16(2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQfSfsp/content/2301.01232v1.pdf'}
+page_content='11)  COX2 MD  44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQfSfsp/content/2301.01232v1.pdf'}
+page_content='98(1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQfSfsp/content/2301.01232v1.pdf'}
+page_content='80)  50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQfSfsp/content/2301.01232v1.pdf'}
+page_content='11(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQfSfsp/content/2301.01232v1.pdf'}
+page_content='97)  57.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQfSfsp/content/2301.01232v1.pdf'}
+page_content='84(4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQfSfsp/content/2301.01232v1.pdf'}
+page_content='04)  57.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQfSfsp/content/2301.01232v1.pdf'}
+page_content='89(2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQfSfsp/content/2301.01232v1.pdf'}
+page_content='62)  ENZYMES  22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQfSfsp/content/2301.01232v1.pdf'}
+page_content='29(1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQfSfsp/content/2301.01232v1.pdf'}
+page_content='60)  28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQfSfsp/content/2301.01232v1.pdf'}
+page_content='01(1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQfSfsp/content/2301.01232v1.pdf'}
+page_content='83)  25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQfSfsp/content/2301.01232v1.pdf'}
+page_content='72(2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQfSfsp/content/2301.01232v1.pdf'}
+page_content='60)  40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQfSfsp/content/2301.01232v1.pdf'}
+page_content='42(2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQfSfsp/content/2301.01232v1.pdf'}
+page_content='02)  ER MD  70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQfSfsp/content/2301.01232v1.pdf'}
+page_content='36(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQfSfsp/content/2301.01232v1.pdf'}
+page_content='78)  70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQfSfsp/content/2301.01232v1.pdf'}
+page_content='41(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQfSfsp/content/2301.01232v1.pdf'}
+page_content='47)  71.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQfSfsp/content/2301.01232v1.pdf'}
+page_content='01(1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQfSfsp/content/2301.01232v1.pdf'}
+page_content='26)  71.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQfSfsp/content/2301.01232v1.pdf'}
+page_content='05(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQfSfsp/content/2301.01232v1.pdf'}
+page_content='83)  FINGERPRINT  65.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQfSfsp/content/2301.01232v1.pdf'}
+page_content='42(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQfSfsp/content/2301.01232v1.pdf'}
+page_content='49)  65.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQfSfsp/content/2301.01232v1.pdf'}
+page_content='85(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQfSfsp/content/2301.01232v1.pdf'}
+page_content='36)  66.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQfSfsp/content/2301.01232v1.pdf'}
+page_content='19(00.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQfSfsp/content/2301.01232v1.pdf'}
+page_content='84)  66.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQfSfsp/content/2301.01232v1.pdf'}
+page_content='26(4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQfSfsp/content/2301.01232v1.pdf'}
+page_content='29)  IMDB-BINARY  64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQfSfsp/content/2301.01232v1.pdf'}
+page_content='09(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQfSfsp/content/2301.01232v1.pdf'}
+page_content='34)  68.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQfSfsp/content/2301.01232v1.pdf'}
+page_content='71(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQfSfsp/content/2301.01232v1.pdf'}
+page_content='59)  68.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQfSfsp/content/2301.01232v1.pdf'}
+page_content='14(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQfSfsp/content/2301.01232v1.pdf'}
+page_content='71)  67.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQfSfsp/content/2301.01232v1.pdf'}
+page_content='60(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQfSfsp/content/2301.01232v1.pdf'}
+page_content='75)  MUTAG  86.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQfSfsp/content/2301.01232v1.pdf'}
+page_content='41(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQfSfsp/content/2301.01232v1.pdf'}
+page_content='33)  85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQfSfsp/content/2301.01232v1.pdf'}
+page_content='58(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQfSfsp/content/2301.01232v1.pdf'}
+page_content='59)  85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQfSfsp/content/2301.01232v1.pdf'}
+page_content='64(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQfSfsp/content/2301.01232v1.pdf'}
+page_content='78)  84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQfSfsp/content/2301.01232v1.pdf'}
+page_content='46(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQfSfsp/content/2301.01232v1.pdf'}
+page_content='44)  NCI1  63.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQfSfsp/content/2301.01232v1.pdf'}
+page_content='61(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQfSfsp/content/2301.01232v1.pdf'}
+page_content='00)  62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQfSfsp/content/2301.01232v1.pdf'}
+page_content='79(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQfSfsp/content/2301.01232v1.pdf'}
+page_content='00)  63.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQfSfsp/content/2301.01232v1.pdf'}
+page_content='59(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQfSfsp/content/2301.01232v1.pdf'}
+page_content='17)  63.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQfSfsp/content/2301.01232v1.pdf'}
+page_content='11(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQfSfsp/content/2301.01232v1.pdf'}
+page_content='93)  PROTEINS  66.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQfSfsp/content/2301.01232v1.pdf'}
+page_content='88(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQfSfsp/content/2301.01232v1.pdf'}
+page_content='22)  66.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQfSfsp/content/2301.01232v1.pdf'}
+page_content='14(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQfSfsp/content/2301.01232v1.pdf'}
+page_content='48)  65.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQfSfsp/content/2301.01232v1.pdf'}
+page_content='73(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQfSfsp/content/2301.01232v1.pdf'}
+page_content='69)  66.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQfSfsp/content/2301.01232v1.pdf'}
+page_content='16(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQfSfsp/content/2301.01232v1.pdf'}
+page_content='76)  PTC FM  53.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQfSfsp/content/2301.01232v1.pdf'}
+page_content='84(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQfSfsp/content/2301.01232v1.pdf'}
+page_content='96)  52.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQfSfsp/content/2301.01232v1.pdf'}
+page_content='45(1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQfSfsp/content/2301.01232v1.pdf'}
+page_content='78)  59.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQfSfsp/content/2301.01232v1.pdf'}
+page_content='14(1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQfSfsp/content/2301.01232v1.pdf'}
+page_content='72)  56.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQfSfsp/content/2301.01232v1.pdf'}
+page_content='25(2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQfSfsp/content/2301.01232v1.pdf'}
+page_content='04)  D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQfSfsp/content/2301.01232v1.pdf'}
+page_content='  Complexity analysis  In this section we discuss, in addition to the time and  space complexity, the sample complexity of our algo-  rithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQfSfsp/content/2301.01232v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQfSfsp/content/2301.01232v1.pdf'}
+page_content='  Sample Complexity  Since the ni’s for binary detection events can be either  0 or 1 we can think of the detection outcomes as binary  strings of length M with at most M ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQfSfsp/content/2301.01232v1.pdf'}
+page_content=' The number  of binary strings of length M with exactly i ones is  �M  i  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQfSfsp/content/2301.01232v1.pdf'}
+page_content='  So the number of possible binary detection events, the  number of binary strings of length M with at most M  7  ones, is given by  |Ω| =  M  �  i=0  �M  i  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQfSfsp/content/2301.01232v1.pdf'}
+page_content='  (14)  We can show this function grows like 2M using the bino-  mial expansion  2M = (1 + 1)M =  M  �  i=0  �M  i  �  1M−i1i =  M  �  i=0  �M  i  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQfSfsp/content/2301.01232v1.pdf'}
+page_content='  (15)  Therefore we could not simply use the probability of the  individual detection events as features without coarse-  graining even when using binary detectors as we would  still need a prohibitively large number of samples to ap-  proximate their probabilities to within a constant error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQfSfsp/content/2301.01232v1.pdf'}
+page_content='  This was the reason for introducing the ν and µ coarse-  graining strategies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQfSfsp/content/2301.01232v1.pdf'}
+page_content='  Since the number of outcomes of the µ distribution  scales linearly with N which is ≤ M the sample com-  plexity of approximating the µ coarse-grained probability  distribution is  O  �M + ln( 1  δ )  ϵ2  �  (16)  which reduces to O(M) for constant ϵ and δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQfSfsp/content/2301.01232v1.pdf'}
+page_content=' The sample  complexity of approximating the ν coarse-grained prob-  ability distribution is  O  �32 + ln( 1  δ )  ϵ2  �  (17)  which reduces to O(1) for constant ϵ and δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQfSfsp/content/2301.01232v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQfSfsp/content/2301.01232v1.pdf'}
+page_content='  Space Complexity  The size of the ν feature vectors is constant with re-  spect to the graph size so the space required is O(1) and  for the µ feature vectors the size grows linearly with N  which is ≤ M so the space required is O(M).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQfSfsp/content/2301.01232v1.pdf'}
+page_content='  How-  ever storing the adjacency matrix of the graphs requires  O(M 2) space complexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQfSfsp/content/2301.01232v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQfSfsp/content/2301.01232v1.pdf'}
+page_content='  Time Complexity  The time complexity is determined by the most com-  putationally time intensive step of the algorithm which  is encoding the adjacency matrix into the GBS device.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQfSfsp/content/2301.01232v1.pdf'}
+page_content='  This is the case because the encoding process requires  taking the Takagi decomposition of the matrix A which  for a M × M matrix has time complexity O(M 3) as it  is a special case of the singular value decomposition [30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQfSfsp/content/2301.01232v1.pdf'}
+page_content='  However there do exist quantum algorithms for comput-  ing the singular value decomposition of a matrix with  complexity that is polylogarithmic in the size of the ma-  trix [31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQfSfsp/content/2301.01232v1.pdf'}
+page_content=' In particular the quantum singular value esti-  mation algorithm for a m × n matrix presented in [32]  has complexity O(polylog(mn)/ϵ) where ϵ is an additive  error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQfSfsp/content/2301.01232v1.pdf'}
+page_content='  E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQfSfsp/content/2301.01232v1.pdf'}
+page_content='  Feature analysis & comparison to classical  kernels  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQfSfsp/content/2301.01232v1.pdf'}
+page_content=' 4 shows the results of performing a principal com-  ponent analysis on the feature vectors generated using  the ν coarse-graining strategy for various datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQfSfsp/content/2301.01232v1.pdf'}
+page_content=' The  analysis shows that the feature associated with vacuum  [0, 0, 0, 0, 0] contributes by far the most in the support of  the first principal component.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQfSfsp/content/2301.01232v1.pdf'}
+page_content=' The analysis also suggests  that in some cases the first 10 or so features contribute  the most to the support of all of the first four principal  components but in other cases, such as with FINGER-  PRINT, most features contribute more or less equally.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQfSfsp/content/2301.01232v1.pdf'}
+page_content='  Our graph kernel has a time complexity that is equiva-  lent to the random walk kernel and better than the short-  est path kernel by a factor of M while outperforming both  on most data sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQfSfsp/content/2301.01232v1.pdf'}
+page_content=' Furthermore the time complexity of  our kernel is not exponential in the size of the subgraphs  we are probing like the subgraph matching kernel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQfSfsp/content/2301.01232v1.pdf'}
+page_content=' The  graphlet sampling kernel does have a more favorable com-  plexity of O(Mdk−1) for graphs with maximum degree  d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQfSfsp/content/2301.01232v1.pdf'}
+page_content=' However it’s important to note that many real world  graphs are what are called ‘scale-free networks’ and from  the network science literature [33] the maximum degree  of these graphs grows polynomially with the graph size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQfSfsp/content/2301.01232v1.pdf'}
+page_content='  Therefore it is possible that the the maximum degree  of these graphs grows linearly with the graph size i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQfSfsp/content/2301.01232v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQfSfsp/content/2301.01232v1.pdf'}
+page_content='  d ∈ O(M) which would lead to a complexity of O(M k)  for the graphlet sampling kernel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQfSfsp/content/2301.01232v1.pdf'}
+page_content=' What is also interesting  is that GBS kernels seems to provide more distinguish-  ing power than some classical kernels for graphs with no  vertex and edge labels like those used in our simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQfSfsp/content/2301.01232v1.pdf'}
+page_content='  Take for example the ENZYMES dataset for which the  binary GBS kernel achieves a classification accuracy of  ≈ 48% while the shortest path kernel reaches about 23%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQfSfsp/content/2301.01232v1.pdf'}
+page_content='  If we instead choose to not ignore vertex labels we found  the shortest path kernel gives a classification accuracy of  about 50%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQfSfsp/content/2301.01232v1.pdf'}
+page_content=' Since the GBS features are related to Haf-  nians this suggests that features related to the number  of perfect matchings of a graph could be more useful for  distinguishing graphs of different classes when one has no  information about the attributes of the graph nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQfSfsp/content/2301.01232v1.pdf'}
+page_content='  IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQfSfsp/content/2301.01232v1.pdf'}
+page_content='  CONCLUSION  We proposed a variation of an algorithm for the ma-  chine learning task of classification with graph-structured  data that uses a Gaussian boson sampler utilizing only  binary detectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQfSfsp/content/2301.01232v1.pdf'}
+page_content=' We show that our algorithm out per-  forms four classical graph kernels for the task of graph  8  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQfSfsp/content/2301.01232v1.pdf'}
+page_content=' 4: Results of the principal component analysis (PCA) on the ν feature vector entries for the ENZYMES,  MUTAG, IMDB BINARY and FINGERPRINT datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQfSfsp/content/2301.01232v1.pdf'}
+page_content=' The heatmaps show the weight/coefficient associated with  each feature with regard to the first four principal components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQfSfsp/content/2301.01232v1.pdf'}
+page_content='  classification on many data sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQfSfsp/content/2301.01232v1.pdf'}
+page_content=' This is most evident  with regard to the ENZYMES data set where the ν fea-  ture map outperforms all methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQfSfsp/content/2301.01232v1.pdf'}
+page_content=' The feature corre-  sponding to detecting vacuum in the first 5 modes plays  a particularity important role as shown by the princi-  pal component analysis as it is related to the Hafnian of  all possible subgraphs of G with their first 5 vertices re-  moved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQfSfsp/content/2301.01232v1.pdf'}
+page_content=' We also show that it is sample efficient, a major  issue for applications of GBS, and has a time complexity  that is comparable with the classical strategies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQfSfsp/content/2301.01232v1.pdf'}
+page_content='  The fact that a GBS kernel using only binary detec-  tors produces such accuracies suggests that technologi-  cally more feasible—binary detectors such as SPADs do  not operate at cryogenic temperatures such as supercon-  ducting PNR ones—GBS devices could have useful appli-  cations for machine learning with graph-structured data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQfSfsp/content/2301.01232v1.pdf'}
+page_content='  We believe that GBS with PNR detectors should also be  explored more for this application with particular atten-  tion given to coarse-graining strategies that both reduce  the sample complexity as well as provide features that  capture useful information about the graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQfSfsp/content/2301.01232v1.pdf'}
+page_content='  A number of questions remain open for investigation  such as how vertex and edge labels can be encoded into  the GBS device.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQfSfsp/content/2301.01232v1.pdf'}
+page_content=' Also as stated earlier it is known that  the existence of a polynomial-time classical algorithm for  exact sampling from the output probability distribution  of a boson sampling or Gaussian boson sampling device  would imply the collapse of the polynomial hierarchy to  the third level and thus the existence of such an algorithm  is believed to be very unlikely [34].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQfSfsp/content/2301.01232v1.pdf'}
+page_content=' This result can also  be extended to GBS with binary detectors [22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQfSfsp/content/2301.01232v1.pdf'}
+page_content=' However  it is not known, although some work has been done in  this area [21], if such arguments exist for algorithms that  sample from coarse-grained versions of these probability  distributions such as those defined in [3] or our work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQfSfsp/content/2301.01232v1.pdf'}
+page_content=' It is  vital to know if such arguments exist as they would imply  these quantum kernels are also likely hard to simulate  classically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQfSfsp/content/2301.01232v1.pdf'}
+page_content='  ACKNOWLEDGMENTS  We thank Maria Schuld, Kamil Br´adler, Scott Aaron-  son, Ignacio Cirac, Miller Eaton, Nicol´as Quesada, An-  drew Blance, Shreyas Murthy and Sefonias Maereg for  useful advice and discussions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQfSfsp/content/2301.01232v1.pdf'}
+page_content=' We thank Research Com-  puting at the University of Virginia for providing access  to, and support with, the Rivanna computing cluster.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQfSfsp/content/2301.01232v1.pdf'}
+page_content='  ENZYMES  FINGERPRNT  Feautures  1  1  01 - [0, 0, 0, 0, 0]  02 - [1, 0, 0, 0, 0]  5 -  5 -  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQfSfsp/content/2301.01232v1.pdf'}
+page_content='00  03 - [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQfSfsp/content/2301.01232v1.pdf'}
+page_content=' 1, 0, 0, 0  04 - [0, 0, 1, 0, 0]  9 -  6  05 - [0, 0, 0, 1, 0]  tures  eautures  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQfSfsp/content/2301.01232v1.pdf'}
+page_content='75  06 - 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQfSfsp/content/2301.01232v1.pdf'}
+page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQfSfsp/content/2301.01232v1.pdf'}
+page_content=' 0, 0, 1  13  13  07 - [1, 1, 0, 0, 0]  08 - [1, 0, 1, 0, 0]  17  17  eau  09 - [1, 0, 0, 1, 0]  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQfSfsp/content/2301.01232v1.pdf'}
+page_content='50  10 - [1, 0, 0, 0, 1]  21  21  11 - [0, 1, 1, 0, 0]  25 -  25  12 - [0, 1, 0, 1, 0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQfSfsp/content/2301.01232v1.pdf'}
+page_content='25  13 - [0, 1, 0, 0, 1]  29 -  29  14 - [0, 0, 1, 1, 0]  15 - [0, 0, 1, 0, 1]  16 - [0, 0, 0, 1, 1]  PC1  PC2  PC3  PC4  PC1  PC2  PC3  PC4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQfSfsp/content/2301.01232v1.pdf'}
+page_content='00  MUTAG  17 - [1, 1, 1, 0, 0]  IMDBBINARY  18 - [1, 1, 0, 1, 0]  1  19 - [1, 1, 0, 0, 1]  5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQfSfsp/content/2301.01232v1.pdf'}
+page_content='25  5 -  20 - [1, 0, 1, 1, 0]  21 - [1, 0, 1, 0, 1]  22 - [1, 0, 0, 1, 1  23 - [0, 1, 1, 1, 0]  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQfSfsp/content/2301.01232v1.pdf'}
+page_content='50  es  13  13  24 - [0, 1, 1, 0, 1]  eautur  25 - [0, 1, 0, 1, 1]  17  17  26 - [0, 0, 1, 1, 1]  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQfSfsp/content/2301.01232v1.pdf'}
+page_content='75  27 - [1, 1, 1, 1, 0  21  E 21  28 - [1, 1, 1, 0, 1]  29 - [1, 1, 0, 1, 1  25 -  25  30 - [1, 0, 1, 1, 1]  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQfSfsp/content/2301.01232v1.pdf'}
+page_content='00  31 - [0, 1, 1, 1, 1]  29 -  29 -  32 - [1, 1, 1, 1, 1]  PC1  PC2  PC3  PC4  PC1  PC2  PC3  PC49  This work was supported by NSF grant PHY-2112867.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQfSfsp/content/2301.01232v1.pdf'}
+page_content='  Appendix A: Reminders about standard GBS  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQfSfsp/content/2301.01232v1.pdf'}
+page_content='  GBS with PNR detectors  There has been substantial work done already on the  connection between graph theory and Gaussain boson  sampling with PNR detectors [18, 35, 36].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQfSfsp/content/2301.01232v1.pdf'}
+page_content='  Here we  present the important concepts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQfSfsp/content/2301.01232v1.pdf'}
+page_content=' Any undirected graph  G with no self-loops and |V | = M vertices can be en-  coded into a M-mode GBS setup consisting of a set of  M squeezers followed by an interferometer of beamsplit-  ters according to its adjacency matrix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQfSfsp/content/2301.01232v1.pdf'}
+page_content=' Once the graph  is encoded into the GBS device the probability of detect-  ing a specific detection event n = (n1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQfSfsp/content/2301.01232v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQfSfsp/content/2301.01232v1.pdf'}
+page_content=', nM) is equal  to  p(n) =  1  �  det(Q)  Haf( ˜An)  n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQfSfsp/content/2301.01232v1.pdf'}
+page_content='  =  1  �  det(Q)  Haf2(An)  n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQfSfsp/content/2301.01232v1.pdf'}
+page_content='  (A1)  with  Q = (I2M − X ˜A)−1,  X =  �  0 I  I 0  �  ,  (A2)  n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQfSfsp/content/2301.01232v1.pdf'}
+page_content=' = n1!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQfSfsp/content/2301.01232v1.pdf'}
+page_content='×.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQfSfsp/content/2301.01232v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQfSfsp/content/2301.01232v1.pdf'}
+page_content='×nM!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQfSfsp/content/2301.01232v1.pdf'}
+page_content=', ˜A = (A⊕A) and Haf() denoting the  Hafnian of a 2M × 2M matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQfSfsp/content/2301.01232v1.pdf'}
+page_content=' The Hafnian is a matrix  function defined mathematically as  Haf(A) =  �  π∈SM  �  (u,v)∈π  Au,v,  (A3)  where SM is the partition of the set {1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQfSfsp/content/2301.01232v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQfSfsp/content/2301.01232v1.pdf'}
+page_content=', 2M} into  unordered disjoint pairs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQfSfsp/content/2301.01232v1.pdf'}
+page_content='  For example if M = 2 then  SM = ({(1, 2), (3, 4)}, {(1, 4), (2, 3)}, {(1, 3), (2, 4)}).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQfSfsp/content/2301.01232v1.pdf'}
+page_content='  If  A is the adjacency matrix of a unweighted graph then  the Hafnian is equal to the number of perfect matchings  of the vertices of the graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQfSfsp/content/2301.01232v1.pdf'}
+page_content='  A perfect matching is a  partition of the vertex set of a graph into pairs such that  each vertex is connected to exactly one edge from the  edge set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQfSfsp/content/2301.01232v1.pdf'}
+page_content=' All perfect matchings of a complete 4-vertex  graph are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQfSfsp/content/2301.01232v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQfSfsp/content/2301.01232v1.pdf'}
+page_content='  An is the n×n submatrix of A induced according to the  photon detection event n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQfSfsp/content/2301.01232v1.pdf'}
+page_content=' An is obtained by repeating  the ith row and column according to the measurement  pattern n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQfSfsp/content/2301.01232v1.pdf'}
+page_content=' If ni = 0 then the ith row and column are  deleted from A but if ni > 0 then the ith row and col-  umn are repeated ni times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQfSfsp/content/2301.01232v1.pdf'}
+page_content=' For example the probability  of detecting the event where each mode has exactly one  photon n = (1, 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQfSfsp/content/2301.01232v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQfSfsp/content/2301.01232v1.pdf'}
+page_content=', 1) would be proportional to the Haf-  nian of the original matrix A since An = A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQfSfsp/content/2301.01232v1.pdf'}
+page_content=' What this  means in terms of the graph is that vertex i and all its  edges are either deleted if ni = 0 or duplicated ni times  if ni > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQfSfsp/content/2301.01232v1.pdf'}
+page_content=' Therefore the probability of a detection event  n is proportional to the squared Hafnian of the subgraph  Gn corresponding to the induced adjacency matrix An.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQfSfsp/content/2301.01232v1.pdf'}
+page_content='  1  2  3  4  (a) Complete graph of 4  vertices  1  2  3  4  1  2  3  4  1  2  3  4  (b) The three perfect matchings of  the complete 4-vertex graph  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQfSfsp/content/2301.01232v1.pdf'}
+page_content=' 5: The complete graph of 4 vertices and its  corresponding perfect matching  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQfSfsp/content/2301.01232v1.pdf'}
+page_content=' 6: Table of different photon detection events n  and the corresponding subgraphs Gn they induce and  the value of the squared Hafnians of those subgraphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQfSfsp/content/2301.01232v1.pdf'}
+page_content='  The probability of the detection event where each  detector detects one photon corresponds to the Hafnian  of the graph encoded into the GBS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQfSfsp/content/2301.01232v1.pdf'}
+page_content=' We can see in the  third graph from the top when a detector detects 2  photons the corresponding vertex is duplicated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQfSfsp/content/2301.01232v1.pdf'}
+page_content='  Examples of different detection events and their corre-  sponding induced subgraphs are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQfSfsp/content/2301.01232v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQfSfsp/content/2301.01232v1.pdf'}
+page_content='  These induced subgraphs are of even size since the  number of photons detected is always even due to the  fact that the inputs are squeezed states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQfSfsp/content/2301.01232v1.pdf'}
+page_content=' However when  displacement is applied to the modes of the GBS the  probability of detecting an odd number of photons is in  general not zero anymore and the probability of individ-  ual detection events is characterized by the loop Hafnian  lHaf() as opposed to the Hafnian [37, 38].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQfSfsp/content/2301.01232v1.pdf'}
+page_content=' We don’t apply  displacement for the experiments done in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQfSfsp/content/2301.01232v1.pdf'}
+page_content='  Gn  Haf(An)  n  2  4  (1,1,1,1)  3  4  (1, 0, 0, 1)  1  4  2  (1,1,1,2)  0  4  3  410  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQfSfsp/content/2301.01232v1.pdf'}
+page_content='  Encoding a graph into a GBS device  To map a graph to a feature vector we must first pro-  gram the GBS device, by setting the squeezing parame-  ters and beamsplitter angles of the device, according to  the adjacency matrix A of the graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQfSfsp/content/2301.01232v1.pdf'}
+page_content=' Any adjacency ma-  trix A ∈ RM×M of an undirected graph of M vertices can  be mapped to a symmetric, positive definite 2M ×2M co-  variance matrix Σ of a pure Gaussian state of M modes  via the following procedure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQfSfsp/content/2301.01232v1.pdf'}
+page_content=' First a doubled adjacency  matrix ˜A is constructed,  ˜A = c  �  A 0  0 A  �  = c(A ⊕ A),  (A4)  where c is a rescaling constant chosen such that 0 < c <  1/λmax where λmax is the maximum singular value of  A [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQfSfsp/content/2301.01232v1.pdf'}
+page_content=' We use ˜A as, unlike A, it is guaranteed to map  to a covariance matrix of a pure Gaussian state which  is easier to prepare than a mixed one [35].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQfSfsp/content/2301.01232v1.pdf'}
+page_content='  This also  has the advantage of allowing us to utilize the identity  Haf(A ⊕ A) = Haf2(A) to relate ˜A to A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQfSfsp/content/2301.01232v1.pdf'}
+page_content='  To map ˜A  to a covariance matrix Σ we use the following matrix  equations  Σ = Q−I2M/2, with Q = (I2M −X ˜A)−1,  X =  �  0 I  I 0  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQfSfsp/content/2301.01232v1.pdf'}
+page_content='  (A5)  To program the GBS device to sample from the probabil-  ity distribution corresponding to the covariance matrix Σ  of the pure Gaussian state we need the unitary matrix  U that characterizes the interferometer of the device as  well as the squeezing parameters r1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQfSfsp/content/2301.01232v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQfSfsp/content/2301.01232v1.pdf'}
+page_content=', rM of each the  M squeezers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQfSfsp/content/2301.01232v1.pdf'}
+page_content=' We can obtain these values by taking the  Takagi decomposition of A which is of the form  A = Udiag(λ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQfSfsp/content/2301.01232v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQfSfsp/content/2301.01232v1.pdf'}
+page_content=', λM)U T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQfSfsp/content/2301.01232v1.pdf'}
+page_content='  (A6)  The squeezing parameters are determined by the sin-  gular values λ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQfSfsp/content/2301.01232v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQfSfsp/content/2301.01232v1.pdf'}
+page_content=', λM and c via the relationship ri =  tanh−1(cλi).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQfSfsp/content/2301.01232v1.pdf'}
+page_content='  The singular values and c also uniquely  determine the mean photon number ¯n of the device ac-  cording to  ¯n =  M  �  i=1  (cλi)2  1 − (cλi)2 =  M  �  i=1  sinh2(ri).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQfSfsp/content/2301.01232v1.pdf'}
+page_content='  (A7)  The rescaling constant c can be used to adjust ¯n as multi-  plying A by c scales it’s singular values without changing  the structure of the graph other than scaling all edge  weights by c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQfSfsp/content/2301.01232v1.pdf'}
+page_content=' The matrix U can be decomposed to give  the parameters of the beamsplitter gates of the interfer-  ometer [39].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQfSfsp/content/2301.01232v1.pdf'}
+page_content='  The GBS device, if using PNR detectors, now samples  from the probability distribution  p(n) =  1  �  det(Q)  Haf( ˜An)  n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQfSfsp/content/2301.01232v1.pdf'}
+page_content='  =  1  �  det(Q)  Haf2(An)  n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQfSfsp/content/2301.01232v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQfSfsp/content/2301.01232v1.pdf'}
+page_content=' (A8)  Appendix B: Super Exponential Growth of GBS  Detection Events for M ∈ O(n2)  Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQfSfsp/content/2301.01232v1.pdf'}
+page_content='  (n+M−1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQfSfsp/content/2301.01232v1.pdf'}
+page_content='  n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQfSfsp/content/2301.01232v1.pdf'}
+page_content='(M−1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQfSfsp/content/2301.01232v1.pdf'}
+page_content=' ∈ ω(  √  M  √  M) for n = ⌊  √  M⌋  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQfSfsp/content/2301.01232v1.pdf'}
+page_content='  (n + M − 1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQfSfsp/content/2301.01232v1.pdf'}
+page_content='  n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQfSfsp/content/2301.01232v1.pdf'}
+page_content=' (M − 1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQfSfsp/content/2301.01232v1.pdf'}
+page_content='  n=⌊  √  M⌋  −−−−−−→ (⌊  √  M⌋ + M − 1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQfSfsp/content/2301.01232v1.pdf'}
+page_content='  (⌊  √  M⌋)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQfSfsp/content/2301.01232v1.pdf'}
+page_content=' (M − 1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQfSfsp/content/2301.01232v1.pdf'}
+page_content='  (⌊  √  M⌋ + M − 1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQfSfsp/content/2301.01232v1.pdf'}
+page_content='  (⌊  √  M⌋)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQfSfsp/content/2301.01232v1.pdf'}
+page_content=' (M − 1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQfSfsp/content/2301.01232v1.pdf'}
+page_content='  = [�⌊  √  M⌋  i=1  (M − 1 + i)](M − 1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQfSfsp/content/2301.01232v1.pdf'}
+page_content='  [�⌊  √  M⌋  i=1  i](M − 1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQfSfsp/content/2301.01232v1.pdf'}
+page_content='  = [�⌊  √  M⌋  i=1  (M − 1 + i)]  [�⌊  √  M⌋  i=1  i]  =  ⌊  √  M⌋  �  i=1  [M − 1  i  + 1]  >  ⌊  √  M⌋  �  i=1  [M − 1  ⌊  √  M⌋  + 1]  = (M − 1  ⌊  √  M⌋  + 1)⌊  √  M⌋  = (  M  ⌊  √  M⌋  + 1 −  1  ⌊  √  M⌋  )⌊  √  M⌋  ≥ ⌊  √  M⌋  ⌊  √  M⌋  Therefore (n+M−1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQfSfsp/content/2301.01232v1.pdf'}
+page_content='  n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQfSfsp/content/2301.01232v1.pdf'}
+page_content='(M−1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQfSfsp/content/2301.01232v1.pdf'}
+page_content=' ∈ ω(  √  M  √  M) for n = ⌊  √  M⌋.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQfSfsp/content/2301.01232v1.pdf'}
+page_content=' We  drop the floor function for simplicity as ⌊x⌋ ∈ Θ(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQfSfsp/content/2301.01232v1.pdf'}
+page_content='  Appendix C: Induction Proof for  �n  k  �  ∈ Θ(nk)  Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQfSfsp/content/2301.01232v1.pdf'}
+page_content='  �n  k  �  ∈ Θ(nk)  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQfSfsp/content/2301.01232v1.pdf'}
+page_content=' Base Case: k = 2  �n  2  �  = n(n − 1)  2!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQfSfsp/content/2301.01232v1.pdf'}
+page_content='  lim  n→∞  n(n−1)  2!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQfSfsp/content/2301.01232v1.pdf'}
+page_content='  n2  = 1  2!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQfSfsp/content/2301.01232v1.pdf'}
+page_content='  0 < 1  2!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQfSfsp/content/2301.01232v1.pdf'}
+page_content=' < ∞  ∴  �n  2  �  ∈ Θ(n2)  Assume result holds up to k = ℓ  �n  ℓ  �  = n(n − 1)(n − 2) · · · (n − ℓ − 1)  ℓ!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQfSfsp/content/2301.01232v1.pdf'}
+page_content='  ∈ Θ(nℓ)  11  Inductive Step: k = ℓ + 1  � n  ℓ + 1  �  = n(n − 1)(n − 2) · · · (n − (ℓ + 2))  (ℓ + 1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQfSfsp/content/2301.01232v1.pdf'}
+page_content='  lim  n→∞  n(n−1)(n−2)···(n−ℓ−2)  (ℓ+1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQfSfsp/content/2301.01232v1.pdf'}
+page_content='  nℓ+1  = lim  n→∞  n(n−1)(n−2)···(n−ℓ−1)  ℓ!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQfSfsp/content/2301.01232v1.pdf'}
+page_content='  nℓ  (n−ℓ−2)  ℓ+1  n  = lim  n→∞  n(n−1)(n−2)···(n−ℓ−1)  ℓ!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQfSfsp/content/2301.01232v1.pdf'}
+page_content='  nℓ  lim  n→∞  (n−ℓ−2)  ℓ+1  n  ≡ 1  ℓ!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQfSfsp/content/2301.01232v1.pdf'}
+page_content='  1  ℓ + 1  =  1  (ℓ + 1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQfSfsp/content/2301.01232v1.pdf'}
+page_content='  0 <  1  (ℓ + 1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQfSfsp/content/2301.01232v1.pdf'}
+page_content=' < ∞  ∴  � n  ℓ + 1  �  ∈ Θ(nℓ+1)  [1] G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQfSfsp/content/2301.01232v1.pdf'}
+page_content=' Nikolentzos, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQfSfsp/content/2301.01232v1.pdf'}
+page_content=' Siglidis, and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQfSfsp/content/2301.01232v1.pdf'}
+page_content=' Vazirgiannis, Graph  kernels: A survey, Journal of Artificial Intelligence Re-  search 72, 943 (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQfSfsp/content/2301.01232v1.pdf'}
+page_content='  [2] N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQfSfsp/content/2301.01232v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQfSfsp/content/2301.01232v1.pdf'}
+page_content=' Kriege, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQfSfsp/content/2301.01232v1.pdf'}
+page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQfSfsp/content/2301.01232v1.pdf'}
+page_content=' Johansson, and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQfSfsp/content/2301.01232v1.pdf'}
+page_content=' Morris, A survey  on graph kernels, Applied Network Science 5, 1 (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQfSfsp/content/2301.01232v1.pdf'}
+page_content='  [3] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQfSfsp/content/2301.01232v1.pdf'}
+page_content=' Schuld, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQfSfsp/content/2301.01232v1.pdf'}
+page_content=' Br´adler, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQfSfsp/content/2301.01232v1.pdf'}
+page_content=' Israel, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQfSfsp/content/2301.01232v1.pdf'}
+page_content=' Su, and B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQfSfsp/content/2301.01232v1.pdf'}
+page_content=' Gupt,  Measuring the similarity of graphs with a gaussian boson  sampler, Physical Review A 101, 032314 (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQfSfsp/content/2301.01232v1.pdf'}
+page_content='  [4] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQfSfsp/content/2301.01232v1.pdf'}
+page_content=' Aaronson and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQfSfsp/content/2301.01232v1.pdf'}
+page_content=' Arkhipov, The computational com-  plexity of linear optics, Electronic Colloquium on Com-  putational Complexity Report No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQfSfsp/content/2301.01232v1.pdf'}
+page_content=' 170, 1 (2010).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQfSfsp/content/2301.01232v1.pdf'}
+page_content='  [5] L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQfSfsp/content/2301.01232v1.pdf'}
+page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQfSfsp/content/2301.01232v1.pdf'}
+page_content=' Valiant, The complexity of computing the perma-  nent, Theoretical computer science 8, 189 (1979).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQfSfsp/content/2301.01232v1.pdf'}
+page_content='  [6] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQfSfsp/content/2301.01232v1.pdf'}
+page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQfSfsp/content/2301.01232v1.pdf'}
+page_content=' Hamilton, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQfSfsp/content/2301.01232v1.pdf'}
+page_content=' Kruse, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQfSfsp/content/2301.01232v1.pdf'}
+page_content=' Sansoni, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQfSfsp/content/2301.01232v1.pdf'}
+page_content=' Barkhofen,  C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQfSfsp/content/2301.01232v1.pdf'}
+page_content=' Silberhorn, and I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQfSfsp/content/2301.01232v1.pdf'}
+page_content=' Jex, Gaussian boson sampling,  Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQfSfsp/content/2301.01232v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQfSfsp/content/2301.01232v1.pdf'}
+page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQfSfsp/content/2301.01232v1.pdf'}
+page_content=' 119, 170501 (2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQfSfsp/content/2301.01232v1.pdf'}
+page_content='  [7] H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQfSfsp/content/2301.01232v1.pdf'}
+page_content='-A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQfSfsp/content/2301.01232v1.pdf'}
+page_content='  Bachor  and  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQfSfsp/content/2301.01232v1.pdf'}
+page_content='  C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQfSfsp/content/2301.01232v1.pdf'}
+page_content='  Ralph,  A Guide to Experiments in Quantum Optics,  3rd  ed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQfSfsp/content/2301.01232v1.pdf'}
+page_content=' (Wiley-VCH, 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQfSfsp/content/2301.01232v1.pdf'}
+page_content='  [8] H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQfSfsp/content/2301.01232v1.pdf'}
+page_content='-S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQfSfsp/content/2301.01232v1.pdf'}
+page_content=' Zhong, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQfSfsp/content/2301.01232v1.pdf'}
+page_content=' Wang, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQfSfsp/content/2301.01232v1.pdf'}
+page_content='-H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQfSfsp/content/2301.01232v1.pdf'}
+page_content=' Deng, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQfSfsp/content/2301.01232v1.pdf'}
+page_content='-C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQfSfsp/content/2301.01232v1.pdf'}
+page_content=' Chen, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQfSfsp/content/2301.01232v1.pdf'}
+page_content='-C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQfSfsp/content/2301.01232v1.pdf'}
+page_content='  Peng, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQfSfsp/content/2301.01232v1.pdf'}
+page_content='-H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQfSfsp/content/2301.01232v1.pdf'}
+page_content=' Luo, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQfSfsp/content/2301.01232v1.pdf'}
+page_content=' Qin, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQfSfsp/content/2301.01232v1.pdf'}
+page_content=' Wu, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQfSfsp/content/2301.01232v1.pdf'}
+page_content=' Ding, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQfSfsp/content/2301.01232v1.pdf'}
+page_content=' Hu,  P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQfSfsp/content/2301.01232v1.pdf'}
+page_content=' Hu, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQfSfsp/content/2301.01232v1.pdf'}
+page_content='-Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQfSfsp/content/2301.01232v1.pdf'}
+page_content=' Yang, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQfSfsp/content/2301.01232v1.pdf'}
+page_content='-J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQfSfsp/content/2301.01232v1.pdf'}
+page_content=' Zhang, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQfSfsp/content/2301.01232v1.pdf'}
+page_content=' Li, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQfSfsp/content/2301.01232v1.pdf'}
+page_content=' Li, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQfSfsp/content/2301.01232v1.pdf'}
+page_content=' Jiang,  L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQfSfsp/content/2301.01232v1.pdf'}
+page_content=' Gan, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQfSfsp/content/2301.01232v1.pdf'}
+page_content=' Yang, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQfSfsp/content/2301.01232v1.pdf'}
+page_content=' You, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQfSfsp/content/2301.01232v1.pdf'}
+page_content=' Wang, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQfSfsp/content/2301.01232v1.pdf'}
+page_content=' Li, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQfSfsp/content/2301.01232v1.pdf'}
+page_content='-L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQfSfsp/content/2301.01232v1.pdf'}
+page_content=' Liu,  C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQfSfsp/content/2301.01232v1.pdf'}
+page_content='-Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQfSfsp/content/2301.01232v1.pdf'}
+page_content=' Lu, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQfSfsp/content/2301.01232v1.pdf'}
+page_content='-W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQfSfsp/content/2301.01232v1.pdf'}
+page_content=' Pan, Quantum computational  advantage using photons, Science 370, 1460 (2020),  https://science.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQfSfsp/content/2301.01232v1.pdf'}
+page_content='sciencemag.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQfSfsp/content/2301.01232v1.pdf'}
+page_content='org/content/370/6523/1460.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQfSfsp/content/2301.01232v1.pdf'}
+page_content='full.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQfSfsp/content/2301.01232v1.pdf'}
+page_content='pdf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQfSfsp/content/2301.01232v1.pdf'}
+page_content='  [9] L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQfSfsp/content/2301.01232v1.pdf'}
+page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQfSfsp/content/2301.01232v1.pdf'}
+page_content=' Madsen, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQfSfsp/content/2301.01232v1.pdf'}
+page_content=' Laudenbach, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQfSfsp/content/2301.01232v1.pdf'}
+page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQfSfsp/content/2301.01232v1.pdf'}
+page_content=' Askarani, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQfSfsp/content/2301.01232v1.pdf'}
+page_content=' Rortais,  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQfSfsp/content/2301.01232v1.pdf'}
+page_content=' Vincent, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQfSfsp/content/2301.01232v1.pdf'}
+page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AzT4oBgHgl3EQfSfsp/content/2301.01232v1.pdf'}
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+Draft version January 10, 2023
+Typeset using LATEX twocolumn style in AASTeX631
+3D modeling of the molecular gas kinematics in optically-selected jellyfish galaxies
+Cecilia Bacchini
+,1 Matilde Mingozzi
+,2 Bianca M. Poggianti
+,1 Alessia Moretti
+,1
+Marco Gullieuszik
+,1 Antonino Marasco
+,1 Bernardo Cervantes Sodi
+,3 Osbaldo S´anchez-Garc´ıa
+,3
+Benedetta Vulcani
+,1 Ariel Werle
+,1 Rosita Paladino
+,4 and Mario Radovich
+1
+1INAF - Osservatorio Astronomico di Padova, vicolo dell’Osservatorio 5, IT-35122 Padova, Italy
+2Space Telescope Science Institute, 3700 San Martin Drive, Baltimore, MD 21218, USA
+3Istituto de Radioastronom´ıa y Astrof´ısica, Universidad Nacional Aut´onoma de M´exico, Campus Morelia, A.P. 3-72, C.P. 58089,
+Michoac´an, M´exico
+4INAF - Istituto di Radioastronomia, via P. Gobetti 101, I-40129 Bologna, Italy
+ABSTRACT
+Cluster galaxies are subject to the ram pressure exerted by the intracluster medium, which can
+perturb or even strip away their gas while leaving the stars unperturbed. We model the distribution
+and kinematics of the stars and the molecular gas in four late-type cluster galaxies (JO201, JO204,
+JO206, and JW100), which show tails of atomic and ionized gas indicative of ongoing ram pressure
+stripping. We analyze MUSE@VLT data and CO data from ALMA searching for signatures of radial
+gas flows, ram pressure stripping, and other perturbations. We find that all galaxies, with the possible
+exception of JW100, host stellar bars. Signatures of ram pressure are found in JO201 and JO206,
+which also shows clear indications of ongoing stripping in the molecular disk outskirts. The stripping
+affects the whole molecular gas disk of JW100. The molecular gas kinematics in JO204 is instead
+dominated by rotation rather than ram pressure. We also find indications of enhanced turbulence of
+the molecular gas compared to field galaxies. Large-scale radial flows of molecular gas are present in
+JO204 and JW100, but more uncertain in JO201 and JO206. We show that our galaxy sample follows
+the molecular gas mass-size relation, confirming that it is essentially independent of environment even
+for the most extreme cases of stripping. Our findings are consistent with the molecular gas being
+affected by ram pressure on different timescales and less severely than the atomic and ionized gas
+phases, likely because the molecular gas is denser and more gravitationally bound to the galaxy.
+1. INTRODUCTION
+In dense environments, such as groups and clusters,
+galaxies are affected by various physical mechanisms
+that can significantly influence their properties and evo-
+lution (Nulsen 1982; Boselli & Gavazzi 2006; Cortese
+et al. 2021). These processes are usually divided into
+gravitational and hydrodynamical interactions (Boselli
+et al. 2022a).
+Gravitational perturbations can be in-
+duced by tidal forces due to the potential of other clus-
+ter/group members (Merritt 1983) or the large scale
+structure itself (Byrd & Valtonen 1990), but also by fly-
+by encounters and mergers (Barnes & Hernquist 1992;
+Kronberger et al. 2006). Gravitational interactions af-
+fect both the stellar and the gaseous components of
+galaxies. Instead, the hydrodynamical interactions be-
+Corresponding author: Cecilia Bacchini
+cecilia.bacchini@inaf.it
+tween the galactic interstellar medium (ISM) and the in-
+tracluster medium (ICM) are expected to influence only
+the gaseous components of galaxies. These mechanisms
+are the thermal evaporation of the cold (T ≲ 104 K) ISM
+due to the interaction of the hot (T ≈ 107 −108 K) ICM
+(Cowie & Songaila 1977), the removal of the outer ISM
+layer due to the viscosity momentum transfer with the
+ICM (viscous stripping) or instabilities (Nulsen 1982;
+Roediger & Hensler 2008), and the ram pressure strip-
+ping, that is the removal of the ISM due to the pressure
+exerted by the ICM while a galaxy is moving through
+the cluster (Gunn & Gott 1972). In addition, the in-
+teraction with the ICM can heat up or strip away the
+hot (T ≈ 106 K) gas corona surrounding galaxies and
+prevent them from accreting new gas, finally quenching
+star formation (starvation; Larson et al. 1980).
+Ram pressure is often considered among the domi-
+nant mechanisms affecting the ISM in cluster galaxies.
+The ram pressure can be calculated as Pram = ρICMV 2
+gal,
+arXiv:2301.03090v1  [astro-ph.GA]  8 Jan 2023
+
+ID2
+Bacchini et al.
+where ρICM and Vgal are the ICM density and the galaxy
+velocity relative to the cluster (Gunn & Gott 1972).
+Hence, this mechanism is expected to be particularly
+strong for galaxies with high Vgal located close to the
+cluster center, where ρICM is the highest. The ram pres-
+sure can have different effects on the gas distribution
+and kinematics in galaxies. The compression of the gas
+disk can make it morphologically lopsided and asymmet-
+ric (e.g. Mapelli et al. 2008; Kronberger et al. 2008b).
+Depending on its direction with respect to the galaxy
+rotation, the ram pressure can decelerate one side of the
+gas disk and accelerate the other, resulting in a kinemat-
+ically lopsided disk, or even shift the kinematic center
+of the gas disk with respect to the optical center of the
+galaxy (e.g. Kronberger et al. 2008b).
+Typical signa-
+tures of ram pressure stripping are one-sided tails of gas
+extending outside the stellar disk and gas clouds that are
+spatially detached and kinematically decoupled from the
+galaxy (e.g. Chung et al. 2007; Merluzzi et al. 2013; Lee
+et al. 2017). Moreover, gas disks in cluster galaxies are
+sometimes less extended and less massive that those in
+field galaxies (Chamaraux et al. 1980; Haynes et al. 1984;
+Cayatte et al. 1990; Schr¨oder et al. 2001; Chung et al.
+2009). Both truncation and gas deficiency are proper-
+ties ascribed to ram pressure stripping, being relatively
+common in both low- (e.g. Chung et al. 2007; Gavazzi
+et al. 2018) and intermediate-redshift cluster galaxies
+(e.g. Cortese et al. 2007; Boselli et al. 2019; Moretti et al.
+2022). The efficiency of ram pressure stripping depends
+on the gas properties, being more effective on a diffuse
+medium than on dense gas clumps, and on the gravita-
+tional pull caused by the galactic potential, that weak-
+ens with increasing galactocentric distance and height
+above the midplane (e.g. Gunn & Gott 1972; Tonnesen
+& Bryan 2009; K¨oppen et al. 2018).
+The atomic gas in galaxies is relatively diffuse and
+typically distributed in a disk that is very extended (up
+to twice the stellar disk diameter; see e.g. Verheijen
+& Sancisi 2001; Wang et al. 2016; Lelli et al. 2016a)
+and thick (up to ≈1 kpc; see e.g. Olling 1996; Yim
+et al. 2011, 2014; Marasco et al. 2017; Bacchini et al.
+2019a,b, 2020b), being very susceptible to ram pressure.
+Indeed, cluster galaxies often contain less atomic gas
+than expected from their optical size or stellar mass and
+have truncated and/or asymmetric HI discs (Chama-
+raux et al. 1980; Haynes et al. 1984; Giovanelli & Haynes
+1985; Cayatte et al. 1990; Solanes et al. 2001; Schr¨oder
+et al. 2001; Waugh et al. 2002; Chung et al. 2009; Loni
+et al. 2021; Zabel et al. 2022). Morever, long tails of
+atomic gas are commonly observed in cluster galaxies
+(Bravo-Alfaro et al. 2000; Kenney et al. 2004; Chung
+et al. 2007; Scott et al. 2010; Sorgho et al. 2017; Ramat-
+soku et al. 2019, 2020; Deb et al. 2020; Healy et al. 2021;
+Deb et al. 2022).
+The molecular gas is typically denser than the atomic
+gas (e.g. Leroy et al. 2008) and its distribution is also
+less extended in both the radial (up to the stellar disk
+diameter; Davis et al. 2013; Brown et al. 2021; Zabel
+et al. 2022) and vertical (up to ≈ 0.3 kpc; Yim et al.
+2011, 2014; Marasco et al. 2017; Bacchini et al. 2019a,b)
+directions. Hence, it is expected that the molecular gas
+is more resilient to ram pressure than the atomic gas
+(e.g. Lee et al. 2017; Brown et al. 2021; Zabel et al.
+2022; Boselli et al. 2022a). Nevertheless, there is grow-
+ing observational evidence that the ram pressure actu-
+ally influences the molecular gas in cluster galaxies, as
+indicated by signatures of compression, kinematic lop-
+sidedness, and shifts between the optical and kinematic
+center (Lee et al. 2017; Zabel et al. 2019; Cramer et al.
+2020). Direct observations of molecular gas stripping by
+ram pressure are limited, but tails and blobs of molec-
+ular gas far from the stellar disk have been observed in
+some cluster galaxies (Vollmer et al. 2008; J´achym et al.
+2014, 2017; Lee et al. 2017; Moretti et al. 2018, 2020a),
+as well as truncated molecular gas disks and H2-deficient
+galaxies (Fumagalli et al. 2009; Boselli et al. 2014; Zabel
+et al. 2019, 2022; Lee et al. 2022). However, a few au-
+thors have found that cluster galaxies can also host a
+normal (both in size and mass) or even enhanced reser-
+voir of molecular gas (Fumagalli et al. 2009; Moretti
+et al. 2020b; Brown et al. 2021; Zabel et al. 2022), pos-
+sible indication that ram pressure can increase the effi-
+ciency of the HI-to-H2 conversion (Moretti et al. 2020a).
+In this work, we analyze the molecular gas distribution
+and kinematics in four cluster galaxies observed with
+the Atacama Large Millimeter Array (ALMA). These
+objects are part of the sample of 114 galaxies observed
+within the Large Program ”GAs Stripping Phenomena
+in galaxies with MUSE” (GASP), which is a survey
+carried out with integral-field Multi Unit Spectroscopic
+Explorer (MUSE) at the Very Large Telescope (VLT).
+The GASP survey aims at understanding the impact of
+environment on the evolution of galaxies by studying
+their stellar and ionized gas emission. Recently, follow-
+up programs have provided multi-wavelength observa-
+tions for a few galaxies in the GASP sample, allowing
+to study other ISM components, such as atomic gas (Ra-
+matsoku et al. 2019, 2020; Deb et al. 2020; Healy et al.
+2021; Deb et al. 2022; Luber et al. 2022), the molec-
+ular gas (Moretti et al. 2018, 2020a,b), and magnetic
+fields (M¨uller et al. 2021), and also young stellar pop-
+ulations (George et al. 2018). An unexpected result of
+the GASP project was the high fraction of active galac-
+tic nuclei (AGN) among ram pressure-stripped galaxies
+
+Molecular gas kinematics in jellyfish galaxies
+3
+(Poggianti et al. 2017a; Peluso et al. 2022; Poggianti &
+the GASP team 2022). This result was interpreted as an
+indication that ram pressure can drive gas flows towards
+the center and feed the AGN activity (e.g. Ricarte et al.
+2020).
+The galaxies analyzed in this work (hereafter
+referred as the GASP-ALMA sample) were studied by
+Poggianti et al. (2017a) and host indeed an AGN. This
+paper aims at answering the following open questions
+about the GASP-ALMA galaxies: What is the impact
+of ram pressure on the distribution and kinematics of
+the molecular gas? Can we detect inflows of molecular
+gas that may feed the AGN?
+This paper is organized as follows. Section 2 presents
+the GASP-ALMA sample and summarizes the relevant
+pieces of information obtained by previous studies. We
+describe the data and methods used to carry out the
+analysis in Sects. 3 and 4, respectively. For each galaxy,
+we present and discuss the results in Sect. 5. In Sect. 6,
+we compare our findings with other works in the litera-
+ture. Section 7 this work and its conclusions.
+We adopt standard cosmological parameters (h = 0.7,
+ΩM = 0.3, and Ωλ = 0.7) and a Chabrier (2003) initial
+mass function.
+2. THE GALAXY SAMPLE
+The GASP-ALMA sample consists of four late-type
+galaxies, i.e.
+JO201, JO204, JO206, and JW100, lo-
+cated in different clusters at redshift 0.04 ≲ z ≲ 0.06 and
+with relatively high stellar mass (see Table 1 and refer-
+ences therein). These galaxies are classified as “jellyfish”
+because they show one-sided tails of ionized gas longer
+than the stellar disk diameter (Poggianti et al. 2016).
+Thanks to the wealth of information provided by the
+GASP project and the availability of multi-wavelength
+observations, these galaxies have been extensively stud-
+ied in the literature (for a brief review, see Poggianti &
+the GASP team 2022). Thus, we summarize some of the
+previous works that are relevant for our analysis.
+The galaxies in our sample are moving through the
+ICM with either super-sonic or transonic line-of-sight ve-
+locities and are located close to the cluster center (Gul-
+lieuszik et al. 2020). These properties indicate that the
+galaxies are in favorable conditions for strong ram pres-
+sure and move on very radial orbits, suggesting that
+they have recently entered into the cluster for the first
+time (Yoon et al. 2017; Jaff´e et al. 2018). While JO204
+and JO206 are relatively isolated for being cluster mem-
+bers (Gullieuszik et al. 2017; Biviano et al. 2017), JO201
+and JW100 belong to a substructure of four and three
+galaxies, respectively (Bellhouse et al. 2017; Poggianti
+et al. 2019). Previous works show that, in all the GASP-
+ALMA galaxies, the stellar kinematics appears to be
+quite regular, while the ionized gas kinematics is very
+perturbed, as expected for galaxies undergoing ram pres-
+sure stripping (Bellhouse et al. 2017; Gullieuszik et al.
+2017; Poggianti et al. 2017b; Jaff´e et al. 2018; Poggianti
+et al. 2019).
+Recent works showed that these galax-
+ies host strongly asymmetric HI disks with long tails
+of atomic gas, and also have significantly reduced the
+HI content with respect to field galaxies (≳ 50 %, Ra-
+matsoku et al. 2019, 2020; Deb et al. 2020; Healy et al.
+2021; Deb et al. 2022). This HI deficiency is however
+not coupled with a deficiency in the molecular gas reser-
+voir, as these galaxies have H2 masses that are 4-5 times
+higher than expected for galaxies with similar stellar
+mass (Moretti et al. 2020a,b).
+As mentioned above, the galaxies in the GASP-ALMA
+sample host an AGN (Poggianti et al. 2017a; Radovich
+et al. 2019; Peluso et al. 2022). It has been shown that
+the AGN is the main source of gas ionization in the cen-
+tral regions of these galaxies and, except for JO206, it
+also causes a low-velocity (≈ 250 − 320 km s−1) wind
+of ionized gas (Radovich et al. 2019). Poggianti et al.
+(2017a) proposed that the AGN activity in these galax-
+ies is triggered by the ram pressure, which can decrease
+the angular momentum of the gas and favor its inflow
+toward the center. In JO201, George et al. (2019) ob-
+served a cavity of about 8.6 kpc with reduced ultravi-
+olet and CO flux around the AGN (see also Radovich
+et al. 2019). By combining optical (MUSE) and sub-
+mm (ALMA) spectroscopic observations, these authors
+proposed that the cavity is due to AGN feedback that is
+either ionizing or sweeping away the gas, possibly reduc-
+ing the star formation activity in the central regions. In
+JO204, Deb et al. (2020) found a redshifted absorption
+feature in the HI global profile, which could be ascribed
+to either a clumpy and fast rotating HI disc seen in front
+of the central radio continuum source or an inflow of
+atomic gas towards the central AGN.
+3. DATA
+This section describes the multi-wavelength observa-
+tions and data products that were used in this work,
+which primarily focuses on the molecular gas. Since the
+ram pressure can influence the kinematics and geometry
+of the molecular gas disk, we analyze the stellar kine-
+matics and use it as a reference. Sections 3.1 and 3.2
+describe the data used to study the molecular gas and
+stellar component, respectively.
+3.1. ALMA data
+We used CO(1–0) and CO(2–1) emission line obser-
+vations obtained with ALMA during Cycle 5 (project
+
+4
+Bacchini et al.
+Table 1. Properties of the galaxy sample.
+Property
+Galaxy
+JO201
+JO204
+JO206
+JW100
+Alternative names
+KAZ 364
+2MASX J10134686-0054514
+2MASX J21134738+0228347
+IC 5337
+Morphological type
+Sab
+Sab
+Sb
+Sa
+Cluster
+Abell 85
+Abell 957
+IIZW108
+Abell 2626
+zclu
+0.05568
+0.04496
+0.04889
+0.05509
+zgal
+0.044631
+0.042372
+0.051089
+0.061891
+Vgal [km s−1]
+-3138
+-743
+629
+1932
+|Vgal/σclu|
+3.7
+1.2
+0.9
+3.0
+Rclu/R200,cl
+0.18
+0.09
+0.29
+0.06
+Distance [Mpc]
+189.1
+179.8
+216.3
+261.4
+Physical scale [kpc/arcsec]
+0.88
+0.84
+1.00
+1.19
+Center R.A. [J2000]
+00:41:30.30
+10:13:46.84
+21:13:47.41
+23:36:25.05
+Center DEC [J2000]
+-09:15:45.9
+-00:54:51.27
++02:28:35.50
++21:09:02.64
+M⋆ [1010 M⊙]
+6.2 ± 0.8
+4.1 ± 0.6
+9.1 ± 0.9
+29 ± 7
+MHI [109 M⊙]
+1.7 ± 0.5
+≳ 1.3 ± 0.1
+3.2 ± 0.9
+2.8 ± 0.8
+MH2 [109 M⊙]
+11.5 ± 5.8
+5.7 ± 2.9
+5.6 ± 2.8
+16.5 ± 8.3
+Note—Galaxy names are in GASP convention; alternative names are also provided. Morphological types are from Fasano et al.
+(2012).
+Cluster redshifts (zclu) are from Biviano et al. (2017).
+Galaxy redshifts (zgal) and distances from the cluster center
+(Rclu/R200,cl) are from Bellhouse et al. (2017); Gullieuszik et al. (2017); Poggianti et al. (2017b, 2019). Galaxy velocities relative
+to cluster are calculated as Vgal = c(zgal − zclu)/(1 + zclu). The optical center coordinates are from Poggianti et al. (2017a).
+Stellar masses (M⋆) and effective radii (Reff) are from Vulcani et al. (2018) and Franchetto et al. (2020), respectively. Atomic gas
+masses (MHI) are from Ramatsoku et al. (2019, 2020) and Deb et al. (2020, 2022); a conservative uncertainty of 30% is assumed.
+Molecular gas masses (MH2) are from Moretti et al. (2020a); a conservative uncertainty of 50% is assumed.
+2017.1.00496.S; PI: Poggianti). These observations were
+already used by Moretti et al. (2020b) to study the
+molecular gas content of the GASP-ALMA galaxies.
+The datacubes used in this work are different from those
+used by Moretti et al. (2020b), as the imaging proce-
+dure was re-performed to increase the spectral resolu-
+tion (Mingozzi et al. to be submitted). Here, we use
+ALMA datacubes with spatial resolution of ≈ 1−2′′(see
+Fig. 1) and velocity resolution of 10 km s−1. Figure 1
+shows, for each galaxy, the moment maps of the CO(1–
+0) datacubes.
+These maps were obtained from the
+ALMA datacubes by applying a mask made by all pixels
+with signal-to-noise ratio (S/N) above 3 in a datacube
+smoothed by a factor 2 (i.e. in which each channel map
+is convolved with a beam 2 times larger than the original
+one).
+We note that Moretti et al. (2020a,b) detected faint
+CO emission coming from the regions outside the stel-
+lar disk and coinciding with the ionized gas tails. The
+emission is not visible in the maps used in this work
+(Fig. 1), despite we used the same ALMA observations.
+This difference is due to the fact that our datacubes
+have better velocity resolution (∆υ = 10 km s−1) but
+lower S/N than those used by Moretti et al. (2020a,b),
+which have ∆υ = 20 km s−1and to the different masking
+procedure adopted in the two works. For this work, we
+used the datacubes with ∆υ = 10 km s−1, being the best
+compromise to have good velocity resolution and S/N in
+the regions within (or close to) the stellar disk. We note
+that we do not find significant differences in mass and
+size of the molecular gas disk with respect to the results
+obtained by Moretti et al. (2020a,b).
+3.2. MUSE data
+To analyze the stellar component, we used the MUSE
+observations obtained by the GASP survey (Poggianti
+et al. 2017b). The data reduction and processing is de-
+tailed in Poggianti et al. (2017b). The wavelength cov-
+erage and spectral resolution of the final datacubes are
+4800 ˚A < λ < 9300˚A and 1770 < R < 3590, respec-
+tively. The pixel size is 0.2 ′′×0.2 ′′with a natural seeing
+of ≈ 1 ′′. In this work, we use the I-band images and
+the stellar velocity fields extracted from the MUSE dat-
+acubes.
+The I-band images are very useful to identify stellar
+substructures, such as bulges and bars. These are typ-
+
+Molecular gas kinematics in jellyfish galaxies
+5
+0h41m31.0s30.5s
+30.0s
+29.5s
+-9°15'35"
+40"
+45"
+50"
+55"
+16'00"
+RA (ICRS)
+Dec (ICRS)
+JO201
+CO(1-0) total intensity map
+5 kpc
+0.001
+0.002
+0.003
+0.004
+0.005
+0.006
+0.007
+FCO(1
+0) [Jy beam
+1 km/s]
+0h41m31.0s30.5s
+30.0s
+29.5s
+-9°15'35"
+40"
+45"
+50"
+55"
+16'00"
+RA (ICRS)
+JO201
+2.04"x1.6"
+CO(1-0) velocity field
+250
+200
+150
+100
+50
+0
+50
+100
+150
+VLOS [km/s]
+0h41m31.0s30.5s
+30.0s
+29.5s
+-9°15'35"
+40"
+45"
+50"
+55"
+16'00"
+RA (ICRS)
+JO201
+CO(1-0) velocity dispersion map
+10
+15
+20
+25
+30
+35
+40
+45
+50
+CO [km/s]
+10h13m47.5s 47.0s
+46.5s
+46.0s
+-0°54'40"
+45"
+50"
+55"
+55'00"
+RA (ICRS)
+Dec (ICRS)
+JO204
+CO(1-0) total intensity map
+5 kpc
+0.002
+0.004
+0.006
+0.008
+0.010
+FCO(1
+0) [Jy beam
+1 km/s]
+10h13m47.5s 47.0s
+46.5s
+46.0s
+-0°54'40"
+45"
+50"
+55"
+55'00"
+RA (ICRS)
+JO204
+1.66"x1.39"
+CO(1-0) velocity field
+200
+100
+0
+100
+200
+VLOS [km/s]
+10h13m47.5s 47.0s
+46.5s
+46.0s
+-0°54'40"
+45"
+50"
+55"
+55'00"
+RA (ICRS)
+JO204
+CO(1-0) velocity dispersion map
+10
+20
+30
+40
+50
+60
+CO [km/s]
+21h13m48.0s47.3s
+46.7s
+46.0s
+2°28'50"
+40"
+30"
+20"
+RA (ICRS)
+Dec (ICRS)
+JO206
+CO(1-0) total intensity map
+5 kpc
+0.002
+0.003
+0.004
+0.005
+0.006
+0.007
+FCO(1
+0) [Jy beam
+1 km/s]
+21h13m48.0s47.3s
+46.7s
+46.0s
+2°28'50"
+40"
+30"
+20"
+RA (ICRS)
+JO206
+1.87"x1.73"
+CO(1-0) velocity field
+100
+0
+100
+200
+300
+VLOS [km/s]
+21h13m48.0s47.3s
+46.7s
+46.0s
+2°28'50"
+40"
+30"
+20"
+RA (ICRS)
+JO206
+CO(1-0) velocity dispersion map
+10
+15
+20
+25
+30
+35
+40
+45
+50
+CO [km/s]
+23h36m25.3s
+24.7s
+24.0s
+21°09'15"
+10"
+05"
+00"
+08'55"
+50"
+RA (ICRS)
+Dec (ICRS)
+JW100
+CO(1-0) total intensity map
+5 kpc
+0.0050
+0.0075
+0.0100
+0.0125
+0.0150
+0.0175
+0.0200
+FCO(1
+0) [Jy beam
+1 km/s]
+23h36m25.3s
+24.7s
+24.0s
+21°09'15"
+10"
+05"
+00"
+08'55"
+50"
+RA (ICRS)
+JW100
+2.09"x1.78"
+CO(1-0) velocity field
+200
+100
+0
+100
+200
+300
+400
+500
+VLOS [km/s]
+23h36m25.3s
+24.7s
+24.0s
+21°09'15"
+10"
+05"
+00"
+08'55"
+50"
+RA (ICRS)
+JW100
+CO(1-0) velocity dispersion map
+10
+20
+30
+40
+50
+60
+70
+80
+CO [km/s]
+Figure 1. Total intensity map (left column), velocity field (central column) and velocity dispersion map (right column) obtained
+from the CO(1-0) emission line datacubes for the four galaxies in the sample. The stars and the white dashed ellipse indicate the
+kinematic center and the region used for modelling the gas kinematics, respectively. In JW100, the grey cross shows the optical
+center. In the total intensity map, the red contours are at 2nσtot, where σtot is the noise in the total map (Lelli et al. 2014;
+Iorio et al. 2017), n = 1...10, and σtot = 0.9, 1.4, 1.4, 2.7 mJy/beam km s−1for JO201, JO204, JO206, and JW100, respectively.
+In the velocity field, the black curves are the iso-velocity contours, with the thick one being at the galaxy systemic velocity (see
+Sect. 4.2 for details). The white dotted line in the velocity field is the kinematic major axis. The bar and the ellipse in the
+bottom left corner respectively show the physical scale and beam of the observations. The grey contours show the stellar disc.
+East is to the left and North to the top.
+
+6
+Bacchini et al.
+ically dominated by intermediate-age (> 1 Gyr) stellar
+populations and hence generally brighter in I-band than
+at short wavelengths (e.g. Knapen et al. 2000).
+Fig-
+ure 2 shows, for each galaxy, the I-band images. These
+were obtained by Franchetto et al. (2020) from the inte-
+grated MUSE datacubes using the Cousins I-band filter
+response curve.
+Franchetto et al. (2020) also derived
+the center coordinates, the position angle, and the incli-
+nation of these galaxies by fitting the galaxy isophotes
+with a series of concentric ellipses using the iraf task
+ELLIPSE (Jedrzejewski 1987).
+The stellar velocity fields are used to analyze the stel-
+lar kinematics, which is useful to interpret the kinemat-
+ics of the molecular gas.
+The stellar kinematics was
+extracted from the MUSE datacube using the Penal-
+ized Pixel-Fitting (pPXF) code (Cappellari & Emsellem
+2004).
+As a preliminary step, the observations were
+masked to remove spurious sources, such as stars and
+background galaxies. Spaxels in the MUSE data were
+binned through the Voronoi algorithm in order to reach
+S/N> 10 in each bin. The observed spectra were fitted
+with the stellar population templates by Vazdekis et al.
+(2010) and using of single stellar populations. More de-
+tails on the procedure can be found in Poggianti et al.
+(2017b) and Moretti et al. (2018).
+4. METHOD
+Our approach relies on the software 3DBarolo1 (v.6.1
+Di Teodoro & Fraternali 2015; Di Teodoro & Peek 2021),
+which simulates galaxy observations assuming a tilted-
+ring model. This consists of a series of concentric annuli
+described by a set of geometric and kinematic parame-
+ters, which can all vary with the galactocentric distance
+R. The geometrical parameters are the coordinates of
+the center x0 and y0, the position angle φ, and the disc
+inclination i. The kinematic parameters are the systemic
+velocity Vsys, the rotation velocity Vrot, the velocity dis-
+persion σ, and the radial velocity in the disc plane Vrad.
+The observed line-of-sight velocity is then (e.g. Begeman
+1987)
+Vlos = Vsys + (Vrot cos θ + Vrad sin θ) sin i ;
+(1)
+where θ is the azimuthal angle in the plane of the disc.
+3DBarolo (hereafter 3DB) was mainly designed to
+fit emission line observations working in 3D, meaning
+that the model is fitted to the datacube channel-by-
+channel. This approach allows us to use all the infor-
+mation in the datacube and to take into account both
+the spatial resolution and the spectral resolution of the
+1 https://editeodoro.github.io/Bbarolo/
+instrument. In a step prior to the fitting, 3DB convolves
+the model with the point spread function (PSF) or the
+beam of the instrument, while the instrumental spec-
+tral broadening is included in the model construction.
+The convolution with the PSF is required to correct for
+the so-called “beam smearing” (Bosma 1981; Begeman
+1987; Di Teodoro & Fraternali 2015). The finite size of
+the PSF smears the line emission on adjacent regions
+where the emitting material has different line-of-sight
+velocity, causing an artificial broadening of the profile.
+As a consequence, the rotation velocity and the velocity
+dispersion can be respectively underestimated and over-
+estimated, if beam smearing is not correctly accounted
+for. This effect is particularly important if the angu-
+lar resolution of the observations is low and where there
+are strong velocity gradients, as in the case of the inner
+regions of massive galaxies with steeply rising rotation
+curve. Moreover, the beam smearing effect is expected
+to become more and more relevant as the inclination an-
+gle of the galaxy increases. 3DB normalizes the model
+using either the flux in each pixel of the total intensity
+map or the azimuthally-averaged flux in each ring. Fi-
+nally, the model is fitted to the observations in order
+to find the set of free parameters that minimizes the
+residuals.
+With respect to 2D methods, which fit the velocity
+field, this 3D procedure not only corrects for the beam
+smearing effect, but also breaks the degeneracy between
+the rotation velocity and the velocity dispersion (e.g.
+Bosma 1981; Begeman 1987; Di Teodoro & Fraternali
+2015). The 3DB task 3DFIT is designed to model emis-
+sion line datacubes working in 3D. The software also
+includes the task 2DFIT, which can be used to model
+the 2D velocity fields. In this work, we use 3DFIT and
+2DFIT to model the kinematics of the molecular gas disk
+and the stellar disk, respectively. For each component,
+we adopted an ad-hoc methodology, that is described in
+Sects. 4.1 and 4.2.
+Before proceeding with the methodology presentation,
+a brief disclaimer is due. We stress that 3DB, like sev-
+eral other kinematic modelling software (e.g. Begeman
+1987; Kamphuis et al. 2015), is specifically designed to
+model radially symmetric gas flows in discs. However,
+the galaxies studied in this work are subject to various
+local disturbances due to internal (bar, AGN feedback)
+and external (ram pressure) mechanisms, which are ex-
+pected to produce deviations from this idealised kine-
+matics.
+Our strategy here is to use 3DB to quantify
+the large-scale ordered motions (i.e., rotation and radial
+flows) in the molecular gas component, and to interpret
+possible deviations from such simple kinematics in terms
+of internal or external mechanisms.
+
+Molecular gas kinematics in jellyfish galaxies
+7
+0h41m32s
+31s
+30s
+-9°15'30"
+45"
+16'00"
+RA (ICRS)
+Dec (ICRS)
+1"
+JO201 I-band
+5 kpc
+20
+40
+60
+80
+100
+F  [10
+20 erg/s/cm2/Angstrom]
+10h13m48.0s47.5s
+47.0s
+46.5s
+46.0s
+45.5s
+-0°54'30"
+40"
+50"
+55'00"
+RA (ICRS)
+Dec (ICRS)
+1"
+JO204 I-band
+5 kpc
+20
+40
+60
+80
+100
+F  [10
+20 erg/s/cm2/Angstrom]
+21h13m48s
+47s
+46s
+2°28'45"
+30"
+15"
+RA (ICRS)
+Dec (ICRS)
+1"
+JO206 I-band
+5 kpc
+20
+40
+60
+80
+100
+F  [10
+20 erg/s/cm2/Angstrom]
+23h36m27s
+26s
+25s
+24s
+23s
+21°09'30"
+15"
+00"
+08'45"
+RA (ICRS)
+Dec (ICRS)
+1"
+JW100 I-band
+5 kpc
+20
+40
+60
+80
+100
+F  [10
+20 erg/s/cm2/Angstrom]
+Figure 2. I-band images, extracted from the MUSE observations. The red contours are at 2n with n going from 1 to 20
+with steps of 0.5 (same units as colorbars). The white stars show the galaxy center. For JO201 and JO206, the white dotted
+ellipses indicate the regions influenced by the bar (see text). The light-blue contour shows the most external isophote (≈ 1.5σ
+above the background) encompassing the Hα emission traced by MUSE, and it indicates the stellar disk defined by Gullieuszik
+et al. (2020). The black dot in the bottom right corner shows the angular resolution of the MUSE observations. The inset
+in the JO204 panel shows a zoom-in of the central regions of the galaxy, with the white circle showing the resolution of the
+observations. East is to the left and north to the top.
+4.1. Modeling the stellar kinematics
+We model the stellar kinematics using the task 2DFIT
+on the velocity field (see Sect. 3.2). We fixed the kine-
+matic center at the optical center reported in Poggianti
+et al. (2017a) and Vrad = 0 km s−1. Since stars are not
+subject to the effect of ram pressure, we expect this to
+be a good approximation everywhere in the galaxy with
+the possible exception of the bar region (but see Sect. 5).
+We adopt the following three-step approach.
+1. We performed a preliminary run with φ, i, Vsys,
+and Vrot as free parameters. The initial values of
+φ and i were taken from Franchetto et al. (2020).
+2. We made a second run with φ, i, and Vrot as free
+parameters, fixing Vsys at the median of the best-
+fit values from the first step.
+3. We run again 3DB with Vrot as the free parameter,
+while φ and i are regularized using a polynomial
+function with degree from zero to three, in order
+to avoid numerical oscillations.
+The ring spacing is fixed to 1′′, which approximately
+corresponds to the spatial resolution of the MUSE ob-
+servations. This choice is also reasonable based on the
+size of the Voronoi bins. In all 3DB runs, we chose to
+give more weight to the regions close to the disc major
+
+8
+Bacchini et al.
+axis (i.e. wfunc=2), in order to maximize the signal from
+the rotational motion.
+We recall that the results obtained with the 2D ap-
+proach are affected by beam-smearing. The angular res-
+olution of the MUSE observations is about 1 ′′, corre-
+sponding to about 1 kpc in our galaxy sample. Hence,
+we expect that the PSF smearing has a mild effect on the
+GASP-ALMA galaxies, except JW100. In this galaxy,
+the PSF smearing is likely important due to its high
+inclination with respect to the line of sight.
+We also note that the assumption of circular orbits
+might be inappropriate for the innermost regions of
+barred galaxies, as the stars in the bar move along
+elongated orbits (e.g. Sellwood & Wilkinson 1993; Ko-
+rmendy & Kennicutt 2004). However, only galaxies for
+which the bar is inclined to both the projected ma-
+jor and minor axes show non-circular motions clearly
+(Sellwood & Wilkinson 1993). Hence, we do not expect
+visible signatures of non-circular motions in JO201 and
+JO206. In Sanchez-Garcia et al. (in preparation), the
+stellar velocity field of the galaxies in the GASP sample
+is fitted using an ad-hoc approach to include large-scale
+non-circular motions induced by bars. Preliminary re-
+sults show that, for the GASP-ALMA galaxies, the re-
+covered stellar rotation velocity obtained by Sanchez-
+Garcia et al. is overall consistent with ours, suggesting
+that the non-circular motions are small compared to ro-
+tation.
+4.2. Modeling the molecular gas kinematics
+We model the molecular gas kinematics using the task
+3DFIT on the ALMA datacubes (see Sect. 3.1). To re-
+duce the free parameters in the model, we first fixed
+the kinematic center at the optical center reported in
+Poggianti et al. (2017a).
+However, since the interac-
+tion with the ICM can displace the kinematic center of
+the gas from that of the stars (e.g. Kronberger et al.
+2008b; Boselli et al. 2022b), we adjusted the kinematic
+center of the molecular gas when necessary.
+We also
+set Vsys at the value obtained from the global profile of
+the emission line. When necessary, Vsys was refined by
+a few km s−1after inspecting the position-velocity dia-
+grams (see Sect. 5).
+1. We performed a first run with Vrad = 0 km s−1and
+leaving free the geometrical and kinematical pa-
+rameters. By setting the 3DB parameter wfunc=2,
+we chose to give more weight to the emission along
+the disc major axis, where most of the information
+on rotational motions lies (θ = 0° in Eq. 1).
+2. We made a second run (i.e. twostage=True) in
+which the geometrical parameters are regularized
+using either a suitable function or the median
+value.
+3. Vrad is left free in the last run, while the other pa-
+rameters are fixed to the best-fit values obtained
+previously.
+By setting wfunc=-2, we give more
+weight to the emission along the disc minor axis,
+where the contribution of radial motions is the
+strongest (θ = 90° in Eq. 1).
+This procedure is substantially based on the approach
+developed by Di Teodoro & Peek (2021), who used
+3DB to model the atomic gas kinematics in a sample of
+nearby galaxies in order to measure gas radial motions
+and mass flows. These authors used 21-cm observations
+with higher spatial resolution and better velocity reso-
+lution than our ALMA data. Radial motions are pos-
+sibly stronger and easier to detect for galaxies affected
+by the ram pressure than in the case of Di Teodoro &
+Peek (2021)’s galaxies, in which radial motions are of
+the order of a few km s−1. We stress that the approach
+adopted in this work takes into account the radial mo-
+tions within the galaxy disk, while motions perpendicu-
+lar to the disk midplane are not considered (Di Teodoro
+& Peek 2021).
+We used the 3DB task ELLPROF to derive the
+azimuthally-averaged radial profiles of the CO surface
+brightness. These profile were adopted for the normal-
+ization procedure of 3DB models and to derive the H2
+surface density ΣH2.
+We also used the 3DB task spacepar to fully explore
+the parameter space for Vrot and σ. This test is useful
+to check whether the model fitting converges to a good
+minimum of the parameter space. We anticipate that,
+while the best-fit Vrot is generally well constrained, it
+is not always the case for σ. This is likely due to the
+complex shape of the emission line profiles.
+A possible caveat in our methodology is that the
+tilted-ring model is based on the assumption of concen-
+tric orbits, which might not be valid for the gas in galax-
+ies affected by strong ram pressure or in an advanced
+stripping stage (e.g. Kronberger et al. 2008b). In these
+cases, the results of our analysis are very uncertain and
+should be taken with caution. However, if stripping is
+not too dramatic, modeling the gas kinematics using the
+tilted-ring approach may be possible for the disk regions
+where some or most of the gas has preserved its original
+motion. Stellar bars are also expected to induce non-
+circular motions due to the gas streaming along the bar
+(e.g. Sellwood & Wilkinson 1993). Indeed, the gas kine-
+matics in barred galaxies is usually modeled using tools
+that are specifically designed to take into account non-
+axisymmetric distortions in the 2D velocity field (e.g.
+
+Molecular gas kinematics in jellyfish galaxies
+9
+Schoenmakers 1999; Spekkens & Sellwood 2007). How-
+ever, these methods fail when the bar is perpendicular
+to or parallel the disk major axis, being unable to break
+the degeneracy between the tangential and radial ve-
+locity components (e.g. Sellwood & S´anchez 2010; Ran-
+driamampandry et al. 2015). We thus decided to adopt
+the tilted-ring approach also in the case of JO201 and
+JO206, which host stellar bars aligned with the disk ma-
+jor axis.
+5. RESULTS AND DISCUSSION
+In this section, we present the best-fit models for the
+molecular gas kinematics and we then compare the stel-
+lar and molecular gas rotation curves. We discuss each
+galaxy individually in Sects. 5.1–5.4 and summarize our
+findings in Sect. 5.5. We analyzed both the CO(1–0)
+and the CO(2–1) datacubes, obtaining essentially the
+same results. Thus, we show the best-fit models for the
+CO(1–0) data, as they have a S/N and angular resolu-
+tion more suitable for modeling the kinematics. From
+here on, CO indicates CO(1–0) unless otherwise stated.
+Since the focus of this work is on the molecular gas, we
+show the best-fit model for the stellar kinematics only
+for JO201 in Fig 3, while the models for the rest of the
+sample can be found in Appendix A.
+5.1. JO201
+The I-band image in Fig. 2 shows that JO201 has
+a stellar bulge. Moreover, the elongated shape of the
+isophotes in the inner regions suggests that JO201 hosts
+a stellar bar, as reported by George et al. (2019).
+Sanchez-Garcia et al. (submitted) estimated that the
+bar length is ≈ 4.6 kpc. We also note that the stellar
+disc of JO201 seems morphologically lopsided, being the
+east side slightly more extended than the west one.
+The top panels in Fig 3 show, from left to right, the
+observed stellar velocity field, the best-fit model, and
+the map of the residuals between the data and the best-
+fit model. The bottom panels display the radial profile
+of the best-fit rotation velocity (left), inclination (cen-
+ter), and PA (right). The stellar velocity field is very
+well reproduced by the model. The residuals in the disk
+outskirts, where the Voronoi bins are the largest, tend
+to be higher than in the inner regions, but still within
+the velocity resolution of the MUSE observations, that
+is ∆v ≈ 50 km s−1. We note that, for R ≲ 5 kpc, the ro-
+tation velocity is much lower than expected for a galaxy
+with stellar mass M⋆ ≈ 9 × 1010 M⊙ and hosting a stel-
+lar bulge. This feature can be explained by the fact that
+the stellar bar is aligned along the disk major axis. In
+a scenario where a large fraction of the stars in the bar
+move on elliptical orbits aligned parallel to the bar (so
+called x1 type; Sellwood 2014), the velocity component
+along the line of sight has its minimum at the apocentre
+and then increases along the major axis. This can result
+in an underestimation of the rotation velocity in the re-
+gions influenced by the bar (e.g. Dicaire et al. 2008; Sell-
+wood & S´anchez 2010; Randriamampandry et al. 2015).
+The total CO intensity map (top left panel in Fig. 1)
+gives useful indications about the effect and direction of
+ram pressure. In JO201, the west side of the disk shows
+compressed contours and regions with bright CO emis-
+sion, possibly suggesting the ram pressure compressed
+this part of the disk (Bellhouse et al. 2017). The most
+evident feature in Fig. 1 is arguably the presence of the
+ring-like structure surrounding the hole in the CO dis-
+tribution in the innermost ≈ 3 kpc (see also George
+et al. 2018, 2019). The ring-like structure is also visible
+in the MUSE images shown by Bellhouse et al. (2017).
+This feature can be explained by the presence of the bar
+driving the formation of a molecular gas ring around the
+co-rotation radius (i.e. where the bar pattern equals the
+angular frequency of circular motions; see Sellwood &
+Wilkinson 1993; Kormendy & Kennicutt 2004). At radii
+well inside co-rotation, gas is expected to fall toward
+the center.
+The molecular gas distribution in barred
+galaxies is typically very concentrated in the center (e.g.
+Kormendy & Kennicutt 2004), while Figure 1 clearly
+shows the lack of CO emission in the innermost regions
+of JO201. George et al. (2019) attribute the CO cavity
+to AGN feedback, which ionizes the molecular hydro-
+gen (i.e. radiative feedback) and sweeps the gas from
+the center (i.e. mechanical feedback). The connection
+between nuclear activity and the gas distribution and
+kinematics is specifically tackled in the companion pa-
+per (Mingozzi et al. to be submitted).
+The CO velocity field of JO201 (2nd panel in the top
+row of Fig. 1) shows that the galaxy is kinematically lop-
+sided, meaning that the velocity gradient in the receding
+and approaching sides of the disc are significantly differ-
+ent from each other (e.g. Richter & Sancisi 1994; Swa-
+ters et al. 1999; Schoenmakers 1999; Shafi et al. 2015).
+For this reason, we modeled the approaching side and
+receding side separately. We compare the observations
+with our best-fit models in Fig. 4, where the left and the
+right panels are for the approaching and receding sides
+of the disc, respectively. The first and second rows in
+Fig. 4 are the position-velocity diagrams (PVDs) along
+the major and minor axis of the disc, respectively. Our
+rotating disc model can reproduce reasonably well the
+observations, indicating that the molecular gas in the
+disk preserved its original rotation, despite the interac-
+tion with the ICM. There is however some gas, which is
+
+10
+Bacchini et al.
+0
+2
+4
+6
+8
+10
+12
+14
+16
+R [kpc]
+0
+50
+100
+150
+200
+250
+Rotation velocity [km/s]
+Bar region
+2nd fit
+3rd fit
+0
+2
+4
+6
+8
+10
+12
+14
+16
+R [kpc]
+20
+30
+40
+50
+60
+70
+80
+Inclination [degrees]
+Bar region
+2nd fit
+Median
+3rd fit
+± MAD
+0
+2
+4
+6
+8
+10
+12
+14
+16
+R [kpc]
+160
+170
+180
+190
+200
+210
+Position angle [degrees]
+Bar region
+2nd fit
+Median
+3rd fit
+± MAD
+0h41m32s
+31s
+30s
+29s
+-9°15'30"
+45"
+16'00"
+RA (ICRS)
+Dec (ICRS)
+1"
+JO201 - Data
+5 kpc
+150
+100
+50
+0
+50
+100
+150
+VLOS [km/s]
+0h41m32s
+31s
+30s
+29s
+RA (ICRS)
+JO201 - Model
+150
+100
+50
+0
+50
+100
+150
+VLOS [km/s]
+0h41m32s
+31s
+30s
+29s
+RA (ICRS)
+JO201 - Residuals
+40
+20
+0
+20
+40
+Data-Model [km/s]
+0.0
+2.5
+5.0
+7.5
+10.0
+12.5
+15.0
+17.5
+R [arcsec]
+0.0
+2.5
+5.0
+7.5
+10.0
+12.5
+15.0
+17.5
+R [arcsec]
+0.0
+2.5
+5.0
+7.5
+10.0
+12.5
+15.0
+17.5
+R [arcsec]
+Figure 3. Top row: stellar velocity field (left), its best-fit model (center), and residual map (right) for JO201. The white
+star indicates the disc center. The black curves are the iso-velocity contours with steps of 50 km s−1. The thick black contour
+indicates Vsys. The white contours in the left panel shows the best-fit model on the data. The bar and the circle in the bottom
+left and right corners respectively show the physical scale and the PSF of the observations. Bottom row: rotation velocity (left),
+inclination (center) , and position angle (right) as a function of the galactocentric distance for the best-fit models of the stellar
+velocity field. The grey dashed area indicates the region influenced by the stellar bar. The empty circles and the red points are
+for the 2nd and the 3rd steps of our procedure (see Sect. 4.1), respectively. The dashed black lines and the grey area indicate
+the median and the median absolute deviation, respectively.
+indicated by the red arrow in Fig. 4, moving with lower
+velocities than those predicted by the model. Since this
+gas is located at galactocentric distances smaller than
+the bar length, its anomalous kinematics is plausibly
+due to the bar influence.
+By exploring the parameter space, we found that
+the best-fit value of the CO velocity dispersion is not
+well-constrained for the outermost ring, likely because
+of the low S/N. For R ≲ 5 kpc, we obtain σCO ≈
+25 − 40 km s−1, which can be explained by the non-
+circular motions due to the stellar bar. Outside the bar
+regions, we find σCO ≈ 20 km s−1, which is about a fac-
+tor 2 higher than the typical values of the molecular gas
+velocity dispersion in local isolated, unbarred galaxies
+(e.g. Bacchini et al. 2020a). This enhancement of σCO
+may be due to ram pressure increasing the molecular gas
+turbulence, either directly or by enhancing the star for-
+mation rate (SFR; see Sect. 5.5 for further discussion).
+We note that the best-fit values of radial velocity are
+consistent with zero, suggesting that the inclusion of
+radial motions does not significantly improve the fit.
+Hence, these values should be taken with caution. The
+direction (either inward or outward) of these radial flows
+cannot be determined unless the near/far sides of the
+galaxy are known. This can be inferred by assuming that
+spiral arms trail the galaxy rotation. Based on the RGB
+image shown by (Bellhouse et al. 2017), the direction of
+spiral arms indicate that JO201 rotates clockwise. Then,
+in 3DB’s convention (Di Teodoro & Peek 2021), radial
+motions with Vrad < 0 point inward, while those with
+Vrad > 0 point outward. Taken at face value, the inflow
+radial velocities at R ≲ 5 kpc are Vrad ≳ −10 km s−1,
+which is comparable with the average values measured in
+the inner regions of nearby spiral galaxies (Di Teodoro &
+Peek 2021). Beyond the bar region, the radial outflow
+with Vrad ≳ 20 km s−1is consistent with being caused
+by ram pressure. However, since non-circular motions
+can be induced by any perturbation of the gravitational
+potential, we cannot exclude a different origin (e.g. Sell-
+wood & S´anchez 2010).
+In Fig. 5 (top left), we compare the circular velocities
+inferred from the kinematics of the stellar and molecular
+gas disks. The stellar circular velocity was obtained from
+the rotation velocity shown in Fig. 3 by correcting for
+the contribution of pressure support (asymmetric drift
+
+Molecular gas kinematics in jellyfish galaxies
+11
+10
+5
+0
+5
+10
+Offset ["]
+300
+200
+100
+0
+100
+VLOS [km/s]
+= 178°
+Model fitted on the approaching side
+10
+5
+0
+5
+10
+Offset ["]
+300
+200
+100
+0
+100
+VLOS [km/s]
+= 180°
+Model fitted on the receding side
+10.0
+7.5
+5.0
+2.5
+0.0
+2.5
+5.0
+7.5
+10.0
+Offset ["]
+300
+200
+100
+0
+100
+VLOS [km/s]
+= 268°
+10.0
+7.5
+5.0
+2.5
+0.0
+2.5
+5.0
+7.5
+10.0
+Offset ["]
+300
+200
+100
+0
+100
+VLOS [km/s]
+= 270°
+0
+2
+4
+6
+8
+40
+60
+i [deg]
+0
+2
+4
+6
+8
+40
+60
+i [deg]
+0
+2
+4
+6
+8
+180
+200
+PA [deg]
+0
+2
+4
+6
+8
+180
+200
+PA [deg]
+0
+2
+4
+6
+8
+R [kpc]
+0
+50
+Vrad [km/s]
+Outflow
+Inflow
+0
+2
+4
+6
+8
+R [kpc]
+0
+50
+Vrad [km/s]
+Outflow
+Inflow
+200
+100
+0
+100
+200
+300
+VLOS (km/s)
+10
+5
+0
+5
+10
+R [kpc]
+200
+100
+0
+100
+200
+300
+VLOS (km/s)
+7.5
+5.0
+2.5
+0.0
+2.5
+5.0
+7.5
+R [kpc]
+200
+100
+0
+100
+200
+300
+VLOS (km/s)
+10
+5
+0
+5
+10
+R [kpc]
+200
+100
+0
+100
+200
+300
+VLOS (km/s)
+7.5
+5.0
+2.5
+0.0
+2.5
+5.0
+7.5
+R [kpc]
+JO201
+Figure 4. Best-fit models of the molecular gas kinematics for JO201 using CO(1–0) emission line observations. The left and the
+right panels are for the approaching and the receding sides of the disc (the other side is shaded), respectively. The first and the
+second rows show the PVD along the major and the minor axis, respectively. The observed CO(1–0) emission is shown in blue
+with black and grey contours, while the red contours show the best-fit model. All the contours are at 2nσch with n = 1, ..., 10
+and σch = 0.7 mJy/beam. The yellow points indicate the projected rotation curves of the best-fit models. The vertical blue
+dotted lines and the red arrows indicate the bar extent and the gas with anomalous kinematics (see text), respectively. In the
+last three rows, the panels show the profiles of inclination, PA, and radial velocity of the best-fit models. The red points are
+the parameters from the 1st fitting step and the dark-red lines are the regularized profiles (see Sect. 4.2). The orange and green
+areas in the bottom panels indicate whether positive/negative values for Vrad mean radial gas outflow/inflow.
+
+12
+Bacchini et al.
+0
+5
+10
+15
+0
+100
+200
+300
+Vcirc [km/s]
+JO201
+CO, app. side
+CO, rec. side
+Stars
+5
+10
+0
+100
+200
+300
+JO204
+CO
+Stars
+0
+5
+10
+15
+20
+R [kpc]
+0
+100
+200
+300
+Vcirc [km/s]
+JO206
+CO
+Stars
+0
+10
+20
+R [kpc]
+0
+100
+200
+300
+400
+JW100
+CO, app. side
+CO, rec. side
+Stars
+Figure 5. Comparison of the stellar (yellow stars) and the
+CO (darkred points) circular velocities for each galaxy in
+our sample.
+When the approaching side and the receding
+side of the disc are modelled separately, the resulting pro-
+files are shown by the blue squares and the red diamonds,
+respectively.
+correction)2. Within R ≈ 5 kpc, the molecular gas and
+stellar circular velocities essentially coincide, but the ve-
+locity gradient is too shallow for a massive galaxy with a
+bulge such as JO201. As mentioned above, this is likely
+due to the stellar bar aligned along the major axis (e.g.
+Dicaire et al. 2008; Sellwood & S´anchez 2010; Randria-
+mampandry et al. 2015). Beyond the bar regions, the
+stellar velocity field of JO201 (Fig. 3) does not show any
+indication of the kinematic lopsidedness, contrary to the
+molecular gas. The receding side of the CO disc reaches
+slightly higher rotation velocities than the stellar disc,
+while the approaching side shows a lower velocity gradi-
+ent. The kinematic lopsidedness in disc galaxies is typ-
+ically ascribed to a triaxial potential, as in the presence
+of a stellar bar (e.g. Swaters et al. 1999; Schoenmakers
+1999; Rhee et al. 2004). However, the regular kinematics
+of the stellar disk seems to suggest that the molecular
+gas kinematics may be perturbed by some mechanisms
+that does not affect the stars, like ram pressure. The
+distortions appear in the outer parts of the galaxy and
+in a symmetric way, as expected for face-on ram pressure
+(Kronberger et al. 2008b; Bellhouse et al. 2017, 2019).
+Indeed, JO201 is moving towards the observer at very
+2 We used Eq. A1 from Posti et al. (2018) for the asymmetric
+drift velocity and Eq. 3 from Mancera Pi˜na et al. (2021a) for the
+central velocity dispersion. We note that the asymmetric drift
+correction is essentially negligible for JO201 and all the other
+galaxies, as expected given their high rotation velocities.
+high velocity (see Table 1), implying that the approach-
+ing and receding sides of the disk move in the opposite
+and the same direction as the ram pressure, respectively.
+The ram pressure is thus expected to decelerate the ap-
+proaching side of the molecular disk and accelerate the
+receding side (Kronberger et al. 2008b), which is consis-
+tent with our results.
+We conclude that the molecular gas kinematics in the
+inner regions of JO201 is mainly dominated by the per-
+turbations due to the stellar bar.
+In the outer parts
+of the molecular gas disk, the kinematic lopsidedness
+and radial motions (although rather uncertain) seem to
+suggest that the molecular gas in JO201 is affected by
+face-on ram pressure, despite other mechanisms cannot
+be ruled out.
+5.2. JO204
+Before focusing on the molecular gas kinematics, it
+is worth noting two features of the stellar component.
+First, the innermost isophotes in the I-band image
+(Fig. 2) show a boxy shape that might indicate the pres-
+ence of a bar seen with high inclination with respect to
+the line-of-sight (e.g. Combes et al. 1990; Bettoni & Gal-
+letta 1994; Kuijken & Merrifield 1995; Bureau & Free-
+man 1999; Merrifield & Kuijken 1999). Unfortunately,
+dust obscuration and projection effects hamper any at-
+tempt to estimate the bar length from the optical im-
+ages. The second feature is visible in the stellar velocity
+field, which shows slightly distorted iso-velocity contours
+in the innermost regions (see Fig. 11). This S-shaped
+feature indicates the presence of non-circular motions
+and, possibly, of a stellar bar (e.g. Bettoni 1989; Vau-
+terin & Dejonghe 1997; Kormendy & Kennicutt 2004;
+Cort´es et al. 2015, Sanchez-Garcia et al. in preparation).
+Indeed, the top right panel of Fig. 11 shows residuals of
+a few tens of km s−1in the regions close to the disc mi-
+nor axis, indicating that a model based on circular orbits
+cannot fully reproduce the observations.
+In Fig. 1, the CO total intensity map shows that the
+molecular gas distribution is strongly concentrated in
+the center and two arm-like structures. Both features
+are typical of barred galaxies (e.g. Athanassoula 1992a;
+Bureau & Freeman 1999; Merrifield & Kuijken 1999; Ko-
+rmendy & Kennicutt 2004; Hogarth et al. 2021). The
+iso-velocity contours in the CO velocity field (Fig. 1) are
+visibly distorted in the inner regions, which typically in-
+dicates the presence of non-circular motions. The CO
+velocity dispersion is also quite high in the inner regions
+of disk.
+To model the CO kinematics, we modified the pro-
+cedure described in Sec. 4.2.
+After various tests, we
+decided to fit the data by fixing all the geometrical pa-
+
+Molecular gas kinematics in jellyfish galaxies
+13
+rameters and leaving free Vrot, Vrad, and σCO, as this
+choice improved the comparison between the model and
+the data. We run 3DB using the reverse option, which
+performs the fit starting from the most external ring
+and then moving inward. This algorithm was designed
+to improve the fit for galaxies seen with high inclination
+(i ≳ 70°). Fig. 6 compares the best-fit model with the
+observations. In Fig. 6, the PVD along the major axis
+shows that, overall, the model reproduces well the ob-
+servations, except for two features. The first feature is
+indicated by the red arrow and consists in gas moving at
+VLOS ≈ 270 km s−1at about 1 kpc from the center. One
+possibility is that this CO emission is probing the inner
+rise of the rotation curve of the molecular disk if the nu-
+clear CO distribution is asymmetric between approach-
+ing and receding sides (see for example Lelli et al. 2022).
+Alternatively, this central emission can be ascribed to
+complex non-circular motions caused by the stellar bar
+(the so-called x2 orbits aligned perpendicular to the bar;
+see Sancisi et al. 1979; Kormendy & Kennicutt 2004;
+Randriamampandry et al. 2015) or even feedback from
+stars or the AGN (e.g. Stuber et al. 2021). Our finding
+is in agreement with the results obtained by Deb et al.
+(2020), who revealed an absorption feature in the HI
+global profile that could be explained by high-velocity
+gas seen in front of the continuum emission from the
+AGN. The second feature that is not reproduced by the
+model is indicated by the magenta arrows. This emis-
+sion comes from gas that moves at lower velocities than
+those predicted by our model. A possible explanation
+is that this gas is decelerated by the ram pressure com-
+ponent in the sky plane. We note that the PVD along
+the major axis (top panel of Fig. 6) shows the charac-
+teristic X-shaped pattern, that is typical of a bar seen
+at high inclination along the line of sight (e.g. Bureau
+& Freeman 1999; Merrifield & Kuijken 1999; Kormendy
+& Kennicutt 2004; Alatalo et al. 2013; Hogarth et al.
+2021).
+In Fig. 6, the PVD along the minor axis shows ex-
+tended gas emission in the lower left quadrant, which
+can only be reproduced by a model with strong radial
+motions of Vrad ≳ 50 km s−1. However, the same feature
+is not observed in the upper right quadrant, indicating
+that the CO distribution (or kinematics) is asymmetric.
+Unfortunately, JO204 does not have visible spiral arms
+and the dust lanes in the optical MUSE and Hubble
+Space Telescope (HST) images (Gullieuszik et al. 2017,
+Gullieuszik et al., submitted) do not allow to clearly
+identify the nearest side of the disk.
+Hence, we can-
+not infer the direction of rotation and radial motions.
+Since these non-circular motions are detected in the in-
+ner parts of the galaxy, one may speculate that they
+10
+5
+0
+5
+10
+Offset ["]
+300
+200
+100
+0
+100
+200
+300
+VLOS [km/s]
+= 327°
+JO204
+10.0
+7.5
+5.0
+2.5
+0.0
+2.5
+5.0
+7.5
+10.0
+Offset ["]
+300
+200
+100
+0
+100
+200
+300
+VLOS [km/s]
+= 237°
+1
+2
+3
+4
+5
+6
+7
+8
+R [kpc]
+0
+100
+Vrad [km/s]
+300
+200
+100
+0
+100
+200
+300
+VLOS (km/s)
+10
+5
+0
+5
+10
+R [kpc]
+300
+200
+100
+0
+100
+200
+300
+VLOS (km/s)
+7.5
+5.0
+2.5
+0.0
+2.5
+5.0
+7.5
+R [kpc]
+Figure 6. Same as Fig. 4 but for JO204. The model fitting is
+performed on the approaching and receding sides at the same
+time, and the inclination and PA are fixed to the values of
+75° and 327°, respectively. The arrows indicate the gas with
+anomalous kinematics (see text). Here σch = 0.9 mJy/beam.
+are inflows driven by a stellar bar. The magnitude of
+radial motions in JO204 is consistent with the values es-
+timated in simulated barred galaxies (see Randriamam-
+pandry et al. 2015), but about 2 times higher than those
+measured by Di Teodoro & Peek (2021) in the atomic
+gas of real barred galaxies.
+The blue arrow indicates
+another feature that is not reproduced by our model.
+However, since this emission is very faint, it is unclear
+whether this is real emission from the galaxy. We find
+σCO ≈ 10 km s−1, but this value is rather uncertain
+based on the inspection of the parameters space.
+
+14
+Bacchini et al.
+Overall, our model can reproduce the molecular gas
+kinematics in JO204 reasonably well, despite the com-
+plexities due to the stellar bar and/or ram pressure. Fig-
+ure 5 (top right panel) shows that the CO circular veloc-
+ity is compatible within the uncertainties with the stel-
+lar circular velocity for R ≳ 3 kpc, indicating that the
+molecular gas has retained most of its original motion.
+The difference in the innermost regions is likely due to
+a combination of dust extinction and resolution effects,
+which may smooth the gradient of the stellar rotation
+curve (see Sect. 4). Similarly to the case of JO201, we
+conclude that the molecular gas kinematics is rotation-
+dominated in JO204.
+Then, the stellar bar plays an
+important role in perturbing the gas kinematics in the
+inner regions and driving radial gas flows, while the ram
+pressure may be a possible explanation for the gas with
+anomalous kinematics in the outskirts of JO204.
+5.3. JO206
+The I-band images in Fig. 2 shows that JO206 hosts
+a stellar bulge. In addition, the elongated shape of the
+isophotes in the inner regions indicate the presence of
+a stellar bar aligned with the disk major axis, as for
+JO201.
+The CO total intensity map in Fig. 1 shows
+that the morphology of the molecular gas distribution is
+asymmetric, suggesting that the ram pressure is directed
+towards south-east. Hence, we expect the molecular gas
+kinematics to be particularly complex in JO206, as a
+consequence of the combined effects of bar perturbations
+and ram pressure stripping. Indeed, the iso-velocity con-
+tours in the CO velocity field (2nd panel in the third row
+in Fig. 1) are even more distorted than those of JO204,
+indicating stronger perturbations.
+In the light of these considerations, we modeled only
+the regions of the molecular gas disk within R ≈ 6 ′′,
+which essentially corresponds to the extent of its re-
+ceding side (Fig. 1). We also adjusted the position of
+the kinematic center with respect to the optical center
+by applying a small shift of ≈ 0.23 ′′. Figure 7 shows
+that our model is able to reproduce reasonably well the
+observations, except for the molecular gas with anoma-
+lous kinematics. The CO emission indicated by the blue
+arrow (offset ≈ 10 − 20 ′′and -200 km s−1≲ VLOS ≲
+−100 km s−1) belongs to the tail of stripped gas (see also
+Fig. 1). Probably, this portion of gas disc was detached
+from the approaching side of the disc and decelerated by
+ram pressure. Another possibility is that the ram pres-
+sure displaced a portion of the disc at larger radii, thus
+its rotation velocity is lowered by conservation of angular
+momentum. The red arrow indicates the receding side
+of the molecular gas disk that has not been stripped. In
+the PVD along the minor axis (second panel in Fig. 7),
+the CO emission indicated by the orange arrow belongs
+to the molecular gas in the stripped tail left behind by
+the galaxy.
+By exploring the parameter space, we found that the
+best-fit value of σCO is well constrained only for the
+2nd and 3rd rings, where we obtained σCO ≈ 30 −
+40 km s−1(see also Fig. 1). These values are higher than
+those typically measured in nearby galaxies using CO
+observations with similar resolution (e.g. Bacchini et al.
+2020a), which is not surprising given the complex kine-
+matics of the molecular gas in JO206.
+We tentatively detect radial motions of ≈ −25 km s−1.
+However, including radial motions does not improve the
+best-fit model, as indicated by the fact that radial veloc-
+ities are consistent with zero. In JO206, the spiral arms
+can be identified from the MUSE optical images (Pog-
+gianti et al. 2017b; Bellhouse et al. 2019).
+Assuming
+trailing spiral arms, we can infer that the galaxy ro-
+tates clockwise, implying that Vrad < 0 for inflows and
+Vrad > 0 for outflows.
+The bottom left panel in Fig. 5 shows that the stellar
+and CO circular velocities coincide within R ≈ 6.5 kpc.
+As in the case of JO201 (see Sect. 5.1), the slow rise of
+the inner rotation curve is plausibly due to the fact that
+the stellar bar is aligned parallel to the disk major axis.
+This suggest that the kinematics of both the molecular
+gas and the stars is dominated by the stellar bar in these
+regions3. We also note that the position and velocity of
+the molecular gas emission indicated by the red arrow
+(Fig. 7, top panel) is perfectly compatible with the cir-
+cular velocity profile of the stars (see Fig. 5). On the
+contrary, the stripped tail indicated by the blue arrow
+(Fig. 7, top panel) is decelerated of about 70km s−1with
+respect to the stars at the same galactocentric distance,
+suggesting that this material is decoupled from the disk
+rotation. The asymmetric perturbations on the molec-
+ular gas kinematics and the displaced kinematic center
+are signatures of edge-on ram pressure stripping (Kron-
+berger et al. 2008b).
+Overall, we conclude that the molecular gas kinemat-
+ics is mainly perturbed by the stellar bar.
+Taken at
+face value, the radial motion in JO206 can be inter-
+preted as a gas inflows driven by the bar, as they are
+within its region of influence. We also find clear indi-
+cations of edge-on ram pressure stripping based on the
+presence of molecular gas emission detached from the
+galaxy and with kinematics decoupled from the main
+disk. This suggests that the ram pressure has a stronger
+3 This result is consistent with the preliminary estimate of the bar
+length obtain by Sanchez-Garcia et al. (in preparation), that is
+approximately 6.7 kpc.
+
+Molecular gas kinematics in jellyfish galaxies
+15
+10
+0
+10
+20
+Offset ["]
+300
+200
+100
+0
+100
+200
+300
+400
+VLOS [km/s]
+= 120°
+JO206
+10
+5
+0
+5
+10
+Offset ["]
+300
+200
+100
+0
+100
+200
+300
+400
+VLOS [km/s]
+= 210°
+0
+1
+2
+3
+4
+5
+6
+7
+60
+80
+i [deg]
+0
+1
+2
+3
+4
+5
+6
+7
+100
+150
+PA [deg]
+0
+1
+2
+3
+4
+5
+6
+7
+R [kpc]
+50
+0
+Vrad [km/s]
+Outflow
+Inflow
+300
+200
+100
+0
+100
+200
+300
+VLOS (km/s)
+10
+0
+10
+20
+R [kpc]
+300
+200
+100
+0
+100
+200
+300
+VLOS (km/s)
+10
+5
+0
+5
+10
+R [kpc]
+Figure 7. Same as Fig. 4 but for JO206. The model fitting
+is performed on both the approaching and receding sides, at
+the same time. Here σch = 0.8 mJy/beam.
+effect on the molecular gas disk in JO206 than in JO201
+and JO204.
+5.4. JW100
+The I-band image in Fig. 2 seems to suggest that
+JW100 hosts a stellar bulge. Moreover, despite the fact
+that JW100 is strongly affected by projection effects and
+dust obscuration, we can tentatively identify the pres-
+ence of a warp in the stellar disk based on the S-shape
+of the isophotes. Regarding the stellar kinematics, our
+model can successfully reproduce the observations and
+recover the stellar rotation curve (see Fig. 13 in Append-
+inx A). However, we found quite high residuals in a ring
+at R ≈ 6 ′′and in the disk outskirts. After various trails,
+we found no significant improvement in the residual map
+using different geometrical parameters and allowing for
+radial motions. This can be due to the combined effects
+of low S/N of the observations in the disk outskirts,
+strong projection effects due to the radial variation in
+disk inclination and PA, and asymmetric dust lanes (see
+Gullieuszik et al., submitted). Since JW100 belongs to
+a substructure of three galaxies in Abel 2626, we cannot
+rule out that the stellar kinematics is perturbed by tidal
+interaction.
+Figure 1 clearly shows that the distribution and kine-
+matics of the molecular gas in JW100 are strongly dis-
+turbed, suggesting that the ram pressure component in
+the sky plane is directed westward and contributes in
+pushing the gas outside the stellar disk.
+The case of
+JW100 may seem surprising, as the high mass of this
+galaxy is expected to produce a strong gravitational pull
+that can efficiently contrast the ram pressure stripping.
+However, the supersonic speed and the close proximity
+to the cluster center (see Table 1) indicate that JW100 is
+in the most favorable conditions for experiencing strong
+ram pressure. Indeed, the iso-velocity contours in the
+CO velocity field (2nd panel in the last row in Fig. 1) are
+even more distorted than the rest of the GASP-ALMA
+sample. Also the 2nd moment map suggests that the
+molecular gas velocity dispersion is very high through-
+out the disk, indicating very complex line profiles.
+We attempt to model the gas kinematics with the aim
+of understanding whether some gas has retained its orig-
+inal rotation. Hence, we run 3DB using the reverse
+option for highly inclined galaxies.
+We fixed the in-
+clination and PA at the values obtained for the stellar
+disc and shifted the kinematic center ≈ 1.6′′westward
+from the optical center.
+We performed the fitting on
+the approaching and receding sides separately, as the
+PVD along the major axis is asymmetric with respect
+to Vsys. The resulting best-fit models are shown in the
+left and right panels of Fig. 8, respectively. The models
+well reproduce the observations, except for the emission
+indicated by the blue arrow in the minor axis PVD. This
+emission comes from the molecular gas in the tail that is
+left behind by JW100 as it falls into the cluster receding
+from the observer. Indeed, the bottom panel in Fig. 8
+shows the profiles of the radial velocity, which reaches
+Vrad ≈ 50 − 100 km s−1in the disk outskirts. Taken at
+face value, the radial velocities are larger than the Vrad
+values of a few km s−1that are typically measured in
+
+16
+Bacchini et al.
+nearby galaxies (e.g. Di Teodoro & Peek 2021). Also
+the skewed shape of the CO emission in the PVD along
+the minor axis (blue arrows in Fig. 8) seem to suggest
+the presence of radial motions. We note that the emis-
+sion from the stripped gas indicated by the blue arrow
+reaches even higher velocities (∆VLOS ≈ −200 km s−1)
+than the model emission. The dust lanes in the HST im-
+ages (Gullieuszik et al., submitted) seem to suggest that
+the west side of JW100 is the nearest one and the galaxy
+is rotating clockwise. This implies that Vrad > 0 indi-
+cates an outward radial flow. This is in agreement with
+the morphology of the molecular gas disk, that clearly
+suggests an ongoing large-scale removal of molecular gas
+by ram pressure. We obtained σCO ≈ 30 − 60 km s−1,
+possibly indicating that the molecular gas is highly tur-
+bulent (see also Fig. 1).
+This is not surprising given
+the strong perturbations affecting the molecular gas in
+JW100.
+The bottom right panel in Fig. 5 compares the circu-
+lar velocity profile of the stellar disk and the molecular
+gas in JW100. We recall that different kinematic centers
+were used for the stellar and molecular gas components.
+Interestingly, the circular velocity of the approaching
+side of the molecular gas disk coincides with that of
+the stellar disk, flattening at Vcirc ≈ 300 km s−1. On
+the contrary, the circular velocity of the receding side
+keeps on growing and reaches Vcirc ≈ 400 km s−1. Sim-
+ilarly to JO206, the asymmetric perturbations on the
+molecular gas disk and the displaced kinematic center
+are signatures of edge-on ram pressure stripping (Kro-
+nberger et al. 2008b). This is consistent with the fact
+that JW100 is falling into the cluster at very high ve-
+locity and its disk is seen at high inclination by the ob-
+server. We note that the circular velocity of JW100 rises
+less steeply than what is typically found in galaxies with
+similar stellar mass. Moreover, Figure 2 seems to sug-
+gest that JW100 potentially hosts a stellar bulge, which
+is expected to produce high circular velocities in the in-
+nermost regions of the galaxy. The shallow and rather
+unusual gradient of the circular velocity might be ex-
+plained by either the presence of a stellar bar aligned
+with the major axis or a dark matter halo with lower-
+than-average concentration (Randriamampandry et al.
+2015).
+Disentangling between these two possibilities
+would require a dedicated mass modelling of the system
+which goes beyond the purpose of this study.
+In conclusion, our results indicate that the molecular
+gas disk of JW100 is dramatically affected by ram pres-
+sure. The morphology and kinematics of the molecular
+gas disk indicate strong ram pressure both in the sky
+plane and along the line of sight. Gravitational inter-
+actions with other members in the same substructure
+may play a role, but we speculate that these effects are
+milder than ram pressure, as the stellar component is
+not as strongly perturbed as the molecular gas.
+5.5. Summary
+In Sect. 5.1, we showed that the molecular gas kine-
+matics in JO201 is dominated by the bar for R ≲ 5 kpc.
+At larger galactocentric distances, the rotation curve
+gradient is modified by some physical mechanisms, that
+is possibly face-on ram pressure. We note that, since
+JO201 belongs to a cluster substructure, we cannot ex-
+clude a different origin (e.g.
+tidal interactions).
+We
+do not find clear signature of ram pressure stripping
+(i.e. molecular gas removed from the main disk), but
+we tentatively identify outward radial flow of gas plau-
+sibly due to ram pressure. Both within and beyond the
+region influenced by the bar, the velocity dispersion of
+the molecular gas is enhanced with respect to the typ-
+ical values measured in field galaxies (Bacchini et al.
+2020b), suggesting strong turbulence motions. Beyond
+the bar region, this enhancement is about a factor 2
+(σCO ≈20 km s−1), which can be either a direct or indi-
+rect consequence of ram pressure increasing (or a com-
+bination of both). Indeed, the ram pressure can directly
+increase the gas kinetic energy, but it can also enhance
+the SFR (e.g. Kronberger et al. 2008a) and thus the ve-
+locity dispersion due to the transferring of the supernova
+energy to the gas (Bacchini et al. 2020b). The SFR of
+JO201 is about 2 times higher than field galaxies with
+similar stellar mass, which supports the second scenario.
+In Sect. 5.2, we found clear signatures of the pres-
+ence of a bar in JO204 based on the molecular gas dis-
+tribution (central concentration, arm-like overdensities)
+and kinematics (PVD shape, radial motions), and the
+stellar kinematics (velocity field). Radial motions are
+clearly present, but we cannot identify their direction
+with the available observations. A bar-driven inflow is a
+reasonable hypothesis. The molecular gas kinematics is
+dominated by rotation, while the ram pressure plays a
+secondary role and we do not find signatures of ram pres-
+sure stripping. We detect molecular gas with anomalous
+kinematics that is compatible with being decelerated by
+face-on ram pressure. We also find some molecular gas
+with high velocity in the central regions of the galaxy,
+but its origin is unclear. The estimated values of the
+molecular gas velocity dispersion (σCO ≈10 km s−1) are
+rather uncertain, but overall consistent with those typ-
+ical of field galaxies (Bacchini et al. 2020b). This is in
+line with the fact that JO204 does not show enhanced
+SFR with respect to field galaxies (Vulcani et al. 2018).
+Similarly to JO201, the molecular gas kinematics in
+JO206 is dominated by the bar for R ≲ 6 kpc (Sect. 5.3).
+
+Molecular gas kinematics in jellyfish galaxies
+17
+15
+10
+5
+0
+5
+10
+15
+Offset ["]
+200
+0
+200
+400
+600
+VLOS [km/s]
+= 269°
+Model fitted on the approaching side
+15
+10
+5
+0
+5
+10
+15
+Offset ["]
+200
+0
+200
+400
+600
+VLOS [km/s]
+= 269°
+Model fitted on the receding side
+10
+5
+0
+5
+10
+15
+Offset ["]
+200
+0
+200
+400
+600
+VLOS [km/s]
+= 179°
+10
+5
+0
+5
+10
+15
+Offset ["]
+200
+0
+200
+400
+600
+VLOS [km/s]
+= 179°
+0.0
+2.5
+5.0
+7.5
+10.0
+12.5
+R [kpc]
+0
+100
+Vrad [km/s]
+Outflow
+Inflow
+0.0
+2.5
+5.0
+7.5
+10.0
+12.5
+R [kpc]
+0
+100
+Outflow
+Inflow
+600
+400
+200
+0
+200
+400
+VLOS (km/s)
+10
+0
+10
+R [kpc]
+600
+400
+200
+0
+200
+400
+VLOS (km/s)
+10
+5
+0
+5
+10
+15
+600
+400
+200
+0
+200
+400
+VLOS (km/s)
+10
+0
+10
+R [kpc]
+600
+400
+200
+0
+200
+400
+VLOS (km/s)
+10
+5
+0
+5
+10
+15
+JW100
+Figure 8. Same as Fig. 4 but for JW100. In the left and right panels, the model fitting is respectively done for the approaching
+and receding sides, and the kinematic center is ≈ 1.6′′westward from the optical center. The inclination and PA are fixed to the
+values of 77° and 179°, respectively. Here σch = 1.1 mJy/beam.
+This seems also to drive inward radial flows of molecular
+gas. We find clear signatures of edge-on ram pressure
+stripping for R > 6 kpc.
+The velocity dispersion of
+the molecular gas is significantly enhanced (σCO ≈30-
+40 km s−1), a likely consequence of the complex motions
+due to the combined influence of the bar and the ram
+pressure.
+In JW100 (Sect. 5.4), the molecular gas distribution
+and kinematics indicate ongoing ram pressure stripping.
+Since JW100 belongs to a cluster substructure and prob-
+ably hosts a warped stellar disk, we cannot exclude that
+gravitational interactions may also play a role. We de-
+tect radial motions that are compatible with an outward
+gas flow. The shallow gradient of the circular velocity
+in the inner regions of JW100 may be explained by a
+stellar bar aligned with the disk major axis, although
+other possibilities cannot be excluded (e.g. combination
+of resolution effects and ram pressure, low-concentration
+dark matter halo). The velocity dispersion of the molec-
+ular gas is quite enhanced (σCO ≈30-60 km s−1), but the
+SFR of JW100 is about 2.5 times lower than field galax-
+ies with similar stellar mass (Vulcani et al. 2018). These
+properties favours the scenario in which the gas turbu-
+lence is directly enhanced by the ram pressure.
+6. COMPARISON WITH PREVIOUS WORKS
+6.1. The connection between stellar bars and AGN
+At least three out of four galaxies in the GASP-ALMA
+sample host a stellar bar. In the case of JW100, the
+presence of the bar is difficult to confirm but arguably
+plausible, given that about 60% of the disk galaxies with
+10 ≲ log(M⋆/M⊙) ≲ 11 host a stellar bar (Aguerri et al.
+2009; Masters et al. 2012; D´ıaz-Garc´ıa et al. 2016). The
+fraction of barred galaxies may be even higher in the
+central regions of clusters (e.g. Andersen 1996; Barazza
+et al. 2009; M´endez-Abreu et al. 2012; Lansbury et al.
+2014; Alonso et al. 2014), but this is likely due to the
+increase of early-type galaxies (which are less likely to
+
+18
+Bacchini et al.
+host a bar) with decreasing clustercentric distance (e.g.
+Tawfeek et al. 2022).
+The non-axisymmetric potential of a stellar bar can
+trigger radial inflow of gas by inducing torques and
+shocks (e.g. Athanassoula 1992b,a; Sellwood & Wilkin-
+son 1993; Sellwood 2014; Marasco et al. 2018), often en-
+hancing the molecular gas concentration in the central
+regions of galaxies (e.g. Sheth et al. 2005; Regan et al.
+2006; Yu et al. 2022).
+Bar-driven inflows of gas may
+play an important role in fueling the central black hole
+and triggering the AGN activity (e.g. Alonso et al. 2013,
+2018; Rosas-Guevara et al. 2020; Silva-Lima et al. 2022).
+However, this topic is debated and some authors showed,
+for instance, that the excess of AGN-hosts among barred
+galaxies vanishes when the dependence on the galaxy
+stellar mass and color are taken into account (e.g. Ho
+et al. 1997; Lee et al. 2012). Moreover, there are indica-
+tions that the bar alone is not always sufficient to feed
+the black hole in an efficient way, requiring the contribu-
+tion of other mechanisms (Combes 2008; Sellwood 2014;
+Fanali et al. 2015; Galloway et al. 2015). This is because,
+in order to feed the black hole, the molecular gas in the
+disk needs to lose almost all its angular momentum (e.g.
+Sellwood & Wilkinson 1993; Krolik 1999; Sellwood 2014;
+Capelo et al. 2022). Using a sample of spiral galaxies,
+Alonso et al. (2014) found that their location within the
+group or cluster influences both the AGN and bar frac-
+tion. This result suggests that the external mechanisms
+affecting galaxies in dense environments may give a sig-
+nificant contribution to triggering the AGN activity. For
+the specific case of jellyfish galaxies, this external mech-
+anism might be the interaction with the ICM (Poggianti
+et al. 2017a; Peluso et al. 2022). Indeed, the ram pres-
+sure can not only compress the gas in the disk, but it
+can also make the gas lose its angular momentum and
+eventually move inward (Ramos-Mart´ınez et al. 2018;
+Ricarte et al. 2020; Farber et al. 2022, Akerman et al.
+submitted).
+Thus, the gas could easily reach the re-
+gion influenced by the bar, which may drag it further
+inward, perhaps reaching the black hole. This picture is
+in agreement with the enhanced fraction of AGN in ram
+pressure stripped galaxies (e.g. Poggianti et al. 2017a;
+Peluso et al. 2022, but see Roman-Oliveira et al. 2019
+for a different conclusion). The relative importance of
+internal mechanisms, such as bars, and external pro-
+cesses, such as ram pressure, in fueling the AGN activity
+is a compelling and debated topic (Alonso et al. 2018;
+Kim & Choi 2020; Boselli et al. 2022a), which would
+require higher statistics than the four galaxies studied
+here. This task goes beyond the scope of this paper and
+we leave it to future studies.
+6.2. Comparison with Virgo galaxies
+There is growing evidence that the ram pressure can
+affect the molecular gas in cluster galaxies. In this sec-
+tion, we compare the GASP-ALMA sample with the
+galaxies in Virgo cluster, in order to increase the statis-
+tics. Other cases of ram pressure affecting the molecu-
+lar disk have been found in Coma (J´achym et al. 2017),
+Norma (J´achym et al. 2014), and Fornax (Zabel et al.
+2019), just to mention some examples.
+Lee et al. (2017) studied the molecular gas kinemat-
+ics in three disk galaxies.
+These author did not find
+clear signs of molecular gas stripping, but they showed
+that the morphological and kinematical disturbances in
+the molecular and atomic gas disks are closely related
+to each other, suggesting that the molecular gas can be
+also affected by strong ram pressure even if it is not
+globally stripped. They also ascribed the perturbation
+in the innermost regions of their galaxies to the pres-
+ence of a stellar bar, rather than to ram pressure. As
+discussed in Sects. 5.1 and 5.2, our results for JO201
+and JO204 are consistent with Lee et al. (2017)’s find-
+ings. Interestingly, all the molecular gas disks in Lee
+et al. (2017) sample are kinematically lopsided, at least
+to some degree, indicating that the molecular gas was
+either accelerated or decelerated by ram pressure (see
+also Cramer et al. 2020). They also found CO clumps
+that are kinematically decoupled from the molecular gas
+disk, suggesting that this gas was displaced by the ram
+pressure, as in the case of JO206 (see Sect. 5.3).
+Recently, Brown et al. (2021) presented the first re-
+sults of the Virgo Environment Traced in CO (VER-
+TICO) survey, which maps CO emission in 51 galaxies
+in Virgo cluster using ALMA. This authors derived the
+mass-size relation for the molecular gas disk for VER-
+TICO galaxies.
+They showed that the scatter in the
+relation is minimized if the disk size is defined as the ra-
+dius where the azimuthally-averaged H2 surface density
+reaches ΣH2 = 5 M⊙pc−2 (R5). As a control sample,
+Brown et al. (2021) used the field galaxies in the Het-
+erodyne Receiver Array CO Line Extragalactic Survey
+(HERACLES, Leroy et al. 2009). Brown et al. (2021)
+found that the best-fit relations for the galaxies in Virgo
+and in the field are consistent. They concluded that R5-
+MH2 relation does not depend on the environment, in
+agreement with the studies on the HI size–mass relation
+(Wang et al. 2016; Stevens et al. 2019), and that galax-
+ies affected by environmental processes move along the
+size-mass relation rather than deviating from it.
+In Fig. 9, we compare our galaxies with the R5-MH2 re-
+lation from Brown et al. (2021). We assumed the Milky
+Way CO-to-H2 conversion factor for consistency with
+Brown et al. (2021). Our galaxies are within the scatter
+
+Molecular gas kinematics in jellyfish galaxies
+19
+of the R5-MH2 relation derived by Brown et al. (2021),
+confirming that this scaling relation does not show any
+clear dependence on environment, even in extreme ram
+pressure cases as the galaxies of our sample. We note
+that the GASP-ALMA sample tend to be slightly below
+the relation, suggesting that the molecular gas distri-
+bution is more centrally concentrated than the average
+for these samples.
+This can be due to the combined
+effect of stellar bars, which tend to increase the gas den-
+sity in the inner regions of the disk (e.g. Kormendy &
+Kennicutt 2004), and ram pressure, which compresses
+the molecular disk. The GASP-ALMA galaxies stand
+out against the other two samples because of their high
+MH2, being up to ≈ 0.5 dex more massive than the Virgo
+and control samples (see also Moretti et al. 2020a,b). On
+the other hand, it has been shown that our galaxies are
+up to 50% deficient in HI with respect to field galax-
+ies with similar mass and size (Ramatsoku et al. 2019,
+2020; Deb et al. 2020; Healy et al. 2021; Deb et al. 2022).
+Taken together, these results suggest an unusually ef-
+ficient conversion of HI to H2 (Moretti et al. 2020b).
+These properties are in agreement with the recent re-
+sults by Zabel et al. (2022) for Virgo galaxies.
+They
+found that the galaxies showing clear signs of ongoing
+ram pressure stripping affecting the HI disk are from
+H2-normal to H2-rich. This was interpreted as an indi-
+cation that ram pressure stripping is not effective at re-
+ducing global molecular gas fractions on the timescales
+in which such features are still clearly visible. This is
+likely because the stripping is less severe on H2 than on
+HI, as the molecular gas is denser and more gravitation-
+ally bound to the galaxy than the atomic gas (Lee et al.
+2017; Boselli et al. 2022a). The atomic gas disk of our
+galaxies show indeed signs of truncation and the ram
+pressure stripping is much more dramatic than for the
+molecular gas disk (Ramatsoku et al. 2019, 2020; Deb
+et al. 2020, 2022)
+6.3. Baryonic Tully-Fisher relation
+Rotation curves of disk galaxies are typically used
+to derive fundamental scaling relations (e.g. Verheijen
+2001; Lelli et al. 2016b; Ponomareva et al. 2017; Io-
+rio et al. 2017; Posti et al. 2018; Mancera Pi˜na et al.
+2021a,b; Di Teodoro & Peek 2021). In particular, the
+baryonic Tully-Fisher relation (hereafter BTFR) is a
+very tight correlation between the mass of baryons and
+the rotation velocity of galaxies, being a useful test-case
+to check the robustness of the stellar rotation derived
+in this work.
+The BTFR is usually derived using HI
+rotation curves, as the atomic gas disk is the most ex-
+tended baryonic component, allowing to probe the flat
+part of the galaxy rotation curve. In jellyfish galaxies,
+6.5
+7.0
+7.5
+8.0
+8.5
+9.0
+9.5
+10.0
+10.5
+11.0
+log[MH2/M
+]
+1.0
+0.5
+0.0
+0.5
+1.0
+1.5
+log[R5/kpc]
+Molecular gas size-mass relation using R5
+Brown+2021
+±
+JO201
+JO204
+JO206
+JW100
+Virgo
+Field
+Figure 9.
+Molecular gas mass-size relation based on R5
+(see Sect. 6.2).
+Each galaxy in the GASP-ALMA sample
+is indicated by a colored symbol. Grey diamonds and pink
+points show galaxies in the Virgo cluster and nearby field
+galaxies (see text), respectively.
+The best-fit relation ob-
+tained by Brown et al. (2021) for the VERTICO and HER-
+ACLES samples is shown by the dash-dotted line, while its
+scatter is represented by the shaded area.
+the atomic gas disk is stripped or truncated by the ram
+pressure and the HI kinematics is strongly perturbed,
+hampering the usage of HI observations to study scal-
+ing relations.
+The ionized gas is not a good alterna-
+tive to HI, as not only it is less spatially extended but
+also more diffuse and thus easier to strip. The results
+of this work suggest that the molecular gas is more re-
+silient to ram pressure, but its spatial extend is still very
+limited. Therefore, the stellar component is likely the
+best way to derive scaling relations in the case of jelly-
+fish galaxies, provided that observations with high spa-
+tial resolution and sensitivity are available. The GASP
+sample is ideal to perform this exercise, thanks to the
+high spatial resolution and sensitivity of the MUSE ob-
+servations.
+In Fig. 10, we show that the galaxies in
+the GASP-ALMA sample closely follow the BTFR de-
+rived by Di Teodoro et al. (2021) using a sample of
+about 200 galaxies from high-mass disks to dwarf galax-
+ies (Lelli et al. 2019). We calculated the velocity in the
+flat part of the rotation curve (Vflat) as the average of
+the outermost 5 measurements of the stellar rotation ve-
+locity (see Fig. 5). The baryonic mass was calculated as
+Mbar = M⋆ + 1.33 (MHI + MH2), where the masses of
+atomic gas (MHI) and molecular gas (MH2) are taken
+from Table 1 and the multiplicative factor 1.36 accounts
+for the Helium content.
+We checked that considering
+only the gas mass within the stellar disk or the total gas
+mass (including the gas in the stripped tail) does not
+
+20
+Bacchini et al.
+1.00
+1.25
+1.50
+1.75
+2.00
+2.25
+2.50
+2.75
+3.00
+log[Vflat/(km/s)]
+7
+8
+9
+10
+11
+12
+13
+log[Mbar/M
+]
+Baryonic Tully-Fisher relation
+Best-fit (Di Teodoro+21)
+y = 3.6x + 2.49
+JO201
+JO204
+JO206
+JW100
+= 0.1 dex
+Lelli+19
+Di Teodoro+21
+Figure 10.
+Baryonic Tully-Fisher relation for the four
+galaxies in the GASP-ALMA sample (triangles, diamond,
+and cross). The grey points show the spiral and dwarf galax-
+ies from Lelli et al. (2019), while the pink stars are for the
+massive disks from Di Teodoro et al. (2021).
+The dashed
+line is their best-fit relation with the shaded area showing
+the orthogonal intrinsic scatter.
+change the results, as the gas mass is largely dominated
+by molecular gas component which is mostly concen-
+trated within the galaxy disk.
+We also checked that
+our galaxies fall on the stellar Tully-Fisher relation (not
+shown here), which is not surprising given that the bary-
+onic mass is largely dominated by the stellar component.
+These simple tests indicate that the GASP sample can
+be used to study important scaling relations of baryons
+and, potentially, dark matter (e.g. Lelli et al. 2016b;
+Posti et al. 2018; Mancera Pi˜na et al. 2021a; Di Teodoro
+et al. 2022). This will be addressed in future work by
+fully exploiting the richness and quality of the MUSE
+observations obtained with the GASP survey (Bacchini
+et al., in preparation).
+7. SUMMARY AND CONCLUSIONS
+Galaxies in dense environments, such as clusters, can
+be affected by the ram pressure due to the interaction
+with the ICM. This process leaves the stellar disk es-
+sentially unperturbed, but it can have a strong impact
+on the morphology, kinematics and overall gas content,
+with important consequences on the evolution of galax-
+ies (Cortese et al. 2021). In this context, jellyfish galax-
+ies are ideal cases to study the impact of ram pressure
+on the gas components. In this work, we have studied
+the distribution and kinematics of the molecular gas in
+a sample of four jellyfish galaxies in the GASP sample
+(Poggianti et al. 2017b). These galaxies were observed
+with ALMA to detect the CO(1–0) and CO(2–1) emis-
+sion Moretti et al. (2020a,b). Thanks to the wealth of
+information obtained from MUSE and ALMA observa-
+tions provided by the GASP survey, we could analyze
+the stellar and CO distribution and kinematics. We used
+the software 3DB based on the tilted-ring approach to
+model the stellar velocity field and the CO emission line
+datacubes. We identified the gas with anomalous veloc-
+ity that cannot be explained by a rotation disk and used
+the information on the stellar distribution and kinemat-
+ics to understand the origin of this anomalous gas. We
+reached the following conclusions.
+1. At least three (JO201, JO204, and JO206) out
+of four galaxies in the GASP-ALMA sample are
+barred.
+In JO201 and JO206, the bars aligned
+with the disk major axis are visible in the I-band
+images and explain the shallow gradient of the cir-
+cular velocity in the inner regions of these galax-
+ies. In JO204, various bar signatures are found in
+the distribution of the molecular gas and the kine-
+matics of both the molecular gas and the stars. In
+JW100, the disk inclination and dust obscuration
+do not allow us to unambiguously identify a bar.
+2. The molecular gas kinematics in JO201 and JO206
+is mainly dominated by non-circular motions in
+the region influenced by the bar, while the ram
+pressure becomes important at larger galactocen-
+tric distance. The ram pressure plays a secondary
+role for the molecular gas kinematics of JO204,
+which is mainly rotation-dominated. Clear indi-
+cations of molecular gas stripping are found in
+two galaxies, JO206 and JW100. In JO206, some
+molecular gas is detached and kinematically de-
+coupled from the main disk. In JW100, the molec-
+ular gas disk is displaced with respect to the stel-
+lar disk and its kinematics is strongly perturbed.
+Since JO201 and JW100 belong to cluster sub-
+structures, other mechanisms than ram pressure
+might be also at play.
+3. Radial flows of molecular gas are manifestly
+present in two galaxies (JO204 and JW100), but
+this is less clear in the other two objects (JO201
+and JO206).
+These gas flows are consistent
+with being bar-driven inflows in JO206 and ram
+pressure-driven outflows in JO201 and JW100.
+The direction of radial motions remains unclear
+for JO204.
+4. The molecular gas velocity dispersion in JO201,
+JO206, and JW100 tends to be enhanced with re-
+spect to field galaxies, suggesting that the gas is
+very turbulent. In the case of JO201 and JO206,
+
+Molecular gas kinematics in jellyfish galaxies
+21
+this can be explained by the complex motions in-
+duced by the bar within its region of influence
+or, beyond the bar region, by the the ram pres-
+sure, which can enhance the gas turbulence di-
+rectly and/or by increasing the SFR. In the case
+of JW100, the most likely scenario is that the gas
+turbulence is directly enhanced by the ram pres-
+sure.
+5. Our galaxies fall within the scatter of the molec-
+ular gas mass-size relation derived for field and
+Virgo galaxies by (Brown et al. 2021), confirming
+that the relation is essentially independent of en-
+vironment.
+Overall, our results are consistent with a scenario in
+which the molecular gas is affected by ram pressure on
+different timescales and less severely than the atomic
+and ionized gas, likely because the molecular gas is
+denser and more gravitationally bound to the galaxy
+than the other gas phases. The galaxies in the GASP-
+ALMA sample host an AGN (Poggianti et al. 2017a;
+Peluso et al. 2022). Both stellar bars and ram pressure
+can contribute to efficiently drive molecular gas towards
+the galaxy center, possibly feeding the central black hole
+and triggering the nuclear activity. Since the relative im-
+portance of bars and ram pressure in fueling the AGN
+has not been fully understood yet, we hope that our
+work may foster future studies. In this work, we have
+shown that high-resolution observations of the molecular
+gas emission can be very useful in identifying stellar bars
+and radial flows. Future effort will be devoted to fur-
+ther study the bar-AGN connection by expanding the
+GASP-ALMA sample. Moreover, we have shown that
+the GASP sample is potentially very useful to investi-
+gate the impact of environment on scaling relations of
+galaxies. In future work, we plan to address this topic
+by fully exploiting the richness and quality of the MUSE
+observations obtained with the GASP survey.
+This paper makes use of the following ALMA data:
+ADS/JAO.ALMA#2017.1.00496.S. ALMA is a partner-
+ship of ESO (representing its member states), NSF
+(USA) and NINS (Japan), together with NRC (Canada)
+and NSC and ASIAA (Taiwan), in cooperation with the
+Republic of Chile. The Joint ALMA Observatory is op-
+erated by ESO, AUI/NRAO and NAOJ. CB acknowl-
+edges financial support from the European Research
+Council (ERC) under the European Union’s Horizon
+2020 research and innovation programme (grant agree-
+ment No.
+833824).
+CB would like to thank E. Di
+Teodoro, F. Rizzo, and F. Fraternali, for useful discus-
+sions and the help with the kinematic modeling.
+Facility: ALMA, MUSE@VLT
+Software:
+3DBarolo (Di Teodoro & Fraternali
+2015), APLpy (Robitaille & Bressert 2012), Astropy (As-
+tropy Collaboration et al. 2013, 2018).
+APPENDIX
+A. BEST-FIT MODELS OF THE STELLAR VELOCITY FIELD
+This section provides the best-fit model of the stellar disk for JO204, JO206, and JW100.
+The top panels in
+Figs. 11, 12, and 13 show the observed stellar velocity field, the best-fit model, and the map of the residuals. The
+bottom panels display the stellar rotation curve and the radial profiles of the PA and inclination. Overall, the stellar
+kinematics is well reproduced by our models. However, the residual map of JO204 clearly shows a pattern in the
+central regions of the disk, which likely indicates the presence of a stellar bar (see also Sect. 5.2). The residual map
+of JW100 highlights some regions where the model does not fully reproduces the observations. The origin of these
+differences is tricky to understand and may be due to asymmetric dust observation or the warp along the line of sight,
+or both (see also Sect. 5.4).
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+
+22
+Bacchini et al.
+0
+2
+4
+6
+8
+10
+12
+R [kpc]
+0
+50
+100
+150
+200
+250
+Rotation velocity [km/s]
+2nd fit
+3rd fit
+0
+2
+4
+6
+8
+10
+12
+R [kpc]
+70
+72
+74
+76
+78
+80
+Inclination [degrees]
+2nd fit
+Median
+3rd fit
+± MAD
+0
+2
+4
+6
+8
+10
+12
+R [kpc]
+318
+320
+322
+324
+326
+328
+Position angle [degrees]
+2nd fit
+Median
+3rd fit
+± MAD
+10h13m48.0s 47.5s
+47.0s
+46.5s
+46.0s
+-0°54'35"
+40"
+45"
+50"
+55"
+55'00"
+05"
+RA (ICRS)
+Dec (ICRS)
+1"
+JO204 - Data
+5 kpc
+200
+150
+100
+50
+0
+50
+100
+150
+200
+VLOS [km/s]
+10h13m48.0s 47.5s
+47.0s
+46.5s
+46.0s
+RA (ICRS)
+JO204 - Model
+200
+150
+100
+50
+0
+50
+100
+150
+200
+VLOS [km/s]
+10h13m48.0s 47.5s
+47.0s
+46.5s
+46.0s
+RA (ICRS)
+JO204 - Residuals
+40
+20
+0
+20
+40
+Data-Model [km/s]
+0
+2
+4
+6
+8
+10
+12
+14
+16
+R [arcsec]
+0
+2
+4
+6
+8
+10
+12
+14
+16
+R [arcsec]
+0
+2
+4
+6
+8
+10
+12
+14
+16
+R [arcsec]
+Figure 11. Same as Fig. 3 but for JO204.
+0
+5
+10
+15
+20
+R [kpc]
+0
+50
+100
+150
+200
+250
+Rotation velocity [km/s]
+2nd fit
+3rd fit
+0.0
+2.5
+5.0
+7.5
+10.0
+12.5
+15.0
+17.5
+20.0
+R [kpc]
+35
+40
+45
+50
+55
+60
+65
+70
+Inclination [degrees]
+2nd fit
+Median
+3rd fit
+± MAD
+0.0
+2.5
+5.0
+7.5
+10.0
+12.5
+15.0
+17.5
+20.0
+R [kpc]
+106
+108
+110
+112
+114
+116
+118
+120
+122
+Position angle [degrees]
+2nd fit
+Median
+3rd fit
+± MAD
+21h13m49s
+48s
+47s
+46s
+2°28'45"
+30"
+15"
+RA (ICRS)
+Dec (ICRS)
+1"
+JO206 - Data
+5 kpc
+200
+150
+100
+50
+0
+50
+100
+150
+200
+VLOS [km/s]
+21h13m49s
+48s
+47s
+46s
+RA (ICRS)
+JO206 - Model
+200
+150
+100
+50
+0
+50
+100
+150
+200
+VLOS [km/s]
+21h13m49s
+48s
+47s
+46s
+RA (ICRS)
+JO206 - Residuals
+40
+20
+0
+20
+40
+Data-Model [km/s]
+0
+5
+10
+15
+20
+R [arcsec]
+0
+5
+10
+15
+20
+R [arcsec]
+0
+5
+10
+15
+20
+R [arcsec]
+Figure 12. Same as Fig. 3 but for JO206.
+
+Molecular gas kinematics in jellyfish galaxies
+23
+0
+5
+10
+15
+20
+25
+R [kpc]
+0
+50
+100
+150
+200
+250
+300
+350
+400
+Rotation velocity [km/s]
+2nd fit
+3rd fit
+0
+5
+10
+15
+20
+25
+R [kpc]
+67.5
+70.0
+72.5
+75.0
+77.5
+80.0
+82.5
+Inclination [degrees]
+2nd fit
+Median
+3rd fit
+± MAD
+0
+5
+10
+15
+20
+25
+R [kpc]
+177
+178
+179
+180
+181
+182
+183
+184
+Position angle [degrees]
+2nd fit
+Median
+3rd fit
+± MAD
+23h36m26s
+25s
+24s
+21°09'15"
+00"
+08'45"
+RA (ICRS)
+Dec (ICRS)
+1"
+JW100 - Data
+5 kpc
+300
+200
+100
+0
+100
+200
+300
+VLOS [km/s]
+23h36m26s
+25s
+24s
+RA (ICRS)
+JW100 - Model
+300
+200
+100
+0
+100
+200
+300
+VLOS [km/s]
+23h36m26s
+25s
+24s
+RA (ICRS)
+JW100 - Residuals
+40
+20
+0
+20
+40
+Data-Model [km/s]
+0
+5
+10
+15
+20
+R [arcsec]
+0
+5
+10
+15
+20
+R [arcsec]
+0
+5
+10
+15
+20
+R [arcsec]
+Figure 13. Same as Fig. 3 but for JW100.
+
+24
+Bacchini et al.
+Astropy Collaboration, Robitaille, T. P., Tollerud, E. J.,
+et al. 2013, A&A, 558, A33,
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+page_content='Draft version January 10, 2023  Typeset using LATEX twocolumn style in AASTeX631  3D modeling of the molecular gas kinematics in optically-selected jellyfish galaxies  Cecilia Bacchini  ,1 Matilde Mingozzi  ,2 Bianca M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
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+page_content='1 Alessia Moretti  ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='1  Marco Gullieuszik  ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='1 Antonino Marasco  ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='1 Bernardo Cervantes Sodi  ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='3 Osbaldo S´anchez-Garc´ıa  ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='3  Benedetta Vulcani  ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='1 Ariel Werle  ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='1 Rosita Paladino  ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='4 and Mario Radovich  1  1INAF - Osservatorio Astronomico di Padova,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' vicolo dell’Osservatorio 5,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
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+page_content=' Italy  2Space Telescope Science Institute,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' 3700 San Martin Drive,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' Baltimore,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' MD 21218,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' USA  3Istituto de Radioastronom´ıa y Astrof´ısica,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' Universidad Nacional Aut´onoma de M´exico,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' Campus Morelia,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' 3-72, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' 58089,  Michoac´an, M´exico  4INAF - Istituto di Radioastronomia, via P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' Gobetti 101, I-40129 Bologna, Italy  ABSTRACT  Cluster galaxies are subject to the ram pressure exerted by the intracluster medium, which can  perturb or even strip away their gas while leaving the stars unperturbed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' We model the distribution  and kinematics of the stars and the molecular gas in four late-type cluster galaxies (JO201, JO204,  JO206, and JW100), which show tails of atomic and ionized gas indicative of ongoing ram pressure  stripping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' We analyze MUSE@VLT data and CO data from ALMA searching for signatures of radial  gas flows, ram pressure stripping, and other perturbations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' We find that all galaxies, with the possible  exception of JW100, host stellar bars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' Signatures of ram pressure are found in JO201 and JO206,  which also shows clear indications of ongoing stripping in the molecular disk outskirts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' The stripping  affects the whole molecular gas disk of JW100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' The molecular gas kinematics in JO204 is instead  dominated by rotation rather than ram pressure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' We also find indications of enhanced turbulence of  the molecular gas compared to field galaxies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' Large-scale radial flows of molecular gas are present in  JO204 and JW100, but more uncertain in JO201 and JO206.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' We show that our galaxy sample follows  the molecular gas mass-size relation, confirming that it is essentially independent of environment even  for the most extreme cases of stripping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' Our findings are consistent with the molecular gas being  affected by ram pressure on different timescales and less severely than the atomic and ionized gas  phases, likely because the molecular gas is denser and more gravitationally bound to the galaxy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' INTRODUCTION  In dense environments, such as groups and clusters,  galaxies are affected by various physical mechanisms  that can significantly influence their properties and evo-  lution (Nulsen 1982;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' Boselli & Gavazzi 2006;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' Cortese  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' These processes are usually divided into  gravitational and hydrodynamical interactions (Boselli  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' 2022a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='  Gravitational perturbations can be in-  duced by tidal forces due to the potential of other clus-  ter/group members (Merritt 1983) or the large scale  structure itself (Byrd & Valtonen 1990), but also by fly-  by encounters and mergers (Barnes & Hernquist 1992;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='  Kronberger et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' 2006).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' Gravitational interactions af-  fect both the stellar and the gaseous components of  galaxies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' Instead, the hydrodynamical interactions be-  Corresponding author: Cecilia Bacchini  cecilia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='bacchini@inaf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='it  tween the galactic interstellar medium (ISM) and the in-  tracluster medium (ICM) are expected to influence only  the gaseous components of galaxies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' These mechanisms  are the thermal evaporation of the cold (T ≲ 104 K) ISM  due to the interaction of the hot (T ≈ 107 −108 K) ICM  (Cowie & Songaila 1977), the removal of the outer ISM  layer due to the viscosity momentum transfer with the  ICM (viscous stripping) or instabilities (Nulsen 1982;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='  Roediger & Hensler 2008), and the ram pressure strip-  ping, that is the removal of the ISM due to the pressure  exerted by the ICM while a galaxy is moving through  the cluster (Gunn & Gott 1972).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' In addition, the in-  teraction with the ICM can heat up or strip away the  hot (T ≈ 106 K) gas corona surrounding galaxies and  prevent them from accreting new gas, finally quenching  star formation (starvation;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' Larson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' 1980).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='  Ram pressure is often considered among the domi-  nant mechanisms affecting the ISM in cluster galaxies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='  The ram pressure can be calculated as Pram = ρICMV 2  gal,  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='03090v1  [astro-ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='GA]  8 Jan 2023  ID2  Bacchini et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='  where ρICM and Vgal are the ICM density and the galaxy  velocity relative to the cluster (Gunn & Gott 1972).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='  Hence, this mechanism is expected to be particularly  strong for galaxies with high Vgal located close to the  cluster center, where ρICM is the highest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' The ram pres-  sure can have different effects on the gas distribution  and kinematics in galaxies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' The compression of the gas  disk can make it morphologically lopsided and asymmet-  ric (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' Mapelli et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' 2008;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' Kronberger et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' 2008b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='  Depending on its direction with respect to the galaxy  rotation, the ram pressure can decelerate one side of the  gas disk and accelerate the other, resulting in a kinemat-  ically lopsided disk, or even shift the kinematic center  of the gas disk with respect to the optical center of the  galaxy (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' Kronberger et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' 2008b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='  Typical signa-  tures of ram pressure stripping are one-sided tails of gas  extending outside the stellar disk and gas clouds that are  spatially detached and kinematically decoupled from the  galaxy (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' Chung et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' 2007;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' Merluzzi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' 2013;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' Lee  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' Moreover, gas disks in cluster galaxies are  sometimes less extended and less massive that those in  field galaxies (Chamaraux et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' 1980;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' Haynes et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' 1984;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='  Cayatte et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' 1990;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' Schr¨oder et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' 2001;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' Chung et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='  2009).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' Both truncation and gas deficiency are proper-  ties ascribed to ram pressure stripping, being relatively  common in both low- (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' Chung et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' 2007;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' Gavazzi  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' 2018) and intermediate-redshift cluster galaxies  (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' Cortese et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' 2007;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' Boselli et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' Moretti et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='  2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' The efficiency of ram pressure stripping depends  on the gas properties, being more effective on a diffuse  medium than on dense gas clumps, and on the gravita-  tional pull caused by the galactic potential, that weak-  ens with increasing galactocentric distance and height  above the midplane (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' Gunn & Gott 1972;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' Tonnesen  & Bryan 2009;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' K¨oppen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='  The atomic gas in galaxies is relatively diffuse and  typically distributed in a disk that is very extended (up  to twice the stellar disk diameter;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' Verheijen  & Sancisi 2001;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' Lelli et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' 2016a)  and thick (up to ≈1 kpc;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' Olling 1996;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' Yim  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' 2011, 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' Marasco et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' Bacchini et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='  2019a,b, 2020b), being very susceptible to ram pressure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='  Indeed, cluster galaxies often contain less atomic gas  than expected from their optical size or stellar mass and  have truncated and/or asymmetric HI discs (Chama-  raux et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' 1980;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' Haynes et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' 1984;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' Giovanelli & Haynes  1985;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' Cayatte et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' 1990;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' Solanes et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' 2001;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' Schr¨oder  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' 2001;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' Waugh et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' 2002;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' Chung et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' 2009;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' Loni  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' Zabel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' Morever, long tails of  atomic gas are commonly observed in cluster galaxies  (Bravo-Alfaro et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' 2000;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' Kenney et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' 2004;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' Chung  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' 2007;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' Scott et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' 2010;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' Sorgho et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' Ramat-  soku et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' 2019, 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' Deb et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' Healy et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='  Deb et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='  The molecular gas is typically denser than the atomic  gas (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' Leroy et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' 2008) and its distribution is also  less extended in both the radial (up to the stellar disk  diameter;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' Davis et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' 2013;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' Brown et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' Zabel  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' 2022) and vertical (up to ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='3 kpc;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' Yim et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='  2011, 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' Marasco et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' Bacchini et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' 2019a,b)  directions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' Hence, it is expected that the molecular gas  is more resilient to ram pressure than the atomic gas  (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' Lee et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' Brown et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' Zabel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='  2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' Boselli et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' 2022a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' Nevertheless, there is grow-  ing observational evidence that the ram pressure actu-  ally influences the molecular gas in cluster galaxies, as  indicated by signatures of compression, kinematic lop-  sidedness, and shifts between the optical and kinematic  center (Lee et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' Zabel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' Cramer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='  2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' Direct observations of molecular gas stripping by  ram pressure are limited, but tails and blobs of molec-  ular gas far from the stellar disk have been observed in  some cluster galaxies (Vollmer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' 2008;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' J´achym et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='  2014, 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' Lee et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' Moretti et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' 2018, 2020a),  as well as truncated molecular gas disks and H2-deficient  galaxies (Fumagalli et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' 2009;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' Boselli et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' Zabel  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' 2019, 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' Lee et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' However, a few au-  thors have found that cluster galaxies can also host a  normal (both in size and mass) or even enhanced reser-  voir of molecular gas (Fumagalli et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' 2009;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' Moretti  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' 2020b;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' Brown et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' Zabel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' 2022), pos-  sible indication that ram pressure can increase the effi-  ciency of the HI-to-H2 conversion (Moretti et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' 2020a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='  In this work, we analyze the molecular gas distribution  and kinematics in four cluster galaxies observed with  the Atacama Large Millimeter Array (ALMA).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' These  objects are part of the sample of 114 galaxies observed  within the Large Program ”GAs Stripping Phenomena  in galaxies with MUSE” (GASP), which is a survey  carried out with integral-field Multi Unit Spectroscopic  Explorer (MUSE) at the Very Large Telescope (VLT).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='  The GASP survey aims at understanding the impact of  environment on the evolution of galaxies by studying  their stellar and ionized gas emission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' Recently, follow-  up programs have provided multi-wavelength observa-  tions for a few galaxies in the GASP sample, allowing  to study other ISM components, such as atomic gas (Ra-  matsoku et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' 2019, 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' Deb et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' Healy et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='  2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' Deb et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' Luber et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' 2022), the molec-  ular gas (Moretti et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' 2018, 2020a,b), and magnetic  fields (M¨uller et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' 2021), and also young stellar pop-  ulations (George et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' An unexpected result of  the GASP project was the high fraction of active galac-  tic nuclei (AGN) among ram pressure-stripped galaxies  Molecular gas kinematics in jellyfish galaxies  3  (Poggianti et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' 2017a;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' Peluso et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' Poggianti &  the GASP team 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' This result was interpreted as an  indication that ram pressure can drive gas flows towards  the center and feed the AGN activity (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' Ricarte et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='  2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='  The galaxies analyzed in this work (hereafter  referred as the GASP-ALMA sample) were studied by  Poggianti et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' (2017a) and host indeed an AGN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' This  paper aims at answering the following open questions  about the GASP-ALMA galaxies: What is the impact  of ram pressure on the distribution and kinematics of  the molecular gas?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' Can we detect inflows of molecular  gas that may feed the AGN?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='  This paper is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' Section 2 presents  the GASP-ALMA sample and summarizes the relevant  pieces of information obtained by previous studies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' We  describe the data and methods used to carry out the  analysis in Sects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' 3 and 4, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' For each galaxy,  we present and discuss the results in Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' In Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' 6,  we compare our findings with other works in the litera-  ture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' Section 7 this work and its conclusions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='  We adopt standard cosmological parameters (h = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='7,  ΩM = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='3, and Ωλ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='7) and a Chabrier (2003) initial  mass function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' THE GALAXY SAMPLE  The GASP-ALMA sample consists of four late-type  galaxies, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='  JO201, JO204, JO206, and JW100, lo-  cated in different clusters at redshift 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='04 ≲ z ≲ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='06 and  with relatively high stellar mass (see Table 1 and refer-  ences therein).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' These galaxies are classified as “jellyfish”  because they show one-sided tails of ionized gas longer  than the stellar disk diameter (Poggianti et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' 2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='  Thanks to the wealth of information provided by the  GASP project and the availability of multi-wavelength  observations, these galaxies have been extensively stud-  ied in the literature (for a brief review, see Poggianti &  the GASP team 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' Thus, we summarize some of the  previous works that are relevant for our analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='  The galaxies in our sample are moving through the  ICM with either super-sonic or transonic line-of-sight ve-  locities and are located close to the cluster center (Gul-  lieuszik et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' These properties indicate that the  galaxies are in favorable conditions for strong ram pres-  sure and move on very radial orbits, suggesting that  they have recently entered into the cluster for the first  time (Yoon et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' Jaff´e et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' While JO204  and JO206 are relatively isolated for being cluster mem-  bers (Gullieuszik et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' Biviano et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' 2017), JO201  and JW100 belong to a substructure of four and three  galaxies, respectively (Bellhouse et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' Poggianti  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' Previous works show that, in all the GASP-  ALMA galaxies, the stellar kinematics appears to be  quite regular, while the ionized gas kinematics is very  perturbed, as expected for galaxies undergoing ram pres-  sure stripping (Bellhouse et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' Gullieuszik et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='  2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' Poggianti et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' 2017b;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' Jaff´e et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' Poggianti  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='  Recent works showed that these galax-  ies host strongly asymmetric HI disks with long tails  of atomic gas, and also have significantly reduced the  HI content with respect to field galaxies (≳ 50 %, Ra-  matsoku et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' 2019, 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' Deb et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' Healy et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='  2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' Deb et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' This HI deficiency is however  not coupled with a deficiency in the molecular gas reser-  voir, as these galaxies have H2 masses that are 4-5 times  higher than expected for galaxies with similar stellar  mass (Moretti et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' 2020a,b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='  As mentioned above, the galaxies in the GASP-ALMA  sample host an AGN (Poggianti et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' 2017a;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' Radovich  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' Peluso et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' It has been shown that  the AGN is the main source of gas ionization in the cen-  tral regions of these galaxies and, except for JO206, it  also causes a low-velocity (≈ 250 − 320 km s−1) wind  of ionized gas (Radovich et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' Poggianti et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='  (2017a) proposed that the AGN activity in these galax-  ies is triggered by the ram pressure, which can decrease  the angular momentum of the gas and favor its inflow  toward the center.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' In JO201, George et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' (2019) ob-  served a cavity of about 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='6 kpc with reduced ultravi-  olet and CO flux around the AGN (see also Radovich  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' By combining optical (MUSE) and sub-  mm (ALMA) spectroscopic observations, these authors  proposed that the cavity is due to AGN feedback that is  either ionizing or sweeping away the gas, possibly reduc-  ing the star formation activity in the central regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' In  JO204, Deb et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' (2020) found a redshifted absorption  feature in the HI global profile, which could be ascribed  to either a clumpy and fast rotating HI disc seen in front  of the central radio continuum source or an inflow of  atomic gas towards the central AGN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' DATA  This section describes the multi-wavelength observa-  tions and data products that were used in this work,  which primarily focuses on the molecular gas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' Since the  ram pressure can influence the kinematics and geometry  of the molecular gas disk, we analyze the stellar kine-  matics and use it as a reference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' Sections 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='1 and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='2  describe the data used to study the molecular gas and  stellar component, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' ALMA data  We used CO(1–0) and CO(2–1) emission line obser-  vations obtained with ALMA during Cycle 5 (project  4  Bacchini et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='  Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' Properties of the galaxy sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='  Property  Galaxy  JO201  JO204  JO206  JW100  Alternative names  KAZ 364  2MASX J10134686-0054514  2MASX J21134738+0228347  IC 5337  Morphological type  Sab  Sab  Sb  Sa  Cluster  Abell 85  Abell 957  IIZW108  Abell 2626  zclu  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='05568  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='04496  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='04889  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='05509  zgal  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='044631  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='042372  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='051089  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='061891  Vgal [km s−1]  3138  743  629  1932  |Vgal/σclu|  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='7  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='9  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='0  Rclu/R200,cl  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='18  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='09  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='29  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='06  Distance [Mpc]  189.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='1  179.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='8  216.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='3  261.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='4  Physical scale [kpc/arcsec]  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='88  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='84  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='00  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='19  Center R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' [J2000]  00:41:30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='30  10:13:46.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='84  21:13:47.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='41  23:36:25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='05  Center DEC [J2000]  09:15:45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='9  00:54:51.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='27  +02:28:35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='50  +21:09:02.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='64  M⋆ [1010 M⊙]  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='2 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='8  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='1 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='6  9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='1 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='9  29 ± 7  MHI [109 M⊙]  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='7 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='5  ≳ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='3 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='1  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='2 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='9  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='8 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='8  MH2 [109 M⊙]  11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='5 ± 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='8  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='7 ± 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='9  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='6 ± 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='8  16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='5 ± 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='3  Note—Galaxy names are in GASP convention;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' alternative names are also provided.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' Morphological types are from Fasano et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='  (2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='  Cluster redshifts (zclu) are from Biviano et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' (2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='  Galaxy redshifts (zgal) and distances from the cluster center  (Rclu/R200,cl) are from Bellhouse et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' (2017);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' Gullieuszik et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' (2017);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' Poggianti et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' (2017b, 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' Galaxy velocities relative  to cluster are calculated as Vgal = c(zgal − zclu)/(1 + zclu).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' The optical center coordinates are from Poggianti et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' (2017a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='  Stellar masses (M⋆) and effective radii (Reff) are from Vulcani et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' (2018) and Franchetto et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' (2020), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' Atomic gas  masses (MHI) are from Ramatsoku et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' (2019, 2020) and Deb et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' (2020, 2022);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' a conservative uncertainty of 30% is assumed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='  Molecular gas masses (MH2) are from Moretti et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' (2020a);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' a conservative uncertainty of 50% is assumed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='  2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='00496.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='S;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' PI: Poggianti).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' These observations were  already used by Moretti et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' (2020b) to study the  molecular gas content of the GASP-ALMA galaxies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='  The datacubes used in this work are different from those  used by Moretti et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' (2020b), as the imaging proce-  dure was re-performed to increase the spectral resolu-  tion (Mingozzi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' to be submitted).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' Here, we use  ALMA datacubes with spatial resolution of ≈ 1−2′′(see  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' 1) and velocity resolution of 10 km s−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' Figure 1  shows, for each galaxy, the moment maps of the CO(1–  0) datacubes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='  These maps were obtained from the  ALMA datacubes by applying a mask made by all pixels  with signal-to-noise ratio (S/N) above 3 in a datacube  smoothed by a factor 2 (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' in which each channel map  is convolved with a beam 2 times larger than the original  one).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='  We note that Moretti et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' (2020a,b) detected faint  CO emission coming from the regions outside the stel-  lar disk and coinciding with the ionized gas tails.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' The  emission is not visible in the maps used in this work  (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' 1), despite we used the same ALMA observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='  This difference is due to the fact that our datacubes  have better velocity resolution (∆υ = 10 km s−1) but  lower S/N than those used by Moretti et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' (2020a,b),  which have ∆υ = 20 km s−1and to the different masking  procedure adopted in the two works.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' For this work, we  used the datacubes with ∆υ = 10 km s−1, being the best  compromise to have good velocity resolution and S/N in  the regions within (or close to) the stellar disk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' We note  that we do not find significant differences in mass and  size of the molecular gas disk with respect to the results  obtained by Moretti et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' (2020a,b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' MUSE data  To analyze the stellar component, we used the MUSE  observations obtained by the GASP survey (Poggianti  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' 2017b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' The data reduction and processing is de-  tailed in Poggianti et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' (2017b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' The wavelength cov-  erage and spectral resolution of the final datacubes are  4800 ˚A < λ < 9300˚A and 1770 < R < 3590, respec-  tively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' The pixel size is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='2 ′′×0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='2 ′′with a natural seeing  of ≈ 1 ′′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' In this work, we use the I-band images and  the stellar velocity fields extracted from the MUSE dat-  acubes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='  The I-band images are very useful to identify stellar  substructures, such as bulges and bars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' These are typ-  Molecular gas kinematics in jellyfish galaxies  5  0h41m31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
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+page_content=' Total intensity map (left column), velocity field (central column) and velocity dispersion map (right column) obtained  from the CO(1-0) emission line datacubes for the four galaxies in the sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' The stars and the white dashed ellipse indicate the  kinematic center and the region used for modelling the gas kinematics, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' In JW100, the grey cross shows the optical  center.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' In the total intensity map, the red contours are at 2nσtot, where σtot is the noise in the total map (Lelli et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='  Iorio et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' 2017), n = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='10, and σtot = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='9, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='4, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='4, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='7 mJy/beam km s−1for JO201, JO204, JO206, and JW100, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='  In the velocity field, the black curves are the iso-velocity contours, with the thick one being at the galaxy systemic velocity (see  Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='2 for details).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' The white dotted line in the velocity field is the kinematic major axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' The bar and the ellipse in the  bottom left corner respectively show the physical scale and beam of the observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' The grey contours show the stellar disc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='  East is to the left and North to the top.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='  6  Bacchini et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='  ically dominated by intermediate-age (> 1 Gyr) stellar  populations and hence generally brighter in I-band than  at short wavelengths (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' Knapen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' 2000).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='  Fig-  ure 2 shows, for each galaxy, the I-band images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' These  were obtained by Franchetto et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' (2020) from the inte-  grated MUSE datacubes using the Cousins I-band filter  response curve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='  Franchetto et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' (2020) also derived  the center coordinates, the position angle, and the incli-  nation of these galaxies by fitting the galaxy isophotes  with a series of concentric ellipses using the iraf task  ELLIPSE (Jedrzejewski 1987).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='  The stellar velocity fields are used to analyze the stel-  lar kinematics, which is useful to interpret the kinemat-  ics of the molecular gas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='  The stellar kinematics was  extracted from the MUSE datacube using the Penal-  ized Pixel-Fitting (pPXF) code (Cappellari & Emsellem  2004).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='  As a preliminary step, the observations were  masked to remove spurious sources, such as stars and  background galaxies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' Spaxels in the MUSE data were  binned through the Voronoi algorithm in order to reach  S/N> 10 in each bin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' The observed spectra were fitted  with the stellar population templates by Vazdekis et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='  (2010) and using of single stellar populations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' More de-  tails on the procedure can be found in Poggianti et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='  (2017b) and Moretti et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' METHOD  Our approach relies on the software 3DBarolo1 (v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='1  Di Teodoro & Fraternali 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' Di Teodoro & Peek 2021),  which simulates galaxy observations assuming a tilted-  ring model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' This consists of a series of concentric annuli  described by a set of geometric and kinematic parame-  ters, which can all vary with the galactocentric distance  R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' The geometrical parameters are the coordinates of  the center x0 and y0, the position angle φ, and the disc  inclination i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' The kinematic parameters are the systemic  velocity Vsys, the rotation velocity Vrot, the velocity dis-  persion σ, and the radial velocity in the disc plane Vrad.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='  The observed line-of-sight velocity is then (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' Begeman  1987)  Vlos = Vsys + (Vrot cos θ + Vrad sin θ) sin i ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='  (1)  where θ is the azimuthal angle in the plane of the disc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='  3DBarolo (hereafter 3DB) was mainly designed to  fit emission line observations working in 3D, meaning  that the model is fitted to the datacube channel-by-  channel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' This approach allows us to use all the infor-  mation in the datacube and to take into account both  the spatial resolution and the spectral resolution of the  1 https://editeodoro.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='io/Bbarolo/  instrument.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' In a step prior to the fitting, 3DB convolves  the model with the point spread function (PSF) or the  beam of the instrument, while the instrumental spec-  tral broadening is included in the model construction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='  The convolution with the PSF is required to correct for  the so-called “beam smearing” (Bosma 1981;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' Begeman  1987;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' Di Teodoro & Fraternali 2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' The finite size of  the PSF smears the line emission on adjacent regions  where the emitting material has different line-of-sight  velocity, causing an artificial broadening of the profile.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='  As a consequence, the rotation velocity and the velocity  dispersion can be respectively underestimated and over-  estimated, if beam smearing is not correctly accounted  for.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' This effect is particularly important if the angu-  lar resolution of the observations is low and where there  are strong velocity gradients, as in the case of the inner  regions of massive galaxies with steeply rising rotation  curve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' Moreover, the beam smearing effect is expected  to become more and more relevant as the inclination an-  gle of the galaxy increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' 3DB normalizes the model  using either the flux in each pixel of the total intensity  map or the azimuthally-averaged flux in each ring.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' Fi-  nally, the model is fitted to the observations in order  to find the set of free parameters that minimizes the  residuals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='  With respect to 2D methods, which fit the velocity  field, this 3D procedure not only corrects for the beam  smearing effect, but also breaks the degeneracy between  the rotation velocity and the velocity dispersion (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='  Bosma 1981;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' Begeman 1987;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' Di Teodoro & Fraternali  2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' The 3DB task 3DFIT is designed to model emis-  sion line datacubes working in 3D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' The software also  includes the task 2DFIT, which can be used to model  the 2D velocity fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' In this work, we use 3DFIT and  2DFIT to model the kinematics of the molecular gas disk  and the stellar disk, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' For each component,  we adopted an ad-hoc methodology, that is described in  Sects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='1 and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='  Before proceeding with the methodology presentation,  a brief disclaimer is due.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' We stress that 3DB, like sev-  eral other kinematic modelling software (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' Begeman  1987;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' Kamphuis et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' 2015), is specifically designed to  model radially symmetric gas flows in discs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' However,  the galaxies studied in this work are subject to various  local disturbances due to internal (bar, AGN feedback)  and external (ram pressure) mechanisms, which are ex-  pected to produce deviations from this idealised kine-  matics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='  Our strategy here is to use 3DB to quantify  the large-scale ordered motions (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=', rotation and radial  flows) in the molecular gas component, and to interpret  possible deviations from such simple kinematics in terms  of internal or external mechanisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='  Molecular gas kinematics in jellyfish galaxies  7  0h41m32s  31s  30s  9°15\'30"  45"  16\'00"  RA (ICRS)  Dec (ICRS)  1"  JO201 I-band  5 kpc  20  40  60  80  100  F  [10  20 erg/s/cm2/Angstrom]  10h13m48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='0s47.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='5s  47.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='0s  46.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='5s  46.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='0s  45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='5s  0°54\'30"  40"  50"  55\'00"  RA (ICRS)  Dec (ICRS)  1"  JO204 I-band  5 kpc  20  40  60  80  100  F  [10  20 erg/s/cm2/Angstrom]  21h13m48s  47s  46s  2°28\'45"  30"  15"  RA (ICRS)  Dec (ICRS)  1"  JO206 I-band  5 kpc  20  40  60  80  100  F  [10  20 erg/s/cm2/Angstrom]  23h36m27s  26s  25s  24s  23s  21°09\'30"  15"  00"  08\'45"  RA (ICRS)  Dec (ICRS)  1"  JW100 I-band  5 kpc  20  40  60  80  100  F  [10  20 erg/s/cm2/Angstrom]  Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' I-band images, extracted from the MUSE observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' The red contours are at 2n with n going from 1 to 20  with steps of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='5 (same units as colorbars).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' The white stars show the galaxy center.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' For JO201 and JO206, the white dotted  ellipses indicate the regions influenced by the bar (see text).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' The light-blue contour shows the most external isophote (≈ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='5σ  above the background) encompassing the Hα emission traced by MUSE, and it indicates the stellar disk defined by Gullieuszik  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' The black dot in the bottom right corner shows the angular resolution of the MUSE observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' The inset  in the JO204 panel shows a zoom-in of the central regions of the galaxy, with the white circle showing the resolution of the  observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' East is to the left and north to the top.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' Modeling the stellar kinematics  We model the stellar kinematics using the task 2DFIT  on the velocity field (see Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' We fixed the kine-  matic center at the optical center reported in Poggianti  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' (2017a) and Vrad = 0 km s−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' Since stars are not  subject to the effect of ram pressure, we expect this to  be a good approximation everywhere in the galaxy with  the possible exception of the bar region (but see Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' 5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='  We adopt the following three-step approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' We performed a preliminary run with φ, i, Vsys,  and Vrot as free parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' The initial values of  φ and i were taken from Franchetto et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' We made a second run with φ, i, and Vrot as free  parameters, fixing Vsys at the median of the best-  fit values from the first step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' We run again 3DB with Vrot as the free parameter,  while φ and i are regularized using a polynomial  function with degree from zero to three, in order  to avoid numerical oscillations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='  The ring spacing is fixed to 1′′, which approximately  corresponds to the spatial resolution of the MUSE ob-  servations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' This choice is also reasonable based on the  size of the Voronoi bins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' In all 3DB runs, we chose to  give more weight to the regions close to the disc major  8  Bacchini et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='  axis (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' wfunc=2), in order to maximize the signal from  the rotational motion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='  We recall that the results obtained with the 2D ap-  proach are affected by beam-smearing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' The angular res-  olution of the MUSE observations is about 1 ′′, corre-  sponding to about 1 kpc in our galaxy sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' Hence,  we expect that the PSF smearing has a mild effect on the  GASP-ALMA galaxies, except JW100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' In this galaxy,  the PSF smearing is likely important due to its high  inclination with respect to the line of sight.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='  We also note that the assumption of circular orbits  might be inappropriate for the innermost regions of  barred galaxies, as the stars in the bar move along  elongated orbits (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' Sellwood & Wilkinson 1993;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' Ko-  rmendy & Kennicutt 2004).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' However, only galaxies for  which the bar is inclined to both the projected ma-  jor and minor axes show non-circular motions clearly  (Sellwood & Wilkinson 1993).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' Hence, we do not expect  visible signatures of non-circular motions in JO201 and  JO206.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' In Sanchez-Garcia et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' (in preparation), the  stellar velocity field of the galaxies in the GASP sample  is fitted using an ad-hoc approach to include large-scale  non-circular motions induced by bars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' Preliminary re-  sults show that, for the GASP-ALMA galaxies, the re-  covered stellar rotation velocity obtained by Sanchez-  Garcia et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' is overall consistent with ours, suggesting  that the non-circular motions are small compared to ro-  tation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' Modeling the molecular gas kinematics  We model the molecular gas kinematics using the task  3DFIT on the ALMA datacubes (see Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' To re-  duce the free parameters in the model, we first fixed  the kinematic center at the optical center reported in  Poggianti et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' (2017a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='  However, since the interac-  tion with the ICM can displace the kinematic center of  the gas from that of the stars (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' Kronberger et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='  2008b;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' Boselli et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' 2022b), we adjusted the kinematic  center of the molecular gas when necessary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='  We also  set Vsys at the value obtained from the global profile of  the emission line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' When necessary, Vsys was refined by  a few km s−1after inspecting the position-velocity dia-  grams (see Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' 5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' We performed a first run with Vrad = 0 km s−1and  leaving free the geometrical and kinematical pa-  rameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' By setting the 3DB parameter wfunc=2,  we chose to give more weight to the emission along  the disc major axis, where most of the information  on rotational motions lies (θ = 0° in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' We made a second run (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' twostage=True) in  which the geometrical parameters are regularized  using either a suitable function or the median  value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' Vrad is left free in the last run, while the other pa-  rameters are fixed to the best-fit values obtained  previously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='  By setting wfunc=-2, we give more  weight to the emission along the disc minor axis,  where the contribution of radial motions is the  strongest (θ = 90° in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='  This procedure is substantially based on the approach  developed by Di Teodoro & Peek (2021), who used  3DB to model the atomic gas kinematics in a sample of  nearby galaxies in order to measure gas radial motions  and mass flows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' These authors used 21-cm observations  with higher spatial resolution and better velocity reso-  lution than our ALMA data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' Radial motions are pos-  sibly stronger and easier to detect for galaxies affected  by the ram pressure than in the case of Di Teodoro &  Peek (2021)’s galaxies, in which radial motions are of  the order of a few km s−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' We stress that the approach  adopted in this work takes into account the radial mo-  tions within the galaxy disk, while motions perpendicu-  lar to the disk midplane are not considered (Di Teodoro  & Peek 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='  We used the 3DB task ELLPROF to derive the  azimuthally-averaged radial profiles of the CO surface  brightness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' These profile were adopted for the normal-  ization procedure of 3DB models and to derive the H2  surface density ΣH2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='  We also used the 3DB task spacepar to fully explore  the parameter space for Vrot and σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' This test is useful  to check whether the model fitting converges to a good  minimum of the parameter space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' We anticipate that,  while the best-fit Vrot is generally well constrained, it  is not always the case for σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' This is likely due to the  complex shape of the emission line profiles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='  A possible caveat in our methodology is that the  tilted-ring model is based on the assumption of concen-  tric orbits, which might not be valid for the gas in galax-  ies affected by strong ram pressure or in an advanced  stripping stage (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' Kronberger et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' 2008b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' In these  cases, the results of our analysis are very uncertain and  should be taken with caution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' However, if stripping is  not too dramatic, modeling the gas kinematics using the  tilted-ring approach may be possible for the disk regions  where some or most of the gas has preserved its original  motion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' Stellar bars are also expected to induce non-  circular motions due to the gas streaming along the bar  (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' Sellwood & Wilkinson 1993).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' Indeed, the gas kine-  matics in barred galaxies is usually modeled using tools  that are specifically designed to take into account non-  axisymmetric distortions in the 2D velocity field (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='  Molecular gas kinematics in jellyfish galaxies  9  Schoenmakers 1999;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' Spekkens & Sellwood 2007).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' How-  ever, these methods fail when the bar is perpendicular  to or parallel the disk major axis, being unable to break  the degeneracy between the tangential and radial ve-  locity components (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' Sellwood & S´anchez 2010;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' Ran-  driamampandry et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' 2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' We thus decided to adopt  the tilted-ring approach also in the case of JO201 and  JO206, which host stellar bars aligned with the disk ma-  jor axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' RESULTS AND DISCUSSION  In this section, we present the best-fit models for the  molecular gas kinematics and we then compare the stel-  lar and molecular gas rotation curves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' We discuss each  galaxy individually in Sects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='1–5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='4 and summarize our  findings in Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' We analyzed both the CO(1–0)  and the CO(2–1) datacubes, obtaining essentially the  same results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' Thus, we show the best-fit models for the  CO(1–0) data, as they have a S/N and angular resolu-  tion more suitable for modeling the kinematics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' From  here on, CO indicates CO(1–0) unless otherwise stated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='  Since the focus of this work is on the molecular gas, we  show the best-fit model for the stellar kinematics only  for JO201 in Fig 3, while the models for the rest of the  sample can be found in Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' JO201  The I-band image in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' 2 shows that JO201 has  a stellar bulge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' Moreover, the elongated shape of the  isophotes in the inner regions suggests that JO201 hosts  a stellar bar, as reported by George et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='  Sanchez-Garcia et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' (submitted) estimated that the  bar length is ≈ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='6 kpc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' We also note that the stellar  disc of JO201 seems morphologically lopsided, being the  east side slightly more extended than the west one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='  The top panels in Fig 3 show, from left to right, the  observed stellar velocity field, the best-fit model, and  the map of the residuals between the data and the best-  fit model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' The bottom panels display the radial profile  of the best-fit rotation velocity (left), inclination (cen-  ter), and PA (right).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' The stellar velocity field is very  well reproduced by the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' The residuals in the disk  outskirts, where the Voronoi bins are the largest, tend  to be higher than in the inner regions, but still within  the velocity resolution of the MUSE observations, that  is ∆v ≈ 50 km s−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' We note that, for R ≲ 5 kpc, the ro-  tation velocity is much lower than expected for a galaxy  with stellar mass M⋆ ≈ 9 × 1010 M⊙ and hosting a stel-  lar bulge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' This feature can be explained by the fact that  the stellar bar is aligned along the disk major axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' In  a scenario where a large fraction of the stars in the bar  move on elliptical orbits aligned parallel to the bar (so  called x1 type;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' Sellwood 2014), the velocity component  along the line of sight has its minimum at the apocentre  and then increases along the major axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' This can result  in an underestimation of the rotation velocity in the re-  gions influenced by the bar (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' Dicaire et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' 2008;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' Sell-  wood & S´anchez 2010;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' Randriamampandry et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' 2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='  The total CO intensity map (top left panel in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' 1)  gives useful indications about the effect and direction of  ram pressure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' In JO201, the west side of the disk shows  compressed contours and regions with bright CO emis-  sion, possibly suggesting the ram pressure compressed  this part of the disk (Bellhouse et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' The most  evident feature in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' 1 is arguably the presence of the  ring-like structure surrounding the hole in the CO dis-  tribution in the innermost ≈ 3 kpc (see also George  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' 2018, 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' The ring-like structure is also visible  in the MUSE images shown by Bellhouse et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' (2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='  This feature can be explained by the presence of the bar  driving the formation of a molecular gas ring around the  co-rotation radius (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' where the bar pattern equals the  angular frequency of circular motions;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' see Sellwood &  Wilkinson 1993;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' Kormendy & Kennicutt 2004).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' At radii  well inside co-rotation, gas is expected to fall toward  the center.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='  The molecular gas distribution in barred  galaxies is typically very concentrated in the center (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='  Kormendy & Kennicutt 2004), while Figure 1 clearly  shows the lack of CO emission in the innermost regions  of JO201.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' George et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' (2019) attribute the CO cavity  to AGN feedback, which ionizes the molecular hydro-  gen (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' radiative feedback) and sweeps the gas from  the center (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' mechanical feedback).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' The connection  between nuclear activity and the gas distribution and  kinematics is specifically tackled in the companion pa-  per (Mingozzi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' to be submitted).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='  The CO velocity field of JO201 (2nd panel in the top  row of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' 1) shows that the galaxy is kinematically lop-  sided, meaning that the velocity gradient in the receding  and approaching sides of the disc are significantly differ-  ent from each other (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' Richter & Sancisi 1994;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' Swa-  ters et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' 1999;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' Schoenmakers 1999;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' Shafi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' 2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='  For this reason, we modeled the approaching side and  receding side separately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' We compare the observations  with our best-fit models in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' 4, where the left and the  right panels are for the approaching and receding sides  of the disc, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' The first and second rows in  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' 4 are the position-velocity diagrams (PVDs) along  the major and minor axis of the disc, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' Our  rotating disc model can reproduce reasonably well the  observations, indicating that the molecular gas in the  disk preserved its original rotation, despite the interac-  tion with the ICM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' There is however some gas, which is  10  Bacchini et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='0  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='4  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='6  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='8  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='10  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='12  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='14  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='16  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='R [kpc]  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='0  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='50  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='100  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='150  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='200  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='250  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='Rotation velocity [km/s]  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='Bar region  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='2nd fit  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='3rd fit  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='0  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='4  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='6  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='8  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='10  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='12  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
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+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='1), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' The dashed black lines and the grey area indicate  the median and the median absolute deviation, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='  indicated by the red arrow in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' 4, moving with lower  velocities than those predicted by the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' Since this  gas is located at galactocentric distances smaller than  the bar length, its anomalous kinematics is plausibly  due to the bar influence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='  By exploring the parameter space, we found that  the best-fit value of the CO velocity dispersion is not  well-constrained for the outermost ring, likely because  of the low S/N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' For R ≲ 5 kpc, we obtain σCO ≈  25 − 40 km s−1, which can be explained by the non-  circular motions due to the stellar bar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' Outside the bar  regions, we find σCO ≈ 20 km s−1, which is about a fac-  tor 2 higher than the typical values of the molecular gas  velocity dispersion in local isolated, unbarred galaxies  (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' Bacchini et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' 2020a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' This enhancement of σCO  may be due to ram pressure increasing the molecular gas  turbulence, either directly or by enhancing the star for-  mation rate (SFR;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' see Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='5 for further discussion).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='  We note that the best-fit values of radial velocity are  consistent with zero, suggesting that the inclusion of  radial motions does not significantly improve the fit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='  Hence, these values should be taken with caution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' The  direction (either inward or outward) of these radial flows  cannot be determined unless the near/far sides of the  galaxy are known.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' This can be inferred by assuming that  spiral arms trail the galaxy rotation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' Based on the RGB  image shown by (Bellhouse et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' 2017), the direction of  spiral arms indicate that JO201 rotates clockwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' Then,  in 3DB’s convention (Di Teodoro & Peek 2021), radial  motions with Vrad < 0 point inward, while those with  Vrad > 0 point outward.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' Taken at face value, the inflow  radial velocities at R ≲ 5 kpc are Vrad ≳ −10 km s−1,  which is comparable with the average values measured in  the inner regions of nearby spiral galaxies (Di Teodoro &  Peek 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' Beyond the bar region, the radial outflow  with Vrad ≳ 20 km s−1is consistent with being caused  by ram pressure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' However, since non-circular motions  can be induced by any perturbation of the gravitational  potential, we cannot exclude a different origin (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' Sell-  wood & S´anchez 2010).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='  In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' 5 (top left), we compare the circular velocities  inferred from the kinematics of the stellar and molecular  gas disks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' The stellar circular velocity was obtained from  the rotation velocity shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' 3 by correcting for  the contribution of pressure support (asymmetric drift  Molecular gas kinematics in jellyfish galaxies  11  10  5  0  5  10  Offset ["]  300  200  100  0  100  VLOS [km/s]  = 178°  Model fitted on the approaching side  10  5  0  5  10  Offset ["]  300  200  100  0  100  VLOS [km/s]  = 180°  Model fitted on the receding side  10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='0  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='5  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='0  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='0  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='5  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='0  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='5  10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='0  Offset ["]  300  200  100  0  100  VLOS [km/s]  = 268°  10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='0  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='5  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='0  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='0  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='5  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='0  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='5  10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='0  Offset ["]  300  200  100  0  100  VLOS [km/s]  = 270°  0  2  4  6  8  40  60  i [deg]  0  2  4  6  8  40  60  i [deg]  0  2  4  6  8  180  200  PA [deg]  0  2  4  6  8  180  200  PA [deg]  0  2  4  6  8  R [kpc]  0  50  Vrad [km/s]  Outflow  Inflow  0  2  4  6  8  R [kpc]  0  50  Vrad [km/s]  Outflow  Inflow  200  100  0  100  200  300  VLOS (km/s)  10  5  0  5  10  R [kpc]  200  100  0  100  200  300  VLOS (km/s)  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='5  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='0  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='0  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='5  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='0  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='5  R [kpc]  200  100  0  100  200  300  VLOS (km/s)  10  5  0  5  10  R [kpc]  200  100  0  100  200  300  VLOS (km/s)  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='5  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='0  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='0  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='5  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='0  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='5  R [kpc]  JO201  Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' Best-fit models of the molecular gas kinematics for JO201 using CO(1–0) emission line observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' The left and the  right panels are for the approaching and the receding sides of the disc (the other side is shaded), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' The first and the  second rows show the PVD along the major and the minor axis, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' The observed CO(1–0) emission is shown in blue  with black and grey contours, while the red contours show the best-fit model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' All the contours are at 2nσch with n = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=', 10  and σch = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='7 mJy/beam.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' The yellow points indicate the projected rotation curves of the best-fit models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' The vertical blue  dotted lines and the red arrows indicate the bar extent and the gas with anomalous kinematics (see text), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' In the  last three rows, the panels show the profiles of inclination, PA, and radial velocity of the best-fit models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' The red points are  the parameters from the 1st fitting step and the dark-red lines are the regularized profiles (see Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' The orange and green  areas in the bottom panels indicate whether positive/negative values for Vrad mean radial gas outflow/inflow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='  12  Bacchini et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='  0  5  10  15  0  100  200  300  Vcirc [km/s]  JO201  CO, app.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' side  CO, rec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' side  Stars  5  10  0  100  200  300  JO204  CO  Stars  0  5  10  15  20  R [kpc]  0  100  200  300  Vcirc [km/s]  JO206  CO  Stars  0  10  20  R [kpc]  0  100  200  300  400  JW100  CO, app.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' side  CO, rec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' side  Stars  Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' Comparison of the stellar (yellow stars) and the  CO (darkred points) circular velocities for each galaxy in  our sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='  When the approaching side and the receding  side of the disc are modelled separately, the resulting pro-  files are shown by the blue squares and the red diamonds,  respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='  correction)2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' Within R ≈ 5 kpc, the molecular gas and  stellar circular velocities essentially coincide, but the ve-  locity gradient is too shallow for a massive galaxy with a  bulge such as JO201.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' As mentioned above, this is likely  due to the stellar bar aligned along the major axis (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='  Dicaire et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' 2008;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' Sellwood & S´anchez 2010;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' Randria-  mampandry et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' 2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' Beyond the bar regions, the  stellar velocity field of JO201 (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' 3) does not show any  indication of the kinematic lopsidedness, contrary to the  molecular gas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' The receding side of the CO disc reaches  slightly higher rotation velocities than the stellar disc,  while the approaching side shows a lower velocity gradi-  ent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' The kinematic lopsidedness in disc galaxies is typ-  ically ascribed to a triaxial potential, as in the presence  of a stellar bar (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' Swaters et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' 1999;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' Schoenmakers  1999;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' Rhee et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' 2004).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' However, the regular kinematics  of the stellar disk seems to suggest that the molecular  gas kinematics may be perturbed by some mechanisms  that does not affect the stars, like ram pressure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' The  distortions appear in the outer parts of the galaxy and  in a symmetric way, as expected for face-on ram pressure  (Kronberger et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' 2008b;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' Bellhouse et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' 2017, 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='  Indeed, JO201 is moving towards the observer at very  2 We used Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' A1 from Posti et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' (2018) for the asymmetric  drift velocity and Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' 3 from Mancera Pi˜na et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' (2021a) for the  central velocity dispersion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' We note that the asymmetric drift  correction is essentially negligible for JO201 and all the other  galaxies, as expected given their high rotation velocities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='  high velocity (see Table 1), implying that the approach-  ing and receding sides of the disk move in the opposite  and the same direction as the ram pressure, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='  The ram pressure is thus expected to decelerate the ap-  proaching side of the molecular disk and accelerate the  receding side (Kronberger et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' 2008b), which is consis-  tent with our results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='  We conclude that the molecular gas kinematics in the  inner regions of JO201 is mainly dominated by the per-  turbations due to the stellar bar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='  In the outer parts  of the molecular gas disk, the kinematic lopsidedness  and radial motions (although rather uncertain) seem to  suggest that the molecular gas in JO201 is affected by  face-on ram pressure, despite other mechanisms cannot  be ruled out.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' JO204  Before focusing on the molecular gas kinematics, it  is worth noting two features of the stellar component.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='  First, the innermost isophotes in the I-band image  (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' 2) show a boxy shape that might indicate the pres-  ence of a bar seen with high inclination with respect to  the line-of-sight (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' Combes et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' 1990;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' Bettoni & Gal-  letta 1994;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' Kuijken & Merrifield 1995;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' Bureau & Free-  man 1999;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' Merrifield & Kuijken 1999).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' Unfortunately,  dust obscuration and projection effects hamper any at-  tempt to estimate the bar length from the optical im-  ages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' The second feature is visible in the stellar velocity  field, which shows slightly distorted iso-velocity contours  in the innermost regions (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' 11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' This S-shaped  feature indicates the presence of non-circular motions  and, possibly, of a stellar bar (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' Bettoni 1989;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' Vau-  terin & Dejonghe 1997;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' Kormendy & Kennicutt 2004;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='  Cort´es et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' 2015, Sanchez-Garcia et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' in preparation).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='  Indeed, the top right panel of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' 11 shows residuals of  a few tens of km s−1in the regions close to the disc mi-  nor axis, indicating that a model based on circular orbits  cannot fully reproduce the observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='  In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' 1, the CO total intensity map shows that the  molecular gas distribution is strongly concentrated in  the center and two arm-like structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' Both features  are typical of barred galaxies (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' Athanassoula 1992a;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='  Bureau & Freeman 1999;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' Merrifield & Kuijken 1999;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' Ko-  rmendy & Kennicutt 2004;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' Hogarth et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' The  iso-velocity contours in the CO velocity field (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' 1) are  visibly distorted in the inner regions, which typically in-  dicates the presence of non-circular motions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' The CO  velocity dispersion is also quite high in the inner regions  of disk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='  To model the CO kinematics, we modified the pro-  cedure described in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='  After various tests, we  decided to fit the data by fixing all the geometrical pa-  Molecular gas kinematics in jellyfish galaxies  13  rameters and leaving free Vrot, Vrad, and σCO, as this  choice improved the comparison between the model and  the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' We run 3DB using the reverse option, which  performs the fit starting from the most external ring  and then moving inward.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' This algorithm was designed  to improve the fit for galaxies seen with high inclination  (i ≳ 70°).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' 6 compares the best-fit model with the  observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' 6, the PVD along the major axis  shows that, overall, the model reproduces well the ob-  servations, except for two features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' The first feature is  indicated by the red arrow and consists in gas moving at  VLOS ≈ 270 km s−1at about 1 kpc from the center.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' One  possibility is that this CO emission is probing the inner  rise of the rotation curve of the molecular disk if the nu-  clear CO distribution is asymmetric between approach-  ing and receding sides (see for example Lelli et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='  Alternatively, this central emission can be ascribed to  complex non-circular motions caused by the stellar bar  (the so-called x2 orbits aligned perpendicular to the bar;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='  see Sancisi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' 1979;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' Kormendy & Kennicutt 2004;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='  Randriamampandry et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' 2015) or even feedback from  stars or the AGN (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' Stuber et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' Our finding  is in agreement with the results obtained by Deb et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='  (2020), who revealed an absorption feature in the HI  global profile that could be explained by high-velocity  gas seen in front of the continuum emission from the  AGN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' The second feature that is not reproduced by the  model is indicated by the magenta arrows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' This emis-  sion comes from gas that moves at lower velocities than  those predicted by our model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' A possible explanation  is that this gas is decelerated by the ram pressure com-  ponent in the sky plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' We note that the PVD along  the major axis (top panel of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' 6) shows the charac-  teristic X-shaped pattern, that is typical of a bar seen  at high inclination along the line of sight (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' Bureau  & Freeman 1999;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' Merrifield & Kuijken 1999;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' Kormendy  & Kennicutt 2004;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' Alatalo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' 2013;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' Hogarth et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='  2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='  In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' 6, the PVD along the minor axis shows ex-  tended gas emission in the lower left quadrant, which  can only be reproduced by a model with strong radial  motions of Vrad ≳ 50 km s−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' However, the same feature  is not observed in the upper right quadrant, indicating  that the CO distribution (or kinematics) is asymmetric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='  Unfortunately, JO204 does not have visible spiral arms  and the dust lanes in the optical MUSE and Hubble  Space Telescope (HST) images (Gullieuszik et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' 2017,  Gullieuszik et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=', submitted) do not allow to clearly  identify the nearest side of the disk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='  Hence, we can-  not infer the direction of rotation and radial motions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='  Since these non-circular motions are detected in the in-  ner parts of the galaxy, one may speculate that they  10  5  0  5  10  Offset ["]  300  200  100  0  100  200  300  VLOS [km/s]  = 327°  JO204  10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='0  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='5  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='0  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='0  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='5  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='0  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='5  10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='0  Offset ["]  300  200  100  0  100  200  300  VLOS [km/s]  = 237°  1  2  3  4  5  6  7  8  R [kpc]  0  100  Vrad [km/s]  300  200  100  0  100  200  300  VLOS (km/s)  10  5  0  5  10  R [kpc]  300  200  100  0  100  200  300  VLOS (km/s)  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='5  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='0  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='0  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='5  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='0  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='5  R [kpc]  Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' Same as Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' 4 but for JO204.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' The model fitting is  performed on the approaching and receding sides at the same  time, and the inclination and PA are fixed to the values of  75° and 327°, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' The arrows indicate the gas with  anomalous kinematics (see text).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' Here σch = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='9 mJy/beam.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='  are inflows driven by a stellar bar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' The magnitude of  radial motions in JO204 is consistent with the values es-  timated in simulated barred galaxies (see Randriamam-  pandry et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' 2015), but about 2 times higher than those  measured by Di Teodoro & Peek (2021) in the atomic  gas of real barred galaxies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='  The blue arrow indicates  another feature that is not reproduced by our model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='  However, since this emission is very faint, it is unclear  whether this is real emission from the galaxy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' We find  σCO ≈ 10 km s−1, but this value is rather uncertain  based on the inspection of the parameters space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='  14  Bacchini et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='  Overall, our model can reproduce the molecular gas  kinematics in JO204 reasonably well, despite the com-  plexities due to the stellar bar and/or ram pressure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' Fig-  ure 5 (top right panel) shows that the CO circular veloc-  ity is compatible within the uncertainties with the stel-  lar circular velocity for R ≳ 3 kpc, indicating that the  molecular gas has retained most of its original motion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='  The difference in the innermost regions is likely due to  a combination of dust extinction and resolution effects,  which may smooth the gradient of the stellar rotation  curve (see Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' Similarly to the case of JO201, we  conclude that the molecular gas kinematics is rotation-  dominated in JO204.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='  Then, the stellar bar plays an  important role in perturbing the gas kinematics in the  inner regions and driving radial gas flows, while the ram  pressure may be a possible explanation for the gas with  anomalous kinematics in the outskirts of JO204.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' JO206  The I-band images in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' 2 shows that JO206 hosts  a stellar bulge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' In addition, the elongated shape of the  isophotes in the inner regions indicate the presence of  a stellar bar aligned with the disk major axis, as for  JO201.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='  The CO total intensity map in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' 1 shows  that the morphology of the molecular gas distribution is  asymmetric, suggesting that the ram pressure is directed  towards south-east.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' Hence, we expect the molecular gas  kinematics to be particularly complex in JO206, as a  consequence of the combined effects of bar perturbations  and ram pressure stripping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' Indeed, the iso-velocity con-  tours in the CO velocity field (2nd panel in the third row  in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' 1) are even more distorted than those of JO204,  indicating stronger perturbations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='  In the light of these considerations, we modeled only  the regions of the molecular gas disk within R ≈ 6 ′′,  which essentially corresponds to the extent of its re-  ceding side (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' We also adjusted the position of  the kinematic center with respect to the optical center  by applying a small shift of ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='23 ′′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' Figure 7 shows  that our model is able to reproduce reasonably well the  observations, except for the molecular gas with anoma-  lous kinematics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' The CO emission indicated by the blue  arrow (offset ≈ 10 − 20 ′′and -200 km s−1≲ VLOS ≲  −100 km s−1) belongs to the tail of stripped gas (see also  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' Probably, this portion of gas disc was detached  from the approaching side of the disc and decelerated by  ram pressure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' Another possibility is that the ram pres-  sure displaced a portion of the disc at larger radii, thus  its rotation velocity is lowered by conservation of angular  momentum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' The red arrow indicates the receding side  of the molecular gas disk that has not been stripped.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' In  the PVD along the minor axis (second panel in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' 7),  the CO emission indicated by the orange arrow belongs  to the molecular gas in the stripped tail left behind by  the galaxy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='  By exploring the parameter space, we found that the  best-fit value of σCO is well constrained only for the  2nd and 3rd rings, where we obtained σCO ≈ 30 −  40 km s−1(see also Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' These values are higher than  those typically measured in nearby galaxies using CO  observations with similar resolution (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' Bacchini et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='  2020a), which is not surprising given the complex kine-  matics of the molecular gas in JO206.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='  We tentatively detect radial motions of ≈ −25 km s−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='  However, including radial motions does not improve the  best-fit model, as indicated by the fact that radial veloc-  ities are consistent with zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' In JO206, the spiral arms  can be identified from the MUSE optical images (Pog-  gianti et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' 2017b;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' Bellhouse et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='  Assuming  trailing spiral arms, we can infer that the galaxy ro-  tates clockwise, implying that Vrad < 0 for inflows and  Vrad > 0 for outflows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='  The bottom left panel in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' 5 shows that the stellar  and CO circular velocities coincide within R ≈ 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='5 kpc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='  As in the case of JO201 (see Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='1), the slow rise of  the inner rotation curve is plausibly due to the fact that  the stellar bar is aligned parallel to the disk major axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='  This suggest that the kinematics of both the molecular  gas and the stars is dominated by the stellar bar in these  regions3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' We also note that the position and velocity of  the molecular gas emission indicated by the red arrow  (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' 7, top panel) is perfectly compatible with the cir-  cular velocity profile of the stars (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' 5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' On the  contrary, the stripped tail indicated by the blue arrow  (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' 7, top panel) is decelerated of about 70km s−1with  respect to the stars at the same galactocentric distance,  suggesting that this material is decoupled from the disk  rotation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' The asymmetric perturbations on the molec-  ular gas kinematics and the displaced kinematic center  are signatures of edge-on ram pressure stripping (Kron-  berger et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' 2008b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='  Overall, we conclude that the molecular gas kinemat-  ics is mainly perturbed by the stellar bar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='  Taken at  face value, the radial motion in JO206 can be inter-  preted as a gas inflows driven by the bar, as they are  within its region of influence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' We also find clear indi-  cations of edge-on ram pressure stripping based on the  presence of molecular gas emission detached from the  galaxy and with kinematics decoupled from the main  disk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' This suggests that the ram pressure has a stronger  3 This result is consistent with the preliminary estimate of the bar  length obtain by Sanchez-Garcia et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' (in preparation), that is  approximately 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='7 kpc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='Molecular gas kinematics in jellyfish galaxies  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='15  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='10  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='0  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='10  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='20  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='Offset ["]  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='300  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='200  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='100  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='0  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='100  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='200  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='300  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='400  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
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+page_content='= 120°  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='JO206  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
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+page_content='5  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='0  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='5  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='10  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='Offset ["]  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='300  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='200  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='100  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
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+page_content='100  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='200  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='300  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='400  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='VLOS [km/s]  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='= 210°  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='0  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='3  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='4  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='5  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='6  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='7  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='60  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='80  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='i [deg]  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='0  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='3  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='4  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='5  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='6  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='7  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='100  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='150  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='PA [deg]  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='0  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='3  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='4  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='5  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='6  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='7  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='R [kpc]  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='50  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='0  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='Vrad [km/s]  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='Outflow  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='Inflow  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='300  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='200  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='100  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='0  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='100  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='200  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='300  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='VLOS (km/s)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='10  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='0  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='10  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='20  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='R [kpc]  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='300  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='200  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='100  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='0  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='100  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='200  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='300  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='VLOS (km/s)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='10  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='5  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='0  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='5  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='10  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='R [kpc]  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='Figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' Same as Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' 4 but for JO206.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' The model fitting  is performed on both the approaching and receding sides, at  the same time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' Here σch = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='8 mJy/beam.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='  effect on the molecular gas disk in JO206 than in JO201  and JO204.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' JW100  The I-band image in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' 2 seems to suggest that  JW100 hosts a stellar bulge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' Moreover, despite the fact  that JW100 is strongly affected by projection effects and  dust obscuration, we can tentatively identify the pres-  ence of a warp in the stellar disk based on the S-shape  of the isophotes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' Regarding the stellar kinematics, our  model can successfully reproduce the observations and  recover the stellar rotation curve (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' 13 in Append-  inx A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' However, we found quite high residuals in a ring  at R ≈ 6 ′′and in the disk outskirts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' After various trails,  we found no significant improvement in the residual map  using different geometrical parameters and allowing for  radial motions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' This can be due to the combined effects  of low S/N of the observations in the disk outskirts,  strong projection effects due to the radial variation in  disk inclination and PA, and asymmetric dust lanes (see  Gullieuszik et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=', submitted).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' Since JW100 belongs to  a substructure of three galaxies in Abel 2626, we cannot  rule out that the stellar kinematics is perturbed by tidal  interaction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='  Figure 1 clearly shows that the distribution and kine-  matics of the molecular gas in JW100 are strongly dis-  turbed, suggesting that the ram pressure component in  the sky plane is directed westward and contributes in  pushing the gas outside the stellar disk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='  The case of  JW100 may seem surprising, as the high mass of this  galaxy is expected to produce a strong gravitational pull  that can efficiently contrast the ram pressure stripping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='  However, the supersonic speed and the close proximity  to the cluster center (see Table 1) indicate that JW100 is  in the most favorable conditions for experiencing strong  ram pressure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' Indeed, the iso-velocity contours in the  CO velocity field (2nd panel in the last row in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' 1) are  even more distorted than the rest of the GASP-ALMA  sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' Also the 2nd moment map suggests that the  molecular gas velocity dispersion is very high through-  out the disk, indicating very complex line profiles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='  We attempt to model the gas kinematics with the aim  of understanding whether some gas has retained its orig-  inal rotation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' Hence, we run 3DB using the reverse  option for highly inclined galaxies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='  We fixed the in-  clination and PA at the values obtained for the stellar  disc and shifted the kinematic center ≈ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='6′′westward  from the optical center.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='  We performed the fitting on  the approaching and receding sides separately, as the  PVD along the major axis is asymmetric with respect  to Vsys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' The resulting best-fit models are shown in the  left and right panels of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' 8, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' The models  well reproduce the observations, except for the emission  indicated by the blue arrow in the minor axis PVD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' This  emission comes from the molecular gas in the tail that is  left behind by JW100 as it falls into the cluster receding  from the observer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' Indeed, the bottom panel in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' 8  shows the profiles of the radial velocity, which reaches  Vrad ≈ 50 − 100 km s−1in the disk outskirts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' Taken at  face value, the radial velocities are larger than the Vrad  values of a few km s−1that are typically measured in  16  Bacchini et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='  nearby galaxies (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' Di Teodoro & Peek 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' Also  the skewed shape of the CO emission in the PVD along  the minor axis (blue arrows in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' 8) seem to suggest  the presence of radial motions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' We note that the emis-  sion from the stripped gas indicated by the blue arrow  reaches even higher velocities (∆VLOS ≈ −200 km s−1)  than the model emission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' The dust lanes in the HST im-  ages (Gullieuszik et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=', submitted) seem to suggest that  the west side of JW100 is the nearest one and the galaxy  is rotating clockwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' This implies that Vrad > 0 indi-  cates an outward radial flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' This is in agreement with  the morphology of the molecular gas disk, that clearly  suggests an ongoing large-scale removal of molecular gas  by ram pressure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' We obtained σCO ≈ 30 − 60 km s−1,  possibly indicating that the molecular gas is highly tur-  bulent (see also Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='  This is not surprising given  the strong perturbations affecting the molecular gas in  JW100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='  The bottom right panel in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' 5 compares the circu-  lar velocity profile of the stellar disk and the molecular  gas in JW100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' We recall that different kinematic centers  were used for the stellar and molecular gas components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='  Interestingly, the circular velocity of the approaching  side of the molecular gas disk coincides with that of  the stellar disk, flattening at Vcirc ≈ 300 km s−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' On  the contrary, the circular velocity of the receding side  keeps on growing and reaches Vcirc ≈ 400 km s−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' Sim-  ilarly to JO206, the asymmetric perturbations on the  molecular gas disk and the displaced kinematic center  are signatures of edge-on ram pressure stripping (Kro-  nberger et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' 2008b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' This is consistent with the fact  that JW100 is falling into the cluster at very high ve-  locity and its disk is seen at high inclination by the ob-  server.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' We note that the circular velocity of JW100 rises  less steeply than what is typically found in galaxies with  similar stellar mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' Moreover, Figure 2 seems to sug-  gest that JW100 potentially hosts a stellar bulge, which  is expected to produce high circular velocities in the in-  nermost regions of the galaxy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' The shallow and rather  unusual gradient of the circular velocity might be ex-  plained by either the presence of a stellar bar aligned  with the major axis or a dark matter halo with lower-  than-average concentration (Randriamampandry et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='  2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='  Disentangling between these two possibilities  would require a dedicated mass modelling of the system  which goes beyond the purpose of this study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='  In conclusion, our results indicate that the molecular  gas disk of JW100 is dramatically affected by ram pres-  sure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' The morphology and kinematics of the molecular  gas disk indicate strong ram pressure both in the sky  plane and along the line of sight.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' Gravitational inter-  actions with other members in the same substructure  may play a role, but we speculate that these effects are  milder than ram pressure, as the stellar component is  not as strongly perturbed as the molecular gas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' Summary  In Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='1, we showed that the molecular gas kine-  matics in JO201 is dominated by the bar for R ≲ 5 kpc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='  At larger galactocentric distances, the rotation curve  gradient is modified by some physical mechanisms, that  is possibly face-on ram pressure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' We note that, since  JO201 belongs to a cluster substructure, we cannot ex-  clude a different origin (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='  tidal interactions).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='  We  do not find clear signature of ram pressure stripping  (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' molecular gas removed from the main disk), but  we tentatively identify outward radial flow of gas plau-  sibly due to ram pressure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' Both within and beyond the  region influenced by the bar, the velocity dispersion of  the molecular gas is enhanced with respect to the typ-  ical values measured in field galaxies (Bacchini et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='  2020b), suggesting strong turbulence motions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' Beyond  the bar region, this enhancement is about a factor 2  (σCO ≈20 km s−1), which can be either a direct or indi-  rect consequence of ram pressure increasing (or a com-  bination of both).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' Indeed, the ram pressure can directly  increase the gas kinetic energy, but it can also enhance  the SFR (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' Kronberger et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' 2008a) and thus the ve-  locity dispersion due to the transferring of the supernova  energy to the gas (Bacchini et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' 2020b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' The SFR of  JO201 is about 2 times higher than field galaxies with  similar stellar mass, which supports the second scenario.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='  In Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='2, we found clear signatures of the pres-  ence of a bar in JO204 based on the molecular gas dis-  tribution (central concentration, arm-like overdensities)  and kinematics (PVD shape, radial motions), and the  stellar kinematics (velocity field).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' Radial motions are  clearly present, but we cannot identify their direction  with the available observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' A bar-driven inflow is a  reasonable hypothesis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' The molecular gas kinematics is  dominated by rotation, while the ram pressure plays a  secondary role and we do not find signatures of ram pres-  sure stripping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' We detect molecular gas with anomalous  kinematics that is compatible with being decelerated by  face-on ram pressure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' We also find some molecular gas  with high velocity in the central regions of the galaxy,  but its origin is unclear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' The estimated values of the  molecular gas velocity dispersion (σCO ≈10 km s−1) are  rather uncertain, but overall consistent with those typ-  ical of field galaxies (Bacchini et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' 2020b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' This is in  line with the fact that JO204 does not show enhanced  SFR with respect to field galaxies (Vulcani et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='  Similarly to JO201, the molecular gas kinematics in  JO206 is dominated by the bar for R ≲ 6 kpc (Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='  Molecular gas kinematics in jellyfish galaxies  17  15  10  5  0  5  10  15  Offset ["]  200  0  200  400  600  VLOS [km/s]  = 269°  Model fitted on the approaching side  15  10  5  0  5  10  15  Offset ["]  200  0  200  400  600  VLOS [km/s]  = 269°  Model fitted on the receding side  10  5  0  5  10  15  Offset ["]  200  0  200  400  600  VLOS [km/s]  = 179°  10  5  0  5  10  15  Offset ["]  200  0  200  400  600  VLOS [km/s]  = 179°  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='0  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='5  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='0  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='5  10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='0  12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='5  R [kpc]  0  100  Vrad [km/s]  Outflow  Inflow  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='0  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='5  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='0  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='5  10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='0  12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='5  R [kpc]  0  100  Outflow  Inflow  600  400  200  0  200  400  VLOS (km/s)  10  0  10  R [kpc]  600  400  200  0  200  400  VLOS (km/s)  10  5  0  5  10  15  600  400  200  0  200  400  VLOS (km/s)  10  0  10  R [kpc]  600  400  200  0  200  400  VLOS (km/s)  10  5  0  5  10  15  JW100  Figure 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' Same as Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' 4 but for JW100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' In the left and right panels, the model fitting is respectively done for the approaching  and receding sides, and the kinematic center is ≈ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='6′′westward from the optical center.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' The inclination and PA are fixed to the  values of 77° and 179°, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' Here σch = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='1 mJy/beam.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='  This seems also to drive inward radial flows of molecular  gas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' We find clear signatures of edge-on ram pressure  stripping for R > 6 kpc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='  The velocity dispersion of  the molecular gas is significantly enhanced (σCO ≈30-  40 km s−1), a likely consequence of the complex motions  due to the combined influence of the bar and the ram  pressure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='  In JW100 (Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='4), the molecular gas distribution  and kinematics indicate ongoing ram pressure stripping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='  Since JW100 belongs to a cluster substructure and prob-  ably hosts a warped stellar disk, we cannot exclude that  gravitational interactions may also play a role.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' We de-  tect radial motions that are compatible with an outward  gas flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' The shallow gradient of the circular velocity  in the inner regions of JW100 may be explained by a  stellar bar aligned with the disk major axis, although  other possibilities cannot be excluded (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' combination  of resolution effects and ram pressure, low-concentration  dark matter halo).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' The velocity dispersion of the molec-  ular gas is quite enhanced (σCO ≈30-60 km s−1), but the  SFR of JW100 is about 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='5 times lower than field galax-  ies with similar stellar mass (Vulcani et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' These  properties favours the scenario in which the gas turbu-  lence is directly enhanced by the ram pressure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' COMPARISON WITH PREVIOUS WORKS  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' The connection between stellar bars and AGN  At least three out of four galaxies in the GASP-ALMA  sample host a stellar bar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' In the case of JW100, the  presence of the bar is difficult to confirm but arguably  plausible, given that about 60% of the disk galaxies with  10 ≲ log(M⋆/M⊙) ≲ 11 host a stellar bar (Aguerri et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='  2009;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' Masters et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' 2012;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' D´ıaz-Garc´ıa et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' 2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' The  fraction of barred galaxies may be even higher in the  central regions of clusters (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' Andersen 1996;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' Barazza  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' 2009;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' M´endez-Abreu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' 2012;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' Lansbury et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='  2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' Alonso et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' 2014), but this is likely due to the  increase of early-type galaxies (which are less likely to  18  Bacchini et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='  host a bar) with decreasing clustercentric distance (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='  Tawfeek et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='  The non-axisymmetric potential of a stellar bar can  trigger radial inflow of gas by inducing torques and  shocks (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' Athanassoula 1992b,a;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' Sellwood & Wilkin-  son 1993;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' Sellwood 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' Marasco et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' 2018), often en-  hancing the molecular gas concentration in the central  regions of galaxies (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' Sheth et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' 2005;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' Regan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='  2006;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' Yu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='  Bar-driven inflows of gas may  play an important role in fueling the central black hole  and triggering the AGN activity (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' Alonso et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' 2013,  2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' Rosas-Guevara et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' Silva-Lima et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='  However, this topic is debated and some authors showed,  for instance, that the excess of AGN-hosts among barred  galaxies vanishes when the dependence on the galaxy  stellar mass and color are taken into account (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' Ho  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' 1997;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' Lee et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' 2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' Moreover, there are indica-  tions that the bar alone is not always sufficient to feed  the black hole in an efficient way, requiring the contribu-  tion of other mechanisms (Combes 2008;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' Sellwood 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='  Fanali et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' Galloway et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' 2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' This is because,  in order to feed the black hole, the molecular gas in the  disk needs to lose almost all its angular momentum (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='  Sellwood & Wilkinson 1993;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' Krolik 1999;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' Sellwood 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='  Capelo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' Using a sample of spiral galaxies,  Alonso et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' (2014) found that their location within the  group or cluster influences both the AGN and bar frac-  tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' This result suggests that the external mechanisms  affecting galaxies in dense environments may give a sig-  nificant contribution to triggering the AGN activity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' For  the specific case of jellyfish galaxies, this external mech-  anism might be the interaction with the ICM (Poggianti  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' 2017a;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' Peluso et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' Indeed, the ram pres-  sure can not only compress the gas in the disk, but it  can also make the gas lose its angular momentum and  eventually move inward (Ramos-Mart´ınez et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='  Ricarte et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' Farber et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' 2022, Akerman et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='  submitted).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='  Thus, the gas could easily reach the re-  gion influenced by the bar, which may drag it further  inward, perhaps reaching the black hole.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' This picture is  in agreement with the enhanced fraction of AGN in ram  pressure stripped galaxies (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' Poggianti et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' 2017a;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='  Peluso et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' 2022, but see Roman-Oliveira et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' 2019  for a different conclusion).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' The relative importance of  internal mechanisms, such as bars, and external pro-  cesses, such as ram pressure, in fueling the AGN activity  is a compelling and debated topic (Alonso et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='  Kim & Choi 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' Boselli et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' 2022a), which would  require higher statistics than the four galaxies studied  here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' This task goes beyond the scope of this paper and  we leave it to future studies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' Comparison with Virgo galaxies  There is growing evidence that the ram pressure can  affect the molecular gas in cluster galaxies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' In this sec-  tion, we compare the GASP-ALMA sample with the  galaxies in Virgo cluster, in order to increase the statis-  tics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' Other cases of ram pressure affecting the molecu-  lar disk have been found in Coma (J´achym et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' 2017),  Norma (J´achym et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' 2014), and Fornax (Zabel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='  2019), just to mention some examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='  Lee et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' (2017) studied the molecular gas kinemat-  ics in three disk galaxies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='  These author did not find  clear signs of molecular gas stripping, but they showed  that the morphological and kinematical disturbances in  the molecular and atomic gas disks are closely related  to each other, suggesting that the molecular gas can be  also affected by strong ram pressure even if it is not  globally stripped.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' They also ascribed the perturbation  in the innermost regions of their galaxies to the pres-  ence of a stellar bar, rather than to ram pressure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' As  discussed in Sects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='1 and 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='2, our results for JO201  and JO204 are consistent with Lee et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' (2017)’s find-  ings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' Interestingly, all the molecular gas disks in Lee  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' (2017) sample are kinematically lopsided, at least  to some degree, indicating that the molecular gas was  either accelerated or decelerated by ram pressure (see  also Cramer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' They also found CO clumps  that are kinematically decoupled from the molecular gas  disk, suggesting that this gas was displaced by the ram  pressure, as in the case of JO206 (see Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='  Recently, Brown et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' (2021) presented the first re-  sults of the Virgo Environment Traced in CO (VER-  TICO) survey, which maps CO emission in 51 galaxies  in Virgo cluster using ALMA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' This authors derived the  mass-size relation for the molecular gas disk for VER-  TICO galaxies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='  They showed that the scatter in the  relation is minimized if the disk size is defined as the ra-  dius where the azimuthally-averaged H2 surface density  reaches ΣH2 = 5 M⊙pc−2 (R5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' As a control sample,  Brown et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' (2021) used the field galaxies in the Het-  erodyne Receiver Array CO Line Extragalactic Survey  (HERACLES, Leroy et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' 2009).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' Brown et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' (2021)  found that the best-fit relations for the galaxies in Virgo  and in the field are consistent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' They concluded that R5-  MH2 relation does not depend on the environment, in  agreement with the studies on the HI size–mass relation  (Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' Stevens et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' 2019), and that galax-  ies affected by environmental processes move along the  size-mass relation rather than deviating from it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='  In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' 9, we compare our galaxies with the R5-MH2 re-  lation from Brown et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' We assumed the Milky  Way CO-to-H2 conversion factor for consistency with  Brown et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' Our galaxies are within the scatter  Molecular gas kinematics in jellyfish galaxies  19  of the R5-MH2 relation derived by Brown et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' (2021),  confirming that this scaling relation does not show any  clear dependence on environment, even in extreme ram  pressure cases as the galaxies of our sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' We note  that the GASP-ALMA sample tend to be slightly below  the relation, suggesting that the molecular gas distri-  bution is more centrally concentrated than the average  for these samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='  This can be due to the combined  effect of stellar bars, which tend to increase the gas den-  sity in the inner regions of the disk (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' Kormendy &  Kennicutt 2004), and ram pressure, which compresses  the molecular disk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' The GASP-ALMA galaxies stand  out against the other two samples because of their high  MH2, being up to ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='5 dex more massive than the Virgo  and control samples (see also Moretti et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' 2020a,b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' On  the other hand, it has been shown that our galaxies are  up to 50% deficient in HI with respect to field galax-  ies with similar mass and size (Ramatsoku et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' 2019,  2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' Deb et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' Healy et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' Deb et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='  Taken together, these results suggest an unusually ef-  ficient conversion of HI to H2 (Moretti et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' 2020b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='  These properties are in agreement with the recent re-  sults by Zabel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' (2022) for Virgo galaxies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='  They  found that the galaxies showing clear signs of ongoing  ram pressure stripping affecting the HI disk are from  H2-normal to H2-rich.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' This was interpreted as an indi-  cation that ram pressure stripping is not effective at re-  ducing global molecular gas fractions on the timescales  in which such features are still clearly visible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' This is  likely because the stripping is less severe on H2 than on  HI, as the molecular gas is denser and more gravitation-  ally bound to the galaxy than the atomic gas (Lee et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='  2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' Boselli et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' 2022a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' The atomic gas disk of our  galaxies show indeed signs of truncation and the ram  pressure stripping is much more dramatic than for the  molecular gas disk (Ramatsoku et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' 2019, 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' Deb  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' 2020, 2022)  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' Baryonic Tully-Fisher relation  Rotation curves of disk galaxies are typically used  to derive fundamental scaling relations (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' Verheijen  2001;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' Lelli et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' 2016b;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' Ponomareva et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' Io-  rio et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' Posti et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' Mancera Pi˜na et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='  2021a,b;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' Di Teodoro & Peek 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' In particular, the  baryonic Tully-Fisher relation (hereafter BTFR) is a  very tight correlation between the mass of baryons and  the rotation velocity of galaxies, being a useful test-case  to check the robustness of the stellar rotation derived  in this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='  The BTFR is usually derived using HI  rotation curves, as the atomic gas disk is the most ex-  tended baryonic component, allowing to probe the flat  part of the galaxy rotation curve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' In jellyfish galaxies,  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='5  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='0  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='5  8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='0  8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='5  9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='0  9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='5  10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='0  10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='5  11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='0  log[MH2/M  ]  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='5  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='0  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='5  log[R5/kpc]  Molecular gas size-mass relation using R5  Brown+2021  ±  JO201  JO204  JO206  JW100  Virgo  Field  Figure 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='  Molecular gas mass-size relation based on R5  (see Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='  Each galaxy in the GASP-ALMA sample  is indicated by a colored symbol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' Grey diamonds and pink  points show galaxies in the Virgo cluster and nearby field  galaxies (see text), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='  The best-fit relation ob-  tained by Brown et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' (2021) for the VERTICO and HER-  ACLES samples is shown by the dash-dotted line, while its  scatter is represented by the shaded area.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='  the atomic gas disk is stripped or truncated by the ram  pressure and the HI kinematics is strongly perturbed,  hampering the usage of HI observations to study scal-  ing relations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='  The ionized gas is not a good alterna-  tive to HI, as not only it is less spatially extended but  also more diffuse and thus easier to strip.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' The results  of this work suggest that the molecular gas is more re-  silient to ram pressure, but its spatial extend is still very  limited.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' Therefore, the stellar component is likely the  best way to derive scaling relations in the case of jelly-  fish galaxies, provided that observations with high spa-  tial resolution and sensitivity are available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' The GASP  sample is ideal to perform this exercise, thanks to the  high spatial resolution and sensitivity of the MUSE ob-  servations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='  In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' 10, we show that the galaxies in  the GASP-ALMA sample closely follow the BTFR de-  rived by Di Teodoro et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' (2021) using a sample of  about 200 galaxies from high-mass disks to dwarf galax-  ies (Lelli et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' We calculated the velocity in the  flat part of the rotation curve (Vflat) as the average of  the outermost 5 measurements of the stellar rotation ve-  locity (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' 5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' The baryonic mass was calculated as  Mbar = M⋆ + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='33 (MHI + MH2), where the masses of  atomic gas (MHI) and molecular gas (MH2) are taken  from Table 1 and the multiplicative factor 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='36 accounts  for the Helium content.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='  We checked that considering  only the gas mass within the stellar disk or the total gas  mass (including the gas in the stripped tail) does not  20  Bacchini et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='00  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='25  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='50  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='75  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='00  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='25  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='50  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='75  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='00  log[Vflat/(km/s)]  7  8  9  10  11  12  13  log[Mbar/M  ]  Baryonic Tully-Fisher relation  Best-fit (Di Teodoro+21)  y = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='6x + 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='49  JO201  JO204  JO206  JW100  = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='1 dex  Lelli+19  Di Teodoro+21  Figure 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='  Baryonic Tully-Fisher relation for the four  galaxies in the GASP-ALMA sample (triangles, diamond,  and cross).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' The grey points show the spiral and dwarf galax-  ies from Lelli et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' (2019), while the pink stars are for the  massive disks from Di Teodoro et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='  The dashed  line is their best-fit relation with the shaded area showing  the orthogonal intrinsic scatter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='  change the results, as the gas mass is largely dominated  by molecular gas component which is mostly concen-  trated within the galaxy disk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='  We also checked that  our galaxies fall on the stellar Tully-Fisher relation (not  shown here), which is not surprising given that the bary-  onic mass is largely dominated by the stellar component.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='  These simple tests indicate that the GASP sample can  be used to study important scaling relations of baryons  and, potentially, dark matter (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' Lelli et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' 2016b;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='  Posti et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' Mancera Pi˜na et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' 2021a;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' Di Teodoro  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' This will be addressed in future work by  fully exploiting the richness and quality of the MUSE  observations obtained with the GASP survey (Bacchini  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=', in preparation).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' SUMMARY AND CONCLUSIONS  Galaxies in dense environments, such as clusters, can  be affected by the ram pressure due to the interaction  with the ICM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' This process leaves the stellar disk es-  sentially unperturbed, but it can have a strong impact  on the morphology, kinematics and overall gas content,  with important consequences on the evolution of galax-  ies (Cortese et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' In this context, jellyfish galax-  ies are ideal cases to study the impact of ram pressure  on the gas components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' In this work, we have studied  the distribution and kinematics of the molecular gas in  a sample of four jellyfish galaxies in the GASP sample  (Poggianti et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' 2017b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' These galaxies were observed  with ALMA to detect the CO(1–0) and CO(2–1) emis-  sion Moretti et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' (2020a,b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' Thanks to the wealth of  information obtained from MUSE and ALMA observa-  tions provided by the GASP survey, we could analyze  the stellar and CO distribution and kinematics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' We used  the software 3DB based on the tilted-ring approach to  model the stellar velocity field and the CO emission line  datacubes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' We identified the gas with anomalous veloc-  ity that cannot be explained by a rotation disk and used  the information on the stellar distribution and kinemat-  ics to understand the origin of this anomalous gas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' We  reached the following conclusions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' At least three (JO201, JO204, and JO206) out  of four galaxies in the GASP-ALMA sample are  barred.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='  In JO201 and JO206, the bars aligned  with the disk major axis are visible in the I-band  images and explain the shallow gradient of the cir-  cular velocity in the inner regions of these galax-  ies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' In JO204, various bar signatures are found in  the distribution of the molecular gas and the kine-  matics of both the molecular gas and the stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' In  JW100, the disk inclination and dust obscuration  do not allow us to unambiguously identify a bar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' The molecular gas kinematics in JO201 and JO206  is mainly dominated by non-circular motions in  the region influenced by the bar, while the ram  pressure becomes important at larger galactocen-  tric distance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' The ram pressure plays a secondary  role for the molecular gas kinematics of JO204,  which is mainly rotation-dominated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' Clear indi-  cations of molecular gas stripping are found in  two galaxies, JO206 and JW100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' In JO206, some  molecular gas is detached and kinematically de-  coupled from the main disk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' In JW100, the molec-  ular gas disk is displaced with respect to the stel-  lar disk and its kinematics is strongly perturbed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='  Since JO201 and JW100 belong to cluster sub-  structures, other mechanisms than ram pressure  might be also at play.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' Radial flows of molecular gas are manifestly  present in two galaxies (JO204 and JW100), but  this is less clear in the other two objects (JO201  and JO206).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='  These gas flows are consistent  with being bar-driven inflows in JO206 and ram  pressure-driven outflows in JO201 and JW100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='  The direction of radial motions remains unclear  for JO204.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' The molecular gas velocity dispersion in JO201,  JO206, and JW100 tends to be enhanced with re-  spect to field galaxies, suggesting that the gas is  very turbulent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' In the case of JO201 and JO206,  Molecular gas kinematics in jellyfish galaxies  21  this can be explained by the complex motions in-  duced by the bar within its region of influence  or, beyond the bar region, by the the ram pres-  sure, which can enhance the gas turbulence di-  rectly and/or by increasing the SFR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' In the case  of JW100, the most likely scenario is that the gas  turbulence is directly enhanced by the ram pres-  sure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' Our galaxies fall within the scatter of the molec-  ular gas mass-size relation derived for field and  Virgo galaxies by (Brown et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' 2021), confirming  that the relation is essentially independent of en-  vironment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='  Overall, our results are consistent with a scenario in  which the molecular gas is affected by ram pressure on  different timescales and less severely than the atomic  and ionized gas, likely because the molecular gas is  denser and more gravitationally bound to the galaxy  than the other gas phases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' The galaxies in the GASP-  ALMA sample host an AGN (Poggianti et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' 2017a;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='  Peluso et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' Both stellar bars and ram pressure  can contribute to efficiently drive molecular gas towards  the galaxy center, possibly feeding the central black hole  and triggering the nuclear activity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' Since the relative im-  portance of bars and ram pressure in fueling the AGN  has not been fully understood yet, we hope that our  work may foster future studies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' In this work, we have  shown that high-resolution observations of the molecular  gas emission can be very useful in identifying stellar bars  and radial flows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' Future effort will be devoted to fur-  ther study the bar-AGN connection by expanding the  GASP-ALMA sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' Moreover, we have shown that  the GASP sample is potentially very useful to investi-  gate the impact of environment on scaling relations of  galaxies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' In future work, we plan to address this topic  by fully exploiting the richness and quality of the MUSE  observations obtained with the GASP survey.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='  This paper makes use of the following ALMA data:  ADS/JAO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='ALMA#2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='00496.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' ALMA is a partner-  ship of ESO (representing its member states), NSF  (USA) and NINS (Japan), together with NRC (Canada)  and NSC and ASIAA (Taiwan), in cooperation with the  Republic of Chile.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' The Joint ALMA Observatory is op-  erated by ESO, AUI/NRAO and NAOJ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' CB acknowl-  edges financial support from the European Research  Council (ERC) under the European Union’s Horizon  2020 research and innovation programme (grant agree-  ment No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='  833824).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='  CB would like to thank E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' Di  Teodoro, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' Rizzo, and F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' Fraternali, for useful discus-  sions and the help with the kinematic modeling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='  Facility: ALMA, MUSE@VLT  Software:  3DBarolo (Di Teodoro & Fraternali  2015), APLpy (Robitaille & Bressert 2012), Astropy (As-  tropy Collaboration et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' 2013, 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='  APPENDIX  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' BEST-FIT MODELS OF THE STELLAR VELOCITY FIELD  This section provides the best-fit model of the stellar disk for JO204, JO206, and JW100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='  The top panels in  Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' 11, 12, and 13 show the observed stellar velocity field, the best-fit model, and the map of the residuals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' The  bottom panels display the stellar rotation curve and the radial profiles of the PA and inclination.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' Overall, the stellar  kinematics is well reproduced by our models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' However, the residual map of JO204 clearly shows a pattern in the  central regions of the disk, which likely indicates the presence of a stellar bar (see also Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' The residual map  of JW100 highlights some regions where the model does not fully reproduces the observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' The origin of these  differences is tricky to understand and may be due to asymmetric dust observation or the warp along the line of sight,  or both (see also Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
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+page_content='1051/0004-6361/201832796  Alonso, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
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+page_content=' 2014, A&A, 572,  A86, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='1051/0004-6361/201424523  Andersen, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' 1996, AJ, 111, 1805, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='1086/117918  22  Bacchini et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='  0  2  4  6  8  10  12  R [kpc]  0  50  100  150  200  250  Rotation velocity [km/s]  2nd fit  3rd fit  0  2  4  6  8  10  12  R [kpc]  70  72  74  76  78  80  Inclination [degrees]  2nd fit  Median  3rd fit  ± MAD  0  2  4  6  8  10  12  R [kpc]  318  320  322  324  326  328  Position angle [degrees]  2nd fit  Median  3rd fit  ± MAD  10h13m48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='0s 47.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='5s  47.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='0s  46.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='5s  46.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='0s  0°54\'35"  40"  45"  50"  55"  55\'00"  05"  RA (ICRS)  Dec (ICRS)  1"  JO204 - Data  5 kpc  200  150  100  50  0  50  100  150  200  VLOS [km/s]  10h13m48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='0s 47.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
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+page_content='5s  46.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='0s  RA (ICRS)  JO204 - Model  200  150  100  50  0  50  100  150  200  VLOS [km/s]  10h13m48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='0s 47.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='5s  47.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='0s  46.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='5s  46.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='0s  RA (ICRS)  JO204 - Residuals  40  20  0  20  40  Data-Model [km/s]  0  2  4  6  8  10  12  14  16  R [arcsec]  0  2  4  6  8  10  12  14  16  R [arcsec]  0  2  4  6  8  10  12  14  16  R [arcsec]  Figure 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' Same as Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' 3 but for JO204.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='  0  5  10  15  20  R [kpc]  0  50  100  150  200  250  Rotation velocity [km/s]  2nd fit  3rd fit  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
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+page_content='0  12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='5  15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='0  17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
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+page_content='0  R [kpc]  35  40  45  50  55  60  65  70  Inclination [degrees]  2nd fit  Median  3rd fit  ± MAD  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
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+page_content='21h13m49s  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
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+page_content='2°28\'45"  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='30"  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
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+page_content='Dec (ICRS)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='1"  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
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+page_content='5 kpc  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='200  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='150  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='100  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
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+page_content='VLOS [km/s]  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
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+page_content='JO206 - Model  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='200  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='150  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='100  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
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+page_content='100  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='150  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='200  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='VLOS [km/s]  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='21h13m49s  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='48s  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='47s  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='46s  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='RA (ICRS)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='JO206 - Residuals  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='40  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='20  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
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+page_content='40  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='Data-Model [km/s]  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='0  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
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+page_content='15  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='20  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='R [arcsec]  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
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+page_content='15  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='20  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='R [arcsec]  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='0  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
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+page_content='10  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='15  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='20  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='R [arcsec]  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='Figure 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' Same as Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' 3 but for JO206.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='  Molecular gas kinematics in jellyfish galaxies  23  0  5  10  15  20  25  R [kpc]  0  50  100  150  200  250  300  350  400  Rotation velocity [km/s]  2nd fit  3rd fit  0  5  10  15  20  25  R [kpc]  67.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='5  70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='0  72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='5  75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='0  77.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='5  80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='0  82.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='5  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='Inclination [degrees]  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='2nd fit  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='Median  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='3rd fit  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='± MAD  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='0  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='5  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='10  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='15  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='20  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='25  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='R [kpc]  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='177  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='178  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='179  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='180  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='181  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='182  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='183  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='184  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='Position angle [degrees]  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='2nd fit  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='Median  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='3rd fit  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='± MAD  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='23h36m26s  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='25s  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='24s  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='21°09\'15"  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='00"  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='08\'45"  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='RA (ICRS)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
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+page_content=' Same as Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
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+page_content='  Astropy Collaboration, Robitaille, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=', Tollerud, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=',  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' 2013, A&A, 558, A33,  doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
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+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=', Sip˝ocz, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=',  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' 2018, AJ, 156, 123, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
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+page_content=' 1992a, MNRAS, 259, 345,  doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
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+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
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+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='328  Bacchini, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
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+page_content=', Iorio, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=', & Pezzulli, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' 2019a,  A&A, 622, A64, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
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+page_content=', Fraternali, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
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+page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' 2020a, A&A,  641, A70, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
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+page_content=', Fraternali, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=', Pezzulli, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=', & Marasco, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='  2020b, A&A, 644, A125,  doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='1051/0004-6361/202038962  Bacchini, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=', Fraternali, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=', Pezzulli, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' 2019b, A&A,  632, A127, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='1051/0004-6361/201936559  Barazza, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=', Jablonka, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=', Desai, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' 2009, A&A,  497, 713, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='1051/0004-6361/200810352  Barnes, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=', & Hernquist, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' 1992, ARA&A, 30, 705,  doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='1146/annurev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='aa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content='090192.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
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+page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' 1987, PhD thesis, -  Bellhouse, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=', Jaff´e, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=', Hau, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
+page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQfUgM7/content/2301.03090v1.pdf'}
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+IRONFORGE: An Open, Secure, Fair, Decentralized
+Federated Learning
+Guangsheng Yu∗, Xu Wang†, Caijun Sun§, Qin Wang∗, Ping Yu‡,
+Wei Ni∗, Renping Liu†, Xiwei Xu∗
+∗CSIRO Data61, Australia
+†University of Technology Sydney, Australia
+‡Harbin University of Technology, China
+§Zhejiang Lab, China
+Abstract—Federated learning (FL) provides an effective ma-
+chine learning (ML) architecture to protect data privacy in a
+distributed manner. However, the inevitable network asynchrony,
+the over-dependence on a central coordinator, and the lack of an
+open and fair incentive mechanism collectively hinder its further
+development. We propose IRONFORGE, a new generation of FL
+framework, that features a Directed Acyclic Graph (DAG)-based
+data structure and eliminates the need for central coordinators
+to achieve fully decentralized operations. IRONFORGE runs in
+a public and open network, and launches a fair incentive
+mechanism by enabling state consistency in the DAG, so that the
+system fits in networks where training resources are unevenly
+distributed. In addition, dedicated defense strategies against
+prevalent FL attacks on incentive fairness and data privacy are
+presented to ensure the security of IRONFORGE. Experimental
+results based on a newly developed testbed FLSim highlight the
+superiority of IRONFORGE to the existing prevalent FL frame-
+works under various specifications in performance, fairness, and
+security. To the best of our knowledge, IRONFORGE is the first
+secure and fully decentralized FL framework that can be applied
+in open networks with realistic network and training settings.
+Index Terms—Federated Learning, DAG, Blockchain
+I. INTRODUCTION
+Federated learning (FL), officially introduced by Google
+in 2017 [1], has become the preference to aggregate data
+from distributed ends without breaching data privacy [1],
+[2]. By aggregating huge data with comprehensive extracted
+features in FL, critical issues such as model overfitting can be
+significantly addressed [3]. However, Πthe inevitable network
+asynchrony,  the over-dependence on a central coordinator,
+and Ž the lack of an open and fair incentive mechanism hinder
+the further development of FL in large and open scenarios [4].
+Traditional FL considers no or low delay throughout an ag-
+gregation process, namely, synchronous FL. However, network
+synchrony is unrealistic due to the inevitable capacity limit
+of computation, bandwidth, and storage, as well as the im-
+balanced capacities among the distributed participants. Thus,
+recent studies propose pseudo-asynchronous FL [5] and asyn-
+chronous FL [6]. The aggregation of pseudo-asynchronous
+FL allows a short interval for collecting the model caches in
+order to ensure that the number of models aggregated can be
+sufficiently large, while the central coordinator immediately
+updates the global model once receiving a new local model
+from any idle participants in asynchronous FL.
+Neither pseudo-asynchronous FL nor asynchronous FL can
+tolerate the single-point-of-failure (SPoF) of the central coor-
+dinator or even a malicious and corrupted coordinator (issue-
+). The over-dependence on the central coordinator could
+potentially degrade the system availability and the training
+flexibility in the sense that an FL network may be confined to
+specific training domains or tasks determined by the coordina-
+tor. Participants in many existing studies [7]–[9], once opting
+in an FL network, would have to obey the defined training
+target with no flexibility to go for different tasks at will.
+In addition to the weak training flexibility, the lack of an
+open and fair incentive mechanism results in participants who
+have fewer resources and a weaker capacity not willing to
+contribute their resources to the global aggregation. This issue
+deteriorates particularly in FL networks where resources are
+not evenly distributed, and potentially leads to the model over-
+fitting and weak generality against contingencies. Although
+the authors of [10] survey the incentive mechanisms in FL,
+all mentioned frameworks require a central coordinator, also
+leading to issue-.
+Existing studies propose to replace the central coordinator
+with a committee running a consensus process in a blockchain
+network to prevent the SPoF or a corrupted coordinator. Mean-
+while, by sharing the model collection during the consensus
+in the committee, pseudo-asynchronous FL can be achieved in
+a decentralized manner, i.e., BlockFL [7], [11]–[14]. Consid-
+ering only issue- being solved and issue-Œ being partially
+solved by BlockFL, the authors of [9] introduce a Directed
+Acyclic Graph (DAG)-based FL where both issue-Πand issue-
+ are solved using the concept of asynchronous FL [6] to
+fully decentralize the FL process. However, the paper [9] only
+considers an ideal network in which the training resources are
+evenly distributed. Moreover, the approach to enabling state
+consistency for a secure and fair incentive mechanism (issue-
+Ž) is missing in [9], which results in difficulty in adopting the
+mechanism in a public and open network.
+We propose IRONFORGE that is an open, secure, fair,
+and decentralized FL system. IRONFORGE solves the above
+mentioned pain points at one time. Openness: It features a
+DAG-based data structure in an open network. Decentral-
+ization: The need for a central coordinator is eliminated
+throughout the process by IRONFORGE, inheriting from the
+concept of asynchronous FL. As a result, the models are
+1
+arXiv:2301.04006v1  [cs.LG]  7 Jan 2023
+
+TABLE I: Qualitative comparisons between the proposed IRONFORGE and the existing FL frameworks
+FL Framework
+Data Structure
+Data Asynchrony
+Decentralization
+Openness
+Incentive
+Security
+Google FL [2]
+Isolated models
+Synchronous
+Centralized
+Private
+�
+�
+Asynchronous FL [6]
+Isolated models
+Asynchronous
+Centralized
+Private
+�
+�
+Block FL [15]
+Blockchain
+Synchronous
+Decentralized
+Private
+Reward
+�
+DAG FL [9]
+DAG
+Asynchronous
+Decentralized
+Public
+Reward
+Poisoning/Backdoor/Lazy
+IRONFORGE
+DAG
+Asynchronous
+Decentralized
+Public
+Reward, Penalty
+Poisoning/Backdoor/Stealing*/Collusion
+* The stealing attack considered in this paper includes the traditional lazy attack.
+The difference is that stealing attackers not only upload their previous models, but also fake the ownership of others’ previous models.
+� Lack of corresponding designs.
+maintained in a decentralized manner by all participants.
+Fairness: IRONFORGE considers a practical scenario, where
+resources are unevenly distributed among users. Each user,
+based on its resource amount, selects several existing models,
+verifies the correctness and evaluates the model accuracy over
+the local dataset, and conducts the aggregation. IRONFORGE
+also enables state consistency, by using which an open and
+fair incentive mechanism can be established to motivate more
+participants. Security: Moreover, dedicated defense strategies
+against malicious attacks on incentive fairness, and against
+dataset privacy breaching are presented to ensure the security
+of IRONFORGE. The key contributions are as follows.
+⊲ We propose a fully decentralized FL framework, namely,
+IRONFORGE, which features a DAG-based data structure.
+IRONFORGE addresses the network asynchrony typically
+undergone in an FL process, and improves the motivation
+of agents participating in the process in an open envi-
+ronment by enabling reliable token rewards with strong
+consistency and model prediction accuracy.
+⊲ We specifically design a new validation mechanism
+guarding against well-known FL attacks, including model
+poisoning attacks, backdoor attacks, lazy attacks, and
+model stealing attacks, among which the model of steal-
+ing attack has never been considered in any existing FL
+frameworks. By making use of noise-enabled Proof-of-
+Learning (PoL) to validate the gradient descent process,
+any malicious behaviors, such as faking the ownership
+or directly using the existing models, or embezzling the
+rewards for their conspirator by claiming a falsified source
+list, can be captured and given punishments.
+⊲ We build a flexible and efficient testbed, named FLSim,
+to simulate the workflow across all considered FL frame-
+works in this paper, including the proposed IRONFORGE.
+We conduct comprehensive experiments based on FLSim,
+comparing the system performance, security, and fairness
+between the existing FL frameworks and IRONFORGE.
+Insights are shed to provide guidelines on how to select
+strategies in IRONFORGE to meet different requirements.
+Extensive experiments corroborate that IRONFORGE outper-
+forms the prevalent FL frameworks with and without attacks
+leveraged, which highlights the holistic solution to the network
+asynchrony (issue-Œ) and the over-dependence on the central
+coordinators (issue-). Strictly and approximately monotonic
+increases of rewards are observed in experiments with increas-
+ing CPU cores, memory capacity, and bandwidth in different
+incentive settings. This indicates that fairness (issue-Ž) can be
+ensured in IRONFORGE under various definitions of fairness.
+The rest of the paper is organized as follows. Section I
+gives the introduction, followed by related works in Section II.
+Section III provides the system overview and Section IV
+details the design of IRONFORGE. Section V presents our
+implementation based on a new testbed with comprehensive
+experimental results. Section VI discusses system security and
+properties. Finally, Section VII concludes this work.
+II. RELATED WORK
+A conventional synchronous FL framework is constructed
+by a central coordinator and numbers of nodes, which main-
+tains the global model and perform FL iterations, respec-
+tively [2]. The coordinator periodically distributes the latest
+global model to the nodes, and then the nodes independently
+train the model with their local data and upload the trained
+local models to the coordinator [16]. After receiving updated
+models from nodes, the coordinator aggregates all the local
+models as a new global model. Such synchronous FL frame-
+work can hardly be adapted to large-scale and heterogeneous
+networks, where asynchrony is non-negligible.
+The issue of data asynchrony is tackled by the asyn-
+chronous FL enabling nodes to train the global model from
+central coordinators at any time, and the coordinators can
+update the global model immediately when any local model
+is collected. In [14], the authors introduced a cache layer
+between the coordinator and local nodes. Each node trains
+the global model with its local data and uploads its model to
+the cache. The coordinator periodically aggregates the local
+model in the cache and generates a new global model. Semi-
+asynchronous FL protocols address the problems in FL such
+as low round efficiency and poor convergence rate happened
+in asynchronous FL. The system [5] incorporates a client se-
+lection algorithm decoupling the coordinator and the selected
+clients for a reduction of average round time. The authors
+of [17] proposed an asynchronous federating-based detection
+approach for end devices. A pre-shared data training strategy
+for non-independent-and-identically-distributed (non-IID) data
+is developed to avoid convergence divergence under the non-
+IID patterns. After the collaborative model training procedure,
+each client further conducts an additional local training process
+to fit respective patterns.
+The aforementioned FL frameworks require central coor-
+dinators to schedule model training and aggregate models.
+The centralized architecture suffers inherent security risks,
+such as SPoF and malicious central coordinator, and limited
+2
+
+scalability with the bottleneck of the central coordinator. The
+most recent Distributed Ledger Technology (DLT) holds the
+potential to decentralize FL systems [18], [19]. Two key
+technologies in DLT are blockchain and DAG. In blockchain,
+a group of miners run the consensus protocol to generate hash-
+chained data blocks, which are assembled from transactional
+data, and synchronize the chained blocks. Blockchain assures
+strong consistency among blockchain nodes and enables smart
+contracts to be executed across the blockchain network in a
+consistent and trustworthy way. In DAG, transactions from
+decentralized DAG users are organized in a DAG structure
+where directed edges indicate the reference relationship be-
+tween the transactions. DAG can achieve high throughput with
+short latency compared with blockchain [20].
+DLT has been developed to remove the central coordinator
+and decentralize FL networks [9], [15], [21]. In BlockFL [15],
+[22], [23], decentralized blockchain miners conduct model
+verification and aggregation. To be specific, miners obtain
+trained local models from working nodes and other miners.
+After verification, miners aggregate local models for the
+updated global models and conduct Proof-of-Work (PoW) to
+create valid blocks containing the new global models. Then,
+the blocks are propagated to all miners to start the next FL
+iteration. The BlockFL relies on the resource-intensive PoW
+consensus protocol to slow down the system and keep miners
+synchronized. To reduce overhead and improve scalability,
+DAG technology [24] is introduced to FL networks [9], [21],
+where trained models are updated to a DAG topology by
+working nodes without any coordination. Working nodes can
+learn the latest local models in the DAG by exchanging data
+with other nodes. By themselves, working nodes select and
+verify aggregate local models and train the models using local
+datasets. Next, working nodes publish their trained models to
+the DAG with directed edges indicating the model reference.
+Existing works only consider homogeneous networks where
+the training resources are evenly distributed and thus lack
+open and fair incentive mechanisms. IRONFORGE proposed
+by this paper, on the other hand, improves the motivation of
+participants with rewards for training contributions and penal-
+ties for dishonest behaviors. IRONFORGE also tackles new
+vulnerabilities in open FL networks, including model stealing
+attacks where attackers steal models from others and claim
+rewards from the plagiarized model, and collusion attacks
+where attackers claim trained models are from conspirators.
+III. SYSTEM OVERVIEW
+In this section, we describe IRONFORGE from the aspects
+of its architecture, workflow, and system assumptions.
+A. System Overview
+We first introduce the roles that participate in the system
+and present our high-level design.
+Architecture. IRONFORGE is a decentralized FL system that
+features a DAG-based network structure to tackle the incon-
+sistency in the decentralized FL process, excessive reliance
+on central coordination, and ineffective motivation of con-
+tributing the learning resources at the same time. Specifically,
+Fig. 1. System model of IRONFORGE
+IRONFORGE builds a hybrid architecture (cf. Fig. 1) that
+involves two types of DAG, namely, Task-DAG and Global-
+DAG (details refer to Fig. 2 and Fig. 3, respectively). The
+training processes in both Task-DAG and Global-DAG are
+traceable owing to the DAG data structure. A DAG node
+published by a participant consists of a model update and the
+directed edges of the node indicate the aggregating relationship
+with existing models during the update, hence no central
+coordinator required to conduct the training processes.
+Global-DAG contains a variety of models adopted by all
+participants, which can be viewed as a “unique” and public
+model resource pool. No consistent testing dataset is given
+in Global-DAG. Each user comes to Global-DAG and hunts
+for models that uniquely meet its own local testing dataset.
+Without central coordinators, any user can fetch models from
+the pool for direct uses, release his task requests, or make con-
+tributions, such as training on Global-DAG or on uncompleted
+training tasks, or verifying the tasks.
+Each training task is managed by a Task-DAG, while IRON-
+FORGE can contain multiple Task-DAGs at the same time
+to handle a range of different training tasks (see the right-
+hand side in Fig. 1). Task-DAGs are task-specific and are
+released by users who aim at improving their local model
+prediction accuracy by virtue of the computational powers
+and resources of others. Within a task, the Task-DAG network
+contains multiple contributors who have the same training
+target provided by the publisher. The trained models for each
+task are broadcast and stored in the corresponding Task-DAG,
+and await the check and verification. The satisfied model of a
+task, observed by the publisher, is subsequently merged into
+Global-DAG, increasing the exposure to the public users. As a
+result, parallel learning on our hybrid DAG networks becomes
+possible, and the resultant models can be collected by Global-
+DAG for further involvement.
+Roles. In IRONFORGE, the users can take different roles:
+viewer, task publisher, verifier, and contributor. A user is a
+participant in the network. Each user can select one or multiple
+roles to perform specific functional activities (see the left-hand
+side in Fig. 1). Specifically, a viewer can directly fetch models
+from the public resource pool without further actions. The
+task publisher aims to propose new tasks and the proposed
+tasks are broadcast and await others’ contributions. In order
+3
+
+Global Network
+User
+Global DAG
+Contributor
+User
+Task DAG
+Global DAG
+Contributor
+Task DAG
+User
+Global DAG
+Contributor
+Task DAG
+Viewer
+Fetching 
+Evaluating
+Models
+Selected Models
+Satisfied Mode
+Task Publisher
+Aggregating
+Verification Committee
+Aggregated Model
+Local dataset
+Contributor
+Training
+Trained Model
+Selected Models
+Task Publisher
+sk
+Aggregated Model
+Verification Committee
+%
+Trained Model
+Task
+WorkNerifyTask
+Community
+……
+……
+t
+① Register and Release a task
+Publisher
+Workers
+② Announce
+③ Observe
+……
+b. Sync the DAG and validate DAG nodes
+c. Evaluate some models with local test dataset
+d. Pick up the best ones and aggregate them
+e. Start training with local training dataset
+f. Publish the model to the DAG
+④ Train and publish models
+Task-model node
+Header:
+Sender: …
+Timestamp: …
+Sources: […]
+Evaluations: […]
+Payload:
+Weights: ipfs_uri+hash
+④
+④
+④
+Local
+dataset
+Local
+dataset
+Local
+dataset
+Task-termination node
+Sender: Publisher
+Timestamp: …
+Winner: …
+Public testing dataset: Dtest
+Accuracy: …
+Balance update: …
+Task-genesis node
+Header:
+Sender: Publisher
+Timestamp: …
+Target: <e.g., 95%>
+Commitment of Dtest: …
+Prize: …
+Contest strategy: …
+Penalty strategy: …
+Verification committee:
+Payload:
+Weights: ipfs_uri+hash
+⑤ End a task by selecting the winner with the highest accuracy
+a. Register on the global FL-DAG
+V
+P
+Verify
+Fig. 2. Task-DAG. The figure illustrates an overview of starting a new DAG-based FL task, also known as Task-DAG. One
+with aims to improve his model accuracy to a certain target with the help of the community can release a task as the task
+publisher. Some amount of token is deposited as the prize which will be subsequently awarded to all eligible participants
+until the winning model is found and selected by the task publisher. The balance update of each participant is recorded in a
+task-termination node published by the task publisher, and can be subsequently settled by the Global-DAG network.
+to reap profits, a user can become a contributor to process a
+training process by selecting, aggregating and training models.
+He can either start the work on Global-DAG or enroll in others’
+published work from the uncompleted tasks. Also, a verifier in
+the system is to verify existing tasks in the resource pool. He
+can contribute or verify either one favorable task or multiple
+tasks in parallel for a higher profit. In short, the four roles
+cover all potential functional activities within IRONFORGE.
+B. Workflow Overview
+Then, we provide an overview of the workflow of IRON-
+FORGE. We focus on the procedures of task establishment and
+task processing by presenting the interactive steps of a user
+between the Task-DAG and Global-DAG.
+Step-1. The user registers a task in the Global-DAG network by
+depositing the committed prize. He obtains a task identifier
+and then broadcasts the task to the network. We assume
+that another user has accepted the proposed task prior and
+worked on the task as a task contributor.
+Step-2. The contributor enters the procedure of training mod-
+els. He first evaluates several existing models from the pool
+and selects a series of models for the shortlist.
+Step-3. Based on the selected models, the contributor aggre-
+gates all the short-listed models and integrates them with
+local datasets to train the model according to requirements.
+Step-4. Once completing the training, the contributor submits
+the trained model to the Task-DAG. Meanwhile, peer con-
+tributors may also work on the same task and generate com-
+petitively trained models. All these models are propagated
+within the Task-DAG network.
+Step-5. The publisher who obtains the trained model termi-
+nates the task by marking it with a termination tag. Once
+selected by users who are conducting the training process
+in Global-DAG, the trained model is deemed to be formally
+synchronized into Global-DAG.
+Notably, a user in the Global-DAG network can either
+contribute to other tasks proposed by peer users, or personally
+publish a task by himself. All the procedures follow similar
+steps, as described from Step-2 to Step-5.
+C. System Assumptions
+In this section, we list our assumptions on the network,
+security, and threat models of IRONFORGE.
+Resource assumption. We do not assume any resource dis-
+tribution in our work. The resource distribution in the entire
+network is random. This means different participants, with a
+high probability, hold different computing resources, including
+computing power, network bandwidth, memory space, storage
+capability, and training dataset quality. Addressing the system
+heterogeneity is one of the core contributions in this work, as
+we weaken the long-existing implicit assumption in previous
+work [9]: the even resource distribution. IRONFORGE enables
+any distribution of shares of any type of resources among the
+participants, making the system practical.
+User behavior assumption. We have two assumptions on user
+behaviors. First, the participants in the network are rational,
+meaning that they can select an arbitrary task, switch to
+others, or quit existing tasks for better profits. Second, different
+participants can focus on different training targets, including
+both task bundles (one task has dependency on another) and
+orthogonal tasks (one task is independent of the others). This
+enables the processing of multiple tasks in parallel, greatly
+improving the system’s overall scalability and performance.
+Security assumption. We assume that the honest nodes
+always conduct honest behaviors, where they obey all the
+policies during the model selection, model aggregation, model
+4
+
+Task-1
+(terminated)
+Task-2
+(terminated)
+Task-N
+(ongoing)
+G
+1
+t1
+t2
+tn
+2
+V
+P
+τ1
+3
+Local
+dataset
+Local
+dataset
+Local
+dataset
+……
+Community
+Publish
+Publish
+Publish
+Contribute
+Contribute
+VRF-
+consensus
+Settlement node
+Sender: <based on consensus>
+Timestamp: …
+PoL results: …
+Unverified PoL: …
+Balance: …
+Fig. 3. Global-DAG. This figure illustrates an overview of the Global-DAG network. With the absence of a centralized
+coordinator, each participant trains a model by selecting and aggregating as many models (including the outcomes of terminated
+tasks) published by others as possible (based on the local capability). Model targets are not unique and according to different
+needs, the network can be treated as a global resource pool containing a variety of models. Ones can either find a model which
+satisfies his local testing dataset from the pool, or make contributions to the pool and obtain token rewards by improving
+existing models. Token balances are periodically settled (endorsed by a verifiable random function (VRF)-driven consensus)
+by settlement nodes that employ a chain structure to achieve strong consistency.
+training, task verification, and other operations related to the
+defense strategies against adversaries. The adversaries have the
+ability to delay the model convergence and lower the model
+accuracy by leveraging popular FL attacks, including lazy
+attacks [9], poisoning attacks [25], and backdoor attacks [26].
+The adversaries also have the ability to breach the incentive
+fairness by leveraging model stealing attacks [27]–[30], and
+compromising the privacy of others’ training datasets. Adver-
+saries not only can upload their previous models (traditional
+lazy attacks), but also fake the ownership of others’ previous
+models or fake their own training process to embezzle rewards
+for their conspirators. These two faking types are defined as
+stealing attacks and collusion attacks, respectively, and both
+belong to the context of model stealing attacks in this paper.
+IV. DECENTRALIZED FEDERATED LEARNING
+IRONFORGE involves four novel mechanisms, i.e., the re-
+lease of Task-DAG networks, the decentralized model training,
+the defense strategy, and the incentive mechanism. IRON-
+FORGE features two types of network, public Global-DAG and
+task-specific Task-DAG. A training task can be outsourced to
+communities by releasing a Task-DAG network and following
+four steps including preparation, initialization, monitoring, and
+finalization. The decentralized training processes of Task-DAG
+and Global-DAG are specifically defined by the new decen-
+tralized model training mechanism, in which users aggregate
+existing models, train the aggregated models and publish the
+model updates to the network in a decentralized way. The
+training processes are guarded by a new defense strategy
+against model stealing attacks in the decentralized setting that
+has never been considered in existing studies. The crafted
+incentive mechanism assures the state consistency of networks
+and enables smart-contract-enhanced incentives including both
+rewards and penalties. Table II summarizes the notations.
+A. Managing Task-DAG
+Any user can outsource an FL training task by managing
+a DAG, as shown in Algo. 1. An FL training task can
+be described with a training target, i.e., the model to be
+trained, the targeted accuracy, and the testing dataset. To build
+incentives to motivate distributed workers and deter malicious
+workers, we design reward, penalty, and verification schemes
+for Task-DAG networks.
+1) Preparation: A training task Task𝑚 can be described
+with an initial model, i.e., 𝑊𝑚,𝑔, and an accuracy target 𝛼𝑚 of
+the model tested on the dataset 𝐷test
+𝑚
+from the task publisher
+𝑈𝑚,𝑝. To reduce the storage and bandwidth overhead, the
+model weights can be stored in external infrastructures, e.g.,
+the InterPlanetary File System (IPFS). The Uniform Resource
+Identifiers (URIs) and hash codes to the weights are embedded
+in DAG nodes for access and verification. The testing dataset
+𝐷test
+𝑚
+is committed in the genesis node of the Task-DAG by
+embedding the hash code, and is revealed by the end of the
+training for model verification. This commit-and-reveal design
+prevents direct access to 𝐷test
+𝑚 during the training process and
+ensures that the final selected model can be publicly verified.
+To run a Task-DAG with an incentive in a secure way, the
+task publisher 𝑈𝑚,𝑝 needs to design a contest strategy Φ𝑚 to
+allocate reward 𝜈𝑚 to contributors and a penalty strategy Υ𝑚
+to suppress the flooding of excessive trivial models and other
+malicious behaviors.
+Some examples of the plug-and-play contest strategy in-
+clude an egalitarian strategy where the prize is divided equally
+among contributors along the traversal of the final winner
+node, or an implementation of “to each according to his
+5
+
+TABLE II: Notation Definition
+Notation
+Definition
+𝑈𝑘
+The 𝑘-th user
+𝐵𝑘
+The balance of 𝑘-th user
+𝐷train
+𝑘
+The local training dataset of 𝑈𝑘
+𝐷test
+𝑘
+The local testing dataset of 𝑈𝑘
+𝛽
+The number of candidate weights
+𝜎
+The number of aggregated weights
+Task-DAG
+Task𝑚
+The 𝑚-th Task-DAG network
+𝛼𝑚
+The accuracy target of Task𝑚
+𝜈𝑚
+The committed prize of Task𝑚
+𝑈𝑚,𝑝
+The publisher of Task𝑚
+𝑁𝑚,𝑔
+The genesis node of Task𝑚
+𝑇𝑚,𝑔
+The creation timestamp of 𝑁𝑚,𝑔
+𝑊𝑚,𝑔
+The initial model weights of Task𝑚
+𝑁𝑚,𝑘,𝑖
+The 𝑖-th node published by the 𝑘-th user in Task𝑚
+M𝑚,𝑘,𝑖
+The source list of 𝑁𝑚,𝑘,𝑖
+𝐸𝑚,𝑘,𝑖
+The evaluation result for 𝑁𝑚,𝑘,𝑖 over 𝐷test
+𝑘
+E𝑚,𝑘,𝑖
+The evaluation result for M𝑚,𝑘,𝑖 over 𝐷test
+𝑘
+𝑊 ∗
+𝑚,𝑘,𝑖
+The model weights aggregated by M𝑚,𝑘,𝑖 before the local
+training
+𝑊𝑚,𝑘,𝑖
+The model weights after the local training
+𝜌𝑚,𝑘,𝑖
+The setting for training 𝑊 ∗
+𝑚,𝑘,𝑖 into 𝑊𝑚,𝑘,𝑖
+𝑇𝑚,𝑘,𝑖
+The creation timestamp of 𝑁𝑚,𝑘,𝑖
+𝑁𝑚,𝑒
+The task-termination node of Task𝑚
+𝑇𝑚,𝑒
+The creation timestamp of 𝑁𝑚,𝑒
+Ψ𝑚
+The prize allocation of Task𝑚
+Φ𝑚
+The contest strategy of Task𝑚
+Υ𝑚
+The penalty strategy for malicious attacks of Task𝑚
+Ω𝑚
+The verification committee of Task𝑚
+Global-DAG
+𝑁𝑔
+The genesis node of Global-DAG
+𝑁𝑘,𝑖
+The 𝑖-th node published by the 𝑘-th user in Global-DAG
+𝑇𝑘,𝑖
+The creation timestamp of 𝑁𝑘,𝑖
+𝑆ℎ
+The ℎ-th settlement node in the settlement sets S
+𝜆ℎ
+The subtree that is aggregated by 𝑆ℎ
+𝑉𝑘,𝑘′,𝑖
+A PoL-challenge raised by 𝑈𝑘′ for 𝑁𝑘,𝑖 where 𝑘 ≠ 𝑘′
+𝜋𝑘,𝑘′,𝑖
+The deposit to raise 𝑉𝑘,𝑘′,𝑖
+𝜖PoL
+The threshold of PoL-verification
+𝑃𝑘,𝑘′,𝑖
+The PoL-response replied by the publisher 𝑈𝑘 of 𝑁𝑘,𝑖 for
+𝑉𝑘,𝑘′,𝑖 raised by 𝑈𝑘′ where 𝑘 ≠ 𝑘′
+𝑅𝑘, ˆ𝑘,𝑖
+The PoL-result sent from 𝑈 ˆ𝑘 on 𝑃𝑘,𝑘′,𝑖 where 𝑘 ≠ ˆ𝑘
+𝜏ℎ
+The timeout for 𝑃𝑘,𝑘′,𝑖 to be published after 𝑉𝑘′,𝑘,𝑖 has
+been published and confirmed by 𝑆ℎ
+Θℎ
+The committee elected to conduct consensus for 𝑆ℎ
+contribution” whereby the prize is allocated based on the
+amount of contribution, or striking a balance in-between. Some
+examples of the penalty strategy include an implementation
+of S-Index
+H-Index (i.e., preventing excessive self-citations) [31] or the
+occupation ratio along the traversal of the final winner node.
+The publisher 𝑈𝑚,𝑝 also invokes the election to nominate a
+set of 𝑈𝑘 that constitute a task committee Ω𝑚 of Task𝑚 for
+evaluating trained models and conducting PoL-verification.
+The committee is elected via a Verifiable Random Function
+(VRF) [32] upon the balances 𝐵𝑘 of eligible 𝑈𝑘.
+2) Initialization: To initialize Task𝑚, the publisher 𝑈𝑚,𝑝
+firstly registers Task𝑚 to the task management smart contract
+Algorithm 1: Manage Task-DAG
+⊲ Initialize a training task
+1 𝑈𝑚,𝑝.Deposit(Task𝑚, 𝜈𝑚)
+2 Ω𝑚 ← 𝑈𝑚,𝑝.VRF(Task𝑚)
+⊲ Elect the nominated verification committee to Task𝑚
+3 𝑁𝑚,𝑝 ← {𝐻 (𝑊𝑚,𝑔), URI(𝑊𝑚,𝑔), 𝐻 (𝐷test
+𝑚 ), 𝛼𝑚,
+𝜈𝑚, Φ𝑚, Υ𝑚, Ω𝑚, 𝑇𝑚,𝑔 }
+4 𝑁𝑚,𝑝 ← 𝑈𝑚,𝑝.Sign(𝑁𝑚,𝑝)
+5 𝑈𝑚,𝑝.Announce(𝑁𝑚,𝑝)
+⊲ Broadcast to the network
+⊲ Observe the training
+6 while True do
+7
+𝑁𝑚,𝑘,𝑖 ← 𝑈𝑚,𝑝.Monitor(Task𝑚)
+8
+if 𝑁𝑚,𝑘,𝑖 breaches Υ𝑚 then
+9
+𝑈𝑚,𝑝.ApplyPenalty(Υ𝑚, 𝑁𝑚,𝑘,𝑖, 𝑈𝑘)
+10
+if 𝐸𝑚,𝑘,𝑖 > 𝛼𝑚 then
+11
+ˆ𝐸𝑚,𝑘,𝑖 ← 𝑈𝑚,𝑝.Evaluate(𝑊𝑚,𝑘,𝑖, 𝐷test
+𝑚 )
+12
+if ˆ𝐸𝑚,𝑘,𝑖 > 𝛼𝑚 then
+13
+ˆ𝑁𝑚,𝑘,𝑖 ← 𝑁𝑚,𝑘,𝑖
+14
+break
+⊲ Finalize the training task
+15 Ψ𝑚 ← 𝑈𝑚,𝑝.AllocatePrize(Φ𝑚, ˆ𝑁𝑚,𝑘,𝑖)
+16 𝑁𝑚,𝑒 ← { ˆ𝑁𝑚,𝑘,𝑖,
+ˆ𝑈𝑘, URI(𝐷test
+𝑚 ),
+ˆ𝐸𝑚,𝑘,𝑖, Ψ𝑚, 𝑇𝑚,𝑒 }
+𝑆𝐶𝑇 on the Global-DAG network by depositing the committed
+prize 𝜈𝑚. Next, 𝑈𝑚,𝑝 prepares a genesis node 𝑁𝑚,𝑔 including
+the commitment and the URI of the initial model to be trained,
+i.e., 𝐻(𝑊𝑚,𝑔) and URI(𝑊𝑚,𝑔), the model accuracy target 𝛼𝑚,
+the commitment of the public testing dataset 𝐻(𝐷test
+𝑚 ), the
+committed prize 𝜈𝑚, the contest strategy Φ𝑚, the penalty
+strategy Υ𝑚, the nominated verification committee Ω𝑚, and
+the creation timestamp 𝑇𝑚,𝑔1. Then, 𝑈𝑚,𝑝 can sign the genesis
+node 𝑁𝑚,𝑔 and announce the genesis node.
+3) Monitoring: Upon these operations, training starts in
+Task𝑚, and the publisher 𝑈𝑚,𝑝 observes the progress until the
+model becomes mature enough. Any 𝑈𝑘, who is interested in
+contributing the computational resources and competing for
+the prize 𝜈𝑚, continues training models and publishing node
+𝑁𝑚,𝑘,𝑖, i.e., the 𝑖-th node released by 𝑈𝑘 in Task𝑚, as a worker.
+If any nodes breaching the penalty strategy Υ𝑚 are found,
+𝑈𝑚,𝑝 issues fines and updates the balance of the publisher of
+the breaching nodes. Note that the balance change in regard
+to the penalty has yet to be finalized at this stage.
+4) Finalization: When the claim of reaching the targeted
+model accuracy 𝛼𝑚 is realized, 𝑈𝑚,𝑝 evaluates the model
+over the testing dataset 𝐷test
+𝑚 . Once the accuracy is surely
+met by the winner node
+ˆ𝑁𝑚,𝑘,𝑖, 𝑈𝑚,𝑝 executes the contest
+strategy Φ𝑚 and obtains the prize allocation Ψ𝑚. Next, 𝑈𝑚,𝑝
+terminates Task𝑚 by creating a task-termination node 𝑁𝑚,𝑒 that
+points to the winner node ˆ𝑁𝑚,𝑘,𝑖 and contains the winner’s
+address ˆ𝑈𝑘, the URI to the testing dataset URI(𝐷test
+𝑚 ), the
+achieved testing accuracy ˆ𝐸𝑚,𝑘,𝑖, the prize allocation Ψ𝑚, and
+the creation timestamp 𝑇𝑚,𝑒. The task publisher 𝑈𝑚,𝑝 then
+signs 𝑁𝑚,𝑒 and broadcasts 𝑁𝑚,𝑒 to the Global-DAG network.
+The balance change in regard to both the prize allocation Ψ𝑚
+and the penalty is subsequently finalized by settlement nodes
+once the termination node is revealed to the public and is
+1The trustworthiness of the timestamp is guaranteed by trusted timestamp-
+ing services.
+6
+
+Algorithm 2: Train Models
+1 for 𝑈𝑘 parallelly do
+⊲ Worker registration
+2
+𝑈𝑘.Register(Task𝑚)
+3
+while Task𝑚 is ongoing do
+⊲ Sync, verify and select nodes
+4
+while unsynchronized do
+5
+𝑁𝑚,𝑘′,𝑖′ ← 𝑈𝑘.SyncNodes(Task𝑚)
+6
+𝑈𝑘.VerifySignature(𝑁𝑚,𝑘′,𝑖′)
+7
+𝑈𝑘.VerifyRegistration(𝑈𝑘′)
+8
+𝑈𝑘.VerifyBalance(𝑈𝑘′)
+9
+if verification passes then
+10
+𝑈𝑘.Propagate(𝑁𝑚,𝑘′,𝑖′)
+11
+for 𝛽 number of 𝑁𝑚,𝑘′,𝑖′ ∈ Task𝑚 parallelly do
+12
+𝐸′
+𝑚,𝑘′,𝑖′ ← 𝑈𝑘.Evaluate(𝑊𝑚,𝑘′,𝑖′, 𝐷test
+𝑘 )
+13
+M𝑚,𝑘,𝑖, E𝑚,𝑘,𝑖 ← 𝑈𝑘.Select({𝑁𝑚,𝑘′,𝑖′, 𝐸′
+𝑚,𝑘′,𝑖′ },
+𝜎)
+⊲ Aggregate, train and contribute nodes
+14
+𝑊 ∗
+𝑚,𝑘,𝑖 ← 𝑈𝑘.Aggregate(M𝑚,𝑘,𝑖)
+15
+𝑊𝑚,𝑘,𝑖 ← 𝑈𝑘.T𝑚,𝑘 (𝑊 ∗
+𝑚,𝑘,𝑖, 𝜌𝑚,𝑘,𝑖, 𝐷train
+𝑘
+)
+16
+𝐸𝑚,𝑘,𝑖 ← 𝑈𝑘.Evaluate(𝑊𝑚,𝑘,𝑖, 𝐷test
+𝑘 )
+17
+𝑁𝑚,𝑘,𝑖 ← {M𝑚,𝑘,𝑖, E𝑚,𝑘,𝑖, 𝜌𝑚,𝑘,𝑖,
+𝐻 (𝑊𝑚,𝑘,𝑖), URI(𝑊𝑚,𝑘,𝑖), 𝐸𝑚,𝑘,𝑖, 𝑇𝑚,𝑘,𝑖}
+18
+𝑁𝑚,𝑘,𝑖 ← 𝑈𝑘.Sign(𝑁𝑚,𝑘,𝑖)
+19
+𝑈𝑘.Announce(𝑁𝑚,𝑘,𝑖)
+referred by any future model in Global-DAG; see details in
+Section IV-D.
+B. Decentralized Model Training
+As an open system, IRONFORGE allows the workers to
+contribute to the decentralized training in both a Task𝑚 and
+Global-DAG. The training process in a Task-DAG is given
+in Algo. 2, while the training process in Global-DAG shares
+the same algorithm except that there does not exist a public
+shared testing dataset 𝐷test
+𝑚 that decides the stopping point (i.e.,
+the training accuracy target 𝛼) in the Global-DAG network.
+Global-DAG acts as a public resource pool of diversified
+models, allowing for free hunting of models that uniquely
+meet customized training targets upon local testing datasets
+𝐷test
+𝑘
+for each 𝑘-th user 𝑈𝑘.
+Task-DAG Training. Any user 𝑈𝑘, who is interested in com-
+peting for the training rewards in a Task𝑚, needs to admit the
+contest and penalty strategies specified in 𝑁𝑚,𝑔 and registers to
+the task management smart contract 𝑆𝐶𝑇 on the Global-DAG
+network by depositing a certain amount of tokens. This can
+suppress Sybil attacks, distributed denial-of-service (DDoS)
+attacks, and other malicious behaviors under the regulation of
+the penalty strategy Υ𝑚.
+While Task𝑚 is running, 𝑈𝑘 can synchronize the view of
+Task𝑚 and obtain the latest nodes 𝑁𝑚,𝑘′,𝑖′ with (𝑘 ≠ 𝑘′ ∨
+𝑖 ≠ 𝑖′) ∧ (𝑇𝑚,𝑘′,𝑖′ < 𝑇𝑚,𝑘,𝑖). 𝑈𝑘 then verifies the signature
+of the nodes and confirms that the corresponding worker 𝑈𝑘′
+is registered and has enough balance from 𝑆𝐶𝑇 . Worker 𝑈𝑘
+drops the nodes that do not pass verification and propagates
+the success nodes to other workers.
+After verification, 𝑈𝑘 randomly evaluates several 𝑁𝑚,𝑘′,𝑖′
+over its local testing dataset 𝐷test
+𝑘
+for the testing accuracy
+𝐸 ′
+𝑚,𝑘′,𝑖′ until it collects 𝛽 candidated weights (Lines 11-13
+of Algo. 2). Next, 𝑈𝑘 picks up the top 𝜎 models constituting
+the source list M𝑚,𝑘,𝑖 which are then aggregated into the pre-
+trained model 𝑊∗
+𝑚,𝑘,𝑖 using a weighted aggregation function,
+as given by
+𝑊∗
+𝑚,𝑘,𝑖 =
+∑︁
+𝑊𝑝 in M𝑚,𝑘,𝑖
+𝐸𝑝 in E𝑚,𝑘,𝑖
+𝐸 𝑝
+�
+𝐸𝑞 ∈E𝑚,𝑘,𝑖 𝐸𝑞
+𝑊𝑝.
+(1)
+After that, 𝑈𝑘 trains the aggregated model 𝑊∗
+𝑚,𝑘,𝑖 over its local
+training dataset 𝐷train
+𝑘
+with training settings 𝜌𝑚,𝑘,𝑖 for a trained
+model 𝑊𝑚,𝑘,𝑖 , as given by
+𝑊𝑚,𝑘,𝑖 = T𝑚,𝑘 (𝑊∗
+𝑚,𝑘,𝑖, 𝜌𝑚,𝑘,𝑖, 𝐷train
+𝑘
+),
+(2)
+where T𝑚,𝑘 is the training function of 𝑈𝑘 for Task𝑚.
+Once the training is done, 𝑈𝑘 evaluates the model over
+its local testing dataset 𝐷test
+𝑘
+and obtains the testing accuracy
+𝐸𝑚,𝑘,𝑖. Next, 𝑈𝑘 prepares a model update node 𝑁𝑚,𝑘,𝑖 with the
+source list M𝑚,𝑘,𝑖, the corresponding accuracy list E𝑚,𝑘,𝑖, the
+hash code and URI to the trained model, i.e., 𝐻(𝑊𝑚,𝑘,𝑖) and
+URI(𝑊𝑚,𝑘,𝑖), the testing accuracy 𝐸𝑚,𝑘,𝑖, the training settings
+𝜌𝑚,𝑘,𝑖, and the creation timestamp 𝑇𝑚,𝑘,𝑖. Worker 𝑈𝑘 then
+signs 𝑁𝑚,𝑘,𝑖 and broadcasts the signed node. Note that the
+training settings 𝜌𝑚,𝑘,𝑖, such as the learning rate and batch
+size, are embedded in nodes as a record of the training process
+for any upcoming PoL processes.
+Global-DAG Training. Training in Global-DAG also goes
+through Algo. 2, except that there exists neither a public shared
+testing dataset 𝐷test
+𝑚 , nor a unique training target that decides
+the stopping point, hence an indefinitely growing DAG. The
+testing accuracy 𝐸 ′
+𝑚,𝑘′,𝑖′ is also removed in rewarding con-
+tributors who upload models to Global DAG. 𝑈𝑘 receives a
+reward whenever one of its models gets referred by any other
+subsequent models in the network, which becomes the default
+contest strategy for Global-DAG. Users conducting training
+in Global-DAG need to be responsible for their own training
+processes, including preparing their own goals and local
+testing datasets and hunting for appropriate models across the
+whole network. Note that, the task-termination nodes of each
+Task𝑚 are also included in Global-DAG, which enables tasks
+to be advertised to the wider public and to be further evolvable
+along with diversified models in Global-DAG.
+C. Proof-of-Learning: Defense against Model Stealing Attacks
+We design a privacy-preserving PoL scheme to prove the
+computing-extensive training work and suppress model steal-
+ing attacks [27]–[30]. The idea of the privacy-preserving PoL
+is based on the reproducibility of training and the PoL in [33]
+in which the training process from the same starting point
+over the same training dataset with the same training settings
+results in the same trained model or bounded differences
+among the trained models. In the proposed privacy-preserving
+PoL, provers can provide obfuscated training dataset and
+give an estimated bound of model differences. We propose
+a new dataset obfuscation process to protect the privacy of
+the training dataset. The proposed privacy-preserving PoL
+scheme in the Global-DAG network is given in Algo. 3. The
+privacy-preserving PoL scheme for Task-DAG networks can be
+7
+
+Algorithm 3: Proof-of-Learning
+⊲ Raise a PoL-challenge
+1 𝑉𝑘,𝑘′,𝑖 ← 𝑈𝑘′.Challenge(𝑁𝑘,𝑖 | 𝑘 ≠ 𝑘′ ∧ not been challenged)
+2 𝑉𝑘,𝑘′,𝑖 ← 𝑈𝑘′.Deposit(𝜋𝑘,𝑘′,𝑖)
+⊲ Broadcast to the network
+3 𝑉𝑘,𝑘′,𝑖 ∈ 𝜆ℎ is subsequently settled by 𝑆ℎ.
+⊲ 𝜏ℎ countdown starts
+⊲ Reply with a PoL-proof
+4
+�
+𝐷train
+𝑘
+←𝑈𝑘.Obfuscate(𝐷train
+𝑘
+)
+5 𝑃𝑘,𝑘′,𝑖 ← 𝑈𝑘.Prove(𝑉𝑘,𝑘′,𝑖, �
+𝐷train
+𝑘
+))
+⊲ Broadcast to the network
+6 𝑃𝑘,𝑘′,𝑖 ∈ 𝜆ℎ+𝑛 is subsequently settled by 𝑆ℎ+𝑛.
+⊲ Verify the PoL-proof
+7 if 𝑃𝑘,𝑘′,𝑖 presents before the timeout then
+8
+if 𝑇𝑘,𝑘′,𝑖 ≤ 𝑇ℎ + 𝜏ℎ then
+9
+for 𝑈 ˆ𝑘 ∈ Θℎ+𝑛+1 parallelly do
+10
+𝑊 replay
+𝑘,𝑖
+←𝑈 ˆ𝑘.LearningReplay(𝑃𝑘,𝑘′,𝑖.(𝑊 ∗
+𝑘,𝑖, �
+𝐷train
+𝑘
+,
+𝜌𝑘,𝑖))
+11
+if ∥ 𝑊 replay
+𝑘,𝑖
+− 𝑁𝑘,𝑖.𝑊𝑘,𝑖 ∥2< 𝜖PoL then
+12
+𝑅𝑘, ˆ𝑘,𝑖 roots for 𝑃𝑘,𝑘′,𝑖
+13
+R𝑘,𝑖 ←Consensus(𝑅𝑘, ˆ𝑘,𝑖 | 𝑈 ˆ𝑘 ∈ Θℎ+𝑛+1)
+14
+if R𝑘,𝑖 roots for 𝑃𝑘,𝑘′,𝑖 then
+15
+emit Challenge fails
+⊲ Learning proved
+16
+goto Finalization
+17 emit Challenge succeeds
+⊲ Learning invalidated
+⊲ Finalization
+18 if Challenge fails then
+19
+𝑈𝑘′ ←Refund(𝛽𝜋𝑘,𝑘′,𝑖 | 𝛽 ∈ (0, 1))
+20 else
+21
+𝑈𝑘′ ←Refund(𝜋𝑘,𝑘′,𝑖)
+22
+𝑈𝑘′ ←Penalty(𝑈𝑘, 𝜋𝑘,𝑘′,𝑖)
+23
+WithdrawReward(𝑁𝑘,𝑖)
+24 𝑆ℎ+𝑛+1 is generated via Algo. 4
+⊲ Notice
+The Consensus is conducted by the nominated
+verification committee Ω𝑚 when the PoL is
+done in a Task𝑚.
+conducted in the same way where the PoL-proof is verified by
+the nominated task committee Ω𝑚.
+1) Challenge: If a node 𝑁𝑘,𝑖 has not been challenged
+before, any worker 𝑁𝑘′ can raise a PoL challenge against the
+node as a challenger. 𝑈𝑘′ needs to deposit a certain amount
+of tokens 𝜋𝑘,𝑘′,𝑖 to the PoL smart contract 𝑆𝐶𝑃𝑜𝐿 for the
+challenging node 𝑉𝑘,𝑘′,𝑖. Then, 𝑈𝑘′ signs and broadcasts the
+challenging node 𝑉𝑘,𝑘′,𝑖 to the network and starts a countdown.
+2) Response: The publisher of 𝑁𝑘,𝑖, i.e., 𝑈𝑘, needs to reply
+to the challenge 𝑉𝑘,𝑘′,𝑖 as a prover within the PoL timeout 𝜏.
+𝑈𝑘 firstly obtains the obfuscated dataset �𝐷train
+𝑘
+by applying
+the noise 𝛿𝑘,𝑘′,𝑖 to its local training dataset 𝐷train
+𝑘
+. Next, 𝑈𝑘
+prepares a PoL proof node 𝑃𝑘,𝑘′,𝑖 with the hash code and
+URI to the obfuscated dataset, i.e., 𝐻( �𝐷train
+𝑘
+) and URI( �𝐷train
+𝑘
+).
+Then, 𝑈𝑘 signs 𝑃𝑘,𝑘′,𝑖 and broadcasts it to the network.
+3) Verification: At the PoL verification stage, the commit-
+tee Θℎ (see details in Section IV-D for the use of Θℎ) verifies
+𝑃𝑘,𝑘′,𝑖 in parallel if the timestamp of the prover node is within
+the timeout 𝜏ℎ. To be specific, a verifier 𝑈 ˆ𝑘 can fetch the
+obfuscated dataset �𝐷train
+𝑘
+with the URI given in 𝑃𝑘,𝑘′,𝑖 and
+confirm its integrity. Next, 𝑈 ˆ𝑘 conducts a training task with
+the starting model described in 𝑁𝑘,𝑖, the training settings 𝜌𝑘,𝑖
+embedded in 𝑁𝑘,𝑖, and the training dataset �𝐷train
+𝑘
+, i.e.,
+𝑊∗
+𝑘,𝑖 =
+∑︁
+𝑊𝑝 in M𝑘,𝑖
+𝐸𝑝 in E𝑘,𝑖
+𝐸 𝑝
+�
+𝐸𝑞 ∈E𝑘,𝑖 𝐸𝑞
+𝑊𝑝
+𝑊replay
+𝑘,𝑖
+= Tˆ𝑘 (𝑊∗
+𝑘,𝑖, 𝜌𝑘,𝑖, �𝐷train
+𝑘
+).
+(3)
+The verifier 𝑈 ˆ𝑘 then calculates the Frobenius-Norm (F-Norm)
+between trained 𝑊replay
+𝑘,𝑖
+and 𝑊𝑘,𝑖 in 𝑁𝑘,𝑖 as the PoL re-
+sult 𝑅𝑘, ˆ𝑘,𝑖 for 𝑃𝑘,𝑘′,𝑖 [34]. If 𝑅𝑘, ˆ𝑘,𝑖 is within the 𝜖PoL, the
+verification on 𝑈 ˆ𝑘 is success and 𝑅𝑘, ˆ𝑘,𝑖 roots for 𝑃𝑘,𝑘′,𝑖.
+All versifiers in the committee Θℎ run consensus algorithms,
+e.g., Practical Byzantine Fault Tolerance (PBFT) [35], on
+PoL results and get the final committee decision R𝑘,𝑖 as
+a proving node in the network. Notice that, to prevent the
+spoofing attacks against the PoL verification and improve
+the security and robustness [36], 𝜖PoL can be dynamically
+adjustable from the early stage to the later stage to circumvent
+the consistent F-norm-based model distance during a PoL
+spoofing attack. Also, any PoL prover 𝑃𝑘,𝑘′,𝑖, ∀𝑘, 𝑘′, 𝑖 could
+alternatively conduct Verifiable Computation (VC) by showing
+an additional VC-proof to guarantee the identity of 𝑊replay
+𝑘,𝑖
+.
+4) Clearing: At the finalization stage, the challenger 𝑈𝑘′
+can only get 𝛽𝜋𝑘,𝑘′,𝑖 from the challenge deposit, 𝛽 ∈ (0, 1),
+if the challenge fails (i.e., the learning is proved). Otherwise,
+𝑈𝑘′ can get a full refund of the challenge deposit and also
+receive a reward from the penalty on 𝑈𝑘. The reward to 𝑈𝑘
+from 𝑁𝑘,𝑖 is revoked as well if the learning cannot be proved.
+D. Achieving State Consistency: Incentive Basis
+IRONFORGE features settlement nodes to achieve state con-
+sistency for the Global-DAG network, enabling smart contracts
+in DAG and recording consistent account states. The process
+is shown in Algo. 4.
+Global-DAG periodically, with the time interval Δ𝑇 , elects
+the settlement committee Θ among which consensus is reached
+to generate settlement nodes. At the beginning of the ℎ-th
+interval, VRF is used to elect a committee securely Θℎ for
+the settlement node 𝑆ℎ and the committee leader 𝑈 ¯ℎ. The
+probability that any worker 𝑈𝑘 is selected for the committee
+depends on the balance of 𝑈𝑘, i.e., 𝐵𝑘; see (4) below,
+VRF-hash(prv𝑘, 𝑠𝑒𝑒𝑑) ∈ Λ𝑘,
+(4)
+where Λ𝑘 is the area portion occupied by 𝑈𝑘 in a hash ring,
+and Λ𝑘 ∝ 𝐵𝑘.
+The committee can then settle the status of Global-DAG.
+To be specific, the leader of the committee 𝑈 ¯𝑘 synchronizes
+the view of a particular preceding moment of Global-DAG
+via consensus with others, and identifies all the tip nodes
+in the ℎ-th interval, which do not have any successor in the
+current interval. From each of the tip nodes, 𝑈 ¯𝑘 traverses back
+according to every preceding node list M𝑘,𝑖 of 𝑁𝑘,𝑖 along the
+path. The search stops and the subtree 𝜆ℎ is obtained when all
+specific nodes are met, i.e., the first visible node which does
+belong to the previous subtree 𝜆ℎ−1 in each path or the genesis
+node 𝑁𝑔. Based on all nodes in 𝜆ℎ, 𝑈 ¯𝑘 updates the balances
+of involved workers according to the training contributions,
+PoL challenges, PoL proofs, and smart contract executions. In
+8
+
+Algorithm 4: Node Generation in the Settlement Set
+⊲ Generating a settlement node upon consensus
+1 while True do
+2
+if 𝑇
+mod Δ𝑇 = 0 then
+3
+Θℎ ← VRF(Balance(𝑈𝑘 | ∀𝑘))).
+⊲ Election
+4
+while Θℎ.Consensus(𝑈𝑘.view | ∀𝑘 ∧ (𝑈𝑘 ∈ Θℎ)) do
+5
+if consensus is reached then
+6
+break and obtain 𝑆ℎ
+7
+S.Append(𝑆ℎ)
+⊲ The latest balance can be found in S[latest]
+⊲ Consensus
+8 for 𝑈𝑘 in Θℎ parallelly do
+9
+Tips ← Prune({𝑁𝑘,𝑖 | 𝑇𝑘,𝑖 > 𝑇 })
+10
+while Traverse(start ←Tips) do
+11
+if (Path-𝑝 reaches 𝑁𝑔 OR 𝑁𝑘,𝑖 ∈ 𝜆ℎ−1) is True then
+12
+Stop Path-𝑝
+13
+if ALL paths have stopped then
+14
+break and obtain 𝜆ℎ
+15
+𝜆ℎ ← Prune(Tips)
+⊲ Obtain the subtree 𝜆ℎ for the creation of 𝑆ℎ
+16
+¯𝑆𝑘,ℎ ← Form(𝜆ℎ.balance, 𝜆ℎ.PoL,
+{𝑁𝑚,𝑒 | never been collected by S})
+⊲ Tips are excluded in balance calculation
+17
+if 𝑈𝑘 is the leader 𝑈 ¯𝑘 of Θℎ then
+18
+¯𝑆leader ← ¯𝑆𝑘,ℎ
+19
+Broadcast( ¯𝑆leader)
+20
+else
+21
+Verify( ¯𝑆leader, ¯𝑆𝑘,ℎ)
+22
+if verification passes then
+23
+emit Consensus is reached
+24
+else
+25
+Elect a new Θℎ ← Θ
+′
+ℎ and redo Consensus
+Global-DAG, each valid reference from 𝑁𝑘′,𝑖 awards the owner
+𝑈𝑘 (𝑘 ≠ 𝑘′) of the referred model 𝑁𝑘,𝑖 with a certain amount
+of tokens. Next, 𝑈 ¯𝑘 proposes a new settlement node ¯𝑆𝑘,ℎ
+covering the updated balances and PoL results. The settlement
+committee Θℎ verifies and votes ¯𝑆𝑘,ℎ. The committee Θℎ
+endorses ¯𝑆𝑘,ℎ as 𝑆ℎ if the committee reaches consensus, or
+elects a new leader otherwise.
+V. IMPLEMENTATION AND EVALUATION
+In this section, we conduct comparisons between the pro-
+posed FL system, IRONFORGE, and other popular frameworks,
+including GoogleFL [2], AsyncFL [6], and BlockFL [15]. We
+experimentally assess IRONFORGE in terms of the model per-
+formance and expected amount of rewards that can be earned
+under a variety of different environment settings, including
+different aggregation strategies, different sizes of hardware and
+software resources, and different types and levels of malicious
+attacks such as lazy attacks [9], poisoning attacks [25], back-
+door attacks [26], and model stealing attacks [27].
+A. Experimental Configurations
+1) Hardware settings: The experiments are conducted on
+6 servers listed as follows.
+Type-A (#1-3):
+• CPU. 2 × Intel(R) Xeon(R) Gold 6230R CPU @
+2.10GHz, 2 × 52 cores
+• GPU. 1 × Quadro RTX 4000, 1 × 8GB
+• Memory. 528GB
+• Bandwidth. 1000Mb/s
+Type-B (#4-6):
+• CPU. 2 × Intel(R) Xeon(R) Gold 6138 CPU @ 2.00GHz,
+2 × 40 cores
+• GPU. 8 × NVIDIA PCIe A100, 8 × 40GB
+• Memory. 250GB
+• Bandwidth. 1000Mb/s
+2) Software settings: We carry out the experiments upon
+Ubuntu 18.04.6 LTS with Keras 2.7 in Python 3.7.13 and
+Docker 20.10.12. We use FastDFS as the distributed file
+system with 15TB storage space for the model weights.
+3) A new testbed - FLSim: To benchmark the considered FL
+frameworks, we build an FL testbed named FLSim as shown
+in Fig. 4. FLSim is docker-containerized upon our servers (#1–
+6). Choosing different FL frameworks is flexibly plug-and-play
+in FLSim via three generic interfaces, i.e., the event emitter,
+model channel, and capability configuration.
+Event emitter. FLSim is event-driven where all events are
+delivered through Redis which serves as a message queue.
+Each runner can receive events in the network in real-time
+by the subscription function of Redis, and can broadcast
+corresponding events according to their role.
+Fig. 4. Architecture of the new FL testbed FLSim
+Model channel. Indexing models are done via MySQL where
+properties such as the URIs of weights are included, while
+the actual model weights are stored in FastDFS. As a result,
+the query efficiency can be significantly improved with no
+need of retaining the large weights unless they are required
+for evaluation or aggregation.
+Capability configuration. The tasks are trained on runners
+deployed on docker clusters. Each docker container represents
+9
+
+GoogleFL
+BlockFL
+FL Framework
+AsyncFL
+IronForge
+Interface
+Event emitter
+Redis
+Model channel
+CapabilityCofig
+MySQL+FastDF
+Master
+Operational Layer
+Minier
+Worker
+Runner
+Virtualisation Layer
+Docker Engine
+Docker Orchestration
+Physical Layer
+Server 1
+Server nTABLE III: Hyper-parameter settings
+Notation
+Definition
+Value (unit)
+P
+idle probability
+0.1
+E
+global epoch
+2000
+𝑒
+default local epoch
+5
+𝑙
+learning rate
+0.002
+𝜂
+sampled weights
+30
+𝛽
+default number of candidate weights
+6
+𝜎
+default number of aggregated weights
+5
+B
+default batch size
+100
+V
+validation set size
+100
+a runner with different resource settings, such as CPU, mem-
+ory, and bandwidth. Specifications of the containers are craft-
+specified with strong scalability and flexibility by defining the
+capability configuration to simulate various scenarios, such
+as the resource imbalance considered in the experiments.
+Moreover, each runner is categorized into different roles based
+on which FL framework has been plugged in, e.g., “workers”
+in all considered frameworks, “masters” in GoogleFL and
+AsyncFL, and “miners” in BlockFL.
+4) Training settings: The tuned hyper-parameters of the
+experiments are summarized in Table III. We perform training
+over the MNIST with 60,000 data samples and a Convolutional
+Neural Network (CNN) model illustrated in Fig. 5.
+Totally 60,000 MNIST samples are randomly split into two
+parts, 48,000 samples are used as the training set and 12,000
+samples are used as the testing set. We create non-IID training
+shards for contributors from the training set to simulate a
+practical network condition. The training set is divided into
+two subsets, i.e., 24,000 for each. The samples in the first
+subset are sorted by labels, and are subsequently distributed
+into 120 shards, 200 for each shard. Thus, the sample labels
+in each shard are relatively concentrated. The other half of
+the samples are randomly selected and distributed into 120
+shards, i.e., 200 samples for each shard. Thus, the sample
+labels in each shard are relatively uniform. Finally, we repeat
+the sampling operation 120 times, and each time we take a
+shard from each of the two subsets for merging. We end up
+obtaining 120 new shards each of which contains 400 samples.
+5) Environment settings: To demonstrate the state-of-the-
+art of IRONFORGE, a comprehensive comparison between
+IRONFORGE and the other three FL frameworks are conducted
+in our experiments, i.e., the synchronous GoogleFL, AsyncFL,
+and BlockFL. We launch 120 runners as workers to train
+models with a probability of P. We also define two events
+that represent receiving two types of intermediate models:
+• GLOBAL MODEL UPLOADED EVENT
+(GMUE):
+the model acquired by aggregating the uploaded local
+models prior to training.
+• LOCAL MODEL UPLOADED EVENT (LMUE): the
+model trained by workers with their local datasets
+These events are broadcast to notify each runner associated
+with the next step to take. Note that the genesis model of a
+task is tagged as a global model in order to initiate any selected
+FL framework. This indicates that emitting GMUE is used to
+notify the network when the task is published.
+Fig. 5. The CNN model is a lightweight version of the model
+in [2]. It contains one convolution layer of which the filter
+size is 32 and the kernel size is 5 × 5, one 2 × 2 max-pooling
+layers, one fully connected layer with 256 units, and ReLu
+activation. The output is processed by a fully connected layer
+with 10 units and softmax activation.
+Synchronous GoogleFL. One additional runner is launched
+as the master to aggregate local model weights. For GoogleFL,
+workers are activated by GMUE and the master is activated
+by LMUE. The workers, when receiving a GMUE, train on
+top of a global model downloaded from the master with
+their own local datasets. The master receives an LMUE when
+the workers upload the trained models, and subsequently
+aggregates all collected local models after a timeout, followed
+by uploading the aggregated result as the global model. The
+above iterating process continues until the task reached the
+max iteration threshold E.
+AsyncFL. One additional runner is launched as the master to
+aggregate local models. AsyncFL shares the same procedure
+as GoogleFL, except that the master, when receiving an
+LMUE during its idle period, creates a new global model by
+aggregating the most recent global model and newly-collected
+local models with an identical weighting factor.
+BlockFL. Five additional runners are launched as miners for
+BlockFL. In each iteration, the miners behave the same way
+as the master does in GoogleFL or AsyncFL. An additional
+step is that the miners compute the nonce to finalize the block
+and compete for the rewards with a synchronized lock being
+used in Redis to ensure the mining order.
+IRONFORGE. No additional runners are launched for IRON-
+FORGE. Each runner acts as a worker and a master at the same
+time, i.e., it supports both aggregating and training operations,
+thus receiving both GMU and LMUE during each iteration.
+6) Security settings: We implement five types of con-
+tributors: normal contributors, poison contributors, backdoor
+contributors, and stealing and colluding contributors.
+Normal contributors act honestly and independently across
+all phases.
+10
+
+input:
+[(None, 28, 28, 1)]
+conv2d 2 input
+InputLayer
+output:
+[(None, 28, 28, 1)]
+input:
+(None,28,28,1)
+conv2d 2
+Conv2D
+output:
+(None,24,24,32)
+input:
+(None,24,24,32)
+max_pooling2d_2
+MaxPooling2D
+output:
+(None,12,12,32)
+input:
+(None, 12,12,32)
+flatten 2
+Flatten
+output:
+(None,4608)
+input:
+(None,4608)
+dense_4
+Dense
+output:
+(None, 256)
+input:
+(None, 256)
+dense 5
+Dense
+output:
+(None, 10)Poisoning contributors aim to undermine the integrity and
+availability of the global model by crafting local poisoning
+models [25]. In this paper, we simulate poison contributors by
+adopting the label-flipping strategy that fakes labels and then
+conducting training on the forged datasets.
+Backdoor contributors aim to fail the global model on
+targeted tasks, typically by adhering crafted triggers to training
+samples, conducting training on the amended samples, and
+then uploading the attack models [26]. In this paper, backdoor
+contributors layer 5 × 5 white patches to training samples and
+change the label of the manipulated samples to a fixed one.
+Stealing contributors aim to gain rewards by stealing model
+weights trained by others and uploading the plagiarized
+weights as their own work [9]. In this paper, a stealing
+contributor 𝑘′ selects and directly uploads one of the existing
+weights by simply changing the ownership from 𝑘 to 𝑘′.
+Colluding contributors aim to embezzle training rewards for
+their conspirators by performing honest training processes but
+claiming source list M from the conspirators. In this paper, a
+certain proportion of contributors tamper the source list M in
+their uploaded weights, with a certain probability.
+B. Results and Evaluations
+Several experiments are conducted from two perspectives,
+i.e., the performance comparison between IRONFORGE and
+others with and without attacks, and the fairness comparison
+between different resource levels in terms of rewards.
+1) Performance - with and without attacks: Fig. 6 shows
+that the proposed IRONFORGE outperforms AsyncFL and
+BlockFL with only slightly slower convergence than the base-
+line GoogleFL across 2,000 iterations with no attacks lever-
+aged. It is worth noting that the red curve increases sharply
+with as few oscillations as that of the baseline, particularly
+highlighting the stability of IRONFORGE.
+Fig. 6. Comparison between IRONFORGE and others in terms
+of accuracy with no attacks leveraged
+The performance comparison with different levels of attack
+behaviors being applied to IRONFORGE is shown in Fig. 7. It
+is realized that IRONFORGE is resistant to the stealing attack
+the most, followed by the resistance to the backdoor attacks
+and poisoning attacks. It is worth noting that the performance
+of a stealing ratio of 20% can be as good as that of others.
+This is because the native validation process in IRONFORGE
+can capture and eliminate the plagiarized models which are
+reused or whose ownerships are fake. The remaining 80% of
+models are still sufficient for contributors to aggregate and
+train by offering strong diversity of the non-IID data samples.
+On the other hand, an evident degradation of the accuracy,
+around 5%, is shown in both poisoning (cf. Fig. 7(b)) and
+backdoor (cf. Fig. 7(c)) contexts. Nevertheless, it can be found
+from Fig. 8(a) and 8(b) that IRONFORGE outperforms all
+the other FL frameworks with either 20% ratio of poisoning
+attackers or 20% ratio of backdoor attackers. This significantly
+highlights the superiority of IRONFORGE in terms of its strong
+resistance to malicious model updating. Fig. 8(c) highlights
+the resistance to stealing attacks and collusion attacks of
+IRONFORGE. Note that only BlockFL is considered in the
+comparison as GoogleFL and AsyncFL do not support incen-
+tives natively. It is found that, by using the native validation
+process in IRONFORGE, including the PoL verification, none
+of the dishonest contributors who leverage either the stealing
+attack or collusion attack can gain rewards. This prevents the
+malicious contributors from faking the ownership of or directly
+using the existing models, or embezzling the rewards for their
+conspirators by claiming a falsified source list.
+Fig. 9(a) shows the accuracy ranges of adjusting the candi-
+date size (𝛽) for different levels of aggregation sizes (𝜎), e.g.,
+the blue band representing the accuracy ranged from 1-of-2
+to 1-of-8 with 𝜎 = 1 and 𝛽 ∈ [2, 8]. The results reveal that
+increasing the aggregation size 𝜎 stabilizes the performance
+in the beginning stage, and allows for an increasingly higher
+convergence point. The effect of increasing the candidate size
+for a certain aggregation size becomes gradually weakened
+as the aggregation size increases, as you can see that the
+width of each band turns more and more narrow. The effect
+of increasing the aggregation size also becomes weakened, as
+you can see from Fig. 9(a) that the performance of 5-of-6
+is as good as that of 7-of-8. The same insight can also be
+shed by observing the performance comparison of adjusting
+the aggregation sizes (𝜎) for different levels of candidate size
+(𝛽) is shown in Fig. 9(b), e.g., the blue band representing the
+accuracy ranged from 1-of-3 to 2-of-3. The bottom line of the
+red band for 𝛽 = 9 performs as poorly as the 2-of-3 strategy
+does, while the top line of the red band performs the best
+among the kinds. It can thus be concluded that knowing that
+no attacks are leveraged, aggregating more weights is more
+beneficial for obtaining high accuracy than merely aggregating
+a low number of weights from more candidate weights.
+Fig. 9(c) shows the running time of different aggregation
+strategies all the way from downloading the weights to up-
+loading a new model. There is a stronger effect on the running
+time when adjusting the candidate size (𝛽) than adjusting the
+aggregation size (𝜎). An implication can be realized based on
+this observation that the bandwidth could be the bottleneck in
+IRONFORGE. We design dedicated experiments that learn this
+phenomenon, as explained in the following Section V-B2.
+2) Fairness - earn rewards: The fairness is investigated in
+terms of the difference in rewards between different levels
+of hardware specifications. The bottleneck of gaining more
+11
+
+0.95
+06'0
+08'0
+BlockFL
+0.75
+AsyncFL
+GoogleFL
+IRONFORGE
+0.70
+0
+250
+500
+750
+1000
+1250
+1500
+1750
+2000
+Iteration(a) Stealing attacks
+(b) Poisoning attacks
+(c) Backdoor attacks
+Fig. 7. Comparison between different levels of stealing attacks, poisoning attacks, and backdoor attacks applied to IRONFORGE
+in terms of accuracy
+(a) Poisoning attacks with a poisoning ratio of 20% (b) Backdoor attacks with a poisoning ratio of 20%
+(c) Stealing and collusion attacks
+Fig. 8. Comparison between IRONFORGE and others in terms of accuracy or rewards with a certain level of poisoning attacks,
+backdoor attacks, and stealing and colluding attacks
+rewards in IRONFORGE can also be realized. We select two
+different contest strategies, i.e., Immediate settlement and
+Winner traverse. The immediate-settlement is the strategy used
+in Global-DAG by default, rewarding every model every time
+it gets referred by others. The winner-traverse could be one of
+the main options used in a Task-DAG, rewarding every model
+that exists in the traversal path all the way from the winner
+node to the genesis node (excluded).
+According to Fig. 10, A monotonic increase of the rewards
+can be realized for the immediate-settlement with increasing
+CPU cores, memory capacity, and bandwidth, and for the
+winner-traverse only with increasing bandwidth (the winner-
+traverse appears to have similar characteristics to BlockFL
+which also fails to offer a pure monotonic increase of rewards
+in all specifications). This is because the winner-traverse strat-
+egy could include moderate models being aggregated during
+each iteration in the winner-traversal path while the winner
+could be highly random when the competition is intense
+and the convergence is near. Therefore, many models with
+high performance could be excluded by the unique winner
+traversal path at the end. On the contrary, the immediate-
+settlement allows every model not to be missed so long as
+a valid reference relationship is confirmed. Nevertheless, the
+result difference between these two strategies does not tell
+the superiority of fairness. Different requirements may lead
+to different principles of fairness. Rooting for an egalitarian
+strategy, or “to each according to his contribution”, or striking
+a balance in between is a flexible option that the proposed
+IRONFORGE offers back to users without a harsh setting.
+On the other hand, the monotonic increase in the band-
+width comparison for both strategies, as shown in Fig. 10(c),
+highlights the bandwidth being the most critical effect for
+earning rewards in IRONFORGE. That is to say, relatively
+poorer users being more active in uploading models to the
+network with higher frequency can help them be more likely
+to share the rewards, rather than spending much time on a
+strongly performant model.
+Fig. 11 learns the effect of data quality upon the rewards by
+adding a random perturbation to 50% of the training samples
+of half of the users with a mean of 0 and a standard deviation
+of 1. We define the affected nodes as “Poor” nodes, while
+“Excellent” nodes own normal data samples only. The result
+shows that the poor nodes that use the immediate-settlement
+strategy enjoy a narrower range of rewards compared to that
+of the winner-traverse strategy and BlockFL. This highlights
+that poor nodes earning rewards via the immediate-settlement
+12
+
+0.9
+0.8
+0.7
+Accuracy
+0.6
+0.5
+0.4
+BlockFL
+0.3
+AsyncFL
+0.2
+GoogleFL
+IRONFORGE
+0.1
+0
+500
+1000
+1500
+2000
+2500
+3000
+3500
+4000
+Iteration0.8
+Accuracy
+0.6
+0.4
+BlockFL
+AsyncFL
+0.2
+GoogleFL
+IRONFORGE
+0
+500
+1000
+1500
+2000
+2500
+3000
+3500
+4000
+IterationBlockFL
+30
+IRONFORGE
+25
+20
+Rewards
+15
+10
+5
+0
+Normal
+Dishonest
+ContributorType,<Normal>/<Stealing/Colluded>1.0
+0.8
+0.6
+0.4
+Stealing Ratio:0%
+StealingRatio:5%
+0.2
+StealingRatio:10%
+Stealing Ratio:20%
+0
+500
+1000
+1500
+2000
+2500
+3000
+3500
+4000
+Iteration1.0
+0.8
+Accuracy
+0.6
+0.4
+Poisoning Ratio:0%
+Poisoning Ratio: 5%
+0.2
+PoisoningRatio:10%
+PoisoningRatio:20%
+0
+500
+1000
+1500
+2000
+2500
+3000
+3500
+4000
+Iteration1.0
+0.8
+0.6
+0.4
+BackdoorRatio:0%
+BackdoorRatio:5%
+0.2
+BackdoorRatio:10%
+BackdoorRatio:20%
+0
+500
+1000
+1500
+2000
+2500
+3000
+3500
+4000
+Iteration(a) Accuracy difference between aggregation sizes
+(b) Accuracy difference between candidate sizes
+(c) Time difference between aggregation strategies
+Fig. 9. Comparison between different combination of the aggregation strategies (𝜎-of-𝛽) applied to IRONFORGE in terms of
+accuracy and execution time
+(a) Reward difference between different CPU cores
+(b) Reward difference between memory capacity
+(c) Reward difference between bandwidths
+Fig. 10. Comparison between different levels of the CPU core, memory capacity, and bandwidth in terms of rewards
+Fig. 11. Comparison between IRONFORGE and others in terms
+of rewards under the influence of the data quality issue
+in IRONFORGE can be more stable and predictable than
+BlockFL and the winner-traverse, and can be less affected by
+unexpected data degradation or network noise in unreliable
+channels. Excellent nodes have more opportunities to earn
+higher rewards than poor nodes while the median value and
+the minima of rewards remain as high as that of poor nodes.
+This reflects the fairness between excellent and poor nodes,
+i.e., offering stable rewards to the poor while the excellent are
+given chances to make a great fortune.
+VI. DISCUSSION AND ANALYSIS
+This analysis focuses on the security of IRONFORGE. Every
+role in the system is involved in the attack model except that
+the timestamp in IRONFORGE is considered synchronous via
+external trustworthy servers.
+Adversaries target to break the state consistency in Global-
+DAG so that operations such as incentive and consensus fail to
+be executed. The adversaries also target to leverage the model
+stealing attack in order to:
+• forge the “amount of work” by simply stealing others’
+models with no more effort being put into the training;
+• collude with attackers by creating a model referring to
+models from colluded attackers.
+In addition, the adversaries can target on breaching the dataset
+privacy during the dataset sharing in a PoL process. At the
+same time, the adversaries can also unbalance the competition
+by abusing others’ datasets to enrich local resources.
+State consistency: The state consistency is guaranteed over a
+sufficiently long period Δ𝑇 as long as the seeds being used in
+each VRF process are secure and the lower bounds of faulty
+tolerance of consensus protocol (e.g., 33% for PBFT) are
+satisfied in the VRF-elected committees. We consider the time
+gap between two settlement nodes Δ𝑆ℎ,𝑆ℎ−1 is sufficiently large
+in IRONFORGE to expect that each user who gets registered
+13
+
+0.9
+0.970
+0.8
+0.965
+Accuracy
+0.960
+Aggregation Size(α)
+0.950
+0.7
+0.945
+1
+2
+0.940
+0.935
+3
+0.930
+4
+0.6
+1000
+1200
+1400
+1600
+1800
+2000
+5
+7
+0
+250
+500
+750
+1000
+1250
+1500
+1750
+2000
+Iteration0.96
+0.94
+Accurao
+0.92
+0.90
+Candidate Size (β)
+3
+0.88
+5
+7
+9
+250
+500
+750
+1000
+1250
+1500
+1750
+2000
+Iteration1-of-3
+2-of-3
+1-of-5
+2-of-5
+3-of-5
+(o-of-
+4-of-5
+Strategy(
+1-of-7
+3-of-7
+5-of-7
+1-of-9
+3-of-9
+5-of-9
+7-of-9
+0
+20
+40
+60
+80
+Running Time (s)600
+BlockFL
+IronForge(Immediatesettlement)
+500
+IronForge(Winnertraverse)
+400
+Rewards
+300
+200
+100
+0
+2
+6
+8
+CPUCores400
+BlockFL
+350
+IronForge(lmmediatesettlement)
+ronForge(Winnertraverse
+300
+250
+rds
+Rewar
+200
+150
+100
+50
+0
+4
+6
+8
+10
+Memory(GB)400
+BlockFL
+IronForge(immediatesettlement)
+350
+IronForge(Winnertraverse)
+300
+Rewards
+250
+200
+150
+100
+50
+0
+2
+4
+8
+12
+Bandwidth(MB/s)BlockFL
+25
+IronForge(lmmediate settlement)
+IronForge (Winner traverse
+20
+Rewards
+15
+10
+5
+0
+Excellent
+Poor
+Data Quality,Poor/Excellentfor committee election has an identical “view” of Global-DAG
+starting from 𝑇ℎ−1 to 𝑇ℎ−1 + Δ𝑇 . This prevents the consensus
+process in the committee from being trapped into an indefinite
+disagreement due to the network asynchrony. On the other
+hand, an unbiased and unpredictable random seed is crucial for
+a fair VRF process where (4) cannot be manipulated. This can
+be achieved by implementing existing randomness generators
+such as RANDAO [37] or RandHound-VRF [38].
+Model stealing attack: Offering the proposed incentive mech-
+anism in a decentralized FL attracts attackers to leverage
+model stealing attacks, by either stealing the model ownership
+or faking the training processes. Attackers can, with no effort
+on local training, steal others’ models and fake ownership with
+ease. Attackers can alternatively fake the source lists upon an
+honest local training process so that their accomplices, who are
+instead placed in the source list, can reap profits against other
+honest users. These two types of model stealing attacks are
+used for misleading those who wish to ensure the necessary
+training overhead and the efforts in the source list. They are
+particularly useful when attackers intend to steal the rewards
+and share them with their accomplices. IRONFORGE enables
+PoL-challenge where users can choose to challenge a model
+via idle resources, and the model owner requires to provide
+the valid PoL-proof in time for a public verification during
+the consensus process. By the committee replaying parts of the
+training from scratch and reaching the consensus, the “amount
+of work” and the source list M can be explicitly determined.
+Dataset privacy and model melting: This security metric is
+an implementation of our work [34]. Dataset obfuscation helps
+to preserve dataset privacy when datasets require to be publicly
+shared for PoL-challenge. Experimental results in [34] show
+that an obfuscated dataset satisfies PoL verification without
+sacrificing the privacy level while being able to decrease the
+model utility against the abuse of collecting provers’ obfus-
+cated datasets, namely, model melting. In addition, applying
+training over different data samples or using non-IID noise
+significantly can reduce the risks of privacy decline when a
+sufficient number of challenges against the same model owner
+are deliberately raised by attackers.
+VII. CONCLUSION
+This paper proposed IRONFORGE, a new generation of FL
+framework constructed by DAG-based structure, which for
+the first time eliminates the need for the central coordinator
+to solve the issues of network asynchrony and the excessive
+reliance on the central coordinator while at the same time
+enabling an open and fair incentive mechanism to encourage
+more participants, particularly in networks where training
+resources are unevenly distributed. Experimental results based
+on a newly developed testbed FLSim along with the security
+analysis highlight the superiority of IRONFORGE over the
+existing prevalent FL frameworks under various specifica-
+tions regarding performance, fairness, and security. To our
+knowledge, this is the first paper proposing a secure and
+fully decentralized FL framework that can be applied in open
+networks with realistic network and training settings.
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+
diff --git a/A9E2T4oBgHgl3EQfnQhW/content/tmp_files/load_file.txt b/A9E2T4oBgHgl3EQfnQhW/content/tmp_files/load_file.txt
new file mode 100644
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--- /dev/null
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@@ -0,0 +1,1200 @@
+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf,len=1199
+page_content='IRONFORGE: An Open, Secure, Fair, Decentralized  Federated Learning  Guangsheng Yu∗, Xu Wang†, Caijun Sun§, Qin Wang∗, Ping Yu‡,  Wei Ni∗, Renping Liu†, Xiwei Xu∗  ∗CSIRO Data61, Australia  †University of Technology Sydney, Australia  ‡Harbin University of Technology, China  §Zhejiang Lab, China  Abstract—Federated learning (FL) provides an effective ma-  chine learning (ML) architecture to protect data privacy in a  distributed manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content=' However, the inevitable network asynchrony,  the over-dependence on a central coordinator, and the lack of an  open and fair incentive mechanism collectively hinder its further  development.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content=' We propose IRONFORGE, a new generation of FL  framework, that features a Directed Acyclic Graph (DAG)-based  data structure and eliminates the need for central coordinators  to achieve fully decentralized operations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content=' IRONFORGE runs in  a public and open network, and launches a fair incentive  mechanism by enabling state consistency in the DAG, so that the  system fits in networks where training resources are unevenly  distributed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content=' In addition, dedicated defense strategies against  prevalent FL attacks on incentive fairness and data privacy are  presented to ensure the security of IRONFORGE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content=' Experimental  results based on a newly developed testbed FLSim highlight the  superiority of IRONFORGE to the existing prevalent FL frame-  works under various specifications in performance, fairness, and  security.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content=' To the best of our knowledge, IRONFORGE is the first  secure and fully decentralized FL framework that can be applied  in open networks with realistic network and training settings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content='  Index Terms—Federated Learning, DAG, Blockchain  I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content=' INTRODUCTION  Federated learning (FL), officially introduced by Google  in 2017 [1], has become the preference to aggregate data  from distributed ends without breaching data privacy [1],  [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content=' By aggregating huge data with comprehensive extracted  features in FL, critical issues such as model overfitting can be  significantly addressed [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content=' However, \x8c the inevitable network  asynchrony, \x8d the over-dependence on a central coordinator,  and \x8e the lack of an open and fair incentive mechanism hinder  the further development of FL in large and open scenarios [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content='  Traditional FL considers no or low delay throughout an ag-  gregation process, namely, synchronous FL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content=' However, network  synchrony is unrealistic due to the inevitable capacity limit  of computation, bandwidth, and storage, as well as the im-  balanced capacities among the distributed participants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content=' Thus,  recent studies propose pseudo-asynchronous FL [5] and asyn-  chronous FL [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content=' The aggregation of pseudo-asynchronous  FL allows a short interval for collecting the model caches in  order to ensure that the number of models aggregated can be  sufficiently large, while the central coordinator immediately  updates the global model once receiving a new local model  from any idle participants in asynchronous FL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content='  Neither pseudo-asynchronous FL nor asynchronous FL can  tolerate the single-point-of-failure (SPoF) of the central coor-  dinator or even a malicious and corrupted coordinator (issue-  \x8d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content=' The over-dependence on the central coordinator could  potentially degrade the system availability and the training  flexibility in the sense that an FL network may be confined to  specific training domains or tasks determined by the coordina-  tor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content=' Participants in many existing studies [7]–[9], once opting  in an FL network, would have to obey the defined training  target with no flexibility to go for different tasks at will.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content='  In addition to the weak training flexibility, the lack of an  open and fair incentive mechanism results in participants who  have fewer resources and a weaker capacity not willing to  contribute their resources to the global aggregation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content=' This issue  deteriorates particularly in FL networks where resources are  not evenly distributed, and potentially leads to the model over-  fitting and weak generality against contingencies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content=' Although  the authors of [10] survey the incentive mechanisms in FL,  all mentioned frameworks require a central coordinator, also  leading to issue-\x8d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content='  Existing studies propose to replace the central coordinator  with a committee running a consensus process in a blockchain  network to prevent the SPoF or a corrupted coordinator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content=' Mean-  while, by sharing the model collection during the consensus  in the committee, pseudo-asynchronous FL can be achieved in  a decentralized manner, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content=', BlockFL [7], [11]–[14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content=' Consid-  ering only issue-\x8d being solved and issue-\x8c being partially  solved by BlockFL, the authors of [9] introduce a Directed  Acyclic Graph (DAG)-based FL where both issue-\x8c and issue-  \x8d are solved using the concept of asynchronous FL [6] to  fully decentralize the FL process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content=' However, the paper [9] only  considers an ideal network in which the training resources are  evenly distributed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content=' Moreover, the approach to enabling state  consistency for a secure and fair incentive mechanism (issue-  \x8e) is missing in [9], which results in difficulty in adopting the  mechanism in a public and open network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content='  We propose IRONFORGE that is an open, secure, fair,  and decentralized FL system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content=' IRONFORGE solves the above  mentioned pain points at one time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content=' Openness: It features a  DAG-based data structure in an open network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content=' Decentral-  ization: The need for a central coordinator is eliminated  throughout the process by IRONFORGE, inheriting from the  concept of asynchronous FL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content=' As a result, the models are  1  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content='04006v1  [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content='LG]  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content='7 Jan 2023  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content='TABLE I: Qualitative comparisons between the proposed IRONFORGE and the existing FL frameworks  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content='FL Framework  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content='Data Structure  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content='Data Asynchrony  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content='Decentralization  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content='Openness  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content='Incentive  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content='Security  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content='Google FL [2]  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content='Isolated models  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content='Synchronous  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content='Centralized  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content='Private  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content='Asynchronous FL [6]  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content='Isolated models  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content='Asynchronous  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content='Centralized  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content='Private  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content='Block FL [15]  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content='Blockchain  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content='Synchronous  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content='Decentralized  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content='Private  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content='Reward  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content='DAG FL [9]  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content='DAG  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content='Asynchronous  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content='Decentralized  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content='Public  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content='Reward  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content='Poisoning/Backdoor/Lazy  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content='IRONFORGE  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content='DAG  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content='Asynchronous  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content='Decentralized  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content='Public  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content='Reward,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content=' Penalty  Poisoning/Backdoor/Stealing*/Collusion  The stealing attack considered in this paper includes the traditional lazy attack.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content='  The difference is that stealing attackers not only upload their previous models, but also fake the ownership of others’ previous models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content='  � Lack of corresponding designs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content='  maintained in a decentralized manner by all participants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content='  Fairness: IRONFORGE considers a practical scenario, where  resources are unevenly distributed among users.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content=' Each user,  based on its resource amount, selects several existing models,  verifies the correctness and evaluates the model accuracy over  the local dataset, and conducts the aggregation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content=' IRONFORGE  also enables state consistency, by using which an open and  fair incentive mechanism can be established to motivate more  participants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content=' Security: Moreover, dedicated defense strategies  against malicious attacks on incentive fairness, and against  dataset privacy breaching are presented to ensure the security  of IRONFORGE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content=' The key contributions are as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content='  ⊲ We propose a fully decentralized FL framework, namely,  IRONFORGE, which features a DAG-based data structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content='  IRONFORGE addresses the network asynchrony typically  undergone in an FL process, and improves the motivation  of agents participating in the process in an open envi-  ronment by enabling reliable token rewards with strong  consistency and model prediction accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content='  ⊲ We specifically design a new validation mechanism  guarding against well-known FL attacks, including model  poisoning attacks, backdoor attacks, lazy attacks, and  model stealing attacks, among which the model of steal-  ing attack has never been considered in any existing FL  frameworks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content=' By making use of noise-enabled Proof-of-  Learning (PoL) to validate the gradient descent process,  any malicious behaviors, such as faking the ownership  or directly using the existing models, or embezzling the  rewards for their conspirator by claiming a falsified source  list, can be captured and given punishments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content='  ⊲ We build a flexible and efficient testbed, named FLSim,  to simulate the workflow across all considered FL frame-  works in this paper, including the proposed IRONFORGE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content='  We conduct comprehensive experiments based on FLSim,  comparing the system performance, security, and fairness  between the existing FL frameworks and IRONFORGE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content='  Insights are shed to provide guidelines on how to select  strategies in IRONFORGE to meet different requirements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content='  Extensive experiments corroborate that IRONFORGE outper-  forms the prevalent FL frameworks with and without attacks  leveraged, which highlights the holistic solution to the network  asynchrony (issue-\x8c) and the over-dependence on the central  coordinators (issue-\x8d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content=' Strictly and approximately monotonic  increases of rewards are observed in experiments with increas-  ing CPU cores, memory capacity, and bandwidth in different  incentive settings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content=' This indicates that fairness (issue-\x8e) can be  ensured in IRONFORGE under various definitions of fairness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content='  The rest of the paper is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content=' Section I  gives the introduction, followed by related works in Section II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content='  Section III provides the system overview and Section IV  details the design of IRONFORGE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content=' Section V presents our  implementation based on a new testbed with comprehensive  experimental results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content=' Section VI discusses system security and  properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content=' Finally, Section VII concludes this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content='  II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content=' RELATED WORK  A conventional synchronous FL framework is constructed  by a central coordinator and numbers of nodes, which main-  tains the global model and perform FL iterations, respec-  tively [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content=' The coordinator periodically distributes the latest  global model to the nodes, and then the nodes independently  train the model with their local data and upload the trained  local models to the coordinator [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content=' After receiving updated  models from nodes, the coordinator aggregates all the local  models as a new global model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content=' Such synchronous FL frame-  work can hardly be adapted to large-scale and heterogeneous  networks, where asynchrony is non-negligible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content='  The issue of data asynchrony is tackled by the asyn-  chronous FL enabling nodes to train the global model from  central coordinators at any time, and the coordinators can  update the global model immediately when any local model  is collected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content=' In [14], the authors introduced a cache layer  between the coordinator and local nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content=' Each node trains  the global model with its local data and uploads its model to  the cache.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content=' The coordinator periodically aggregates the local  model in the cache and generates a new global model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content=' Semi-  asynchronous FL protocols address the problems in FL such  as low round efficiency and poor convergence rate happened  in asynchronous FL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content=' The system [5] incorporates a client se-  lection algorithm decoupling the coordinator and the selected  clients for a reduction of average round time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content=' The authors  of [17] proposed an asynchronous federating-based detection  approach for end devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content=' A pre-shared data training strategy  for non-independent-and-identically-distributed (non-IID) data  is developed to avoid convergence divergence under the non-  IID patterns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content=' After the collaborative model training procedure,  each client further conducts an additional local training process  to fit respective patterns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content='  The aforementioned FL frameworks require central coor-  dinators to schedule model training and aggregate models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content='  The centralized architecture suffers inherent security risks,  such as SPoF and malicious central coordinator, and limited  2  scalability with the bottleneck of the central coordinator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content=' The  most recent Distributed Ledger Technology (DLT) holds the  potential to decentralize FL systems [18], [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content=' Two key  technologies in DLT are blockchain and DAG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content=' In blockchain,  a group of miners run the consensus protocol to generate hash-  chained data blocks, which are assembled from transactional  data, and synchronize the chained blocks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content=' Blockchain assures  strong consistency among blockchain nodes and enables smart  contracts to be executed across the blockchain network in a  consistent and trustworthy way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content=' In DAG, transactions from  decentralized DAG users are organized in a DAG structure  where directed edges indicate the reference relationship be-  tween the transactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content=' DAG can achieve high throughput with  short latency compared with blockchain [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content='  DLT has been developed to remove the central coordinator  and decentralize FL networks [9], [15], [21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content=' In BlockFL [15],  [22], [23], decentralized blockchain miners conduct model  verification and aggregation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content=' To be specific, miners obtain  trained local models from working nodes and other miners.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content='  After verification, miners aggregate local models for the  updated global models and conduct Proof-of-Work (PoW) to  create valid blocks containing the new global models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content=' Then,  the blocks are propagated to all miners to start the next FL  iteration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content=' The BlockFL relies on the resource-intensive PoW  consensus protocol to slow down the system and keep miners  synchronized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content=' To reduce overhead and improve scalability,  DAG technology [24] is introduced to FL networks [9], [21],  where trained models are updated to a DAG topology by  working nodes without any coordination.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content=' Working nodes can  learn the latest local models in the DAG by exchanging data  with other nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content=' By themselves, working nodes select and  verify aggregate local models and train the models using local  datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content=' Next, working nodes publish their trained models to  the DAG with directed edges indicating the model reference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content='  Existing works only consider homogeneous networks where  the training resources are evenly distributed and thus lack  open and fair incentive mechanisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content=' IRONFORGE proposed  by this paper, on the other hand, improves the motivation of  participants with rewards for training contributions and penal-  ties for dishonest behaviors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content=' IRONFORGE also tackles new  vulnerabilities in open FL networks, including model stealing  attacks where attackers steal models from others and claim  rewards from the plagiarized model, and collusion attacks  where attackers claim trained models are from conspirators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content='  III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content=' SYSTEM OVERVIEW  In this section, we describe IRONFORGE from the aspects  of its architecture, workflow, and system assumptions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content='  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content=' System Overview  We first introduce the roles that participate in the system  and present our high-level design.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content='  Architecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content=' IRONFORGE is a decentralized FL system that  features a DAG-based network structure to tackle the incon-  sistency in the decentralized FL process, excessive reliance  on central coordination, and ineffective motivation of con-  tributing the learning resources at the same time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content=' Specifically,  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content=' System model of IRONFORGE  IRONFORGE builds a hybrid architecture (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content=' 1) that  involves two types of DAG, namely, Task-DAG and Global-  DAG (details refer to Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content=' 2 and Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content=' 3, respectively).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content=' The  training processes in both Task-DAG and Global-DAG are  traceable owing to the DAG data structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content=' A DAG node  published by a participant consists of a model update and the  directed edges of the node indicate the aggregating relationship  with existing models during the update, hence no central  coordinator required to conduct the training processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content='  Global-DAG contains a variety of models adopted by all  participants, which can be viewed as a “unique” and public  model resource pool.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content=' No consistent testing dataset is given  in Global-DAG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content=' Each user comes to Global-DAG and hunts  for models that uniquely meet its own local testing dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content='  Without central coordinators, any user can fetch models from  the pool for direct uses, release his task requests, or make con-  tributions, such as training on Global-DAG or on uncompleted  training tasks, or verifying the tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content='  Each training task is managed by a Task-DAG, while IRON-  FORGE can contain multiple Task-DAGs at the same time  to handle a range of different training tasks (see the right-  hand side in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content=' 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content=' Task-DAGs are task-specific and are  released by users who aim at improving their local model  prediction accuracy by virtue of the computational powers  and resources of others.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content=' Within a task, the Task-DAG network  contains multiple contributors who have the same training  target provided by the publisher.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content=' The trained models for each  task are broadcast and stored in the corresponding Task-DAG,  and await the check and verification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content=' The satisfied model of a  task, observed by the publisher, is subsequently merged into  Global-DAG, increasing the exposure to the public users.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content=' As a  result, parallel learning on our hybrid DAG networks becomes  possible, and the resultant models can be collected by Global-  DAG for further involvement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content='  Roles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content=' In IRONFORGE, the users can take different roles:  viewer, task publisher, verifier, and contributor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content=' A user is a  participant in the network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content=' Each user can select one or multiple  roles to perform specific functional activities (see the left-hand  side in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content=' 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content=' Specifically, a viewer can directly fetch models  from the public resource pool without further actions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content=' The  task publisher aims to propose new tasks and the proposed  tasks are broadcast and await others’ contributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content=' In order  3  Global Network  User  Global DAG  Contributor  User  Task DAG  Global DAG  Contributor  Task DAG  User  Global DAG  Contributor  Task DAG  Viewer  Fetching  Evaluating  Models  Selected Models  Satisfied Mode  Task Publisher  Aggregating  Verification Committee  Aggregated Model  Local dataset  Contributor  Training  Trained Model  Selected Models  Task Publisher  sk  Aggregated Model  Verification Committee  %  Trained Model  Task  WorkNerifyTask  Community  ……' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content='  ……' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content='  t  ① Register and Release a task  Publisher  Workers  ② Announce  ③ Observe  ……' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content='  b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content=' Sync the DAG and validate DAG nodes  c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content=' Evaluate some models with local test dataset  d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content=' Pick up the best ones and aggregate them  e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content=' Start training with local training dataset  f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content=' Publish the model to the DAG  ④ Train and publish models  Task-model node  Header:  Sender: …  Timestamp: …  Sources: […]  Evaluations: […]  Payload:  Weights: ipfs_uri+hash  ④  ④  ④  Local  dataset  Local  dataset  Local  dataset  Task-termination node  Sender: Publisher  Timestamp: …  Winner: …  Public testing dataset: Dtest  Accuracy: …  Balance update: …  Task-genesis node  Header:  Sender: Publisher  Timestamp: …  Target: <e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content=', 95%>  Commitment of Dtest: …  Prize: …  Contest strategy: …  Penalty strategy: …  Verification committee:  Payload:  Weights: ipfs_uri+hash  ⑤ End a task by selecting the winner with the highest accuracy  a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content=' Register on the global FL-DAG  V  P  Verify  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content=' Task-DAG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content=' The figure illustrates an overview of starting a new DAG-based FL task, also known as Task-DAG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content=' One  with aims to improve his model accuracy to a certain target with the help of the community can release a task as the task  publisher.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content=' Some amount of token is deposited as the prize which will be subsequently awarded to all eligible participants  until the winning model is found and selected by the task publisher.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content=' The balance update of each participant is recorded in a  task-termination node published by the task publisher, and can be subsequently settled by the Global-DAG network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content='  to reap profits, a user can become a contributor to process a  training process by selecting, aggregating and training models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content='  He can either start the work on Global-DAG or enroll in others’  published work from the uncompleted tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content=' Also, a verifier in  the system is to verify existing tasks in the resource pool.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content=' He  can contribute or verify either one favorable task or multiple  tasks in parallel for a higher profit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content=' In short, the four roles  cover all potential functional activities within IRONFORGE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content='  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content=' Workflow Overview  Then, we provide an overview of the workflow of IRON-  FORGE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content=' We focus on the procedures of task establishment and  task processing by presenting the interactive steps of a user  between the Task-DAG and Global-DAG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content='  Step-1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content=' The user registers a task in the Global-DAG network by  depositing the committed prize.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content=' He obtains a task identifier  and then broadcasts the task to the network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content=' We assume  that another user has accepted the proposed task prior and  worked on the task as a task contributor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content='  Step-2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content=' The contributor enters the procedure of training mod-  els.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content=' He first evaluates several existing models from the pool  and selects a series of models for the shortlist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content='  Step-3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content=' Based on the selected models, the contributor aggre-  gates all the short-listed models and integrates them with  local datasets to train the model according to requirements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content='  Step-4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content=' Once completing the training, the contributor submits  the trained model to the Task-DAG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content=' Meanwhile, peer con-  tributors may also work on the same task and generate com-  petitively trained models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content=' All these models are propagated  within the Task-DAG network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content='  Step-5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content=' The publisher who obtains the trained model termi-  nates the task by marking it with a termination tag.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content=' Once  selected by users who are conducting the training process  in Global-DAG, the trained model is deemed to be formally  synchronized into Global-DAG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content='  Notably, a user in the Global-DAG network can either  contribute to other tasks proposed by peer users, or personally  publish a task by himself.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content=' All the procedures follow similar  steps, as described from Step-2 to Step-5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content='  C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content=' System Assumptions  In this section, we list our assumptions on the network,  security, and threat models of IRONFORGE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content='  Resource assumption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content=' We do not assume any resource dis-  tribution in our work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content=' The resource distribution in the entire  network is random.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content=' This means different participants, with a  high probability, hold different computing resources, including  computing power, network bandwidth, memory space, storage  capability, and training dataset quality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content=' Addressing the system  heterogeneity is one of the core contributions in this work, as  we weaken the long-existing implicit assumption in previous  work [9]: the even resource distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content=' IRONFORGE enables  any distribution of shares of any type of resources among the  participants, making the system practical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content='  User behavior assumption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content=' We have two assumptions on user  behaviors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content=' First, the participants in the network are rational,  meaning that they can select an arbitrary task, switch to  others, or quit existing tasks for better profits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content=' Second, different  participants can focus on different training targets, including  both task bundles (one task has dependency on another) and  orthogonal tasks (one task is independent of the others).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content=' This  enables the processing of multiple tasks in parallel, greatly  improving the system’s overall scalability and performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content='  Security assumption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content=' We assume that the honest nodes  always conduct honest behaviors, where they obey all the  policies during the model selection, model aggregation, model  4  Task-1  (terminated)  Task-2  (terminated)  Task-N  (ongoing)  G  1  t1  t2  tn  2  V  P  τ1  3  Local  dataset  Local  dataset  Local  dataset  ……' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content='  Community  Publish  Publish  Publish  Contribute  Contribute  VRF-  consensus  Settlement node  Sender: <based on consensus>  Timestamp: …  PoL results: …  Unverified PoL: …  Balance: …  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content=' Global-DAG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content=' This figure illustrates an overview of the Global-DAG network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content=' With the absence of a centralized  coordinator, each participant trains a model by selecting and aggregating as many models (including the outcomes of terminated  tasks) published by others as possible (based on the local capability).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content=' Model targets are not unique and according to different  needs, the network can be treated as a global resource pool containing a variety of models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content=' Ones can either find a model which  satisfies his local testing dataset from the pool, or make contributions to the pool and obtain token rewards by improving  existing models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content=' Token balances are periodically settled (endorsed by a verifiable random function (VRF)-driven consensus)  by settlement nodes that employ a chain structure to achieve strong consistency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content='  training, task verification, and other operations related to the  defense strategies against adversaries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content=' The adversaries have the  ability to delay the model convergence and lower the model  accuracy by leveraging popular FL attacks, including lazy  attacks [9], poisoning attacks [25], and backdoor attacks [26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content='  The adversaries also have the ability to breach the incentive  fairness by leveraging model stealing attacks [27]–[30], and  compromising the privacy of others’ training datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content=' Adver-  saries not only can upload their previous models (traditional  lazy attacks), but also fake the ownership of others’ previous  models or fake their own training process to embezzle rewards  for their conspirators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content=' These two faking types are defined as  stealing attacks and collusion attacks, respectively, and both  belong to the context of model stealing attacks in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content='  IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content=' DECENTRALIZED FEDERATED LEARNING  IRONFORGE involves four novel mechanisms, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content=', the re-  lease of Task-DAG networks, the decentralized model training,  the defense strategy, and the incentive mechanism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content=' IRON-  FORGE features two types of network, public Global-DAG and  task-specific Task-DAG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content=' A training task can be outsourced to  communities by releasing a Task-DAG network and following  four steps including preparation, initialization, monitoring, and  finalization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content=' The decentralized training processes of Task-DAG  and Global-DAG are specifically defined by the new decen-  tralized model training mechanism, in which users aggregate  existing models, train the aggregated models and publish the  model updates to the network in a decentralized way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content=' The  training processes are guarded by a new defense strategy  against model stealing attacks in the decentralized setting that  has never been considered in existing studies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content=' The crafted  incentive mechanism assures the state consistency of networks  and enables smart-contract-enhanced incentives including both  rewards and penalties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content=' Table II summarizes the notations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content='  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content=' Managing Task-DAG  Any user can outsource an FL training task by managing  a DAG, as shown in Algo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content=' An FL training task can  be described with a training target, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content=', the model to be  trained, the targeted accuracy, and the testing dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content=' To build  incentives to motivate distributed workers and deter malicious  workers, we design reward, penalty, and verification schemes  for Task-DAG networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content='  1) Preparation: A training task Task𝑚 can be described  with an initial model, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content=', 𝑊𝑚,𝑔, and an accuracy target 𝛼𝑚 of  the model tested on the dataset 𝐷test  𝑚  from the task publisher  𝑈𝑚,𝑝.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content=' To reduce the storage and bandwidth overhead, the  model weights can be stored in external infrastructures, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content=',  the InterPlanetary File System (IPFS).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content=' The Uniform Resource  Identifiers (URIs) and hash codes to the weights are embedded  in DAG nodes for access and verification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content=' The testing dataset  𝐷test  𝑚  is committed in the genesis node of the Task-DAG by  embedding the hash code, and is revealed by the end of the  training for model verification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content=' This commit-and-reveal design  prevents direct access to 𝐷test  𝑚 during the training process and  ensures that the final selected model can be publicly verified.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content='  To run a Task-DAG with an incentive in a secure way, the  task publisher 𝑈𝑚,𝑝 needs to design a contest strategy Φ𝑚 to  allocate reward 𝜈𝑚 to contributors and a penalty strategy Υ𝑚  to suppress the flooding of excessive trivial models and other  malicious behaviors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content='  Some examples of the plug-and-play contest strategy in-  clude an egalitarian strategy where the prize is divided equally  among contributors along the traversal of the final winner  node,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content=' or an implementation of “to each according to his  5  TABLE II: Notation Definition  Notation  Definition  𝑈𝑘  The 𝑘-th user  𝐵𝑘  The balance of 𝑘-th user  𝐷train  𝑘  The local training dataset of 𝑈𝑘  𝐷test  𝑘  The local testing dataset of 𝑈𝑘  𝛽  The number of candidate weights  𝜎  The number of aggregated weights  Task-DAG  Task𝑚  The 𝑚-th Task-DAG network  𝛼𝑚  The accuracy target of Task𝑚  𝜈𝑚  The committed prize of Task𝑚  𝑈𝑚,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content='𝑝  The publisher of Task𝑚  𝑁𝑚,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content='𝑔  The genesis node of Task𝑚  𝑇𝑚,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content='𝑔  The creation timestamp of 𝑁𝑚,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content='𝑔  𝑊𝑚,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content='𝑔  The initial model weights of Task𝑚  𝑁𝑚,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content='𝑘,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content='𝑖  The 𝑖-th node published by the 𝑘-th user in Task𝑚  M𝑚,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content='𝑘,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content='𝑖  The source list of 𝑁𝑚,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content='𝑘,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content='𝑖  𝐸𝑚,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content='𝑘,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content='𝑖  The evaluation result for 𝑁𝑚,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content='𝑘,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content='𝑖 over 𝐷test  𝑘  E𝑚,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content='𝑘,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content='𝑖  The evaluation result for M𝑚,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content='𝑘,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content='𝑖 over 𝐷test  𝑘  𝑊 ∗  𝑚,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content='𝑘,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content='𝑖  The model weights aggregated by M𝑚,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content='𝑘,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content='𝑖 before the local  training  𝑊𝑚,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content='𝑘,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content='𝑖  The model weights after the local training  𝜌𝑚,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content='𝑘,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content='𝑖  The setting for training 𝑊 ∗  𝑚,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content='𝑘,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content='𝑖 into 𝑊𝑚,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content='𝑘,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content='𝑖  𝑇𝑚,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content='𝑘,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content='𝑖  The creation timestamp of 𝑁𝑚,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content='𝑘,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content='𝑖  𝑁𝑚,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content='𝑒  The task-termination node of Task𝑚  𝑇𝑚,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content='𝑒  The creation timestamp of 𝑁𝑚,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content='𝑒  Ψ𝑚  The prize allocation of Task𝑚  Φ𝑚  The contest strategy of Task𝑚  Υ𝑚  The penalty strategy for malicious attacks of Task𝑚  Ω𝑚  The verification committee of Task𝑚  Global-DAG  𝑁𝑔  The genesis node of Global-DAG  𝑁𝑘,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content='𝑖  The 𝑖-th node published by the 𝑘-th user in Global-DAG  𝑇𝑘,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content='𝑖  The creation timestamp of 𝑁𝑘,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content='𝑖  𝑆ℎ  The ℎ-th settlement node in the settlement sets S  𝜆ℎ  The subtree that is aggregated by 𝑆ℎ  𝑉𝑘,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content='𝑘′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content='𝑖  A PoL-challenge raised by 𝑈𝑘′ for 𝑁𝑘,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content='𝑖 where 𝑘 ≠ 𝑘′  𝜋𝑘,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content='𝑘′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content='𝑖  The deposit to raise 𝑉𝑘,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content='𝑘′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content='𝑖  𝜖PoL  The threshold of PoL-verification  𝑃𝑘,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content='𝑘′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content='𝑖  The PoL-response replied by the publisher 𝑈𝑘 of 𝑁𝑘,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content='𝑖 for  𝑉𝑘,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content='𝑘′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content='𝑖 raised by 𝑈𝑘′ where 𝑘 ≠ 𝑘′  𝑅𝑘,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content=' ˆ𝑘,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content='𝑖  The PoL-result sent from 𝑈 ˆ𝑘 on 𝑃𝑘,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content='𝑘′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content='𝑖 where 𝑘 ≠ ˆ𝑘  𝜏ℎ  The timeout for 𝑃𝑘,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content='𝑘′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content='𝑖 to be published after 𝑉𝑘′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content='𝑘,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content='𝑖 has  been published and confirmed by 𝑆ℎ  Θℎ  The committee elected to conduct consensus for 𝑆ℎ  contribution” whereby the prize is allocated based on the  amount of contribution,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content=' or striking a balance in-between.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content=' Some  examples of the penalty strategy include an implementation  of S-Index  H-Index (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content=', preventing excessive self-citations) [31] or the  occupation ratio along the traversal of the final winner node.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content='  The publisher 𝑈𝑚,𝑝 also invokes the election to nominate a  set of 𝑈𝑘 that constitute a task committee Ω𝑚 of Task𝑚 for  evaluating trained models and conducting PoL-verification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content='  The committee is elected via a Verifiable Random Function  (VRF) [32] upon the balances 𝐵𝑘 of eligible 𝑈𝑘.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content='  2) Initialization: To initialize Task𝑚, the publisher 𝑈𝑚,𝑝  firstly registers Task𝑚 to the task management smart contract  Algorithm 1: Manage Task-DAG  ⊲ Initialize a training task  1 𝑈𝑚,𝑝.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content='Deposit(Task𝑚, 𝜈𝑚)  2 Ω𝑚 ← 𝑈𝑚,𝑝.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content='VRF(Task𝑚)  ⊲ Elect the nominated verification committee to Task𝑚  3 𝑁𝑚,𝑝 ← {𝐻 (𝑊𝑚,𝑔), URI(𝑊𝑚,𝑔), 𝐻 (𝐷test  𝑚 ), 𝛼𝑚,  𝜈𝑚, Φ𝑚, Υ𝑚, Ω𝑚, 𝑇𝑚,𝑔 }  4 𝑁𝑚,𝑝 ← 𝑈𝑚,𝑝.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content='Sign(𝑁𝑚,𝑝)  5 𝑈𝑚,𝑝.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content='Announce(𝑁𝑚,𝑝)  ⊲ Broadcast to the network  ⊲ Observe the training  6 while True do  7  𝑁𝑚,𝑘,𝑖 ← 𝑈𝑚,𝑝.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content='Monitor(Task𝑚)  8  if 𝑁𝑚,𝑘,𝑖 breaches Υ𝑚 then  9  𝑈𝑚,𝑝.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content='ApplyPenalty(Υ𝑚, 𝑁𝑚,𝑘,𝑖, 𝑈𝑘)  10  if 𝐸𝑚,𝑘,𝑖 > 𝛼𝑚 then  11  ˆ𝐸𝑚,𝑘,𝑖 ← 𝑈𝑚,𝑝.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content='Evaluate(𝑊𝑚,𝑘,𝑖, 𝐷test  𝑚 )  12  if ˆ𝐸𝑚,𝑘,𝑖 > 𝛼𝑚 then  13  ˆ𝑁𝑚,𝑘,𝑖 ← 𝑁𝑚,𝑘,𝑖  14  break  ⊲ Finalize the training task  15 Ψ𝑚 ← 𝑈𝑚,𝑝.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content='AllocatePrize(Φ𝑚, ˆ𝑁𝑚,𝑘,𝑖)  16 𝑁𝑚,𝑒 ← { ˆ𝑁𝑚,𝑘,𝑖,  ˆ𝑈𝑘, URI(𝐷test  𝑚 ),  ˆ𝐸𝑚,𝑘,𝑖, Ψ𝑚, 𝑇𝑚,𝑒 }  𝑆𝐶𝑇 on the Global-DAG network by depositing the committed  prize 𝜈𝑚.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content=' Next, 𝑈𝑚,𝑝 prepares a genesis node 𝑁𝑚,𝑔 including  the commitment and the URI of the initial model to be trained,  i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content=', 𝐻(𝑊𝑚,𝑔) and URI(𝑊𝑚,𝑔), the model accuracy target 𝛼𝑚,  the commitment of the public testing dataset 𝐻(𝐷test  𝑚 ), the  committed prize 𝜈𝑚, the contest strategy Φ𝑚, the penalty  strategy Υ𝑚, the nominated verification committee Ω𝑚, and  the creation timestamp 𝑇𝑚,𝑔1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content=' Then, 𝑈𝑚,𝑝 can sign the genesis  node 𝑁𝑚,𝑔 and announce the genesis node.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content='  3) Monitoring: Upon these operations, training starts in  Task𝑚, and the publisher 𝑈𝑚,𝑝 observes the progress until the  model becomes mature enough.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content=' Any 𝑈𝑘, who is interested in  contributing the computational resources and competing for  the prize 𝜈𝑚, continues training models and publishing node  𝑁𝑚,𝑘,𝑖, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content=', the 𝑖-th node released by 𝑈𝑘 in Task𝑚, as a worker.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content='  If any nodes breaching the penalty strategy Υ𝑚 are found,  𝑈𝑚,𝑝 issues fines and updates the balance of the publisher of  the breaching nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content=' Note that the balance change in regard  to the penalty has yet to be finalized at this stage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content='  4) Finalization: When the claim of reaching the targeted  model accuracy 𝛼𝑚 is realized, 𝑈𝑚,𝑝 evaluates the model  over the testing dataset 𝐷test  𝑚 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content=' Once the accuracy is surely  met by the winner node  ˆ𝑁𝑚,𝑘,𝑖, 𝑈𝑚,𝑝 executes the contest  strategy Φ𝑚 and obtains the prize allocation Ψ𝑚.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content=' Next, 𝑈𝑚,𝑝  terminates Task𝑚 by creating a task-termination node 𝑁𝑚,𝑒 that  points to the winner node ˆ𝑁𝑚,𝑘,𝑖 and contains the winner’s  address ˆ𝑈𝑘, the URI to the testing dataset URI(𝐷test  𝑚 ), the  achieved testing accuracy ˆ𝐸𝑚,𝑘,𝑖, the prize allocation Ψ𝑚, and  the creation timestamp 𝑇𝑚,𝑒.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content=' The task publisher 𝑈𝑚,𝑝 then  signs 𝑁𝑚,𝑒 and broadcasts 𝑁𝑚,𝑒 to the Global-DAG network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content='  The balance change in regard to both the prize allocation Ψ𝑚  and the penalty is subsequently finalized by settlement nodes  once the termination node is revealed to the public and is  1The trustworthiness of the timestamp is guaranteed by trusted timestamp-  ing services.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content='  6  Algorithm 2: Train Models  1 for 𝑈𝑘 parallelly do  ⊲ Worker registration  2  𝑈𝑘.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content='Register(Task𝑚)  3  while Task𝑚 is ongoing do  ⊲ Sync, verify and select nodes  4  while unsynchronized do  5  𝑁𝑚,𝑘′,𝑖′ ← 𝑈𝑘.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content='SyncNodes(Task𝑚)  6  𝑈𝑘.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content='VerifySignature(𝑁𝑚,𝑘′,𝑖′)  7  𝑈𝑘.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content='VerifyRegistration(𝑈𝑘′)  8  𝑈𝑘.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content='VerifyBalance(𝑈𝑘′)  9  if verification passes then  10  𝑈𝑘.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content='Propagate(𝑁𝑚,𝑘′,𝑖′)  11  for 𝛽 number of 𝑁𝑚,𝑘′,𝑖′ ∈ Task𝑚 parallelly do  12  𝐸′  𝑚,𝑘′,𝑖′ ← 𝑈𝑘.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content='Evaluate(𝑊𝑚,𝑘′,𝑖′, 𝐷test  𝑘 )  13  M𝑚,𝑘,𝑖, E𝑚,𝑘,𝑖 ← 𝑈𝑘.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content='Select({𝑁𝑚,𝑘′,𝑖′, 𝐸′  𝑚,𝑘′,𝑖′ },  𝜎)  ⊲ Aggregate, train and contribute nodes  14  𝑊 ∗  𝑚,𝑘,𝑖 ← 𝑈𝑘.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content='Aggregate(M𝑚,𝑘,𝑖)  15  𝑊𝑚,𝑘,𝑖 ← 𝑈𝑘.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content='T𝑚,𝑘 (𝑊 ∗  𝑚,𝑘,𝑖, 𝜌𝑚,𝑘,𝑖, 𝐷train  𝑘  )  16  𝐸𝑚,𝑘,𝑖 ← 𝑈𝑘.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content='Evaluate(𝑊𝑚,𝑘,𝑖, 𝐷test  𝑘 )  17  𝑁𝑚,𝑘,𝑖 ← {M𝑚,𝑘,𝑖, E𝑚,𝑘,𝑖, 𝜌𝑚,𝑘,𝑖,  𝐻 (𝑊𝑚,𝑘,𝑖), URI(𝑊𝑚,𝑘,𝑖), 𝐸𝑚,𝑘,𝑖, 𝑇𝑚,𝑘,𝑖}  18  𝑁𝑚,𝑘,𝑖 ← 𝑈𝑘.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content='Sign(𝑁𝑚,𝑘,𝑖)  19  𝑈𝑘.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content='Announce(𝑁𝑚,𝑘,𝑖)  referred by any future model in Global-DAG;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content=' see details in  Section IV-D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content='  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content=' Decentralized Model Training  As an open system, IRONFORGE allows the workers to  contribute to the decentralized training in both a Task𝑚 and  Global-DAG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content=' The training process in a Task-DAG is given  in Algo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content=' 2, while the training process in Global-DAG shares  the same algorithm except that there does not exist a public  shared testing dataset 𝐷test  𝑚 that decides the stopping point (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content=',  the training accuracy target 𝛼) in the Global-DAG network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content='  Global-DAG acts as a public resource pool of diversified  models, allowing for free hunting of models that uniquely  meet customized training targets upon local testing datasets  𝐷test  𝑘  for each 𝑘-th user 𝑈𝑘.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content='  Task-DAG Training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content=' Any user 𝑈𝑘, who is interested in com-  peting for the training rewards in a Task𝑚, needs to admit the  contest and penalty strategies specified in 𝑁𝑚,𝑔 and registers to  the task management smart contract 𝑆𝐶𝑇 on the Global-DAG  network by depositing a certain amount of tokens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content=' This can  suppress Sybil attacks, distributed denial-of-service (DDoS)  attacks, and other malicious behaviors under the regulation of  the penalty strategy Υ𝑚.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content='  While Task𝑚 is running, 𝑈𝑘 can synchronize the view of  Task𝑚 and obtain the latest nodes 𝑁𝑚,𝑘′,𝑖′ with (𝑘 ≠ 𝑘′ ∨  𝑖 ≠ 𝑖′) ∧ (𝑇𝑚,𝑘′,𝑖′ < 𝑇𝑚,𝑘,𝑖).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content=' 𝑈𝑘 then verifies the signature  of the nodes and confirms that the corresponding worker 𝑈𝑘′  is registered and has enough balance from 𝑆𝐶𝑇 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content=' Worker 𝑈𝑘  drops the nodes that do not pass verification and propagates  the success nodes to other workers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content='  After verification, 𝑈𝑘 randomly evaluates several 𝑁𝑚,𝑘′,𝑖′  over its local testing dataset 𝐷test  𝑘  for the testing accuracy  𝐸 ′  𝑚,𝑘′,𝑖′ until it collects 𝛽 candidated weights (Lines 11-13  of Algo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content=' 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content=' Next, 𝑈𝑘 picks up the top 𝜎 models constituting  the source list M𝑚,𝑘,𝑖 which are then aggregated into the pre-  trained model 𝑊∗  𝑚,𝑘,𝑖 using a weighted aggregation function,  as given by  𝑊∗  𝑚,𝑘,𝑖 =  ∑︁  𝑊𝑝 in M𝑚,𝑘,𝑖  𝐸𝑝 in E𝑚,𝑘,𝑖  𝐸 𝑝  �  𝐸𝑞 ∈E𝑚,𝑘,𝑖 𝐸𝑞  𝑊𝑝.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content='  (1)  After that, 𝑈𝑘 trains the aggregated model 𝑊∗  𝑚,𝑘,𝑖 over its local  training dataset 𝐷train  𝑘  with training settings 𝜌𝑚,𝑘,𝑖 for a trained  model 𝑊𝑚,𝑘,𝑖 , as given by  𝑊𝑚,𝑘,𝑖 = T𝑚,𝑘 (𝑊∗  𝑚,𝑘,𝑖, 𝜌𝑚,𝑘,𝑖, 𝐷train  𝑘  ),  (2)  where T𝑚,𝑘 is the training function of 𝑈𝑘 for Task𝑚.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content='  Once the training is done, 𝑈𝑘 evaluates the model over  its local testing dataset 𝐷test  𝑘  and obtains the testing accuracy  𝐸𝑚,𝑘,𝑖.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content=' Next, 𝑈𝑘 prepares a model update node 𝑁𝑚,𝑘,𝑖 with the  source list M𝑚,𝑘,𝑖, the corresponding accuracy list E𝑚,𝑘,𝑖, the  hash code and URI to the trained model, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content=', 𝐻(𝑊𝑚,𝑘,𝑖) and  URI(𝑊𝑚,𝑘,𝑖), the testing accuracy 𝐸𝑚,𝑘,𝑖, the training settings  𝜌𝑚,𝑘,𝑖, and the creation timestamp 𝑇𝑚,𝑘,𝑖.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content=' Worker 𝑈𝑘 then  signs 𝑁𝑚,𝑘,𝑖 and broadcasts the signed node.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content=' Note that the  training settings 𝜌𝑚,𝑘,𝑖, such as the learning rate and batch  size, are embedded in nodes as a record of the training process  for any upcoming PoL processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content='  Global-DAG Training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content=' Training in Global-DAG also goes  through Algo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content=' 2, except that there exists neither a public shared  testing dataset 𝐷test  𝑚 , nor a unique training target that decides  the stopping point, hence an indefinitely growing DAG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content=' The  testing accuracy 𝐸 ′  𝑚,𝑘′,𝑖′ is also removed in rewarding con-  tributors who upload models to Global DAG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content=' 𝑈𝑘 receives a  reward whenever one of its models gets referred by any other  subsequent models in the network, which becomes the default  contest strategy for Global-DAG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content=' Users conducting training  in Global-DAG need to be responsible for their own training  processes, including preparing their own goals and local  testing datasets and hunting for appropriate models across the  whole network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content=' Note that, the task-termination nodes of each  Task𝑚 are also included in Global-DAG, which enables tasks  to be advertised to the wider public and to be further evolvable  along with diversified models in Global-DAG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content='  C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content=' Proof-of-Learning: Defense against Model Stealing Attacks  We design a privacy-preserving PoL scheme to prove the  computing-extensive training work and suppress model steal-  ing attacks [27]–[30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content=' The idea of the privacy-preserving PoL  is based on the reproducibility of training and the PoL in [33]  in which the training process from the same starting point  over the same training dataset with the same training settings  results in the same trained model or bounded differences  among the trained models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content=' In the proposed privacy-preserving  PoL, provers can provide obfuscated training dataset and  give an estimated bound of model differences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content=' We propose  a new dataset obfuscation process to protect the privacy of  the training dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content=' The proposed privacy-preserving PoL  scheme in the Global-DAG network is given in Algo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content=' The  privacy-preserving PoL scheme for Task-DAG networks can be  7  Algorithm 3: Proof-of-Learning  ⊲ Raise a PoL-challenge  1 𝑉𝑘,𝑘′,𝑖 ← 𝑈𝑘′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content='Challenge(𝑁𝑘,𝑖 | 𝑘 ≠ 𝑘′ ∧ not been challenged)  2 𝑉𝑘,𝑘′,𝑖 ← 𝑈𝑘′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content='Deposit(𝜋𝑘,𝑘′,𝑖)  ⊲ Broadcast to the network  3 𝑉𝑘,𝑘′,𝑖 ∈ 𝜆ℎ is subsequently settled by 𝑆ℎ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content='  ⊲ 𝜏ℎ countdown starts  ⊲ Reply with a PoL-proof  4  �  𝐷train  𝑘  ←𝑈𝑘.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content='Obfuscate(𝐷train  𝑘  )  5 𝑃𝑘,𝑘′,𝑖 ← 𝑈𝑘.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content='Prove(𝑉𝑘,𝑘′,𝑖, �  𝐷train  𝑘  ))  ⊲ Broadcast to the network  6 𝑃𝑘,𝑘′,𝑖 ∈ 𝜆ℎ+𝑛 is subsequently settled by 𝑆ℎ+𝑛.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content='  ⊲ Verify the PoL-proof  7 if 𝑃𝑘,𝑘′,𝑖 presents before the timeout then  8  if 𝑇𝑘,𝑘′,𝑖 ≤ 𝑇ℎ + 𝜏ℎ then  9  for 𝑈 ˆ𝑘 ∈ Θℎ+𝑛+1 parallelly do  10  𝑊 replay  𝑘,𝑖  ←𝑈 ˆ𝑘.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content='LearningReplay(𝑃𝑘,𝑘′,𝑖.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content=' (𝑊 ∗  𝑘,𝑖, �  𝐷train  𝑘  ,  𝜌𝑘,𝑖))  11  if ∥ 𝑊 replay  𝑘,𝑖  − 𝑁𝑘,𝑖.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content='𝑊𝑘,𝑖 ∥2< 𝜖PoL then  12  𝑅𝑘, ˆ𝑘,𝑖 roots for 𝑃𝑘,𝑘′,𝑖  13  R𝑘,𝑖 ←Consensus(𝑅𝑘, ˆ𝑘,𝑖 | 𝑈 ˆ𝑘 ∈ Θℎ+𝑛+1)  14  if R𝑘,𝑖 roots for 𝑃𝑘,𝑘′,𝑖 then  15  emit Challenge fails  ⊲ Learning proved  16  goto Finalization  17 emit Challenge succeeds  ⊲ Learning invalidated  ⊲ Finalization  18 if Challenge fails then  19  𝑈𝑘′ ←Refund(𝛽𝜋𝑘,𝑘′,𝑖 | 𝛽 ∈ (0, 1))  20 else  21  𝑈𝑘′ ←Refund(𝜋𝑘,𝑘′,𝑖)  22  𝑈𝑘′ ←Penalty(𝑈𝑘, 𝜋𝑘,𝑘′,𝑖)  23  WithdrawReward(𝑁𝑘,𝑖)  24 𝑆ℎ+𝑛+1 is generated via Algo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content=' 4  ⊲ Notice  The Consensus is conducted by the nominated  verification committee Ω𝑚 when the PoL is  done in a Task𝑚.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content='  conducted in the same way where the PoL-proof is verified by  the nominated task committee Ω𝑚.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content='  1) Challenge: If a node 𝑁𝑘,𝑖 has not been challenged  before, any worker 𝑁𝑘′ can raise a PoL challenge against the  node as a challenger.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content=' 𝑈𝑘′ needs to deposit a certain amount  of tokens 𝜋𝑘,𝑘′,𝑖 to the PoL smart contract 𝑆𝐶𝑃𝑜𝐿 for the  challenging node 𝑉𝑘,𝑘′,𝑖.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content=' Then, 𝑈𝑘′ signs and broadcasts the  challenging node 𝑉𝑘,𝑘′,𝑖 to the network and starts a countdown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content='  2) Response: The publisher of 𝑁𝑘,𝑖, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content=', 𝑈𝑘, needs to reply  to the challenge 𝑉𝑘,𝑘′,𝑖 as a prover within the PoL timeout 𝜏.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content='  𝑈𝑘 firstly obtains the obfuscated dataset �𝐷train  𝑘  by applying  the noise 𝛿𝑘,𝑘′,𝑖 to its local training dataset 𝐷train  𝑘  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content=' Next, 𝑈𝑘  prepares a PoL proof node 𝑃𝑘,𝑘′,𝑖 with the hash code and  URI to the obfuscated dataset, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content=', 𝐻( �𝐷train  𝑘  ) and URI( �𝐷train  𝑘  ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content='  Then, 𝑈𝑘 signs 𝑃𝑘,𝑘′,𝑖 and broadcasts it to the network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content='  3) Verification: At the PoL verification stage, the commit-  tee Θℎ (see details in Section IV-D for the use of Θℎ) verifies  𝑃𝑘,𝑘′,𝑖 in parallel if the timestamp of the prover node is within  the timeout 𝜏ℎ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content=' To be specific, a verifier 𝑈 ˆ𝑘 can fetch the  obfuscated dataset �𝐷train  𝑘  with the URI given in 𝑃𝑘,𝑘′,𝑖 and  confirm its integrity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content=' Next, 𝑈 ˆ𝑘 conducts a training task with  the starting model described in 𝑁𝑘,𝑖, the training settings 𝜌𝑘,𝑖  embedded in 𝑁𝑘,𝑖, and the training dataset �𝐷train  𝑘  , i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content=',  𝑊∗  𝑘,𝑖 =  ∑︁  𝑊𝑝 in M𝑘,𝑖  𝐸𝑝 in E𝑘,𝑖  𝐸 𝑝  �  𝐸𝑞 ∈E𝑘,𝑖 𝐸𝑞  𝑊𝑝  𝑊replay  𝑘,𝑖  = Tˆ𝑘 (𝑊∗  𝑘,𝑖, 𝜌𝑘,𝑖, �𝐷train  𝑘  ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content='  (3)  The verifier 𝑈 ˆ𝑘 then calculates the Frobenius-Norm (F-Norm)  between trained 𝑊replay  𝑘,𝑖  and 𝑊𝑘,𝑖 in 𝑁𝑘,𝑖 as the PoL re-  sult 𝑅𝑘, ˆ𝑘,𝑖 for 𝑃𝑘,𝑘′,𝑖 [34].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content=' If 𝑅𝑘, ˆ𝑘,𝑖 is within the 𝜖PoL, the  verification on 𝑈 ˆ𝑘 is success and 𝑅𝑘, ˆ𝑘,𝑖 roots for 𝑃𝑘,𝑘′,𝑖.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content='  All versifiers in the committee Θℎ run consensus algorithms,  e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content=', Practical Byzantine Fault Tolerance (PBFT) [35], on  PoL results and get the final committee decision R𝑘,𝑖 as  a proving node in the network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content=' Notice that, to prevent the  spoofing attacks against the PoL verification and improve  the security and robustness [36], 𝜖PoL can be dynamically  adjustable from the early stage to the later stage to circumvent  the consistent F-norm-based model distance during a PoL  spoofing attack.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content=' Also, any PoL prover 𝑃𝑘,𝑘′,𝑖, ∀𝑘, 𝑘′, 𝑖 could  alternatively conduct Verifiable Computation (VC) by showing  an additional VC-proof to guarantee the identity of 𝑊replay  𝑘,𝑖  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content='  4) Clearing: At the finalization stage, the challenger 𝑈𝑘′  can only get 𝛽𝜋𝑘,𝑘′,𝑖 from the challenge deposit, 𝛽 ∈ (0, 1),  if the challenge fails (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content=', the learning is proved).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content=' Otherwise,  𝑈𝑘′ can get a full refund of the challenge deposit and also  receive a reward from the penalty on 𝑈𝑘.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content=' The reward to 𝑈𝑘  from 𝑁𝑘,𝑖 is revoked as well if the learning cannot be proved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content='  D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content=' Achieving State Consistency: Incentive Basis  IRONFORGE features settlement nodes to achieve state con-  sistency for the Global-DAG network, enabling smart contracts  in DAG and recording consistent account states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content=' The process  is shown in Algo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content='  Global-DAG periodically, with the time interval Δ𝑇 , elects  the settlement committee Θ among which consensus is reached  to generate settlement nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content=' At the beginning of the ℎ-th  interval, VRF is used to elect a committee securely Θℎ for  the settlement node 𝑆ℎ and the committee leader 𝑈 ¯ℎ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content=' The  probability that any worker 𝑈𝑘 is selected for the committee  depends on the balance of 𝑈𝑘, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content=', 𝐵𝑘;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content=' see (4) below,  VRF-hash(prv𝑘, 𝑠𝑒𝑒𝑑) ∈ Λ𝑘,  (4)  where Λ𝑘 is the area portion occupied by 𝑈𝑘 in a hash ring,  and Λ𝑘 ∝ 𝐵𝑘.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content='  The committee can then settle the status of Global-DAG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content='  To be specific, the leader of the committee 𝑈 ¯𝑘 synchronizes  the view of a particular preceding moment of Global-DAG  via consensus with others, and identifies all the tip nodes  in the ℎ-th interval, which do not have any successor in the  current interval.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content=' From each of the tip nodes, 𝑈 ¯𝑘 traverses back  according to every preceding node list M𝑘,𝑖 of 𝑁𝑘,𝑖 along the  path.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content=' The search stops and the subtree 𝜆ℎ is obtained when all  specific nodes are met, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content=', the first visible node which does  belong to the previous subtree 𝜆ℎ−1 in each path or the genesis  node 𝑁𝑔.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content=' Based on all nodes in 𝜆ℎ, 𝑈 ¯𝑘 updates the balances  of involved workers according to the training contributions,  PoL challenges, PoL proofs, and smart contract executions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content=' In  8  Algorithm 4: Node Generation in the Settlement Set  ⊲ Generating a settlement node upon consensus  1 while True do  2  if 𝑇  mod Δ𝑇 = 0 then  3  Θℎ ← VRF(Balance(𝑈𝑘 | ∀𝑘))).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content='  ⊲ Election  4  while Θℎ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content='Consensus(𝑈𝑘.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content='view | ∀𝑘 ∧ (𝑈𝑘 ∈ Θℎ)) do  5  if consensus is reached then  6  break and obtain 𝑆ℎ  7  S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content='Append(𝑆ℎ)  ⊲ The latest balance can be found in S[latest]  ⊲ Consensus  8 for 𝑈𝑘 in Θℎ parallelly do  9  Tips ← Prune({𝑁𝑘,𝑖 | 𝑇𝑘,𝑖 > 𝑇 })  10  while Traverse(start ←Tips) do  11  if (Path-𝑝 reaches 𝑁𝑔 OR 𝑁𝑘,𝑖 ∈ 𝜆ℎ−1) is True then  12  Stop Path-𝑝  13  if ALL paths have stopped then  14  break and obtain 𝜆ℎ  15  𝜆ℎ ← Prune(Tips)  ⊲ Obtain the subtree 𝜆ℎ for the creation of 𝑆ℎ  16  ¯𝑆𝑘,ℎ ← Form(𝜆ℎ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content='balance, 𝜆ℎ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content='PoL,  {𝑁𝑚,𝑒 | never been collected by S})  ⊲ Tips are excluded in balance calculation  17  if 𝑈𝑘 is the leader 𝑈 ¯𝑘 of Θℎ then  18  ¯𝑆leader ← ¯𝑆𝑘,ℎ  19  Broadcast( ¯𝑆leader)  20  else  21  Verify( ¯𝑆leader, ¯𝑆𝑘,ℎ)  22  if verification passes then  23  emit Consensus is reached  24  else  25  Elect a new Θℎ ← Θ  ′  ℎ and redo Consensus  Global-DAG, each valid reference from 𝑁𝑘′,𝑖 awards the owner  𝑈𝑘 (𝑘 ≠ 𝑘′) of the referred model 𝑁𝑘,𝑖 with a certain amount  of tokens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content=' Next, 𝑈 ¯𝑘 proposes a new settlement node ¯𝑆𝑘,ℎ  covering the updated balances and PoL results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content=' The settlement  committee Θℎ verifies and votes ¯𝑆𝑘,ℎ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content=' The committee Θℎ  endorses ¯𝑆𝑘,ℎ as 𝑆ℎ if the committee reaches consensus, or  elects a new leader otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content='  V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content=' IMPLEMENTATION AND EVALUATION  In this section, we conduct comparisons between the pro-  posed FL system, IRONFORGE, and other popular frameworks,  including GoogleFL [2], AsyncFL [6], and BlockFL [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content=' We  experimentally assess IRONFORGE in terms of the model per-  formance and expected amount of rewards that can be earned  under a variety of different environment settings, including  different aggregation strategies, different sizes of hardware and  software resources, and different types and levels of malicious  attacks such as lazy attacks [9], poisoning attacks [25], back-  door attacks [26], and model stealing attacks [27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content='  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content=' Experimental Configurations  1) Hardware settings: The experiments are conducted on  6 servers listed as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content='  Type-A (#1-3):  CPU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content=' 2 × Intel(R) Xeon(R) Gold 6230R CPU @  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content='10GHz, 2 × 52 cores  GPU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content=' 1 × Quadro RTX 4000, 1 × 8GB  Memory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content=' 528GB  Bandwidth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content=' 1000Mb/s  Type-B (#4-6):  CPU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content=' 2 × Intel(R) Xeon(R) Gold 6138 CPU @ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content='00GHz,  2 × 40 cores  GPU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content=' 8 × NVIDIA PCIe A100, 8 × 40GB  Memory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content=' 250GB  Bandwidth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content=' 1000Mb/s  2) Software settings: We carry out the experiments upon  Ubuntu 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content='04.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content='6 LTS with Keras 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content='7 in Python 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content='13 and  Docker 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content=' We use FastDFS as the distributed file  system with 15TB storage space for the model weights.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content='  3) A new testbed - FLSim: To benchmark the considered FL  frameworks, we build an FL testbed named FLSim as shown  in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content=' FLSim is docker-containerized upon our servers (#1–  6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content=' Choosing different FL frameworks is flexibly plug-and-play  in FLSim via three generic interfaces, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content=', the event emitter,  model channel, and capability configuration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content='  Event emitter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content=' FLSim is event-driven where all events are  delivered through Redis which serves as a message queue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content='  Each runner can receive events in the network in real-time  by the subscription function of Redis, and can broadcast  corresponding events according to their role.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content='  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content=' Architecture of the new FL testbed FLSim  Model channel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content=' Indexing models are done via MySQL where  properties such as the URIs of weights are included, while  the actual model weights are stored in FastDFS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content=' As a result,  the query efficiency can be significantly improved with no  need of retaining the large weights unless they are required  for evaluation or aggregation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content='  Capability configuration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content=' The tasks are trained on runners  deployed on docker clusters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content=' Each docker container represents  9  GoogleFL  BlockFL  FL Framework  AsyncFL  IronForge  Interface  Event emitter  Redis  Model channel  CapabilityCofig  MySQL+FastDF  Master  Operational Layer  Minier  Worker  Runner  Virtualisation Layer  Docker Engine  Docker Orchestration  Physical Layer  Server 1  Server nTABLE III: Hyper-parameter settings  Notation  Definition  Value (unit)  P  idle probability  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content='1  E  global epoch  2000  𝑒  default local epoch  5  𝑙  learning rate  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content='002  𝜂  sampled weights  30  𝛽  default number of candidate weights  6  𝜎  default number of aggregated weights  5  B  default batch size  100  V  validation set size  100  a runner with different resource settings, such as CPU, mem-  ory, and bandwidth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content=' Specifications of the containers are craft-  specified with strong scalability and flexibility by defining the  capability configuration to simulate various scenarios, such  as the resource imbalance considered in the experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content='  Moreover, each runner is categorized into different roles based  on which FL framework has been plugged in, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content=', “workers”  in all considered frameworks, “masters” in GoogleFL and  AsyncFL, and “miners” in BlockFL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content='  4) Training settings: The tuned hyper-parameters of the  experiments are summarized in Table III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content=' We perform training  over the MNIST with 60,000 data samples and a Convolutional  Neural Network (CNN) model illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content='  Totally 60,000 MNIST samples are randomly split into two  parts, 48,000 samples are used as the training set and 12,000  samples are used as the testing set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content=' We create non-IID training  shards for contributors from the training set to simulate a  practical network condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content=' The training set is divided into  two subsets, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content=', 24,000 for each.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content=' The samples in the first  subset are sorted by labels, and are subsequently distributed  into 120 shards, 200 for each shard.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content=' Thus, the sample labels  in each shard are relatively concentrated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content=' The other half of  the samples are randomly selected and distributed into 120  shards, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content=', 200 samples for each shard.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content=' Thus, the sample  labels in each shard are relatively uniform.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content=' Finally, we repeat  the sampling operation 120 times, and each time we take a  shard from each of the two subsets for merging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content=' We end up  obtaining 120 new shards each of which contains 400 samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content='  5) Environment settings: To demonstrate the state-of-the-  art of IRONFORGE, a comprehensive comparison between  IRONFORGE and the other three FL frameworks are conducted  in our experiments, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content=', the synchronous GoogleFL, AsyncFL,  and BlockFL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content=' We launch 120 runners as workers to train  models with a probability of P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content=' We also define two events  that represent receiving two types of intermediate models:  GLOBAL MODEL UPLOADED EVENT  (GMUE):  the model acquired by aggregating the uploaded local  models prior to training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content='  LOCAL MODEL UPLOADED EVENT (LMUE): the  model trained by workers with their local datasets  These events are broadcast to notify each runner associated  with the next step to take.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content=' Note that the genesis model of a  task is tagged as a global model in order to initiate any selected  FL framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content=' This indicates that emitting GMUE is used to  notify the network when the task is published.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content='  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content=' The CNN model is a lightweight version of the model  in [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content=' It contains one convolution layer of which the filter  size is 32 and the kernel size is 5 × 5, one 2 × 2 max-pooling  layers, one fully connected layer with 256 units, and ReLu  activation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content=' The output is processed by a fully connected layer  with 10 units and softmax activation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content='  Synchronous GoogleFL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content=' One additional runner is launched  as the master to aggregate local model weights.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content=' For GoogleFL,  workers are activated by GMUE and the master is activated  by LMUE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content=' The workers, when receiving a GMUE, train on  top of a global model downloaded from the master with  their own local datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content=' The master receives an LMUE when  the workers upload the trained models, and subsequently  aggregates all collected local models after a timeout, followed  by uploading the aggregated result as the global model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content=' The  above iterating process continues until the task reached the  max iteration threshold E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content='  AsyncFL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content=' One additional runner is launched as the master to  aggregate local models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content=' AsyncFL shares the same procedure  as GoogleFL, except that the master, when receiving an  LMUE during its idle period, creates a new global model by  aggregating the most recent global model and newly-collected  local models with an identical weighting factor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content='  BlockFL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content=' Five additional runners are launched as miners for  BlockFL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content=' In each iteration, the miners behave the same way  as the master does in GoogleFL or AsyncFL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content=' An additional  step is that the miners compute the nonce to finalize the block  and compete for the rewards with a synchronized lock being  used in Redis to ensure the mining order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content='  IRONFORGE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content=' No additional runners are launched for IRON-  FORGE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content=' Each runner acts as a worker and a master at the same  time, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content=', it supports both aggregating and training operations,  thus receiving both GMU and LMUE during each iteration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content='  6) Security settings: We implement five types of con-  tributors: normal contributors, poison contributors, backdoor  contributors, and stealing and colluding contributors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content='  Normal contributors act honestly and independently across  all phases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content='  10  input:  [(None,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content=' 28,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content=' 28,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content=' 1)]  conv2d 2 input  InputLayer  output:  [(None,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content=' 28,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content=' 28,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content=' 1)]  input:  (None,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content='28,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content='28,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content='1)  conv2d 2  Conv2D  output:  (None,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content='24,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content='24,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content='32)  input:  (None,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content='24,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content='24,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content='32)  max_pooling2d_2  MaxPooling2D  output:  (None,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content='12,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content='12,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content='32)  input:  (None,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content=' 12,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content='12,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content='32)  flatten 2  Flatten  output:  (None,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content='4608)  input:  (None,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content='4608)  dense_4  Dense  output:  (None,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content=' 256)  input:  (None,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content=' 256)  dense 5  Dense  output:  (None,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content=' 10)Poisoning contributors aim to undermine the integrity and  availability of the global model by crafting local poisoning  models [25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content=' In this paper, we simulate poison contributors by  adopting the label-flipping strategy that fakes labels and then  conducting training on the forged datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content='  Backdoor contributors aim to fail the global model on  targeted tasks, typically by adhering crafted triggers to training  samples, conducting training on the amended samples, and  then uploading the attack models [26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content=' In this paper, backdoor  contributors layer 5 × 5 white patches to training samples and  change the label of the manipulated samples to a fixed one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content='  Stealing contributors aim to gain rewards by stealing model  weights trained by others and uploading the plagiarized  weights as their own work [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content=' In this paper, a stealing  contributor 𝑘′ selects and directly uploads one of the existing  weights by simply changing the ownership from 𝑘 to 𝑘′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content='  Colluding contributors aim to embezzle training rewards for  their conspirators by performing honest training processes but  claiming source list M from the conspirators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content=' In this paper, a  certain proportion of contributors tamper the source list M in  their uploaded weights, with a certain probability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content='  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content=' Results and Evaluations  Several experiments are conducted from two perspectives,  i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content=', the performance comparison between IRONFORGE and  others with and without attacks, and the fairness comparison  between different resource levels in terms of rewards.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content='  1) Performance - with and without attacks: Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content=' 6 shows  that the proposed IRONFORGE outperforms AsyncFL and  BlockFL with only slightly slower convergence than the base-  line GoogleFL across 2,000 iterations with no attacks lever-  aged.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content=' It is worth noting that the red curve increases sharply  with as few oscillations as that of the baseline, particularly  highlighting the stability of IRONFORGE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content='  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content=' Comparison between IRONFORGE and others in terms  of accuracy with no attacks leveraged  The performance comparison with different levels of attack  behaviors being applied to IRONFORGE is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content=' It  is realized that IRONFORGE is resistant to the stealing attack  the most, followed by the resistance to the backdoor attacks  and poisoning attacks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content=' It is worth noting that the performance  of a stealing ratio of 20% can be as good as that of others.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content='  This is because the native validation process in IRONFORGE  can capture and eliminate the plagiarized models which are  reused or whose ownerships are fake.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content=' The remaining 80% of  models are still sufficient for contributors to aggregate and  train by offering strong diversity of the non-IID data samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content='  On the other hand, an evident degradation of the accuracy,  around 5%, is shown in both poisoning (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content=' 7(b)) and  backdoor (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content=' 7(c)) contexts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content=' Nevertheless, it can be found  from Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content=' 8(a) and 8(b) that IRONFORGE outperforms all  the other FL frameworks with either 20% ratio of poisoning  attackers or 20% ratio of backdoor attackers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content=' This significantly  highlights the superiority of IRONFORGE in terms of its strong  resistance to malicious model updating.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content=' 8(c) highlights  the resistance to stealing attacks and collusion attacks of  IRONFORGE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content=' Note that only BlockFL is considered in the  comparison as GoogleFL and AsyncFL do not support incen-  tives natively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content=' It is found that, by using the native validation  process in IRONFORGE, including the PoL verification, none  of the dishonest contributors who leverage either the stealing  attack or collusion attack can gain rewards.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content=' This prevents the  malicious contributors from faking the ownership of or directly  using the existing models, or embezzling the rewards for their  conspirators by claiming a falsified source list.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content='  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content=' 9(a) shows the accuracy ranges of adjusting the candi-  date size (𝛽) for different levels of aggregation sizes (𝜎), e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content=',  the blue band representing the accuracy ranged from 1-of-2  to 1-of-8 with 𝜎 = 1 and 𝛽 ∈ [2, 8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content=' The results reveal that  increasing the aggregation size 𝜎 stabilizes the performance  in the beginning stage, and allows for an increasingly higher  convergence point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content=' The effect of increasing the candidate size  for a certain aggregation size becomes gradually weakened  as the aggregation size increases, as you can see that the  width of each band turns more and more narrow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content=' The effect  of increasing the aggregation size also becomes weakened, as  you can see from Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content=' 9(a) that the performance of 5-of-6  is as good as that of 7-of-8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content=' The same insight can also be  shed by observing the performance comparison of adjusting  the aggregation sizes (𝜎) for different levels of candidate size  (𝛽) is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content=' 9(b), e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content=', the blue band representing the  accuracy ranged from 1-of-3 to 2-of-3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content=' The bottom line of the  red band for 𝛽 = 9 performs as poorly as the 2-of-3 strategy  does, while the top line of the red band performs the best  among the kinds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content=' It can thus be concluded that knowing that  no attacks are leveraged, aggregating more weights is more  beneficial for obtaining high accuracy than merely aggregating  a low number of weights from more candidate weights.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content='  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content=' 9(c) shows the running time of different aggregation  strategies all the way from downloading the weights to up-  loading a new model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content=' There is a stronger effect on the running  time when adjusting the candidate size (𝛽) than adjusting the  aggregation size (𝜎).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content=' An implication can be realized based on  this observation that the bandwidth could be the bottleneck in  IRONFORGE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content=' We design dedicated experiments that learn this  phenomenon, as explained in the following Section V-B2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content='  2) Fairness - earn rewards: The fairness is investigated in  terms of the difference in rewards between different levels  of hardware specifications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content=' The bottleneck of gaining more  11  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content="95  06'0  08'0  BlockFL  0." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content='75  AsyncFL  GoogleFL  IRONFORGE  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content='70  0  250  500  750  1000  1250  1500  1750  2000  Iteration(a) Stealing attacks  (b) Poisoning attacks  (c) Backdoor attacks  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content=' Comparison between different levels of stealing attacks, poisoning attacks, and backdoor attacks applied to IRONFORGE  in terms of accuracy  (a) Poisoning attacks with a poisoning ratio of 20% (b) Backdoor attacks with a poisoning ratio of 20%  (c) Stealing and collusion attacks  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content=' Comparison between IRONFORGE and others in terms of accuracy or rewards with a certain level of poisoning attacks,  backdoor attacks, and stealing and colluding attacks  rewards in IRONFORGE can also be realized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content=' We select two  different contest strategies, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content=', Immediate settlement and  Winner traverse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content=' The immediate-settlement is the strategy used  in Global-DAG by default, rewarding every model every time  it gets referred by others.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content=' The winner-traverse could be one of  the main options used in a Task-DAG, rewarding every model  that exists in the traversal path all the way from the winner  node to the genesis node (excluded).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content='  According to Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content=' 10, A monotonic increase of the rewards  can be realized for the immediate-settlement with increasing  CPU cores, memory capacity, and bandwidth, and for the  winner-traverse only with increasing bandwidth (the winner-  traverse appears to have similar characteristics to BlockFL  which also fails to offer a pure monotonic increase of rewards  in all specifications).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content=' This is because the winner-traverse strat-  egy could include moderate models being aggregated during  each iteration in the winner-traversal path while the winner  could be highly random when the competition is intense  and the convergence is near.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content=' Therefore, many models with  high performance could be excluded by the unique winner  traversal path at the end.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content=' On the contrary, the immediate-  settlement allows every model not to be missed so long as  a valid reference relationship is confirmed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content=' Nevertheless, the  result difference between these two strategies does not tell  the superiority of fairness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content=' Different requirements may lead  to different principles of fairness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content=' Rooting for an egalitarian  strategy, or “to each according to his contribution”, or striking  a balance in between is a flexible option that the proposed  IRONFORGE offers back to users without a harsh setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content='  On the other hand, the monotonic increase in the band-  width comparison for both strategies, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content=' 10(c),  highlights the bandwidth being the most critical effect for  earning rewards in IRONFORGE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content=' That is to say, relatively  poorer users being more active in uploading models to the  network with higher frequency can help them be more likely  to share the rewards, rather than spending much time on a  strongly performant model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content='  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content=' 11 learns the effect of data quality upon the rewards by  adding a random perturbation to 50% of the training samples  of half of the users with a mean of 0 and a standard deviation  of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content=' We define the affected nodes as “Poor” nodes, while  “Excellent” nodes own normal data samples only.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content=' The result  shows that the poor nodes that use the immediate-settlement  strategy enjoy a narrower range of rewards compared to that  of the winner-traverse strategy and BlockFL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content=' This highlights  that poor nodes earning rewards via the immediate-settlement  12  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content='9  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content='8  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content='7  Accuracy  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content='4  BlockFL  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content='3  AsyncFL  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content='2  GoogleFL  IRONFORGE  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content='1  0  500  1000  1500  2000  2500  3000  3500  4000  Iteration0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content='8  Accuracy  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content='4  BlockFL  AsyncFL  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content='2  GoogleFL  IRONFORGE  0  500  1000  1500  2000  2500  3000  3500  4000  IterationBlockFL  30  IRONFORGE  25  20  Rewards  15  10  5  0  Normal  Dishonest  ContributorType,<Normal>/<Stealing/Colluded>1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content='8  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content='4  Stealing Ratio:0%  StealingRatio:5%  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content='2  StealingRatio:10%  Stealing Ratio:20%  0  500  1000  1500  2000  2500  3000  3500  4000  Iteration1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content='8  Accuracy  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content='4  Poisoning Ratio:0%  Poisoning Ratio: 5%  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content='2  PoisoningRatio:10%  PoisoningRatio:20%  0  500  1000  1500  2000  2500  3000  3500  4000  Iteration1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content='8  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content='4  BackdoorRatio:0%  BackdoorRatio:5%  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content='2  BackdoorRatio:10%  BackdoorRatio:20%  0  500  1000  1500  2000  2500  3000  3500  4000  Iteration(a) Accuracy difference between aggregation sizes  (b) Accuracy difference between candidate sizes  (c) Time difference between aggregation strategies  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content=' Comparison between different combination of the aggregation strategies (𝜎-of-𝛽) applied to IRONFORGE in terms of  accuracy and execution time  (a) Reward difference between different CPU cores  (b) Reward difference between memory capacity  (c) Reward difference between bandwidths  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content=' 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content=' Comparison between different levels of the CPU core, memory capacity, and bandwidth in terms of rewards  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content=' 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content=' Comparison between IRONFORGE and others in terms  of rewards under the influence of the data quality issue  in IRONFORGE can be more stable and predictable than  BlockFL and the winner-traverse, and can be less affected by  unexpected data degradation or network noise in unreliable  channels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content=' Excellent nodes have more opportunities to earn  higher rewards than poor nodes while the median value and  the minima of rewards remain as high as that of poor nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content='  This reflects the fairness between excellent and poor nodes,  i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content=', offering stable rewards to the poor while the excellent are  given chances to make a great fortune.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content='  VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content=' DISCUSSION AND ANALYSIS  This analysis focuses on the security of IRONFORGE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content=' Every  role in the system is involved in the attack model except that  the timestamp in IRONFORGE is considered synchronous via  external trustworthy servers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content='  Adversaries target to break the state consistency in Global-  DAG so that operations such as incentive and consensus fail to  be executed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content=' The adversaries also target to leverage the model  stealing attack in order to:  forge the “amount of work” by simply stealing others’  models with no more effort being put into the training;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content='  collude with attackers by creating a model referring to  models from colluded attackers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content='  In addition, the adversaries can target on breaching the dataset  privacy during the dataset sharing in a PoL process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content=' At the  same time, the adversaries can also unbalance the competition  by abusing others’ datasets to enrich local resources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content='  State consistency: The state consistency is guaranteed over a  sufficiently long period Δ𝑇 as long as the seeds being used in  each VRF process are secure and the lower bounds of faulty  tolerance of consensus protocol (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content=', 33% for PBFT) are  satisfied in the VRF-elected committees.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content=' We consider the time  gap between two settlement nodes Δ𝑆ℎ,𝑆ℎ−1 is sufficiently large  in IRONFORGE to expect that each user who gets registered  13  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content='9  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content='970  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content='8  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content='965  Accuracy  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content='960  Aggregation Size(α)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content='950  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content='7  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content='945  1  2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content='940  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content='935  3  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content='930  4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content='6  1000  1200  1400  1600  1800  2000  5  7  0  250  500  750  1000  1250  1500  1750  2000  Iteration0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content='96  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content='94  Accurao  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content='92  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content='90  Candidate Size (β)  3  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content='88  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content='5  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
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+page_content='500  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content='750  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
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+page_content='Iteration1-of-3  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content='2-of-3  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
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+page_content='BlockFL  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
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+page_content='BlockFL  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content='IronForge(immediatesettlement)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content='350  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content='IronForge(Winnertraverse)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content='300  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content='Rewards  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content='250  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content='200  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content='150  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content='100  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content='50  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content='0  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content='2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content='4  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content='8  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content='12  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content='Bandwidth(MB/s)BlockFL  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content='25  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content='IronForge(lmmediate settlement)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content='IronForge (Winner traverse  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content='20  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content='Rewards  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content='15  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content='10  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content='5  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content='0  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content='Excellent  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content='Poor  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content='Data Quality,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content='Poor/Excellentfor committee election has an identical “view” of Global-DAG  starting from 𝑇ℎ−1 to 𝑇ℎ−1 + Δ𝑇 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content=' This prevents the consensus  process in the committee from being trapped into an indefinite  disagreement due to the network asynchrony.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content=' On the other  hand, an unbiased and unpredictable random seed is crucial for  a fair VRF process where (4) cannot be manipulated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content=' This can  be achieved by implementing existing randomness generators  such as RANDAO [37] or RandHound-VRF [38].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content='  Model stealing attack: Offering the proposed incentive mech-  anism in a decentralized FL attracts attackers to leverage  model stealing attacks, by either stealing the model ownership  or faking the training processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content=' Attackers can, with no effort  on local training, steal others’ models and fake ownership with  ease.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content=' Attackers can alternatively fake the source lists upon an  honest local training process so that their accomplices, who are  instead placed in the source list, can reap profits against other  honest users.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content=' These two types of model stealing attacks are  used for misleading those who wish to ensure the necessary  training overhead and the efforts in the source list.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content=' They are  particularly useful when attackers intend to steal the rewards  and share them with their accomplices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content=' IRONFORGE enables  PoL-challenge where users can choose to challenge a model  via idle resources, and the model owner requires to provide  the valid PoL-proof in time for a public verification during  the consensus process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content=' By the committee replaying parts of the  training from scratch and reaching the consensus, the “amount  of work” and the source list M can be explicitly determined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content='  Dataset privacy and model melting: This security metric is  an implementation of our work [34].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content=' Dataset obfuscation helps  to preserve dataset privacy when datasets require to be publicly  shared for PoL-challenge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content=' Experimental results in [34] show  that an obfuscated dataset satisfies PoL verification without  sacrificing the privacy level while being able to decrease the  model utility against the abuse of collecting provers’ obfus-  cated datasets, namely, model melting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content=' In addition, applying  training over different data samples or using non-IID noise  significantly can reduce the risks of privacy decline when a  sufficient number of challenges against the same model owner  are deliberately raised by attackers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content='  VII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content=' CONCLUSION  This paper proposed IRONFORGE, a new generation of FL  framework constructed by DAG-based structure, which for  the first time eliminates the need for the central coordinator  to solve the issues of network asynchrony and the excessive  reliance on the central coordinator while at the same time  enabling an open and fair incentive mechanism to encourage  more participants, particularly in networks where training  resources are unevenly distributed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content=' Experimental results based  on a newly developed testbed FLSim along with the security  analysis highlight the superiority of IRONFORGE over the  existing prevalent FL frameworks under various specifica-  tions regarding performance, fairness, and security.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
+page_content=' To our  knowledge, this is the first paper proposing a secure and  fully decentralized FL framework that can be applied in open  networks with realistic network and training settings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQfnQhW/content/2301.04006v1.pdf'}
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+OPTIMIZATION PERSPECTIVES ON SHELLSORT
+Oscar Skean
+Department of Computer Science
+University of Kentucky
+oscar.skean@uky.edu
+Richard Ehrenborg
+Department of Mathematics
+University of Kentucky
+Jerzy W. Jaromczyk
+Department of Computer Science
+University of Kentucky
+ABSTRACT
+Shellsort is a sorting method that is attractive due to its simplicity, yet it takes effort to analyze its
+efficiency. The heart of the algorithm is the gap sequence chosen a priori and used during sorting.
+The selection of this gap sequence affects the efficiency of Shellsort, and thus drives both its theo-
+retical and experimental analysis. We contribute to Shellsort by identifying efficient gap sequences
+based on new parameterized functions. Specifically, a parameter grid-search identifies optimal pa-
+rameters for different input sizes for sorting by observing minimal overhead in three categories:
+number of comparisons, number of exchanges, and running time. We report that our method finds
+sequences that outperform state-of-the-art gap sequences concerning the number of comparisons for
+chosen small array sizes. Additionally, our function-based sequences outperform the running time
+of the Tokuda sequences for chosen large array sizes. However, no substantial improvements were
+observed when minimizing the number of exchanges.
+1
+Introduction
+The Shellsort algorithm is a sorting method that was among the first to be discovered. Published in 1959 [1], it saw
+early interest due to its low memory requirements and simple implementation. Despite this, its analysis is difficult and
+remains incomplete. The algorithm has found practical use today in memory-constrained environments, embedded
+systems, and the bzip2 compressor. Recently it has also found use in data-oblivious sorting [2] and in fully homomor-
+phic encryption [3].
+Shellsort is an in-place comparison sort and can be viewed as a generalization of insertion sort. For a data array
+A of size N, Shellsort operates using a predetermined gap sequence 1 = k1 < · · · < km < N. The algorithm
+performs m passes over A: starting with the largest km and ending with k1. During pass j, for a given gap km−j,
+insertion sort occurs for the km−j subarrays consisting of the data elements A(i), A(i + km−j), A(i + 2 · km−j), . . .
+for i = 0, . . . , km−j − 1. We say a k-inversion is a pair (i, i + k) such that the inequality A(i) > A(i + k) holds.
+After pass j, all km−j-inversions that were originally in A have been solved. We say an array with no k-inversions
+is k-sorted. Note that the final pass with k1 is equivalent to insertion sort and is necessary to guarantee sortedness.
+Therefore, the purpose of the gap sequence is to presort A as much as possible before the expensive final insertion sort
+pass.
+The main results of this paper are:
+1. New efficient Shellsort sequences derived from experimentally optimizing sequence-generating functions.
+For prescribed array sizes, these sequences outperform well-known efficient sequences (e.g. Tokuda, Ciura)
+with respect to the number of comparisons. These sequences also outperform the running time of the Tokuda
+sequence, making them the fastest function-based sequence on the tested array sizes.
+2. We demonstrate results of experimental analysis comparing our proposed approaches with well-known se-
+quences by measuring the number of comparisons, exchange operations, and running time needed to sort
+randomized permutations.
+Traditionally, improvements for Shellsort have come from finding gap sequences with theoretical properties. We
+discuss some particularly important sequences in Section 2. Then in Section 3, we introduce parameterized sequence-
+arXiv:2301.00316v1  [cs.DS]  1 Jan 2023
+
+generating functions that generate a Shellsort sequence. The parameters are then optimized in a grid-search finding
+the best possible sequence that can be produced from that function for a chosen array size. In Section 4 we discuss our
+experimental methodology to compare the performance of the optimized template sequences to the baseline sequences
+mentioned in Section 2.
+2
+Background
+The selection of a good gap sequence is critical to the performance of Shellsort. There has been a plethora of work
+focused on selecting good sequences [4, 5, 6]. Some of the earliest proposed sequences were based on powers of
+2 [7, 1]. Then Pratt showed that the sequence of 2p3q obtains a number of inversions that is Θ(Nlog2N) in the
+worst case [8]. We call this sequence Pratt-23 in Table 1. This sequence still has the best known asymptotic time
+complexity for any Shellsort sequence. However, it has a very large constant factor which spurred the development of
+new sequences.
+The proof technique used to show the time complexity of Pratt-23 was based on counting the inversions of a sequence
+that has already been 2-sorted and 3-sorted. A natural extension of this is to apply the Frobenius problem to place
+bounds on what has already been sorted in prior passes. A typical formulation of the Frobenius problem is as follows:
+Suppose that you have k coins of denominations u1, u2, . . . , uk. What is the largest value which cannot be made
+with a nonnegative linear combination of these coins? This largest value is known as the Frobenius number [9]. In
+the context of Shellsort, the coins can be equated to gap size and the Frobenius number can be equated to the largest
+remaining inversion after sorting with the gaps. Using the Frobenius problem, several sequences were proposed that
+had a lower constant factor than Pratt-23 despite having a worse time-complexity [10, 11].
+Following those sequences, the focus of gap sequence selection shifted from finding theoretically good sequences
+to finding experimentally good ones. For example, one property that was observed was that a geometric sequence
+with a growth of 2.25 often performed well in practice. This observation was the basis of the Tokuda sequence [6].
+See Table 1 for a functional form. To the best of our knowledge, this remains the most competitive function in the
+literature.
+The next improvement came from Ciura in 2002 [4]. The Ciura sequence disregarded the idea of a function-based
+sequence, and instead searched for the best set of gap elements themselves. Ciura found the best sequence for array
+sizes of 128 and 1000, as well as a sequence that was conjectured to perform better for much larger array sizes. We
+call these Ciura-128, Ciura-1000, and Ciura-Large in Table 1.
+The Ciura sequences also marked a transition in how Shellsort performance was measured. Previously, most works
+counted the number of exchanges used by the algorithm [8]. This partly because some proof techniques relied on
+counting the number of inversions. Ciura instead focused on optimizing the number of comparisons, which was found
+to be more directly related to the computation time of Shellsort. A comparison is defined as checking if a pair of array
+elements are inverted. An exchange is typically defined as the variable swap used to fix an inversion. Minimizing the
+number of comparisons is especially beneficial when comparisons are expensive to make, such as when sorting large
+satellite data. Similarly, minimizing exchanges is beneficial in memory-constrained systems. In this work, we make
+clear distinctions in Section 4 about what measurement we’re optimizing for. For a full treatment of the history of gap
+sequences, we point the reader to [5].
+3
+Parameterized Template Functions
+The approach of directly optimizing the gap sequence, as in [4], grows in computational cost very quickly as N
+increases. This growth is due the fact that as N increases, both the expected number of sequence elements and their
+possible range of values increases. To help alleviate this, the authors of [4] found a suitable sequence prefix and
+optimize the extension of it by a few values. However, at very large N even finding this prefix would be very costly.
+Here, we formulate the problem of finding the optimal sequence as optimizing the parameters of a pre-defined
+sequence-generating function. Guided by principles that we have seen in sequences that perform well, we define
+two functions as follows.
+kA(i) = ⌊(a⌊ i
+b ⌋ · c⌊ i
+d ⌋)f + e⌋
+(1)
+kB(i) = ⌊(a · b⌊ i
+c ⌋) + d⌋
+(2)
+We refer to (1) as Ours-A and (2) as Ours-B in Table 1. Both formats contain the floor function in exponents. We
+found this to allow the function to express a more ”chaotic” sequence, which helped improve performance. The unique
+characteristic of Ours-A is the parameter f, an exponent that helps regulate growth. The parameter a of Ours-B was
+2
+
+Sequence Name
+Function
+Optimized for N
+Parameters
+Initial Terms
+Ciura [4]
+-
+128
+-
+1 4 9 24 85 126
+-
+1000
+-
+1 4 10 23 57 156 409 995
+-
+Large
+-
+1 4 10 23 57 132 301 701 1750
+Tokuda [6]
+⌈ (9/4)k−1
+(9/4)−1 ⌉
+-
+-
+1 4 9 20 46 103 233 525 . . .
+Ours A
+⌊(a⌊ i
+b ⌋ · c⌊ i
+d ⌋)f + e⌋
+128 (Comp)
+Table 2
+1 4 9 24 85 150 . . .
+1000 (Comp)
+Table 2
+1 4 10 23 57 153 400 . . .
+1000 (Time)
+Table 2
+1 3 7 16 33 85 179 472 . . .
+Ours B
+⌊(a · b⌊ i
+c ⌋) + d⌋
+10000 (Comp)
+Table 2
+1 4 10 27 72 187 488 . . .
+Pratt-23 [8]
+Ordered 2p · 3q
+-
+-
+1 2 3 4 6 8 9 . . .
+Pratt-25
+Ordered 2p · 5q
+-
+-
+1 2 4 5 8 10 15 16 . . .
+Pratt-34
+Ordered 3p · 4q
+-
+1 3 4 9 12 16 24 . . .
+Table 1: Gap sequences that are compared during experiments
+Template
+a
+b
+c
+d
+e
+f
+Ours-A128-Comp
+2.6321
+1.6841
+2.1570
+0.7360
+3
+0.7630
+Ours-A1000-Comp
+3.5789
+2.6316
+3.8158
+2.1579
+3
+0.7632
+Ours-A1000-Time
+2.75
+2.75
+3.7142
+2.4286
+2
+0.7429
+Ours-B10000-Comp
+4.0816
+8.5714
+2.2449
+0
+-
+-
+Table 2: Optimized parameters for template functions
+designed to have a similar purpose, albeit via multiplication. Ours-B contains fewer parameters which allows for
+quicker optimization. For conciseness, we only optimize Ours-A for array sizes 128 and 1000, and Ours-B for 10000.
+Furthermore, we denote as Ours-A1000-Comp if optimizing Format A for number of comparisons on arrays of size
+1000. In the following section, we discuss the experimental procedure for optimizing (1) and (2), as well as for
+comparing them to other baseline sequences.
+4
+Experimental Procedure
+Because the function is highly non-convex, it is difficult to utilize efficient techniques such as gradient descent. In-
+stead, we employ a grid-search approach. One benefit of using a function grid-search, as opposed to direct sequence
+optimization, is that the size of the search space has no relation to N. It depends only on the granularity and bounds
+of the search. This is in constrast to the methodology used by Ciura, in which the number of tested sequences grows
+with N [4].
+For Ours-A, we define the grid for parameters a, b, c, d, f as 20 linearly spaced values between 0.5 and 5. We also
+allow e to be an integer value between 0 and 10, including both endpoints. Because Ours-B has fewer parameters,
+we can take a more fine-grained approach. For parameters a, b, c, we test 50 linearly spaced values from 0 to 10. For
+parameter d, we constrain it to be the same as e in Ours-A.
+The data array that we test on contains N distinct values 1 through N, and we shuffle it with the Fischer-Yates shuffle.
+For each set of parameters in the grid-space, we compute the mean cost over 1000 iterations. We then take the set of
+parameters producing the lowest mean cost as optimal. This cost can be defined as number of comparisons, number
+of exchanges, or time.
+Because Ciura sequences are optimized for specific array sizes with no means of extending them, it would be inap-
+propriate to directly compare them with any function-based sequence at large array sizes. One method of extending a
+Ciura sequence is by starting a geometric series on its last term with a ratio of 2.25. We adopt this method of extension
+when measuring the performance of Ciura sequences.
+3
+
+Sequence
+N=20
+N=128
+N=200
+µCO
+µEX
+µCO
+µEX
+µCO
+µEX
+Ours-A128-Comp
+76 ± 6
+38 ± 6
+998 ± 33
+531 ± 33
+1786 ± 46
+948 ± 48
+Ours-A1000-Comp
+76 ± 6
+39 ± 7
+1004 ± 32
+516 ± 31
+1787 ± 44
+919 ± 45
+Ours-A1000-Time
+79 ± 5
+39 ± 7
+1035 ± 26
+468 ± 27
+1832 ± 38
+846 ± 39
+Ours-B10000-Comp
+76 ± 7
+33 ± 5
+1096 ± 52
+535 ± 36
+1775 ± 49
+960 ± 49
+Ciura-128
+76 ± 6
+37 ± 6
+998 ± 32
+531 ± 33
+1800 ± 46
+970 ± 49
+Ciura-1000
+76 ± 7
+39 ± 7
+1006 ± 31
+519 ± 34
+1787 ± 45
+920 ± 44
+Ciura-Long
+76 ± 7
+39 ± 7
+1004 ± 32
+516 ± 32
+1794 ± 44
+907 ± 42
+Tokuda
+76 ± 6
+37 ± 6
+1020 ± 28
+490 ± 28
+1808 ± 42
+891 ± 43
+Pratt-25
+111 ± 4
+27 ± 4
+1732 ± 16
+345 ± 17
+3207 ± 21
+610 ± 24
+Pratt-23
+136 ± 3
+25 ± 4
+2209 ± 13
+333 ± 15
+4095 ± 19
+589 ± 21
+Pratt-34
+95 ± 4
+29 ± 4
+1424 ± 16
+374 ± 19
+2593 ± 25
+660 ± 26
+Table 3: Number of operations to sort small arrays averaged over 1000 random array permutations. µCO denotes the
+number of comparisons, µEX denotes the number of exchanges.
+Sequence
+N=1000
+N=2000
+N=5000
+µCO
+µEX
+µCO
+µEX
+µCO
+µEX
+Ours-A128-Comp
+13250 ± 203
+7847 ± 199
+30530 ± 378
+18611 ± 384
+91122 ± 973
+57728 ± 904
+Ours-A1000-Comp
+12941 ± 167
+7004 ± 155
+29596 ± 293
+16234 ± 282
+86821 ± 768
+50349 ± 770
+Ours-A1000-Time
+13193 ± 144
+6461 ± 146
+30120 ± 263
+14913 ± 257
+87455 ± 548
+44305 ± 552
+Ours-B10000-Comp
+12980 ± 186
+7245 ± 177
+29643 ± 305
+17241 ± 325
+86514 ± 617
+57388 ± 817
+Ciura-128
+13300 ± 166
+7003 ± 168
+30359 ± 318
+15987 ± 310
+88193 ± 629
+46689 ± 627
+Ciura-1000
+12918 ± 161
+7002 ± 155
+29534 ± 282
+16138 ± 274
+86641 ± 757
+47852 ± 751
+Ciura-Long
+13035 ± 142
+6701 ± 149
+29567 ± 246
+15427 ± 261
+86232 ± 502
+45347 ± 496
+Tokuda
+13116 ± 143
+6556 ± 142
+29888 ± 241
+14952 ± 228
+86838 ± 454
+44116 ± 472
+Pratt-25
+26211 ± 68
+4318 ± 72
+62722 ± 122
+9755 ± 131
+194196 ± 263
+28195 ± 278
+Pratt-23
+34380 ± 64
+4253 ± 69
+82785 ± 106
+9669 ± 116
+259088 ± 242
+28354 ± 257
+Pratt-34
+20974 ± 89
+4671 ± 87
+50038 ± 153
+10543 ± 160
+154298 ± 372
+30448 ± 372
+Table 4: Number of operations to sort medium-sized arrays averaged over 1000 random array permutations
+4.1
+Filtering
+There are two techniques we employ to reduce our grid-search space.
+First, we notice that different sets of parameters could produce the same sequence. For example, for the template
+function Ours-A, the ordering of (a, b) and (c, d) do not matter. We precalculate all of the sequences produced in the
+grid and only experiment on the unique ones. For example, for N = 10000, this reduces the grid search space from
+over 1.5 million sets of parameters to about 1 million.
+Second, we use sequential analysis to act as a low-pass filter for screening out obviously poor sequences. This statis-
+tical approach was first applied to Shellsort in [4]. Given bounds for the mean and an upper bound for the variance,
+sequential analysis is able to tell in just a few repetitions whether or not a sample mean falls below the mean bounds
+with a certain confidence. Sequential analysis allows us to quickly accept good gap sequences that have a low mean
+number of comparisons. Any sequence that is accepted by the filter is then run for the full 1000 iterations to obtain a
+more accurate estimate of the mean. We adopt the same setup as in [4].
+The Ciura sequences were optimized with respect to the number of comparisons, and because they are well-known to
+be some of the most practical sequences, we optimize our template functions with respect to the number of comparisons
+as well.
+4.2
+Hardware
+The experiments were performed on a Ubuntu machine with an 8-core Intel Xeon W-3225. Experiments counting
+number of comparisons and exchanges were multithreaded. Experiments involving measurement of time were done
+4
+
+Sequence
+N=10000
+µCO
+µEX
+Ours-A128-Comp
+206356 ± 1796
+132351 ± 1797
+Ours-A1000-Comp
+196336 ± 1707
+119012 ± 1710
+Ours-A1000-Time
+194052 ± 879
+98952 ± 883
+Ours-B10000-Comp
+192029 ± 992
+209292 ± 1293
+Ciura-128
+195256 ± 1106
+105544 ± 1109
+Ciura-1000
+193778 ± 1895
+111338 ± 1897
+Ciura-Long
+191435 ± 892
+101680 ± 897
+Tokuda
+192574 ± 795
+98071 ± 796
+Pratt-25
+450131 ± 516
+62191 ± 526
+Pratt-23
+604502 ± 451
+66923 ± 725
+Pratt-34
+355382 ± 723
+63272 ± 462
+Table 5: Number of operations to sort an array size of 10000 averaged over 1000 random array permutations
+Sequence
+Running Time (ms)
+Ours-A128-Comp
+3.15 ± 0.08
+Ours-A1000-Comp
+3.02 ± 0.06
+Ours-A1000-Time
+3.01 ± 0.06
+Ours-B10000-Comp
+3.04 ± 0.07
+Ciura-128
+3.07 ± 0.06
+Ciura-1000
+3.01 ± 0.06
+Ciura-Long
+3.04 ± 0.07
+Tokuda
+3.06 ± 0.08
+Pratt-25
+5.00 ± 0.09
+Pratt-23
+6.35 ± 0.11
+Pratt-34
+4.17 ± 0.08
+Table 6: Time to sort an array size of 1000 averaged over 1000 random array permutations
+Figure 1: (Left) For varying array sizes, shows the difference in number of comparisons between baseline sequences
+and Ours-A128. A positive value means Ours-A128 uses fewer comparisons. (Right) Number of comparisons for
+varying array sizes larger than what Ours-A128 was optimized for.
+5
+
+Difference inMeanNumberof Comparisons
+35
+Ciura-128
+Ciura-Large
+30 
+Tokuda
+25
+differences
+20
+15
+10
+5
+0
+25
+50
+75
+100
+125
+150
+175
+200
+Array SizeMeanNumberofComparisonsvsArraySize
+1800
+Ours-A128
+Ciura-128
+Ciura-Large
+1700
+Tokuda
+NumberofComparisons
+1600
+1500
+1400
+1300
+1200
+150
+160
+170
+180
+190
+200
+Array Sizesingle threaded, as any multithreaded applications could cause discrepancies in time measurement. All code was
+written in Python.
+4.3
+Results
+The best parameters that we found for Ours-A and Ours-B are in Table 2. Additionally, the first few terms of the
+sequences are shown in Table 1.
+For array size 200, we have found that Ours-B10000-Comp outperforms all other tested sequences in terms of number
+of comparisons, as shown in Table 3. Figure 1 is a graphical aid to this table. These graphs show that sequences
+generated by template functions can still perform well for array sizes larger, but not significantly so, than what they
+were optimized for. Furthermore, we found that both Ours-A128-Comp and Ours-A1000-Comp met the number of
+comparisons of, but did not surpass, the Ciura and Tokuda sequences. It’s interesting to note that several of the initial
+terms are equivalent between Ciura-128 and Ours-A128-Comp.
+We also test medium and large arrays, with results shown in Tables 4 and 5. For a graphical representation of Table 5,
+see Figure 1 in the Appendix. Our new sequences approach the performance of Ciura sequences without surpassing
+them. However, Ours-B10000 still surpasses the Tokuda sequence for all array sizes that we have tested here. Recall
+that the Tokuda sequence is currently the best known sequence to be generated from a function. Therefore, we have
+shown a new function-based sequence that outperforms other function-based sequences in terms of the number of
+comparisons. As mentioned previously, this is particularly useful if comparisons are a dominant operation such as
+when sorting large satellite data.
+On the other hand, our experiments with optimizing sequences to minimize running time are shown in Table 6. The
+relevant sequence, Ours-A1000-Time, takes a similar running time to the Ciura-1000 sequence. Both are faster than
+any other tested sequence. The sequence Ours-A1000-Time is particularly interesting because its first few elements (1
+3 7) are different than most other fast sequences (typically starting with 1 4 9). This difference may imply that while
+sequences beginning with (1 4 9) may be very good at minimizing number of comparisons, it does not guarantee that
+they have a good overall running time.
+5
+Conclusion
+Improvements for Shellsort traditionally come from finding gap sequences with better theoretical properties. Here
+we introduced an experimental framework to find improved gap sequences, following in the footsteps of [4]. Our
+generated gap sequences outperformed all well-known gap sequences in terms of number of comparisons on prescribed
+array sizes. Furthermore, the sequence Ours-A1000-Time is, to our knowledge, the function-based sequence with the
+quickest running time. However, it meets the performance of the Ciura sequence but does not surpass it. This may be
+improved with different sequence-generating functions or experimental setup, which we leave to future work. While
+the sequences presented here were optimized for chosen array sizes, the optimization may be repeated for any array
+size of interest.
+References
+[1] D. L. Shell, “A high-speed sorting procedure,” Communications of the ACM, vol. 2, no. 7, pp. 30–32, 1959.
+[2] M. T. Goodrich, “Randomized shellsort: A simple data-oblivious sorting algorithm,” Journal of the ACM (JACM), vol. 58,
+no. 6, pp. 1–26, 2011.
+[3] J.-W. Lee, Y.-S. Kim, and J.-S. No, “Analysis of modified shell sort for fully homomorphic encryption,” IEEE Access, vol. 9,
+pp. 126198–126215, 2021.
+[4] M. Ciura, “Best increments for the average case of shellsort,” in International Symposium on Fundamentals of Computation
+Theory, pp. 106–117, Springer, 2001.
+[5] R. Sedgewick, “Analysis of shellsort and related algorithms,” in European Symposium on Algorithms, pp. 1–11, Springer,
+1996.
+[6] N. Tokuda, “An improved shellsort,” in Proceedings of the IFIP 12th World Computer Congress on Algorithms, Software,
+Architecture-Information Processing’92, Volume 1-Volume I, pp. 449–457, 1992.
+[7] T. N. Hibbard, “An empirical study of minimal storage sorting,” Communications of the ACM, vol. 6, no. 5, pp. 206–213,
+1963.
+[8] V. R. Pratt, “Shellsort and sorting networks,” tech. rep., STANFORD UNIV CA DEPT OF COMPUTER SCIENCE, 1972.
+[9] E. S. Selmer, “On the linear diophantine problem of Frobenius,” 1977.
+6
+
+[10] J. Incerpi and R. Sedgewick, “Improved upper bounds on shellsort,” Journal of Computer and System Sciences, vol. 31, no. 2,
+pp. 210–224, 1985.
+[11] E. S. Selmer, “On shellsort and the Frobenius problem,” 1987.
+7
+
diff --git a/BtAyT4oBgHgl3EQfePgk/content/tmp_files/load_file.txt b/BtAyT4oBgHgl3EQfePgk/content/tmp_files/load_file.txt
new file mode 100644
index 0000000000000000000000000000000000000000..1773ba1ef9d924524af4e5bac6903097f4f0633b
--- /dev/null
+++ b/BtAyT4oBgHgl3EQfePgk/content/tmp_files/load_file.txt
@@ -0,0 +1,508 @@
+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf,len=507
+page_content='OPTIMIZATION PERSPECTIVES ON SHELLSORT  Oscar Skean  Department of Computer Science  University of Kentucky  oscar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content='skean@uky.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content='edu  Richard Ehrenborg  Department of Mathematics  University of Kentucky  Jerzy W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content=' Jaromczyk  Department of Computer Science  University of Kentucky  ABSTRACT  Shellsort is a sorting method that is attractive due to its simplicity, yet it takes effort to analyze its  efficiency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content=' The heart of the algorithm is the gap sequence chosen a priori and used during sorting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content='  The selection of this gap sequence affects the efficiency of Shellsort, and thus drives both its theo-  retical and experimental analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content=' We contribute to Shellsort by identifying efficient gap sequences  based on new parameterized functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content=' Specifically, a parameter grid-search identifies optimal pa-  rameters for different input sizes for sorting by observing minimal overhead in three categories:  number of comparisons, number of exchanges, and running time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content=' We report that our method finds  sequences that outperform state-of-the-art gap sequences concerning the number of comparisons for  chosen small array sizes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content=' Additionally, our function-based sequences outperform the running time  of the Tokuda sequences for chosen large array sizes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content=' However, no substantial improvements were  observed when minimizing the number of exchanges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content='  1  Introduction  The Shellsort algorithm is a sorting method that was among the first to be discovered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content=' Published in 1959 [1], it saw  early interest due to its low memory requirements and simple implementation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content=' Despite this, its analysis is difficult and  remains incomplete.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content=' The algorithm has found practical use today in memory-constrained environments, embedded  systems, and the bzip2 compressor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content=' Recently it has also found use in data-oblivious sorting [2] and in fully homomor-  phic encryption [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content='  Shellsort is an in-place comparison sort and can be viewed as a generalization of insertion sort.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content=' For a data array  A of size N, Shellsort operates using a predetermined gap sequence 1 = k1 < · · · < km < N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content=' The algorithm  performs m passes over A: starting with the largest km and ending with k1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content=' During pass j, for a given gap km−j,  insertion sort occurs for the km−j subarrays consisting of the data elements A(i), A(i + km−j), A(i + 2 · km−j), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content='  for i = 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content=' , km−j − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content=' We say a k-inversion is a pair (i, i + k) such that the inequality A(i) > A(i + k) holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content='  After pass j, all km−j-inversions that were originally in A have been solved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content=' We say an array with no k-inversions  is k-sorted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content=' Note that the final pass with k1 is equivalent to insertion sort and is necessary to guarantee sortedness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content='  Therefore, the purpose of the gap sequence is to presort A as much as possible before the expensive final insertion sort  pass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content='  The main results of this paper are:  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content=' New efficient Shellsort sequences derived from experimentally optimizing sequence-generating functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content='  For prescribed array sizes, these sequences outperform well-known efficient sequences (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content=' Tokuda, Ciura)  with respect to the number of comparisons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content=' These sequences also outperform the running time of the Tokuda  sequence, making them the fastest function-based sequence on the tested array sizes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content=' We demonstrate results of experimental analysis comparing our proposed approaches with well-known se-  quences by measuring the number of comparisons, exchange operations, and running time needed to sort  randomized permutations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content='  Traditionally, improvements for Shellsort have come from finding gap sequences with theoretical properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content=' We  discuss some particularly important sequences in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content=' Then in Section 3, we introduce parameterized sequence-  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content='00316v1  [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content='DS]  1 Jan 2023  generating functions that generate a Shellsort sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content=' The parameters are then optimized in a grid-search finding  the best possible sequence that can be produced from that function for a chosen array size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content=' In Section 4 we discuss our  experimental methodology to compare the performance of the optimized template sequences to the baseline sequences  mentioned in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content='  2  Background  The selection of a good gap sequence is critical to the performance of Shellsort.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content=' There has been a plethora of work  focused on selecting good sequences [4, 5, 6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content=' Some of the earliest proposed sequences were based on powers of  2 [7, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content=' Then Pratt showed that the sequence of 2p3q obtains a number of inversions that is Θ(Nlog2N) in the  worst case [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content=' We call this sequence Pratt-23 in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content=' This sequence still has the best known asymptotic time  complexity for any Shellsort sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content=' However, it has a very large constant factor which spurred the development of  new sequences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content='  The proof technique used to show the time complexity of Pratt-23 was based on counting the inversions of a sequence  that has already been 2-sorted and 3-sorted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content=' A natural extension of this is to apply the Frobenius problem to place  bounds on what has already been sorted in prior passes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content=' A typical formulation of the Frobenius problem is as follows:  Suppose that you have k coins of denominations u1, u2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content=' , uk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content=' What is the largest value which cannot be made  with a nonnegative linear combination of these coins?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content=' This largest value is known as the Frobenius number [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content=' In  the context of Shellsort, the coins can be equated to gap size and the Frobenius number can be equated to the largest  remaining inversion after sorting with the gaps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content=' Using the Frobenius problem, several sequences were proposed that  had a lower constant factor than Pratt-23 despite having a worse time-complexity [10, 11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content='  Following those sequences, the focus of gap sequence selection shifted from finding theoretically good sequences  to finding experimentally good ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content=' For example, one property that was observed was that a geometric sequence  with a growth of 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content='25 often performed well in practice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content=' This observation was the basis of the Tokuda sequence [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content='  See Table 1 for a functional form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content=' To the best of our knowledge, this remains the most competitive function in the  literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content='  The next improvement came from Ciura in 2002 [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content=' The Ciura sequence disregarded the idea of a function-based  sequence, and instead searched for the best set of gap elements themselves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content=' Ciura found the best sequence for array  sizes of 128 and 1000, as well as a sequence that was conjectured to perform better for much larger array sizes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content=' We  call these Ciura-128, Ciura-1000, and Ciura-Large in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content='  The Ciura sequences also marked a transition in how Shellsort performance was measured.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content=' Previously, most works  counted the number of exchanges used by the algorithm [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content=' This partly because some proof techniques relied on  counting the number of inversions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content=' Ciura instead focused on optimizing the number of comparisons, which was found  to be more directly related to the computation time of Shellsort.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content=' A comparison is defined as checking if a pair of array  elements are inverted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content=' An exchange is typically defined as the variable swap used to fix an inversion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content=' Minimizing the  number of comparisons is especially beneficial when comparisons are expensive to make, such as when sorting large  satellite data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content=' Similarly, minimizing exchanges is beneficial in memory-constrained systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content=' In this work, we make  clear distinctions in Section 4 about what measurement we’re optimizing for.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content=' For a full treatment of the history of gap  sequences, we point the reader to [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content='  3  Parameterized Template Functions  The approach of directly optimizing the gap sequence, as in [4], grows in computational cost very quickly as N  increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content=' This growth is due the fact that as N increases, both the expected number of sequence elements and their  possible range of values increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content=' To help alleviate this, the authors of [4] found a suitable sequence prefix and  optimize the extension of it by a few values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content=' However, at very large N even finding this prefix would be very costly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content='  Here, we formulate the problem of finding the optimal sequence as optimizing the parameters of a pre-defined  sequence-generating function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content=' Guided by principles that we have seen in sequences that perform well, we define  two functions as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content='  kA(i) = ⌊(a⌊ i  b ⌋ · c⌊ i  d ⌋)f + e⌋  (1)  kB(i) = ⌊(a · b⌊ i  c ⌋) + d⌋  (2)  We refer to (1) as Ours-A and (2) as Ours-B in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content=' Both formats contain the floor function in exponents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content=' We  found this to allow the function to express a more ”chaotic” sequence, which helped improve performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content=' The unique  characteristic of Ours-A is the parameter f, an exponent that helps regulate growth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content=' The parameter a of Ours-B was  2  Sequence Name  Function  Optimized for N  Parameters  Initial Terms  Ciura [4] 128 1 4 9 24 85 126 1000 1 4 10 23 57 156 409 995 Large 1 4 10 23 57 132 301 701 1750  Tokuda [6]  ⌈ (9/4)k−1  (9/4)−1 ⌉ 1 4 9 20 46 103 233 525 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content='  Ours A  ⌊(a⌊ i  b ⌋ · c⌊ i  d ⌋)f + e⌋  128 (Comp)  Table 2  1 4 9 24 85 150 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content='  1000 (Comp)  Table 2  1 4 10 23 57 153 400 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content='  1000 (Time)  Table 2  1 3 7 16 33 85 179 472 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content='  Ours B  ⌊(a · b⌊ i  c ⌋) + d⌋  10000 (Comp)  Table 2  1 4 10 27 72 187 488 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content='  Pratt-23 [8]  Ordered 2p · 3q 1 2 3 4 6 8 9 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content='  Pratt-25  Ordered 2p · 5q 1 2 4 5 8 10 15 16 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content='  Pratt-34  Ordered 3p · 4q 1 3 4 9 12 16 24 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content='  Table 1: Gap sequences that are compared during experiments  Template  a  b  c  d  e  f  Ours-A128-Comp  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content='6321  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content='6841  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content='1570  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content='7360  3  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content='7630  Ours-A1000-Comp  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content='5789  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content='6316  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content='8158  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content='1579  3  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content='7632  Ours-A1000-Time  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content='75  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content='75  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content='7142  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content='4286  2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content='7429  Ours-B10000-Comp  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content='0816  8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content='5714  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content='2449  0 Table 2: Optimized parameters for template functions  designed to have a similar purpose, albeit via multiplication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content=' Ours-B contains fewer parameters which allows for  quicker optimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content=' For conciseness, we only optimize Ours-A for array sizes 128 and 1000, and Ours-B for 10000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content='  Furthermore, we denote as Ours-A1000-Comp if optimizing Format A for number of comparisons on arrays of size  1000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content=' In the following section, we discuss the experimental procedure for optimizing (1) and (2), as well as for  comparing them to other baseline sequences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content='  4  Experimental Procedure  Because the function is highly non-convex, it is difficult to utilize efficient techniques such as gradient descent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content=' In-  stead, we employ a grid-search approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content=' One benefit of using a function grid-search, as opposed to direct sequence  optimization, is that the size of the search space has no relation to N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content=' It depends only on the granularity and bounds  of the search.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content=' This is in constrast to the methodology used by Ciura, in which the number of tested sequences grows  with N [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content='  For Ours-A, we define the grid for parameters a, b, c, d, f as 20 linearly spaced values between 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content='5 and 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content=' We also  allow e to be an integer value between 0 and 10, including both endpoints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content=' Because Ours-B has fewer parameters,  we can take a more fine-grained approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content=' For parameters a, b, c, we test 50 linearly spaced values from 0 to 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content=' For  parameter d, we constrain it to be the same as e in Ours-A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content='  The data array that we test on contains N distinct values 1 through N, and we shuffle it with the Fischer-Yates shuffle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content='  For each set of parameters in the grid-space, we compute the mean cost over 1000 iterations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content=' We then take the set of  parameters producing the lowest mean cost as optimal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content=' This cost can be defined as number of comparisons, number  of exchanges, or time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content='  Because Ciura sequences are optimized for specific array sizes with no means of extending them, it would be inap-  propriate to directly compare them with any function-based sequence at large array sizes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content=' One method of extending a  Ciura sequence is by starting a geometric series on its last term with a ratio of 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content='25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content=' We adopt this method of extension  when measuring the performance of Ciura sequences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content='  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content='3  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content='Sequence  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content='N=20  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content='N=128  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content='N=200  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content='µCO  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content='µEX  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content='µCO  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content='µEX  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content='µCO  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content='µEX  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content='Ours-A128-Comp  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content='76 ± 6  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content='38 ± 6  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content='998 ± 33  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content='531 ± 33  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content='1786 ± 46  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content='948 ± 48  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content='Ours-A1000-Comp  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content='76 ± 6  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content='39 ± 7  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content='1004 ± 32  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content='516 ± 31  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content='1787 ± 44  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content='919 ± 45  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content='Ours-A1000-Time  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content='79 ± 5  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content='39 ± 7  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content='1035 ± 26  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content='468 ± 27  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content='1832 ± 38  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content='846 ± 39  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content='Ours-B10000-Comp  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content='76 ± 7  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content='33 ± 5  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content='1096 ± 52  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content='535 ± 36  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content='1775 ± 49  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content='960 ± 49  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content='Ciura-128  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content='76 ± 6  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content='37 ± 6  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content='998 ± 32  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content='531 ± 33  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content='1800 ± 46  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content='970 ± 49  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content='Ciura-1000  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content='76 ± 7  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content='39 ± 7  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content='1006 ± 31  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content='519 ± 34  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content='1787 ± 45  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content='920 ± 44  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content='Ciura-Long  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content='76 ± 7  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content='39 ± 7  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content='1004 ± 32  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content='516 ± 32  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content='1794 ± 44  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content='907 ± 42  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content='Tokuda  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content='76 ± 6  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content='37 ± 6  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content='1020 ± 28  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content='490 ± 28  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content='1808 ± 42  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content='891 ± 43  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content='Pratt-25  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content='111 ± 4  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content='27 ± 4  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content='1732 ± 16  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content='345 ± 17  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content='3207 ± 21  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content='610 ± 24  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content='Pratt-23  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content='136 ± 3  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content='25 ± 4  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content='2209 ± 13  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content='333 ± 15  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content='4095 ± 19  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content='589 ± 21  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content='Pratt-34  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content='95 ± 4  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content='29 ± 4  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content='1424 ± 16  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content='374 ± 19  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content='2593 ± 25  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content='660 ± 26  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content='Table 3: Number of operations to sort small arrays averaged over 1000 random array permutations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content=' µCO denotes the  number of comparisons, µEX denotes the number of exchanges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content='  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content='Sequence  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content='N=1000  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content='N=2000  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content='N=5000  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content='µCO  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content='µEX  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content='µCO  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content='µEX  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content='µCO  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content='µEX  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content='Ours-A128-Comp  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content='13250 ± 203  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content='7847 ± 199  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content='30530 ± 378  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content='18611 ± 384  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content='91122 ± 973  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content='57728 ± 904  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content='Ours-A1000-Comp  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content='12941 ± 167  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content='7004 ± 155  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content='29596 ± 293  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content='16234 ± 282  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content='86821 ± 768  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content='50349 ± 770  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content='Ours-A1000-Time  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content='13193 ± 144  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content='6461 ± 146  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content='30120 ± 263  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content='14913 ± 257  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content='87455 ± 548  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content='44305 ± 552  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content='Ours-B10000-Comp  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content='12980 ± 186  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content='7245 ± 177  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content='29643 ± 305  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content='17241 ± 325  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content='86514 ± 617  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content='57388 ± 817  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content='Ciura-128  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content='13300 ± 166  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content='7003 ± 168  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content='30359 ± 318  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content='15987 ± 310  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content='88193 ± 629  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content='46689 ± 627  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content='Ciura-1000  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content='12918 ± 161  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content='7002 ± 155  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content='29534 ± 282  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content='16138 ± 274  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content='86641 ± 757  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content='47852 ± 751  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content='Ciura-Long  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content='13035 ± 142  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content='6701 ± 149  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content='29567 ± 246  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content='15427 ± 261  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content='86232 ± 502  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content='45347 ± 496  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content='Tokuda  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content='13116 ± 143  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content='6556 ± 142  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content='29888 ± 241  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content='14952 ± 228  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content='86838 ± 454  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content='44116 ± 472  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content='Pratt-25  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content='26211 ± 68  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content='4318 ± 72  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content='62722 ± 122  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content='9755 ± 131  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content='194196 ± 263  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content='28195 ± 278  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content='Pratt-23  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content='34380 ± 64  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content='4253 ± 69  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content='82785 ± 106  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content='9669 ± 116  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content='259088 ± 242  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content='28354 ± 257  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content='Pratt-34  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content='20974 ± 89  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content='4671 ± 87  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content='50038 ± 153  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content='10543 ± 160  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content='154298 ± 372  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content='30448 ± 372  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content='Table 4: Number of operations to sort medium-sized arrays averaged over 1000 random array permutations  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content='1  Filtering  There are two techniques we employ to reduce our grid-search space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content='  First, we notice that different sets of parameters could produce the same sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content=' For example, for the template  function Ours-A, the ordering of (a, b) and (c, d) do not matter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content=' We precalculate all of the sequences produced in the  grid and only experiment on the unique ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content=' For example, for N = 10000, this reduces the grid search space from  over 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content='5 million sets of parameters to about 1 million.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content='  Second, we use sequential analysis to act as a low-pass filter for screening out obviously poor sequences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content=' This statis-  tical approach was first applied to Shellsort in [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content=' Given bounds for the mean and an upper bound for the variance,  sequential analysis is able to tell in just a few repetitions whether or not a sample mean falls below the mean bounds  with a certain confidence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content=' Sequential analysis allows us to quickly accept good gap sequences that have a low mean  number of comparisons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content=' Any sequence that is accepted by the filter is then run for the full 1000 iterations to obtain a  more accurate estimate of the mean.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content=' We adopt the same setup as in [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content='  The Ciura sequences were optimized with respect to the number of comparisons, and because they are well-known to  be some of the most practical sequences, we optimize our template functions with respect to the number of comparisons  as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content='2  Hardware  The experiments were performed on a Ubuntu machine with an 8-core Intel Xeon W-3225.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content=' Experiments counting  number of comparisons and exchanges were multithreaded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content=' Experiments involving measurement of time were done  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content='4  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content='Sequence  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content='N=10000  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content='µCO  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content='µEX  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content='Ours-A128-Comp  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content='206356 ± 1796  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content='132351 ± 1797  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content='Ours-A1000-Comp  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content='196336 ± 1707  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content='119012 ± 1710  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content='Ours-A1000-Time  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content='194052 ± 879  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content='98952 ± 883  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content='Ours-B10000-Comp  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content='192029 ± 992  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content='209292 ± 1293  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content='Ciura-128  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content='195256 ± 1106  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content='105544 ± 1109  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content='Ciura-1000  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content='193778 ± 1895  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content='111338 ± 1897  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content='Ciura-Long  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content='191435 ± 892  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content='101680 ± 897  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content='Tokuda  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content='192574 ± 795  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content='98071 ± 796  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content='Pratt-25  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content='450131 ± 516  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content='62191 ± 526  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content='Pratt-23  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content='604502 ± 451  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content='66923 ± 725  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content='Pratt-34  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content='355382 ± 723  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content='63272 ± 462  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content='Table 5: Number of operations to sort an array size of 10000 averaged over 1000 random array permutations  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content='Sequence  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content='Running Time (ms)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content='Ours-A128-Comp  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content='15 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content='08  Ours-A1000-Comp  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content='02 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content='06  Ours-A1000-Time  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content='01 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content='06  Ours-B10000-Comp  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content='04 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content='07  Ciura-128  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content='07 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content='06  Ciura-1000  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content='01 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content='06  Ciura-Long  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content='04 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content='07  Tokuda  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content='06 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content='08  Pratt-25  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content='00 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content='09  Pratt-23  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content='35 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content='11  Pratt-34  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content='17 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content='08  Table 6: Time to sort an array size of 1000 averaged over 1000 random array permutations  Figure 1: (Left) For varying array sizes, shows the difference in number of comparisons between baseline sequences  and Ours-A128.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content=' A positive value means Ours-A128 uses fewer comparisons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content=' (Right) Number of comparisons for  varying array sizes larger than what Ours-A128 was optimized for.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content='  5  Difference inMeanNumberof Comparisons  35  Ciura-128  Ciura-Large  30  Tokuda  25  differences  20  15  10  5  0  25  50  75  100  125  150  175  200  Array SizeMeanNumberofComparisonsvsArraySize  1800  Ours-A128  Ciura-128  Ciura-Large  1700  Tokuda  NumberofComparisons  1600  1500  1400  1300  1200  150  160  170  180  190  200  Array Sizesingle threaded, as any multithreaded applications could cause discrepancies in time measurement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content=' All code was  written in Python.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content='3  Results  The best parameters that we found for Ours-A and Ours-B are in Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content=' Additionally, the first few terms of the  sequences are shown in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content='  For array size 200, we have found that Ours-B10000-Comp outperforms all other tested sequences in terms of number  of comparisons, as shown in Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content=' Figure 1 is a graphical aid to this table.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content=' These graphs show that sequences  generated by template functions can still perform well for array sizes larger, but not significantly so, than what they  were optimized for.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content=' Furthermore, we found that both Ours-A128-Comp and Ours-A1000-Comp met the number of  comparisons of, but did not surpass, the Ciura and Tokuda sequences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content=' It’s interesting to note that several of the initial  terms are equivalent between Ciura-128 and Ours-A128-Comp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content='  We also test medium and large arrays, with results shown in Tables 4 and 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content=' For a graphical representation of Table 5,  see Figure 1 in the Appendix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content=' Our new sequences approach the performance of Ciura sequences without surpassing  them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content=' However, Ours-B10000 still surpasses the Tokuda sequence for all array sizes that we have tested here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content=' Recall  that the Tokuda sequence is currently the best known sequence to be generated from a function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content=' Therefore, we have  shown a new function-based sequence that outperforms other function-based sequences in terms of the number of  comparisons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content=' As mentioned previously, this is particularly useful if comparisons are a dominant operation such as  when sorting large satellite data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content='  On the other hand, our experiments with optimizing sequences to minimize running time are shown in Table 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content=' The  relevant sequence, Ours-A1000-Time, takes a similar running time to the Ciura-1000 sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content=' Both are faster than  any other tested sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content=' The sequence Ours-A1000-Time is particularly interesting because its first few elements (1  3 7) are different than most other fast sequences (typically starting with 1 4 9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content=' This difference may imply that while  sequences beginning with (1 4 9) may be very good at minimizing number of comparisons, it does not guarantee that  they have a good overall running time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content='  5  Conclusion  Improvements for Shellsort traditionally come from finding gap sequences with better theoretical properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content=' Here  we introduced an experimental framework to find improved gap sequences, following in the footsteps of [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content=' Our  generated gap sequences outperformed all well-known gap sequences in terms of number of comparisons on prescribed  array sizes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content=' Furthermore, the sequence Ours-A1000-Time is, to our knowledge, the function-based sequence with the  quickest running time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content=' However, it meets the performance of the Ciura sequence but does not surpass it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content=' This may be  improved with different sequence-generating functions or experimental setup, which we leave to future work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content=' While  the sequences presented here were optimized for chosen array sizes, the optimization may be repeated for any array  size of interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
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+Bifurcation instructed design of multistate machines
+Teaya Yang,1 David Hathcock,1 Yuchao Chen,1 Paul
+McEuen,2 James P. Sethna,1 Itai Cohen,2 and Itay Griniasty1
+1Laboratory of Atomic and Solid State Physics,
+Cornell University, Ithaca, New York 14853-2501, USA
+2Laboratory of Atomic and Solid State Physics,
+Cornell University, Ithaca, New York 14853-2501,
+USA and Kavli Institute at Cornell for Nanoscale Science, Cornell University, Ithaca, NY, USA
+(Dated: January 5, 2023)
+We propose a novel design paradigm for multi-state machines where transitions from one state
+to another are organized by bifurcations of multiple equilibria of the energy landscape describing
+the collective interactions of the machine components. This design paradigm is attractive since,
+near bifurcations, small variations in a few control parameters can result in large changes to the
+system’s state providing an emergent lever mechanism. Further, the topological configuration of
+transitions between states near such bifurcations ensures robust operation, making the machine less
+sensitive to fabrication errors and noise. To design such machines, we develop and implement a
+new efficient algorithm that searches for interactions between the machine components that give
+rise to energy landscapes with these bifurcation structures. We demonstrate a proof of concept for
+this approach by designing magneto elastic machines whose motions are primarily guided by their
+magnetic energy landscapes and show that by operating near bifurcations we can achieve multiple
+transition pathways between states. This proof of concept demonstration illustrates the power of
+this approach, which could be especially useful for soft robotics and at the microscale where typical
+macroscale designs are difficult to implement.
+Systems composed of a large number of interact-
+ing elements such as meta-materials, elastic mem-
+branes, and proteins can exhibit emergent behaviors
+that arise from the collaborative interaction of the
+system components. Designing functionality in such
+systems is a formidable task that requires searches
+in a high dimensional parameter space of the sys-
+tem components and their interactions. Developing
+organizing principles for effectively designing such
+systems remains an outstanding problem in the field
+[1–6]. Here, we propose that designing multi-state
+machines around bifurcations of multiple equilibria
+is a powerful paradigm that can be used to system-
+atically organize such searches.
+Bifurcations, where a single equilibrium configu-
+ration splits into multiple equilibria as a function of
+a control parameter is a canonical dynamical sys-
+tems structure that has been used to explain vari-
+ous natural phenomena ranging from phase transi-
+tions [7] to the operation of simple machines. Exam-
+ples of simple machines include Venus flytraps and
+hummingbird beaks that have been shown to open
+smoothly and then snap shut by operating about
+a cusp bifurcation where three equilibria converge
+[8]. Designing systems to operate near such bifur-
+cations provides several advantages. Since the split-
+ting of the equilibria has a power law dependence
+on the control parameters [4, 7], operating near bi-
+furcations automatically provides a lever mechanism
+by which small variations in the control parameters
+lead to large changes in the system state [12, 13].
+In the case of the Venus fly trap, slight changes in
+hydrostatic pressure can drive large motions of the
+trap. Similarly in hummingbirds, slight twisting of
+the jaw bones enables rapid closing of a wide open
+beak.
+Further, such bifurcations organize a topo-
+logically protected structure of saddle node mani-
+folds. As such, provided that the system trajectory
+encircles the cusp bifurcation where the saddle node
+manifolds meet, the system is guaranteed to exhibit
+a smooth change in state followed by a snap.
+In
+the Venus fly trap and hummingbird examples, this
+topological protection guarantees that the opening
+and snapping of the trap or beak is robust against
+variations in the applied hydrostatic or muscle forces
+driving the transitions in the system state.
+Here,
+we propose that moving beyond cusp bifurcations
+to design systems that operate near bifurcations of
+arbitrarily many equilibria preserves the lever ad-
+vantage and topological protection of cusp bifurca-
+tions.
+Such systems can be driven by only a few
+control parameters to undergo snapping transitions
+between multiple states making the design of ma-
+chines near such bifurcations a powerful paradigm
+for organizing complex functions. To develop and
+demonstrate this paradigm, we experimentally in-
+vestigate increasingly sophisticated magneto elastic
+machines whose function is organized by such bifur-
+cations.
+We start by constructing a simple magneto elas-
+tic machine consisting of a control panel that can
+be translated in the x − y plane and a second panel
+arXiv:2301.01507v1  [cond-mat.soft]  4 Jan 2023
+
+2
+FIG. 1.
+Magneto-Elastic
+machine
+capable
+of
+adopting multiple configurations due to operat-
+ing near a cusp bifurcation (a.) System: Panels P1
+and P2 are decorated with identical magnets. Panel P1 is
+actuated externally to translate in the x and y directions,
+in response Panel P2 rotates about a hinge, the dynam-
+ics are over-damped.
+(b.)
+Magnetic potential energy
+landscapes: We plot the potential for dx = −0.09 and
+dy ∈ [0.28, 0.17, 0.02], where dx and dy are deviations
+from the cusp’s position. Varying y we cross two saddle
+node bifurcations where the number of extrema of the
+magneto elastic landscape changes. (c.)
+Equilibrium
+manifold: The system’s equilibria θ(dx, dy) are plotted
+as a function of the deviation of the parameters, color
+signifies the value of θ. Brown curve marks saddle node
+bifurcations where the number of equilibria change, and
+the light red curve denotes the experimental trajectory.
+(d.) Experimentally observed snap-through transition:
+The system follows a parametric trajectory marked by a
+red curve and colored tube whose color denotes the pre-
+dicted state θ, around a cusp bifurcation. The colored
+disks represent the experimentally measured state θ. As
+expected a single snap through transition at a saddle
+node bifurcation (curves colored according to the bifur-
+cating state θ and converging at the cusp) is observed.
+that is free to rotate about a hinge connecting the
+two panels (Fig. 1a and experimental apparatus
+schematic Fig.S1). The state of the system, is given
+by the angle θ between the panels. By decorating
+the panels with magnets, we are able to design a
+magneto elastic landscape with different numbers of
+minima as a function of the parameters x and y
+(Fig. 1b).
+Transitions between these minima cor-
+respond to changes in the state of the system. To
+understand the various pathways for making such
+transitions we construct the manifold defined by the
+local equilibria as a function of the parameters x and
+y. For this particular arrangement of magnets, we
+calculate (see SI) that the resulting manifold has a
+domain with multiple solutions delineated by saddle
+node bifurcation curves (Brown). These curves in-
+tersect and terminate at a cusp bifurcation beyond
+which there is only a single equilibrium state. By
+translating the control panel in the x-y plane, the
+system can undergo either smooth or abrupt changes
+in θ. For example, starting the system at point (i)
+and moving through points (ii-v), the hinge angle
+increases smoothly. A further slight increase in the
+control parameter y, however, leads to an abrupt
+transition from a high to a low angle, correspond-
+ing to points (v) and (vi) respectively. These pre-
+dictions are born out by the experiments (Fig. 1d),
+which also show a smooth increase in θ for a path-
+way that encircles the cusp (i-v) and an abrupt tran-
+sition in θ when crossing a saddle node curve (v-vi).
+In this 2D representation the system makes a transi-
+tion when the color of the path (yellow) matches the
+color denoting the state associated with the saddle
+node curve (yellow). This magneto elastic mecha-
+nism is reminiscent of the cocking and snapping of
+a Venus flytrap or a humming bird’s beak.
+In addition to providing a mechanism for abrupt
+transitions, operating near a cusp bifurcation creates
+a lever mechanism where small variations in the con-
+trol parameters lead to large variations in the system
+state. This mechanism resolves the generic problem
+that creating large variations in the system state of-
+ten requires unfeasibly large variations in the control
+parameters. Lever mechanisms are generic near bi-
+furcations of equilibria since the magnitude of the
+transition in the system state is typically propor-
+tional to the square root of the parameter distance
+from the bifurcation.
+To characterize this lever mechanism in our exper-
+iment, we map the snapping transition curves asso-
+ciated with the saddle node bifurcations.
+Specifi-
+cally, for a given value of y (or x) we toggle x (y)
+so that the system snaps back and forth, and record
+the values of the control parameters x and y, and θ
+immediately after each transition (Fig. 2a).
+To test the scaling relations, we define the normal
+
+a)
+HingeAxis
+N
+s
+Parameter
+SystemState
+X-ControlParameter
+b)
+[nrun qe] A
+L
+2.2
+2.952.2
+2.95
+2.2
+2.95
+e[rad]-SystemState
+c)
+111
+2.9
+ [rad] - System State
+Snap-through
+S
+2.5
+d)
+LS
+Cusp Point
+0.03
+0.3
+0.15
+-0.06
+0
+dy [cm]-Control Parameter
+-0.15
+-0.093
+FIG. 2.
+Parametric levers The change in the state
+of the system after a snap through transition near a
+cusp bifurcation scales sub linearly with the normal
+form parameters. This sublinear scaling leads to large
+variation of the state in response to small variation
+of the system parameters.
+a) Measurements of snap
+through transitions near a cusp: The blue points mark
+the state of the magneto elastic system of Fig. 1 af-
+ter a snap through transition.
+The dashed curve is a
+fit of snap through transitions near a cusp bifurcation
+to the data, derived from the normal form potential
+˜V = δθ4 + a2δθ2 + a1δθ.
+The normal form parame-
+ters a1 and a2 are locally given by re-scaled rotations
+of dx and dy, which are the deviations of the parame-
+ters away from the cusp. b) Scaling laws near a cusp:
+The predicted scaling laws are demonstrated by project-
+ing the measurements and fit onto log-log plots. Near
+the cusp the system response to a1 acts as a giant lever,
+∂δθ/∂a1 ∼ 50.
+form parameters a1 and a2 as rotations of the dis-
+placement of the parameters x and y from the cusp.
+We then fit the predicted scaling form ∆θ ∝ √a2
+and a1 ∝ a3/2
+2
+near a cusp to determine the cusp’s
+position and the rotation of the normal form param-
+eters. The fitted model then predicts that ∆θ ∝ a1/3
+1
+(see SI). Because the scaling exponents for ∆θ are
+fractions of unity, small variations of the parameters
+along a1 and a2 lead to large variations of the sys-
+tem’s state.
+For example, in our experiments the
+range of actuation for panel 1’s position is approx-
+imately 1 cm and the range of angles accessible to
+panel 2 is 180◦ or π radians. Near the bifurcation
+a translation along a1 of 0.1% of its range (∼ 10µ
+m) leads to a snap that changes θ by ∼ 5% of it
+range (∼ 0.1 rad) providing a lever advantage of
+∼50 (Fig. 2b ).
+The complexity of the actions achieved by such
+magneto elastic mechanisms is dictated by the range
+and number of stable states that the system can ac-
+cess. This complexity can be achieved by designing
+the magneto elastic potentials such that the system
+operates near bifurcations between multiple states.
+For example, working near a hypothetical symmetric
+butterfly bifurcation associated with the potential
+V = θ6 + a4θ4 + a2θ2 + a1θ should enable smooth
+and abrupt transitions between three stable states
+in any order depending on the chosen trajectory for
+the control parameters. In Fig. 3a we show a cut
+through parameter space of the saddle node surfaces
+near this butterfly bifurcation. If the system starts
+in the S (Small) state and moves along the depicted
+trajectory (black arrows), it would first snap to the
+M (Medium) state when the system crosses the pur-
+ple saddle node bifurcation and then the L (Large)
+state when it crosses the green curve. For the return
+path, however, the system would transition from
+the L minimum directly to the S minimum when
+it crosses the yellow saddle node bifurcation curve.
+Moreover, by working near the bifurcation, the lever
+mechanism should allow for transitioning between
+these distinct states within an accessible range of
+experimental control parameters.
+SEARCH ALGORITHM FOR
+BIFURCATIONS OF MULTIPLE
+EQUILIBRIA
+To design parametric configurations correspond-
+ing to bifurcations of multiple equilibria we develop
+a search gradient continuation algorithm that takes
+advantage of their nested structure. Bifurcations as-
+sociated with k equilibria (minima plus maxima) are
+degenerate singularities where the first k derivatives
+of the potential vanish. Thus they can be found iter-
+atively by searching for singularities of the potential
+with increasing order, solving for one constraint at a
+time. We find that this method is especially efficient
+in finding experimentally realizable parametric con-
+figurations corresponding to bifurcations of multiple
+equilibria. Moreover, this method naturally extends
+to searching for bifurcations with desired properties
+by introducing further constraints, for example opti-
+mizing the robustness of the bifurcation’s associated
+states to external noise.
+For ease of illustration we describe how to use this
+approach to find the symmetrized butterfly bifurca-
+tion described above with parameters a1, a2, a4 and
+variable θ. For a random combination of parame-
+ters we find an equilibrium angle where dV/dθ = 0.
+Generically, this point is part of a smooth man-
+ifold over which this constraint holds.
+We then
+vary a1, a2, a4 and θ within this manifold to min-
+imize the next constraint |d2V/dθ2|.
+The trajec-
+tory follows the gradient of the second constraint as
+closely as possible while maintaining the first con-
+straint dV/dθ = 0 until we reach a point on a sad-
+
+a)
+b)
+0.09 h
+ [rad]
+System State
+0.05
+0.06
+0.04
+0
+0.05
+0.1
+0.18
+-0.05
+a2 [cm] - Normal Form Parameter
+] - System State
+0.09
+0
+- 0.02
+dx [cm]
+0.05.
+0.06
+ControlParameter
+73
+0.1
+80 [rad]
+0.04
+0.15
+1×10-4
+3×10-4
+0.001
+dy [cm] - Control Parameter
+a,[cm] - Normal Form Parameter4
+dle node surface, which is a manifold where both
+the first and second constraints hold.1 Minimizing
+the third derivative within the saddle node manifold
+maintains the first two constraints and allows for
+finding a cusp bifurcation associated with two sta-
+ble equilibria. Successive iterations allow for identi-
+fying bifurcations between an increasing number of
+equilibria and eventually the butterfly bifurcation.
+Our gradient continuation algorithm adapts stan-
+dard algorithms from the dynamical systems liter-
+ature [1, 2, 15] and retools them to locally follow
+the gradient of the unsatisfied constraint (see SI for
+further details). We depict the resulting search path
+in Fig. 3b, which highlights the fact that, indepen-
+dent of the number of parameters, the search algo-
+rithm follows a 1D trajectory, which is organized
+by the nested structure of the intermediate bifurca-
+tions. These properties enable the algorithm to find
+realizable bifurcations for systems with hundreds of
+parameters.
+THREE STATES AND THE BUTTERFLY
+BIFURCATION
+As a proof of concept for our approach we demon-
+strate the construction and operation of a magneto
+elastic machine with 3 stable states operating near
+a bifurcation of multiple equilibria. The first step in
+designing such a machine is to implement our gra-
+dient continuation algorithm to design a magneto
+elastic potential with a butterfly bifurcation between
+three stable states. To realize a system operating
+near such a bifurcation where only three control pa-
+rameters (x,y,z positions of panel 1) are actively var-
+ied, we allowed the algorithm to also determine the
+x,y positions of two of the nine magnets on panel
+1.2 With these seven parameters, the algorithm was
+able to identify multiple butterfly bifurcations that
+satisfied these criteria (See SI for details).
+Having found an appropriate butterfly bifurca-
+tion, we use standard dynamical systems continua-
+tion algorithms[1, 17] to compute and plot the saddle
+node surfaces in the control parameter space (x,y,z)
+near the bifurcation (Fig. 4). We find multiple dis-
+tinct surfaces where the color denotes the angle θ
+1 A local minimum of |∂2
+θV | with respect to variation of all
+parameters, that lies on the fixed point manifold will throw
+the algorithm off, but this is a co-dimension m point for a
+system with m parameters, and so highly unlikely.
+2 Typically, a butterfly bifurcation requires four control pa-
+rameters to navigate between all of the stable states. Here,
+we have identified a nonlinear mapping of the three active
+control parameters (x,y,z) onto the four dimensional space,
+which enables transitions between arbitrary minima.
+FIG. 3.
+Bifurcations of multiple equilibria.
+a)
+Work cycle near a butterfly: A system operating near a
+hypothetical symmetrized butterfly bifurcation can cycle
+between three states. The bifurcation is associated with
+a potential V = θ6+a4θ4+a2θ2+a1θ and three accessible
+states denoted by large (L), medium (M) and small (S).
+As the system follows the trajectory denoted by black ar-
+rows with colored background marking its state θ, it cy-
+cles between the three states snapping from S to M to L
+and back to S by changing a2 and a1 while a4 = 0.1. The
+snaps occur at saddle node bifurcations (colored curves)
+whose color signifies the state θ of the minima that is an-
+nihilated at each boundary. b) Gradient Continuation
+algorithm: The search algorithm finds bifurcations of
+multiple equilibria by following a one dimensional curve.
+Starting from a bifurcation of k equilibria the algorithm
+searches for a bifurcation of k+1 equilibria by following a
+curve in the augmented parameter space, tangent to the
+gradient of |V k+1| in the kth bifurcation manifold. We
+draw a search for a butterfly bifurcation in its symmet-
+ric normal form potential.
+The entire volume denotes
+the equilibrium manifold. Starting from a fixed point,
+the algorithm finds a saddle node bifurcation (along the
+white curve), Parameters are then varied on the saddle
+node surface (yellow), and cusp surface (thin lines) to re-
+spectively find a cusp bifurcation (along the gray curve)
+and a swallow tail bifurcation (along the black curve)
+near a butterfly bifurcation (black point).
+at which the saddle node bifurcation occurs3. In-
+structed by these surfaces, we design a cyclic path
+3 There is further local data in the potential at a saddle node
+
+D[a.u.]
+a1
+1.0
+L
+s
+0.5
+SL
+SM
+M
+0
+M
+SML
+12
+SL
+SL
+-0.5
+L
+-1.0
+Butterfly
+ Saddle node
+Cusp
+Equilibrium
+a1
+a4
+a25
+through the parameter space such that the system
+snaps between the large, medium, and small min-
+ima. The path color at each point denotes the sys-
+tem state, θ.
+As with the cusp and symmetrized
+butterfly bifurcations depictions in Figs. 1d and 3a,
+transitions occur at intersections of the path and
+saddle node surfaces where their colors match. We
+note that for the generic butterfly bifurcation, the
+surface structure can be quite complicated as shown
+by the two projections in Fig. 4a,b.
+In contrast
+to the symmetrized butterfly bifurcation structure
+(Fig. 2a), this complicated structure necessitate us-
+ing all three control parameters x, y, and z, to design
+a pathway that cycles between the three states. Im-
+portantly, despite the surface complexity the design
+is robust. Specifically, since the trajectory crosses
+surfaces, slight deviations in the control parameters
+should still lead to similar snaps, snap sequences,
+and ultimately the resulting complex actions of the
+entire magneto elastic machine.
+Using the design parameters determined by our
+search algorithm, we built a magneto elastic machine
+similar to that depicted in Fig. 1a, but with a dif-
+ferent magnetic dipole pattern and with two of the
+magnets in panel 1 displaced in the panel plane (See
+SI). By following the theoretically predicted path, we
+found three snap through transitions from small to
+large, large to medium, and medium to small (Fig. 4c
+and Movie S1). Two of the transitions occurred at
+the predicted locations, while the large to medium
+transition was displaced by 0.4 cm from its predicted
+location. In addition, we found excellent fidelity be-
+tween the predicted and measured angles θ for the
+equilibrium states. Using the same magneto elastic
+machine, we also designed and demonstrated cycli-
+cal paths with two transitions (See Fig.S3 and Movie
+S3). Finally, when the system was taken apart and
+reassembled, we were able to reliably reproduce the
+transitions associated with the designed trajectories.
+DISCUSSION
+The
+experimental
+validation
+of
+this
+design
+paradigm with a butterfly bifurcation of 5 equilib-
+ria strongly supports the conjecture that this frame-
+work could be extended to design systems perform-
+ing increasingly sophisticated functions by operating
+surface that can instruct the design of a trajectory.
+For
+example the sign of the third derivative of the potential
+signals whether the state’s angle will increase or decrease as
+it bifurcates. Moreover, the merging of saddle node surfaces
+can also be delineated by plotting the cusp bifurcations.
+Here, we do not include this additional information for ease
+of viewing
+near bifurcations with a growing number of equilib-
+ria. Potential energies with these increasingly rare
+bifurcations can be found efficiently, because the gra-
+dient continuation algorithm follows a one dimen-
+sional search path. Moreover, the associated lever
+mechanisms provide a design feature where the op-
+eration of the machine will likely be confined to a
+small parameter volume, enabling the execution of
+these actions by realizable machines.
+Microscopic magneto-elastic machines could prove
+to be a useful instance of design instructed by bifur-
+cations of multiple equilibria: An important emerg-
+ing strategy for manufacturing microscopic and soft
+machines is fabricating them using two dimensional
+lithographic and printing techniques [18–22]. Such
+fabrication techniques, however, restrict the imple-
+mentation of compound mechanisms composed of
+springs, cogs, screws etc. that are used to achieve
+complex actions in traditional macroscale machines.
+These lever mechanisms could be replaced with mag-
+neto elastic mechanisms with lever advantages in-
+duced by bifurcations. Magnetic interactions are es-
+pecially well suited for this purpose since they are
+long ranged and not easily screened. This long range
+allows for global changes to the conformation in re-
+sponse to local actuation of system components.
+Importantly, since bifurcations of multiple equi-
+libria are notoriously sensitive to variations of pa-
+rameters, there is a concern that a machine oper-
+ating near such bifurcations will be very sensitive
+to environmental noise, such as thermal vibrations,
+as well as to fabrication precision. Indeed, close to
+a bifurcation the sensitivity of the system to varia-
+tions of certain combinations of the system param-
+eters grows exponentially as the number of asso-
+ciated equilibria increases.
+Mathematically this is
+captured by mapping the potential to a canonical
+normal form via a change of coordinates [4, 5, Sec.
+36.6] (see SI for derivation). Practically, however,
+this increased sensitivity is often blunted outside of
+the infinitesimal environment of the bifurcation. At
+a finite distance from the bifurcation the mapping
+to the normal form or its linearization will often
+cease to be valid because of other singularities of
+the potential or the nonlinear fall off in the poten-
+tial. This non-linearity is especially pronounced in
+keplerian potentials such as that of magnetic inter-
+actions. Critically, the saddle node manifolds coa-
+lescing at the bifurcation are generically preserved
+outside this radius of convergence as they are topo-
+logically protected and can only annihilate at a cusp
+or a bifurcation of more equilibria. Thus, operating
+a machine near a bifurcation of multiple equilibria,
+but at a finite distance from it, allows the design
+of trajectories that take advantage of the multiple
+saddle node transitions associated with it, and their
+
+6
+FIG. 4.
+3-state Cycle Near Butterfly Bifurcation Point (a.) Theory The saddle node surfaces of a magneto-
+elastic system with three active control parameters, x,y and z are plotted, their color denotes the angle θ at which
+the snap occurs. The system’s magnetic pattern is designed using the gradient continuation algorithm such that it
+operates near a butterfly bifurcation where multiple saddle node surfaces coalesce, enabling multiple snap-through
+transitions at the surfaces. A trajectory (colored tube with white arrows) is chosen such that the system snaps in
+cycles between three states Large (L), Medium (M) and Small (S) angles. The system’s predicted state is denoted by
+the tube’s color. At intersections of the trajectory with a surface where their colors match the system is predicted
+to snap to a new state. (b.) Experimental demonstration: The colored dots mark the experimental value of the
+system’s state as it follows the designed trajectory. We observe three distinct transitions as predicted.
+lever advantages, while avoiding the local exponen-
+tial sensitivity.
+Similarly, the sensitivity of a system designed near
+a bifurcation of multiple equilibria to external noise
+grows exponentially with the number of associated
+states.
+This growth in sensitivity arises from the
+decrease in the potential barriers between adjacent
+states. For example in a potential with k equilibria
+where all the potential barriers are of equal height,
+and the minima are equally deep (which is propor-
+tional to a Chebyshev polynomial of the first kind
+of order k + 1) the barrier heights decay as 2−k.
+This sensitivity seems prohibitive as we imagine im-
+plementing this design principle to create systems
+cycling between multiple states.
+Despite this in-
+creased sensitivity, however, we estimate that the
+strength of magnetic interactions assures that mag-
+neto elastic systems are robust to thermal noise at
+the microscale. Specifically, in magneto elastic sys-
+tems the potential is proportional to the dipole-
+dipole interaction strength µ0µ2L6/R3 of two mag-
+nets with magnetic dipole densities µ panel size L
+and typical distance between dipoles R.
+Thermal
+noise is then comparable to the magneto elastic po-
+tential barrier height when the number of equilibria
+k ∼ log2
+�
+µ0µ2L3/(R/L)3
+kbT
+�
+. The magnetic dipole den-
+sities µ are of order 106A/m at the microscale [24].
+The smallest two state door (equivalent to the de-
+vice in Fig. 1a) that is robust to thermal noise is
+then ∼ .1µm in size, approaching the size limit of
+30nm for fabricating stable magnetic domains [25].
+Conversely, a 100 µm machine will become sensitive
+to thermal noise near a bifurcation of ∼ 40 equilib-
+ria, that is 20 distinct states compressed in a span
+of 100 degrees.
+Finally, the designs that we have implemented in
+this paper assume operation in a low Reynolds num-
+ber regime where inertia can be neglected. In the
+macroscale implementation this was achieved by at-
+taching a damping panel immersed in a solution of
+glycerol. We expect our designs to work even bet-
+ter as these machines are implemented at smaller
+scales since the importance of inertia drops quadrat-
+ically with the system size. Operation of a 100 µm
+scale machine in water, for example, would enable
+the system to be in the low Re regime while operat-
+ing at rates that are 1000 fold faster than those in
+the macroscale experiment.
+CONCLUSIONS
+We have shown that the operation of multi-
+parameter machines near bifurcations of multiple
+equilibria allows them to efficiently and robustly cy-
+cle between multiple conformation.
+Moreover, we
+developed a generic step-by-step framework to de-
+sign and implement systems that operate near such
+bifurcations.
+Specifically, we: 1) created a search
+algorithm that optimizes over fabrication and other
+system parameters to enable operation near such bi-
+furcations; 2) mapped the manifold of saddle node
+bifurcations to determine a useful trajectory for the
+machine operation and; 3) demonstrated the ro-
+bustness of this approach by constructing and op-
+erating a magneto elastic machine that can cycle
+and robustly snap between multiple distinct con-
+figurations in response to small variations of a few
+control parameters.
+Importantly, this design ap-
+proach and step-by-step implementation is generic
+and could be applied to many complex systems with
+
+z [cm] - Control
+Parameter
+-0.2
+L
+-0.4 -
+M
+-0.4
+-0.2
+-0.6 
+-0.4 -
+-0.8
+2
+-0.8 
+-0.6 -
+-0.8 -
+-
+[rad]
+2.0-
+0
+1
+1.5
+MO
+-1.0
+-0.5
+y
+-0.5
+-1.0
+-0.5
+y [cm] - Control Parameter
+-3
+-1.0
+-2
+X
+-2.0
+1.5
+X7
+multiple interacting components ranging from arti-
+ficial proteins, where the interactions are electro-
+static, to neural networks (both biological and syn-
+thetic) where the interactions are governed by net-
+work topology.
+Cycling between transitions in mechanical imple-
+mentations of such systems can generate work or lo-
+comotion. If the system is over-damped, as is often
+the case in microscopic systems operating in fluids,
+work and locomotion can be achieved by coupling
+the system to mechanisms that break time reversal
+symmetry.
+These mechanisms include ratchets or
+cilia-like flexible rods [26]. In the case of the mag-
+neto elastic hinge described here, time reversal sym-
+metry is broken by combining the smooth transla-
+tions of the control panel with abrupt transitions in
+the state of the dynamic panel. In systems where
+the control variable is not a mechanical parameter
+time reversal symmetry can be broken by using the
+angle as an effective dynamical variable governing a
+system with multiple degrees of freedom such as is
+often used to parameterize robot locomotion.
+More broadly, it is interesting to consider the ex-
+tension of our work to systems with a larger number
+of dynamical variables (θ1, θ2, . . .). Here, we envision
+that by working near bifurcations of multiple vari-
+ables (e.g. elliptic umbilic bifurcations) one could
+organize snaps between states separated along mul-
+tiple variables. Such designs require extending our
+search algorithm to multiple variables while main-
+taining its low dimensional search path.
+Alterna-
+tively, one could design mechanisms based on multi-
+ple local bifurcations that are weakly coupled across
+the machine.
+For example, one bifurcation of n
+states could be used to control θ1 while a second
+bifurcation of m states organizes the dynamics of
+the variable θ2. By weakly coupling the panels, and
+hence the variables θ1 and θ2, the machine can trans-
+form between n × m states in a coordinated fashion.
+Indeed this approach is already being implemented
+for bifurcations with two states [27–29]. Increasing
+the number of states associated with each variable
+would enable a similarly rich landscape for machine
+design with far fewer mechanical elements or panels.
+Finally, it is interesting to consider whether this
+design paradigm can be used to understand natural
+systems beyond the Venus fly trap and humming-
+bird beak. For example, molecular machines such
+as proteins often transition between different config-
+urations. It is interesting to consider whether such
+transitions can be thought of as snaps organized by
+bifurcations of many states [3, 6]. As another ex-
+ample, bifurcation theory has been implemented to
+identify and explain epigenetic dynamics of cell dif-
+ferentiation [30–32]. These approaches often focus
+on consecutive 2-state bifurcations. The results pre-
+sented here however, suggest that a comparably sim-
+ple evolutionary pathway could entail development
+of multi-state bifurcations. Such a structures could
+allow the addition of new states while maintaining
+the existing configuration through an evolutionary
+process, similar to the path taken by the gradient
+continuation algorithm.
+I.
+MATERIALS AND METHODS:
+Construction of experimental hinge system
+Panel P1 is constrained to a set of linear trans-
+lation stages that allow its position to be adjusted
+manually to any x or y coordinates near the cusp.
+For experiments near the butterfly bifurcation point,
+an extra translation stage is attached to Panel P1 to
+allow adjustment of its z coordinate. Panel P2 is
+attached to an OVA friction-less thrust air bushing
+with a 13mm shaft. The air bushing is attached to
+a fixed metal housing to limit Panel P2 to its ro-
+tational degree of freedom. A T-shaped paddle is
+attached to the bottom of the shaft and immersed
+in glycerol to introduce damping to the system. Ad-
+ditionally, we position a Basler Ace aca3088-57um
+area scan camera above the center of the air bushing
+to take top-view images of the air bushing which are
+then used to calculate the angle response of Panel
+P2 to high precision.
+A.
+Panels for experiments near cusp point
+Each magnetic panel is constructed using two 1/16
+in thick laser-cut acrylic pieces and nine grade N48
+neodymium magnets of diameter 1/16 in and height
+1/8 in.
+Magnets are arranged in a 3-by-3 square
+lattice with lattice constant of 2.5 cm.
+B.
+Panels for experiments near butterfly point
+Each magnetic panel is constructed using two 1/16
+in thick laser-cut acrylic pieces and nine grade N48
+neodymium magnets of diameter 1/8 in and height
+1/8 in. Magnets are arranged in a 3-by-3 square lat-
+tice with lattice constant of 2.5 cm. In panel P1 the
+x, y position of two of the magnets is displaced ac-
+cording to the design determined by the search algo-
+rithm. The two magnets whose position is offset are
+the magnet in the bottom row on the right column,
+whose offsets are dx1 = 1.418cm, dy1 = −0.273cm,
+and the magnet in the middle row on the left column,
+with offsets dx2 = −0.826cm, dy2 = −0.986cm. A
+
+8
+technical drawing illustrating the panels used for the
+butterfly experiment is included in the SI.
+FIG. 5. Experimental Setup Sketch of the experimen-
+tal system used for demonstration of cycles and angle
+measurements. Panel P1 is attached to a set of transla-
+tion stages which allows us to implement the spatial con-
+trol parameters in all experiments. Panel P2 is attached
+to an air bushing that is fixed in space. An attachment
+submerged in glycerol is added to the base of Panel P2
+to introduce damping to the system.
+Angle measurements
+A marker is attached to the top of the air bush-
+ing, and a camera records the location of the marker
+during the experiment. At each given time, the mea-
+sured angle is the determined by three points: cur-
+rent marker location, location of the center of rota-
+tion, and marker location at θ = 0. We calibrate
+the system by recording the location of the pixel at
+θ = 0 and several other distinct angles. The pixel
+location corresponding to the center of rotation is
+obtained using a fitted circle through the calibra-
+tion data points.
+The resulting angle is then de-
+duced from the three measured points. This data
+collection process is conducted in MATLAB.
+Acknowledgments We thank Michael Brenner,
+Chrisy Xiyu Du, Yan Yang, Robert Distasio, and
+John Guckenheimer for inspiring discussions. This
+work was financially supported primarily by NSF
+Grant DMREF-89228, NSF Grant EFRI-1935252,
+NSF Grant CBET-2010118, Cornell Center for Ma-
+terials Research DMR-1719875, and by Air Force
+Office of Scientific Research Grant MURI: FA9550-
+16-1-0031. I.G was also supported by the Cornell
+Laboratory of Atomic and Solid State Physics. D.H
+was supported by an NSF Graduate Research Fel-
+lowship Grant No. DGE-2139899.
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+Supplemental material - Bifurcation instructed design of multistate machines
+II.
+CALCULATION OF THE POTENTIAL ENERGY LANDSCAPE
+To model the dynamics of our experimental hinge system, we compute the potential energy landscape
+arising from the dipole-dipole interactions between the magnets embedded in each panel. The magnets used
+in our experiments are well approximated by perfect dipoles. Therefore, the potential energy for the system
+is a sum of dipole-dipole interaction energies
+V = −
+�
+i∈P 1
+�
+j∈P 2
+µ0m2
+4π|rij|3 [3( ˆmi · ˆrij)( ˆmj · ˆrij) − ˆmi · ˆmj] ,
+(S1)
+where µ0 is the vacuum permeability, m is the dipole strength (identical for all magnets), mi is the orientation
+of dipole i, and rij is the distance between magnets i and j. Note that the interaction energy for dipoles in
+the same panel is constant, so we can restrict the sum to pairs of dipoles in different panels.
+To derive the θ dependence of the energy landscape, we must write the dipole orientations and positions
+in terms of our control parameters x, y, and z and the dynamical variable θ. The dipoles on P1 are always
+oriented in the z-direction, while the dipoles on the rotating panel P2 have orientation that changes with θ:
+ˆmi = δiˆz
+ˆmj = δj{sin θ , 0, − cos θ},
+(S2)
+where δi = ±1 is the orientation of magnet i with respect to panel P1 (similar for δj). The positions of
+individual dipoles are given by
+ri = {xi, yi, 0} + {x, y, z}
+rj = Rθ{xj, yj, 0},
+(S3)
+leading to interdipole distance rij = ri − rj. Here xi and yi are the x − y positions of dipole i in panel
+P1 (similar for xj, yj), x, y, and z are the coordinates of the control panel, and Rθ is the rotation matrix
+corresponding to a rotation by angle θ about the y-axis.
+Together Eqs. (S1-S3) give the potential energy in terms of the hinge angle θ, our control parameters x,
+y, and z, and design parameters xi, yi, δi, xj, yj, and δj. Since the hinge experiment is heavily damped, θ
+follows gradient dynamics ˙θ = ∂θV and the stable equilibrium angles are given by the local minima of the
+potential landscape V .
+
+2
+III.
+CUSP EXPERIMENTS
+In the cusp experiments, Panel P1’s x and y positions are measured as displacements from their value
+when the panels are 180◦ open, and are aligned along z and y such that the panel’s backs and bottoms
+are parallel. The magnets closest to the hinge axis are removed from it by 0.75cm on both panels. The
+back of the cylindrical magnets are aligned with the panel’s backs. The damping paddle used in the cusp
+experiments have dimensions 1.5cm by 3.0cm.
+A.
+Experimental estimation of the cusp point
+We estimate the location of the cusp point as the bifurcation of the two measured saddle node curves. We
+map the saddle node curve by toggling x (y), for a given value of y (or x), so that the system snaps back
+and forth, and record the values of the control parameters x and y, and θ immediately after each transition
+(Fig. S2). Moreover, to verify the position of the cusp we record the angle θ of the system before and after
+snapping, and observe that the change in angle upon snapping disappears at the cusp point.
+Finally, we inspect all data collected along the bifurcation curves as shown in Fig. 2a in the main text,
+and use a spline fit for the saddle-node bifurcations from L to S and the saddle-node bifurcations from S to
+L. We define the cusp point as the intersection of the two splines.
+B.
+Single snap experiment
+The mangeto elastic potential calculated for the experiment predicts a cusp at a slightly removed para-
+metric position. The discrepency between the experimentally measured and theoretically predicted cusps
+could be due to fabrication errors. To effectively compare theory and experiment in this section only, we
+parameterize the system as a function of its displacement from the cusp for both theory and experiment
+using using dx and dy. We then follow the predicted path by controlling panel P1’s x and y positions using
+the translation stages. We begin the experiment by letting the system maintain its equilibrium at the initial
+dx, dy position. We then change the position of Panel P1 at a slow and steady rate. Angle measurements
+are recorded at various locations in the loop as shown in Fig. S1(a) (see also Fig. 1 in the main text), and
+the change in position is paused once the transition happens at point vi in order to let the system settle
+down and obtain an accurate angle measurement. We confirm that the system returns to its original state
+once we return to the starting dx, dy position.
+C.
+Scaling experiment
+To fit the scaling relations, we use the the same section of the data set used for determining the location
+of the experimental cusp point. We neglect data in the nonlinear region of the saddle-node curves far away
+from the cusp point, as well as data too close to the cusp point, where the errors due to measurement noise
+are comparable to the distance to the cusp. The data points used for the scaling relations are highlighted in
+Fig. S2(a). The state parameter values used in the scaling analysis correspond to the angle measurements
+obtained at the points right after the snap through transitions.
+IV.
+ONE DIMENSIONAL BIFURCATIONS OF EQUILIBRIA: NORMAL FORM AND
+SCALING
+The ability to design magneto elastic machines and control parameter pathways that robustly lead to
+complex actions corroborates the validity of a new design paradigm: operation near bifurcations of multiple
+equilibria. The demonstrated trajectories take advantage of the structure of available dynamics near bifur-
+cations of equilibira. These bifurcations are the loci of multiple distinct coalescing saddle node manifolds,
+as illustrated for the idealized symmetric butterfly bifurcation (Fig 3b in the main text). By weaving a
+trajectory that crosses and avoids chosen saddle node bifurcations we design a pathway that leads to com-
+plex actions. The system then cycles through multiple states via small variations of the control parameters,
+
+3
+taking advantage of the multiple accessible lever mechanisms associated with these saddle node surfaces.
+The sensitivity of the realized design increases as the number of equilibria associated with the bifurcation
+grows.
+Butterfly, cusp and saddle node bifurcations are the first in a series of bifurcations of equilibria in
+one-dimensional gradient systems.
+More generally, in systems with a single degree of freedom x, bifur-
+cations of k equilibria are points in parameter space where the first k derivatives of the potential vanish,
+{dV/dx, d2V/dx2, . . . , dkV/dxk} = ⃗0. That is, they are equilibrium points satisfying k − 1 equations beyond
+that of mechanical equilibrium dV/dx = 0 and therefore lie on a manifold of co-dimension k − 1 within the
+equilibrium manifold. The sensitivity of a bifurcation of k equilibria to variation in its parameters can be
+estimated through the topological equivalence of the dynamics near it to those in a normal form potential
+�V = ϕk+1 +
+k−1
+�
+i=1
+aiϕi,
+(S4)
+where the variable ϕ(θ) and normal form parameters ai(p) are coordinate transformations of the angle θ and
+parameters p respectively. The normal form describes the unfolding of the Taylor expansion of the potential
+at the bifurcation V ∼ xk+1 by variations of the parameters [S1–S3]. The unfolded normal form potential
+demonstrates that the parameteric environment of a codimension k bifurcation includes domains with 1 to
+⌈(k + 1)/2⌉ minima delineated by k saddle-node manifolds which coalesce at the bifurcation. Moreover, it
+implies scaling relations between the variation in the system’s state upon a snap through transition induced
+by crossing a saddle node bifurcation associated with a codimension k − 1 bifurcation and the variation of a
+normal form parameter that causes the snap:
+δϕ ∝ a1/(k−m+1)
+m
+,
+m < k.
+(S5)
+Heuristically the scaling can be derived from the normal form by noting that near the bifurcation the kth
+derivative of the potential must still vanish, and so δϕ2 ∼ ak. Similarly the next k − 1 derivatives must
+progressively vanish, setting the scaling of am. An explicit proof is given in [S4] and summarized in [S5,
+Sec. 36.6]. These scaling relations carry over to the original variable and parameters near the bifurcation
+where the maps ϕ(x) and am(p) are approximately linear. Indeed, the scaling relations we experimentally
+observed near a the cusp bifurcations are those of the systems state with the normal form parameters near
+a bifurcation of three equilibria, i.e., a cusp [S4, S5].
+These scaling relations imply that the sensitivity of the system to variations of parameters grows expo-
+nentially with the number of associated equilibria. A system designed near a bifurcation of k equilibria can
+toggle its state between order unity separated states, δϕ ∼ 1/2, in response to variations of the linear normal
+form coefficient a1 of order 1/2k. That is, both the potential lever advantage and the sensitivity to noise in
+the parameters grow as the number of associated equilibria grows. However, the parametric domain in which
+the mapping to the normal form is linear is often very small. The nonlinearity of the mapping often blunts
+the sensitivity of the response. Thus, the increased lever advantage near bifurcations of multiple equilibria
+is often not experimentally accessible. Conversely the system is not so sensitive to parametric noise when
+operated at a small parametric distance from the bifurcation about which it is designed, as demonstrated
+by the reproducibility of the experimental three state system, which was easily constructed twice.
+V.
+CONTINUATION ALGORITHMS
+To find bifurcations of multiple equilibria in the dynamics of our model system and to map out the saddle
+node structure in the vicinity of the high-order point, we use a series of continuation algorithms. In one
+dimension, a codimension k bifurcation point is defined by the vanishing of the first k derivatives of the
+potential: ∂j
+θV (θ∗, {ξi}) = 0 for j = 1, 2, . . . , k. These constraints define a codimenion k manifold in the
+space of dynamical variables and parameters (θ∗, {ξi}).
+A.
+Traditional continuation
+Standard continuation algorithms compute bifurcation curves by varying a small number of parameters,
+and then projecting onto the bifurcation manifold [S1]. For example, suppose we have found a co-dimension
+
+4
+k bifurcation. This requires the first k derivatives of the potential vanish, fixing θ∗ and k − 1 parame-
+ters ξ1, ξ2, . . . , ξk−1. Varying an additional parameter ξk produces a line emanating from our initial point
+(θ∗, {ξ}) = p. The continuation algorithm maps out this line by (i) taking a step along the tangent vec-
+tor Tk(p) to the curve, which is the null-vector of the gradient of the first k derivatives of the potential
+Tk(p) ≡
+�
+⃗v ∈ Rk+1 | ∀j ∈ (1, 2, . . . , k), ⃗v · ∇θ,ξ1,ξ2,...,ξk∂j
+θV = 0
+�
+and (ii) correcting this step using a Newton-
+Raphson algorithm4 to search perpendicular to the step for a point where the first k derivatives of the po-
+tential vanish. This approach can be used to progressively search for higher order bifurcation points. For
+example, a fixed-point can be continued until ∂2V (θ∗, {ξi})/∂θ2 vanishes, indicating a saddle node bifurca-
+tion. Continuing the saddle-node can lead to a cusp bifurcation, which in turn might lead to a swallowtail
+bifurcation. In this way, progressively adding parameters and performing continuations of one-dimensional
+curves can lead toward high-codimension bifurcation points. Once we have found a high-order bifurcation
+point, we use this algorithm to map out the saddle node surfaces nearby. The surfaces can in turn be used
+to design cycles in control parameters that cause the system to perform desired snapping transitions.
+The standard continuation approach, however, has limitations for microscopic machine design. In partic-
+ular, it has limited utility for finding the high-order bifurcation points near which our machine will operate.
+In our model system we have many free parameters, including the positions of each of the magnets embed-
+ded in the panels. Varying a given experimental parameter does not guarantee we will find the next order
+bifurcation point. Instead we want to vary many parameters simultaneously, which greatly improves the
+likelihood that a higher-order bifurcation point is contained within the search space and allows for a more
+efficient approach toward that point. We have developed a gradient continuation algorithm to carry out this
+multi-parameter search.
+B.
+Design algorithm: Gradient continuation
+The gradient continuation algorithm works as follows. Suppose we have N parameters ξi in our system,
+plus the degree-of-freedom θ.
+A point, p, where the first k derivatives of the potential vanish belongs
+to a co-dimension k manifold in the full (N + 1)-dimensional augmented parameter space, composed of
+the equilibrium state and control parameters, (θ∗, {ξi}). Starting from the point p, take a step along the
+gradient of the k+1 derivative of the potential ∇θ,ξ1,ξ2,...,ξN ∂k+1
+θ
+V , projected onto the tangent surface to the
+manifold at p. The tangent surface is the null-space of the gradient of the first k derivatives of the potential5,
+Tk,N(p) ≡
+�
+⃗v ∈ RN+1 | ∀j ∈ (1, 2, . . . , k), ⃗v · ∇θ,ξ1,ξ2,...,ξN ∂j
+θV = 0
+�
+. This procedure finds the step within the
+co-dimension k manifold that maximizes the change in ∂k+1
+θ
+V , which we need to vanish in order to find the
+next order bifurcation. After the step, the algorithm performs a corrective Newton-Raphson search [S6],
+constrained to the hyperplane T ⊥
+k,N(p) perpendicular to the null-space, which returns to the codimension
+k manifold on which the first k derivatives of the potential vanish. As in the standard continuation, this
+approach is repeated to progressively find higher order bifurcation points. A visualization of the gradient
+search algorithm, applied to the potential V = θ6 + a4θ4 + a2θ2 + a1θ, is shown in Fig. 3b in the main text.
+VI.
+BUTTERFLY EXPERIMENTS
+A.
+Butterfly panels
+In the butterfly experiments, Panel P1’s x, y and z positions are measured as displacements from their
+value when the panels are 180◦ open, the magnets closest to the hinge axis are removed from it by 2.5cm
+on both panels, the panels are aligned vertically, and the back of the cylindrical magnets on Panel P1 are
+aligned with the center of the magnets on Panel P2. This small change in magnet alignment (compared
+4 Newton-Raphson(f, Ω, p) [S6] searches for the roots of the
+functions f over the space Ω starting at the point p.
+5 Notice that this algorithm uses all N parameters ξ1, . . . , ξN
+to search for a codimension k bifurcation, while the stan-
+dard continuation in the previous section only used k pa-
+rameters ξ1, . . . , ξk. The null-space Tk,N(p) has dimension
+(N − k + 1).
+
+5
+to the single snap experiment) is found to reduce the discrepancy between experiment and prediction. An
+illustration for the panels is shown in Fig. S3. The damping paddle has dimensions 8.0cm by 2.5cm for the
+butterfly experiments. The position of the magnets on panel P1 was changed such that the system operates
+next to a butterfly bifurcation, as specified in the main text and in the following sections.
+B.
+Application of the continuation algorithm
+To find an experimentally feasible path and magnetic pattern, we implement the continuation algorithm
+by first finding a butterfly point in parameter space, then validating the resulting pattern against known
+experimental constraints (e.g. we require physically realizable panel angles and magnet positions). Before
+each search using the continuation algorithm, we first randomly generate orientations of the 18 magnetic
+dipoles on the two panels. We also randomly select two magnets on Panel P1 to be displaced from their
+lattice positions, by (dx1, dy1) and (dx2, dy2) respectively. The search algorithm is always initialized with
+the values {θ, dx, dy, dz, dx1, dy1, dx2, dy2} = {1.1rad, 0.5cm, −0.25cm, 0, 0, 0, 0, 0}.
+Next,
+we
+let
+the
+algorithm
+try
+to
+find
+a
+butterfly
+bifurcation
+point.
+If
+no
+but-
+terfly
+point
+can
+be
+found,
+we
+repeat
+the
+initialization
+process
+and
+repeat
+the
+search
+with
+a
+new
+randomly
+generated
+magnetic
+pattern.
+The
+butterfly
+point
+corresponding
+to
+the
+pattern
+we
+used
+in
+our
+experiments
+is
+located
+at
+{θ, dx, dy, dz, dx1, dy1, dx2, dy2}
+=
+{2.131rad, −0.355cm, −0.304cm, −0.824cm, 0.918cm, −0.698cm, −0.326cm, −0.486cm}.
+If the butterfly point is found, we investigate the potential plots at various points in parameter space near
+the bifurcation point. Specifically, we offset one or more of the 6 search parameters by ±0.2 and find the
+number of minima that exist between 0 to 180 degrees at each of these locations. The potential plots at
+locations with three minima are then inspected to decide the experimental feasibility of the pattern. Ideally,
+all three minima are at least 5 degrees apart, and the smallest minimum is at least 5 degrees (for z = 0)
+to prevent the panels from touching during experiment. We also look for patterns with large triple-minima
+regions, for example if three visibly deep minima can be observed when at least one parameter is changed
+by ±0.5cm.
+After an experimentally feasible pattern is discovered, we manipulate the three experimentally controllable
+parameters (x,y,z) continuously around the point with deepest triple minima and observe changes in our
+model of the potential landscape. The design of the control path is guided by visualization of the saddle-node
+surfaces mapped out using the standard continuation algorithm detailed above. Several paths are tested in
+the model to obtain the desired sequence of bifurcations and to optimize various properties of the transitions
+(e.g. the magnitude of the snaps and depth of the minima).
+C.
+Experiments for trajectories near a butterfly point
+We set up the experiment by laser-cutting the holes for magnets at the exact locations corresponding to the
+found dx1, dy1, dx2, dy2 values, which were 1.418cm, −0.273cm, −0.826cm, and −0.986cm respectively. We
+also add a translation stage to control Panel P1’s z position. We begin the experiment by following the exact
+coordinates provided by the theoretically designed path. In the event that a predicted transition cannot be
+seen using the predicted path coordinates (due to fabrication or calibration errors shifting the surface), we
+translate the system further from the original predicted path to determine a more robust path that may
+account for some shifting in coordinates due to experimental errors (for example see Fig. 4b in the main
+text). Once an experimental path is shown to demonstrate the predicted behavior with the desired number
+of state transitions, we record the locations for state transitions in experiment, and repeat the experiment
+while slowing down the rate of change in x,y positions near the transitions to give the system enough time to
+respond in the presence of large damping. Those experiments show excellent qualitative agreement with the
+theoretically designed paths, although the locations at which transitions happen and the equilibrium angle
+of the panel are often shifted by a small amount due to experimental error.
+
+6
+D.
+Additional operation mode: double-snap trajectories
+The intricate saddle-node surface structure near the butterfly bifurcation enables a variety of snapping
+behaviors with the same panel design, beyond the 3-state cycle presented in the main text. Here we present
+a second snapping sequence that was measured experimentally.
+By using the same trajectory in parameter space as the three-snap sequence in the main text, but traverses
+the path in the reverse direction, we observe a two-snap sequence between small (S) and large (L) angles.
+Fig. S4a shows this trajectory together with the same saddle surfaces from Fig. 4a in the main text. The
+experimentally measured angles along this backward cycle are shown in Fig. S4b (see also Movie S2). Besides
+a minor systematic shift in the angles of the L state, we find excellent fidelity between the predicted and
+measured angles. The snapping transitions occur almost exactly at the predicted locations.
+Our example trajectories demonstrate that the saddle-node structure in the vicinity of a butterfly bifurca-
+tions enables a great deal of flexibility in controlling state transitions of a mechanical system. For practical
+applications, further fine-tuning of the control trajectory can be used to optimize features the system’s
+behavior (e.g., the positions of the steady states and their lifetimes in the presence of environmental noise).
+VII.
+GENERALIZATIONS: MULTIDIMENSIONAL BIFURCATIONS AND SUPPLEMENTAL
+SCALING BEHAVIOURS
+A.
+Stopping conditions in higher dimensions
+While our proof-of-concept experiment is limited to a hinge with a single degree of freedom (the opening
+angle), our approach and gradient continuation algorithm are straightforward to apply to systems with
+multiple degrees of freedom, e.g. a microscopic robot with multiple panels connected by elastic hinges. The
+cuspoidal bifurcations discussed in this paper also naturally appear in higher-dimensional gradient systems.
+However, the analytic criteria to classify them is somewhat more complicated: we can not simply search
+for points where higher order derivatives of the potential vanish. In this section we will discuss stopping
+criteria in higher dimensions, i.e. what quantities should we follow with the gradient continuation algorithm
+to search for bifurcations of increasing order?
+With two or more degrees of freedom, a saddle-node bifurcation occurs when a fixed-point (stable or
+unstable) collides with a saddle point, resulting in mutual annihilation. This occurs when an eigenvalue of
+the Hessian of the potential Aij = −∂θi∂θjV crosses 0 (here θi are the dynamical variables). For the purposes
+of applying gradient continuation starting from a fixed point, it is therefore convenient to use det A as the
+stopping criteria, since the determinant vanishes when an eigenvalue does.
+Near a saddle-node bifurcation, the state space can be decomposed (by the Center Manifold Theorem)
+into (i) the invariant center manifold emanating from the fixed point along the direction of the critical
+eigenvector (with eigenvalue 0) and (ii) a stable/unstable manifold on which the flows exponentially grow or
+decay (for the purposes of machine design we generally want only stable directions). Due to the vanishing
+eigenvalue, the dynamics on the center manifold are nonlinear at lowest order.
+These dynamics can be
+determined perturbatively by expanding the gradient of the potential, projecting onto the center manifold
+and enforcing the invariance of the center manifold [S1]. Higher-order bifurcations occur when the center
+manifold expansion coefficients vanish. For example, vanishing quadratic term indicates a cusp bifurcation,
+vanishing cubic term indicates a swallowtail, and so on. Thus these coefficients replace the higher-order
+derivatives of the potential as the stopping criteria in the gradient continuation algorithm. Below we give
+explicit expressions for these expansion coefficients.
+Suppose we have an n-dimensional system θ ∈ Rn that undergoes a saddle node bifurcation at θ = 0.
+Near this point, the dynamics can be expanded as follows,
+˙θ = A θ + F(θ),
+(S6)
+where A is the Hessian of the potential (which has a zero eigenvalue) and F(θ) collects all quadratic and
+
+7
+higher-order terms in multilinear forms,
+F(θ) = 1
+2B(θ, θ) + 1
+6C(θ, θ, θ) + 1
+24D(θ, θ, θ, θ) + O(||θ||5)
+= 1
+2
+n
+�
+i,j=1
+∂2F(φ)
+∂φi∂φj
+����
+φ=0
+θiθj + 1
+6
+n
+�
+i,j,k=1
+∂3F(φ)
+∂φi∂φj∂φk
+����
+φ=0
+θiθjθk
++ 1
+24
+n
+�
+i,j,k,l=1
+∂4F(φ)
+∂φi∂φj∂φk∂φl
+����
+φ=0
+θiθjθkθl + O(||θ||5).
+(S7)
+Let ψ and ϕ be the right and left eigenvectors corresponding to the zero eigenvalue: Aψ = 0 and AT ϕ = 0.
+The projection of θ onto the center manifold ϑ = ϕ · θ has dynamics
+˙ϑ = a2ϑ2 + a3ϑ3 + O(ϑ4).
+(S8)
+Following Kuznetsov, we derive the coefficients up to fourth order (third order is given in Ref. [S1]),
+a2 = 1
+2ϕ · B(ψ, ψ)
+a3 = 1
+6ϕ · C(ψ, ψ, ψ) + 1
+2ϕ · B(ψ, b2)
+a4 = 1
+24ϕ · D(ψ, ψ, ψ, ψ) + 1
+4ϕ · C(ψ, ψ, b2) + 1
+8ϕ · B(b2, b2) + 1
+6ϕ · B(ψ, b3),
+(S9)
+where
+b2 = A−1
+su
+�
+ψ[ϕ · B(ψ, ψ)] − B(ψ, ψ)
+�
+b3 = A−1
+su
+�
+ψ[ϕ · C(ψ, ψ, ψ) + 3ϕ · B(ψ, b2)] + 3b2[ϕ · B(q, q)] − C(ψ, ψ, ψ) − 3B(ψ, b2)
+�
+(S10)
+and A−1
+su is the inverse of A restricted to the stable/unstable subspace (which doesn’t have zero eigenvalues).
+As mentioned above, vanishing a2 indicates a cusp, if a3 also vanishes we have a swallowtail, and if all three
+coefficients are zero we have a butterfly. The vectors b2 and b3 describe the curvature of the center manifold
+in the full θ space, θ = qϑ + b2ϑ2/2 + b3ϑ3/6. While these bifurcations are one dimensional (they occur on
+the one-dimensional invariant center manifold), the curvature of the center manifold as we move further from
+the bifurcation point could allow snapping between states with reasonable separation in multiple dimensions.
+In principle, this would enable machines to carry out work cycles near a butterfly bifurcation.
+B.
+Scaling for the Thom’s seven: hyperbolic and elliptic umbilics
+Beyond the quasi-one-dimensional bifurcations there are also cuspoidal bifurcations that are genuinely mul-
+tidimensional. In two dimensions, for example, we have elliptic umbilic, hyperbolic umbilic, and parabolic
+umbilic catastrophes (these together with the four one-dimensional bifurcations saddle-node, cusp, swallow-
+tail, and butterfly make up the Thom seven). Like the cusp and butterfly bifurcations, the unfolding of the
+normal form predicts and intricate saddle-surface structure describing how fixed-points and saddle-points
+come together and collide in the vicinity of the bifurcation point. These higher-dimensional bifurcations also
+obey advantageous scaling laws, relating the changes in state to the variation of control parameters. For
+example, the normal form potentials for the elliptic and hyperbolic umbilics are
+Velliptic = θ3
+1
+3 − θ1θ2
+2 + a(θ2
+1 + θ2
+2) + bθ1 + cθ2
+Vhyperbolic = θ3
+1 + θ3
+2 + aθ1θ2 + bθ1 + cθ2
+(S11)
+from which the follow scaling can be derived [S4],
+δθ1, δθ2 ∼ a
+b, c ∼ a2.
+(S12)
+Increasing the dimension further leads to even more cuspoidal bifurcations; these have been enumerated
+by Arnold using an ADE classification [S7]. While the search criteria for such bifurcations is increasingly
+complicated, they provide a rich design space for multi-component machines.
+
+8
+C.
+Reynolds number scaling
+The magnetic decorations in our experiments are arranged in each panel about a square lattice with
+unit separation of 2.5cm. To explore over-damped, gradient dynamics, that are ubiquitous in microscopic
+mechanisms, the rotating panel is attached to a paddle moving through a glycerol bath. The results of our
+experiments then hold also for smaller systems in fluid with comparable kinematic viscosity. If the system
+is smaller by a factor Ω ≪ 1, the time ∆t it takes our macroscopic over-damped system, of typical size L,
+to traverse an angular expanse ∆θ is equal to the time it takes a microscopic system, of size ΩL to traverse
+the same angular expanse in the same liquid. This comes about because both the viscous drag force and
+the magnetic force between dipoles of magnetization M1 and M2, FDrag ∼ L2 ˙γ, Fdipole ∼ M1M2/R4, are
+quadratic in the typical system sizes. For over-damped dynamics this results in a length-scale independent
+strain-rate, ˙γ. The system is over-damped if its Reynolds number Re = L2 ˙γ/ν, is smaller then 1, where ν
+is the fluid’s kinematic viscosity. The Reynold’s number of a miniaturized system is therefore smaller by a
+factor of Ω2. Reducing the system’s size can compensate for changes in the system’s composition, such as
+embedding it in water rather than glycerol, or the growth of magnetic dipole strength density as the system
+size decreases.
+[S1] Y. A. Kuznetsov, “Topological equivalence, bifurcations, and structural stability of dynamical systems,” in
+Elements of Applied Bifurcation Theory (Springer New York, New York, NY, 2004)
+.
+[S2] J. Guckenheimer and P. Holmes, Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields
+(Springer New York, New York, NY, 1983)
+.
+[S3] J. W. Bruce and P. J. Giblin, Curves and Singularities: A Geometrical Introduction to Singularity Theory, 2nd
+ed. (Cambridge University Press, 1992)
+.
+[S4] M. V. Berry, Journal of Physics A: Mathematical and General 10, 2061 (1977)
+.
+[S5] DLMF, “NIST Digital Library of Mathematical Functions,” http://dlmf.nist.gov/, Release 1.1.4 of 2022-01-15
+(2022), f. W. J. Olver, A. B. Olde Daalhuis, D. W. Lozier, B. I. Schneider, R. F. Boisvert, C. W. Clark, B. R.
+Miller, B. V. Saunders, H. S. Cohl, and M. A. McClain, eds.
+[S6] W.
+H.
+Press,
+S.
+A.
+Teukolsky,
+W.
+T.
+Vetterling,
+and
+B.
+P.
+Flannery,
+Numerical recipes 3rd edition: The art of scientific computing (Cambridge university press, 2007)
+.
+[S7] V. I. Arnol’d, “Bifurcations of equilibria,” in Dynamical Systems V, edited by V. I. Arnol’d (Springer Berlin
+Heidelberg, Berlin, Heidelberg, 1994) pp. 10–38
+.
+[S8] M. Smith, G. Yanega, and A. Ruina, Journal of theoretical biology 282, 41 (2011)
+.
+
+9
+FIG. S1. Single snap-through mechanism (a.) As we vary the control parameters along a loop around the cusp
+point as shown, we expect to see a single snap-through buckling behavior (point v to point vi) for each cycle, akin
+to how hummingbirds use their beak to capture prey [S8]. (b.) The predicted potential energy curves for points
+labeled from i to vi are presented. The saddle-node bifurcation occurs between v and vi as indicated by the arrow
+in v. (c.) We experimentally observe the predicted snap-through behavior. Due to experimental errors, the location
+of the cusp point is shifted, but we see excellent agreement between the theory and measurements after shifting the
+coordinates to align the theoretical and experimental cusp points.
+FIG. S2. Snap Through transitions near a cusp. These plots show the equilibrium angle recorded in experiments
+following a snap-through transition. The corresponding (x, y) denote the values of the control parameters at which
+the snap-through occurred. (a.) Highlights the the data points used to fit the cusp scaling. We exclude data far from
+the cusp, where higher order terms in the normal form are non-negligible, and close to the cusp, where measurement
+and fabrication error are comparable to the distance from the cusp. (b.) Highlights the data corresponding to the
+upper and lower saddle-node curves.
+
+a)
+b)
+c)
+ty
+i
+vi
+iii
++
+anel
+-y
+L 2.9
+-y
+ [rad] - System State
+Snap-through
+i:  = 2.606
+vi: 0 = 2.653
+c+↑
+Snap-through ↑
++c
+Snap-through
+v
+ii
+S
+2.5
+ii: θ = 2.841 
+v: θ = 2.830
+LSI
+h+ ↑
+-y 
+L!
+S
+h+↑
+yT
+Cusp Point
+iii
+iv
+-c
+-0.03
+0.3
+m
+0.15
+-0.06
+0
+dy [cm] - Control Parameter
+-0.15
+-0.09
+iii: 0 = 2.854 
+iv: 0 = 2.864a)
+b)
+·ScalingDataPoints
+· Upper Saddle Node Curve
+·UnusedDataPoints
+O
+·
+LowerSaddleNodeCurve
+·EstimatedCuspPoint
+·EstimatedCuspPoint
+2.85
+2.85
+[rad] - System State
+2.8
+[rad] - System State
+2.8
+2.75
+2.75
+2.7
+2.7.
+2.65
+2.65
+0
+2.6
+0.15
+2.6
+0.15
+0.2
+Q
+0.2
+2.55
+0.25
+2.55
+0.25
+0.8
+0.3
+0.8
+0.3
+0.35
+0.75
+,0.75
+0.35
+x [cm]
+x [cm]
+Control
+y [cm] -
+Control
+Parameter
+Parameter10
+FIG. S3. Butterfly panels: In the butterfly experiments, Panel P1’s x, y and z positions are measured as displacements
+from their value when the panels are 180◦ open, the magnets closest to the hinge axis are removed from it by 2.5cm
+on both panels, the panels are aligned vertically and the back of the cylindrical magnets on Panel P1 are aligned with
+the center of the magnets on Panel P2. This small change in magnet alignment is found to reduce the discrepancy
+between experiment and prediction.
+FIG. S4.
+2-state Cycle Near Butterfly Bifurcation Point (a.) Theory The saddle node surfaces of a magneto-
+elastic system with three active control parameters, x,y and z are plotted, their color denotes the angle θ at which
+the snap occurs. The system’s magnetic pattern is designed using the gradient continuation algorithm such that it
+operates near a butterfly bifurcation where multiple saddle node surfaces coalesce, enabling multiple snap-through
+transitions at the surfaces. A trajectory (colored tube with white arrows) is chosen such that the system snaps back
+and forth between two states with Large (L) and Small (S) angles. This trajectory is identical to that for the 3-state
+cycle in Fig. 4 in the main text, but the path is traversed in the opposite direction. The system’s predicted state is
+denoted by the tube’s color. At intersections of the trajectory with a surface where their colors match the system is
+predicted to snap to a new state. (b.) Experimental demonstration: The colored dots mark the experimental value
+of the system’s state as it follows the designed trajectory. We observe two distinct transitions as predicted.
+
+Hinge Axis
+11
+2..
+X
+dx2
+Z
+7.5
+5
+2.5
+0
+[cm]
+x [cm][cm] - Control
+Parameter
+-0.2 -
+-0.4
+-0.4 
+Stat
+-0.2
+7
+-0.6
+-0.4 
+tem
+-0.8 -
+-0.8
+-0.6 -
+-0.8 -
+0
+-2.0】
+rad
+-1.5
+.1
+-1.0 >
+2
+-0.5
+y
+-0.5
+S
+-1.0
+-0.5
+-1.0
+3
+y [cm] - C
+Parameter
+1.5
+x
+-2.0
+1.5
+x
\ No newline at end of file
diff --git a/FdAzT4oBgHgl3EQfi_0V/content/tmp_files/load_file.txt b/FdAzT4oBgHgl3EQfi_0V/content/tmp_files/load_file.txt
new file mode 100644
index 0000000000000000000000000000000000000000..c2c8028b8eec560acc8c2409dc0fa464cfbce8ce
--- /dev/null
+++ b/FdAzT4oBgHgl3EQfi_0V/content/tmp_files/load_file.txt
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+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf,len=1067
+page_content='Bifurcation instructed design of multistate machines  Teaya Yang,1 David Hathcock,1 Yuchao Chen,1 Paul  McEuen,2 James P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content=' Sethna,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content='1 Itai Cohen,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content='2 and Itay Griniasty1  1Laboratory of Atomic and Solid State Physics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content='  Cornell University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content=' Ithaca,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content=' New York 14853-2501,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content=' USA  2Laboratory of Atomic and Solid State Physics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content='  Cornell University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content=' Ithaca,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content=' New York 14853-2501,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content='  USA and Kavli Institute at Cornell for Nanoscale Science,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content=' Cornell University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content=' Ithaca,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content=' NY,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content=' USA  (Dated: January 5,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content=' 2023)  We propose a novel design paradigm for multi-state machines where transitions from one state  to another are organized by bifurcations of multiple equilibria of the energy landscape describing  the collective interactions of the machine components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content=' This design paradigm is attractive since,  near bifurcations, small variations in a few control parameters can result in large changes to the  system’s state providing an emergent lever mechanism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content=' Further, the topological configuration of  transitions between states near such bifurcations ensures robust operation, making the machine less  sensitive to fabrication errors and noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content=' To design such machines, we develop and implement a  new efficient algorithm that searches for interactions between the machine components that give  rise to energy landscapes with these bifurcation structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content=' We demonstrate a proof of concept for  this approach by designing magneto elastic machines whose motions are primarily guided by their  magnetic energy landscapes and show that by operating near bifurcations we can achieve multiple  transition pathways between states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content=' This proof of concept demonstration illustrates the power of  this approach, which could be especially useful for soft robotics and at the microscale where typical  macroscale designs are difficult to implement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content='  Systems composed of a large number of interact-  ing elements such as meta-materials, elastic mem-  branes, and proteins can exhibit emergent behaviors  that arise from the collaborative interaction of the  system components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content=' Designing functionality in such  systems is a formidable task that requires searches  in a high dimensional parameter space of the sys-  tem components and their interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content=' Developing  organizing principles for effectively designing such  systems remains an outstanding problem in the field  [1–6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content=' Here, we propose that designing multi-state  machines around bifurcations of multiple equilibria  is a powerful paradigm that can be used to system-  atically organize such searches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content='  Bifurcations, where a single equilibrium configu-  ration splits into multiple equilibria as a function of  a control parameter is a canonical dynamical sys-  tems structure that has been used to explain vari-  ous natural phenomena ranging from phase transi-  tions [7] to the operation of simple machines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content=' Exam-  ples of simple machines include Venus flytraps and  hummingbird beaks that have been shown to open  smoothly and then snap shut by operating about  a cusp bifurcation where three equilibria converge  [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content=' Designing systems to operate near such bifur-  cations provides several advantages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content=' Since the split-  ting of the equilibria has a power law dependence  on the control parameters [4, 7], operating near bi-  furcations automatically provides a lever mechanism  by which small variations in the control parameters  lead to large changes in the system state [12, 13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content='  In the case of the Venus fly trap, slight changes in  hydrostatic pressure can drive large motions of the  trap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content=' Similarly in hummingbirds, slight twisting of  the jaw bones enables rapid closing of a wide open  beak.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content='  Further, such bifurcations organize a topo-  logically protected structure of saddle node mani-  folds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content=' As such, provided that the system trajectory  encircles the cusp bifurcation where the saddle node  manifolds meet, the system is guaranteed to exhibit  a smooth change in state followed by a snap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content='  In  the Venus fly trap and hummingbird examples, this  topological protection guarantees that the opening  and snapping of the trap or beak is robust against  variations in the applied hydrostatic or muscle forces  driving the transitions in the system state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content='  Here,  we propose that moving beyond cusp bifurcations  to design systems that operate near bifurcations of  arbitrarily many equilibria preserves the lever ad-  vantage and topological protection of cusp bifurca-  tions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content='  Such systems can be driven by only a few  control parameters to undergo snapping transitions  between multiple states making the design of ma-  chines near such bifurcations a powerful paradigm  for organizing complex functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content=' To develop and  demonstrate this paradigm, we experimentally in-  vestigate increasingly sophisticated magneto elastic  machines whose function is organized by such bifur-  cations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content='  We start by constructing a simple magneto elas-  tic machine consisting of a control panel that can  be translated in the x − y plane and a second panel  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content='01507v1  [cond-mat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content='soft]  4 Jan 2023  2  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content='  Magneto-Elastic  machine  capable  of  adopting multiple configurations due to operat-  ing near a cusp bifurcation (a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content=') System: Panels P1  and P2 are decorated with identical magnets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content=' Panel P1 is  actuated externally to translate in the x and y directions,  in response Panel P2 rotates about a hinge, the dynam-  ics are over-damped.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content='  (b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content=')  Magnetic potential energy  landscapes: We plot the potential for dx = −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content='09 and  dy ∈ [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content='28, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content='17, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content='02], where dx and dy are deviations  from the cusp’s position.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content=' Varying y we cross two saddle  node bifurcations where the number of extrema of the  magneto elastic landscape changes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content=' (c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content=')  Equilibrium  manifold: The system’s equilibria θ(dx, dy) are plotted  as a function of the deviation of the parameters, color  signifies the value of θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content=' Brown curve marks saddle node  bifurcations where the number of equilibria change, and  the light red curve denotes the experimental trajectory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content='  (d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content=') Experimentally observed snap-through transition:  The system follows a parametric trajectory marked by a  red curve and colored tube whose color denotes the pre-  dicted state θ, around a cusp bifurcation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content=' The colored  disks represent the experimentally measured state θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content=' As  expected a single snap through transition at a saddle  node bifurcation (curves colored according to the bifur-  cating state θ and converging at the cusp) is observed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content='  that is free to rotate about a hinge connecting the  two panels (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content=' 1a and experimental apparatus  schematic Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content='S1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content=' The state of the system, is given  by the angle θ between the panels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content=' By decorating  the panels with magnets, we are able to design a  magneto elastic landscape with different numbers of  minima as a function of the parameters x and y  (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content=' 1b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content='  Transitions between these minima cor-  respond to changes in the state of the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content=' To  understand the various pathways for making such  transitions we construct the manifold defined by the  local equilibria as a function of the parameters x and  y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content=' For this particular arrangement of magnets, we  calculate (see SI) that the resulting manifold has a  domain with multiple solutions delineated by saddle  node bifurcation curves (Brown).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content=' These curves in-  tersect and terminate at a cusp bifurcation beyond  which there is only a single equilibrium state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content=' By  translating the control panel in the x-y plane, the  system can undergo either smooth or abrupt changes  in θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content=' For example, starting the system at point (i)  and moving through points (ii-v), the hinge angle  increases smoothly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content=' A further slight increase in the  control parameter y, however, leads to an abrupt  transition from a high to a low angle, correspond-  ing to points (v) and (vi) respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content=' These pre-  dictions are born out by the experiments (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content=' 1d),  which also show a smooth increase in θ for a path-  way that encircles the cusp (i-v) and an abrupt tran-  sition in θ when crossing a saddle node curve (v-vi).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content='  In this 2D representation the system makes a transi-  tion when the color of the path (yellow) matches the  color denoting the state associated with the saddle  node curve (yellow).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content=' This magneto elastic mecha-  nism is reminiscent of the cocking and snapping of  a Venus flytrap or a humming bird’s beak.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content='  In addition to providing a mechanism for abrupt  transitions, operating near a cusp bifurcation creates  a lever mechanism where small variations in the con-  trol parameters lead to large variations in the system  state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content=' This mechanism resolves the generic problem  that creating large variations in the system state of-  ten requires unfeasibly large variations in the control  parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content=' Lever mechanisms are generic near bi-  furcations of equilibria since the magnitude of the  transition in the system state is typically propor-  tional to the square root of the parameter distance  from the bifurcation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content='  To characterize this lever mechanism in our exper-  iment, we map the snapping transition curves asso-  ciated with the saddle node bifurcations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content='  Specifi-  cally, for a given value of y (or x) we toggle x (y)  so that the system snaps back and forth, and record  the values of the control parameters x and y, and θ  immediately after each transition (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content=' 2a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content='  To test the scaling relations, we define the normal  a)  HingeAxis  N  s  Parameter  SystemState  X-ControlParameter  b)  [nrun qe] A  L  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content='2  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content='952.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content='2  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content='95  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content='2  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content='95  e[rad]-SystemState  c)  111  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content='9  [rad] - System State  Snap-through  S  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content='5  d)  LS  Cusp Point  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content='03  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content='3  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content='15  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content='06  0  dy [cm]-Control Parameter  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content='15  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content='093  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content='  Parametric levers The change in the state  of the system after a snap through transition near a  cusp bifurcation scales sub linearly with the normal  form parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content=' This sublinear scaling leads to large  variation of the state in response to small variation  of the system parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content='  a) Measurements of snap  through transitions near a cusp: The blue points mark  the state of the magneto elastic system of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content=' 1 af-  ter a snap through transition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content='  The dashed curve is a  fit of snap through transitions near a cusp bifurcation  to the data, derived from the normal form potential  ˜V = δθ4 + a2δθ2 + a1δθ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content='  The normal form parame-  ters a1 and a2 are locally given by re-scaled rotations  of dx and dy, which are the deviations of the parame-  ters away from the cusp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content=' b) Scaling laws near a cusp:  The predicted scaling laws are demonstrated by project-  ing the measurements and fit onto log-log plots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content=' Near  the cusp the system response to a1 acts as a giant lever,  ∂δθ/∂a1 ∼ 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content='  form parameters a1 and a2 as rotations of the dis-  placement of the parameters x and y from the cusp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content='  We then fit the predicted scaling form ∆θ ∝ √a2  and a1 ∝ a3/2  2  near a cusp to determine the cusp’s  position and the rotation of the normal form param-  eters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content=' The fitted model then predicts that ∆θ ∝ a1/3  1  (see SI).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content=' Because the scaling exponents for ∆θ are  fractions of unity, small variations of the parameters  along a1 and a2 lead to large variations of the sys-  tem’s state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content='  For example, in our experiments the  range of actuation for panel 1’s position is approx-  imately 1 cm and the range of angles accessible to  panel 2 is 180◦ or π radians.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content=' Near the bifurcation  a translation along a1 of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content='1% of its range (∼ 10µ  m) leads to a snap that changes θ by ∼ 5% of it  range (∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content='1 rad) providing a lever advantage of  ∼50 (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content=' 2b ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content='  The complexity of the actions achieved by such  magneto elastic mechanisms is dictated by the range  and number of stable states that the system can ac-  cess.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content=' This complexity can be achieved by designing  the magneto elastic potentials such that the system  operates near bifurcations between multiple states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content='  For example, working near a hypothetical symmetric  butterfly bifurcation associated with the potential  V = θ6 + a4θ4 + a2θ2 + a1θ should enable smooth  and abrupt transitions between three stable states  in any order depending on the chosen trajectory for  the control parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content=' 3a we show a cut  through parameter space of the saddle node surfaces  near this butterfly bifurcation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content=' If the system starts  in the S (Small) state and moves along the depicted  trajectory (black arrows), it would first snap to the  M (Medium) state when the system crosses the pur-  ple saddle node bifurcation and then the L (Large)  state when it crosses the green curve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content=' For the return  path, however, the system would transition from  the L minimum directly to the S minimum when  it crosses the yellow saddle node bifurcation curve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content='  Moreover, by working near the bifurcation, the lever  mechanism should allow for transitioning between  these distinct states within an accessible range of  experimental control parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content='  SEARCH ALGORITHM FOR  BIFURCATIONS OF MULTIPLE  EQUILIBRIA  To design parametric configurations correspond-  ing to bifurcations of multiple equilibria we develop  a search gradient continuation algorithm that takes  advantage of their nested structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content=' Bifurcations as-  sociated with k equilibria (minima plus maxima) are  degenerate singularities where the first k derivatives  of the potential vanish.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content=' Thus they can be found iter-  atively by searching for singularities of the potential  with increasing order, solving for one constraint at a  time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content=' We find that this method is especially efficient  in finding experimentally realizable parametric con-  figurations corresponding to bifurcations of multiple  equilibria.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content=' Moreover, this method naturally extends  to searching for bifurcations with desired properties  by introducing further constraints, for example opti-  mizing the robustness of the bifurcation’s associated  states to external noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content='  For ease of illustration we describe how to use this  approach to find the symmetrized butterfly bifurca-  tion described above with parameters a1, a2, a4 and  variable θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content=' For a random combination of parame-  ters we find an equilibrium angle where dV/dθ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content='  Generically, this point is part of a smooth man-  ifold over which this constraint holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content='  We then  vary a1, a2, a4 and θ within this manifold to min-  imize the next constraint |d2V/dθ2|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content='  The trajec-  tory follows the gradient of the second constraint as  closely as possible while maintaining the first con-  straint dV/dθ = 0 until we reach a point on a sad-  a)  b)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content='09 h  [rad]  System State  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content='05  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content='06  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content='04  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content='05  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content='1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content='18  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content='05  a2 [cm] - Normal Form Parameter  ] - System State  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content='09  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content='02  dx [cm]  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content='05.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content='  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content='06  ControlParameter  73  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content='1  80 [rad]  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content='04  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content='15  1×10-4  3×10-4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content='001  dy [cm] - Control Parameter  a,[cm] - Normal Form Parameter4  dle node surface, which is a manifold where both  the first and second constraints hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content='1 Minimizing  the third derivative within the saddle node manifold  maintains the first two constraints and allows for  finding a cusp bifurcation associated with two sta-  ble equilibria.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content=' Successive iterations allow for identi-  fying bifurcations between an increasing number of  equilibria and eventually the butterfly bifurcation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content='  Our gradient continuation algorithm adapts stan-  dard algorithms from the dynamical systems liter-  ature [1, 2, 15] and retools them to locally follow  the gradient of the unsatisfied constraint (see SI for  further details).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content=' We depict the resulting search path  in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content=' 3b, which highlights the fact that, indepen-  dent of the number of parameters, the search algo-  rithm follows a 1D trajectory, which is organized  by the nested structure of the intermediate bifurca-  tions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content=' These properties enable the algorithm to find  realizable bifurcations for systems with hundreds of  parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content='  THREE STATES AND THE BUTTERFLY  BIFURCATION  As a proof of concept for our approach we demon-  strate the construction and operation of a magneto  elastic machine with 3 stable states operating near  a bifurcation of multiple equilibria.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content=' The first step in  designing such a machine is to implement our gra-  dient continuation algorithm to design a magneto  elastic potential with a butterfly bifurcation between  three stable states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content=' To realize a system operating  near such a bifurcation where only three control pa-  rameters (x,y,z positions of panel 1) are actively var-  ied, we allowed the algorithm to also determine the  x,y positions of two of the nine magnets on panel  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content='2 With these seven parameters, the algorithm was  able to identify multiple butterfly bifurcations that  satisfied these criteria (See SI for details).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content='  Having found an appropriate butterfly bifurca-  tion, we use standard dynamical systems continua-  tion algorithms[1, 17] to compute and plot the saddle  node surfaces in the control parameter space (x,y,z)  near the bifurcation (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content=' 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content=' We find multiple dis-  tinct surfaces where the color denotes the angle θ  1 A local minimum of |∂2  θV | with respect to variation of all  parameters, that lies on the fixed point manifold will throw  the algorithm off, but this is a co-dimension m point for a  system with m parameters, and so highly unlikely.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content='  2 Typically, a butterfly bifurcation requires four control pa-  rameters to navigate between all of the stable states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content=' Here,  we have identified a nonlinear mapping of the three active  control parameters (x,y,z) onto the four dimensional space,  which enables transitions between arbitrary minima.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content='  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content='  Bifurcations of multiple equilibria.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content='  a)  Work cycle near a butterfly: A system operating near a  hypothetical symmetrized butterfly bifurcation can cycle  between three states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content=' The bifurcation is associated with  a potential V = θ6+a4θ4+a2θ2+a1θ and three accessible  states denoted by large (L), medium (M) and small (S).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content='  As the system follows the trajectory denoted by black ar-  rows with colored background marking its state θ, it cy-  cles between the three states snapping from S to M to L  and back to S by changing a2 and a1 while a4 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content=' The  snaps occur at saddle node bifurcations (colored curves)  whose color signifies the state θ of the minima that is an-  nihilated at each boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content=' b) Gradient Continuation  algorithm: The search algorithm finds bifurcations of  multiple equilibria by following a one dimensional curve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content='  Starting from a bifurcation of k equilibria the algorithm  searches for a bifurcation of k+1 equilibria by following a  curve in the augmented parameter space, tangent to the  gradient of |V k+1| in the kth bifurcation manifold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content=' We  draw a search for a butterfly bifurcation in its symmet-  ric normal form potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content='  The entire volume denotes  the equilibrium manifold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content=' Starting from a fixed point,  the algorithm finds a saddle node bifurcation (along the  white curve), Parameters are then varied on the saddle  node surface (yellow), and cusp surface (thin lines) to re-  spectively find a cusp bifurcation (along the gray curve)  and a swallow tail bifurcation (along the black curve)  near a butterfly bifurcation (black point).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content='  at which the saddle node bifurcation occurs3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content=' In-  structed by these surfaces, we design a cyclic path  3 There is further local data in the potential at a saddle node  D[a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content=']  a1  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content='0  L  s  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content='5  SL  SM  M  0  M  SML  12  SL  SL  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content='5  L  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content='0  Butterfly  Saddle node  Cusp  Equilibrium  a1  a4  a25  through the parameter space such that the system  snaps between the large, medium, and small min-  ima.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content=' The path color at each point denotes the sys-  tem state, θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content='  As with the cusp and symmetrized  butterfly bifurcations depictions in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content=' 1d and 3a,  transitions occur at intersections of the path and  saddle node surfaces where their colors match.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content=' We  note that for the generic butterfly bifurcation, the  surface structure can be quite complicated as shown  by the two projections in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content=' 4a,b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content='  In contrast  to the symmetrized butterfly bifurcation structure  (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content=' 2a), this complicated structure necessitate us-  ing all three control parameters x, y, and z, to design  a pathway that cycles between the three states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content=' Im-  portantly, despite the surface complexity the design  is robust.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content=' Specifically, since the trajectory crosses  surfaces, slight deviations in the control parameters  should still lead to similar snaps, snap sequences,  and ultimately the resulting complex actions of the  entire magneto elastic machine.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content='  Using the design parameters determined by our  search algorithm, we built a magneto elastic machine  similar to that depicted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content=' 1a, but with a dif-  ferent magnetic dipole pattern and with two of the  magnets in panel 1 displaced in the panel plane (See  SI).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content=' By following the theoretically predicted path, we  found three snap through transitions from small to  large, large to medium, and medium to small (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content=' 4c  and Movie S1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content=' Two of the transitions occurred at  the predicted locations, while the large to medium  transition was displaced by 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content='4 cm from its predicted  location.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content=' In addition, we found excellent fidelity be-  tween the predicted and measured angles θ for the  equilibrium states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content=' Using the same magneto elastic  machine, we also designed and demonstrated cycli-  cal paths with two transitions (See Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content='S3 and Movie  S3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content=' Finally, when the system was taken apart and  reassembled, we were able to reliably reproduce the  transitions associated with the designed trajectories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content='  DISCUSSION  The  experimental  validation  of  this  design  paradigm with a butterfly bifurcation of 5 equilib-  ria strongly supports the conjecture that this frame-  work could be extended to design systems perform-  ing increasingly sophisticated functions by operating  surface that can instruct the design of a trajectory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content='  For  example the sign of the third derivative of the potential  signals whether the state’s angle will increase or decrease as  it bifurcates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content=' Moreover, the merging of saddle node surfaces  can also be delineated by plotting the cusp bifurcations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content='  Here, we do not include this additional information for ease  of viewing  near bifurcations with a growing number of equilib-  ria.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content=' Potential energies with these increasingly rare  bifurcations can be found efficiently, because the gra-  dient continuation algorithm follows a one dimen-  sional search path.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content=' Moreover, the associated lever  mechanisms provide a design feature where the op-  eration of the machine will likely be confined to a  small parameter volume, enabling the execution of  these actions by realizable machines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content='  Microscopic magneto-elastic machines could prove  to be a useful instance of design instructed by bifur-  cations of multiple equilibria: An important emerg-  ing strategy for manufacturing microscopic and soft  machines is fabricating them using two dimensional  lithographic and printing techniques [18–22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content=' Such  fabrication techniques, however, restrict the imple-  mentation of compound mechanisms composed of  springs, cogs, screws etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content=' that are used to achieve  complex actions in traditional macroscale machines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content='  These lever mechanisms could be replaced with mag-  neto elastic mechanisms with lever advantages in-  duced by bifurcations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content=' Magnetic interactions are es-  pecially well suited for this purpose since they are  long ranged and not easily screened.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content=' This long range  allows for global changes to the conformation in re-  sponse to local actuation of system components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content='  Importantly, since bifurcations of multiple equi-  libria are notoriously sensitive to variations of pa-  rameters, there is a concern that a machine oper-  ating near such bifurcations will be very sensitive  to environmental noise, such as thermal vibrations,  as well as to fabrication precision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content=' Indeed, close to  a bifurcation the sensitivity of the system to varia-  tions of certain combinations of the system param-  eters grows exponentially as the number of asso-  ciated equilibria increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content='  Mathematically this is  captured by mapping the potential to a canonical  normal form via a change of coordinates [4, 5, Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content='  36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content='6] (see SI for derivation).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content=' Practically, however,  this increased sensitivity is often blunted outside of  the infinitesimal environment of the bifurcation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content=' At  a finite distance from the bifurcation the mapping  to the normal form or its linearization will often  cease to be valid because of other singularities of  the potential or the nonlinear fall off in the poten-  tial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content=' This non-linearity is especially pronounced in  keplerian potentials such as that of magnetic inter-  actions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content=' Critically, the saddle node manifolds coa-  lescing at the bifurcation are generically preserved  outside this radius of convergence as they are topo-  logically protected and can only annihilate at a cusp  or a bifurcation of more equilibria.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content=' Thus, operating  a machine near a bifurcation of multiple equilibria,  but at a finite distance from it, allows the design  of trajectories that take advantage of the multiple  saddle node transitions associated with it, and their  6  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content='  3-state Cycle Near Butterfly Bifurcation Point (a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content=') Theory The saddle node surfaces of a magneto-  elastic system with three active control parameters, x,y and z are plotted, their color denotes the angle θ at which  the snap occurs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content=' The system’s magnetic pattern is designed using the gradient continuation algorithm such that it  operates near a butterfly bifurcation where multiple saddle node surfaces coalesce, enabling multiple snap-through  transitions at the surfaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content=' A trajectory (colored tube with white arrows) is chosen such that the system snaps in  cycles between three states Large (L), Medium (M) and Small (S) angles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content=' The system’s predicted state is denoted by  the tube’s color.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content=' At intersections of the trajectory with a surface where their colors match the system is predicted  to snap to a new state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content=' (b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content=') Experimental demonstration: The colored dots mark the experimental value of the  system’s state as it follows the designed trajectory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content=' We observe three distinct transitions as predicted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content='  lever advantages, while avoiding the local exponen-  tial sensitivity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content='  Similarly, the sensitivity of a system designed near  a bifurcation of multiple equilibria to external noise  grows exponentially with the number of associated  states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content='  This growth in sensitivity arises from the  decrease in the potential barriers between adjacent  states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content=' For example in a potential with k equilibria  where all the potential barriers are of equal height,  and the minima are equally deep (which is propor-  tional to a Chebyshev polynomial of the first kind  of order k + 1) the barrier heights decay as 2−k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content='  This sensitivity seems prohibitive as we imagine im-  plementing this design principle to create systems  cycling between multiple states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content='  Despite this in-  creased sensitivity, however, we estimate that the  strength of magnetic interactions assures that mag-  neto elastic systems are robust to thermal noise at  the microscale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content=' Specifically, in magneto elastic sys-  tems the potential is proportional to the dipole-  dipole interaction strength µ0µ2L6/R3 of two mag-  nets with magnetic dipole densities µ panel size L  and typical distance between dipoles R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content='  Thermal  noise is then comparable to the magneto elastic po-  tential barrier height when the number of equilibria  k ∼ log2  �  µ0µ2L3/(R/L)3  kbT  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content=' The magnetic dipole den-  sities µ are of order 106A/m at the microscale [24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content='  The smallest two state door (equivalent to the de-  vice in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content=' 1a) that is robust to thermal noise is  then ∼ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content='1µm in size, approaching the size limit of  30nm for fabricating stable magnetic domains [25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content='  Conversely, a 100 µm machine will become sensitive  to thermal noise near a bifurcation of ∼ 40 equilib-  ria, that is 20 distinct states compressed in a span  of 100 degrees.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content='  Finally, the designs that we have implemented in  this paper assume operation in a low Reynolds num-  ber regime where inertia can be neglected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content=' In the  macroscale implementation this was achieved by at-  taching a damping panel immersed in a solution of  glycerol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content=' We expect our designs to work even bet-  ter as these machines are implemented at smaller  scales since the importance of inertia drops quadrat-  ically with the system size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content=' Operation of a 100 µm  scale machine in water, for example, would enable  the system to be in the low Re regime while operat-  ing at rates that are 1000 fold faster than those in  the macroscale experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content='  CONCLUSIONS  We have shown that the operation of multi-  parameter machines near bifurcations of multiple  equilibria allows them to efficiently and robustly cy-  cle between multiple conformation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content='  Moreover, we  developed a generic step-by-step framework to de-  sign and implement systems that operate near such  bifurcations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content='  Specifically, we: 1) created a search  algorithm that optimizes over fabrication and other  system parameters to enable operation near such bi-  furcations;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content=' 2) mapped the manifold of saddle node  bifurcations to determine a useful trajectory for the  machine operation and;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content=' 3) demonstrated the ro-  bustness of this approach by constructing and op-  erating a magneto elastic machine that can cycle  and robustly snap between multiple distinct con-  figurations in response to small variations of a few  control parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content='  Importantly, this design ap-  proach and step-by-step implementation is generic  and could be applied to many complex systems with  z [cm] - Control  Parameter  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content='2  L  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content='4 -  M  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content='4 -  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content='8  2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content='8  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content='6 -  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content='8 - [rad]  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content='0-  0  1  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content='5  MO  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content='5  y  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content='5  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content='5  y [cm] - Control Parameter  3  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content='0  2  X  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content='0  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content='5  X7  multiple interacting components ranging from arti-  ficial proteins, where the interactions are electro-  static, to neural networks (both biological and syn-  thetic) where the interactions are governed by net-  work topology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content='  Cycling between transitions in mechanical imple-  mentations of such systems can generate work or lo-  comotion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content=' If the system is over-damped, as is often  the case in microscopic systems operating in fluids,  work and locomotion can be achieved by coupling  the system to mechanisms that break time reversal  symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content='  These mechanisms include ratchets or  cilia-like flexible rods [26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content=' In the case of the mag-  neto elastic hinge described here, time reversal sym-  metry is broken by combining the smooth transla-  tions of the control panel with abrupt transitions in  the state of the dynamic panel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content=' In systems where  the control variable is not a mechanical parameter  time reversal symmetry can be broken by using the  angle as an effective dynamical variable governing a  system with multiple degrees of freedom such as is  often used to parameterize robot locomotion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content='  More broadly, it is interesting to consider the ex-  tension of our work to systems with a larger number  of dynamical variables (θ1, θ2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content=' ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content=' Here, we envision  that by working near bifurcations of multiple vari-  ables (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content=' elliptic umbilic bifurcations) one could  organize snaps between states separated along mul-  tiple variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content=' Such designs require extending our  search algorithm to multiple variables while main-  taining its low dimensional search path.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content='  Alterna-  tively, one could design mechanisms based on multi-  ple local bifurcations that are weakly coupled across  the machine.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content='  For example, one bifurcation of n  states could be used to control θ1 while a second  bifurcation of m states organizes the dynamics of  the variable θ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content=' By weakly coupling the panels, and  hence the variables θ1 and θ2, the machine can trans-  form between n × m states in a coordinated fashion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content='  Indeed this approach is already being implemented  for bifurcations with two states [27–29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content=' Increasing  the number of states associated with each variable  would enable a similarly rich landscape for machine  design with far fewer mechanical elements or panels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content='  Finally, it is interesting to consider whether this  design paradigm can be used to understand natural  systems beyond the Venus fly trap and humming-  bird beak.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content=' For example, molecular machines such  as proteins often transition between different config-  urations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content=' It is interesting to consider whether such  transitions can be thought of as snaps organized by  bifurcations of many states [3, 6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content=' As another ex-  ample, bifurcation theory has been implemented to  identify and explain epigenetic dynamics of cell dif-  ferentiation [30–32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content=' These approaches often focus  on consecutive 2-state bifurcations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content=' The results pre-  sented here however, suggest that a comparably sim-  ple evolutionary pathway could entail development  of multi-state bifurcations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content=' Such a structures could  allow the addition of new states while maintaining  the existing configuration through an evolutionary  process, similar to the path taken by the gradient  continuation algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content='  I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content='  MATERIALS AND METHODS:  Construction of experimental hinge system  Panel P1 is constrained to a set of linear trans-  lation stages that allow its position to be adjusted  manually to any x or y coordinates near the cusp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content='  For experiments near the butterfly bifurcation point,  an extra translation stage is attached to Panel P1 to  allow adjustment of its z coordinate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content=' Panel P2 is  attached to an OVA friction-less thrust air bushing  with a 13mm shaft.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content=' The air bushing is attached to  a fixed metal housing to limit Panel P2 to its ro-  tational degree of freedom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content=' A T-shaped paddle is  attached to the bottom of the shaft and immersed  in glycerol to introduce damping to the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content=' Ad-  ditionally, we position a Basler Ace aca3088-57um  area scan camera above the center of the air bushing  to take top-view images of the air bushing which are  then used to calculate the angle response of Panel  P2 to high precision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content='  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content='  Panels for experiments near cusp point  Each magnetic panel is constructed using two 1/16  in thick laser-cut acrylic pieces and nine grade N48  neodymium magnets of diameter 1/16 in and height  1/8 in.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content='  Magnets are arranged in a 3-by-3 square  lattice with lattice constant of 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content='5 cm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content='  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content='  Panels for experiments near butterfly point  Each magnetic panel is constructed using two 1/16  in thick laser-cut acrylic pieces and nine grade N48  neodymium magnets of diameter 1/8 in and height  1/8 in.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content=' Magnets are arranged in a 3-by-3 square lat-  tice with lattice constant of 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content='5 cm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content=' In panel P1 the  x, y position of two of the magnets is displaced ac-  cording to the design determined by the search algo-  rithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content=' The two magnets whose position is offset are  the magnet in the bottom row on the right column,  whose offsets are dx1 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content='418cm, dy1 = −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content='273cm,  and the magnet in the middle row on the left column,  with offsets dx2 = −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content='826cm, dy2 = −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content='986cm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content=' A  8  technical drawing illustrating the panels used for the  butterfly experiment is included in the SI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content='  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content=' Experimental Setup Sketch of the experimen-  tal system used for demonstration of cycles and angle  measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content=' Panel P1 is attached to a set of transla-  tion stages which allows us to implement the spatial con-  trol parameters in all experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content=' Panel P2 is attached  to an air bushing that is fixed in space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content=' An attachment  submerged in glycerol is added to the base of Panel P2  to introduce damping to the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content='  Angle measurements  A marker is attached to the top of the air bush-  ing, and a camera records the location of the marker  during the experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content=' At each given time, the mea-  sured angle is the determined by three points: cur-  rent marker location, location of the center of rota-  tion, and marker location at θ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content=' We calibrate  the system by recording the location of the pixel at  θ = 0 and several other distinct angles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content=' The pixel  location corresponding to the center of rotation is  obtained using a fitted circle through the calibra-  tion data points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content='  The resulting angle is then de-  duced from the three measured points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content=' This data  collection process is conducted in MATLAB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content='  Acknowledgments We thank Michael Brenner,  Chrisy Xiyu Du, Yan Yang, Robert Distasio, and  John Guckenheimer for inspiring discussions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content=' This  work was financially supported primarily by NSF  Grant DMREF-89228, NSF Grant EFRI-1935252,  NSF Grant CBET-2010118, Cornell Center for Ma-  terials Research DMR-1719875, and by Air Force  Office of Scientific Research Grant MURI: FA9550-  16-1-0031.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
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+page_content='  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content='  Vetterling,  and  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content='  P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content='  Flannery,  Numerical recipes 3rd edition: The art of scientific computing  (Cambridge university press, 2007).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content='  Supplemental material - Bifurcation instructed design of multistate machines  II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content='  CALCULATION OF THE POTENTIAL ENERGY LANDSCAPE  To model the dynamics of our experimental hinge system, we compute the potential energy landscape  arising from the dipole-dipole interactions between the magnets embedded in each panel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content=' The magnets used  in our experiments are well approximated by perfect dipoles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content=' Therefore, the potential energy for the system  is a sum of dipole-dipole interaction energies  V = −  �  i∈P 1  �  j∈P 2  µ0m2  4π|rij|3 [3( ˆmi · ˆrij)( ˆmj · ˆrij) − ˆmi · ˆmj] ,  (S1)  where µ0 is the vacuum permeability, m is the dipole strength (identical for all magnets), mi is the orientation  of dipole i, and rij is the distance between magnets i and j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content=' Note that the interaction energy for dipoles in  the same panel is constant, so we can restrict the sum to pairs of dipoles in different panels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content='  To derive the θ dependence of the energy landscape, we must write the dipole orientations and positions  in terms of our control parameters x, y, and z and the dynamical variable θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content=' The dipoles on P1 are always  oriented in the z-direction, while the dipoles on the rotating panel P2 have orientation that changes with θ:  ˆmi = δiˆz  ˆmj = δj{sin θ , 0, − cos θ},  (S2)  where δi = ±1 is the orientation of magnet i with respect to panel P1 (similar for δj).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content=' The positions of  individual dipoles are given by  ri = {xi, yi, 0} + {x, y, z}  rj = Rθ{xj, yj, 0},  (S3)  leading to interdipole distance rij = ri − rj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content=' Here xi and yi are the x − y positions of dipole i in panel  P1 (similar for xj, yj), x, y, and z are the coordinates of the control panel, and Rθ is the rotation matrix  corresponding to a rotation by angle θ about the y-axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content='  Together Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content=' (S1-S3) give the potential energy in terms of the hinge angle θ, our control parameters x,  y, and z, and design parameters xi, yi, δi, xj, yj, and δj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content=' Since the hinge experiment is heavily damped, θ  follows gradient dynamics ˙θ = ∂θV and the stable equilibrium angles are given by the local minima of the  potential landscape V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content='  2  III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content='  CUSP EXPERIMENTS  In the cusp experiments, Panel P1’s x and y positions are measured as displacements from their value  when the panels are 180◦ open, and are aligned along z and y such that the panel’s backs and bottoms  are parallel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content=' The magnets closest to the hinge axis are removed from it by 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content='75cm on both panels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content=' The  back of the cylindrical magnets are aligned with the panel’s backs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content=' The damping paddle used in the cusp  experiments have dimensions 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content='5cm by 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content='0cm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content='  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content='  Experimental estimation of the cusp point  We estimate the location of the cusp point as the bifurcation of the two measured saddle node curves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content=' We  map the saddle node curve by toggling x (y), for a given value of y (or x), so that the system snaps back  and forth, and record the values of the control parameters x and y, and θ immediately after each transition  (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content=' S2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content=' Moreover, to verify the position of the cusp we record the angle θ of the system before and after  snapping, and observe that the change in angle upon snapping disappears at the cusp point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content='  Finally, we inspect all data collected along the bifurcation curves as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content=' 2a in the main text,  and use a spline fit for the saddle-node bifurcations from L to S and the saddle-node bifurcations from S to  L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content=' We define the cusp point as the intersection of the two splines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content='  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content='  Single snap experiment  The mangeto elastic potential calculated for the experiment predicts a cusp at a slightly removed para-  metric position.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content=' The discrepency between the experimentally measured and theoretically predicted cusps  could be due to fabrication errors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content=' To effectively compare theory and experiment in this section only, we  parameterize the system as a function of its displacement from the cusp for both theory and experiment  using using dx and dy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content=' We then follow the predicted path by controlling panel P1’s x and y positions using  the translation stages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content=' We begin the experiment by letting the system maintain its equilibrium at the initial  dx, dy position.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content=' We then change the position of Panel P1 at a slow and steady rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content=' Angle measurements  are recorded at various locations in the loop as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content=' S1(a) (see also Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content=' 1 in the main text), and  the change in position is paused once the transition happens at point vi in order to let the system settle  down and obtain an accurate angle measurement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content=' We confirm that the system returns to its original state  once we return to the starting dx, dy position.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content='  C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content='  Scaling experiment  To fit the scaling relations, we use the the same section of the data set used for determining the location  of the experimental cusp point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content=' We neglect data in the nonlinear region of the saddle-node curves far away  from the cusp point, as well as data too close to the cusp point, where the errors due to measurement noise  are comparable to the distance to the cusp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content=' The data points used for the scaling relations are highlighted in  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content=' S2(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content=' The state parameter values used in the scaling analysis correspond to the angle measurements  obtained at the points right after the snap through transitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content='  IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content='  ONE DIMENSIONAL BIFURCATIONS OF EQUILIBRIA: NORMAL FORM AND  SCALING  The ability to design magneto elastic machines and control parameter pathways that robustly lead to  complex actions corroborates the validity of a new design paradigm: operation near bifurcations of multiple  equilibria.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content=' The demonstrated trajectories take advantage of the structure of available dynamics near bifur-  cations of equilibira.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content=' These bifurcations are the loci of multiple distinct coalescing saddle node manifolds,  as illustrated for the idealized symmetric butterfly bifurcation (Fig 3b in the main text).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content=' By weaving a  trajectory that crosses and avoids chosen saddle node bifurcations we design a pathway that leads to com-  plex actions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content=' The system then cycles through multiple states via small variations of the control parameters,  3  taking advantage of the multiple accessible lever mechanisms associated with these saddle node surfaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content='  The sensitivity of the realized design increases as the number of equilibria associated with the bifurcation  grows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content='  Butterfly, cusp and saddle node bifurcations are the first in a series of bifurcations of equilibria in  one-dimensional gradient systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content='  More generally, in systems with a single degree of freedom x, bifur-  cations of k equilibria are points in parameter space where the first k derivatives of the potential vanish,  {dV/dx, d2V/dx2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content=' , dkV/dxk} = ⃗0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content=' That is, they are equilibrium points satisfying k − 1 equations beyond  that of mechanical equilibrium dV/dx = 0 and therefore lie on a manifold of co-dimension k − 1 within the  equilibrium manifold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content=' The sensitivity of a bifurcation of k equilibria to variation in its parameters can be  estimated through the topological equivalence of the dynamics near it to those in a normal form potential  �V = ϕk+1 +  k−1  �  i=1  aiϕi,  (S4)  where the variable ϕ(θ) and normal form parameters ai(p) are coordinate transformations of the angle θ and  parameters p respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content=' The normal form describes the unfolding of the Taylor expansion of the potential  at the bifurcation V ∼ xk+1 by variations of the parameters [S1–S3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content=' The unfolded normal form potential  demonstrates that the parameteric environment of a codimension k bifurcation includes domains with 1 to  ⌈(k + 1)/2⌉ minima delineated by k saddle-node manifolds which coalesce at the bifurcation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content=' Moreover, it  implies scaling relations between the variation in the system’s state upon a snap through transition induced  by crossing a saddle node bifurcation associated with a codimension k − 1 bifurcation and the variation of a  normal form parameter that causes the snap:  δϕ ∝ a1/(k−m+1)  m  ,  m < k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content='  (S5)  Heuristically the scaling can be derived from the normal form by noting that near the bifurcation the kth  derivative of the potential must still vanish, and so δϕ2 ∼ ak.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content=' Similarly the next k − 1 derivatives must  progressively vanish, setting the scaling of am.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content=' An explicit proof is given in [S4] and summarized in [S5,  Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content=' 36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content='6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content=' These scaling relations carry over to the original variable and parameters near the bifurcation  where the maps ϕ(x) and am(p) are approximately linear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content=' Indeed, the scaling relations we experimentally  observed near a the cusp bifurcations are those of the systems state with the normal form parameters near  a bifurcation of three equilibria, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content=', a cusp [S4, S5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content='  These scaling relations imply that the sensitivity of the system to variations of parameters grows expo-  nentially with the number of associated equilibria.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content=' A system designed near a bifurcation of k equilibria can  toggle its state between order unity separated states, δϕ ∼ 1/2, in response to variations of the linear normal  form coefficient a1 of order 1/2k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content=' That is, both the potential lever advantage and the sensitivity to noise in  the parameters grow as the number of associated equilibria grows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content=' However, the parametric domain in which  the mapping to the normal form is linear is often very small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content=' The nonlinearity of the mapping often blunts  the sensitivity of the response.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content=' Thus, the increased lever advantage near bifurcations of multiple equilibria  is often not experimentally accessible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content=' Conversely the system is not so sensitive to parametric noise when  operated at a small parametric distance from the bifurcation about which it is designed, as demonstrated  by the reproducibility of the experimental three state system, which was easily constructed twice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content='  V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content='  CONTINUATION ALGORITHMS  To find bifurcations of multiple equilibria in the dynamics of our model system and to map out the saddle  node structure in the vicinity of the high-order point, we use a series of continuation algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content=' In one  dimension, a codimension k bifurcation point is defined by the vanishing of the first k derivatives of the  potential: ∂j  θV (θ∗, {ξi}) = 0 for j = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content=' , k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content=' These constraints define a codimenion k manifold in the  space of dynamical variables and parameters (θ∗, {ξi}).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content='  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content='  Traditional continuation  Standard continuation algorithms compute bifurcation curves by varying a small number of parameters,  and then projecting onto the bifurcation manifold [S1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content=' For example, suppose we have found a co-dimension  4  k bifurcation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content=' This requires the first k derivatives of the potential vanish, fixing θ∗ and k − 1 parame-  ters ξ1, ξ2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content=' , ξk−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content=' Varying an additional parameter ξk produces a line emanating from our initial point  (θ∗, {ξ}) = p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content=' The continuation algorithm maps out this line by (i) taking a step along the tangent vec-  tor Tk(p) to the curve, which is the null-vector of the gradient of the first k derivatives of the potential  Tk(p) ≡  �  ⃗v ∈ Rk+1 | ∀j ∈ (1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content=' , k), ⃗v · ∇θ,ξ1,ξ2,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content=',ξk∂j  θV = 0  �  and (ii) correcting this step using a Newton-  Raphson algorithm4 to search perpendicular to the step for a point where the first k derivatives of the po-  tential vanish.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content=' This approach can be used to progressively search for higher order bifurcation points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content=' For  example, a fixed-point can be continued until ∂2V (θ∗, {ξi})/∂θ2 vanishes, indicating a saddle node bifurca-  tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content=' Continuing the saddle-node can lead to a cusp bifurcation, which in turn might lead to a swallowtail  bifurcation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content=' In this way, progressively adding parameters and performing continuations of one-dimensional  curves can lead toward high-codimension bifurcation points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content=' Once we have found a high-order bifurcation  point, we use this algorithm to map out the saddle node surfaces nearby.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content=' The surfaces can in turn be used  to design cycles in control parameters that cause the system to perform desired snapping transitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content='  The standard continuation approach, however, has limitations for microscopic machine design.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content=' In partic-  ular, it has limited utility for finding the high-order bifurcation points near which our machine will operate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content='  In our model system we have many free parameters, including the positions of each of the magnets embed-  ded in the panels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content=' Varying a given experimental parameter does not guarantee we will find the next order  bifurcation point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content=' Instead we want to vary many parameters simultaneously, which greatly improves the  likelihood that a higher-order bifurcation point is contained within the search space and allows for a more  efficient approach toward that point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content=' We have developed a gradient continuation algorithm to carry out this  multi-parameter search.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content='  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content='  Design algorithm: Gradient continuation  The gradient continuation algorithm works as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content=' Suppose we have N parameters ξi in our system,  plus the degree-of-freedom θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content='  A point, p, where the first k derivatives of the potential vanish belongs  to a co-dimension k manifold in the full (N + 1)-dimensional augmented parameter space, composed of  the equilibrium state and control parameters, (θ∗, {ξi}).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content=' Starting from the point p, take a step along the  gradient of the k+1 derivative of the potential ∇θ,ξ1,ξ2,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content=',ξN ∂k+1  θ  V , projected onto the tangent surface to the  manifold at p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content=' The tangent surface is the null-space of the gradient of the first k derivatives of the potential5,  Tk,N(p) ≡  �  ⃗v ∈ RN+1 | ∀j ∈ (1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content=' , k), ⃗v · ∇θ,ξ1,ξ2,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content=',ξN ∂j  θV = 0  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content=' This procedure finds the step within the  co-dimension k manifold that maximizes the change in ∂k+1  θ  V , which we need to vanish in order to find the  next order bifurcation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content=' After the step, the algorithm performs a corrective Newton-Raphson search [S6],  constrained to the hyperplane T ⊥  k,N(p) perpendicular to the null-space, which returns to the codimension  k manifold on which the first k derivatives of the potential vanish.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content=' As in the standard continuation, this  approach is repeated to progressively find higher order bifurcation points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content=' A visualization of the gradient  search algorithm, applied to the potential V = θ6 + a4θ4 + a2θ2 + a1θ, is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content=' 3b in the main text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content='  VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content='  BUTTERFLY EXPERIMENTS  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content='  Butterfly panels  In the butterfly experiments, Panel P1’s x, y and z positions are measured as displacements from their  value when the panels are 180◦ open, the magnets closest to the hinge axis are removed from it by 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content='5cm  on both panels, the panels are aligned vertically, and the back of the cylindrical magnets on Panel P1 are  aligned with the center of the magnets on Panel P2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content=' This small change in magnet alignment (compared  4 Newton-Raphson(f, Ω, p) [S6] searches for the roots of the  functions f over the space Ω starting at the point p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content='  5 Notice that this algorithm uses all N parameters ξ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content=' , ξN  to search for a codimension k bifurcation, while the stan-  dard continuation in the previous section only used k pa-  rameters ξ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content=' , ξk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content=' The null-space Tk,N(p) has dimension  (N − k + 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content='  5  to the single snap experiment) is found to reduce the discrepancy between experiment and prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content=' An  illustration for the panels is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content=' S3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content=' The damping paddle has dimensions 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content='0cm by 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content='5cm for the  butterfly experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content=' The position of the magnets on panel P1 was changed such that the system operates  next to a butterfly bifurcation, as specified in the main text and in the following sections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content='  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content='  Application of the continuation algorithm  To find an experimentally feasible path and magnetic pattern, we implement the continuation algorithm  by first finding a butterfly point in parameter space, then validating the resulting pattern against known  experimental constraints (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content=' we require physically realizable panel angles and magnet positions).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content=' Before  each search using the continuation algorithm, we first randomly generate orientations of the 18 magnetic  dipoles on the two panels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content=' We also randomly select two magnets on Panel P1 to be displaced from their  lattice positions, by (dx1, dy1) and (dx2, dy2) respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content=' The search algorithm is always initialized with  the values {θ, dx, dy, dz, dx1, dy1, dx2, dy2} = {1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content='1rad, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content='5cm, −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content='25cm, 0, 0, 0, 0, 0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content='  Next,  we  let  the  algorithm  try  to  find  a  butterfly  bifurcation  point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content='  If  no  but-  terfly  point  can  be  found,  we  repeat  the  initialization  process  and  repeat  the  search  with  a  new  randomly  generated  magnetic  pattern.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content='  The  butterfly  point  corresponding  to  the  pattern  we  used  in  our  experiments  is  located  at  {θ, dx, dy, dz, dx1, dy1, dx2, dy2}  =  {2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content='131rad, −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content='355cm, −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content='304cm, −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content='824cm, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content='918cm, −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content='698cm, −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content='326cm, −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content='486cm}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content='  If the butterfly point is found, we investigate the potential plots at various points in parameter space near  the bifurcation point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content=' Specifically, we offset one or more of the 6 search parameters by ±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content='2 and find the  number of minima that exist between 0 to 180 degrees at each of these locations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content=' The potential plots at  locations with three minima are then inspected to decide the experimental feasibility of the pattern.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content=' Ideally,  all three minima are at least 5 degrees apart, and the smallest minimum is at least 5 degrees (for z = 0)  to prevent the panels from touching during experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content=' We also look for patterns with large triple-minima  regions, for example if three visibly deep minima can be observed when at least one parameter is changed  by ±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content='5cm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content='  After an experimentally feasible pattern is discovered, we manipulate the three experimentally controllable  parameters (x,y,z) continuously around the point with deepest triple minima and observe changes in our  model of the potential landscape.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content=' The design of the control path is guided by visualization of the saddle-node  surfaces mapped out using the standard continuation algorithm detailed above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content=' Several paths are tested in  the model to obtain the desired sequence of bifurcations and to optimize various properties of the transitions  (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content=' the magnitude of the snaps and depth of the minima).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content='  C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content='  Experiments for trajectories near a butterfly point  We set up the experiment by laser-cutting the holes for magnets at the exact locations corresponding to the  found dx1, dy1, dx2, dy2 values, which were 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content='418cm, −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content='273cm, −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content='826cm, and −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content='986cm respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content=' We  also add a translation stage to control Panel P1’s z position.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content=' We begin the experiment by following the exact  coordinates provided by the theoretically designed path.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content=' In the event that a predicted transition cannot be  seen using the predicted path coordinates (due to fabrication or calibration errors shifting the surface), we  translate the system further from the original predicted path to determine a more robust path that may  account for some shifting in coordinates due to experimental errors (for example see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content=' 4b in the main  text).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content=' Once an experimental path is shown to demonstrate the predicted behavior with the desired number  of state transitions, we record the locations for state transitions in experiment, and repeat the experiment  while slowing down the rate of change in x,y positions near the transitions to give the system enough time to  respond in the presence of large damping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content=' Those experiments show excellent qualitative agreement with the  theoretically designed paths, although the locations at which transitions happen and the equilibrium angle  of the panel are often shifted by a small amount due to experimental error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content='  6  D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content='  Additional operation mode: double-snap trajectories  The intricate saddle-node surface structure near the butterfly bifurcation enables a variety of snapping  behaviors with the same panel design, beyond the 3-state cycle presented in the main text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content=' Here we present  a second snapping sequence that was measured experimentally.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content='  By using the same trajectory in parameter space as the three-snap sequence in the main text, but traverses  the path in the reverse direction, we observe a two-snap sequence between small (S) and large (L) angles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content='  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content=' S4a shows this trajectory together with the same saddle surfaces from Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content=' 4a in the main text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content=' The  experimentally measured angles along this backward cycle are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content=' S4b (see also Movie S2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content=' Besides  a minor systematic shift in the angles of the L state, we find excellent fidelity between the predicted and  measured angles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content=' The snapping transitions occur almost exactly at the predicted locations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content='  Our example trajectories demonstrate that the saddle-node structure in the vicinity of a butterfly bifurca-  tions enables a great deal of flexibility in controlling state transitions of a mechanical system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content=' For practical  applications, further fine-tuning of the control trajectory can be used to optimize features the system’s  behavior (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content=', the positions of the steady states and their lifetimes in the presence of environmental noise).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content='  VII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content='  GENERALIZATIONS: MULTIDIMENSIONAL BIFURCATIONS AND SUPPLEMENTAL  SCALING BEHAVIOURS  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content='  Stopping conditions in higher dimensions  While our proof-of-concept experiment is limited to a hinge with a single degree of freedom (the opening  angle), our approach and gradient continuation algorithm are straightforward to apply to systems with  multiple degrees of freedom, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content=' a microscopic robot with multiple panels connected by elastic hinges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content=' The  cuspoidal bifurcations discussed in this paper also naturally appear in higher-dimensional gradient systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content='  However, the analytic criteria to classify them is somewhat more complicated: we can not simply search  for points where higher order derivatives of the potential vanish.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content=' In this section we will discuss stopping  criteria in higher dimensions, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content=' what quantities should we follow with the gradient continuation algorithm  to search for bifurcations of increasing order?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content='  With two or more degrees of freedom, a saddle-node bifurcation occurs when a fixed-point (stable or  unstable) collides with a saddle point, resulting in mutual annihilation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content=' This occurs when an eigenvalue of  the Hessian of the potential Aij = −∂θi∂θjV crosses 0 (here θi are the dynamical variables).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content=' For the purposes  of applying gradient continuation starting from a fixed point, it is therefore convenient to use det A as the  stopping criteria, since the determinant vanishes when an eigenvalue does.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content='  Near a saddle-node bifurcation, the state space can be decomposed (by the Center Manifold Theorem)  into (i) the invariant center manifold emanating from the fixed point along the direction of the critical  eigenvector (with eigenvalue 0) and (ii) a stable/unstable manifold on which the flows exponentially grow or  decay (for the purposes of machine design we generally want only stable directions).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content=' Due to the vanishing  eigenvalue, the dynamics on the center manifold are nonlinear at lowest order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content='  These dynamics can be  determined perturbatively by expanding the gradient of the potential, projecting onto the center manifold  and enforcing the invariance of the center manifold [S1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content=' Higher-order bifurcations occur when the center  manifold expansion coefficients vanish.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content=' For example, vanishing quadratic term indicates a cusp bifurcation,  vanishing cubic term indicates a swallowtail, and so on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content=' Thus these coefficients replace the higher-order  derivatives of the potential as the stopping criteria in the gradient continuation algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content=' Below we give  explicit expressions for these expansion coefficients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content='  Suppose we have an n-dimensional system θ ∈ Rn that undergoes a saddle node bifurcation at θ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content='  Near this point,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content=' the dynamics can be expanded as follows,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content='  ˙θ = A θ + F(θ),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content='  (S6)  where A is the Hessian of the potential (which has a zero eigenvalue) and F(θ) collects all quadratic and  7  higher-order terms in multilinear forms,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content='  F(θ) = 1  2B(θ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content=' θ) + 1  6C(θ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content=' θ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content=' θ) + 1  24D(θ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content=' θ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content=' θ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content=' θ) + O(||θ||5)  = 1  2  n  �  i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content='j=1  ∂2F(φ)  ∂φi∂φj  ����  φ=0  θiθj + 1  6  n  �  i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content='j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content='k=1  ∂3F(φ)  ∂φi∂φj∂φk  ����  φ=0  θiθjθk  + 1  24  n  �  i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content='j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content='k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content='l=1  ∂4F(φ)  ∂φi∂φj∂φk∂φl  ����  φ=0  θiθjθkθl + O(||θ||5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content='  (S7)  Let ψ and ϕ be the right and left eigenvectors corresponding to the zero eigenvalue: Aψ = 0 and AT ϕ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content='  The projection of θ onto the center manifold ϑ = ϕ · θ has dynamics  ˙ϑ = a2ϑ2 + a3ϑ3 + O(ϑ4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content='  (S8)  Following Kuznetsov, we derive the coefficients up to fourth order (third order is given in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content=' [S1]),  a2 = 1  2ϕ · B(ψ, ψ)  a3 = 1  6ϕ · C(ψ, ψ, ψ) + 1  2ϕ · B(ψ, b2)  a4 = 1  24ϕ · D(ψ, ψ, ψ, ψ) + 1  4ϕ · C(ψ, ψ, b2) + 1  8ϕ · B(b2, b2) + 1  6ϕ · B(ψ, b3),  (S9)  where  b2 = A−1  su  �  ψ[ϕ · B(ψ, ψ)] − B(ψ, ψ)  �  b3 = A−1  su  �  ψ[ϕ · C(ψ, ψ, ψ) + 3ϕ · B(ψ, b2)] + 3b2[ϕ · B(q, q)] − C(ψ, ψ, ψ) − 3B(ψ, b2)  �  (S10)  and A−1  su is the inverse of A restricted to the stable/unstable subspace (which doesn’t have zero eigenvalues).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content='  As mentioned above, vanishing a2 indicates a cusp, if a3 also vanishes we have a swallowtail, and if all three  coefficients are zero we have a butterfly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content=' The vectors b2 and b3 describe the curvature of the center manifold  in the full θ space, θ = qϑ + b2ϑ2/2 + b3ϑ3/6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content=' While these bifurcations are one dimensional (they occur on  the one-dimensional invariant center manifold), the curvature of the center manifold as we move further from  the bifurcation point could allow snapping between states with reasonable separation in multiple dimensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content='  In principle, this would enable machines to carry out work cycles near a butterfly bifurcation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content='  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content='  Scaling for the Thom’s seven: hyperbolic and elliptic umbilics  Beyond the quasi-one-dimensional bifurcations there are also cuspoidal bifurcations that are genuinely mul-  tidimensional.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content=' In two dimensions, for example, we have elliptic umbilic, hyperbolic umbilic, and parabolic  umbilic catastrophes (these together with the four one-dimensional bifurcations saddle-node, cusp, swallow-  tail, and butterfly make up the Thom seven).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content=' Like the cusp and butterfly bifurcations, the unfolding of the  normal form predicts and intricate saddle-surface structure describing how fixed-points and saddle-points  come together and collide in the vicinity of the bifurcation point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content=' These higher-dimensional bifurcations also  obey advantageous scaling laws, relating the changes in state to the variation of control parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content=' For  example, the normal form potentials for the elliptic and hyperbolic umbilics are  Velliptic = θ3  1  3 − θ1θ2  2 + a(θ2  1 + θ2  2) + bθ1 + cθ2  Vhyperbolic = θ3  1 + θ3  2 + aθ1θ2 + bθ1 + cθ2  (S11)  from which the follow scaling can be derived [S4],  δθ1, δθ2 ∼ a  b, c ∼ a2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content='  (S12)  Increasing the dimension further leads to even more cuspoidal bifurcations;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content=' these have been enumerated  by Arnold using an ADE classification [S7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content=' While the search criteria for such bifurcations is increasingly  complicated, they provide a rich design space for multi-component machines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content='  8  C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content='  Reynolds number scaling  The magnetic decorations in our experiments are arranged in each panel about a square lattice with  unit separation of 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content='5cm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content=' To explore over-damped, gradient dynamics, that are ubiquitous in microscopic  mechanisms, the rotating panel is attached to a paddle moving through a glycerol bath.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content=' The results of our  experiments then hold also for smaller systems in fluid with comparable kinematic viscosity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content=' If the system  is smaller by a factor Ω ≪ 1, the time ∆t it takes our macroscopic over-damped system, of typical size L,  to traverse an angular expanse ∆θ is equal to the time it takes a microscopic system, of size ΩL to traverse  the same angular expanse in the same liquid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content=' This comes about because both the viscous drag force and  the magnetic force between dipoles of magnetization M1 and M2, FDrag ∼ L2 ˙γ, Fdipole ∼ M1M2/R4, are  quadratic in the typical system sizes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content=' For over-damped dynamics this results in a length-scale independent  strain-rate, ˙γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content=' The system is over-damped if its Reynolds number Re = L2 ˙γ/ν, is smaller then 1, where ν  is the fluid’s kinematic viscosity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content=' The Reynold’s number of a miniaturized system is therefore smaller by a  factor of Ω2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content=' Reducing the system’s size can compensate for changes in the system’s composition, such as  embedding it in water rather than glycerol, or the growth of magnetic dipole strength density as the system  size decreases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content='  [S1] Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content=' Kuznetsov, “Topological equivalence, bifurcations, and structural stability of dynamical systems,” in  Elements of Applied Bifurcation Theory (Springer New York, New York, NY, 2004)  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content='  [S2] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content=' Guckenheimer and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content=' Holmes, Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields  (Springer New York, New York, NY, 1983)  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
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+page_content=' Bruce and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
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+page_content=' Giblin, Curves and Singularities: A Geometrical Introduction to Singularity Theory, 2nd  ed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
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+page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content=' Berry, Journal of Physics A: Mathematical and General 10, 2061 (1977)  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content='  [S5] DLMF, “NIST Digital Library of Mathematical Functions,” http://dlmf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content='nist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content='gov/, Release 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content='4 of 2022-01-15  (2022), f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content=' Olver, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content=' Olde Daalhuis, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content=' Lozier, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content=' Schneider, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content=' Boisvert, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content=' Clark, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content='  Miller, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content=' Saunders, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content=' Cohl, and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content=' McClain, eds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content='  [S6] W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content='  H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content='  Press,  S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content='  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content='  Teukolsky,  W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content='  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content='  Vetterling,  and  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content='  P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content='  Flannery,  Numerical recipes 3rd edition: The art of scientific computing (Cambridge university press, 2007)  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content='  [S7] V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content=' Arnol’d, “Bifurcations of equilibria,” in Dynamical Systems V, edited by V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content=' Arnol’d (Springer Berlin  Heidelberg, Berlin, Heidelberg, 1994) pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content=' 10–38  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content='  [S8] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content=' Smith, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content=' Yanega, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content=' Ruina, Journal of theoretical biology 282, 41 (2011)  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content='  9  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content=' S1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content=' Single snap-through mechanism (a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content=') As we vary the control parameters along a loop around the cusp  point as shown, we expect to see a single snap-through buckling behavior (point v to point vi) for each cycle, akin  to how hummingbirds use their beak to capture prey [S8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content=' (b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content=') The predicted potential energy curves for points  labeled from i to vi are presented.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content=' The saddle-node bifurcation occurs between v and vi as indicated by the arrow  in v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content=' (c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content=') We experimentally observe the predicted snap-through behavior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content=' Due to experimental errors, the location  of the cusp point is shifted, but we see excellent agreement between the theory and measurements after shifting the  coordinates to align the theoretical and experimental cusp points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content='  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content=' S2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content=' Snap Through transitions near a cusp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content=' These plots show the equilibrium angle recorded in experiments  following a snap-through transition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content=' The corresponding (x, y) denote the values of the control parameters at which  the snap-through occurred.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content=' (a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content=') Highlights the the data points used to fit the cusp scaling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content=' We exclude data far from  the cusp, where higher order terms in the normal form are non-negligible, and close to the cusp, where measurement  and fabrication error are comparable to the distance from the cusp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content=' (b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content=') Highlights the data corresponding to the  upper and lower saddle-node curves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content='  a)  b)  c)  ty  i  vi  iii  +  anel  y  L 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content='9  y  [rad] - System State  Snap-through  i:  = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content='606  vi: 0 = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content='653  c+↑  Snap-through ↑  +c  Snap-through  v  ii  S  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content='5  ii: θ = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content='841  v: θ = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content='830  LSI  h+ ↑  y  L!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content='  S  h+↑  yT  Cusp Point  iii  iv  c  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content='03  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content='3  m  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content='15  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content='06  0  dy [cm] - Control Parameter  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content='15  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content='09  iii: 0 = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content='854  iv: 0 = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content='864a)  b)  ScalingDataPoints  Upper Saddle Node Curve  UnusedDataPoints  O LowerSaddleNodeCurve  EstimatedCuspPoint  EstimatedCuspPoint  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content='85  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content='85  [rad] - System State  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content='8  [rad] - System State  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content='8  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content='75  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content='75  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content='7  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content='65  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content='65  0  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content='15  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content='15  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content='2  Q  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content='2  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content='55  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content='25  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content='55  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content='25  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content='8  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content='3  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content='8  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content='3  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content='35  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content='75  ,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content='75  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content='35  x [cm]  x [cm]  Control  y [cm] -  Control  Parameter  Parameter10  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content=' S3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content=' Butterfly panels: In the butterfly experiments, Panel P1’s x, y and z positions are measured as displacements  from their value when the panels are 180◦ open, the magnets closest to the hinge axis are removed from it by 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content='5cm  on both panels, the panels are aligned vertically and the back of the cylindrical magnets on Panel P1 are aligned with  the center of the magnets on Panel P2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content=' This small change in magnet alignment is found to reduce the discrepancy  between experiment and prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content='  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content=' S4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content='  2-state Cycle Near Butterfly Bifurcation Point (a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content=') Theory The saddle node surfaces of a magneto-  elastic system with three active control parameters, x,y and z are plotted, their color denotes the angle θ at which  the snap occurs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content=' The system’s magnetic pattern is designed using the gradient continuation algorithm such that it  operates near a butterfly bifurcation where multiple saddle node surfaces coalesce, enabling multiple snap-through  transitions at the surfaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content=' A trajectory (colored tube with white arrows) is chosen such that the system snaps back  and forth between two states with Large (L) and Small (S) angles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content=' This trajectory is identical to that for the 3-state  cycle in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content=' 4 in the main text, but the path is traversed in the opposite direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content=' The system’s predicted state is  denoted by the tube’s color.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content=' At intersections of the trajectory with a surface where their colors match the system is  predicted to snap to a new state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content=' (b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content=') Experimental demonstration: The colored dots mark the experimental value  of the system’s state as it follows the designed trajectory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content=' We observe two distinct transitions as predicted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content='  Hinge Axis  11  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content='.  X  dx2  Z  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
+page_content='5  5  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfi_0V/content/2301.01507v1.pdf'}
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+How to Allocate your Label Budget? Choosing between
+Active Learning and Learning to Reject in Anomaly Detection
+Lorenzo Perini,1 Daniele Giannuzzi, Jesse Davis 1
+1 KU Leuven, Department of Computer Science, DTAI & Leuven.AI, B-3000 Leuven, Belgium
+lorenzo.perini@kuleuven.be, danielegiannuzzi1998@gmail.com, jesse.davis@kuleuven.be
+Abstract
+Anomaly detection attempts at finding examples that deviate
+from the expected behaviour. Usually, anomaly detection is
+tackled from an unsupervised perspective because anomalous
+labels are rare and difficult to acquire. However, the lack of
+labels makes the anomaly detector have high uncertainty in
+some regions, which usually results in poor predictive perfor-
+mance or low user trust in the predictions. One can reduce
+such uncertainty by collecting specific labels using Active
+Learning (AL), which targets examples close to the detec-
+tor’s decision boundary. Alternatively, one can increase the
+user trust by allowing the detector to abstain from making
+highly uncertain predictions, which is called Learning to Re-
+ject (LR). One way to do this is by thresholding the detector’s
+uncertainty based on where its performance is low, which re-
+quires labels to be evaluated. Although both AL and LR need
+labels, they work with different types of labels: AL seeks
+strategic labels, which are evidently biased, while LR requires
+i.i.d. labels to evaluate the detector’s performance and set the
+rejection threshold. Because one usually has a unique label
+budget, deciding how to optimally allocate it is challenging.
+In this paper, we propose a mixed strategy that, given a budget
+of labels, decides in multiple rounds whether to use the bud-
+get to collect AL labels or LR labels. The strategy is based
+on a reward function that measures the expected gain when
+allocating the budget to either side. We evaluate our strategy
+on 18 benchmark datasets and compare it to some baselines.
+Introduction
+Anomaly detection is the task of automatically detect-
+ing examples that do not follow expected patterns (Chan-
+dola, Banerjee, and Kumar 2009). These examples, named
+anomalies, are usually indicative of critical events such as
+water leaks in stores (Perini, Vercruyssen, and Davis 2022),
+breakdowns in gas turbines (Zhao, Wen, and Li 2016), or
+failures in the petroleum extraction (Mart´ı et al. 2015). Such
+critical events usually come along with elevated (mainte-
+nance) costs or with substantial natural damages (e.g., dis-
+persion of petroleum or gas). Thus, detecting anomalies in
+time is a relevant task that limits such resource waste.
+Collecting labels, especially for anomalies, is often a
+hard task because anomalies are costly events (e.g., ma-
+chine failures cannot be voluntarily induced), or simply
+time-consuming (e.g., you may need to label 100s of exam-
+ples before getting an anomaly). Thus, anomaly detection is
+often tackled from an unsupervised perspective. However,
+the lack of labels usually forces the unsupervised detector
+to have high uncertainty on specific regions of the example
+space (Perini, Vercruyssen, and Davis 2020). High uncer-
+tainty is undesirable because it is often associated with poor
+predictive performance or reduced trust in the predictions.
+This uncertainty can be tackled in two complementary
+ways. On the one hand, one can try to learn a more accu-
+rate detector by acquiring a limited number of labels using
+Active Learning (AL) (Abe, Zadrozny, and Langford 2006).
+On the other hand, it is possible to increase the user trust
+in the detector’s outputs by allowing the detector to abstain
+from making a prediction when it is highly uncertain, which
+is called Learning to Reject (LR) (Hendrickx et al. 2021;
+De Stefano, Sansone, and Vento 2000). One way to do this
+is to set a rejection threshold on the detector’s uncertainty
+based on where its performance is poor (Cortes, DeSalvo,
+and Mohri 2016). However, evaluating the detector perfor-
+mance requires labels.
+Both of these approaches rely on labeled data. However,
+the types of labels needed for each approach are quite differ-
+ent. Many AL strategies rely on biased sampling strategies
+such as explicitly targeting acquiring labels, for example, for
+which the detector is highly uncertain (i.e., near the detec-
+tor’s current decision boundary) as these are known to yield
+better performance (Pimentel et al. 2020; Culotta and Mc-
+Callum 2005). Alas, using such labels to evaluate the detec-
+tor’s performance, as required when setting the threshold in
+LR, will yield a biased performance estimate and hence a
+sub-optimal threshold (Marrocco, Molinara, and Tortorella
+2007). Thus, if a user has a fixed budget for acquiring la-
+bels there is a tension between collecting (a) strategic labels
+that can be used to train a better detector, or (b) i.i.d. labels
+that can be used to evaluate performance and set a proper
+rejection threshold. Therefore, a data scientist is confronted
+with the challenging question of how they should optimally
+allocate their label budget between these two purposes.
+In this paper, we assume that the label budget can be
+split and allocated in multiple rounds. We introduce BAL-
+LAD (Budget allocation for Active Learning and Learning to
+reject in Anomaly Detection) a novel adaptive strategy that,
+in each allocation round, (1) measures the potential reward
+obtained by assigning the budget to either AL or LR, and (2)
+chooses the highest reward option to collect the labels.
+arXiv:2301.02909v1  [cs.LG]  7 Jan 2023
+
+Preliminaries and Related Work
+Anomaly Detection.
+Let X be a d−dimensional random
+variable with unknown p(X). We are given a dataset D =
+{x1, . . . , xn} with n examples and d features is drawn i.i.d.
+from p(X). Let V = {xn+1, . . . , xm} ∼i.i.d p(X), m >
+n, be a validation set. Let Y be the label random variable,
+such that Y |X = x indicates the class label (1 if anomaly,
+0 if normal) for x ∈ Rd. An anomaly detection problem
+is the task of finding an anomaly score function h: Rd →
+R and a threshold t ∈ R such that Y = h(t)(X), where
+h(t)(x) = 1 if h(x) ≥ t, 0 otherwise. Usually, one sets t
+based on the contamination factor γ, i.e. the proportion of
+anomalies (Perini, Buerkner, and Klami 2022).
+Pool-based Active Learning (AL).
+The goal of pool-
+based AL strategies is to reduce the detector’s uncertainty
+by selecting the most informative training instances. The
+AL approaches can be classified into 3 categories (Monarch
+2021): uncertainty-based sampling strategies aim to select
+the unlabeled data samples with the highest uncertainty (Ha-
+cohen, Dekel, and Weinshall 2022), diversity strategies cap-
+ture the diversity among the training data (Abe, Zadrozny,
+and Langford 2006; Dagan and Engelson 1995), combined
+strategies integrate the advantages of uncertainty-based and
+diversity-based criteria (Ebert, Fritz, and Schiele 2012).
+Learning to Reject (LR).
+The goal of a detector’s re-
+ject option is to abstain from making a prediction when
+a detector is too uncertain about predicting a test exam-
+ple (Hendrickx et al. 2021; Cortes, DeSalvo, and Mohri
+2016). Our goal is to develop a detector-agnostic strategy
+that does ambiguity rejection, as novelty rejection would
+reject all anomalies. Thus, we use a dependent rejector ar-
+chitecture (Chow 1970). We indicate by C(x) the detector’s
+confidence for predicting x ∈ V , and with τ ∈ [0, 1] the re-
+jection threshold. If the confidence is below τ, the prediction
+is rejected ht(x) = ®. Note that for appropriate inference,
+we need to collect validation labels randomly (i.i.d.).
+A strategy to allocate the label budget
+This paper tackles the following problem:
+Given: initially unlabeled training set D and validation set
+V , the dataset’s contamination factor γ, an anomaly de-
+tector h, and a label budget B;
+Do: decide whether, in each allocation round k, to acquire
+labels for D (AL) or for V (LR).
+Both training the detector with more labels (AL) and learn-
+ing a threshold using larger validation data (LR) improve the
+detector’s performance. However, choosing the side to max-
+imize such improvement is challenging for multiple reasons.
+First, it requires measuring the reward of either side, i.e. the
+expected gain in terms of the detector’s improvement. Sec-
+ond, the rewards need to be on a similar scale such that nei-
+ther side is privileged during the process. Third, comparing
+a standard detector to one with the reject option is challeng-
+ing because the latter needs ad-hoc metrics to overcome the
+problem of predicting three classes (anomaly, normal, re-
+ject) (Nadeem, Zucker, and Hanczar 2009).
+In this paper, we introduce BALLAD, a strategy that mea-
+sures the reward of allocating the budget for AL, i.e. collect-
+ing strategic labels on the training set, and for LR, i.e., col-
+lecting random labels on the validation set. Let B = k·b ∈ N
+be our labelling budget. We perform k rounds and the la-
+bels of b examples are queried in each round. We initialize
+the problem by (1) training the detector with no labels and
+setting a default rejection threshold, and (2) collecting b ran-
+dom labels for V (LR) and for D (AL) for a total of 2b labels.
+This allows us to compute the initial rewards by measuring
+how the detector varies from (1) to (2): for LR, we mea-
+sure the variation after re-setting the validation threshold;
+for AL, we measure the variation after re-training the detec-
+tor with the new labels. Then, we start the allocation loop. In
+each round, we allocate the budget to the option (LR or AL)
+with the highest reward, and we update the reward using the
+new labels. We propose two alternative reward functions: 1)
+the entropy reward looks at the detector’s probabilities, ei-
+ther for prediction (AL), or for rejection (LR); 2) the cosine
+reward considers the predicted class labels, either anomaly
+yes/no (AL), or rejection yes/no (LR).
+Measuring the reward
+Because we do not know how beneficial the next label al-
+location would be for the detector, we look at the past and
+measure the effect of the last allocation round. Our challenge
+is to design a reward function that reflects the gain when
+querying the labels. We use the following methods to derive
+the reward for both AL and LR, by using as detector’s prob-
+abilities either the probability of predicting anomaly (AL),
+or the probability of rejecting the example (LR). Similarly
+to Vercruyssen et al. (2022), we consider two scenarios:
+Entropy.
+Adding more labels has the ability to decrease
+the overall uncertainty of the anomaly detector. Thus, we
+measure the variation of the detector’s probabilities as:
+Re(k) = Ex∼X [|H(hk(x)) − H(hk−1(x))|] ,
+(1)
+where H(h(x)) = −p log2 p is the entropy of the detector’s
+probabilities p, and the subscript indicates the query round
+(for k > 2). A large difference in entropy means a large
+detector variation, which indicates a large impact of the new
+labels and, in turn, a large reward Re.
+Cosine.
+More directly, one can measure the impact of the
+labels in terms of variation of class predictions. Given the
+detector’s probabilities, we threshold them at 0.5 and assign
+value 1 to higher probabilities and 0 to lower ones. Thus, we
+measure the cosine similarity between different outputs as
+Rc(k) = ED∼X
+�
+1 −
+hk(D) · hk−1(D)
+∥hk(D)∥ · ∥hk−1(D)∥
+�
+,
+(2)
+where h(D) is a vector containing the outputs (0 or 1) by the
+detector h, and ∥·∥ is the Euclidean norm. This metric is less
+sensitive to little variations in the detector and discriminates
+more in case the new labels change the predicted class.
+
+Deriving the detector’s probabilities
+Measuring the reward needs some probabilities, which are
+not easy to derive due to the partially supervised setting. For
+both prediction and rejection, we exploit the squashing func-
+tion: given a positive real score s ∈ R+ and a threshold
+λ ∈ R+, the squashing function
+Sλ : R+ → (0, 1),
+Sλ(s) = 1 − 2− s2
+λ2
+maps s to a probability > 0.5 if s > λ, and ≤ 0.5 other-
+wise. Roughly speaking, Sλ calibrates the probabilities by
+centering λ as the decision threshold.
+Detector’s posterior probabilities.
+Given the contamina-
+tion factor γ, a common approach to set the threshold t is
+by forcing the detector to have a training positive class rate
+equal to γ. Thus, one can center the probabilities to t by
+transforming the anomaly scores h(x) through the squash-
+ing function:
+P(ht(x) = 1) = St(h(x)) s. t. t = Qh(1 − γ).
+We set t as the 1−γth quantile of the score distribution such
+that only a proportion of γ scores have P(ht(x) = 1) ≥ 0.5.
+Rejection probabilities.
+Given a validation set with some
+labels, we (1) set a specific detector confidence C(x), and (2)
+set the rejection threshold τ ∈ [0, 1]. For the former, we use
+the detector’s posterior probabilities:
+C(x) = 2 ×
+��P(ht(x) = 1) − 0.5
+�� ∈ [0, 1].
+Thus, the closer P(ht(x) = 1) is to 0.5 (high uncertainty),
+the lower the detector confidence. For the latter, we opti-
+mize the threshold τ over the validation set (only the la-
+beled examples) by minimizing a cost function M(ht). Fi-
+nally, we compute the rejection probabilities by centering 1-
+confidence values to the rejection threshold, i.e. by applying
+the squashing function
+P(ht(x) = ®) = Sτ(1 − C(x)).
+The cost-based evaluation metric
+Given a detector with a reject option and a detector without
+it, we cannot compare their performance on the non-rejected
+examples, as they would have different test sets. Thus, we
+introduce a cost-based evaluation metric. Formally, given a
+rejection cost cr > 0, a false positive cost cfp > 0, and a
+false negative cost cfn > 0, the detector is evaluated as:
+Mh = cr · P(ht(X) = ®) + cfp · P(ht(X) = 1|Y = 0)
++ cfn · P(ht(X) = 0|Y = 1).
+Note that we assume cost null for the correct predictions,
+while every misprediction as well as the rejection gets pe-
+nalized. Because rejecting is assumed to be less costly than
+mispredicting, the rejection cost needs to satisfy the inequal-
+ity cr ≤ min{cfp × (1 − γ), cfn × γ}, otherwise one could
+predict either always normal and pay an expected cost of
+cfn × γ, or always anomaly and pay cfp × (1 − γ).
+Experiments
+We experimentally answer the following questions:
+Q1. Does BALLAD result in lower costs when compared to
+using only AL or LR?
+Q2. Which reward metric is better?
+Q3. Is the reward function on a similar scale for AL and LR?
+Q4. How does our strategy behave when varying cfp, cfn ?
+Experimental setup
+Methods.
+We compare BALLAD1 to two baselines: ALL-
+IN-AL allocates all the budget for active learning and sets
+the rejection threshold using the (biased) training labels; on
+the contrary, ALL-IN-LR allocates all the budget for learn-
+ing to reject and uses an unlabeled training set.
+Table 1: Properties of the 18 datasets used.
+Dataset
+# Examples
+# Features
+γ
+ALOI
+12384
+27
+0.0304
+Annthyroid
+7129
+21
+0.0749
+Arrhythmia
+271
+259
+0.0996
+Cardiotocography
+1734
+21
+0.0496
+Glass
+214
+7
+0.0421
+InternetAds
+1682
+1555
+0.0499
+KDDCup99
+48113
+40
+0.0042
+PageBlocks
+5473
+10
+0.1023
+PenDigits
+9868
+16
+0.0020
+Pima
+526
+8
+0.0494
+Shuttle
+1013
+9
+0.0128
+SpamBase
+2661
+57
+0.0499
+Stamps
+340
+9
+0.0912
+WBC
+223
+9
+0.0448
+WDBC
+367
+30
+0.0272
+WPBC
+160
+33
+0.0562
+Waveform
+3443
+21
+0.0290
+Wilt
+4655
+5
+0.0199
+Data.
+We carry out our study on 18 publicly available
+benchmark datasets, which are widely used in the litera-
+ture (Campos et al. 2016). See Table 1 for the properties.
+Setup.
+For each of the 18 benchmark datasets, we go as
+follows: (i) we split the dataset into training, validation and
+test sets using the proportions 40 − 40 − 20 (we have a large
+validation set to better measure the impact of rejection); (ii)
+we fit the anomaly detector on the unlabeled dataset and set
+the rejection threshold to the default value of 0.1; (ii) we
+allocate a budget b to LR and AL by randomly selecting
+the initial examples; (iii) we optimize the rejection thresh-
+old and measure the LR reward; (iv) we train the anomaly
+detector on the partially labeled training set and measure the
+AL reward; (v) we allocate the next round budget b to the
+option with the highest reward and repeat (iii) or (iv) until
+the whole budget B is used. During each of the steps, we
+measure the detector performance on the test set using our
+1Code available at https://github.com/Lorenzo-Perini/Ballad
+
+cost function. We set B to the 30% of the training set’s size,
+and b to 2% of it, such that we run 15 allocation rounds. We
+repeat (i - v) 10 times and report the average results. In total
+we run 18 × 15 × 10 = 2700 experiments.
+Costs and hyperparameters.
+We set cfp = cfn = 1 and
+cr = γ, following the cost inequality. SSDO (Vercruyssen
+et al. 2018) with its default parameters is used as the semi-
+supervised anomaly detector (Soenen et al. 2021). We use
+IFOREST (Liu, Ting, and Zhou 2008) as its unsupervised
+prior. We use Uncertainty Sampling as the active learning
+strategy (Zhan et al. 2021), and the entropy as default re-
+ward. For setting the rejection threshold, we use Bayesian
+Optimization (GP MINIMIZE implemented in SKOPT) with
+20 calls (Frazier 2018) and limit the rejection rate on the
+validation set to 50%.
+Experimental results
+Q1. Comparing BALLAD to the ALL-IN strategies. Fig-
+ure 1 shows the comparison between BALLAD with the en-
+tropy reward and the ALL-IN strategies on the 18 bench-
+mark datasets. On 8 datasets (Arrhythmia, Glass, Kdd-
+Cup99, Pima, SpamBase, Wbc, Wdbc, Wpbc), BALLAD re-
+sults in evident lower costs, although sometimes the differ-
+ence is small. On 5 datasets (Cardiotocography, InternetAds,
+PageBlocks, Stamps, Waveform) BALLAD performs simi-
+lar/worse than ALL-IN-AL. This happens because SSDO has
+an overall high performance and a contained uncertainty in
+the predictions. On the other hand, in 3 cases (Aloi, Annthy-
+roid, Wilt), allocating all the budget for LR has a lower cost.
+This is due to the detector being inaccurate and unable to
+learn from the training labels, which makes learning an opti-
+mal threshold more convenient. As support for this intuition,
+we analyze the plain test AUC of SSDO on the whole test
+set (no rejection) for each of the three previous cases. By ag-
+gregating over the rounds, SSDO obtains an average AUC
+equal to 0.86, 0.88, and 0.57 when the best strategy is, re-
+spectively, BALLAD, ALL-IN-AL, and ALL-IN-LR. Finally,
+BALLAD obtains an overall average cost of 0.043, which is
+≈ 20% lower than the baselines’ average cost (0.055 for
+ALL-IN-AL, 0.054 for ALL-IN-LR).
+Q2. Which reward function works better? We analyze
+both types of reward functions that we introduced in Eq. 1
+and Eq. 2. Table 2 shows the mean and standard deviation
+of the cost, divided by allocation round. Overall, using the
+cosine reward builds a strategy that produces on average low
+costs for little budget (≤ 10%), whereas, for a higher bud-
+get, the entropy reward obtains better average costs. This is
+due to the highly imbalanced choices made by the cosine re-
+ward: the strategy opts for AL in 93% of the cases, which
+usually improves a lot the detector’s performance with few
+labels but tends to produce little effect when enough labels
+are given. On the other hand, the entropy reward is more bal-
+anced and opts for AL in 63% of the cases. This allows the
+detector to keep decreasing the costs while learning during
+the allocation rounds and obtain more steady performance.
+Q3. Is the entropy reward balanced for AL and LR? Fig-
+ure 2 shows the distribution of the difference between AL
+and LR entropy rewards over all the 2700 experiments. Neg-
+Budget
+Entropy Re
+Cosine Rc
+2%
+0.0536 ± 0.0401
+0.0399 ± 0.0303
+4%
+0.0465 ± 0.0330
+0.0411 ± 0.0334
+6%
+0.0443 ± 0.0284
+0.0398 ± 0.0321
+8%
+0.0436 ± 0.0303
+0.0399 ± 0.0306
+10%
+0.0420 ± 0.0299
+0.0411 ± 0.0325
+12%
+0.0416 ± 0.0303
+0.0433 ± 0.0347
+14%
+0.0413 ± 0.0306
+0.0448 ± 0.0367
+16%
+0.0408 ± 0.0288
+0.0456 ± 0.0372
+18%
+0.0403 ± 0.0290
+0.0457 ± 0.0355
+20%
+0.0412 ± 0.0297
+0.0451 ± 0.0363
+22%
+0.0407 ± 0.0301
+0.0451 ± 0.0359
+24%
+0.0421 ± 0.0325
+0.0438 ± 0.0361
+26%
+0.0417 ± 0.0345
+0.0438 ± 0.0363
+28%
+0.0416 ± 0.0332
+0.0438 ± 0.0380
+30%
+0.0418 ± 0.0345
+0.0427 ± 0.0354
+Table 2: Average (± std) cost per test example over the
+datasets grouped by allocation round for each of the two re-
+ward functions. For low budgets, the cosine reward obtains
+lower costs, while not being competitive for high budgets.
+ative values indicate that the LR reward is higher than AL’s
+one, while the opposite holds for positive values. Overall,
+the median is close to 0, which means that there is no clearly
+predominant strategy. Because the left tail of the density is
+larger than the right one, we conclude that LR rewards have
+higher variability (std = 0.07 vs 0.03).
+Q4. The impact of varying cfp, and cfn. In this experi-
+ment, we penalize more false positives and false negatives
+by setting, one at a time, cfp and cfn to 10. We compare
+BALLAD to the two ALL-IN baselines. For cfp = 10, our
+strategy is still the best for low budgets (< 15%), reduc-
+ing the relative cost by between 5% and 25% with respect
+to the runner-up ALL-IN-LR. However, for higher budgets
+(> 15%), ALL-IN-LR becomes the best strategy as it re-
+duces BALLAD’s cost by around 20% and ALL-IN-AL’s cost
+by more than 40%. This happens because the anomaly de-
+tector produces too many false positives, which, if rejected,
+allow us to reduce the cost. For cfn = 10, BALLAD performs
+much better than the baselines, reducing their cost by around
+20% (vs ALL-IN-LR) and 24% (vs ALL-IN-AL).
+Conclusion
+We proposed BALLAD, a novel strategy to decide whether
+to allocate the budget for Active Learning (AL), i.e. labeling
+strategic training instances, or for Learning to Reject (LR),
+i.e. labeling a random validation set. Our key insight is that
+we can measure the expected reward when labeling either set
+and allocate the label in the next round to the option with the
+highest reward. We proposed two reward functions (entropy
+and cosine similarity based). Experimentally, we evaluated
+BALLAD on 18 datasets, and show that it performs better
+than simply allocating all the labels to either AL or LR.
+
+0.03
+0.06
+0.09
+0.12
+Aloi
+0.03
+0.06
+0.09
+0.12
+Ann
+0.03
+0.06
+0.09
+0.12
+Arr
+AL-LR
+All-in AL
+All-in LR
+0.03
+0.06
+0.09
+0.12
+Car
+0.03
+0.06
+0.09
+0.12
+Glass
+0.03
+0.06
+0.09
+0.12
+Int
+0.03
+0.06
+0.09
+0.12
+Cost per test example
+Kdd
+0.03
+0.06
+0.09
+0.12
+Page
+0.03
+0.06
+0.09
+0.12
+Pen
+0.03
+0.06
+0.09
+0.12
+Pima
+0.03
+0.06
+0.09
+0.12
+Shu
+0.03
+0.06
+0.09
+0.12
+Spam
+4
+8
+12
+16
+20
+24
+28
+0.03
+0.06
+0.09
+0.12
+Stam
+4
+8
+12
+16
+20
+24
+28
+0.03
+0.06
+0.09
+0.12
+Wbc
+4
+8
+12
+16
+20
+24
+28
+Allocated budget (% of labeled examples)
+0.03
+0.06
+0.09
+0.12
+Wdbc
+4
+8
+12
+16
+20
+24
+28
+0.03
+0.06
+0.09
+0.12
+Wpbc
+4
+8
+12
+16
+20
+24
+28
+0.03
+0.06
+0.09
+0.12
+Wave
+4
+8
+12
+16
+20
+24
+28
+0.03
+0.06
+0.09
+0.12
+Wilt
+Figure 1: Comparison between BALLAD and the ALL-IN strategies on the 18 benchmarks. The x-axis reports the 15 rounds of
+2% labels each. The y-axis shows the average cost per test example. BALLAD obtains lower costs in the majority of cases.
+0.3
+0.2
+0.1
+0.0
+0.1
+0.2
+0.3
+AL reward - LR reward
+Density
+Median
+Figure 2: Distribution of the difference between AL’s and
+LR’s entropy reward. The median close to 0 indicates the
+absence of a predominant strategy.
+Acknowledgements.
+This work was presented at the 1st
+AAAI Workshop on Uncertainty Reasoning and Quantifica-
+tion in Decision Making (UDM23).
+This research is supported by an FB Ph.D. fellowship by
+FWO-Vlaanderen (grant 1166222N) [LP], the Flemish Gov-
+ernment under the “Onderzoeksprogramma Artifici¨ele Intel-
+ligentie (AI) Vlaanderen” programme [JD], and KUL Re-
+search Fund iBOF/21/075 [JD].
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diff --git a/JNE1T4oBgHgl3EQfGAOI/content/tmp_files/load_file.txt b/JNE1T4oBgHgl3EQfGAOI/content/tmp_files/load_file.txt
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+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfGAOI/content/2301.02909v1.pdf,len=675
+page_content='How to Allocate your Label Budget?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfGAOI/content/2301.02909v1.pdf'}
+page_content=' Choosing between  Active Learning and Learning to Reject in Anomaly Detection  Lorenzo Perini,1 Daniele Giannuzzi, Jesse Davis 1  1 KU Leuven, Department of Computer Science, DTAI & Leuven.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfGAOI/content/2301.02909v1.pdf'}
+page_content='AI, B-3000 Leuven, Belgium  lorenzo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfGAOI/content/2301.02909v1.pdf'}
+page_content='perini@kuleuven.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfGAOI/content/2301.02909v1.pdf'}
+page_content='be, danielegiannuzzi1998@gmail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfGAOI/content/2301.02909v1.pdf'}
+page_content='com, jesse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfGAOI/content/2301.02909v1.pdf'}
+page_content='davis@kuleuven.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfGAOI/content/2301.02909v1.pdf'}
+page_content='be  Abstract  Anomaly detection attempts at finding examples that deviate  from the expected behaviour.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfGAOI/content/2301.02909v1.pdf'}
+page_content=' Usually, anomaly detection is  tackled from an unsupervised perspective because anomalous  labels are rare and difficult to acquire.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfGAOI/content/2301.02909v1.pdf'}
+page_content=' However, the lack of  labels makes the anomaly detector have high uncertainty in  some regions, which usually results in poor predictive perfor-  mance or low user trust in the predictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfGAOI/content/2301.02909v1.pdf'}
+page_content=' One can reduce  such uncertainty by collecting specific labels using Active  Learning (AL), which targets examples close to the detec-  tor’s decision boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfGAOI/content/2301.02909v1.pdf'}
+page_content=' Alternatively, one can increase the  user trust by allowing the detector to abstain from making  highly uncertain predictions, which is called Learning to Re-  ject (LR).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfGAOI/content/2301.02909v1.pdf'}
+page_content=' One way to do this is by thresholding the detector’s  uncertainty based on where its performance is low, which re-  quires labels to be evaluated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfGAOI/content/2301.02909v1.pdf'}
+page_content=' Although both AL and LR need  labels, they work with different types of labels: AL seeks  strategic labels, which are evidently biased, while LR requires  i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfGAOI/content/2301.02909v1.pdf'}
+page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfGAOI/content/2301.02909v1.pdf'}
+page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfGAOI/content/2301.02909v1.pdf'}
+page_content=' labels to evaluate the detector’s performance and set the  rejection threshold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfGAOI/content/2301.02909v1.pdf'}
+page_content=' Because one usually has a unique label  budget, deciding how to optimally allocate it is challenging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfGAOI/content/2301.02909v1.pdf'}
+page_content='  In this paper, we propose a mixed strategy that, given a budget  of labels, decides in multiple rounds whether to use the bud-  get to collect AL labels or LR labels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfGAOI/content/2301.02909v1.pdf'}
+page_content=' The strategy is based  on a reward function that measures the expected gain when  allocating the budget to either side.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfGAOI/content/2301.02909v1.pdf'}
+page_content=' We evaluate our strategy  on 18 benchmark datasets and compare it to some baselines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfGAOI/content/2301.02909v1.pdf'}
+page_content='  Introduction  Anomaly detection is the task of automatically detect-  ing examples that do not follow expected patterns (Chan-  dola, Banerjee, and Kumar 2009).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfGAOI/content/2301.02909v1.pdf'}
+page_content=' These examples, named  anomalies, are usually indicative of critical events such as  water leaks in stores (Perini, Vercruyssen, and Davis 2022),  breakdowns in gas turbines (Zhao, Wen, and Li 2016), or  failures in the petroleum extraction (Mart´ı et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfGAOI/content/2301.02909v1.pdf'}
+page_content=' 2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfGAOI/content/2301.02909v1.pdf'}
+page_content=' Such  critical events usually come along with elevated (mainte-  nance) costs or with substantial natural damages (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfGAOI/content/2301.02909v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfGAOI/content/2301.02909v1.pdf'}
+page_content=', dis-  persion of petroleum or gas).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfGAOI/content/2301.02909v1.pdf'}
+page_content=' Thus, detecting anomalies in  time is a relevant task that limits such resource waste.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfGAOI/content/2301.02909v1.pdf'}
+page_content='  Collecting labels, especially for anomalies, is often a  hard task because anomalies are costly events (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfGAOI/content/2301.02909v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfGAOI/content/2301.02909v1.pdf'}
+page_content=', ma-  chine failures cannot be voluntarily induced), or simply  time-consuming (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfGAOI/content/2301.02909v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfGAOI/content/2301.02909v1.pdf'}
+page_content=', you may need to label 100s of exam-  ples before getting an anomaly).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfGAOI/content/2301.02909v1.pdf'}
+page_content=' Thus, anomaly detection is  often tackled from an unsupervised perspective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfGAOI/content/2301.02909v1.pdf'}
+page_content=' However,  the lack of labels usually forces the unsupervised detector  to have high uncertainty on specific regions of the example  space (Perini, Vercruyssen, and Davis 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfGAOI/content/2301.02909v1.pdf'}
+page_content=' High uncer-  tainty is undesirable because it is often associated with poor  predictive performance or reduced trust in the predictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfGAOI/content/2301.02909v1.pdf'}
+page_content='  This uncertainty can be tackled in two complementary  ways.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfGAOI/content/2301.02909v1.pdf'}
+page_content=' On the one hand, one can try to learn a more accu-  rate detector by acquiring a limited number of labels using  Active Learning (AL) (Abe, Zadrozny, and Langford 2006).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfGAOI/content/2301.02909v1.pdf'}
+page_content='  On the other hand, it is possible to increase the user trust  in the detector’s outputs by allowing the detector to abstain  from making a prediction when it is highly uncertain, which  is called Learning to Reject (LR) (Hendrickx et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfGAOI/content/2301.02909v1.pdf'}
+page_content=' 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfGAOI/content/2301.02909v1.pdf'}
+page_content='  De Stefano, Sansone, and Vento 2000).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfGAOI/content/2301.02909v1.pdf'}
+page_content=' One way to do this  is to set a rejection threshold on the detector’s uncertainty  based on where its performance is poor (Cortes, DeSalvo,  and Mohri 2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfGAOI/content/2301.02909v1.pdf'}
+page_content=' However, evaluating the detector perfor-  mance requires labels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfGAOI/content/2301.02909v1.pdf'}
+page_content='  Both of these approaches rely on labeled data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfGAOI/content/2301.02909v1.pdf'}
+page_content=' However,  the types of labels needed for each approach are quite differ-  ent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfGAOI/content/2301.02909v1.pdf'}
+page_content=' Many AL strategies rely on biased sampling strategies  such as explicitly targeting acquiring labels, for example, for  which the detector is highly uncertain (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfGAOI/content/2301.02909v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfGAOI/content/2301.02909v1.pdf'}
+page_content=', near the detec-  tor’s current decision boundary) as these are known to yield  better performance (Pimentel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfGAOI/content/2301.02909v1.pdf'}
+page_content=' 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfGAOI/content/2301.02909v1.pdf'}
+page_content=' Culotta and Mc-  Callum 2005).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfGAOI/content/2301.02909v1.pdf'}
+page_content=' Alas, using such labels to evaluate the detec-  tor’s performance, as required when setting the threshold in  LR, will yield a biased performance estimate and hence a  sub-optimal threshold (Marrocco, Molinara, and Tortorella  2007).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfGAOI/content/2301.02909v1.pdf'}
+page_content=' Thus, if a user has a fixed budget for acquiring la-  bels there is a tension between collecting (a) strategic labels  that can be used to train a better detector, or (b) i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfGAOI/content/2301.02909v1.pdf'}
+page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfGAOI/content/2301.02909v1.pdf'}
+page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfGAOI/content/2301.02909v1.pdf'}
+page_content=' labels  that can be used to evaluate performance and set a proper  rejection threshold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfGAOI/content/2301.02909v1.pdf'}
+page_content=' Therefore, a data scientist is confronted  with the challenging question of how they should optimally  allocate their label budget between these two purposes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfGAOI/content/2301.02909v1.pdf'}
+page_content='  In this paper, we assume that the label budget can be  split and allocated in multiple rounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfGAOI/content/2301.02909v1.pdf'}
+page_content=' We introduce BAL-  LAD (Budget allocation for Active Learning and Learning to  reject in Anomaly Detection) a novel adaptive strategy that,  in each allocation round, (1) measures the potential reward  obtained by assigning the budget to either AL or LR, and (2)  chooses the highest reward option to collect the labels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfGAOI/content/2301.02909v1.pdf'}
+page_content='  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfGAOI/content/2301.02909v1.pdf'}
+page_content='02909v1  [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfGAOI/content/2301.02909v1.pdf'}
+page_content='LG]  7 Jan 2023  Preliminaries and Related Work  Anomaly Detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfGAOI/content/2301.02909v1.pdf'}
+page_content='  Let X be a d−dimensional random  variable with unknown p(X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfGAOI/content/2301.02909v1.pdf'}
+page_content=' We are given a dataset D =  {x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfGAOI/content/2301.02909v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfGAOI/content/2301.02909v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfGAOI/content/2301.02909v1.pdf'}
+page_content=' , xn} with n examples and d features is drawn i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfGAOI/content/2301.02909v1.pdf'}
+page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfGAOI/content/2301.02909v1.pdf'}
+page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfGAOI/content/2301.02909v1.pdf'}
+page_content='  from p(X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfGAOI/content/2301.02909v1.pdf'}
+page_content=' Let V = {xn+1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfGAOI/content/2301.02909v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfGAOI/content/2301.02909v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfGAOI/content/2301.02909v1.pdf'}
+page_content=' , xm} ∼i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfGAOI/content/2301.02909v1.pdf'}
+page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfGAOI/content/2301.02909v1.pdf'}
+page_content='d p(X), m >  n, be a validation set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfGAOI/content/2301.02909v1.pdf'}
+page_content=' Let Y be the label random variable,  such that Y |X = x indicates the class label (1 if anomaly,  0 if normal) for x ∈ Rd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfGAOI/content/2301.02909v1.pdf'}
+page_content=' An anomaly detection problem  is the task of finding an anomaly score function h: Rd →  R and a threshold t ∈ R such that Y = h(t)(X), where  h(t)(x) = 1 if h(x) ≥ t, 0 otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfGAOI/content/2301.02909v1.pdf'}
+page_content=' Usually, one sets t  based on the contamination factor γ, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfGAOI/content/2301.02909v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfGAOI/content/2301.02909v1.pdf'}
+page_content=' the proportion of  anomalies (Perini, Buerkner, and Klami 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfGAOI/content/2301.02909v1.pdf'}
+page_content='  Pool-based Active Learning (AL).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfGAOI/content/2301.02909v1.pdf'}
+page_content='  The goal of pool-  based AL strategies is to reduce the detector’s uncertainty  by selecting the most informative training instances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfGAOI/content/2301.02909v1.pdf'}
+page_content=' The  AL approaches can be classified into 3 categories (Monarch  2021): uncertainty-based sampling strategies aim to select  the unlabeled data samples with the highest uncertainty (Ha-  cohen, Dekel, and Weinshall 2022), diversity strategies cap-  ture the diversity among the training data (Abe, Zadrozny,  and Langford 2006;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfGAOI/content/2301.02909v1.pdf'}
+page_content=' Dagan and Engelson 1995), combined  strategies integrate the advantages of uncertainty-based and  diversity-based criteria (Ebert, Fritz, and Schiele 2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfGAOI/content/2301.02909v1.pdf'}
+page_content='  Learning to Reject (LR).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfGAOI/content/2301.02909v1.pdf'}
+page_content='  The goal of a detector’s re-  ject option is to abstain from making a prediction when  a detector is too uncertain about predicting a test exam-  ple (Hendrickx et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfGAOI/content/2301.02909v1.pdf'}
+page_content=' 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfGAOI/content/2301.02909v1.pdf'}
+page_content=' Cortes, DeSalvo, and Mohri  2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfGAOI/content/2301.02909v1.pdf'}
+page_content=' Our goal is to develop a detector-agnostic strategy  that does ambiguity rejection, as novelty rejection would  reject all anomalies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfGAOI/content/2301.02909v1.pdf'}
+page_content=' Thus, we use a dependent rejector ar-  chitecture (Chow 1970).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfGAOI/content/2301.02909v1.pdf'}
+page_content=' We indicate by C(x) the detector’s  confidence for predicting x ∈ V , and with τ ∈ [0, 1] the re-  jection threshold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfGAOI/content/2301.02909v1.pdf'}
+page_content=' If the confidence is below τ, the prediction  is rejected ht(x) = ®.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfGAOI/content/2301.02909v1.pdf'}
+page_content=' Note that for appropriate inference,  we need to collect validation labels randomly (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfGAOI/content/2301.02909v1.pdf'}
+page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfGAOI/content/2301.02909v1.pdf'}
+page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfGAOI/content/2301.02909v1.pdf'}
+page_content=' ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfGAOI/content/2301.02909v1.pdf'}
+page_content='  A strategy to allocate the label budget  This paper tackles the following problem:  Given: initially unlabeled training set D and validation set  V , the dataset’s contamination factor γ, an anomaly de-  tector h, and a label budget B;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfGAOI/content/2301.02909v1.pdf'}
+page_content='  Do: decide whether, in each allocation round k, to acquire  labels for D (AL) or for V (LR).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfGAOI/content/2301.02909v1.pdf'}
+page_content='  Both training the detector with more labels (AL) and learn-  ing a threshold using larger validation data (LR) improve the  detector’s performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfGAOI/content/2301.02909v1.pdf'}
+page_content=' However, choosing the side to max-  imize such improvement is challenging for multiple reasons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfGAOI/content/2301.02909v1.pdf'}
+page_content='  First, it requires measuring the reward of either side, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfGAOI/content/2301.02909v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfGAOI/content/2301.02909v1.pdf'}
+page_content=' the  expected gain in terms of the detector’s improvement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfGAOI/content/2301.02909v1.pdf'}
+page_content=' Sec-  ond, the rewards need to be on a similar scale such that nei-  ther side is privileged during the process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfGAOI/content/2301.02909v1.pdf'}
+page_content=' Third, comparing  a standard detector to one with the reject option is challeng-  ing because the latter needs ad-hoc metrics to overcome the  problem of predicting three classes (anomaly, normal, re-  ject) (Nadeem, Zucker, and Hanczar 2009).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfGAOI/content/2301.02909v1.pdf'}
+page_content='  In this paper, we introduce BALLAD, a strategy that mea-  sures the reward of allocating the budget for AL, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfGAOI/content/2301.02909v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfGAOI/content/2301.02909v1.pdf'}
+page_content=' collect-  ing strategic labels on the training set, and for LR, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfGAOI/content/2301.02909v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfGAOI/content/2301.02909v1.pdf'}
+page_content=', col-  lecting random labels on the validation set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfGAOI/content/2301.02909v1.pdf'}
+page_content=' Let B = k·b ∈ N  be our labelling budget.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfGAOI/content/2301.02909v1.pdf'}
+page_content=' We perform k rounds and the la-  bels of b examples are queried in each round.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfGAOI/content/2301.02909v1.pdf'}
+page_content=' We initialize  the problem by (1) training the detector with no labels and  setting a default rejection threshold, and (2) collecting b ran-  dom labels for V (LR) and for D (AL) for a total of 2b labels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfGAOI/content/2301.02909v1.pdf'}
+page_content='  This allows us to compute the initial rewards by measuring  how the detector varies from (1) to (2): for LR, we mea-  sure the variation after re-setting the validation threshold;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfGAOI/content/2301.02909v1.pdf'}
+page_content='  for AL, we measure the variation after re-training the detec-  tor with the new labels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfGAOI/content/2301.02909v1.pdf'}
+page_content=' Then, we start the allocation loop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfGAOI/content/2301.02909v1.pdf'}
+page_content=' In  each round, we allocate the budget to the option (LR or AL)  with the highest reward, and we update the reward using the  new labels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfGAOI/content/2301.02909v1.pdf'}
+page_content=' We propose two alternative reward functions: 1)  the entropy reward looks at the detector’s probabilities, ei-  ther for prediction (AL), or for rejection (LR);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfGAOI/content/2301.02909v1.pdf'}
+page_content=' 2) the cosine  reward considers the predicted class labels, either anomaly  yes/no (AL), or rejection yes/no (LR).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfGAOI/content/2301.02909v1.pdf'}
+page_content='  Measuring the reward  Because we do not know how beneficial the next label al-  location would be for the detector, we look at the past and  measure the effect of the last allocation round.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfGAOI/content/2301.02909v1.pdf'}
+page_content=' Our challenge  is to design a reward function that reflects the gain when  querying the labels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfGAOI/content/2301.02909v1.pdf'}
+page_content=' We use the following methods to derive  the reward for both AL and LR, by using as detector’s prob-  abilities either the probability of predicting anomaly (AL),  or the probability of rejecting the example (LR).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfGAOI/content/2301.02909v1.pdf'}
+page_content=' Similarly  to Vercruyssen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfGAOI/content/2301.02909v1.pdf'}
+page_content=' (2022), we consider two scenarios:  Entropy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfGAOI/content/2301.02909v1.pdf'}
+page_content='  Adding more labels has the ability to decrease  the overall uncertainty of the anomaly detector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfGAOI/content/2301.02909v1.pdf'}
+page_content=' Thus, we  measure the variation of the detector’s probabilities as:  Re(k) = Ex∼X [|H(hk(x)) − H(hk−1(x))|] ,  (1)  where H(h(x)) = −p log2 p is the entropy of the detector’s  probabilities p, and the subscript indicates the query round  (for k > 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfGAOI/content/2301.02909v1.pdf'}
+page_content=' A large difference in entropy means a large  detector variation, which indicates a large impact of the new  labels and, in turn, a large reward Re.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfGAOI/content/2301.02909v1.pdf'}
+page_content='  Cosine.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfGAOI/content/2301.02909v1.pdf'}
+page_content='  More directly, one can measure the impact of the  labels in terms of variation of class predictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfGAOI/content/2301.02909v1.pdf'}
+page_content=' Given the  detector’s probabilities, we threshold them at 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfGAOI/content/2301.02909v1.pdf'}
+page_content='5 and assign  value 1 to higher probabilities and 0 to lower ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfGAOI/content/2301.02909v1.pdf'}
+page_content=' Thus, we  measure the cosine similarity between different outputs as  Rc(k) = ED∼X  �  1 −  hk(D) · hk−1(D)  ∥hk(D)∥ · ∥hk−1(D)∥  �  ,  (2)  where h(D) is a vector containing the outputs (0 or 1) by the  detector h, and ∥·∥ is the Euclidean norm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfGAOI/content/2301.02909v1.pdf'}
+page_content=' This metric is less  sensitive to little variations in the detector and discriminates  more in case the new labels change the predicted class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfGAOI/content/2301.02909v1.pdf'}
+page_content='  Deriving the detector’s probabilities  Measuring the reward needs some probabilities, which are  not easy to derive due to the partially supervised setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfGAOI/content/2301.02909v1.pdf'}
+page_content=' For  both prediction and rejection, we exploit the squashing func-  tion: given a positive real score s ∈ R+ and a threshold  λ ∈ R+, the squashing function  Sλ : R+ → (0, 1),  Sλ(s) = 1 − 2− s2  λ2  maps s to a probability > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfGAOI/content/2301.02909v1.pdf'}
+page_content='5 if s > λ, and ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfGAOI/content/2301.02909v1.pdf'}
+page_content='5 other-  wise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfGAOI/content/2301.02909v1.pdf'}
+page_content=' Roughly speaking, Sλ calibrates the probabilities by  centering λ as the decision threshold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfGAOI/content/2301.02909v1.pdf'}
+page_content='  Detector’s posterior probabilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfGAOI/content/2301.02909v1.pdf'}
+page_content='  Given the contamina-  tion factor γ, a common approach to set the threshold t is  by forcing the detector to have a training positive class rate  equal to γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfGAOI/content/2301.02909v1.pdf'}
+page_content=' Thus, one can center the probabilities to t by  transforming the anomaly scores h(x) through the squash-  ing function:  P(ht(x) = 1) = St(h(x)) s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfGAOI/content/2301.02909v1.pdf'}
+page_content=' t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfGAOI/content/2301.02909v1.pdf'}
+page_content=' t = Qh(1 − γ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfGAOI/content/2301.02909v1.pdf'}
+page_content='  We set t as the 1−γth quantile of the score distribution such  that only a proportion of γ scores have P(ht(x) = 1) ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfGAOI/content/2301.02909v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfGAOI/content/2301.02909v1.pdf'}
+page_content='  Rejection probabilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfGAOI/content/2301.02909v1.pdf'}
+page_content='  Given a validation set with some  labels, we (1) set a specific detector confidence C(x), and (2)  set the rejection threshold τ ∈ [0, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfGAOI/content/2301.02909v1.pdf'}
+page_content=' For the former, we use  the detector’s posterior probabilities:  C(x) = 2 ×  ��P(ht(x) = 1) − 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfGAOI/content/2301.02909v1.pdf'}
+page_content='5  �� ∈ [0, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfGAOI/content/2301.02909v1.pdf'}
+page_content='  Thus, the closer P(ht(x) = 1) is to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfGAOI/content/2301.02909v1.pdf'}
+page_content='5 (high uncertainty),  the lower the detector confidence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfGAOI/content/2301.02909v1.pdf'}
+page_content=' For the latter, we opti-  mize the threshold τ over the validation set (only the la-  beled examples) by minimizing a cost function M(ht).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfGAOI/content/2301.02909v1.pdf'}
+page_content=' Fi-  nally, we compute the rejection probabilities by centering 1-  confidence values to the rejection threshold, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfGAOI/content/2301.02909v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfGAOI/content/2301.02909v1.pdf'}
+page_content=' by applying  the squashing function  P(ht(x) = ®) = Sτ(1 − C(x)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfGAOI/content/2301.02909v1.pdf'}
+page_content='  The cost-based evaluation metric  Given a detector with a reject option and a detector without  it, we cannot compare their performance on the non-rejected  examples, as they would have different test sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfGAOI/content/2301.02909v1.pdf'}
+page_content=' Thus, we  introduce a cost-based evaluation metric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfGAOI/content/2301.02909v1.pdf'}
+page_content=' Formally, given a  rejection cost cr > 0, a false positive cost cfp > 0, and a  false negative cost cfn > 0, the detector is evaluated as:  Mh = cr · P(ht(X) = ®) + cfp · P(ht(X) = 1|Y = 0)  + cfn · P(ht(X) = 0|Y = 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfGAOI/content/2301.02909v1.pdf'}
+page_content='  Note that we assume cost null for the correct predictions,  while every misprediction as well as the rejection gets pe-  nalized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfGAOI/content/2301.02909v1.pdf'}
+page_content=' Because rejecting is assumed to be less costly than  mispredicting, the rejection cost needs to satisfy the inequal-  ity cr ≤ min{cfp × (1 − γ), cfn × γ}, otherwise one could  predict either always normal and pay an expected cost of  cfn × γ, or always anomaly and pay cfp × (1 − γ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfGAOI/content/2301.02909v1.pdf'}
+page_content='  Experiments  We experimentally answer the following questions:  Q1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfGAOI/content/2301.02909v1.pdf'}
+page_content=' Does BALLAD result in lower costs when compared to  using only AL or LR?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfGAOI/content/2301.02909v1.pdf'}
+page_content='  Q2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfGAOI/content/2301.02909v1.pdf'}
+page_content=' Which reward metric is better?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfGAOI/content/2301.02909v1.pdf'}
+page_content='  Q3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfGAOI/content/2301.02909v1.pdf'}
+page_content=' Is the reward function on a similar scale for AL and LR?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfGAOI/content/2301.02909v1.pdf'}
+page_content='  Q4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfGAOI/content/2301.02909v1.pdf'}
+page_content=' How does our strategy behave when varying cfp, cfn ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfGAOI/content/2301.02909v1.pdf'}
+page_content='  Experimental setup  Methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfGAOI/content/2301.02909v1.pdf'}
+page_content='  We compare BALLAD1 to two baselines: ALL-  IN-AL allocates all the budget for active learning and sets  the rejection threshold using the (biased) training labels;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfGAOI/content/2301.02909v1.pdf'}
+page_content=' on  the contrary, ALL-IN-LR allocates all the budget for learn-  ing to reject and uses an unlabeled training set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfGAOI/content/2301.02909v1.pdf'}
+page_content='  Table 1: Properties of the 18 datasets used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfGAOI/content/2301.02909v1.pdf'}
+page_content='  Dataset  # Examples  # Features  γ  ALOI  12384  27  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfGAOI/content/2301.02909v1.pdf'}
+page_content='0304  Annthyroid  7129  21  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfGAOI/content/2301.02909v1.pdf'}
+page_content='0749  Arrhythmia  271  259  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfGAOI/content/2301.02909v1.pdf'}
+page_content='0996  Cardiotocography  1734  21  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfGAOI/content/2301.02909v1.pdf'}
+page_content='0496  Glass  214  7  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfGAOI/content/2301.02909v1.pdf'}
+page_content='0421  InternetAds  1682  1555  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfGAOI/content/2301.02909v1.pdf'}
+page_content='0499  KDDCup99  48113  40  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfGAOI/content/2301.02909v1.pdf'}
+page_content='0042  PageBlocks  5473  10  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfGAOI/content/2301.02909v1.pdf'}
+page_content='1023  PenDigits  9868  16  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfGAOI/content/2301.02909v1.pdf'}
+page_content='0020  Pima  526  8  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfGAOI/content/2301.02909v1.pdf'}
+page_content='0494  Shuttle  1013  9  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfGAOI/content/2301.02909v1.pdf'}
+page_content='0128  SpamBase  2661  57  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfGAOI/content/2301.02909v1.pdf'}
+page_content='0499  Stamps  340  9  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfGAOI/content/2301.02909v1.pdf'}
+page_content='0912  WBC  223  9  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfGAOI/content/2301.02909v1.pdf'}
+page_content='0448  WDBC  367  30  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfGAOI/content/2301.02909v1.pdf'}
+page_content='0272  WPBC  160  33  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfGAOI/content/2301.02909v1.pdf'}
+page_content='0562  Waveform  3443  21  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfGAOI/content/2301.02909v1.pdf'}
+page_content='0290  Wilt  4655  5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfGAOI/content/2301.02909v1.pdf'}
+page_content='0199  Data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfGAOI/content/2301.02909v1.pdf'}
+page_content='  We carry out our study on 18 publicly available  benchmark datasets, which are widely used in the litera-  ture (Campos et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfGAOI/content/2301.02909v1.pdf'}
+page_content=' 2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfGAOI/content/2301.02909v1.pdf'}
+page_content=' See Table 1 for the properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfGAOI/content/2301.02909v1.pdf'}
+page_content='  Setup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfGAOI/content/2301.02909v1.pdf'}
+page_content='  For each of the 18 benchmark datasets, we go as  follows: (i) we split the dataset into training, validation and  test sets using the proportions 40 − 40 − 20 (we have a large  validation set to better measure the impact of rejection);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfGAOI/content/2301.02909v1.pdf'}
+page_content=' (ii)  we fit the anomaly detector on the unlabeled dataset and set  the rejection threshold to the default value of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfGAOI/content/2301.02909v1.pdf'}
+page_content='1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfGAOI/content/2301.02909v1.pdf'}
+page_content=' (ii) we  allocate a budget b to LR and AL by randomly selecting  the initial examples;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfGAOI/content/2301.02909v1.pdf'}
+page_content=' (iii) we optimize the rejection thresh-  old and measure the LR reward;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfGAOI/content/2301.02909v1.pdf'}
+page_content=' (iv) we train the anomaly  detector on the partially labeled training set and measure the  AL reward;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfGAOI/content/2301.02909v1.pdf'}
+page_content=' (v) we allocate the next round budget b to the  option with the highest reward and repeat (iii) or (iv) until  the whole budget B is used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfGAOI/content/2301.02909v1.pdf'}
+page_content=' During each of the steps, we  measure the detector performance on the test set using our  1Code available at https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfGAOI/content/2301.02909v1.pdf'}
+page_content='com/Lorenzo-Perini/Ballad  cost function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfGAOI/content/2301.02909v1.pdf'}
+page_content=' We set B to the 30% of the training set’s size,  and b to 2% of it, such that we run 15 allocation rounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfGAOI/content/2301.02909v1.pdf'}
+page_content=' We  repeat (i - v) 10 times and report the average results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfGAOI/content/2301.02909v1.pdf'}
+page_content=' In total  we run 18 × 15 × 10 = 2700 experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfGAOI/content/2301.02909v1.pdf'}
+page_content='  Costs and hyperparameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfGAOI/content/2301.02909v1.pdf'}
+page_content='  We set cfp = cfn = 1 and  cr = γ, following the cost inequality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfGAOI/content/2301.02909v1.pdf'}
+page_content=' SSDO (Vercruyssen  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfGAOI/content/2301.02909v1.pdf'}
+page_content=' 2018) with its default parameters is used as the semi-  supervised anomaly detector (Soenen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfGAOI/content/2301.02909v1.pdf'}
+page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfGAOI/content/2301.02909v1.pdf'}
+page_content=' We use  IFOREST (Liu, Ting, and Zhou 2008) as its unsupervised  prior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfGAOI/content/2301.02909v1.pdf'}
+page_content=' We use Uncertainty Sampling as the active learning  strategy (Zhan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfGAOI/content/2301.02909v1.pdf'}
+page_content=' 2021), and the entropy as default re-  ward.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfGAOI/content/2301.02909v1.pdf'}
+page_content=' For setting the rejection threshold, we use Bayesian  Optimization (GP MINIMIZE implemented in SKOPT) with  20 calls (Frazier 2018) and limit the rejection rate on the  validation set to 50%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfGAOI/content/2301.02909v1.pdf'}
+page_content='  Experimental results  Q1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfGAOI/content/2301.02909v1.pdf'}
+page_content=' Comparing BALLAD to the ALL-IN strategies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfGAOI/content/2301.02909v1.pdf'}
+page_content=' Fig-  ure 1 shows the comparison between BALLAD with the en-  tropy reward and the ALL-IN strategies on the 18 bench-  mark datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfGAOI/content/2301.02909v1.pdf'}
+page_content=' On 8 datasets (Arrhythmia, Glass, Kdd-  Cup99, Pima, SpamBase, Wbc, Wdbc, Wpbc), BALLAD re-  sults in evident lower costs, although sometimes the differ-  ence is small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfGAOI/content/2301.02909v1.pdf'}
+page_content=' On 5 datasets (Cardiotocography, InternetAds,  PageBlocks, Stamps, Waveform) BALLAD performs simi-  lar/worse than ALL-IN-AL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfGAOI/content/2301.02909v1.pdf'}
+page_content=' This happens because SSDO has  an overall high performance and a contained uncertainty in  the predictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfGAOI/content/2301.02909v1.pdf'}
+page_content=' On the other hand, in 3 cases (Aloi, Annthy-  roid, Wilt), allocating all the budget for LR has a lower cost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfGAOI/content/2301.02909v1.pdf'}
+page_content='  This is due to the detector being inaccurate and unable to  learn from the training labels, which makes learning an opti-  mal threshold more convenient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfGAOI/content/2301.02909v1.pdf'}
+page_content=' As support for this intuition,  we analyze the plain test AUC of SSDO on the whole test  set (no rejection) for each of the three previous cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfGAOI/content/2301.02909v1.pdf'}
+page_content=' By ag-  gregating over the rounds, SSDO obtains an average AUC  equal to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfGAOI/content/2301.02909v1.pdf'}
+page_content='86, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfGAOI/content/2301.02909v1.pdf'}
+page_content='88, and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfGAOI/content/2301.02909v1.pdf'}
+page_content='57 when the best strategy is, re-  spectively, BALLAD, ALL-IN-AL, and ALL-IN-LR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfGAOI/content/2301.02909v1.pdf'}
+page_content=' Finally,  BALLAD obtains an overall average cost of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfGAOI/content/2301.02909v1.pdf'}
+page_content='043, which is  ≈ 20% lower than the baselines’ average cost (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfGAOI/content/2301.02909v1.pdf'}
+page_content='055 for  ALL-IN-AL, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfGAOI/content/2301.02909v1.pdf'}
+page_content='054 for ALL-IN-LR).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfGAOI/content/2301.02909v1.pdf'}
+page_content='  Q2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfGAOI/content/2301.02909v1.pdf'}
+page_content=' Which reward function works better?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfGAOI/content/2301.02909v1.pdf'}
+page_content=' We analyze  both types of reward functions that we introduced in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfGAOI/content/2301.02909v1.pdf'}
+page_content=' 1  and Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfGAOI/content/2301.02909v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfGAOI/content/2301.02909v1.pdf'}
+page_content=' Table 2 shows the mean and standard deviation  of the cost, divided by allocation round.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfGAOI/content/2301.02909v1.pdf'}
+page_content=' Overall, using the  cosine reward builds a strategy that produces on average low  costs for little budget (≤ 10%), whereas, for a higher bud-  get, the entropy reward obtains better average costs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfGAOI/content/2301.02909v1.pdf'}
+page_content=' This is  due to the highly imbalanced choices made by the cosine re-  ward: the strategy opts for AL in 93% of the cases, which  usually improves a lot the detector’s performance with few  labels but tends to produce little effect when enough labels  are given.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfGAOI/content/2301.02909v1.pdf'}
+page_content=' On the other hand, the entropy reward is more bal-  anced and opts for AL in 63% of the cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfGAOI/content/2301.02909v1.pdf'}
+page_content=' This allows the  detector to keep decreasing the costs while learning during  the allocation rounds and obtain more steady performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfGAOI/content/2301.02909v1.pdf'}
+page_content='  Q3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfGAOI/content/2301.02909v1.pdf'}
+page_content=' Is the entropy reward balanced for AL and LR?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfGAOI/content/2301.02909v1.pdf'}
+page_content=' Fig-  ure 2 shows the distribution of the difference between AL  and LR entropy rewards over all the 2700 experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfGAOI/content/2301.02909v1.pdf'}
+page_content=' Neg-  Budget  Entropy Re  Cosine Rc  2%  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfGAOI/content/2301.02909v1.pdf'}
+page_content='0536 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfGAOI/content/2301.02909v1.pdf'}
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+page_content='0354  Table 2: Average (± std) cost per test example over the  datasets grouped by allocation round for each of the two re-  ward functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfGAOI/content/2301.02909v1.pdf'}
+page_content=' For low budgets, the cosine reward obtains  lower costs, while not being competitive for high budgets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfGAOI/content/2301.02909v1.pdf'}
+page_content='  ative values indicate that the LR reward is higher than AL’s  one, while the opposite holds for positive values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfGAOI/content/2301.02909v1.pdf'}
+page_content=' Overall,  the median is close to 0, which means that there is no clearly  predominant strategy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfGAOI/content/2301.02909v1.pdf'}
+page_content=' Because the left tail of the density is  larger than the right one, we conclude that LR rewards have  higher variability (std = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfGAOI/content/2301.02909v1.pdf'}
+page_content='07 vs 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfGAOI/content/2301.02909v1.pdf'}
+page_content='03).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfGAOI/content/2301.02909v1.pdf'}
+page_content='  Q4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfGAOI/content/2301.02909v1.pdf'}
+page_content=' The impact of varying cfp, and cfn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfGAOI/content/2301.02909v1.pdf'}
+page_content=' In this experi-  ment, we penalize more false positives and false negatives  by setting, one at a time, cfp and cfn to 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfGAOI/content/2301.02909v1.pdf'}
+page_content=' We compare  BALLAD to the two ALL-IN baselines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfGAOI/content/2301.02909v1.pdf'}
+page_content=' For cfp = 10, our  strategy is still the best for low budgets (< 15%), reduc-  ing the relative cost by between 5% and 25% with respect  to the runner-up ALL-IN-LR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfGAOI/content/2301.02909v1.pdf'}
+page_content=' However, for higher budgets  (> 15%), ALL-IN-LR becomes the best strategy as it re-  duces BALLAD’s cost by around 20% and ALL-IN-AL’s cost  by more than 40%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfGAOI/content/2301.02909v1.pdf'}
+page_content=' This happens because the anomaly de-  tector produces too many false positives, which, if rejected,  allow us to reduce the cost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfGAOI/content/2301.02909v1.pdf'}
+page_content=' For cfn = 10, BALLAD performs  much better than the baselines, reducing their cost by around  20% (vs ALL-IN-LR) and 24% (vs ALL-IN-AL).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfGAOI/content/2301.02909v1.pdf'}
+page_content='  Conclusion  We proposed BALLAD, a novel strategy to decide whether  to allocate the budget for Active Learning (AL), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfGAOI/content/2301.02909v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfGAOI/content/2301.02909v1.pdf'}
+page_content=' labeling  strategic training instances, or for Learning to Reject (LR),  i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfGAOI/content/2301.02909v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfGAOI/content/2301.02909v1.pdf'}
+page_content=' labeling a random validation set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfGAOI/content/2301.02909v1.pdf'}
+page_content=' Our key insight is that  we can measure the expected reward when labeling either set  and allocate the label in the next round to the option with the  highest reward.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfGAOI/content/2301.02909v1.pdf'}
+page_content=' We proposed two reward functions (entropy  and cosine similarity based).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfGAOI/content/2301.02909v1.pdf'}
+page_content=' Experimentally, we evaluated  BALLAD on 18 datasets, and show that it performs better  than simply allocating all the labels to either AL or LR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfGAOI/content/2301.02909v1.pdf'}
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+page_content='3  AL reward - LR reward  Density  Median  Figure 2: Distribution of the difference between AL’s and  LR’s entropy reward.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfGAOI/content/2301.02909v1.pdf'}
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+page_content='  Acknowledgements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfGAOI/content/2301.02909v1.pdf'}
+page_content='  This work was presented at the 1st  AAAI Workshop on Uncertainty Reasoning and Quantifica-  tion in Decision Making (UDM23).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfGAOI/content/2301.02909v1.pdf'}
+page_content='  This research is supported by an FB Ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfGAOI/content/2301.02909v1.pdf'}
+page_content='D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfGAOI/content/2301.02909v1.pdf'}
+page_content=' fellowship by  FWO-Vlaanderen (grant 1166222N) [LP], the Flemish Gov-  ernment under the “Onderzoeksprogramma Artifici¨ele Intel-  ligentie (AI) Vlaanderen” programme [JD], and KUL Re-  search Fund iBOF/21/075 [JD].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfGAOI/content/2301.02909v1.pdf'}
+page_content='  References  Abe, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfGAOI/content/2301.02909v1.pdf'}
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+page_content=' and Houle, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfGAOI/content/2301.02909v1.pdf'}
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+page_content=' Data mining and  knowledge discovery.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfGAOI/content/2301.02909v1.pdf'}
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+page_content=' IEEE Transactions on information theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfGAOI/content/2301.02909v1.pdf'}
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+page_content=' DeSalvo, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfGAOI/content/2301.02909v1.pdf'}
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+page_content=' and Mohri, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfGAOI/content/2301.02909v1.pdf'}
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+page_content=' Springer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfGAOI/content/2301.02909v1.pdf'}
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+page_content='  Pimentel, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfGAOI/content/2301.02909v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfGAOI/content/2301.02909v1.pdf'}
+page_content=' Monteiro, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfGAOI/content/2301.02909v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfGAOI/content/2301.02909v1.pdf'}
+page_content=' Veloso, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfGAOI/content/2301.02909v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfGAOI/content/2301.02909v1.pdf'}
+page_content=' and Ziviani, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfGAOI/content/2301.02909v1.pdf'}
+page_content='  2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfGAOI/content/2301.02909v1.pdf'}
+page_content=' Deep active learning for anomaly detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfGAOI/content/2301.02909v1.pdf'}
+page_content=' In 2020  International Joint Conference on Neural Networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfGAOI/content/2301.02909v1.pdf'}
+page_content=' IEEE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfGAOI/content/2301.02909v1.pdf'}
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+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfGAOI/content/2301.02909v1.pdf'}
+page_content=' Van Wolputte, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfGAOI/content/2301.02909v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfGAOI/content/2301.02909v1.pdf'}
+page_content=' Perini, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfGAOI/content/2301.02909v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfGAOI/content/2301.02909v1.pdf'}
+page_content=' Vercruyssen, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfGAOI/content/2301.02909v1.pdf'}
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+page_content='  Meert, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfGAOI/content/2301.02909v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfGAOI/content/2301.02909v1.pdf'}
+page_content=' Davis, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfGAOI/content/2301.02909v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfGAOI/content/2301.02909v1.pdf'}
+page_content=' and Blockeel, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfGAOI/content/2301.02909v1.pdf'}
+page_content=' 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfGAOI/content/2301.02909v1.pdf'}
+page_content=' The effect of  hyperparameter tuning on the comparative evaluation of un-  supervised anomaly detection methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfGAOI/content/2301.02909v1.pdf'}
+page_content=' In Proceedings of  the KDD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfGAOI/content/2301.02909v1.pdf'}
+page_content='  Vercruyssen, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfGAOI/content/2301.02909v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfGAOI/content/2301.02909v1.pdf'}
+page_content=' Meert, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfGAOI/content/2301.02909v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfGAOI/content/2301.02909v1.pdf'}
+page_content=' Verbruggen, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfGAOI/content/2301.02909v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfGAOI/content/2301.02909v1.pdf'}
+page_content=' Maes, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfGAOI/content/2301.02909v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfGAOI/content/2301.02909v1.pdf'}
+page_content='  Baumer, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfGAOI/content/2301.02909v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfGAOI/content/2301.02909v1.pdf'}
+page_content=' and Davis, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfGAOI/content/2301.02909v1.pdf'}
+page_content=' 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfGAOI/content/2301.02909v1.pdf'}
+page_content=' Semi-supervised anomaly  detection with an application to water analytics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfGAOI/content/2301.02909v1.pdf'}
+page_content=' In 2018  IEEE international conference on data mining (icdm).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfGAOI/content/2301.02909v1.pdf'}
+page_content=' IEEE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfGAOI/content/2301.02909v1.pdf'}
+page_content='  Vercruyssen, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfGAOI/content/2301.02909v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfGAOI/content/2301.02909v1.pdf'}
+page_content=' Perini, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfGAOI/content/2301.02909v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfGAOI/content/2301.02909v1.pdf'}
+page_content=' Meert, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfGAOI/content/2301.02909v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfGAOI/content/2301.02909v1.pdf'}
+page_content=' and Davis, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfGAOI/content/2301.02909v1.pdf'}
+page_content='  2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfGAOI/content/2301.02909v1.pdf'}
+page_content=' Multi-domain Active Learning for Semi-supervised  Anomaly Detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfGAOI/content/2301.02909v1.pdf'}
+page_content=' ECML 2022 published proceedings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfGAOI/content/2301.02909v1.pdf'}
+page_content='  Zhan, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfGAOI/content/2301.02909v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfGAOI/content/2301.02909v1.pdf'}
+page_content=' Liu, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfGAOI/content/2301.02909v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfGAOI/content/2301.02909v1.pdf'}
+page_content=' Li, Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfGAOI/content/2301.02909v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfGAOI/content/2301.02909v1.pdf'}
+page_content=' and Chan, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfGAOI/content/2301.02909v1.pdf'}
+page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfGAOI/content/2301.02909v1.pdf'}
+page_content=' 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfGAOI/content/2301.02909v1.pdf'}
+page_content=' A Compara-  tive Survey: Benchmarking for Pool-based Active Learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfGAOI/content/2301.02909v1.pdf'}
+page_content='  In IJCAI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfGAOI/content/2301.02909v1.pdf'}
+page_content='  Zhao, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfGAOI/content/2301.02909v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfGAOI/content/2301.02909v1.pdf'}
+page_content=' Wen, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfGAOI/content/2301.02909v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfGAOI/content/2301.02909v1.pdf'}
+page_content=' and Li, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfGAOI/content/2301.02909v1.pdf'}
+page_content=' 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfGAOI/content/2301.02909v1.pdf'}
+page_content=' A review on gas tur-  bine anomaly detection for implementing health manage-  ment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfGAOI/content/2301.02909v1.pdf'}
+page_content=' Turbo Expo: Power for Land, Sea, and Air.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfGAOI/content/2301.02909v1.pdf'}
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+ 
+RESEARCH ARTICLE 
+European Journal of Information Technologies and Computer Science 
+www.ej-compute.org 
+ 
+ 
+ 
+  
+DOI: http://dx.doi.org/10.24018/ejcompute.YEAR.VOL.ISSUE.ID   
+Vol X | Issue Y | Month Year 
+1 
+ 
+I. INTRODUCTION 
+     Hair, made of keratin protein, pertains to beauty and 
+masculinity. Approximately 5 million hair follicles are 
+present throughout our body [1]. Scalp Hair maintains body 
+temperature and protects the brain from external heat. A 
+typical hair growth cycle runs for 2-7 years, according to 
+Patel et al. [2] and Wolff, Fischer, and Blume-Peytavi [3]. A 
+healthy human has 100,000 hairs on the scalp, and 50-100 
+hair loss per day is considered normal. Hair loss is not a 
+present-day issue. The hair-loss treatment was found in 
+ancient Ayurveda scriptures 6000 years ago [2]. However, 
+Hair and scalp-related issues are gaining more recognition 
+nowadays compared to earlier years due to certain factors, 
+such as environmental pollution, hormonal imbalance, 
+autoimmune disease, gut microbiota alteration, elevated 
+physical and mental stress levels in human lifestyle, seasonal 
+change, unhealthy diet, micronutrient deficiency, genetic 
+predisposition, and side-effects of drugs [2], [3]. According 
+to Peyravian et al., 80 million Americans have hair loss-
+related issues to some extent [4]. Although most hair loss 
+diseases are localized, some can spread to other locations. 
+Some 
+diseases 
+require 
+prescribed 
+drugs 
+and 
+hair 
+transplantation. Some diseases are caused by bacterial or 
+fungal infections and require antibiotic treatment. Often, 
+there are genetic and sexual predispositions in hair-scalp 
+diseases. 
+     Alopecia, folliculitis, and psoriasis are some common 
+causes of hair loss. There is a difference between regular hair 
+fall and alopecia; the latter develops coin-sized bald patches 
+all over the scalp area. Alopecia or patchy hair loss can be of 
+different types. Androgenetic alopecia or male-pattern 
+baldness (MPB) is the most common form of alopecia where 
+the hairline starts to recede, following a pattern where the 
+frontal and temple area are most affected. 70% of men and 
+40% of women get this type of hair loss and thinning issue 
+[3]. According to Liu et al., MPB is an X-linked polygenic 
+disease, and males are more genetically prone to develop 
+baldness at a mature age [5]. Topical minoxidil solution 
+thickens the hair by 50% [3]. On the other hand, Alopecia 
+areata (AA) is an autoimmune disease affecting individuals 
+irrespective of age and sex. Primarily affecting the scalp area, 
+AA can also spread in the beard, eyelashes, and eyebrows. In 
+this case, the body’s immune cells cannot recognize hair 
+follicles as ‘self.’ Instead, they consider these follicles as 
+‘foreign,’ which ultimately causes the hair follicles to be 
+Hair and Scalp Disease Detection using Machine 
+Learning and Image Processing 
+ 
+Mrinmoy Roy, Anica Tasnim Protity 
+ABSTRACT 
+ 
+Almost 80 million Americans suffer from hair loss due to aging, 
+stress, medication, or genetic makeup. Hair and scalp-related 
+diseases often go unnoticed in the beginning. Sometimes, a patient 
+cannot differentiate between hair loss and regular hair fall. 
+Diagnosing hair-related diseases is time-consuming as it requires 
+professional dermatologists to perform visual and medical tests. 
+Because of that, the overall diagnosis gets delayed, which worsens 
+the severity of the illness. Due to the image-processing ability, neural 
+network-based applications are used in various sectors, especially 
+healthcare and health informatics, to predict deadly diseases like 
+cancers and tumors. These applications assist clinicians and patients 
+and provide an initial insight into early-stage symptoms. In this 
+study, we used a deep learning approach that successfully predicts 
+three main types of hair loss and scalp-related diseases: alopecia, 
+psoriasis, and folliculitis. However, limited study in this area, 
+unavailability of a proper dataset, and degree of variety among the 
+images scattered over the internet made the task challenging. 150 
+images were obtained from various sources and then preprocessed 
+by denoising, image equalization, enhancement, and data balancing, 
+thereby minimizing the error rate. After feeding the processed data 
+into the 2D convolutional neural network (CNN) model, we obtained 
+overall training accuracy of 96.2%, with a validation accuracy of 
+91.1%. The precision and recall score of alopecia, psoriasis, and 
+folliculitis are 0.895, 0.846, and 1.0, respectively. We also created a 
+dataset of the scalp images for future prospective researchers. 
+Keywords:  Deep Learning, Health Informatics, Machine Learning, 
+Scalp/ Hair Diseases. 
+ 
+ 
+ Published Online:  
+ISSN: 2736-5492 
+DOI 10.24018/ejcompute.YEAR.Vol.Issue.ID 
+ 
+Mrinmoy Roy  
+Department 
+of 
+Computer 
+Science, 
+Northern Illinois University, USA.  
+(e-mail: mrinmoy.cs10 gmail.com)  
+Anica Tasnim Protity  
+Department of Biological Sciences, 
+Northern Illinois University, USA. 
+(e-mail: 
+protity.microbiology@gmail.com) 
+ 
+*Corresponding Author 
+@ 
+
+ 
+RESEARCH ARTICLE 
+European Journal of Information Technologies and Computer Science 
+www.ej-compute.org 
+ 
+ 
+ 
+  
+DOI: http://dx.doi.org/10.24018/ejcompute.YEAR.VOL.ISSUE.ID   
+Vol X | Issue Y | Month Year 
+2 
+ 
+targeted and destroyed by the immune cells. It is an example 
+of a hereditary disease. The study from Benigno et al. 
+reported that, in the US alone, 700,000 individuals suffer 
+from AA [6]. This disease, if diagnosed early, might resolve 
+spontaneously. In severe cases, topical corticosteroid or 
+immune therapy is used [3]. 
+    Sometimes, the hair follicles might get inflamed because 
+of the action of bacterial accumulation. This follicle 
+inflammation 
+is 
+called 
+folliculitis 
+decalvans. 
+The 
+bacterium Staphylococcus aureus damages the follicle and 
+prevents hair growth. Staphylococcus aureus uses hair tufts 
+to 
+enter 
+underneath 
+the 
+follicle, 
+causing 
+chronic 
+inflammation, redness, swelling, scarring, itching, and hair 
+loss. Antibiotic treatment combined with surgical removal of 
+hair tufts and corticosteroids for reducing inflammation are 
+the 
+prescribed 
+treatment 
+for 
+Folliculitis 
+decalvans 
+[3]. Psoriasis is another form of common scalp skin disease. 
+According to [7], 54% of 5600 psoriasis patients had scalp 
+psoriasis. Severe psoriasis may cause significant itching, 
+scaling, and redness in the scalp. The application of topical 
+shampoo and corticosteroids are the treatment options by 
+Chan et al. [8].    
+    Some scalp infections may be treatable if diagnosed 
+early. Some but not all diseases may go on their own. Only 
+an expert physician can detect the illness by visual 
+observation. In some cases, early disease detection is 
+beneficial for dermatologists to initiate the treatment. An 
+early scalp inspection includes a dermatoscopic examination 
+of the scalp for inflammation, itching, localized lesion, 
+dandruff, follicular flakes, louse eggs (nits), and a scalp 
+biopsy. Besides visual observation, the patient can undergo 
+blood and hormone tests to detect the exact disease. 
+Unfortunately, most hair and scalp diseases are diagnosed in 
+advanced stages, which complicate the treatment options. All 
+these factors lengthen the diagnosis and treatment process. 
+Therefore, researchers are putting more effort into developing 
+different mechanisms for the early detection of hair and scalp 
+diseases. 
+    In the 21st century, with all the advancements in 
+computational technology, extensive application of machine 
+learning has made our daily lives simple, comfortable, and 
+secure. The increasing popularity of machine learning and its 
+nature to extract patterns from data are directing researchers 
+to incorporate several machine learning algorithms into 
+health informatics. Especially during the Covid-19 pandemic 
+era, different applications like restraining people from covid-
+19 spread [9], SARS-CoV-2 screening and treatment [10], 
+lock-down control in case of high dimensional input [11] 
+came into play, which made machine learning and healthcare 
+systems inseparable. Overall, adapting, integrating, and 
+developing deep learning-based applications on patients’ 
+information, medical reports, and audio-video feedback make 
+the diagnosis process faster. Nowadays, patients can get at 
+least the initial idea of disease detection by themselves using 
+easily accessible smart devices. All these applications clear 
+their confusion and help them make health-related decisions 
+independently.  
+    The high computational capability of neural networks is, 
+therefore, a breakthrough in healthcare and medical 
+diagnostic organizations. Convolutional neural networks 
+(CNN) have brought revolutionary success in detecting 
+deadly diseases. To date, neural networks are assisting 
+healthcare professionals in the early detection of different 
+types of tumors and cancers, such as skin cancer (melanoma) 
+[12], stomach cancer (adenocarcinoma) [13], and brain 
+tumors (glioblastoma) [14]. Neural networks are applicable 
+in detecting life-threatening dengue fever [15] and covid-19 
+[16] as well. In one study, CNN was used to extract complex 
+temporal dynamic features from heart rate variability (HRV) 
+signals, developing an algorithm that facilitated the early 
+detection of diabetics [17]. Using the image processing ability 
+of the neural networks, we can extract features from hair, skin 
+and scalp images to classify and categorize numerous hair and 
+scalp-related diseases. In this work, due to the importance of 
+early-stage hair disease detection, we applied convolutional 
+neural networks to 3 types of hair diseases and developed a 
+model to detect them successfully. 
+ 
+II. CHALLENGES AND CONTRIBUTIONS 
+    A classic application of computer vision is to detect 
+disease using digital images. Researchers can exploit a pool 
+of digital images obtained from one or more datasets, 
+preprocess the images, feed the images into the neural 
+network, and develop a model to detect the disease. 
+Unfortunately, minimal research has been performed on the 
+machine-learning approach for scalp disease detection. There 
+are several unique challenges behind this. First and foremost, 
+hair diseases are not localized and can spread to different 
+regions of the scalp, beard, eyebrows, eyelashes, and pubic 
+area. Second, every image needs different types of 
+preprocessing before feeding to neural networks. Different 
+scalp skin tones, hair colors, and types around the detection 
+zones make the imaging process more complicated. Third, no 
+proper dataset for scalp diseases is available over the internet, 
+and images taken from the internet differ in size and 
+resolution. Moreover, one must be conscious of minimalizing 
+and correcting the error in disease detection; otherwise, the 
+high false-positive and false-negative rates result in 
+misdiagnosis of the disease and worsening hair loss. 
+    To overcome the challenges, we developed a model 
+which can successfully classify the alopecia, folliculitis, and 
+psoriasis diseases with a minimal false-positive and false-
+negative rate. Though it is challenging to collect images for 
+the diseases from the internet, and the images are varied in 
+color, 
+shape, 
+and 
+resolution, 
+we 
+applied 
+various 
+preprocessing, such as denoising, resizing, enhancement and 
+created a dataset that might help in further scalp diseases 
+research. 
+ 
+III. RELATED WORKS 
+    Disease detection using machine learning approaches is 
+gaining popularity in health informatics. Many skin and 
+scalp-related diseases can be detected using images of 
+infected regions within a few seconds. In one study by 
+Choudhary et al. [18], a framework is developed to 
+differentiate alopecia areata from healthy hair. They obtained 
+200 healthy hair images from the figaro1K dataset and 68 
+alopecia areata hair images from DermNet. After a series of 
+enhancement and segmentation, three key features were 
+
+ 
+RESEARCH ARTICLE 
+European Journal of Information Technologies and Computer Science 
+www.ej-compute.org 
+ 
+ 
+ 
+  
+DOI: http://dx.doi.org/10.24018/ejcompute.YEAR.VOL.ISSUE.ID   
+Vol X | Issue Y | Month Year 
+3 
+ 
+extracted from the images: texture, shape, and color. The 
+researchers divided the dataset into 70%-30% train-test-split 
+and applied a support vector machine (SNM) and k-nearest 
+neighbor (KNN) for the classification task. Overall, they 
+achieved 91.4% and 88.9% accuracy using SVM and KNN, 
+respectively, with a 10-fold cross-validation approach. 
+However, using other machine learning algorithms might 
+increase in the accuracy rate, which should have been 
+discussed. Besides, the application of Histogram Equalization 
+(HE) for image enhancement complicated the process of 
+getting accurate texture features from distorted images, as HE 
+itself adds noise to the output image, distorting the signals. 
+Moreover, this study only shed light on alopecia areata 
+disease, ignoring the inter-class differences between other 
+similar type diseases, which increased the likelihood of 
+inaccurate prediction of other diseases as alopecia areata, 
+thereby making this framework less reliable. 
+    Another study [19] proposed a model for early alopecia 
+detection. They used 100 samples for this research, with 80% 
+as training data and the other 20% as testing data. They 
+looked for four attributes, length of the hair, nail brittleness, 
+amount of damage made to the hair, and hair follicle. Two-
+layer feed-forward network with a back propagation 
+technique was used for detection purposes. The proposed 
+model system consisting of 4 input neurons, 10 hidden 
+neurons, and a linear output neuron, achieved 91% training 
+accuracy with 86.7% validation accuracy. It showed the best 
+performance at epoch 4 with a 0.059687 gradient. However, 
+the study has some pitfalls, too, as they did not mention their 
+data source or differentiate data classes with their respective 
+sample sizes. Also, no image pre-processing was performed 
+on the collected images. Although there is a possibility of 
+overfitting without a proper data balancing technique, this 
+report did not discuss the data balancing between the two 
+classes. Furthermore, they did not calculate the model’s false-
+positive and false-negative rates, which is crucial for a model 
+specially developed for the healthcare system. 
+    Related work [20] was performed on skin disease 
+detection, where machine learning was used to analyze the 
+digital image of the affected skin area for identifying eczema, 
+melanoma, and psoriasis. Their dataset consists of 80 images 
+from different websites specific to skin diseases. By using a 
+convolutional neural network for feature extraction and 
+applying multiclass SVM on those features, they achieved 
+100% accuracy in disease classification. However, they did 
+not explore other essential model performance matrices and 
+overfitting issues. In another skin disease detection-based 
+article [21], the authors proposed a scheme to classify skin 
+lesions into five categories: healthy, acne, eczema, benign, 
+and malignant melanoma, using a pre-trained CNN model, 
+AlexNET for feature extraction and error correcting output 
+codes support vector machine for classification. The dataset 
+consists of 9144 images from different sources and achieved 
+84.21% accuracy using a 10-fold cross-validation technique. 
+    Overall, we observed very few works on hair diseases. 
+The recent related works lack at least one of the following 
+categories – discussion over false positive and false negative 
+rates, ignoring inter-class differences, model reliability, and 
+overfitting problem. In this work, we have attempted to fill 
+these gaps by leveraging a convolutional neural network 
+algorithm on hair disease images while maintaining high 
+accuracy with good precision and recall scores. 
+ 
+IV. DATA DESCRIPTION & DEVICE 
+A. Data Collection 
+    The most challenging part of using visual images for 
+disease prediction and disease classification is data 
+collection. Often, one can get fewer appropriate images for 
+a specific illness found. Moreover, the pictures are scattered 
+over the internet. In this study, the authors extracted the 
+images from different websites, such as DermQuest, 
+DermNet, MedicineNet, DermnetNZ, and various medical 
+professionals. 
+ 
+TABLE I: IMAGES PER DISEASE 
+Disease 
+Quantity 
+Alopecia 
+65 
+Psoriasis 
+45 
+Folliculitis 
+40 
+ 
+ 
+Fig. 1. Image subset of each disease category. 
+ 
+    The image quantity is different for each category. We 
+found more alopecia-related images than other diseases 
+because alopecia is more frequent and severe among the 
+human population. The number of samples in each type of 
+disease is listed in Table I. Randomly selected images in 
+each category are graphically represented in Fig 1. Our 
+dataset is made publicly available on Github [22].  
+B. Device 
+    The research was conducted on Dell Latitude 5520 
+laptop device having 11th generation Intel Core i5 (8 MB 
+cache, 4 cores, 8 threads, up to 4.40 GHz Turbo) and 
+running on Windows 10 Pro operating system. The device 
+has 16 GB, 1 x 16 GB, DDR4, 3200 MHz random access 
+memory (RAM), and 256 GB, M.2 PCIe NVMe, SSD, 
+Class 35 (NVRAM). For the classification of images, we 
+utilized the integrated Intel Iris XE graphics capable with a 
+thunderbolt for I5-1145G7 vPro processor. For the data 
+
+Alopecia
+Psoriasis
+folliculitis 
+RESEARCH ARTICLE 
+European Journal of Information Technologies and Computer Science 
+www.ej-compute.org 
+ 
+ 
+ 
+  
+DOI: http://dx.doi.org/10.24018/ejcompute.YEAR.VOL.ISSUE.ID   
+Vol X | Issue Y | Month Year 
+4 
+ 
+collection, we used iPhone-13 Pro Max having Hexa-core 
+(2x3.23 GHz Avalanche + 4x1.82 GHz Blizzard) CPU and 
+Apple GPU (5-core graphics). We used a mobile device 
+with 128GB 6GB RAM, and a 12 MP triple main camera 
+for the image collection. 
+V. PROPOSED MODEL 
+    In this section, we introduce the system workflow of our 
+model and explain the functions of each module in details. As 
+shown in Fig. 2, first, the captured image is sent to 
+preprocessing steps which are divided into three parts: image 
+equalization, image enhancement, and data balancing. 
+ 
+ 
+Fig. 2. System workflow of hair disease detection model. 
+ 
+Among these three, the first two parts are mainly for 
+increasing image quality, and the last part is for model 
+versatility. After the preprocessing steps, the image is passed 
+to the Neural Network model for the classification task. We 
+used a convolutional neural network that classifies an image 
+successfully into three different classes: alopecia, folliculitis, 
+and psoriasis. 
+A. Denoising 
+ 
+ 
+Fig. 3. Left original image & right non-local means denoised image. 
+ 
+    Noise is the degradation of image signals caused by 
+external sources [23]. Noise introduces random variations of 
+brightness or color information in the captured images. Most 
+of the time, images on the internet have some noise associated 
+with them. As we have collected most of the data samples 
+from different dermatology websites, the noise in our dataset 
+is not homogeneously distributed, which made it more 
+complex. Therefore, we applied additional filters for 
+denoising the collected images. We started with the gaussian 
+filter for a better image classification process. However, after 
+using the gaussian filter, the images become completely 
+blurred, which leads to the loss of important information and 
+damage to the edges. We then applied the median filter, 
+which worked better than the gaussian filter with kernel_size 
+= 3. Though we achieved better accuracy using the bilateral 
+filter, we got the best results while applying the non-local 
+means filter with patch_size = 3 and patch_distance = 5. This 
+non-local means filter preserved all the edges and reduced the 
+noise better than the other filters for our application which is 
+shown in Fig. 3. 
+B. Image Equalization 
+    Often the captured image doesn’t reflect the natural view 
+and needs contrast enhancement to meet the level of realistic 
+view [24]. Especially images with high color depth and after 
+denoising effects need normalization for a better realistic 
+view [25]. First, we applied histogram equalization (HE). 
+However, the HE increases the contrast of the background 
+when used in images with low color depth, and information 
+is lost as the histogram is not confined to the local region. To 
+overcome the problem, we applied CLAHE (Contrast 
+Limited Adaptive Histogram Equalization) by dividing an 
+image into equal size non-overlapping areas and computing a 
+histogram for each region. After clipping the histogram, we 
+distributed the clipped value over the histogram equalization, 
+which gives us control of the over-amplification of the 
+contrast and generates the resultant image shown in Fig. 4. 
+ 
+ 
+Fig. 4. Image equalization using CLAHE. 
+ 
+C. Data Balancing 
+    The overall performance of a machine learning model 
+depends on the balanced dataset because, without it, minority 
+class detection becomes difficult. Balancing a dataset reduces 
+the risk of skewing towards the majority. Imbalanced data 
+might achieve high accuracy, but the results are biased toward 
+the majority class. As alopecia is a common disease, we have 
+more alopecia images than other diseases, which creates an 
+imbalanced dataset for our model. For balancing the dataset, 
+we used data augmentation techniques (re-scaling, random 
+rotating, cropping, vertical and horizontal flipping) and 
+oversampled the infrequent class. 
+D. Neural Network Model 
+    Neural network is the most applied model for visual data 
+analysis. Neural network needs limited human assistance and 
+
+Image
+Image
+Data
+Denoising
+Enhancement
+Augmentation
+Convolutional
+Diseaseclass
+Neural
+Detected
+Network
+Model 
+RESEARCH ARTICLE 
+European Journal of Information Technologies and Computer Science 
+www.ej-compute.org 
+ 
+ 
+ 
+  
+DOI: http://dx.doi.org/10.24018/ejcompute.YEAR.VOL.ISSUE.ID   
+Vol X | Issue Y | Month Year 
+5 
+ 
+can identify complex non-linear relationship between input 
+and output. From global or local scale modeling [26] to 
+diagnosis by medical image classification, neural network is 
+using extensively. Moreover, Facial recognition, image 
+labeling, accurate video subtitles, assisting call centers, 
+automated virtual agents all these things are using neural 
+network. There are 3 types of neural network available: 
+Artificial Neural Networks (ANN), Convolution Neural 
+Networks (CNN) and Recurrent Neural Networks (RNN). 
+Each neural network has mainly 3 components: an input 
+layer, a processing layer, and an output layer.  
+ 
+ 
+Fig. 5. Neural Network Model. 
+ 
+   In this study, CNN is utilized for classification because it 
+takes image’s raw pixel data, trains a model, and extracts the 
+features automatically for better detection. We used autokeras 
+to find the best model for this problem. After trying 25 
+different combinations, we selected 3 hidden layers with 1 
+input and 1 output layer as our final model which is shown in 
+Fig. 5. For training the model, we used batch_size = 16 with 
+50 epochs for each batch. The preprocessed data is divided 
+into 70-30 train-test-split for training and validation purpose. 
+Our model consists of 256 inputs, 3 x 3 square kernel, 3 
+output units and a softmax output. We used ReLU as our 
+activation function to prevent the exponential growth of 
+required computation and to explore the non-linear 
+relationship between input and output variables. After each 
+convolutional layer, input goes through the pooling layer 
+having 2 x 2 kernel size to reduce the dimensions of the 
+features map. Pooling layer summarizes the presented 
+features in a region and helps to prevent the over-fitting 
+problem by down sampling. We also used dropout layer after 
+each pooling layer to prevent neurons in a layer from 
+synchronously optimizing their weights and converging to the 
+same goal. Our model’s dropout rate is 0.3, which means 30% 
+of the neurons of this layer will be randomly dropped in each 
+epoch. 
+    All the resultant 2-D arrays from pooled features map 
+passes through the flatten layer and converted to single 
+dimensional long continuous linear vector in the transition 
+towards the fully connected layer as in Fig. 5. In the fully 
+connected layer, every single output pixel from the 
+convolutional layers is connected to 3 output classes. Though 
+dense layer is computationally expensive, we used 2 dense 
+layers for our classification task. Finally, we used softmax 
+activation function to transform the 3 units of fully connected 
+layer to a probability distribution which is represented by a 
+vector of 3 elements, and the highest probability element 
+selected as the final class. We leveraged adam optimizer for 
+learning purpose and reducing the overall loss by changing 
+the weights and learning rates. We used adam because it can 
+handle sparse gradients on noisy problems and combines the 
+best properties of AgaGrad and RMSProp algorithms. 
+ 
+VI. RESULTS 
+    We trained our CNN model using the optimal 
+hyperparameters selected from the grid search. These 
+hyperparameters are listed in Table II. We divided the dataset 
+into 70%-30% train-test-split where 105 randomly selected 
+images are used for training and 45 random images for 
+testing. After applying the preprocessing steps, we used the 
+training dataset to train the CNN model and evaluated the test 
+dataset using the model. 
+ 
+TABLE II: HYPERPARAMETERS OF CNN MODEL 
+Hyperparameters 
+Values 
+Batch Size 
+16 
+Epoch 
+50 
+Kernel Size 
+3 x 3 
+Optimizer 
+Adam 
+Dropout Rate 
+0.3 
+Pooling Size 
+2 x 2 
+Activation Function 
+ReLU 
+ 
+ 
+Fig. 6. Training and Validation loss for CNN. 
+ 
+
+Conv 1
+Max-Pooling
+Conv 2
+Max-Pooling
+Conv 3
+Max-Pooling
+Convolution
+(2 x 2)
+Convolution
+(2 x 2) Kernel
+Convolution
+(2 x 2) Kernel
+Fully-Connected
+(3 x 3)
+Kernel
+(3 x 3)
+Dropout 0.3
+(3 × 3) Kernel
+Dropout 0.3
+Input
+OutputTraining and validation loss
+1.25
+Training loss
+1.00
+Validation loss
+0.50
+0.25
+0.00
+0
+10
+20
+30
+40
+50
+Epochs 
+RESEARCH ARTICLE 
+European Journal of Information Technologies and Computer Science 
+www.ej-compute.org 
+ 
+ 
+ 
+  
+DOI: http://dx.doi.org/10.24018/ejcompute.YEAR.VOL.ISSUE.ID   
+Vol X | Issue Y | Month Year 
+6 
+ 
+ 
+Fig. 7. Training and Validation Accuracy for CNN. 
+    Our system achieved 96.2% training accuracy and 
+91.1% validation accuracy on the unseen data. Validation and 
+training losses for every epoch are shown in Fig 6. The 
+training losses decreased from 1.1685 to 0.1017, and the 
+validation losses decrease from 1.1260 to 0.3438 while going 
+from epoch 1 to epoch 50. At the same time, Training 
+accuracy and validation accuracy increased to 96.2% and 
+91.1%, respectively, from epoch 1 to epoch 50, shown in Fig. 
+7. 
+ 
+ 
+Fig. 8. Confusion Matrix of Our Model. 
+ 
+ 
+Fig. 9. Fractional Incorrect Prediction of Our Model. 
+ 
+    The confusion matrix in Fig. 8 shows the correct and 
+wrong classification for each category with inter-class 
+classification. Among 45 test images, alopecia (label 0) has 
+19 images, psoriasis (label 1) has 13 images, and folliculitis 
+(label 2) has 13 images. A total of 17 alopecia images were 
+classified as alopecia and the other 2 were incorrectly 
+classified as psoriasis. Again, 11 psoriasis images are 
+classified as psoriasis, but 2 psoriasis images were incorrectly 
+classified as alopecia. All 13 folliculitis images are classified 
+correctly. The fractional incorrect prediction for each class is 
+shown in Fig 9. Our model achieved the precision and recall 
+score of 0.895 for the alopecia disease class, 0.846 for the 
+psoriasis disease class, and 1.0 for the folliculitis disease 
+class. As the precision and recall scores are same in each 
+class, F1 scores are also similar to their respective precision 
+and recall values. 
+ 
+VII. CONCLUSION 
+    Although early-stage detection of hair and scalp-related 
+diseases is the key to the treatment process, hair loss and scalp 
+diseases can often go undetected due to a lack of awareness 
+and a lengthy diagnosis test. An AI-based application might 
+pave the way to facilitate early disease detection. In this 
+study, we developed a machine learning model to accurately 
+predict three hair and scalp-related diseases: alopecia, 
+folliculitis, and psoriasis by feeding 150 preprocessed image 
+data into a 2-D convolutional neural network model. After 
+using 70% of the data to train the model, we analyzed 
+remaining 30% of images for testing our model. After 
+subsequent training, the model gave an overall 96.2% training 
+accuracy on the training data and 91.1% validation accuracy 
+for the test data, with a high precision and recall scores for 
+each disease type. We have also provided our dataset with this 
+study. Our proposed system would assist dermatologists and 
+patients with a better understanding of disease classification 
+and initiating early treatment options for the three most 
+frequently occurred hair and scalp diseases. 
+ 
+FUNDING 
+    No funding was used to write this research paper. 
+ 
+CONFLICT OF INTEREST 
+The authors declare that they do not have any conflicts of 
+interest. 
+ 
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+2018;132:437–43. doi: 10.1016/j.procs.2018.05.157. 
+[20] ALKolifi ALEnezi NS. A method of skin disease detection using image 
+processing and machine learning. Procedia Comput Sci. 2019;163:85–
+92. doi: 10.1016/j.procs.2019.12.090. 
+[21] Hameed N, Shabut AM, Hossain MA. Multi-class skin diseases 
+classification using deep convolutional neural network and support 
+vector machine. In: 2018 12th International Conference on Software, 
+Knowledge, Information Management & Applications (SKIMA). 
+IEEE; 2018. 
+[22] Mrinmoy-Roy. Scalp-Hair-Diseases-Detection [Internet]. 2022 [cited 
+2022 
+Oct 
+30]. 
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+https://github.com/Mrinmoy-
+Roy/Scalp-Hair-Diseases-Detection.git. 
+[23] Swain A. Noise in digital image processing [Internet]. Image Vision. 
+2018 [cited 2022 Oct 29]. Available from: https://medium.com/image-
+vision/noise-in-digital-image-processing-55357c9fab71. 
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+
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+page_content='RESEARCH ARTICLE European Journal of Information Technologies and Computer Science www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfUPel/content/2301.00122v1.pdf'}
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+page_content='YEAR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfUPel/content/2301.00122v1.pdf'}
+page_content='VOL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfUPel/content/2301.00122v1.pdf'}
+page_content='ISSUE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfUPel/content/2301.00122v1.pdf'}
+page_content='ID Vol X | Issue Y | Month Year 1  I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfUPel/content/2301.00122v1.pdf'}
+page_content=' INTRODUCTION Hair, made of keratin protein, pertains to beauty and masculinity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfUPel/content/2301.00122v1.pdf'}
+page_content=' Approximately 5 million hair follicles are present throughout our body [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfUPel/content/2301.00122v1.pdf'}
+page_content=' Scalp Hair maintains body temperature and protects the brain from external heat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfUPel/content/2301.00122v1.pdf'}
+page_content=' A typical hair growth cycle runs for 2-7 years, according to Patel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfUPel/content/2301.00122v1.pdf'}
+page_content=' [2] and Wolff, Fischer, and Blume-Peytavi [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfUPel/content/2301.00122v1.pdf'}
+page_content=' A healthy human has 100,000 hairs on the scalp, and 50-100 hair loss per day is considered normal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfUPel/content/2301.00122v1.pdf'}
+page_content=' Hair loss is not a present-day issue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfUPel/content/2301.00122v1.pdf'}
+page_content=' The hair-loss treatment was found in ancient Ayurveda scriptures 6000 years ago [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfUPel/content/2301.00122v1.pdf'}
+page_content=' However, Hair and scalp-related issues are gaining more recognition nowadays compared to earlier years due to certain factors, such as environmental pollution, hormonal imbalance, autoimmune disease, gut microbiota alteration, elevated physical and mental stress levels in human lifestyle, seasonal change, unhealthy diet, micronutrient deficiency, genetic predisposition, and side-effects of drugs [2], [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfUPel/content/2301.00122v1.pdf'}
+page_content=' According to Peyravian et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfUPel/content/2301.00122v1.pdf'}
+page_content=', 80 million Americans have hair loss- related issues to some extent [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfUPel/content/2301.00122v1.pdf'}
+page_content=' Although most hair loss diseases are localized, some can spread to other locations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfUPel/content/2301.00122v1.pdf'}
+page_content=' Some diseases require prescribed drugs and hair transplantation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfUPel/content/2301.00122v1.pdf'}
+page_content=' Some diseases are caused by bacterial or fungal infections and require antibiotic treatment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfUPel/content/2301.00122v1.pdf'}
+page_content=' Often, there are genetic and sexual predispositions in hair-scalp diseases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfUPel/content/2301.00122v1.pdf'}
+page_content=' Alopecia, folliculitis, and psoriasis are some common causes of hair loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfUPel/content/2301.00122v1.pdf'}
+page_content='  There is a difference between regular hair fall and alopecia;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfUPel/content/2301.00122v1.pdf'}
+page_content=' the latter develops coin-sized bald patches all over the scalp area.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfUPel/content/2301.00122v1.pdf'}
+page_content=' Alopecia or patchy hair loss can be of different types.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfUPel/content/2301.00122v1.pdf'}
+page_content=' Androgenetic alopecia or male-pattern baldness (MPB) is the most common form of alopecia where the hairline starts to recede, following a pattern where the frontal and temple area are most affected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfUPel/content/2301.00122v1.pdf'}
+page_content=' 70% of men and 40% of women get this type of hair loss and thinning issue [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfUPel/content/2301.00122v1.pdf'}
+page_content=' According to Liu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfUPel/content/2301.00122v1.pdf'}
+page_content=', MPB is an X-linked polygenic disease, and males are more genetically prone to develop baldness at a mature age [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfUPel/content/2301.00122v1.pdf'}
+page_content=' Topical minoxidil solution thickens the hair by 50% [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfUPel/content/2301.00122v1.pdf'}
+page_content=' On the other hand, Alopecia areata (AA) is an autoimmune disease affecting individuals irrespective of age and sex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfUPel/content/2301.00122v1.pdf'}
+page_content=' Primarily affecting the scalp area, AA can also spread in the beard, eyelashes, and eyebrows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfUPel/content/2301.00122v1.pdf'}
+page_content=' In this case, the body’s immune cells cannot recognize hair follicles as ‘self.’ Instead, they consider these follicles as ‘foreign,’ which ultimately causes the hair follicles to be Hair and Scalp Disease Detection using Machine Learning and Image Processing  Mrinmoy Roy, Anica Tasnim Protity ABSTRACT  Almost 80 million Americans suffer from hair loss due to aging, stress, medication, or genetic makeup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfUPel/content/2301.00122v1.pdf'}
+page_content=' Hair and scalp-related diseases often go unnoticed in the beginning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfUPel/content/2301.00122v1.pdf'}
+page_content=' Sometimes, a patient cannot differentiate between hair loss and regular hair fall.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfUPel/content/2301.00122v1.pdf'}
+page_content=' Diagnosing hair-related diseases is time-consuming as it requires professional dermatologists to perform visual and medical tests.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfUPel/content/2301.00122v1.pdf'}
+page_content=' Because of that, the overall diagnosis gets delayed, which worsens the severity of the illness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfUPel/content/2301.00122v1.pdf'}
+page_content=' Due to the image-processing ability, neural network-based applications are used in various sectors, especially healthcare and health informatics, to predict deadly diseases like cancers and tumors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfUPel/content/2301.00122v1.pdf'}
+page_content=' These applications assist clinicians and patients and provide an initial insight into early-stage symptoms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfUPel/content/2301.00122v1.pdf'}
+page_content=' In this study, we used a deep learning approach that successfully predicts three main types of hair loss and scalp-related diseases: alopecia, psoriasis, and folliculitis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfUPel/content/2301.00122v1.pdf'}
+page_content=' However, limited study in this area, unavailability of a proper dataset, and degree of variety among the images scattered over the internet made the task challenging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfUPel/content/2301.00122v1.pdf'}
+page_content=' 150 images were obtained from various sources and then preprocessed by denoising, image equalization, enhancement, and data balancing, thereby minimizing the error rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfUPel/content/2301.00122v1.pdf'}
+page_content=' After feeding the processed data into the 2D convolutional neural network (CNN) model, we obtained overall training accuracy of 96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfUPel/content/2301.00122v1.pdf'}
+page_content='2%, with a validation accuracy of 91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfUPel/content/2301.00122v1.pdf'}
+page_content='1%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfUPel/content/2301.00122v1.pdf'}
+page_content='  The precision and recall score of alopecia, psoriasis, and folliculitis are 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfUPel/content/2301.00122v1.pdf'}
+page_content='895, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfUPel/content/2301.00122v1.pdf'}
+page_content='846, and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfUPel/content/2301.00122v1.pdf'}
+page_content='0, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfUPel/content/2301.00122v1.pdf'}
+page_content=' We also created a dataset of the scalp images for future prospective researchers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfUPel/content/2301.00122v1.pdf'}
+page_content=' Keywords:  Deep Learning, Health Informatics, Machine Learning, Scalp/ Hair Diseases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfUPel/content/2301.00122v1.pdf'}
+page_content='  Published Online:  ISSN: 2736 5492  DOI 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfUPel/content/2301.00122v1.pdf'}
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+page_content='YEAR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfUPel/content/2301.00122v1.pdf'}
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+page_content='ID  Mrinmoy Roy  Department  of  Computer  Science,  Northern Illinois University, USA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfUPel/content/2301.00122v1.pdf'}
+page_content='  (e mail: mrinmoy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfUPel/content/2301.00122v1.pdf'}
+page_content='cs10 gmail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfUPel/content/2301.00122v1.pdf'}
+page_content='com)  Anica Tasnim Protity  Department of Biological Sciences,  Northern Illinois University, USA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfUPel/content/2301.00122v1.pdf'}
+page_content='  (e mail:  protity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfUPel/content/2301.00122v1.pdf'}
+page_content='microbiology@gmail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfUPel/content/2301.00122v1.pdf'}
+page_content='com)  Corresponding Author @  RESEARCH ARTICLE European Journal of Information Technologies and Computer Science www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfUPel/content/2301.00122v1.pdf'}
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+page_content='org  DOI: http://dx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfUPel/content/2301.00122v1.pdf'}
+page_content='doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfUPel/content/2301.00122v1.pdf'}
+page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfUPel/content/2301.00122v1.pdf'}
+page_content='24018/ejcompute.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfUPel/content/2301.00122v1.pdf'}
+page_content='YEAR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfUPel/content/2301.00122v1.pdf'}
+page_content='VOL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfUPel/content/2301.00122v1.pdf'}
+page_content='ISSUE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfUPel/content/2301.00122v1.pdf'}
+page_content='ID Vol X | Issue Y | Month Year 2  targeted and destroyed by the immune cells.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfUPel/content/2301.00122v1.pdf'}
+page_content=' It is an example of a hereditary disease.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfUPel/content/2301.00122v1.pdf'}
+page_content=' The study from Benigno et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfUPel/content/2301.00122v1.pdf'}
+page_content=' reported that, in the US alone, 700,000 individuals suffer from AA [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfUPel/content/2301.00122v1.pdf'}
+page_content=' This disease, if diagnosed early, might resolve spontaneously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfUPel/content/2301.00122v1.pdf'}
+page_content=' In severe cases, topical corticosteroid or immune therapy is used [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfUPel/content/2301.00122v1.pdf'}
+page_content=' Sometimes, the hair follicles might get inflamed because of the action of bacterial accumulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfUPel/content/2301.00122v1.pdf'}
+page_content=' This follicle inflammation is called folliculitis decalvans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfUPel/content/2301.00122v1.pdf'}
+page_content=' The bacterium Staphylococcus aureus damages the follicle and prevents hair growth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfUPel/content/2301.00122v1.pdf'}
+page_content=' Staphylococcus aureus uses hair tufts to enter underneath the follicle, causing chronic inflammation, redness, swelling, scarring, itching, and hair loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfUPel/content/2301.00122v1.pdf'}
+page_content=' Antibiotic treatment combined with surgical removal of hair tufts and corticosteroids for reducing inflammation are the prescribed treatment for Folliculitis decalvans [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfUPel/content/2301.00122v1.pdf'}
+page_content=' Psoriasis is another form of common scalp skin disease.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfUPel/content/2301.00122v1.pdf'}
+page_content=' According to [7], 54% of 5600 psoriasis patients had scalp psoriasis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfUPel/content/2301.00122v1.pdf'}
+page_content=' Severe psoriasis may cause significant itching, scaling, and redness in the scalp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfUPel/content/2301.00122v1.pdf'}
+page_content=' The application of topical shampoo and corticosteroids are the treatment options by Chan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfUPel/content/2301.00122v1.pdf'}
+page_content=' [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfUPel/content/2301.00122v1.pdf'}
+page_content=' Some scalp infections may be treatable if diagnosed early.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfUPel/content/2301.00122v1.pdf'}
+page_content=' Some but not all diseases may go on their own.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfUPel/content/2301.00122v1.pdf'}
+page_content=' Only an expert physician can detect the illness by visual observation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfUPel/content/2301.00122v1.pdf'}
+page_content=' In some cases, early disease detection is beneficial for dermatologists to initiate the treatment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfUPel/content/2301.00122v1.pdf'}
+page_content='  An early scalp inspection includes a dermatoscopic examination of the scalp for inflammation, itching, localized lesion, dandruff, follicular flakes, louse eggs (nits), and a scalp biopsy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfUPel/content/2301.00122v1.pdf'}
+page_content=' Besides visual observation, the patient can undergo blood and hormone tests to detect the exact disease.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfUPel/content/2301.00122v1.pdf'}
+page_content=' Unfortunately, most hair and scalp diseases are diagnosed in advanced stages, which complicate the treatment options.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfUPel/content/2301.00122v1.pdf'}
+page_content=' All these factors lengthen the diagnosis and treatment process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfUPel/content/2301.00122v1.pdf'}
+page_content=' Therefore, researchers are putting more effort into developing different mechanisms for the early detection of hair and scalp diseases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfUPel/content/2301.00122v1.pdf'}
+page_content=' In the 21st century, with all the advancements in computational technology, extensive application of machine learning has made our daily lives simple, comfortable, and secure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfUPel/content/2301.00122v1.pdf'}
+page_content=' The increasing popularity of machine learning and its nature to extract patterns from data are directing researchers to incorporate several machine learning algorithms into health informatics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfUPel/content/2301.00122v1.pdf'}
+page_content=' Especially during the Covid-19 pandemic era, different applications like restraining people from covid- 19 spread [9], SARS-CoV-2 screening and treatment [10], lock-down control in case of high dimensional input [11] came into play, which made machine learning and healthcare systems inseparable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfUPel/content/2301.00122v1.pdf'}
+page_content=' Overall, adapting, integrating, and developing deep learning-based applications on patients’ information, medical reports, and audio-video feedback make the diagnosis process faster.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfUPel/content/2301.00122v1.pdf'}
+page_content='  Nowadays, patients can get at least the initial idea of disease detection by themselves using easily accessible smart devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfUPel/content/2301.00122v1.pdf'}
+page_content=' All these applications clear their confusion and help them make health-related decisions independently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfUPel/content/2301.00122v1.pdf'}
+page_content=' The high computational capability of neural networks is, therefore, a breakthrough in healthcare and medical diagnostic organizations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfUPel/content/2301.00122v1.pdf'}
+page_content=' Convolutional neural networks (CNN) have brought revolutionary success in detecting deadly diseases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfUPel/content/2301.00122v1.pdf'}
+page_content=' To date, neural networks are assisting healthcare professionals in the early detection of different types of tumors and cancers, such as skin cancer (melanoma) [12], stomach cancer (adenocarcinoma) [13], and brain tumors (glioblastoma) [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfUPel/content/2301.00122v1.pdf'}
+page_content=' Neural networks are applicable in detecting life-threatening dengue fever [15] and covid-19 [16] as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfUPel/content/2301.00122v1.pdf'}
+page_content=' In one study, CNN was used to extract complex temporal dynamic features from heart rate variability (HRV) signals, developing an algorithm that facilitated the early detection of diabetics [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfUPel/content/2301.00122v1.pdf'}
+page_content=' Using the image processing ability of the neural networks, we can extract features from hair, skin and scalp images to classify and categorize numerous hair and scalp-related diseases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfUPel/content/2301.00122v1.pdf'}
+page_content=' In this work, due to the importance of early-stage hair disease detection, we applied convolutional neural networks to 3 types of hair diseases and developed a model to detect them successfully.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfUPel/content/2301.00122v1.pdf'}
+page_content='  II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfUPel/content/2301.00122v1.pdf'}
+page_content=' CHALLENGES AND CONTRIBUTIONS A classic application of computer vision is to detect disease using digital images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfUPel/content/2301.00122v1.pdf'}
+page_content=' Researchers can exploit a pool of digital images obtained from one or more datasets, preprocess the images, feed the images into the neural network, and develop a model to detect the disease.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfUPel/content/2301.00122v1.pdf'}
+page_content=' Unfortunately, minimal research has been performed on the machine-learning approach for scalp disease detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfUPel/content/2301.00122v1.pdf'}
+page_content=' There are several unique challenges behind this.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfUPel/content/2301.00122v1.pdf'}
+page_content=' First and foremost, hair diseases are not localized and can spread to different regions of the scalp, beard, eyebrows, eyelashes, and pubic area.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfUPel/content/2301.00122v1.pdf'}
+page_content=' Second, every image needs different types of preprocessing before feeding to neural networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfUPel/content/2301.00122v1.pdf'}
+page_content=' Different scalp skin tones, hair colors, and types around the detection zones make the imaging process more complicated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfUPel/content/2301.00122v1.pdf'}
+page_content=' Third, no proper dataset for scalp diseases is available over the internet, and images taken from the internet differ in size and resolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfUPel/content/2301.00122v1.pdf'}
+page_content=' Moreover, one must be conscious of minimalizing and correcting the error in disease detection;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfUPel/content/2301.00122v1.pdf'}
+page_content=' otherwise, the high false-positive and false-negative rates result in misdiagnosis of the disease and worsening hair loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfUPel/content/2301.00122v1.pdf'}
+page_content=' To overcome the challenges, we developed a model which can successfully classify the alopecia, folliculitis, and psoriasis diseases with a minimal false-positive and false- negative rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfUPel/content/2301.00122v1.pdf'}
+page_content='  Though it is challenging to collect images for the diseases from the internet, and the images are varied in color, shape, and resolution, we applied various preprocessing, such as denoising, resizing, enhancement and created a dataset that might help in further scalp diseases research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfUPel/content/2301.00122v1.pdf'}
+page_content='  III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfUPel/content/2301.00122v1.pdf'}
+page_content=' RELATED WORKS Disease detection using machine learning approaches is gaining popularity in health informatics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfUPel/content/2301.00122v1.pdf'}
+page_content=' Many skin and scalp-related diseases can be detected using images of infected regions within a few seconds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfUPel/content/2301.00122v1.pdf'}
+page_content=' In one study by Choudhary et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfUPel/content/2301.00122v1.pdf'}
+page_content=' [18], a framework is developed to differentiate alopecia areata from healthy hair.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfUPel/content/2301.00122v1.pdf'}
+page_content=' They obtained 200 healthy hair images from the figaro1K dataset and 68 alopecia areata hair images from DermNet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfUPel/content/2301.00122v1.pdf'}
+page_content=' After a series of enhancement and segmentation, three key features were  RESEARCH ARTICLE European Journal of Information Technologies and Computer Science www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfUPel/content/2301.00122v1.pdf'}
+page_content='ej-compute.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfUPel/content/2301.00122v1.pdf'}
+page_content='org  DOI: http://dx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfUPel/content/2301.00122v1.pdf'}
+page_content='doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfUPel/content/2301.00122v1.pdf'}
+page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfUPel/content/2301.00122v1.pdf'}
+page_content='24018/ejcompute.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfUPel/content/2301.00122v1.pdf'}
+page_content='YEAR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfUPel/content/2301.00122v1.pdf'}
+page_content='VOL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfUPel/content/2301.00122v1.pdf'}
+page_content='ISSUE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfUPel/content/2301.00122v1.pdf'}
+page_content='ID Vol X | Issue Y | Month Year 3  extracted from the images: texture, shape, and color.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfUPel/content/2301.00122v1.pdf'}
+page_content=' The researchers divided the dataset into 70%-30% train-test-split and applied a support vector machine (SNM) and k-nearest neighbor (KNN) for the classification task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfUPel/content/2301.00122v1.pdf'}
+page_content=' Overall, they achieved 91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfUPel/content/2301.00122v1.pdf'}
+page_content='4% and 88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfUPel/content/2301.00122v1.pdf'}
+page_content='9% accuracy using SVM and KNN, respectively, with a 10-fold cross-validation approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfUPel/content/2301.00122v1.pdf'}
+page_content=' However, using other machine learning algorithms might increase in the accuracy rate, which should have been discussed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfUPel/content/2301.00122v1.pdf'}
+page_content=' Besides, the application of Histogram Equalization (HE) for image enhancement complicated the process of getting accurate texture features from distorted images, as HE itself adds noise to the output image, distorting the signals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfUPel/content/2301.00122v1.pdf'}
+page_content=' Moreover, this study only shed light on alopecia areata disease, ignoring the inter-class differences between other similar type diseases, which increased the likelihood of inaccurate prediction of other diseases as alopecia areata, thereby making this framework less reliable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfUPel/content/2301.00122v1.pdf'}
+page_content=' Another study [19] proposed a model for early alopecia detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfUPel/content/2301.00122v1.pdf'}
+page_content=' They used 100 samples for this research, with 80% as training data and the other 20% as testing data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfUPel/content/2301.00122v1.pdf'}
+page_content=' They looked for four attributes, length of the hair, nail brittleness, amount of damage made to the hair, and hair follicle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfUPel/content/2301.00122v1.pdf'}
+page_content=' Two- layer feed-forward network with a back propagation technique was used for detection purposes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfUPel/content/2301.00122v1.pdf'}
+page_content='  The proposed model system consisting of 4 input neurons, 10 hidden neurons, and a linear output neuron, achieved 91% training accuracy with 86.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfUPel/content/2301.00122v1.pdf'}
+page_content='7% validation accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfUPel/content/2301.00122v1.pdf'}
+page_content=' It showed the best performance at epoch 4 with a 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfUPel/content/2301.00122v1.pdf'}
+page_content='059687 gradient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfUPel/content/2301.00122v1.pdf'}
+page_content=' However, the study has some pitfalls, too, as they did not mention their data source or differentiate data classes with their respective sample sizes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfUPel/content/2301.00122v1.pdf'}
+page_content=' Also, no image pre-processing was performed on the collected images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfUPel/content/2301.00122v1.pdf'}
+page_content=' Although there is a possibility of overfitting without a proper data balancing technique, this report did not discuss the data balancing between the two classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfUPel/content/2301.00122v1.pdf'}
+page_content=' Furthermore, they did not calculate the model’s false- positive and false-negative rates, which is crucial for a model specially developed for the healthcare system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfUPel/content/2301.00122v1.pdf'}
+page_content=' Related work [20] was performed on skin disease detection, where machine learning was used to analyze the digital image of the affected skin area for identifying eczema, melanoma, and psoriasis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfUPel/content/2301.00122v1.pdf'}
+page_content=' Their dataset consists of 80 images from different websites specific to skin diseases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfUPel/content/2301.00122v1.pdf'}
+page_content=' By using a convolutional neural network for feature extraction and applying multiclass SVM on those features, they achieved 100% accuracy in disease classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfUPel/content/2301.00122v1.pdf'}
+page_content=' However, they did not explore other essential model performance matrices and overfitting issues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfUPel/content/2301.00122v1.pdf'}
+page_content='  In another skin disease detection-based article [21], the authors proposed a scheme to classify skin lesions into five categories: healthy, acne, eczema, benign, and malignant melanoma, using a pre-trained CNN model, AlexNET for feature extraction and error correcting output codes support vector machine for classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfUPel/content/2301.00122v1.pdf'}
+page_content=' The dataset consists of 9144 images from different sources and achieved 84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfUPel/content/2301.00122v1.pdf'}
+page_content='21% accuracy using a 10-fold cross-validation technique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfUPel/content/2301.00122v1.pdf'}
+page_content=' Overall, we observed very few works on hair diseases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfUPel/content/2301.00122v1.pdf'}
+page_content=' The recent related works lack at least one of the following categories – discussion over false positive and false negative rates, ignoring inter-class differences, model reliability, and overfitting problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfUPel/content/2301.00122v1.pdf'}
+page_content=' In this work, we have attempted to fill these gaps by leveraging a convolutional neural network algorithm on hair disease images while maintaining high accuracy with good precision and recall scores.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfUPel/content/2301.00122v1.pdf'}
+page_content='  IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfUPel/content/2301.00122v1.pdf'}
+page_content=' DATA DESCRIPTION & DEVICE A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfUPel/content/2301.00122v1.pdf'}
+page_content=' Data Collection The most challenging part of using visual images for disease prediction and disease classification is data collection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfUPel/content/2301.00122v1.pdf'}
+page_content=' Often, one can get fewer appropriate images for a specific illness found.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfUPel/content/2301.00122v1.pdf'}
+page_content=' Moreover, the pictures are scattered over the internet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfUPel/content/2301.00122v1.pdf'}
+page_content=' In this study, the authors extracted the images from different websites, such as DermQuest, DermNet, MedicineNet, DermnetNZ, and various medical professionals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfUPel/content/2301.00122v1.pdf'}
+page_content='  TABLE I: IMAGES PER DISEASE Disease Quantity Alopecia 65 Psoriasis 45 Folliculitis 40  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfUPel/content/2301.00122v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfUPel/content/2301.00122v1.pdf'}
+page_content=' Image subset of each disease category.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfUPel/content/2301.00122v1.pdf'}
+page_content='  The image quantity is different for each category.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfUPel/content/2301.00122v1.pdf'}
+page_content=' We found more alopecia-related images than other diseases because alopecia is more frequent and severe among the human population.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfUPel/content/2301.00122v1.pdf'}
+page_content=' The number of samples in each type of disease is listed in Table I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfUPel/content/2301.00122v1.pdf'}
+page_content=' Randomly selected images in each category are graphically represented in Fig 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfUPel/content/2301.00122v1.pdf'}
+page_content=' Our dataset is made publicly available on Github [22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfUPel/content/2301.00122v1.pdf'}
+page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfUPel/content/2301.00122v1.pdf'}
+page_content=' Device The research was conducted on Dell Latitude 5520 laptop device having 11th generation Intel Core i5 (8 MB cache, 4 cores, 8 threads, up to 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfUPel/content/2301.00122v1.pdf'}
+page_content='40 GHz Turbo) and running on Windows 10 Pro operating system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfUPel/content/2301.00122v1.pdf'}
+page_content=' The device has 16 GB, 1 x 16 GB, DDR4, 3200 MHz random access memory (RAM), and 256 GB, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfUPel/content/2301.00122v1.pdf'}
+page_content='2 PCIe NVMe, SSD, Class 35 (NVRAM).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfUPel/content/2301.00122v1.pdf'}
+page_content=' For the classification of images, we utilized the integrated Intel Iris XE graphics capable with a thunderbolt for I5-1145G7 vPro processor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfUPel/content/2301.00122v1.pdf'}
+page_content=' For the data  Alopecia Psoriasis folliculitis RESEARCH ARTICLE European Journal of Information Technologies and Computer Science www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfUPel/content/2301.00122v1.pdf'}
+page_content='ej-compute.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfUPel/content/2301.00122v1.pdf'}
+page_content='org  DOI: http://dx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfUPel/content/2301.00122v1.pdf'}
+page_content='doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfUPel/content/2301.00122v1.pdf'}
+page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfUPel/content/2301.00122v1.pdf'}
+page_content='24018/ejcompute.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfUPel/content/2301.00122v1.pdf'}
+page_content='YEAR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfUPel/content/2301.00122v1.pdf'}
+page_content='VOL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfUPel/content/2301.00122v1.pdf'}
+page_content='ISSUE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfUPel/content/2301.00122v1.pdf'}
+page_content='ID Vol X | Issue Y | Month Year 4  collection, we used iPhone-13 Pro Max having Hexa-core (2x3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfUPel/content/2301.00122v1.pdf'}
+page_content='23 GHz Avalanche + 4x1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfUPel/content/2301.00122v1.pdf'}
+page_content='82 GHz Blizzard) CPU and Apple GPU (5-core graphics).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfUPel/content/2301.00122v1.pdf'}
+page_content=' We used a mobile device with 128GB 6GB RAM, and a 12 MP triple main camera for the image collection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfUPel/content/2301.00122v1.pdf'}
+page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfUPel/content/2301.00122v1.pdf'}
+page_content=' PROPOSED MODEL In this section, we introduce the system workflow of our model and explain the functions of each module in details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfUPel/content/2301.00122v1.pdf'}
+page_content=' As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfUPel/content/2301.00122v1.pdf'}
+page_content=' 2, first, the captured image is sent to preprocessing steps which are divided into three parts: image equalization, image enhancement, and data balancing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfUPel/content/2301.00122v1.pdf'}
+page_content='  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfUPel/content/2301.00122v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfUPel/content/2301.00122v1.pdf'}
+page_content=' System workflow of hair disease detection model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfUPel/content/2301.00122v1.pdf'}
+page_content='  Among these three, the first two parts are mainly for increasing image quality, and the last part is for model versatility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfUPel/content/2301.00122v1.pdf'}
+page_content=' After the preprocessing steps, the image is passed to the Neural Network model for the classification task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfUPel/content/2301.00122v1.pdf'}
+page_content=' We used a convolutional neural network that classifies an image successfully into three different classes: alopecia, folliculitis, and psoriasis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfUPel/content/2301.00122v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfUPel/content/2301.00122v1.pdf'}
+page_content=' Denoising  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfUPel/content/2301.00122v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfUPel/content/2301.00122v1.pdf'}
+page_content=' Left original image & right non-local means denoised image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfUPel/content/2301.00122v1.pdf'}
+page_content='  Noise is the degradation of image signals caused by external sources [23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfUPel/content/2301.00122v1.pdf'}
+page_content=' Noise introduces random variations of brightness or color information in the captured images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfUPel/content/2301.00122v1.pdf'}
+page_content=' Most of the time, images on the internet have some noise associated with them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfUPel/content/2301.00122v1.pdf'}
+page_content=' As we have collected most of the data samples from different dermatology websites, the noise in our dataset is not homogeneously distributed, which made it more complex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfUPel/content/2301.00122v1.pdf'}
+page_content=' Therefore, we applied additional filters for denoising the collected images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfUPel/content/2301.00122v1.pdf'}
+page_content=' We started with the gaussian filter for a better image classification process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfUPel/content/2301.00122v1.pdf'}
+page_content=' However, after using the gaussian filter, the images become completely blurred, which leads to the loss of important information and damage to the edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfUPel/content/2301.00122v1.pdf'}
+page_content=' We then applied the median filter, which worked better than the gaussian filter with kernel_size = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfUPel/content/2301.00122v1.pdf'}
+page_content=' Though we achieved better accuracy using the bilateral filter, we got the best results while applying the non-local means filter with patch_size = 3 and patch_distance = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfUPel/content/2301.00122v1.pdf'}
+page_content=' This non-local means filter preserved all the edges and reduced the noise better than the other filters for our application which is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfUPel/content/2301.00122v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfUPel/content/2301.00122v1.pdf'}
+page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfUPel/content/2301.00122v1.pdf'}
+page_content=' Image Equalization Often the captured image doesn’t reflect the natural view and needs contrast enhancement to meet the level of realistic view [24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfUPel/content/2301.00122v1.pdf'}
+page_content=' Especially images with high color depth and after denoising effects need normalization for a better realistic view [25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfUPel/content/2301.00122v1.pdf'}
+page_content=' First, we applied histogram equalization (HE).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfUPel/content/2301.00122v1.pdf'}
+page_content='  However, the HE increases the contrast of the background when used in images with low color depth, and information is lost as the histogram is not confined to the local region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfUPel/content/2301.00122v1.pdf'}
+page_content=' To overcome the problem, we applied CLAHE (Contrast Limited Adaptive Histogram Equalization) by dividing an image into equal size non-overlapping areas and computing a histogram for each region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfUPel/content/2301.00122v1.pdf'}
+page_content=' After clipping the histogram, we distributed the clipped value over the histogram equalization, which gives us control of the over-amplification of the contrast and generates the resultant image shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfUPel/content/2301.00122v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfUPel/content/2301.00122v1.pdf'}
+page_content='  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfUPel/content/2301.00122v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfUPel/content/2301.00122v1.pdf'}
+page_content=' Image equalization using CLAHE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfUPel/content/2301.00122v1.pdf'}
+page_content='  C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfUPel/content/2301.00122v1.pdf'}
+page_content=' Data Balancing The overall performance of a machine learning model depends on the balanced dataset because, without it, minority class detection becomes difficult.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfUPel/content/2301.00122v1.pdf'}
+page_content=' Balancing a dataset reduces the risk of skewing towards the majority.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfUPel/content/2301.00122v1.pdf'}
+page_content=' Imbalanced data might achieve high accuracy, but the results are biased toward the majority class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfUPel/content/2301.00122v1.pdf'}
+page_content=' As alopecia is a common disease, we have more alopecia images than other diseases, which creates an imbalanced dataset for our model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfUPel/content/2301.00122v1.pdf'}
+page_content=' For balancing the dataset, we used data augmentation techniques (re-scaling, random rotating, cropping, vertical and horizontal flipping) and oversampled the infrequent class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfUPel/content/2301.00122v1.pdf'}
+page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfUPel/content/2301.00122v1.pdf'}
+page_content=' Neural Network Model Neural network is the most applied model for visual data analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfUPel/content/2301.00122v1.pdf'}
+page_content=' Neural network needs limited human assistance and  Image Image Data Denoising Enhancement Augmentation Convolutional Diseaseclass Neural Detected Network Model RESEARCH ARTICLE European Journal of Information Technologies and Computer Science www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfUPel/content/2301.00122v1.pdf'}
+page_content='ej-compute.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfUPel/content/2301.00122v1.pdf'}
+page_content='org  DOI: http://dx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfUPel/content/2301.00122v1.pdf'}
+page_content='doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfUPel/content/2301.00122v1.pdf'}
+page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfUPel/content/2301.00122v1.pdf'}
+page_content='24018/ejcompute.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfUPel/content/2301.00122v1.pdf'}
+page_content='YEAR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfUPel/content/2301.00122v1.pdf'}
+page_content='VOL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfUPel/content/2301.00122v1.pdf'}
+page_content='ISSUE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfUPel/content/2301.00122v1.pdf'}
+page_content='ID Vol X | Issue Y | Month Year 5  can identify complex non-linear relationship between input and output.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfUPel/content/2301.00122v1.pdf'}
+page_content=' From global or local scale modeling [26] to diagnosis by medical image classification, neural network is using extensively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfUPel/content/2301.00122v1.pdf'}
+page_content=' Moreover, Facial recognition, image labeling, accurate video subtitles, assisting call centers, automated virtual agents all these things are using neural network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfUPel/content/2301.00122v1.pdf'}
+page_content=' There are 3 types of neural network available: Artificial Neural Networks (ANN), Convolution Neural Networks (CNN) and Recurrent Neural Networks (RNN).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfUPel/content/2301.00122v1.pdf'}
+page_content=' Each neural network has mainly 3 components: an input layer, a processing layer, and an output layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfUPel/content/2301.00122v1.pdf'}
+page_content='  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfUPel/content/2301.00122v1.pdf'}
+page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfUPel/content/2301.00122v1.pdf'}
+page_content=' Neural Network Model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfUPel/content/2301.00122v1.pdf'}
+page_content='  In this study, CNN is utilized for classification because it takes image’s raw pixel data, trains a model, and extracts the features automatically for better detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfUPel/content/2301.00122v1.pdf'}
+page_content=' We used autokeras to find the best model for this problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfUPel/content/2301.00122v1.pdf'}
+page_content=' After trying 25 different combinations, we selected 3 hidden layers with 1 input and 1 output layer as our final model which is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfUPel/content/2301.00122v1.pdf'}
+page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfUPel/content/2301.00122v1.pdf'}
+page_content=' For training the model, we used batch_size = 16 with 50 epochs for each batch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfUPel/content/2301.00122v1.pdf'}
+page_content=' The preprocessed data is divided into 70-30 train-test-split for training and validation purpose.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfUPel/content/2301.00122v1.pdf'}
+page_content=' Our model consists of 256 inputs, 3 x 3 square kernel, 3 output units and a softmax output.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfUPel/content/2301.00122v1.pdf'}
+page_content=' We used ReLU as our activation function to prevent the exponential growth of required computation and to explore the non-linear relationship between input and output variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfUPel/content/2301.00122v1.pdf'}
+page_content=' After each convolutional layer, input goes through the pooling layer having 2 x 2 kernel size to reduce the dimensions of the features map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfUPel/content/2301.00122v1.pdf'}
+page_content=' Pooling layer summarizes the presented features in a region and helps to prevent the over-fitting problem by down sampling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfUPel/content/2301.00122v1.pdf'}
+page_content=' We also used dropout layer after each pooling layer to prevent neurons in a layer from synchronously optimizing their weights and converging to the same goal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfUPel/content/2301.00122v1.pdf'}
+page_content=' Our model’s dropout rate is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfUPel/content/2301.00122v1.pdf'}
+page_content='3, which means 30% of the neurons of this layer will be randomly dropped in each epoch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfUPel/content/2301.00122v1.pdf'}
+page_content='  All the resultant 2-D arrays from pooled features map passes through the flatten layer and converted to single dimensional long continuous linear vector in the transition towards the fully connected layer as in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfUPel/content/2301.00122v1.pdf'}
+page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfUPel/content/2301.00122v1.pdf'}
+page_content=' In the fully connected layer, every single output pixel from the convolutional layers is connected to 3 output classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfUPel/content/2301.00122v1.pdf'}
+page_content=' Though dense layer is computationally expensive, we used 2 dense layers for our classification task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfUPel/content/2301.00122v1.pdf'}
+page_content=' Finally, we used softmax activation function to transform the 3 units of fully connected layer to a probability distribution which is represented by a vector of 3 elements, and the highest probability element selected as the final class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfUPel/content/2301.00122v1.pdf'}
+page_content=' We leveraged adam optimizer for learning purpose and reducing the overall loss by changing the weights and learning rates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfUPel/content/2301.00122v1.pdf'}
+page_content=' We used adam because it can handle sparse gradients on noisy problems and combines the best properties of AgaGrad and RMSProp algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfUPel/content/2301.00122v1.pdf'}
+page_content='  VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfUPel/content/2301.00122v1.pdf'}
+page_content=' RESULTS We trained our CNN model using the optimal hyperparameters selected from the grid search.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfUPel/content/2301.00122v1.pdf'}
+page_content=' These hyperparameters are listed in Table II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfUPel/content/2301.00122v1.pdf'}
+page_content=' We divided the dataset into 70%-30% train-test-split where 105 randomly selected images are used for training and 45 random images for testing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfUPel/content/2301.00122v1.pdf'}
+page_content=' After applying the preprocessing steps, we used the training dataset to train the CNN model and evaluated the test dataset using the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfUPel/content/2301.00122v1.pdf'}
+page_content='  TABLE II: HYPERPARAMETERS OF CNN MODEL Hyperparameters Values Batch Size 16 Epoch 50 Kernel Size 3 x 3 Optimizer Adam Dropout Rate 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfUPel/content/2301.00122v1.pdf'}
+page_content='3 Pooling Size 2 x 2 Activation Function ReLU  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfUPel/content/2301.00122v1.pdf'}
+page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfUPel/content/2301.00122v1.pdf'}
+page_content=' Training and Validation loss for CNN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfUPel/content/2301.00122v1.pdf'}
+page_content='  Conv 1 Max-Pooling Conv 2 Max-Pooling Conv 3 Max-Pooling Convolution (2 x 2) Convolution (2 x 2) Kernel Convolution (2 x 2) Kernel Fully-Connected (3 x 3) Kernel (3 x 3) Dropout 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfUPel/content/2301.00122v1.pdf'}
+page_content='3 (3 × 3) Kernel Dropout 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfUPel/content/2301.00122v1.pdf'}
+page_content='3 Input OutputTraining and validation loss 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfUPel/content/2301.00122v1.pdf'}
+page_content='25 Training loss 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfUPel/content/2301.00122v1.pdf'}
+page_content='00 Validation loss 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfUPel/content/2301.00122v1.pdf'}
+page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfUPel/content/2301.00122v1.pdf'}
+page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfUPel/content/2301.00122v1.pdf'}
+page_content='00 0 10 20 30 40 50 Epochs RESEARCH ARTICLE European Journal of Information Technologies and Computer Science www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfUPel/content/2301.00122v1.pdf'}
+page_content='ej-compute.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfUPel/content/2301.00122v1.pdf'}
+page_content='org  DOI: http://dx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfUPel/content/2301.00122v1.pdf'}
+page_content='doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfUPel/content/2301.00122v1.pdf'}
+page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfUPel/content/2301.00122v1.pdf'}
+page_content='24018/ejcompute.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfUPel/content/2301.00122v1.pdf'}
+page_content='YEAR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfUPel/content/2301.00122v1.pdf'}
+page_content='VOL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfUPel/content/2301.00122v1.pdf'}
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+page_content='ID Vol X | Issue Y | Month Year 6  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfUPel/content/2301.00122v1.pdf'}
+page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfUPel/content/2301.00122v1.pdf'}
+page_content=' Training and Validation Accuracy for CNN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfUPel/content/2301.00122v1.pdf'}
+page_content=' Our system achieved 96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfUPel/content/2301.00122v1.pdf'}
+page_content='2% training accuracy and 91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfUPel/content/2301.00122v1.pdf'}
+page_content='1% validation accuracy on the unseen data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfUPel/content/2301.00122v1.pdf'}
+page_content=' Validation and training losses for every epoch are shown in Fig 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfUPel/content/2301.00122v1.pdf'}
+page_content=' The training losses decreased from 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfUPel/content/2301.00122v1.pdf'}
+page_content='1685 to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfUPel/content/2301.00122v1.pdf'}
+page_content='1017, and the validation losses decrease from 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfUPel/content/2301.00122v1.pdf'}
+page_content='1260 to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfUPel/content/2301.00122v1.pdf'}
+page_content='3438 while going from epoch 1 to epoch 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfUPel/content/2301.00122v1.pdf'}
+page_content=' At the same time, Training accuracy and validation accuracy increased to 96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfUPel/content/2301.00122v1.pdf'}
+page_content='2% and 91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfUPel/content/2301.00122v1.pdf'}
+page_content='1%, respectively, from epoch 1 to epoch 50, shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfUPel/content/2301.00122v1.pdf'}
+page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfUPel/content/2301.00122v1.pdf'}
+page_content='  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfUPel/content/2301.00122v1.pdf'}
+page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfUPel/content/2301.00122v1.pdf'}
+page_content=' Confusion Matrix of Our Model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfUPel/content/2301.00122v1.pdf'}
+page_content='  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfUPel/content/2301.00122v1.pdf'}
+page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfUPel/content/2301.00122v1.pdf'}
+page_content=' Fractional Incorrect Prediction of Our Model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfUPel/content/2301.00122v1.pdf'}
+page_content='  The confusion matrix in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfUPel/content/2301.00122v1.pdf'}
+page_content=' 8 shows the correct and wrong classification for each category with inter-class classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfUPel/content/2301.00122v1.pdf'}
+page_content=' Among 45 test images, alopecia (label 0) has 19 images, psoriasis (label 1) has 13 images, and folliculitis (label 2) has 13 images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfUPel/content/2301.00122v1.pdf'}
+page_content=' A total of 17 alopecia images were classified as alopecia and the other 2 were incorrectly classified as psoriasis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfUPel/content/2301.00122v1.pdf'}
+page_content=' Again, 11 psoriasis images are classified as psoriasis, but 2 psoriasis images were incorrectly classified as alopecia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfUPel/content/2301.00122v1.pdf'}
+page_content=' All 13 folliculitis images are classified correctly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfUPel/content/2301.00122v1.pdf'}
+page_content=' The fractional incorrect prediction for each class is shown in Fig 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfUPel/content/2301.00122v1.pdf'}
+page_content=' Our model achieved the precision and recall score of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfUPel/content/2301.00122v1.pdf'}
+page_content='895 for the alopecia disease class, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfUPel/content/2301.00122v1.pdf'}
+page_content='846 for the psoriasis disease class, and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfUPel/content/2301.00122v1.pdf'}
+page_content='0 for the folliculitis disease class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfUPel/content/2301.00122v1.pdf'}
+page_content=' As the precision and recall scores are same in each class, F1 scores are also similar to their respective precision and recall values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfUPel/content/2301.00122v1.pdf'}
+page_content='  VII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfUPel/content/2301.00122v1.pdf'}
+page_content=' CONCLUSION Although early-stage detection of hair and scalp-related diseases is the key to the treatment process, hair loss and scalp diseases can often go undetected due to a lack of awareness and a lengthy diagnosis test.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfUPel/content/2301.00122v1.pdf'}
+page_content=' An AI-based application might pave the way to facilitate early disease detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfUPel/content/2301.00122v1.pdf'}
+page_content=' In this study, we developed a machine learning model to accurately predict three hair and scalp-related diseases: alopecia, folliculitis, and psoriasis by feeding 150 preprocessed image data into a 2-D convolutional neural network model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfUPel/content/2301.00122v1.pdf'}
+page_content=' After using 70% of the data to train the model, we analyzed remaining 30% of images for testing our model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfUPel/content/2301.00122v1.pdf'}
+page_content=' After subsequent training, the model gave an overall 96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfUPel/content/2301.00122v1.pdf'}
+page_content='2% training accuracy on the training data and 91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfUPel/content/2301.00122v1.pdf'}
+page_content='1% validation accuracy for the test data, with a high precision and recall scores for each disease type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfUPel/content/2301.00122v1.pdf'}
+page_content=' We have also provided our dataset with this study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfUPel/content/2301.00122v1.pdf'}
+page_content=' Our proposed system would assist dermatologists and patients with a better understanding of disease classification and initiating early treatment options for the three most frequently occurred hair and scalp diseases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfUPel/content/2301.00122v1.pdf'}
+page_content='  FUNDING No funding was used to write this research paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfUPel/content/2301.00122v1.pdf'}
+page_content='  CONFLICT OF INTEREST The authors declare that they do not have any conflicts of interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfUPel/content/2301.00122v1.pdf'}
+page_content='  REFERENCES [1] Cotsarelis G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfUPel/content/2301.00122v1.pdf'}
+page_content=' Gene expression profiling gets to the root of human hair follicle stem cells.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfUPel/content/2301.00122v1.pdf'}
+page_content=' J Clin Invest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfUPel/content/2301.00122v1.pdf'}
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+page_content=' doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfUPel/content/2301.00122v1.pdf'}
+page_content='1172/JCI27490.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfUPel/content/2301.00122v1.pdf'}
+page_content=' [2] Patel S, Sharma V, Chauhan NS, Thakur M, Dixit VK.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfUPel/content/2301.00122v1.pdf'}
+page_content=' Hair growth: Focus on herbal therapeutic agent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfUPel/content/2301.00122v1.pdf'}
+page_content=' Curr Drug Discov Technol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfUPel/content/2301.00122v1.pdf'}
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+page_content='12(1):21–42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfUPel/content/2301.00122v1.pdf'}
+page_content=' doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfUPel/content/2301.00122v1.pdf'}
+page_content='2174/1570163812666150610115055.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfUPel/content/2301.00122v1.pdf'}
+page_content=' [3] Wolff H, Fischer TW, Blume-Peytavi U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfUPel/content/2301.00122v1.pdf'}
+page_content=' The diagnosis and treatment of hair and scalp diseases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfUPel/content/2301.00122v1.pdf'}
+page_content=' Dtsch Arztebl Int.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfUPel/content/2301.00122v1.pdf'}
+page_content=' 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfUPel/content/2301.00122v1.pdf'}
+page_content=' doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfUPel/content/2301.00122v1.pdf'}
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+page_content='2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfUPel/content/2301.00122v1.pdf'}
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+page_content=' [4] Peyravian N, Deo S, Daunert S, Jimenez JJ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfUPel/content/2301.00122v1.pdf'}
+page_content=' The inflammatory aspect of male and female pattern hair loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfUPel/content/2301.00122v1.pdf'}
+page_content=' J Inflamm Res.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfUPel/content/2301.00122v1.pdf'}
+page_content=' 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfUPel/content/2301.00122v1.pdf'}
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+page_content=' Eur J Hum Genet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfUPel/content/2301.00122v1.pdf'}
+page_content=' 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfUPel/content/2301.00122v1.pdf'}
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+page_content=' [6] Benigno M, Anastassopoulos KP, Mostaghimi A, Udall M, Daniel SR, Cappelleri JC, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfUPel/content/2301.00122v1.pdf'}
+page_content=' A large cross-sectional survey study of the prevalence of alopecia areata in the United States.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfUPel/content/2301.00122v1.pdf'}
+page_content=' Clin Cosmet Investig Dermatol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfUPel/content/2301.00122v1.pdf'}
+page_content=' 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfUPel/content/2301.00122v1.pdf'}
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+page_content=' [7] Farber EM, Nall L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfUPel/content/2301.00122v1.pdf'}
+page_content=' The natural history of psoriasis in 5,600 patients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfUPel/content/2301.00122v1.pdf'}
+page_content=' Dermatology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfUPel/content/2301.00122v1.pdf'}
+page_content=' 1974;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfUPel/content/2301.00122v1.pdf'}
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+page_content=' IEEE;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfUPel/content/2301.00122v1.pdf'}
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+page_content='com/Mrinmoy- Roy/Scalp-Hair-Diseases-Detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfUPel/content/2301.00122v1.pdf'}
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+page_content='com/image- vision/noise-in-digital-image-processing-55357c9fab71.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfUPel/content/2301.00122v1.pdf'}
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+page_content='com/analytics-vidhya/image- equalization-contrast-enhancing-in-python-82600d3b371c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfUPel/content/2301.00122v1.pdf'}
+page_content='  [25] Hiroyasu T, Hayashinuma K, Ichikawa H, Yagi N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfUPel/content/2301.00122v1.pdf'}
+page_content=' Preprocessing with image denoising and histogram equalization for endoscopy image analysis using texture analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfUPel/content/2301.00122v1.pdf'}
+page_content=' Annu Int Conf IEEE Eng Med Biol Soc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfUPel/content/2301.00122v1.pdf'}
+page_content=' 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfUPel/content/2301.00122v1.pdf'}
+page_content='2015:789–92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfUPel/content/2301.00122v1.pdf'}
+page_content=' doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfUPel/content/2301.00122v1.pdf'}
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+page_content=' An exploratory study of Sentinel-1 SAR for rapid urban flood mapping on Google Earth Engine.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfUPel/content/2301.00122v1.pdf'}
+page_content=' Int J Appl Earth Obs Geoinf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfUPel/content/2301.00122v1.pdf'}
+page_content=' 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfUPel/content/2301.00122v1.pdf'}
+page_content='113(103002):103002.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfUPel/content/2301.00122v1.pdf'}
+page_content=' doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfUPel/content/2301.00122v1.pdf'}
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+page_content='2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfUPel/content/2301.00122v1.pdf'}
+page_content='103002.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfUPel/content/2301.00122v1.pdf'}
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+SELF-SUPERVISED LEARNING FOR A NONLINEAR INVERSE
+PROBLEM WITH FORWARD OPERATOR INVOLVING AN
+UNKNOWN FUNCTION ARISING IN PHOTOACOUSTIC
+TOMOGRAPHY
+Gyeongha Hwang1, Gihyeon Jeon2∗, Sunghwan Moon3
+1 Department of Mathematics, Yeungnam University, Gyeongsan 38541, Republic of Korea
+2 School of Mathematics, Kyungpook National University, Daegu 41566, Republic of Korea
+3 Department of Mathematics, Kyungpook National University, Daegu 41566, Republic of Korea
+*Corresponding author E-mails: rydbr6709@knu.ac.kr
+ABSTRACT. In this article, we concern with a nonlinear inverse problem with forward opera-
+tor involving an unknown function. The problem arises in diverse applications and is challenging
+by the presence of the unknown function, which makes it ill-posed. Additionally, the nonlinear
+nature of the problem makes it difficult to use traditional methods and thus the study has
+addressed a simplified version of the problem by either linearizing it or assuming knowledge of
+the unknown function. Here, we propose a self-supervised learning to directly tackle a nonlinear
+inverse problem involving an unknown function. In particular, we focus on an inverse problem
+derived in Photoacoustic Tomograpy (PAT) which is a hybrid medical imaging with high res-
+olution and contrast. PAT can be modelled based on the wave equation. The measured data
+is the solution of the equation restricted to the surface and the initial pressure of the equation
+contains the biological information on the object of interest. The speed of sound wave in the
+equation is unknown. Our goal is to determine the initial pressure and the speed of sound wave
+simultaneously. Under a simple assumption that the sound speed is a function of the initial
+pressure, the problem becomes a nonlinear inverse problem involving an unknown function. The
+experimental results demonstrate that the proposed algorithm performs successfully.
+1
+Introduction
+Inverse problem is to find the cause factor from the observed data, which has applications
+in many fields such as optics, radar, acoustics, communication theory, signal processing,
+medical imaging, computer vision, geophysics, oceanography, and astronomy because it
+tells us about what we cannot directly observe. The forward operator, the inverse of
+the inverse problem, can be modelled as an (non)-linear system and often involves an
+unknown function. Due to the nature of the inverse problem, it is usually very hard to
+know the cause factor. For example, in medical imaging the cause factor is the human
+body section and in seismology, we never know the structure of the earth’s interior.
+In this article, we concern with a nonlinear inverse problem with forward operator involv-
+ing an unknown function. Our goal is to find the unknown function and the inverse opera-
+tor simultaneously from the measurements. The problem is generally ill-posed because of
+1
+arXiv:2301.08693v1  [math.NA]  20 Jan 2023
+
+the unknown function. Additionally, the nonlinearity in the problem makes conventional
+methods difficult to use. To handle the problem, one may simplify the problem linearly
+or assume knowledge of the unknown function. Here we propose a self-supervised frame-
+work to directly tackle a nonlinear inverse problem involving an unknown function. In
+particular, we address an inverse problem derived in Photoacoustic Tomography (PAT).
+Although our proposed framework has been proposed to solve the problem arising in
+PAT, it is generic and can be extended to handle a nonlinear inverse problem involving
+an unknown function.
+The rest of the section is devoted to an introduction of PAT. In section 2, we formulate
+an inverse problem arising in PAT, which is nonlinear and also involves an unknown
+function. The structure and learning method of the proposed framework for the problem
+are described in section 3. The numerical simulation results in section 4 demonstrate that
+the proposed algorithm performs successfully.
+1.1
+Photoacoustic Tomography
+PAT is a hybrid medical imaging that combines the high contrast of optical imaging with
+the high spatial resolution of ultrasound images [1, 2, 3]. The physical basis of PAT is
+the photoacoustic effect discovered by Bell in 1881 [4]. In PAT, when an non-destructive
+testing target object absorbs a non-ionizing laser pulse, it thermally expands and emits
+acoustic waves. The emitted ultrasound contains biological information on the target
+object and is measured by an detector placed around it. The internal image of the target
+object is reconstructed from this measured data. The advantage of PAT is that it is
+economical and less harmful because of non-ionizing radiation use [5].
+The propagation of the emitted ultrasound p(x, t) can be described by the wave equation
+∂2
+t p(x, t) = c(x)2∆xp(x, t)
+on R2 × [0, ∞)
+(1)
+with the initial conditions
+p(x, 0) = f(x)
+∂tp(x, 0) = 0
+on R2.
+(2)
+Here c is the speed of waves and f is the initial pressure which contains biological in-
+formation such as the location of a cancer cells in a physically small tissue. It is natural
+assumption that f has compact support in the bounded domain Ω and the detectors are
+located on the boundary ∂Ω of the domain. Regarding the measurement procedure, the
+point-shaped detector measures the average pressure above ∂Ω where the detectors are
+located and this average pressure is the value of a pressure wave p(x, t). Therefore, one of
+mathematical problems in PAT is reconstructing f from the measured data p|∂Ω×[0,∞),
+which implies obtaining an internal image of the target object.
+It is well-known that given the initial pressure f and the speed c, the solution p is
+determined uniquely. Let us define the wave forward operator W as
+W : (f, c) �→ p|∂Ω×[0,∞),
+i.e.,
+W(f, c) = p|∂Ω×[0,∞).
+Reconstructing problem for f from W(f, c) is studied when speed c is constant [6, 7].
+Okanen, Stefanov and Uhlmann study the explicit reconstruction when the sound speed
+2
+
+is known
+[8, 9]. If c depends on space variable x, the problem become much more
+difficult. A few of researchers have studied the problem with a given variable sound
+speed [10, 11, 12, 13]. Liu and Uhlmann figure out the sufficient conditions for recovering
+f and c [14].
+Recently, the application of deep learning in an medical imaging including PAT has been
+investigated extensively. Roles of deep learning in tomography include forward and in-
+verse operator approximation, image reconstruction from sparse data, and artifact/noise
+removal from reconstructed images [15, 16, 17, 18, 19, 20, 21]. There are also studies on
+limited-view data (see [22, 23]). H. Shan et al. propose an iterative optimization algo-
+rithm which reconstructs f and c simultaneously via a supervised learning [24]. However,
+most works deal with linear inverse problems or inverse problems without involving an
+unknown function [25].
+Many studies on PAT with deep learning are based on a supervised learning. A supervised
+learning exploits a collection of paired data of the boundary data and the initial pres-
+sure. In practical applications, it is difficult to obtain the initial pressure, because initial
+pressure represents internal human body. Therefore, it is necessary to study a learning
+method exploiting the boundary data only. One such method is a self-supervised learning
+which exploits supervised signals that are generated from the input data by leveraging
+its structure [26, 27].
+2
+Problem formulation
+In this section, we formulate the problem precisely. For this, we will make several as-
+sumptions. First we assume f has compact support, since a target object is finite. Sec-
+ondly, c is assumed to be a function of f, namely c(x)2 = Γ(f(x)) for some function
+Γ : [0, 1] → [0, ∞), because the wave speed c depends on the medium. Lastly, we assume
+that Γ(0) and Γ(1) are known, namely Γ(0) = c0 and Γ(1) = c1. The last assumption is
+reasonable because Γ(0) and Γ(1) represent the wave speeds in the air and the highest
+thermal expansion coefficient respectively. Then the equation (1) is rewritten as:
+∂2
+t p(x, t) = Γ(f(x))∆xp(x, t)
+on R2 × [0, ∞).
+(3)
+Let us define WΓ by WΓ(f) = p|∂Ω×[0,∞) where p is the solution of (3) with initial
+conditions (2). Then the inverse problem can be formulated as determining unknown Γ
+and f from a given WΓ(f). However, this problem is ill-posed: for any Γ′ satisfying
+� Γ = Γ′ on Im(f)
+Γ ̸= Γ′ on Dom(Γ) \ Im(f),
+we have WΓ(f) = WΓ′(f). Hence Γ can not be uniquely determined from WΓ(f). There
+is also a possibility that there exist Γ1, Γ2, f1 and f2 such that Γ1 ̸= Γ2, f1 ̸= f2 and
+WΓ1(f1) = WΓ2(f2). Instead, we consider the following inverse problem:
+Problem 1. Let the collection of boundary data BΓ := {WΓ(f) | Γ : [0, 1] → [0, ∞), Γ(0) =
+c0, Γ(1) = c1 and f ∈ L2(R2) has compact support} be given.
+3
+
+1. Determine unknown Γ from BΓ.
+2. For all WΓ(f) ∈ BΓ, determine f.
+Then the uniqueness statements for Problem 1 are
+Hypothesis 1. If Γ1 ̸= Γ2, then BΓ1 ̸= BΓ2.
+and
+Hypothesis 2. For fixed Γ, if f1 ̸= f2, then WΓ(f1) ̸= WΓ(f2).
+In this article, we aim to solve Problem 1 under Hypothesis 1 and 2. The problem is
+difficult to solve because of
+1. (3) involves unknown Γ.
+2. (3) is not linear.
+We are going to solve Problem 1 by exploiting a deep neural network (DNN). Since DNN
+can only handle with finite data, we address the following inverse problem.
+Problem 2. For given {WΓ(fi) | Γ : [0, 1] → [0, ∞), Γ(0) = c0, Γ(1) = c1 and fi ∈
+L2(R2) has compact support, i = 1, · · · , N}, determine Γ and {fi|i = 1, · · · , N}.
+3
+Network Design
+Figure 1: The network architecture
+We propose a self-supervised learning for the problem formulated in section 2. Our goal
+is simultaneously reconstructing {fi}N
+i=1 and Γ from given collection {WΓ(fi)}N
+i=1. The
+proposed framework is depicted in Figure 1. It consists of three components:
+4
+
+input
+output
+Wr(f)
+W (R(Wr(f)))
+R(Wr(f))
+M(R(Wr(f)))
+LOSS = MSEWr(f),W(RWr(f)))1. Reconstruction network R
+2. Mapping network M
+3. Wave forward operator W.
+The reconstruction network R learns to reconstruct the initial data from the measured
+data. The mapping network M approximates the function Γ : [0, 1] → [0, ∞) satisfying
+c(x)2 = Γ(f(x)). The forward operator W assigns to the initial data and the wave
+speed the measured data. Here we adopt the k-space method. If every component in the
+framework functions properly, then output should be same to the input. Thus we define
+the loss function as the difference between the input and the output:
+L = 1
+N
+N
+�
+i=1
+∥WΓ(fi) − WM(R(WΓ(fi))∥2
+∥WΓ(fi)∥2
+.
+Remark. Our method estimates Γ and the inverse operator W−1
+Γ . The estimated inverse
+operator can be used for the fast inference of the initial pressure from the boundary
+measurement.
+Remark. The proposed framework is generic and can be extended to handle a nonlinear
+inverse problem involving an unknown function.
+Now, the detailed structures of each component in the framework are described below.
+3.1
+Reconstruction network R
+The reconstruction network R is a network that reconstruct f from input data WΓ(f).
+Indeed, it approximates the inverse map W−1
+Γ
+: WΓ(f) �→ f. If the speed Γ of the wave
+is constant, it is well-known that the inverse map of (3) is linear [7, 28, 29]. Inspired by
+this fact, we propose the reconstruction network as a perturbation of a linear map:
+R := T1 + U ◦ T2,
+(4)
+where T1, T2 : Rm×m → Rm×m are linear and U : Rm×m → Rm×m is the U-net described
+in Figure 2. U-net is a type of convolutional neural network (CNN) introduced in [30] and
+is used widely in medical imaging. U-net consists of a contracting path and an expansive
+path. The contacting path has a typical CNN structure, where the input data is extracted
+into feature map with small size and large channel. In the expansive path, the size of
+the feature map increases again, and the number of channels decreases. In the end of R,
+since the range of f is [0, 1], we used the clamp function which rounds up values smaller
+than the minimum and round down values larger than the maximum.
+5
+
+Figure 2: U-net architecture for 64 × 64 size
+The proposed reconstruction network show a high performance for the low resolution
+data like 64 × 64 (see section 4.3.1). In case of high resolution data, however the linear
+operators T1 and T2 in the reconstruction network R make some problems, because
+they contains too many parameters. It causes lots of critical points which impede the
+convergence to the global minimum. It also makes a hardware issue and thus for the high
+resolution data, we employ Pixel Shuffle and Pixel Unshuffle which reduce the number of
+parameters contained in linear operators [31]. The Pixel Unshuffle splits one image into
+several images and the Pixel Shuffle merges several images into one image, as illustrated
+in Figure 3. Instead of applying the linear operators (T1 and T2) directly to the high
+resolution data, we process the data as follows (see Figure 4) :
+1. Split the high resolution data (m × m) into four low resolution data ( m
+2 × m
+2 ) by
+exploiting the Pixel Unshuffle.
+2. Apply four different linear operators to each low resolution data.
+3. Merge the outputs of the linear operators by using the Pixel Shuffle.
+6
+
+Figure 3: Pixel Shuffle and Pixel Unshuffle
+Figure 4: Architecture of alternative map to linear for high resolution data
+3.2
+Mapping network M
+We use multilayer perceptron (MLP) to approximate unknown Γ, because MLP can
+approximate any continuous function (a universal approximation theorem, see [32, 33]).
+The proposed network is a simple structure containing only three hidden layers of 10
+nodes. To satisfy the assumption that Γ(0) = c0 and Γ(1) = c1, the output of MLP is
+slightly manipulated as
+M(f) = MLP(f) − MLP(0) ∗ (1 − f) − MLP(1) ∗ f + ((c1 − c0)f + c0),
+7
+
+Linear map
+Linearmap2
+Pixel
+Pixel
+Unshuffle
+Shuffle
+Linear map 3
+Linear map 4so that
+M(0) = c0 and M(1) = c1.
+(5)
+3.3
+Forward problem
+A solution of the initial value problem (3) can be computed by the k-space method
+[34, 35]. The k-space method is a numerical method for computing solution of acoustic
+wave propagation, which uses information in the frequency space to obtain a solution for
+the next time step. For calculating propagation of p(x, t), let w(x, t) =
+1
+Γ(f(x))p(x, t) be
+an auxiliary field. Then we have,
+∂2
+t w(x, t) = ∆x [Γ(f(x))w(x, t)] .
+Taking the Fourier transform Fx for w with respect to x yields
+∂2
+t Fxw(k, t) = −|k|2Fx
+�
+Γ(f(·))w(·, t)
+�
+(k).
+(6)
+Meanwhile, the numerical approximation of the second derivative of Fxw is
+∂2
+t Fxw(k, t) ≈ Fxw(k, t + ∆t) − 2Fxw(k, t) + Fxw(k, t − ∆t)
+(∆t)2
+,
+(7)
+where ∆t is the time step. Then, by combining (6) and (7), we have
+Fxw(k, t + ∆t) = 2Fxw(k, t) − Fxw(k, t − ∆t) − (∆t)2|k|2Fx
+�
+Γ(f(·))w(·, t)
+�
+(k).
+By taking the inverse Fourier transform F−1
+k , we obtain
+w(x, t + ∆t) = 2w(x, t) − w(x, t − ∆t) − F−1
+k
+�
+(∆t)2|·|2Fx
+�
+Γ(f)w
+�
+(·, t)
+�
+(x).
+Here, replacing (∆t)2|k|2 in the third term with 4 sin2 �
+(∆t)|k|
+2
+�
+provides more accurate
+discretization (see [34, 35]). Finally, we have wave propagation formula:
+w(x, t + ∆t) = 2w(x, t) − w(x, t − ∆t) − F−1
+k
+�
+4 sin2
+�(∆t)| · |
+2
+�
+Fx
+�
+Γ(f)w
+�
+(·, t)
+�
+(x),
+or equivalently,
+p(x, t + ∆t) = 2p(x, t) − p(x, t − ∆t) − Γ(f)F−1
+k
+�
+4 sin2
+�(∆t)| · |
+2
+�
+Fx
+�
+p
+�
+(·, t)
+�
+(x).
+4
+Numerical Simulations
+In this section, we present the details of implementation and experimental results when
+Ω is the unit ball.
+8
+
+4.1
+Datasets
+The Shepp-Logan phantom, an artificial image that describes a cross section of the brain
+commonly used for simulation in tomography, contains 10 ellipses [36]. Each ellipse is
+created with 6 parameters: major axis, minor axis, the x-coordinate and the y-coordinate
+of center, rotation angle, and intensity value. The data set of the initial condition f defined
+on [−1.0, 1.0]2 ⊂ R2 is generated by slightly changing these 6 parameters with
+supp(f) ⊂
+�
+(x, y) ∈ R2 :
+x2
+0.692 +
+y2
+0.922 ≤ 1
+�
+.
+We create a set of 2,688 phantoms P = {fi}2688
+i=1 . For Γ, we consider four cases: linear,
+square root, square, and constant
+1. Γ1(f) = 0.3f + 0.7
+2. Γ2(f) = 0.3√f + 0.7
+3. Γ3(f) = 0.3f 2 + 0.7
+4. Γ4(f) = 0.7.
+For 1 ≤ j ≤ 4, we make the collection of data {WΓjfi}2688
+i=1 by using the forward operator
+for P and Γj. Of the 2,688 data, we use 2,048 data for training, 128 data for validation
+and 512 data for testing respectively.
+Figure 5: Examples of phantoms
+9
+
+4.2
+Training
+We use the Adam optimizer based on stochastic gradient descent and adaptive moment
+estimation to train the network
+[37]. There are two neural networks in the proposed
+frameworks: the reconstruction network R and the mapping network M. The learning
+rates for linear term of R, perturbation term of R, and M are chosen to be 10−4,
+10−3, and 10−3, respectively. Momentum parameters of the Adam optimizer were set at
+β1 = 0.9 and β2 = 0.999, respectively.
+We specifically put the batch size to 2. For general tasks, a moderately large batch
+size reduces training time. However, in this problem, a small batch size is advantageous
+because our model must be able to reconstruct an exact image for each data rather than
+an average result.
+4.3
+Results
+In this section, we illustrate experimental results. The overall results are displayed in
+Figure 6, Table 1, Figure 7, Figure 8, Table 2 and Figure 9. Here the losses for f and
+WΓ(f) are respectively defined by
+loss for f = 1
+N
+N
+�
+i=1
+∥fi − R(WΓ(fi))∥2
+∥fi∥2
+,
+and
+loss for WΓf = 1
+N
+N
+�
+i=1
+∥WΓ(fi) − WM(R(WΓ(fi))∥2
+∥WΓ(fi)∥2
+.
+4.3.1
+Low resolution data
+We conduct the simulation utilizing a dataset of images with a size of 64×64. The results
+for the mapping networks are shown in Figure 6. We see that the mapping networks
+accurately approximates Γ. When Γ3 = 0.3f 2 + 0.7, there is a difference between the
+plot of the mapping network M and the plot of Γ. This is because the values of f ∈ P
+almost belong to [0, 0.3]∪1 and so it has little effect on WΓ(f). In all cases, the process of
+training the mapping networks requires approximately 103 iterations. The results of the
+reconstruction networks are illustrated in Table 1 and Figure 7. Table 1 shows the test
+errors. So it can be concluded that the reconstruction networks accurately approximate
+the inverse maps in each case. The training of the reconstruction networks necessitates
+approximately 105 iterations.
+Remark. The assumption on Γ, (5) is crucial. If constraint (5) is not given in M, it
+may take a long time to approximate Γ, or it may fail to find Γ. Under the constraint, M
+can quickly determine Γ. Early determination of Γ helps the learning of reconstruction
+networks.
+10
+
+Figure 6: Comparison of the mapping network M and ground truth Γ for 64 × 64 data
+Assumption
+loss for f
+loss for WΓ(f)
+Γ1 = 0.3f + 0.7
+0.00504
+0.00702
+Γ2 = 0.3√f + 0.7
+0.00537
+0.00947
+Γ3 = 0.3f 2 + 0.7
+0.00557
+0.00634
+Γ4 = 0.7
+0.01373
+0.00456
+Table 1: Test errors for f and WΓ(f) according to Γ after 102,400 iterations for 64 × 64
+data
+11
+
+[2 = 0.3Vf + 0.7
+[1 = 0.3f + 0.7
+1.0
+ground truth
+1.0
+mapping network
+0.7
+0.7
+0.0
+0.5
+1.0
+0.0
+0.5
+1.0
+[3 = 0.3f2 + 0.7
+[4 = 0.7
+1.0
+1.0
+0.7
+0.7
+0.0
+0.5
+1.0
+0.0
+0.5
+1.0Figure 7: Reconstruction results according to Γ for 64 × 64 data
+4.3.2
+High resolution data
+In the simulation for high resolution data, two linear operators T1 and T2 in the recon-
+struction network (4) are replaced by alternative map described in Figure 4. The dataset
+is prepared with images of a size of 96 × 96. Similarly to the low resolution case, the
+mapping network M approximates Γ accurately within 103 iterations (Figure 8). On the
+other hand, the reconstruction networks for each Γ exhibit a slight decrease in perfor-
+mance but still acceptable (Table 2 and Figure 9). We surmise that the slight decrease
+in performance is a result of the reduction in parameters brought about by the Pixel
+Unshuffle and Pixel Shuffle operations.
+12
+
+[1 = 0.3f + 0.7
+[2 = 0.3V f + 0.7
+[3 = 0.3f2 + 0.7
+4 = 0.7
+ground truth
+1.0
+0.8
+0.6
+0.4
+0.2
+0.0Figure 8: Comparison of the mapping network M and ground truth Γ for 96 × 96 data
+Assumption
+loss for f
+loss for WΓ(f)
+Γ1 = 0.3f + 0.7
+0.00860
+0.01293
+Γ2 = 0.3√f + 0.7
+0.01023
+0.01679
+Γ3 = 0.3f 2 + 0.7
+0.00710
+0.01132
+Γ4 = 0.7
+0.00689
+0.69511
+Table 2: Test errors for f and WΓ(f) according to Γ after 102,400 iterations for 96 × 96
+data
+13
+
+[2 = 0.3Vf + 0.7
+[1 = 0.3f + 0.7
+1.0
+ground truth
+1.0
+mapping network
+0.7
+0.7
+0.0
+0.5
+1.0
+0.0
+0.5
+1.0
+[3 = 0.3f2 + 0.7
+[4 = 0.7
+1.0
+1.0
+0.7
+0.7
+0.0
+0.5
+1.0
+0.0
+0.5
+1.0Figure 9: Reconstruction results according to Γ for 96 × 96 data
+5
+Conclusions
+Here, we propose a self-supervised learning for a nonlinear inverse problem with forward
+operator involving an unknown function. In medical imaging such as PAT, the initial
+pressure is mostly untrackable for the measured data. Moreover it is difficult to know the
+wave speed. So it is necessary to reconstruct the initial pressure f and the wave speed
+simultaneously. Under the simple assumption, the problem becomes a nonlinear inverse
+problem involving an unknown function. The experimental results demonstrate the high
+performance of the proposed algorithm. Our framework can be extended to a nonlinear
+inverse problem involving an unknown function, formulated under more complicated
+situations. This can be an interesting line of future research.
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+
diff --git a/ONFAT4oBgHgl3EQfyh6B/content/tmp_files/load_file.txt b/ONFAT4oBgHgl3EQfyh6B/content/tmp_files/load_file.txt
new file mode 100644
index 0000000000000000000000000000000000000000..809f4aff7ef4fd1e6ddfe0e2eef6c76bae7f6e0e
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+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFAT4oBgHgl3EQfyh6B/content/2301.08693v1.pdf,len=469
+page_content='SELF-SUPERVISED LEARNING FOR A NONLINEAR INVERSE  PROBLEM WITH FORWARD OPERATOR INVOLVING AN  UNKNOWN FUNCTION ARISING IN PHOTOACOUSTIC  TOMOGRAPHY  Gyeongha Hwang1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFAT4oBgHgl3EQfyh6B/content/2301.08693v1.pdf'}
+page_content=' Gihyeon Jeon2∗,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFAT4oBgHgl3EQfyh6B/content/2301.08693v1.pdf'}
+page_content=' Sunghwan Moon3  1 Department of Mathematics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFAT4oBgHgl3EQfyh6B/content/2301.08693v1.pdf'}
+page_content=' Yeungnam University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFAT4oBgHgl3EQfyh6B/content/2301.08693v1.pdf'}
+page_content=' Gyeongsan 38541,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFAT4oBgHgl3EQfyh6B/content/2301.08693v1.pdf'}
+page_content=' Republic of Korea  2 School of Mathematics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFAT4oBgHgl3EQfyh6B/content/2301.08693v1.pdf'}
+page_content=' Kyungpook National University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFAT4oBgHgl3EQfyh6B/content/2301.08693v1.pdf'}
+page_content=' Daegu 41566,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFAT4oBgHgl3EQfyh6B/content/2301.08693v1.pdf'}
+page_content=' Republic of Korea  3 Department of Mathematics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFAT4oBgHgl3EQfyh6B/content/2301.08693v1.pdf'}
+page_content=' Kyungpook National University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFAT4oBgHgl3EQfyh6B/content/2301.08693v1.pdf'}
+page_content=' Daegu 41566,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFAT4oBgHgl3EQfyh6B/content/2301.08693v1.pdf'}
+page_content=' Republic of Korea  Corresponding author E-mails: rydbr6709@knu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFAT4oBgHgl3EQfyh6B/content/2301.08693v1.pdf'}
+page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFAT4oBgHgl3EQfyh6B/content/2301.08693v1.pdf'}
+page_content='kr  ABSTRACT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFAT4oBgHgl3EQfyh6B/content/2301.08693v1.pdf'}
+page_content=' In this article, we concern with a nonlinear inverse problem with forward opera-  tor involving an unknown function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFAT4oBgHgl3EQfyh6B/content/2301.08693v1.pdf'}
+page_content=' The problem arises in diverse applications and is challenging  by the presence of the unknown function, which makes it ill-posed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFAT4oBgHgl3EQfyh6B/content/2301.08693v1.pdf'}
+page_content=' Additionally, the nonlinear  nature of the problem makes it difficult to use traditional methods and thus the study has  addressed a simplified version of the problem by either linearizing it or assuming knowledge of  the unknown function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFAT4oBgHgl3EQfyh6B/content/2301.08693v1.pdf'}
+page_content=' Here, we propose a self-supervised learning to directly tackle a nonlinear  inverse problem involving an unknown function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFAT4oBgHgl3EQfyh6B/content/2301.08693v1.pdf'}
+page_content=' In particular, we focus on an inverse problem  derived in Photoacoustic Tomograpy (PAT) which is a hybrid medical imaging with high res-  olution and contrast.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFAT4oBgHgl3EQfyh6B/content/2301.08693v1.pdf'}
+page_content=' PAT can be modelled based on the wave equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFAT4oBgHgl3EQfyh6B/content/2301.08693v1.pdf'}
+page_content=' The measured data  is the solution of the equation restricted to the surface and the initial pressure of the equation  contains the biological information on the object of interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFAT4oBgHgl3EQfyh6B/content/2301.08693v1.pdf'}
+page_content=' The speed of sound wave in the  equation is unknown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFAT4oBgHgl3EQfyh6B/content/2301.08693v1.pdf'}
+page_content=' Our goal is to determine the initial pressure and the speed of sound wave  simultaneously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFAT4oBgHgl3EQfyh6B/content/2301.08693v1.pdf'}
+page_content=' Under a simple assumption that the sound speed is a function of the initial  pressure, the problem becomes a nonlinear inverse problem involving an unknown function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFAT4oBgHgl3EQfyh6B/content/2301.08693v1.pdf'}
+page_content=' The  experimental results demonstrate that the proposed algorithm performs successfully.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFAT4oBgHgl3EQfyh6B/content/2301.08693v1.pdf'}
+page_content='  1  Introduction  Inverse problem is to find the cause factor from the observed data, which has applications  in many fields such as optics, radar, acoustics, communication theory, signal processing,  medical imaging, computer vision, geophysics, oceanography, and astronomy because it  tells us about what we cannot directly observe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFAT4oBgHgl3EQfyh6B/content/2301.08693v1.pdf'}
+page_content=' The forward operator, the inverse of  the inverse problem, can be modelled as an (non)-linear system and often involves an  unknown function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFAT4oBgHgl3EQfyh6B/content/2301.08693v1.pdf'}
+page_content=' Due to the nature of the inverse problem, it is usually very hard to  know the cause factor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFAT4oBgHgl3EQfyh6B/content/2301.08693v1.pdf'}
+page_content=' For example, in medical imaging the cause factor is the human  body section and in seismology, we never know the structure of the earth’s interior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFAT4oBgHgl3EQfyh6B/content/2301.08693v1.pdf'}
+page_content='  In this article, we concern with a nonlinear inverse problem with forward operator involv-  ing an unknown function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFAT4oBgHgl3EQfyh6B/content/2301.08693v1.pdf'}
+page_content=' Our goal is to find the unknown function and the inverse opera-  tor simultaneously from the measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFAT4oBgHgl3EQfyh6B/content/2301.08693v1.pdf'}
+page_content=' The problem is generally ill-posed because of  1  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFAT4oBgHgl3EQfyh6B/content/2301.08693v1.pdf'}
+page_content='08693v1  [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFAT4oBgHgl3EQfyh6B/content/2301.08693v1.pdf'}
+page_content='NA]  20 Jan 2023  the unknown function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFAT4oBgHgl3EQfyh6B/content/2301.08693v1.pdf'}
+page_content=' Additionally, the nonlinearity in the problem makes conventional  methods difficult to use.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFAT4oBgHgl3EQfyh6B/content/2301.08693v1.pdf'}
+page_content=' To handle the problem, one may simplify the problem linearly  or assume knowledge of the unknown function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFAT4oBgHgl3EQfyh6B/content/2301.08693v1.pdf'}
+page_content=' Here we propose a self-supervised frame-  work to directly tackle a nonlinear inverse problem involving an unknown function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFAT4oBgHgl3EQfyh6B/content/2301.08693v1.pdf'}
+page_content=' In  particular, we address an inverse problem derived in Photoacoustic Tomography (PAT).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFAT4oBgHgl3EQfyh6B/content/2301.08693v1.pdf'}
+page_content='  Although our proposed framework has been proposed to solve the problem arising in  PAT, it is generic and can be extended to handle a nonlinear inverse problem involving  an unknown function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFAT4oBgHgl3EQfyh6B/content/2301.08693v1.pdf'}
+page_content='  The rest of the section is devoted to an introduction of PAT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFAT4oBgHgl3EQfyh6B/content/2301.08693v1.pdf'}
+page_content=' In section 2, we formulate  an inverse problem arising in PAT, which is nonlinear and also involves an unknown  function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFAT4oBgHgl3EQfyh6B/content/2301.08693v1.pdf'}
+page_content=' The structure and learning method of the proposed framework for the problem  are described in section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFAT4oBgHgl3EQfyh6B/content/2301.08693v1.pdf'}
+page_content=' The numerical simulation results in section 4 demonstrate that  the proposed algorithm performs successfully.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFAT4oBgHgl3EQfyh6B/content/2301.08693v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFAT4oBgHgl3EQfyh6B/content/2301.08693v1.pdf'}
+page_content='1  Photoacoustic Tomography  PAT is a hybrid medical imaging that combines the high contrast of optical imaging with  the high spatial resolution of ultrasound images [1, 2, 3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFAT4oBgHgl3EQfyh6B/content/2301.08693v1.pdf'}
+page_content=' The physical basis of PAT is  the photoacoustic effect discovered by Bell in 1881 [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFAT4oBgHgl3EQfyh6B/content/2301.08693v1.pdf'}
+page_content=' In PAT, when an non-destructive  testing target object absorbs a non-ionizing laser pulse, it thermally expands and emits  acoustic waves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFAT4oBgHgl3EQfyh6B/content/2301.08693v1.pdf'}
+page_content=' The emitted ultrasound contains biological information on the target  object and is measured by an detector placed around it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFAT4oBgHgl3EQfyh6B/content/2301.08693v1.pdf'}
+page_content=' The internal image of the target  object is reconstructed from this measured data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFAT4oBgHgl3EQfyh6B/content/2301.08693v1.pdf'}
+page_content=' The advantage of PAT is that it is  economical and less harmful because of non-ionizing radiation use [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFAT4oBgHgl3EQfyh6B/content/2301.08693v1.pdf'}
+page_content='  The propagation of the emitted ultrasound p(x, t) can be described by the wave equation  ∂2  t p(x, t) = c(x)2∆xp(x, t)  on R2 × [0, ∞)  (1)  with the initial conditions  p(x, 0) = f(x)  ∂tp(x, 0) = 0  on R2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFAT4oBgHgl3EQfyh6B/content/2301.08693v1.pdf'}
+page_content='  (2)  Here c is the speed of waves and f is the initial pressure which contains biological in-  formation such as the location of a cancer cells in a physically small tissue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFAT4oBgHgl3EQfyh6B/content/2301.08693v1.pdf'}
+page_content=' It is natural  assumption that f has compact support in the bounded domain Ω and the detectors are  located on the boundary ∂Ω of the domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFAT4oBgHgl3EQfyh6B/content/2301.08693v1.pdf'}
+page_content=' Regarding the measurement procedure, the  point-shaped detector measures the average pressure above ∂Ω where the detectors are  located and this average pressure is the value of a pressure wave p(x, t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFAT4oBgHgl3EQfyh6B/content/2301.08693v1.pdf'}
+page_content=' Therefore, one of  mathematical problems in PAT is reconstructing f from the measured data p|∂Ω×[0,∞),  which implies obtaining an internal image of the target object.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFAT4oBgHgl3EQfyh6B/content/2301.08693v1.pdf'}
+page_content='  It is well-known that given the initial pressure f and the speed c, the solution p is  determined uniquely.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFAT4oBgHgl3EQfyh6B/content/2301.08693v1.pdf'}
+page_content=' Let us define the wave forward operator W as  W : (f, c) �→ p|∂Ω×[0,∞),  i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFAT4oBgHgl3EQfyh6B/content/2301.08693v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFAT4oBgHgl3EQfyh6B/content/2301.08693v1.pdf'}
+page_content=',  W(f, c) = p|∂Ω×[0,∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFAT4oBgHgl3EQfyh6B/content/2301.08693v1.pdf'}
+page_content='  Reconstructing problem for f from W(f, c) is studied when speed c is constant [6, 7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFAT4oBgHgl3EQfyh6B/content/2301.08693v1.pdf'}
+page_content='  Okanen, Stefanov and Uhlmann study the explicit reconstruction when the sound speed  2  is known  [8, 9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFAT4oBgHgl3EQfyh6B/content/2301.08693v1.pdf'}
+page_content=' If c depends on space variable x, the problem become much more  difficult.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFAT4oBgHgl3EQfyh6B/content/2301.08693v1.pdf'}
+page_content=' A few of researchers have studied the problem with a given variable sound  speed [10, 11, 12, 13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFAT4oBgHgl3EQfyh6B/content/2301.08693v1.pdf'}
+page_content=' Liu and Uhlmann figure out the sufficient conditions for recovering  f and c [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFAT4oBgHgl3EQfyh6B/content/2301.08693v1.pdf'}
+page_content='  Recently, the application of deep learning in an medical imaging including PAT has been  investigated extensively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFAT4oBgHgl3EQfyh6B/content/2301.08693v1.pdf'}
+page_content=' Roles of deep learning in tomography include forward and in-  verse operator approximation, image reconstruction from sparse data, and artifact/noise  removal from reconstructed images [15, 16, 17, 18, 19, 20, 21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFAT4oBgHgl3EQfyh6B/content/2301.08693v1.pdf'}
+page_content=' There are also studies on  limited-view data (see [22, 23]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFAT4oBgHgl3EQfyh6B/content/2301.08693v1.pdf'}
+page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFAT4oBgHgl3EQfyh6B/content/2301.08693v1.pdf'}
+page_content=' Shan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFAT4oBgHgl3EQfyh6B/content/2301.08693v1.pdf'}
+page_content=' propose an iterative optimization algo-  rithm which reconstructs f and c simultaneously via a supervised learning [24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFAT4oBgHgl3EQfyh6B/content/2301.08693v1.pdf'}
+page_content=' However,  most works deal with linear inverse problems or inverse problems without involving an  unknown function [25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFAT4oBgHgl3EQfyh6B/content/2301.08693v1.pdf'}
+page_content='  Many studies on PAT with deep learning are based on a supervised learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFAT4oBgHgl3EQfyh6B/content/2301.08693v1.pdf'}
+page_content=' A supervised  learning exploits a collection of paired data of the boundary data and the initial pres-  sure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFAT4oBgHgl3EQfyh6B/content/2301.08693v1.pdf'}
+page_content=' In practical applications, it is difficult to obtain the initial pressure, because initial  pressure represents internal human body.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFAT4oBgHgl3EQfyh6B/content/2301.08693v1.pdf'}
+page_content=' Therefore, it is necessary to study a learning  method exploiting the boundary data only.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFAT4oBgHgl3EQfyh6B/content/2301.08693v1.pdf'}
+page_content=' One such method is a self-supervised learning  which exploits supervised signals that are generated from the input data by leveraging  its structure [26, 27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFAT4oBgHgl3EQfyh6B/content/2301.08693v1.pdf'}
+page_content='  2  Problem formulation  In this section, we formulate the problem precisely.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFAT4oBgHgl3EQfyh6B/content/2301.08693v1.pdf'}
+page_content=' For this, we will make several as-  sumptions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFAT4oBgHgl3EQfyh6B/content/2301.08693v1.pdf'}
+page_content=' First we assume f has compact support, since a target object is finite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFAT4oBgHgl3EQfyh6B/content/2301.08693v1.pdf'}
+page_content=' Sec-  ondly, c is assumed to be a function of f, namely c(x)2 = Γ(f(x)) for some function  Γ : [0, 1] → [0, ∞), because the wave speed c depends on the medium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFAT4oBgHgl3EQfyh6B/content/2301.08693v1.pdf'}
+page_content=' Lastly, we assume  that Γ(0) and Γ(1) are known, namely Γ(0) = c0 and Γ(1) = c1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFAT4oBgHgl3EQfyh6B/content/2301.08693v1.pdf'}
+page_content=' The last assumption is  reasonable because Γ(0) and Γ(1) represent the wave speeds in the air and the highest  thermal expansion coefficient respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFAT4oBgHgl3EQfyh6B/content/2301.08693v1.pdf'}
+page_content=' Then the equation (1) is rewritten as:  ∂2  t p(x, t) = Γ(f(x))∆xp(x, t)  on R2 × [0, ∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFAT4oBgHgl3EQfyh6B/content/2301.08693v1.pdf'}
+page_content='  (3)  Let us define WΓ by WΓ(f) = p|∂Ω×[0,∞) where p is the solution of (3) with initial  conditions (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFAT4oBgHgl3EQfyh6B/content/2301.08693v1.pdf'}
+page_content=' Then the inverse problem can be formulated as determining unknown Γ  and f from a given WΓ(f).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFAT4oBgHgl3EQfyh6B/content/2301.08693v1.pdf'}
+page_content=' However, this problem is ill-posed: for any Γ′ satisfying  � Γ = Γ′ on Im(f)  Γ ̸= Γ′ on Dom(Γ) \\ Im(f),  we have WΓ(f) = WΓ′(f).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFAT4oBgHgl3EQfyh6B/content/2301.08693v1.pdf'}
+page_content=' Hence Γ can not be uniquely determined from WΓ(f).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFAT4oBgHgl3EQfyh6B/content/2301.08693v1.pdf'}
+page_content=' There  is also a possibility that there exist Γ1, Γ2, f1 and f2 such that Γ1 ̸= Γ2, f1 ̸= f2 and  WΓ1(f1) = WΓ2(f2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFAT4oBgHgl3EQfyh6B/content/2301.08693v1.pdf'}
+page_content=' Instead, we consider the following inverse problem:  Problem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFAT4oBgHgl3EQfyh6B/content/2301.08693v1.pdf'}
+page_content=' Let the collection of boundary data BΓ := {WΓ(f) | Γ : [0, 1] → [0, ∞), Γ(0) =  c0, Γ(1) = c1 and f ∈ L2(R2) has compact support} be given.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFAT4oBgHgl3EQfyh6B/content/2301.08693v1.pdf'}
+page_content='  3  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFAT4oBgHgl3EQfyh6B/content/2301.08693v1.pdf'}
+page_content=' Determine unknown Γ from BΓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFAT4oBgHgl3EQfyh6B/content/2301.08693v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFAT4oBgHgl3EQfyh6B/content/2301.08693v1.pdf'}
+page_content=' For all WΓ(f) ∈ BΓ, determine f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFAT4oBgHgl3EQfyh6B/content/2301.08693v1.pdf'}
+page_content='  Then the uniqueness statements for Problem 1 are  Hypothesis 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFAT4oBgHgl3EQfyh6B/content/2301.08693v1.pdf'}
+page_content=' If Γ1 ̸= Γ2, then BΓ1 ̸= BΓ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFAT4oBgHgl3EQfyh6B/content/2301.08693v1.pdf'}
+page_content='  and  Hypothesis 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFAT4oBgHgl3EQfyh6B/content/2301.08693v1.pdf'}
+page_content=' For fixed Γ, if f1 ̸= f2, then WΓ(f1) ̸= WΓ(f2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFAT4oBgHgl3EQfyh6B/content/2301.08693v1.pdf'}
+page_content='  In this article, we aim to solve Problem 1 under Hypothesis 1 and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFAT4oBgHgl3EQfyh6B/content/2301.08693v1.pdf'}
+page_content=' The problem is  difficult to solve because of  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFAT4oBgHgl3EQfyh6B/content/2301.08693v1.pdf'}
+page_content=' (3) involves unknown Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFAT4oBgHgl3EQfyh6B/content/2301.08693v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFAT4oBgHgl3EQfyh6B/content/2301.08693v1.pdf'}
+page_content=' (3) is not linear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFAT4oBgHgl3EQfyh6B/content/2301.08693v1.pdf'}
+page_content='  We are going to solve Problem 1 by exploiting a deep neural network (DNN).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFAT4oBgHgl3EQfyh6B/content/2301.08693v1.pdf'}
+page_content=' Since DNN  can only handle with finite data, we address the following inverse problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFAT4oBgHgl3EQfyh6B/content/2301.08693v1.pdf'}
+page_content='  Problem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFAT4oBgHgl3EQfyh6B/content/2301.08693v1.pdf'}
+page_content=' For given {WΓ(fi) | Γ : [0, 1] → [0, ∞), Γ(0) = c0, Γ(1) = c1 and fi ∈  L2(R2) has compact support, i = 1, · · · , N}, determine Γ and {fi|i = 1, · · · , N}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFAT4oBgHgl3EQfyh6B/content/2301.08693v1.pdf'}
+page_content='  3  Network Design  Figure 1: The network architecture  We propose a self-supervised learning for the problem formulated in section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFAT4oBgHgl3EQfyh6B/content/2301.08693v1.pdf'}
+page_content=' Our goal  is simultaneously reconstructing {fi}N  i=1 and Γ from given collection {WΓ(fi)}N  i=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFAT4oBgHgl3EQfyh6B/content/2301.08693v1.pdf'}
+page_content=' The  proposed framework is depicted in Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFAT4oBgHgl3EQfyh6B/content/2301.08693v1.pdf'}
+page_content=' It consists of three components:  4  input  output  Wr(f)  W (R(Wr(f)))  R(Wr(f))  M(R(Wr(f)))  LOSS = MSEWr(f),W(RWr(f)))1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFAT4oBgHgl3EQfyh6B/content/2301.08693v1.pdf'}
+page_content=' Reconstruction network R  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFAT4oBgHgl3EQfyh6B/content/2301.08693v1.pdf'}
+page_content=' Mapping network M  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFAT4oBgHgl3EQfyh6B/content/2301.08693v1.pdf'}
+page_content=' Wave forward operator W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFAT4oBgHgl3EQfyh6B/content/2301.08693v1.pdf'}
+page_content='  The reconstruction network R learns to reconstruct the initial data from the measured  data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFAT4oBgHgl3EQfyh6B/content/2301.08693v1.pdf'}
+page_content=' The mapping network M approximates the function Γ : [0, 1] → [0, ∞) satisfying  c(x)2 = Γ(f(x)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFAT4oBgHgl3EQfyh6B/content/2301.08693v1.pdf'}
+page_content=' The forward operator W assigns to the initial data and the wave  speed the measured data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFAT4oBgHgl3EQfyh6B/content/2301.08693v1.pdf'}
+page_content=' Here we adopt the k-space method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFAT4oBgHgl3EQfyh6B/content/2301.08693v1.pdf'}
+page_content=' If every component in the  framework functions properly, then output should be same to the input.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFAT4oBgHgl3EQfyh6B/content/2301.08693v1.pdf'}
+page_content=' Thus we define  the loss function as the difference between the input and the output:  L = 1  N  N  �  i=1  ∥WΓ(fi) − WM(R(WΓ(fi))∥2  ∥WΓ(fi)∥2  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFAT4oBgHgl3EQfyh6B/content/2301.08693v1.pdf'}
+page_content='  Remark.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFAT4oBgHgl3EQfyh6B/content/2301.08693v1.pdf'}
+page_content=' Our method estimates Γ and the inverse operator W−1  Γ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFAT4oBgHgl3EQfyh6B/content/2301.08693v1.pdf'}
+page_content=' The estimated inverse  operator can be used for the fast inference of the initial pressure from the boundary  measurement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFAT4oBgHgl3EQfyh6B/content/2301.08693v1.pdf'}
+page_content='  Remark.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFAT4oBgHgl3EQfyh6B/content/2301.08693v1.pdf'}
+page_content=' The proposed framework is generic and can be extended to handle a nonlinear  inverse problem involving an unknown function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFAT4oBgHgl3EQfyh6B/content/2301.08693v1.pdf'}
+page_content='  Now, the detailed structures of each component in the framework are described below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFAT4oBgHgl3EQfyh6B/content/2301.08693v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFAT4oBgHgl3EQfyh6B/content/2301.08693v1.pdf'}
+page_content='1  Reconstruction network R  The reconstruction network R is a network that reconstruct f from input data WΓ(f).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFAT4oBgHgl3EQfyh6B/content/2301.08693v1.pdf'}
+page_content='  Indeed, it approximates the inverse map W−1  Γ  : WΓ(f) �→ f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFAT4oBgHgl3EQfyh6B/content/2301.08693v1.pdf'}
+page_content=' If the speed Γ of the wave  is constant, it is well-known that the inverse map of (3) is linear [7, 28, 29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFAT4oBgHgl3EQfyh6B/content/2301.08693v1.pdf'}
+page_content=' Inspired by  this fact, we propose the reconstruction network as a perturbation of a linear map:  R := T1 + U ◦ T2,  (4)  where T1, T2 : Rm×m → Rm×m are linear and U : Rm×m → Rm×m is the U-net described  in Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFAT4oBgHgl3EQfyh6B/content/2301.08693v1.pdf'}
+page_content=' U-net is a type of convolutional neural network (CNN) introduced in [30] and  is used widely in medical imaging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFAT4oBgHgl3EQfyh6B/content/2301.08693v1.pdf'}
+page_content=' U-net consists of a contracting path and an expansive  path.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFAT4oBgHgl3EQfyh6B/content/2301.08693v1.pdf'}
+page_content=' The contacting path has a typical CNN structure, where the input data is extracted  into feature map with small size and large channel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFAT4oBgHgl3EQfyh6B/content/2301.08693v1.pdf'}
+page_content=' In the expansive path, the size of  the feature map increases again, and the number of channels decreases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFAT4oBgHgl3EQfyh6B/content/2301.08693v1.pdf'}
+page_content=' In the end of R,  since the range of f is [0, 1], we used the clamp function which rounds up values smaller  than the minimum and round down values larger than the maximum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFAT4oBgHgl3EQfyh6B/content/2301.08693v1.pdf'}
+page_content='  5  Figure 2: U-net architecture for 64 × 64 size  The proposed reconstruction network show a high performance for the low resolution  data like 64 × 64 (see section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFAT4oBgHgl3EQfyh6B/content/2301.08693v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFAT4oBgHgl3EQfyh6B/content/2301.08693v1.pdf'}
+page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFAT4oBgHgl3EQfyh6B/content/2301.08693v1.pdf'}
+page_content=' In case of high resolution data, however the linear  operators T1 and T2 in the reconstruction network R make some problems, because  they contains too many parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFAT4oBgHgl3EQfyh6B/content/2301.08693v1.pdf'}
+page_content=' It causes lots of critical points which impede the  convergence to the global minimum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFAT4oBgHgl3EQfyh6B/content/2301.08693v1.pdf'}
+page_content=' It also makes a hardware issue and thus for the high  resolution data, we employ Pixel Shuffle and Pixel Unshuffle which reduce the number of  parameters contained in linear operators [31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFAT4oBgHgl3EQfyh6B/content/2301.08693v1.pdf'}
+page_content=' The Pixel Unshuffle splits one image into  several images and the Pixel Shuffle merges several images into one image, as illustrated  in Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFAT4oBgHgl3EQfyh6B/content/2301.08693v1.pdf'}
+page_content=' Instead of applying the linear operators (T1 and T2) directly to the high  resolution data, we process the data as follows (see Figure 4) :  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFAT4oBgHgl3EQfyh6B/content/2301.08693v1.pdf'}
+page_content=' Split the high resolution data (m × m) into four low resolution data ( m  2 × m  2 ) by  exploiting the Pixel Unshuffle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFAT4oBgHgl3EQfyh6B/content/2301.08693v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFAT4oBgHgl3EQfyh6B/content/2301.08693v1.pdf'}
+page_content=' Apply four different linear operators to each low resolution data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFAT4oBgHgl3EQfyh6B/content/2301.08693v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFAT4oBgHgl3EQfyh6B/content/2301.08693v1.pdf'}
+page_content=' Merge the outputs of the linear operators by using the Pixel Shuffle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFAT4oBgHgl3EQfyh6B/content/2301.08693v1.pdf'}
+page_content='  6  Figure 3: Pixel Shuffle and Pixel Unshuffle  Figure 4: Architecture of alternative map to linear for high resolution data  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFAT4oBgHgl3EQfyh6B/content/2301.08693v1.pdf'}
+page_content='2  Mapping network M  We use multilayer perceptron (MLP) to approximate unknown Γ, because MLP can  approximate any continuous function (a universal approximation theorem, see [32, 33]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFAT4oBgHgl3EQfyh6B/content/2301.08693v1.pdf'}
+page_content='  The proposed network is a simple structure containing only three hidden layers of 10  nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFAT4oBgHgl3EQfyh6B/content/2301.08693v1.pdf'}
+page_content=' To satisfy the assumption that Γ(0) = c0 and Γ(1) = c1, the output of MLP is  slightly manipulated as  M(f) = MLP(f) − MLP(0) ∗ (1 − f) − MLP(1) ∗ f + ((c1 − c0)f + c0),  7  Linear map  Linearmap2  Pixel  Pixel  Unshuffle  Shuffle  Linear map 3  Linear map 4so that  M(0) = c0 and M(1) = c1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFAT4oBgHgl3EQfyh6B/content/2301.08693v1.pdf'}
+page_content='  (5)  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFAT4oBgHgl3EQfyh6B/content/2301.08693v1.pdf'}
+page_content='3  Forward problem  A solution of the initial value problem (3) can be computed by the k-space method  [34, 35].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFAT4oBgHgl3EQfyh6B/content/2301.08693v1.pdf'}
+page_content=' The k-space method is a numerical method for computing solution of acoustic  wave propagation, which uses information in the frequency space to obtain a solution for  the next time step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFAT4oBgHgl3EQfyh6B/content/2301.08693v1.pdf'}
+page_content=' For calculating propagation of p(x, t), let w(x, t) =  1  Γ(f(x))p(x, t) be  an auxiliary field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFAT4oBgHgl3EQfyh6B/content/2301.08693v1.pdf'}
+page_content=' Then we have,  ∂2  t w(x, t) = ∆x [Γ(f(x))w(x, t)] .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFAT4oBgHgl3EQfyh6B/content/2301.08693v1.pdf'}
+page_content='  Taking the Fourier transform Fx for w with respect to x yields  ∂2  t Fxw(k, t) = −|k|2Fx  �  Γ(f(·))w(·, t)  �  (k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFAT4oBgHgl3EQfyh6B/content/2301.08693v1.pdf'}
+page_content='  (6)  Meanwhile, the numerical approximation of the second derivative of Fxw is  ∂2  t Fxw(k, t) ≈ Fxw(k, t + ∆t) − 2Fxw(k, t) + Fxw(k, t − ∆t)  (∆t)2  ,  (7)  where ∆t is the time step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFAT4oBgHgl3EQfyh6B/content/2301.08693v1.pdf'}
+page_content=' Then, by combining (6) and (7), we have  Fxw(k, t + ∆t) = 2Fxw(k, t) − Fxw(k, t − ∆t) − (∆t)2|k|2Fx  �  Γ(f(·))w(·, t)  �  (k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFAT4oBgHgl3EQfyh6B/content/2301.08693v1.pdf'}
+page_content='  By taking the inverse Fourier transform F−1  k , we obtain  w(x, t + ∆t) = 2w(x, t) − w(x, t − ∆t) − F−1  k  �  (∆t)2|·|2Fx  �  Γ(f)w  �  (·, t)  �  (x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFAT4oBgHgl3EQfyh6B/content/2301.08693v1.pdf'}
+page_content='  Here, replacing (∆t)2|k|2 in the third term with 4 sin2 �  (∆t)|k|  2  �  provides more accurate  discretization (see [34, 35]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFAT4oBgHgl3EQfyh6B/content/2301.08693v1.pdf'}
+page_content=' Finally, we have wave propagation formula:  w(x, t + ∆t) = 2w(x, t) − w(x, t − ∆t) − F−1  k  �  4 sin2  �(∆t)| · |  2  �  Fx  �  Γ(f)w  �  (·, t)  �  (x),  or equivalently,  p(x, t + ∆t) = 2p(x, t) − p(x, t − ∆t) − Γ(f)F−1  k  �  4 sin2  �(∆t)| · |  2  �  Fx  �  p  �  (·, t)  �  (x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFAT4oBgHgl3EQfyh6B/content/2301.08693v1.pdf'}
+page_content='  4  Numerical Simulations  In this section, we present the details of implementation and experimental results when  Ω is the unit ball.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFAT4oBgHgl3EQfyh6B/content/2301.08693v1.pdf'}
+page_content='  8  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFAT4oBgHgl3EQfyh6B/content/2301.08693v1.pdf'}
+page_content='1  Datasets  The Shepp-Logan phantom, an artificial image that describes a cross section of the brain  commonly used for simulation in tomography, contains 10 ellipses [36].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFAT4oBgHgl3EQfyh6B/content/2301.08693v1.pdf'}
+page_content=' Each ellipse is  created with 6 parameters: major axis, minor axis, the x-coordinate and the y-coordinate  of center, rotation angle, and intensity value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFAT4oBgHgl3EQfyh6B/content/2301.08693v1.pdf'}
+page_content=' The data set of the initial condition f defined  on [−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFAT4oBgHgl3EQfyh6B/content/2301.08693v1.pdf'}
+page_content='0, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFAT4oBgHgl3EQfyh6B/content/2301.08693v1.pdf'}
+page_content='0]2 ⊂ R2 is generated by slightly changing these 6 parameters with  supp(f) ⊂  �  (x, y) ∈ R2 :  x2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFAT4oBgHgl3EQfyh6B/content/2301.08693v1.pdf'}
+page_content='692 +  y2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFAT4oBgHgl3EQfyh6B/content/2301.08693v1.pdf'}
+page_content='922 ≤ 1  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFAT4oBgHgl3EQfyh6B/content/2301.08693v1.pdf'}
+page_content='  We create a set of 2,688 phantoms P = {fi}2688  i=1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFAT4oBgHgl3EQfyh6B/content/2301.08693v1.pdf'}
+page_content=' For Γ, we consider four cases: linear,  square root, square, and constant  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFAT4oBgHgl3EQfyh6B/content/2301.08693v1.pdf'}
+page_content=' Γ1(f) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFAT4oBgHgl3EQfyh6B/content/2301.08693v1.pdf'}
+page_content='3f + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFAT4oBgHgl3EQfyh6B/content/2301.08693v1.pdf'}
+page_content='7  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFAT4oBgHgl3EQfyh6B/content/2301.08693v1.pdf'}
+page_content=' Γ2(f) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFAT4oBgHgl3EQfyh6B/content/2301.08693v1.pdf'}
+page_content='3√f + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFAT4oBgHgl3EQfyh6B/content/2301.08693v1.pdf'}
+page_content='7  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFAT4oBgHgl3EQfyh6B/content/2301.08693v1.pdf'}
+page_content=' Γ3(f) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFAT4oBgHgl3EQfyh6B/content/2301.08693v1.pdf'}
+page_content='3f 2 + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFAT4oBgHgl3EQfyh6B/content/2301.08693v1.pdf'}
+page_content='7  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFAT4oBgHgl3EQfyh6B/content/2301.08693v1.pdf'}
+page_content=' Γ4(f) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFAT4oBgHgl3EQfyh6B/content/2301.08693v1.pdf'}
+page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFAT4oBgHgl3EQfyh6B/content/2301.08693v1.pdf'}
+page_content='  For 1 ≤ j ≤ 4, we make the collection of data {WΓjfi}2688  i=1 by using the forward operator  for P and Γj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFAT4oBgHgl3EQfyh6B/content/2301.08693v1.pdf'}
+page_content=' Of the 2,688 data, we use 2,048 data for training, 128 data for validation  and 512 data for testing respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFAT4oBgHgl3EQfyh6B/content/2301.08693v1.pdf'}
+page_content='  Figure 5: Examples of phantoms  9  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFAT4oBgHgl3EQfyh6B/content/2301.08693v1.pdf'}
+page_content='2  Training  We use the Adam optimizer based on stochastic gradient descent and adaptive moment  estimation to train the network  [37].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFAT4oBgHgl3EQfyh6B/content/2301.08693v1.pdf'}
+page_content=' There are two neural networks in the proposed  frameworks: the reconstruction network R and the mapping network M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFAT4oBgHgl3EQfyh6B/content/2301.08693v1.pdf'}
+page_content=' The learning  rates for linear term of R, perturbation term of R, and M are chosen to be 10−4,  10−3, and 10−3, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFAT4oBgHgl3EQfyh6B/content/2301.08693v1.pdf'}
+page_content=' Momentum parameters of the Adam optimizer were set at  β1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFAT4oBgHgl3EQfyh6B/content/2301.08693v1.pdf'}
+page_content='9 and β2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFAT4oBgHgl3EQfyh6B/content/2301.08693v1.pdf'}
+page_content='999, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFAT4oBgHgl3EQfyh6B/content/2301.08693v1.pdf'}
+page_content='  We specifically put the batch size to 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFAT4oBgHgl3EQfyh6B/content/2301.08693v1.pdf'}
+page_content=' For general tasks, a moderately large batch  size reduces training time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFAT4oBgHgl3EQfyh6B/content/2301.08693v1.pdf'}
+page_content=' However, in this problem, a small batch size is advantageous  because our model must be able to reconstruct an exact image for each data rather than  an average result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFAT4oBgHgl3EQfyh6B/content/2301.08693v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFAT4oBgHgl3EQfyh6B/content/2301.08693v1.pdf'}
+page_content='3  Results  In this section, we illustrate experimental results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFAT4oBgHgl3EQfyh6B/content/2301.08693v1.pdf'}
+page_content=' The overall results are displayed in  Figure 6, Table 1, Figure 7, Figure 8, Table 2 and Figure 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFAT4oBgHgl3EQfyh6B/content/2301.08693v1.pdf'}
+page_content=' Here the losses for f and  WΓ(f) are respectively defined by  loss for f = 1  N  N  �  i=1  ∥fi − R(WΓ(fi))∥2  ∥fi∥2  ,  and  loss for WΓf = 1  N  N  �  i=1  ∥WΓ(fi) − WM(R(WΓ(fi))∥2  ∥WΓ(fi)∥2  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFAT4oBgHgl3EQfyh6B/content/2301.08693v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFAT4oBgHgl3EQfyh6B/content/2301.08693v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFAT4oBgHgl3EQfyh6B/content/2301.08693v1.pdf'}
+page_content='1  Low resolution data  We conduct the simulation utilizing a dataset of images with a size of 64×64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFAT4oBgHgl3EQfyh6B/content/2301.08693v1.pdf'}
+page_content=' The results  for the mapping networks are shown in Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFAT4oBgHgl3EQfyh6B/content/2301.08693v1.pdf'}
+page_content=' We see that the mapping networks  accurately approximates Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFAT4oBgHgl3EQfyh6B/content/2301.08693v1.pdf'}
+page_content=' When Γ3 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFAT4oBgHgl3EQfyh6B/content/2301.08693v1.pdf'}
+page_content='3f 2 + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFAT4oBgHgl3EQfyh6B/content/2301.08693v1.pdf'}
+page_content='7, there is a difference between the  plot of the mapping network M and the plot of Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFAT4oBgHgl3EQfyh6B/content/2301.08693v1.pdf'}
+page_content=' This is because the values of f ∈ P  almost belong to [0, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFAT4oBgHgl3EQfyh6B/content/2301.08693v1.pdf'}
+page_content='3]∪1 and so it has little effect on WΓ(f).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFAT4oBgHgl3EQfyh6B/content/2301.08693v1.pdf'}
+page_content=' In all cases, the process of  training the mapping networks requires approximately 103 iterations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFAT4oBgHgl3EQfyh6B/content/2301.08693v1.pdf'}
+page_content=' The results of the  reconstruction networks are illustrated in Table 1 and Figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFAT4oBgHgl3EQfyh6B/content/2301.08693v1.pdf'}
+page_content=' Table 1 shows the test  errors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFAT4oBgHgl3EQfyh6B/content/2301.08693v1.pdf'}
+page_content=' So it can be concluded that the reconstruction networks accurately approximate  the inverse maps in each case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFAT4oBgHgl3EQfyh6B/content/2301.08693v1.pdf'}
+page_content=' The training of the reconstruction networks necessitates  approximately 105 iterations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFAT4oBgHgl3EQfyh6B/content/2301.08693v1.pdf'}
+page_content='  Remark.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFAT4oBgHgl3EQfyh6B/content/2301.08693v1.pdf'}
+page_content=' The assumption on Γ, (5) is crucial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFAT4oBgHgl3EQfyh6B/content/2301.08693v1.pdf'}
+page_content=' If constraint (5) is not given in M, it  may take a long time to approximate Γ, or it may fail to find Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFAT4oBgHgl3EQfyh6B/content/2301.08693v1.pdf'}
+page_content=' Under the constraint, M  can quickly determine Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFAT4oBgHgl3EQfyh6B/content/2301.08693v1.pdf'}
+page_content=' Early determination of Γ helps the learning of reconstruction  networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFAT4oBgHgl3EQfyh6B/content/2301.08693v1.pdf'}
+page_content='  10  Figure 6: Comparison of the mapping network M and ground truth Γ for 64 × 64 data  Assumption  loss for f  loss for WΓ(f)  Γ1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFAT4oBgHgl3EQfyh6B/content/2301.08693v1.pdf'}
+page_content='3f + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFAT4oBgHgl3EQfyh6B/content/2301.08693v1.pdf'}
+page_content='7  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFAT4oBgHgl3EQfyh6B/content/2301.08693v1.pdf'}
+page_content='00504  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFAT4oBgHgl3EQfyh6B/content/2301.08693v1.pdf'}
+page_content='00702  Γ2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFAT4oBgHgl3EQfyh6B/content/2301.08693v1.pdf'}
+page_content='3√f + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFAT4oBgHgl3EQfyh6B/content/2301.08693v1.pdf'}
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+page_content='00537  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFAT4oBgHgl3EQfyh6B/content/2301.08693v1.pdf'}
+page_content='00947  Γ3 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFAT4oBgHgl3EQfyh6B/content/2301.08693v1.pdf'}
+page_content='3f 2 + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFAT4oBgHgl3EQfyh6B/content/2301.08693v1.pdf'}
+page_content='7  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFAT4oBgHgl3EQfyh6B/content/2301.08693v1.pdf'}
+page_content='00557  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFAT4oBgHgl3EQfyh6B/content/2301.08693v1.pdf'}
+page_content='00634  Γ4 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFAT4oBgHgl3EQfyh6B/content/2301.08693v1.pdf'}
+page_content='7  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFAT4oBgHgl3EQfyh6B/content/2301.08693v1.pdf'}
+page_content='01373  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFAT4oBgHgl3EQfyh6B/content/2301.08693v1.pdf'}
+page_content='00456  Table 1: Test errors for f and WΓ(f) according to Γ after 102,400 iterations for 64 × 64  data  11  [2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFAT4oBgHgl3EQfyh6B/content/2301.08693v1.pdf'}
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+page_content='0  ground truth  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFAT4oBgHgl3EQfyh6B/content/2301.08693v1.pdf'}
+page_content='0  mapping network  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFAT4oBgHgl3EQfyh6B/content/2301.08693v1.pdf'}
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+page_content='0Figure 7: Reconstruction results according to Γ for 64 × 64 data  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFAT4oBgHgl3EQfyh6B/content/2301.08693v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFAT4oBgHgl3EQfyh6B/content/2301.08693v1.pdf'}
+page_content='2  High resolution data  In the simulation for high resolution data, two linear operators T1 and T2 in the recon-  struction network (4) are replaced by alternative map described in Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFAT4oBgHgl3EQfyh6B/content/2301.08693v1.pdf'}
+page_content=' The dataset  is prepared with images of a size of 96 × 96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFAT4oBgHgl3EQfyh6B/content/2301.08693v1.pdf'}
+page_content=' Similarly to the low resolution case, the  mapping network M approximates Γ accurately within 103 iterations (Figure 8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFAT4oBgHgl3EQfyh6B/content/2301.08693v1.pdf'}
+page_content=' On the  other hand, the reconstruction networks for each Γ exhibit a slight decrease in perfor-  mance but still acceptable (Table 2 and Figure 9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFAT4oBgHgl3EQfyh6B/content/2301.08693v1.pdf'}
+page_content=' We surmise that the slight decrease  in performance is a result of the reduction in parameters brought about by the Pixel  Unshuffle and Pixel Shuffle operations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFAT4oBgHgl3EQfyh6B/content/2301.08693v1.pdf'}
+page_content='  12  [1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFAT4oBgHgl3EQfyh6B/content/2301.08693v1.pdf'}
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+page_content='7  ground truth  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFAT4oBgHgl3EQfyh6B/content/2301.08693v1.pdf'}
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+page_content='0Figure 8: Comparison of the mapping network M and ground truth Γ for 96 × 96 data  Assumption  loss for f  loss for WΓ(f)  Γ1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFAT4oBgHgl3EQfyh6B/content/2301.08693v1.pdf'}
+page_content='3f + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFAT4oBgHgl3EQfyh6B/content/2301.08693v1.pdf'}
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+page_content='01293  Γ2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFAT4oBgHgl3EQfyh6B/content/2301.08693v1.pdf'}
+page_content='3√f + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFAT4oBgHgl3EQfyh6B/content/2301.08693v1.pdf'}
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+page_content='01679  Γ3 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFAT4oBgHgl3EQfyh6B/content/2301.08693v1.pdf'}
+page_content='3f 2 + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFAT4oBgHgl3EQfyh6B/content/2301.08693v1.pdf'}
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+page_content='01132  Γ4 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFAT4oBgHgl3EQfyh6B/content/2301.08693v1.pdf'}
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+page_content='69511  Table 2: Test errors for f and WΓ(f) according to Γ after 102,400 iterations for 96 × 96  data  13  [2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFAT4oBgHgl3EQfyh6B/content/2301.08693v1.pdf'}
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+page_content='0Figure 9: Reconstruction results according to Γ for 96 × 96 data  5  Conclusions  Here, we propose a self-supervised learning for a nonlinear inverse problem with forward  operator involving an unknown function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFAT4oBgHgl3EQfyh6B/content/2301.08693v1.pdf'}
+page_content=' In medical imaging such as PAT, the initial  pressure is mostly untrackable for the measured data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFAT4oBgHgl3EQfyh6B/content/2301.08693v1.pdf'}
+page_content=' Moreover it is difficult to know the  wave speed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFAT4oBgHgl3EQfyh6B/content/2301.08693v1.pdf'}
+page_content=' So it is necessary to reconstruct the initial pressure f and the wave speed  simultaneously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFAT4oBgHgl3EQfyh6B/content/2301.08693v1.pdf'}
+page_content=' Under the simple assumption, the problem becomes a nonlinear inverse  problem involving an unknown function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFAT4oBgHgl3EQfyh6B/content/2301.08693v1.pdf'}
+page_content=' The experimental results demonstrate the high  performance of the proposed algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFAT4oBgHgl3EQfyh6B/content/2301.08693v1.pdf'}
+page_content=' Our framework can be extended to a nonlinear  inverse problem involving an unknown function, formulated under more complicated  situations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFAT4oBgHgl3EQfyh6B/content/2301.08693v1.pdf'}
+page_content=' This can be an interesting line of future research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFAT4oBgHgl3EQfyh6B/content/2301.08693v1.pdf'}
+page_content='  References  [1] Huabei Jiang.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFAT4oBgHgl3EQfyh6B/content/2301.08693v1.pdf'}
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+page_content=' CRC Press, 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFAT4oBgHgl3EQfyh6B/content/2301.08693v1.pdf'}
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+page_content=' Photoacoustic tomography: principles  and advances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFAT4oBgHgl3EQfyh6B/content/2301.08693v1.pdf'}
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+page_content='  [3] Peter Kuchment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFAT4oBgHgl3EQfyh6B/content/2301.08693v1.pdf'}
+page_content=' The Radon transform and medical imaging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFAT4oBgHgl3EQfyh6B/content/2301.08693v1.pdf'}
+page_content=' SIAM, 2013.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFAT4oBgHgl3EQfyh6B/content/2301.08693v1.pdf'}
+page_content='  14  [1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFAT4oBgHgl3EQfyh6B/content/2301.08693v1.pdf'}
+page_content='3f + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFAT4oBgHgl3EQfyh6B/content/2301.08693v1.pdf'}
+page_content='7  [2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFAT4oBgHgl3EQfyh6B/content/2301.08693v1.pdf'}
+page_content='3V f + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFAT4oBgHgl3EQfyh6B/content/2301.08693v1.pdf'}
+page_content='7  [3 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFAT4oBgHgl3EQfyh6B/content/2301.08693v1.pdf'}
+page_content='3f2 + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFAT4oBgHgl3EQfyh6B/content/2301.08693v1.pdf'}
+page_content='7  「4 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFAT4oBgHgl3EQfyh6B/content/2301.08693v1.pdf'}
+page_content='7  ground truth  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFAT4oBgHgl3EQfyh6B/content/2301.08693v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFAT4oBgHgl3EQfyh6B/content/2301.08693v1.pdf'}
+page_content='8  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFAT4oBgHgl3EQfyh6B/content/2301.08693v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFAT4oBgHgl3EQfyh6B/content/2301.08693v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFAT4oBgHgl3EQfyh6B/content/2301.08693v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFAT4oBgHgl3EQfyh6B/content/2301.08693v1.pdf'}
+page_content='0[4] Alexander Graham Bell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFAT4oBgHgl3EQfyh6B/content/2301.08693v1.pdf'}
+page_content=' On the production and reproduction of sound by light.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFAT4oBgHgl3EQfyh6B/content/2301.08693v1.pdf'}
+page_content=' In  Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFAT4oBgHgl3EQfyh6B/content/2301.08693v1.pdf'}
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+page_content=' Adv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFAT4oBgHgl3EQfyh6B/content/2301.08693v1.pdf'}
+page_content=' Sci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFAT4oBgHgl3EQfyh6B/content/2301.08693v1.pdf'}
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+page_content='  [5] Idan Steinberg, David M Huland, Ophir Vermesh, Hadas E Frostig, Willemieke S  Tummers, and Sanjiv S Gambhir.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFAT4oBgHgl3EQfyh6B/content/2301.08693v1.pdf'}
+page_content=' Photoacoustic clinical imaging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFAT4oBgHgl3EQfyh6B/content/2301.08693v1.pdf'}
+page_content=' Photoacoustics,  14:77–98, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFAT4oBgHgl3EQfyh6B/content/2301.08693v1.pdf'}
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+page_content=' Photoacoustic tomogra-  phy with direction dependent data: An exact series reconstruction approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFAT4oBgHgl3EQfyh6B/content/2301.08693v1.pdf'}
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+page_content=' Universal back-projection algorithm for photoa-  coustic computed tomography.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFAT4oBgHgl3EQfyh6B/content/2301.08693v1.pdf'}
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+page_content=' Photoacoustic and thermoacoustic tomog-  raphy with an uncertain wave speed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFAT4oBgHgl3EQfyh6B/content/2301.08693v1.pdf'}
+page_content=' Mathematical Research Letters, 2014.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFAT4oBgHgl3EQfyh6B/content/2301.08693v1.pdf'}
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+page_content=' Thermoacoustic tomography with variable  sound speed Inverse Problems, 25(7):075011, 16, 2009.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFAT4oBgHgl3EQfyh6B/content/2301.08693v1.pdf'}
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+page_content=' An efficient  neumann series–based algorithm for thermoacoustic and photoacoustic tomography  with variable sound speed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFAT4oBgHgl3EQfyh6B/content/2301.08693v1.pdf'}
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+page_content='  A direct method for  photoacoustic tomography with inhomogeneous sound speed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFAT4oBgHgl3EQfyh6B/content/2301.08693v1.pdf'}
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+page_content=' Reconstruction and time rever-  sal in thermoacoustic tomography in acoustically homogeneous and inhomogeneous  media.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFAT4oBgHgl3EQfyh6B/content/2301.08693v1.pdf'}
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+page_content='  Singular value decomposition  of the wave forward operator with radial variable coefficients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFAT4oBgHgl3EQfyh6B/content/2301.08693v1.pdf'}
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+page_content=' Determining both sound speed and internal  source in thermo-and photo-acoustic tomography.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFAT4oBgHgl3EQfyh6B/content/2301.08693v1.pdf'}
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+page_content=' Pho-  toacoustic image reconstruction via deep learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFAT4oBgHgl3EQfyh6B/content/2301.08693v1.pdf'}
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+page_content=' SPIE, 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFAT4oBgHgl3EQfyh6B/content/2301.08693v1.pdf'}
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+page_content=' Deep learning techniques for inverse prob-  lems in imaging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFAT4oBgHgl3EQfyh6B/content/2301.08693v1.pdf'}
+page_content=' IEEE Journal on Selected Areas in Information Theory, 1(1):39–56,  2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFAT4oBgHgl3EQfyh6B/content/2301.08693v1.pdf'}
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+Neural-Symbolic Inference for Robust Autoregressive Graph Parsing via
+Compositional Uncertainty Quantification
+Zi Lin
+UC San Diego
+lzi@ucsd.edu
+Jeremiah Liu†‡
+Google Research & Harvard University
+jereliu@google.com
+Jingbo Shang†
+UC San Diego
+jshang@ucsd.edu
+Abstract
+Pre-trained seq2seq models excel at graph se-
+mantic parsing with rich annotated data, but
+generalize worse to out-of-distribution (OOD)
+and long-tail examples. In comparison, sym-
+bolic parsers under-perform on population-
+level metrics, but exhibit unique strength in
+OOD and tail generalization.
+In this work,
+we study compositionality-aware approach to
+neural-symbolic inference informed by model
+confidence, performing fine-grained neural-
+symbolic reasoning at subgraph level (i.e.,
+nodes and edges) and precisely targeting sub-
+graph components with high uncertainty in
+the neural parser.
+As a result, the method
+combines the distinct strength of the neural
+and symbolic approaches in capturing differ-
+ent aspects of the graph prediction, leading to
+well-rounded generalization performance both
+across domains and in the tail.
+We empiri-
+cally investigate the approach in the English
+Resource Grammar (ERG) parsing problem
+on a diverse suite of standard in-domain and
+seven OOD corpora. Our approach leads to
+35.26% and 35.60% error reduction in aggre-
+gated SMATCH score over neural and sym-
+bolic approaches respectively, and 14% abso-
+lute accuracy gain in key tail linguistic cate-
+gories over the neural model, outperforming
+prior state-of-art methods that do not account
+for compositionality or uncertainty.
+1
+Introduction
+A structured account of compositional meaning has
+become a longstanding goal for Natural Language
+Processing. To this end, a number of efforts have
+focused on encoding semantic relationships and at-
+tributes into graph-based meaning representations
+(MRs, see Appendix A for details). In particular,
+graph semantic parsing has been an important task
+in almost every Semantic Evaluation (SemEval)
+exercise since 2014. In recent years, we have wit-
+nessed the burgeoning of applying neural networks
+† Co-senior authors. ‡ Work done at Google.
+to semantic parsing. Pre-trained language model-
+based approaches have led to significant improve-
+ments across different MRs (Oepen et al., 2019,
+2020). However, these models often generalize
+poorly to out-of-distribution (OOD) and tail ex-
+amples (Cheng et al., 2019; Shaw et al., 2021;
+Kim, 2021; Lin et al., 2022), while grammar or
+rule-based parser work relatively robustly across
+different linguistic phenomena and language do-
+mains (Cao et al., 2021; Lin et al., 2022). See
+Section 6 for a review of related work.
+In this paper, we propose a novel compositional
+neural-symbolic inference for graph semantic pars-
+ing, which takes advantage of both uncertainty
+quantification from a seq2seq parser and prior
+knowledge from a symbolic parser at the subgraph
+level (i.e., nodes and edges). We take graph seman-
+tic parsing for English Resource Grammar (ERG)
+as our case study. ERG is a compositional semantic
+representation explicitly coupled with the syntactic
+structure. Compared to other graph-based meaning
+representations like Abstract Meaning Representa-
+tion (AMR), ERG has high coverage of English text
+and strong transferability across domains, render-
+ing itself as an attractive target formalism for auto-
+mated semantic parsing. Furthermore, many years
+of ERG research has led to well-established sym-
+bolic parser and a rich set of carefully constructed
+corpus across different application domains and
+fine-grained linguistic phenomena, making it an
+ideal candidate for studying cross-domain general-
+ization of neural-symbolic methods (Oepen et al.,
+2002; Crysmann and Packard, 2012).
+We start with a novel investigation of the uncer-
+tainty calibration behaviour of a T5-based state-of-
+the-art neural ERG parser (Lin et al., 2022) on the
+subgraph level (Section 3), where we make some
+key observations: (1) the performance of the neu-
+ral parser degrades when it becomes uncertain at
+the subgraph level, while (2) the symbolic parser
+works still robustly when the neural parser is un-
+arXiv:2301.11459v1  [cs.CL]  26 Jan 2023
+
+_the_q<0>
+_want_v_1<2>
+_boy_n_1<1>
+_believe_v_1<6>
+_girl_n_1<0>
+pron<7>
+pronoun_q
+_the_q<3>
+BV
+BV
+BV
+ARG1
+ARG2
+ARG1
+ARG2
+The<0> boy<1> wants<2> the<3> girl<4> to<5> believe<6> him<7>
+Abstract Concepts
+(grammatical function)
+Root
+Token-Node Alignments
+Surface Concepts 
+(related to surface tokens)
+<·>
+(a) EDS Representation
+( _want_v_1
+:ARG1 ( _boy_n_1
+:BV-of ( _the_q ) ) 
+:AGR2 ( _believe_v_1 
+:ARG1 ( _girl_n_1
+:BV-of ( _the_q ) )
+:ARG2 ( pron
+:BV-of ( pronoun_q ) ) ) )
+(b) Variable-free PENMAN notation
+Figure 1: The EDS representation for ERG and the corresponding linearization of the example sentence “The boy
+wants the girl to believe him”.
+certain at the subgraph level. This motivates us to
+develop a compositional neural-symbolic inference
+process where the neural and symbolic parser col-
+laborates at a more fine-grained level and guided
+by model uncertainty, which is an aspect missing in
+the previous neural-symbolic and ensemble parsing
+literature (see Appendix 6).
+We then propose a decision-theoretic criteria to
+allow for neural-symbolic inference at subgraph
+level (i.e., nodes and edges) and incorporates the
+neural parser’s fine-grained uncertainty for each
+graph component prediction (Section 4.1). The
+key to this approach is a meta graph GM that enu-
+merates possible candidates for each node/edge
+prediction, and is constructed by merging multiple
+beam predictions from the neural seq2seq model.
+The core challenge here is how to properly quan-
+tify compositional uncertainty using a seq2seq
+model, i.e., assigning model probability for a node
+or edge prediction. For example, our interest is to
+express the conditional probability of a graph node
+v with respect to its parent p(v|pa(v), x), rather
+than the likelihood of v conditioning on the previ-
+ous tokens in the linearized string. As a result, it
+cannot be achieved by relying on the naive token-
+level autoregressive probabilities from the beam
+search. To address this issue, we introduce a sim-
+ple probabilistic formalism termed Graph Autore-
+gressive Process (GAP) (Section 4.2). GAP adopts
+a dual representation of an autoregressive process
+and a probabilistic graphical model, and can serve
+as a powerful medium for expressing compositional
+uncertainty for seq2seq graph parsing.
+We demonstrate the effectiveness of our ap-
+proach in experiments across a diverse suite of
+eight in-domain and OOD evaluation datasets en-
+compassing domains including Wikipedia entries,
+news articles, email communications, etc (Section
+5).
+We achieve the best results on the overall
+performance across the eight domains, attaining
+35.26% and 35.60% error reduction in the aggre-
+gated SMATCH score over the neural and symbolic
+parser, respectively. Our approach also exhibits sig-
+nificantly stronger robustness in generalization to
+OOD datasets and long-tail linguistic phenomena
+than previous work, while maintaining the state-
+of-the-art performance on in-domain test. Further
+study also shows that the compositionality aspects
+of neural-symbolic inference helps the model to as-
+semble novel graph solution that the original infer-
+ence process (e.g., beam search or symbolic parse)
+fails to provide (Section 5.4).
+In summary, our contributions are four-fold:
+• We present a novel investigation of neural graph
+parser’s uncertainty calibration performance at
+subgraph level (Section 3). Our study confirms
+the seq2seq uncertainty is effective for detecting
+model error even out-of-distribution, establishing
+the first empirical basis for the utility of compo-
+sitional uncertainty in seq2seq graph parsing.
+• We propose a practical and principled framework
+for neural-symbolic graph parsing that utilizes
+model uncertainty and exploits compositionality
+(Section 4.1). The method is fully compatible
+with modern large pre-trained seq2seq network
+using beam decoding, and is general-purpose and
+applicable to any graph semantic parsing task.
+• We propose a simple probabilistic formalism
+(GAP) to express a seq2seq model’s composi-
+tional uncertainty (Section 4.2). GAP allows
+us to go beyond the conventional autoregres-
+sive sequence probability and express long-range
+parent-child conditional probability on the graph,
+serving as a useful medium of compositional un-
+certainty quantification.
+• We conduct a comprehensive study to evalu-
+ate the state-of-the-art graph parsing approaches
+across a diverse suite of in-domain and out-of-
+distribution datasets (Section 5). Our study re-
+veals surprising weakness of previous neural-
+symbolic methods in OOD generalization, and
+confirms the proposed method significantly im-
+
+0.3947
+0.7788
+0.9212
+0.9764
+0.9930
+0.9983
+0.9996
+0.9999
+1.0000
+T5 Model's Probabilities
+0.4
+0.5
+0.6
+0.7
+0.8
+0.9
+1.0
+Node/Edge Accuracies
+tanaka
+T5
+ACE
+0.5810
+0.8004
+0.9150
+0.9692
+0.9879
+0.9967
+0.9991
+0.9998
+1.0000
+T5 Model's Probabilities
+0.4
+0.5
+0.6
+0.7
+0.8
+0.9
+1.0
+Node/Edge Accuracies
+brown
+T5
+ACE
+Figure 2: Bar charts for the predictive accuracies of the T5 parser (blue) and ACE parser (orange) for all the
+node / edge prediction across different uncertainty buckets based on T5 model’s probabilities. The performance
+is evaluated on the Tanaka and Brown datasets. Each bin represents a quantile bucket of the model probability
+(i.e., they contain the same number of examples). Since at most of the subgraphs, the model is pretty certain
+(log P > −1e − 5), we exclude these pretty certain predictions in the figures.
+proves models OOD and tail performance.
+Reproducibility.
+Our
+code
+is
+available
+on
+Github:
+https://github.com/google/
+uncertainty-baselines/tree/main/
+baselines/t5/data/deepbank.
+2
+Background
+2.1
+English Resource Grammar (ERG)
+In this work, we take the representations from En-
+glish Resource Grammar (ERG; Flickinger et al.,
+2014) as our target meaning representations. ERG
+is a broad-coverage computational grammar of En-
+glish that derives underspecified logical-form rep-
+resentations of meaning (Oepen and Flickinger,
+2019). It is rooted in the general linguistic theory
+of Head-driven Phrase Structure Grammar (HPSG;
+Pollard and Sag, 1994).
+ERG can be presented into different types of an-
+notation formalism (Copestake et al., 2005). This
+work focuses on the Elementary Dependency Struc-
+ture (EDS; Oepen and Lønning, 2006) which is
+a compact representation that can be expressed
+as a directed acyclic graph (DAG) and is widely
+adopted in the neural parsing approaches (Buys and
+Blunsom, 2017; Chen et al., 2018). An example is
+shown in Figure 1(a).
+2.2
+Parsing Approaches
+In this section, we review the state-of-the-art sym-
+bolic and neural parsers utilized in our work, i.e.,
+the ACE parser (Crysmann and Packard, 2012) and
+the T5 parser (Lin et al., 2022). Appendix B re-
+views other ERG parsing techniques.
+The symbolic parser: ACE. The ACE parser
+(Crysmann and Packard, 2012) is one of the state-
+of-the-art symbolic parsers. It first decomposes
+sentences into ERG-consistent candidate derivation
+trees, and the parser will rank candidates based on
+the structural features in the nodes of the deriva-
+tion trees via maximum entropy models (Oepen
+and Lønning, 2006; Toutanova et al., 2005). This
+approach fails to parse sentences for which no valid
+derivation is found.
+The neural parser: T5. Lin et al. (2022) pro-
+posed a T5-based ERG parser which achieves the
+best known results on the in-domain DeepBank
+benchmark. It is the first work that successfully
+transfers the ERG parsing problem into a pure end-
+to-end translation problem via compositionality-
+aware tokenization and a variable-free top-down
+graph linearization based on the PENMAN nota-
+tion (Kasper, 1989). Figure 1(b) shows an example
+of the linearized graph string from the original EDS
+graph.
+3
+Motivation: Subgraph-level
+Uncertainty in Seq2seq Graph Parsing
+We hypothesize that when the neural seq2seq
+model is uncertain at the subgraph level, it is more
+likely to make mistakes. Assuming the symbolic
+parser performs more robustly in these situations,
+we can then design a procedure to ask the symbolic
+parser for help when the model is uncertain. To
+validate this hypothesis, we conduct experiments
+to empirically explore the following two questions:
+(1) how does the model perform when it is uncer-
+tain at the subgraph level? and (2) how does the
+symbolic parser perform when the model is uncer-
+tain?
+First, we compute model probabilities for each
+graph element (i.e., node and edge) prediction (see
+Section 4.2 for how to compute these quanitities),
+and identify the corresponding ACE parser pre-
+diction using the graph matching algorithm from
+SMATCH (Cai and Knight, 2013). We then evaluate
+the accuracies of those graph element predictions
+with respect to the gold labels, and compare it to
+
+that of the ACE parser.
+In Figure 2, we plot the bar charts compare
+the neural and symbolic performance in different
+bucket of seq2seq model uncertainties on the two
+largest datasets (e.g., Tanaka and Brown, see Ap-
+pendix G). Results on other datasets can be found in
+the Appendix K. As shown in the figure, low model
+probability generally corresponds to low T5 per-
+formance, while the corresponding ACE parser’s
+accuracies spread relatively stably (e.g., it attains
+> 90% accuracy in the lowest-confidence buck-
+ets, while T5 accuracy is < 50%). This implies
+that when the model is uncertain, the accuracy of
+the neural model tend to be low, while the ACE
+parser still performs well. This has motivated us
+to develop a compositional neural-symbolic infer-
+ence procedure guided the model’s subgraph level
+uncertainty, such that the T5 and ACE parser can
+collaborate at a more fine-grained level via com-
+postional uncertainty quantification (Section 4).
+4
+Methods
+Notation & Problem Statement. For graph se-
+mantic parsing, the input is a natural language ut-
+terance x, and the output is a directed acyclic graph
+(DAG) G = ⟨N, E⟩, where N is the set of nodes
+and E ∈ N × N is the set of edges (e.g., Figure
+1(a)). In the case of seq2seq parsing, G is repre-
+sented as a linearized graph string g = s1s2 · · · sL
+which consists of symbols {sl}L
+l=1 (e.g., Figure
+1(b)). As the graph prediction is probabilistic, each
+of the graph element v ∈ N ∪ E is a random vari-
+able whose values are the symbols si observed
+from the beam outputs, leading to marginal prob-
+abilities p(v = si|x) and conditional probabilities
+p(v = si|v′ = sj, x).
+To this end, our goal is to produce a principled
+inference procedure for graph prediction account-
+ing for model uncertainty on predicting graph ele-
+ments v ∈ G. In the sequel, Section 4.1 presents
+a decision-theoretic criterion that leverages the
+graphical model likelihood p(G|x) to conduct com-
+positional neural-symbolic inference for graph pre-
+diction. To properly express the graphic model
+likelihood p(G|x) = �
+v∈G p(v|pa(v), x) using
+a learned seq2seq model, Section 4.2 introduces
+a simple probabilistic formalism termed Graph
+Autoregressive Process (GAP) to translate the au-
+toregressive sequence probability from the seq2seq
+model to graphical model probability. Appendix E
+discusses some additional extensions.
+4.1
+Compositional Neural-Symbolic
+Inference
+Previously, an uncertainty-aware decision criteria
+was proposed for neural-symbolic inference based
+on the Hurwicz pessimism-optimism criteria
+R(G|x) (Lin et al., 2022). Specifically, the criteria
+is written as:
+R(G|x) = α(x)∗Rp(G|x)+(1−α(x))∗R0(G),
+where R(G|x) = − log p(G|x) is the neural model
+likelihood, R0(G) = log p0(G) is the symbolic
+prior likelihood, and α(x) is a the uncertainty-
+driven trade-off coefficient to balance between
+the optimistic MLE criteria Rp(G|x) and the pes-
+simistic, prior-centered criteria R0(G|x) centered
+around symbolic prediction G0.
+A key drawback of this approach is the lack of
+accounting for the compositionality. This motivates
+us to consider synthesizing the multiple graph pre-
+dictions {Gk}K
+k=1 from the neural parser to form
+a meta graph G 1, where we can leverage the dis-
+entangled uncertainty of p(G|x) to perform fine-
+grained neural-symbolic inference for each graph
+component v ∈ G (i.e., nodes or edges). Specifi-
+cally, we leverage the factorized graphical model
+likelihood p(G|x) = �
+v∈G p(v| pa(v), x) to de-
+compose the overall decision criteria R(G|x) into
+that of individual components R(v|x):
+R(v|x) = α(v|x) ∗ log p(v| pa(v), x)
++ (1 − α(v|x)) ∗ log p0(v),
+(1)
+and the overall criteria is written as R(G|x) =
+�
+v∈G R(v|x). Here pa(v) refers to the parents
+of v in G, and α(v|x) = sigmoid(− 1
+T H(v|x) +
+b) is the component-specific trade-off param-
+eter driven by model uncertainty H(v|x)
+=
+− log p(v| pa(v), x), and (T, b) are scalar calibra-
+tion hyperparameters that can be tuned on the dev
+set.
+Following previous work (Lin et al., 2022), the
+symbolic prior p0 for each graph component v is
+defined as a Boltzmann distribution based on the
+graph output G0 from the symbolic parser, i.e.,
+p0(v = s) ∝ exp(I(s ∈ G0)), so that it is pro-
+portional to the empirical probability of whether
+a symbol s appears in G0. Notice that we have
+1Given a group of candidate graphs {Gk}K
+k=1, well-
+established algorithm exists to synthesize different graph pre-
+dictions into a meta graph G (Cai and Knight, 2013; Hoang
+et al., 2021) (see Appendix F for a more detailed review).
+
+ignored the normalizing constants since they do
+not impact optimization.
+Algorithm 1 summarizes the full algorithm.
+As shown, during inference, the method pro-
+ceeds
+by
+starting
+from
+the
+root
+node
+v0
+and
+selects
+the
+optimal
+prediction
+ˆv0
+=
+arg maxc0∈Candidate(v0) R(c0|x), where c0 are dif-
+ferent candidates for v0 given by the meta graph G.
+The algorithm then recursively performs the same
+neural-symbolic inference procedure for the chil-
+dren of v0 (i.e., ch(v)). The algorithm terminates
+when the optimal candidates for all graph variables
+v ∈ G are determined.
+As a result, the algorithm is able to adap-
+tively combine subgraph predictions across mul-
+tiple beam candidates thanks to the meta graph G,
+and appropriately weight between the local neural
+and symbolic information thanks to the uncertainty-
+aware decision criteria R(v|x). Empirically, this
+also gives the algorithm the ability to synthesize
+novel graph predictions that are distinct from its
+base models (Section 5.4).
+Algorithm 1 Compositional Neural-Symbolic Inference
+Inputs:
+Meta graph G
+Graphical model likelihood log p(G|x)
+Symbolic prior p0
+Output:
+Neural-symbolic graph prediction G
+Initialize:
+v = root(GM); G = GM.
+if G does not contain undecided candidates then return G
+else
+for cv ∈ Candidate(v) do
+Compute decision criteria R(cv|x) (Equation 1)
+Select optimal candidate ˆv = arg maxc R(c|x)
+Remove non-optimal candidates of v from G
+Recursively perform Algorithm 1 for all v′ ∈ ch(v)
+4.2
+Compositional Uncertainty
+Quantification with Graph Autogressive
+Process (GAP)
+To properly model the uncertainty p(G|x) from a
+seq2seq model, we need an intermediate probabilis-
+tic representation to translate the raw token-level
+probability to the distribution over graph elements.
+To this end, we introduce a simple probabilistic
+formalism termed Graph Autoregressive Process
+(GAP), which is a probability distribution assigning
+seq2seq learned probability to the graph elements
+v ∈ G. Specifically, as the seq2seq-predicted graph
+adopts both a sequence-based representation g =
+s1, ..., sL and a graph representation G = ⟨N, E⟩,
+the GAP model adopts both an autoregressive repre-
+sentation p(g|x) = �
+i p(si|s<i, x) (Section 4.2.1),
+and also a probabilistic graphical model repre-
+sentation p(G|x) = �
+v∈G p(v| pa(v), x) (Section
+4.2.2). Both representations share the same set of
+underlying probability measures (i.e., the graphical-
+model likelihood p(G|x) can be derived from the
+autoregressive probabilities p(si|s<i, x)) (Figure
+3), rendering itself a useful medium for princi-
+pled compositional neural-symbolic inference us-
+ing seq2seq probabilities.
+4.2.1
+Autoregressive Representation for
+Linearized Sequence g
+Given an input sequence x and output sequence
+y = y1y2 · · · yN, the token-level autoregressive
+distribution from a seq2seq model is p(y|x) =
+�N
+i=1 p(yi|y<i, x). In the context of graph pars-
+ing, the output sequence describes a linearized
+graph g = s1s2 · · · sL, where each symbol si =
+{yi1yi2 · · · yiNi} represents either a node n ∈
+N or an edge e ∈ E of the graph and corre-
+sponds to a collection of beam-decoded tokens
+{yi1yi2 · · · yiNi}, e.g., the node _the_q in Figure
+1(a) is represented by tokens {_, the, _q}. This
+process is illustrated in follows:
+s1
+s2
+s3
+sL
+g:
+y:
+y1 y2 y3
+y4 y5
+y6 y7 y8
+yN-1 yN
+…
+…
+To
+this
+end,
+the
+Graph
+Autoregressive
+Process
+(GAP)
+assigns
+probability
+to
+each
+linearized
+graph
+g
+=
+s1s2 · · · sL
+autore-
+greesviely
+as
+p(g|x)
+=
+�L
+i=1 p(si|s<i, x),
+and the conditional probability p(si|s<i, x) is
+computed by aggregating the token probability:
+p(si|s<i, x) = p({yi1 · · · yiNi }|s<i, x) =
+Ni
+�
+j=1
+p(yij|yi<j, s<i, x)
+Marginal and Conditional Probability. Impor-
+tantly, GAP allows us to compute the marginal
+and (non-local) conditional probabilities for
+graph elements si.
+Given the input x, the
+marginal probability of si
+is computed as
+p(si|x) =
+�
+s<i
+p(si|s<i, x)p(s<i|x)ds<i
+by integrating over the space of all possible sub-
+sequences s<i prior to the symbol si. Then, the
+(non-local) conditional probability between two
+graph elements (si, sj) with i < j is computed as
+p(sj|si, x) =
+�
+si→j,s<i
+p(si, si→j|si, s<i, x)p(si|s<i, x)p(s<i|x)dsi→jds<i
+
+by integrating over the space of subsequences si→j
+between (si, sj) and the subsequence s<i before
+si. Higher order conditional (e.g., p(sj|(si, sl), x))
+can be computed analogously. Notice this gives us
+the ability to reason about long-range dependencies
+between non-adjacent symbols on the sequence.
+Furthermore, the conditional probability on the
+reverse direction can also be computed using the
+Bayes’ rule: p(si|sj, x) = p(sj|si,x)p(si|x)
+p(sj|x)
+.
+Efficient Estimation Using Beam Outputs. In
+practice, we can estimate p(si|x) and p(sj|si, x)
+efficiently via importance sampling using the
+output from the beam decoding {gk}K
+k=1, where
+K is the beam size (Malinin and Gales, 2020).
+The marginal probability can be computed as
+ˆp(si|x) =
+K
+�
+k=1
+πkp(si|sk,<i, x)
+(2)
+where πk =
+exp( 1
+t log p(gk|x))
+�K
+k=1 exp( 1
+t log p(gk|x)) is the impor-
+tance weight proportional to the beam candidate
+gk’s log likelihoods, and t > 0 is the temperature
+parameter fixed to a small constant (e.g., t = 0.1,
+see Appendix C.1 further discussion) (Malinin and
+Gales, 2020). If the symbol si does not appear in
+the kth beam, we set p(si|sk,<i, x) = 0.
+Then, for two symbols (si, sj) with i < j, we
+can estimate the joint probability as
+ˆp(sj|si, x) =
+K
+�
+k=1
+πi
+kp(sj|si, sk,i→j, sk,<i, x)
+(3)
+where πi
+k =
+exp( 1
+t log p(gk|x))∗I(si∈gk)
+�K
+k=1 exp( 1
+t log p(gk|x))∗I(si∈gk) is the
+importance weight among beam candidates that
+contains si. Notice this is different from Equation
+2 where πk is computed over all beam candidates
+regardless of whether it contains si.
+4.2.2
+Probabilistic Graphical Model
+Representation for G
+So far, we have focused on probability computa-
+tion based on the graph’s linearized representation
+p(g|x) = �
+i p(si|s<i, x). To conduct the compo-
+sitional neural-symbolic inference (Section 4.1),
+we also need to consider GAP’s graphical model
+representation p(G|x) = �
+v∈G p(v| pa(v), x).
+GAP’s graphical model representation G de-
+pends on the meta graph G constructed from K
+candidate graphs {Gk}K
+k=1 (Section 4.1). Figure 3
+shows an example, where ni and ej are the candi-
+dates for the node and edge predictions collected
+from beam sequences. Compared to the sequence-
+based representation g, G provides two advantages:
+it (1) explicitly enumerates different candidates for
+each node and edge prediction (e.g., n2 v.s. n3 for
+predicting the third element), and (2) provides an
+explicit account of the parent-child relationships
+between variables on the graph (e.g., e2 is a child
+node of n1 in the predicted graph, which is not re-
+flected in the autoregressive representation). From
+the probabilistic learning perspective, G describes
+the space of possible graphs (i.e., the support) for
+a graph distribution p(G|x) : G → [0, 1].
+To this end, GAP assigns proper graph-level
+probability p(G|x) to graphs G sampled from the
+meta graph G via the graphical model likelihood:
+p(G|x) =
+�
+v∈G
+p(v| pa(v), x)
+=
+�
+n∈N
+p(n| pa(n), x) ∗
+�
+e∈E
+p(e| pa(e), x)
+where p(v| pa(v), x) is the conditional probability
+for v with respect to their parents pa(v) in
+G.
+Given the candidates graphs {Gk}K
+k=1, we
+can express the likelihood for p(v| pa(v), x)
+by
+writing
+down
+a
+multinomial
+likelihood
+enumerating over different values of pa(v)
+(Murphy, 2012). This in fact leads to a simple
+expression for the model likelihood as a sim-
+ple averaging of the beam-sequence log likelihoods:
+log p(n| pa(n), x) ∝ 1
+K
+K
+�
+k=1
+log p(n| pa(n) = ck)
+(4)
+where ck is the value of pa(n) in kth beam se-
+quence, and the conditional probabilities are
+computed using Equation (3). See Appendix D for
+a detailed derivation.
+Algorithm 2 Graph Autoregressive Process
+Inputs:
+Beam candidates with probabilities {p(gk|x)}K
+k=1
+Meta graph G
+Output:
+Marginal probabilities {p(s|x)}
+Graph model likelihood log p(G|x)
+for v ∈ G do
+Compute marginal likelihood:
+p(v = s|x) (Equation 2)
+Compute graphical model likelihood:
+log p(v = s| pa(v), x) (Equation 4)
+return {p(v|x)}, log p(G|x)) = �
+v∈G log p(v| pa(v), x)
+In summary, for each graph element variable
+v ∈ G, GAP allows us to compute the graphical-
+model conditional likelihood p(v|pa(v), x) via its
+graphical model representation, and also to com-
+pute the marginal probability p(v|x) via its autore-
+gressive presentation. The conditional likelihood is
+
+e1: L-INDEX
+n1: _and_c
+n2: _cathedral_n_1
+e2: R-INDEX
+n4: named "Bazaar"
+g1
+e1: L-INDEX
+n1: _and_c
+n2: _cathedral_n_1
+e2: R-INDEX
+n5: named "bazar"
+e1: L-INDEX
+n1: _and_c
+n3: named "Cathedral"
+e2: R-INDEX
+n6: _bazaar_n_1
+g2
+g3
+Autoregressive Sequential Representation (i.e., Beam Sequences)
+…
+…
+…
+…
+…
+Graphical Model Representation (i.e., Meta Graph)
+v
+n1: _and_c
+e1: L-INDEX
+n2: _cathedral_n_1
+n3: named "Cathedral"
+e2: R-INDEX
+n4: named "Bazaar"
+n5: named "bazar"
+n6: _bazaar_n_1
+GM
+Figure 3: Visual illustration of constructing graphical model representation GM from autorepressive representation
+{gk}K
+k=1. The example here represents the sentence “The Cathedral and the Bazaar” from the Eric Raymond
+Essay dataset. Note that here we have omitted the brackets in g for simplicity (see 1(b)).
+crucial for neural-symbolic inference (Section 4.1),
+and the marginal probability is useful for sparsity
+regularization in global graph structure inference
+(Appendix E). Algorithm 2 summarizes the full
+GAP computation.
+5
+Experiments
+5.1
+Experiment Setup
+Datasets. Consistent with previous ERG works,
+we train the neural model on DeepBank v1.1 an-
+notation of the Wall Stree Journal (WSJ), sections
+00-21 (the same text annotated in the Penn Tree
+Bank) that correspond to ERG version 1214.
+For OOD evaluation, we select 7 diverse datasets
+from the Redwoods Treebank corpus: Wikipedia
+(Wiki), the Brown Corpus (Brown), the Eric
+Raymond Essay (Essay), customer emails (E-
+commerce), meeting/hotel scheduling (Verbmobil),
+Norwegian tourism (LOGON) and the Tanaka Cor-
+pus (Tanaka) (See Appendix G for more details).
+Model. Following Lin et al. (2022), We train a
+T5large using the official T5X finetune pipeline2,
+and use beam search with size K = 5 at inference
+time. Further details are collected in Appendix H.
+Evaluation.
+we use the standard eval metric
+SMATCH (Cai and Knight, 2013), which computes
+the maximum F1-score obtainable from an align-
+ment between the predicted and gold graphs. We
+evaluate the models’ average-case performance on
+all the 8 in-domain and OOD datasets, and also
+conduct fine-grained evaluation of the models’ tail
+generalization performance across 19 important
+linguistic subcategories (Appendix J, Table 2).
+Baselines.
+We compare with two recent state-
+of-the-art approaches from the neural-symbolic
+2https://github.com/google-research/t5x/blob/
+main/t5x/train.py
+#
+T5
+ACE
+Vote
+Collab.
+Ours
+ACE*
+WSJ (in-domain)
+1,437
+96.56
+87.14
+88.22
+97.01
+96.77
+90.94
+Wiki
+1,307
+90.12
+80.25
+80.55
+90.58
+90.04
+90.42
+Brown
+2,182
+92.05
+91.74
+85.46
+93.58
+93.11
+93.20
+Essay
+591
+92.19
+92.64
+83.72
+93.57
+93.76
+93.52
+E-commerce
+1,114
+93.15
+97.25
+87.38
+95.44
+97.37
+98.36
+Verbmobil
+931
+90.06
+95.15
+84.80
+92.24
+96.42
+97.62
+LOGON
+1,895
+87.13
+93.58
+80.11
+92.88
+93.33
+94.17
+Tanaka
+2,796
+95.24
+98.38
+91.03
+96.79
+98.14
+98.55
+Mean w/ in-domain
+-
+92.06
+92.02
+85.16
+94.01
+94.86
+94.60
+Mean w/o in-domain
+-
+91.50
+92.63
+84.72
+93.64
+94.62
+95.05
+Table 1:
+SMATCH for T5, ACE, and collabora-
+tive/compostional inference. # refers to the number of
+sentences in the dataset. ACE* refers to the evaluation
+results only for valid parse. Collab. refers to collabo-
+rative inference from Lin et al. (2022). Vote refers to
+voting strategy from Hoang et al. (2021). The bold and
+underlined refer to the best and the second best results.
+and ensemble graph parsing literature, respectively.
+(see Appendix 6 for a review) (1) Lin et al. (2022)
+is uncertainty-aware neural-symbolic framework
+method attained state-of-the-art performance on the
+in-domain DeepBank test set, and (2) Hoang et al.
+(2021), a majority-voting-based graph ensemble
+method that uses a voting strategy based on beam
+sequences from the T5 model and predictions from
+the ACE parser 3. It doesn’t exploit uncertainty.
+5.2
+Results
+The results are shown in Table 1. Detailed in-
+domain comparision with other previous work is in
+Appendix I. As shown, among the base models, the
+T5 and ACE parser achieve similar overall perfor-
+mance, with T5 strongly outperforms on in-domain
+data but underperforms on the OOD data (see last
+row in Table 1). Our approach achieves best re-
+3We have tried several other variants for the voting can-
+didates, e.g., top K predictions from the T5 parser and top 1
+prediction + ACE prediction. It turns out the best one is using
+top K predictions from the T5 parser and ACE predictions.
+
+Essay
+E-commerce
+Verbmobil
+Type
+#
+ACE
+T5
+Collab.
+Ours
+#
+ACE
+T5
+Collab.
+Ours
+#
+ACE
+T5
+Collab.
+Ours
+Compound
+671
+83.76
+73.39
+76.75
+80.26
+844
+95.50
+67.96
+83.22
+94.94
+308
+86.36
+67.41
+68.13
+87.50*
+Nominal w/ nominalization
+15
+80.00
+80.00
+73.33
+80.00
+6
+100.00
+77.78
+100.00
+100.00
+-
+-
+-
+-
+-
+Nominal w/ noun
+521
+88.68
+76.84
+80.79
+84.56
+682
+95.60
+72.67
+86.93
+95.45
+194
+95.88
+77.80
+83.50
+95.15
+Verbal
+18
+72.22
+57.89
+73.68*
+78.95*
+-
+-
+-
+-
+-
+-
+-
+-
+-
+-
+Named entity
+74
+67.57
+71.05*
+68.42*
+60.53
+28
+92.86
+77.49
+80.00
+93.33*
+80
+62.50
+56.51
+52.50
+67.50*
+Argument structure
+3,314
+87.09
+82.63
+85.52
+85.26
+5,932
+95.79
+83.60
+88.47
+94.73
+4,206
+95.29
+77.52
+86.57
+94.56
+Total verb
+1,616
+83.66
+81.11
+83.78*
+82.56
+4,504
+95.12
+83.48
+87.36
+93.90
+2,330
+95.19
+82.25
+89.36
+94.35
+Basic verb
+895
+83.35
+82.01
+84.71*
+83.50*
+2,910
+94.85
+87.20
+90.14
+92.77
+1,206
+94.36
+89.15
+91.48
+94.48*
+ARG1
+694
+88.90
+88.26
+90.61*
+88.62
+2,494
+96.79
+95.64
+97.08*
+97.31*
+1,168
+96.75
+95.40
+95.27
+96.90*
+ARG2
+708
+88.28
+86.69
+89.04*
+88.77*
+2,660
+97.14
+91.11
+93.36
+97.20*
+876
+95.89
+89.34
+93.91
+95.65
+ARG3
+69
+83.61
+78.57
+78.57
+80.14
+382
+90.05
+70.91
+75.13
+78.07
+62
+93.55
+67.56
+87.50
+96.88*
+Verb-particle
+721
+84.05
+79.99
+65.15
+81.41
+1,592
+95.61
+76.94
+82.31
+95.95*
+1,124
+96.09
+74.14
+87.07
+94.22
+ARG1
+620
+87.90
+84.39
+86.53
+85.58
+1,448
+96.27
+80.77
+84.73
+96.62*
+1,096
+96.53
+80.20
+90.77
+96.90*
+ARG2
+498
+86.14
+84.96
+86.52*
+88.77*
+888
+96.85
+71.30
+81.33
+95.56
+424
+94.34
+66.73
+78.90
+92.66
+ARG3
+62
+79.03
+65.15
+65.15
+74.24
+208
+93.27
+69.05
+83.02
+96.23*
+24
+83.33
+47.17
+58.33
+58.33
+Total noun
+189
+91.53
+82.90
+86.01
+86.49
+90
+100.00
+76.81
+78.26
+97.83
+26
+92.31
+69.00
+93.33*
+93.33*
+Total adjective
+1,336
+90.64
+84.36
+87.13
+88.39
+1,116
+97.67
+84.62
+93.07
+97.34
+1,838
+95.43
+72.54
+82.75
+94.81
+Reentrancy
+850
+80.59
+78.39
+81.26*
+77.01
+1,686
+95.73
+75.83
+81.59
+84.76
+800
+93.25
+60.23
+72.77
+89.20
+passive
+173
+86.71
+83.33
+88.89*
+86.71
+222
+98.20
+85.56
+92.11
+97.37
+12
+100.00
+79.10
+100.00
+100.00
+Table 2: Comparing ACE, Collab. (Lin et al., 2022) and our parsers on fine-grained linguistic categories. All
+scores are reported in accuracy. The gray colored row means long-tail phenomenon (< 500 cases in the training
+set). The bold indicates the best results among neural approaches (T5, Collab. and Ours). * indicates the result is
+better than ACE parser.
+sults on overall performance, which is ∼ 35% er-
+ror reduction in aggregated SMATCH score over the
+T5-based and symbolic approaches.
+We now compare with the previous state-of-the-
+art methods. Though in-domain performance is not
+the focus of this work, our approach is still compa-
+rable to Collab, i.e., the neural-symbolic method
+from Lin et al. (2022). However, on the challenging
+out-of-domain eval sets (e.g., E-commerce, Verb-
+mobil whose topic and style are significantly differ-
+ent from WSJ), the performance of Collab starts
+to deteriorate. In comparison, our neural-symbolic
+approach remains robust out-of-domain. Its perfor-
+mance stays competitive with and even sometimes
+outperforms the ACE parser on difficult domains,
+illustrating the advantage of compositionality.
+We also notice that the voting-based ensemble
+method Vote (Hoang et al., 2021) performs poorly
+in the neural-symbolic setting, despite based a mod-
+erate number of beam sequences. This is likely
+because the majority-voting approach requires a
+large number of diverse predictions from distinct
+models. When there are only two models, the abil-
+ity of quantifying uncertainty becomes important.
+5.3
+Fine-grained Linguistic Evaluation
+ERG provides different levels of linguistic informa-
+tion that can benefit many NLP tasks, e.g., named
+entity recognition and semantic role labeling. This
+rich linguistic annotation provides an oppurtunity
+to evaluate model performance in meaningful pop-
+ulation subgroups. Detailed description of those
+linguistic phenomena is in Appendix J.
+Result is in Table 2. As shown, on OOD datasets,
+the T5 parser underperforms the ACE parser on
+most of the linguistic categories. Our approach
+outperforms both the neural model and the non-
+compositional neural-symbolic method especially
+on long-tail categories (the gray colored rows in the
+table), attaining an > 14% average absolute gain
+compared to the base model. In some categories,
+our method even outperforms the ACE parser while
+all base model underperforms, e.g., ARG3 of basic
+verb on Verbmobil and ARG3 of verb-particle on
+E-commerce.
+5.4
+Case Study: Synthesizing Novel Graphs
+To test if our methods can generate optimal graph
+solution which the base models fail to obtain, we
+further explore the percentage of novel graphs
+(graphs that are not identical to any of the can-
+didate predictions of the neural or symbolic model)
+for each dataset, and compare the corresponding
+SMATCH scores on those novel cases. The results
+are shown in Table 3. We see that our method syn-
+thesize novel graph parses that are in general of
+higher quality than that of the base models, thanks
+to the calibrated uncertainty (Section 4.2). This
+indicates the compositional neural-symbolic infer-
+ence can synthesize evidence across neural and
+symbolic results and produce novel graphs that are
+closer to ground truth.
+
+%
+Top 1
+Top 2
+Top 3
+Top 4
+Top 5
+Collab.
+ACE
+Ours
+In-domain
+31.25
+94.95
+93.01
+91.91
+89.92
+89.58
+95.10
+82.80
+98.44
+Wiki
+32.29
+87.55
+86.54
+85.56
+86.00
+83.90
+88.77
+82.67
+92.24
+Brown
+46.84
+90.54
+89.34
+88.57
+88.10
+87.11
+92.53
+96.15
+96.56
+Essay
+50.93
+90.71
+90.02
+89.31
+89.02
+87.60
+92.41
+95.73
+96.08
+E-commerce
+34.65
+90.03
+88.34
+86.61
+85.56
+82.91
+92.82
+98.96
+97.54
+Verbmobil
+39.96
+85.45
+83.06
+81.54
+79.30
+78.27
+88.42
+97.78
+96.70
+LOGON
+58.10
+90.75
+89.65
+88.20
+87.90
+86.95
+92.50
+96.70
+97.06
+Tanaka
+24.89
+89.35
+87.46
+85.60
+83.55
+83.16
+92.30
+98.23
+98.27
+All
+38.76
+90.57
+89.18
+88.01
+87.24
+86.13
+92.29
+93.93
+96.28
+Table 3: SMATCH performance on novel graphs, where the results of our inference process are not identical to any
+of the candidates from the base model.
+6
+Related Work
+In this section we introduce related work for neural-
+symbolic and ensemble learning for graph semantic
+parsing. For a broader context of graph semantic
+parsing, please refer to Appendix B.
+Neural-Symbolic
+Graph
+Semantic
+Parsing.
+Though neural models excel at semantic parsing,
+they have been shown to struggle with out-of-
+distribution compositional generalization, while
+grammar or rule-based approaches work relatively
+robustly. This has motivated the work in neural-
+symbolic parsing where symbolic approaches are
+imported as inductive bias (Shaw et al., 2021; Kim,
+2021; Cheng et al., 2019; Cole et al., 2021). For
+graph meaning representations, importing induc-
+tive bias into neural model was somehow difficult
+due to the much more complicated structure com-
+pared to pure syntactic rules or logical formalism
+(Peng et al., 2015; Peng and Gildea, 2016). To
+address this, Lin et al. (2022) proposes a collabora-
+tive framework by designing a decision criterion for
+beam search that incorporates the prior knowledge
+from a symbolic parser and accounts for model un-
+certainty, which achieves the state-of-the-art results
+on the in-domain test set.
+Ensemble Learning for Graph Parsing. Ensem-
+ble learning is a popular machine learning approach
+that combines predictions from multiple candidates
+to create a new one that is more robust and ac-
+curate than individual predictions. Previous stud-
+ies have explored various ensemble learning ap-
+proaches for graph parsing (Green and Žabokrtský,
+2012; Barzdins and Gosko, 2016). Specifically, for
+graph semantic parsing at subgraph level, Hoang
+et al. (2021) make use of checkpoints from models
+of different architectures, and mining the largest
+graph that is the most supported by a collection of
+graph predictions. They then propose a heuristic
+algorithm to approximate the optimal solution.
+Compare to the previous ensemble work, our
+work differ in three ways: (1) Our decision rule is
+based on neural model confidence, so the decision
+is driven not by model consensus, but by model
+confidence which indicates when the main (neural)
+result is untrustworthy and needs to be comple-
+mented by symbolic result. Model consensus is
+effective when there exists a large number of candi-
+date models. However, in the neural-symbolic set-
+ting when there are only two models, the ability of
+quantifying model uncertainty becomes important.
+(2) A secondary contribution of our work is to pro-
+duce an parsing approach for the ERG community
+that not only exhibits strong average-case perfor-
+mance on in-domain and OOD environments, but
+also generalizes robustly in important categories of
+tail linguistic phenomena. Therefore, our investi-
+gation goes beyond average-case performance and
+evaluates in tail generalization as well. (3) We re-
+veal a more nuance picture of neural models’ OOD
+performance: a neural model’s top K parses in fact
+often contains subgraphs that generalize well to
+OOD scenarios, but the vanilla MLE-based infer-
+ence fails to select them (see Section 5.4 for more
+details).
+7
+Conclusions
+We have shown how to perform accurate and ro-
+bust semantic parsing across a diverse range of gen-
+res and linguistic categories for English Resource
+Grammar. We achieve this by taking the advan-
+tage of both the symbolic parser (ACE) and the
+neural parser (T5) at a fine-grained subgraph level
+using compositional uncertainty, an aspect miss-
+ing in the previous neural-symbolic or ensemble
+parsing work. Our approach attains the best known
+result on the aggregated SMATCH score across
+eight evaluation corpus from Redwoods Treebank,
+attaining 35.26% and 35.60% error reduction over
+
+the neural and symbolic parser, respectively.
+Acknowledgement
+Our work is sponsored in part by National Sci-
+ence Foundation Convergence Accelerator under
+award OIA-2040727 as well as generous gifts
+from Google, Adobe, and Teradata. Any opinions,
+findings, and conclusions or recommendations ex-
+pressed herein are those of the authors and should
+not be interpreted as necessarily representing the
+views, either expressed or implied, of the U.S. Gov-
+ernment. The U.S. Government is authorized to
+reproduce and distribute reprints for government
+purposes not withstanding any copyright annota-
+tion hereon. We thank Du Phan, Panupong Pasupat,
+Jie Ren, Balaji Lakshminarayanan and Deepak Ra-
+machandran for helpful discussion.
+Limitation
+Here we discuss a potential limitations of the cur-
+rent study:
+Problem domain
+In this work, we have selected
+English Resource Grammar as the target formalism.
+This is a deliberate choice based on the availabil-
+ity of (1) realistic out-of-distribution evaluation
+corpus, and (2) well-established, high-quality sym-
+bolic parser. This is a common setting in indus-
+trial applications, where an practitioner is tempted
+to combine large pre-trained neural model with
+expert-developed symbolic rules to improve perfor-
+mance for a new domain. Unfortunately, we are not
+aware of another popular meaning representation
+for which both resources are available. To over-
+come this challenge, we may consider studying
+collaborative inference between a standard seq2seq
+model and some indirect symbolic supervision, e.g.,
+syntactic parser or CCG parser (Steedman, 2001),
+which is an interesting direction for future work.
+Uncertainty estimation techniques
+The vanilla
+seq2seq model is known to under-estimate the true
+probability of the high-likelihood output sequences,
+wasting a considerable amount of probability mass
+towards the space of improbable outputs (Ott et al.,
+2018; LeBrun et al., 2022). This systematic un-
+derestimation of neural likelihood may lead to a
+conservative neural-symbolic procedure that im-
+plicitly favors the information from the symbolic
+prior. It may also negatively impact calibration
+quality, leading the model to under-detect wrong
+predictions. To this end, it is interesting to ask
+if a more advanced seq2seq uncertainty method
+(e.g., Monte Carlo dropout or Gaussian process
+(Gal and Ghahramani, 2016; Liu et al., 2020)) can
+provide systematically better uncertainty quantifi-
+cation, and consequently improved downstream
+performance.
+Graphical model specification
+The GAP model
+presented
+in
+this
+work
+considers
+a
+classi-
+cal
+graphical
+model
+likelihood
+p(G|x)
+=
+�
+v∈G p(v| pa(v), x) , which leads to a clean fac-
+torization between graph elements v and fast prob-
+ability computation. However, it also assumes a
+local Markov property that v is conditional inde-
+pendent to its ancestors given the parent pa(v). In
+theory, the probability learned by a seq2seq model
+is capable of modeling higher order conditionals
+between arbitrary elements on the graph. Therefore
+it is interesting to ask if a more sophisticated graph-
+ical model with higher-order dependency structure
+can lead to better performance in practice while
+maintaining reasonable computational complexity.
+Understanding different types of uncertainty
+There exists many different types of uncertainties
+occur in a machine learning system (Hüllermeier
+and Waegeman, 2021). This includes data uncer-
+tainty (e.g., erroneously annotated training labels,
+ill-formedness of the input sentence, or inherent
+ambiguity in the example-to-label mapping), and
+also model uncertainty which occurs the test ex-
+ample not containing familiar patterns the model
+learned from the training data. In this work, we
+quantifies uncertainty using mean log likelihood,
+which broadly captures both types of uncertainty
+and does not make a distinction between these dif-
+ferent subtypes. As different source of uncertainty
+may lead to different strategy in neural-symbolic
+parsing, the future work should look into more
+fine-grained uncertainty signal that can decompose
+these different sources of error and uncertainty, and
+propose adaptive strategy to handle different sce-
+narios.
+Ethical Consideration
+This paper focused on neural-symbolic semantic
+parsing for the English Resource Grammar (ERG).
+Our architecture are built based on open-source
+models and datasets (all available online). We do
+not anticipate any major ethical concerns.
+
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+A
+Graph-based Meaning Representation
+Considerable NLP research has been devoted to
+the transformation of natural language utterances
+into a desired linguistically motivated semantic rep-
+resentation. Such a representation can be under-
+stood as a class of discrete structures that describe
+lexical, syntactic, semantic, pragmatic, as well as
+many other aspects of the phenomenon of human
+language. In this domain, graph-based representa-
+tions provide a light-weight yet effective way to en-
+code rich semantic information of natural language
+sentences and have been receiving heightened at-
+tention in recent years. Popular frameworks un-
+der this umbrella includes Bi-lexical Semantic De-
+pendency Graphs (SDG; Bos et al., 2004; Ivanova
+et al., 2012; Oepen et al., 2015), Abstract Mean-
+ing Representation (AMR; Banarescu et al., 2013),
+Graph-based Representations for English Resource
+Grammar (ERG; Oepen and Lønning, 2006; Copes-
+take, 2009), and Universal Conceptual Cognitive
+Annotation (UCCA; Abend and Rappoport, 2013).
+B
+Literature Review on Graph Semantic
+Parsing
+In this section, we present a summary of different
+parsing technologies for graph-based meaning rep-
+resentations in addition to the ones discussed in
+2.2, with a focus on English Resource Grammar
+(ERG).
+Grammar-based approach
+In this type of ap-
+proach, a semantic graph is derived according to
+a set of lexical and syntactico-semantic rules. For
+ERG parsing, sentences are parsed to HPSG deriva-
+tions consistent with ERG. The nodes in the deriva-
+tion trees are feature structures, from which MRS
+is extracted through unification. The parser has a
+default parse ranking procedure trained on a tree-
+bank, where maximum entropy models are used
+to score the derivations in order to find the most
+likely parse. However, this approach fails to parse
+sentences for which no valid derivation is found
+(Toutanova et al., 2005). There are two main exist-
+ing grammar-based parsers for ERG parsing: the
+PET system (Callmeier, 2000) and the ACE system
+(Crysmann and Packard, 2012). The core algo-
+rithms implemented by both systems are the same,
+but ACE is faster in certain common configurations.
+We choose ACE as the symbolic parser in our work.
+Factorization-based approach
+This type of ap-
+proach is inspired by graph-based dependency tree
+parsing (McDonald, 2006). A factorization-based
+parser explicitly models the target semantic struc-
+tures by defining a score function that can eval-
+uate the probability of any candidate graph. For
+ERG parsing, Cao et al. (2021) implemented a two-
+step pipeline architecture that identifies the concept
+nodes and dependencies by solving two optimiza-
+tion problems, where prediction of the first step is
+utilized as the input for the second step. Chen et al.
+(2019) presented a four-stage pipeline to incremen-
+tally construct an ERG graph, whose core idea is
+
+similar to previous work.
+Transition-based approach
+In these parsing
+systems, the meaning representations graph is gen-
+erated via a series of actions, in a process that is
+very similar to dependency tree parsing (Yamada
+and Matsumoto, 2003; Nivre, 2008), with the dif-
+ference being that the actions for graph parsing
+need to allow reentrancies. For ERG parsing, Buys
+and Blunsom (2017) proposed a neural encoder-
+decoder transition-based parser, which uses stack-
+based embedding features to predict graphs jointly
+with unlexicalized predicates and their token align-
+ments.
+Composition-based approach
+Following a prin-
+ciple of compositionality, a semantic graph can
+be viewed as the result of a derivation process, in
+which a set of lexical and syntactico-semantic rules
+are iteratively applied and evaluated. For ERG pars-
+ing, based on Chen et al. (2018), Chen et al. (2019)
+proposed a composition-based parser whose core
+engine is a graph rewriting system that explicitly
+explores the syntactico-semantic recursive deriva-
+tions that are governed by a synchronous SHRG.
+Translation-based approach
+This type of ap-
+proach is inspired by the success of seq2seq mod-
+els which are the heart of modern Neural Machine
+Translation. A translation-based parser encodes
+and views a target semantic graph as a string from
+another language. In a broader context of graph
+semantic parsing, simply applying seq2seq models
+is not successful, in part because effective lineariza-
+tion (encoding graphs as linear sequences) and data
+sparsity were thought to pose significant challenges
+(Konstas et al., 2017). Alternatively, some specifi-
+cally designed preprocessing procedures for vocab-
+ulary and entities can help to address these issues
+(Konstas et al., 2017; Peng et al., 2017). These pre-
+processing procedures are very specific to a certain
+type of meaning representation and are difficult
+to transfer to others. To address this, Lin et al.
+(2022) propose a variable-free top-down lineariza-
+tion and a compositionality-aware tokenization for
+ERG graph preprocessing, and successfully trans-
+fer the ERG parsing into a translation problem that
+can be solved by a state-of-the-art seq2seq model
+T5 (Raffel et al., 2020). The parser achieves the
+best known results on the in-domain test set from
+the DeepBank benchmark.
+C
+Additional Methods Discussions
+C.1
+Efficient Probability Estimation Using
+Beam Outputs
+The marginalized probability ˆp(si|x) provides a
+way to reason about the global importance of si by
+integrating the probabilistic evidence p(si|sk,<i, x)
+over the whole beam-sampled posterior space. It is
+able to capture the cases of spurious graph elements
+si with high local probability p(si|sk,<i, x) but low
+global likelihood (i.e., only appear in a few low-
+probability beam candidates), which is useful for
+inferring sparse global structures for the meta graph
+(Appendix E).
+In the importance weight πk, the temperature pa-
+rameter t controls how evidence for p(si|x) is ag-
+gregated across beam samples {gk}K
+k=1. When t →
+0, the above is equivalent to selecting p(si|sk,<i, x)
+from the most probable subsequence sk,<i; when
+t → ∞, the above is equivalent to simple averaging
+of p(si|sk,<i, x) from all beam candidates. In the
+experiments, we find that the value of t does not
+have a significant impact on the final performance.
+In general, we recommend fixing it to a small value
+(e.g., t = 0.1) to suitably downweighting the con-
+tribution from improbable beam candidates.
+D
+Simplified Expression for Graphical
+Model Likelihood
+Given the candidates graphs {Gk}K
+k=1, we can
+express the likelihood for p(v| pa(v), x) by writing
+down a multinomial likelihood enumerating over
+different values of pa(v) (Murphy, 2012). For
+example, say pa(n) = (e1, e2) which represents
+a subgraph of two edges (e1, e2) pointing into
+a node n.
+Then the conditional probability
+p(n| pa(n), x) can be computed by enumer-
+ating over the observed values of (e1, e2) pair:
+p(n| pa(n), x) = p(n|(e1, e2), x)
+∝
+�
+c∈Candidate(e1,e2)
+p(n|(e1, e2) = c, x)Kc
+where Candidate(e) is the collection of possi-
+ble symbols s the variable e can take, and
+Kc is the number of times (e1, e2) takes a
+particular value c
+∈
+Candidate(e1, e2)
+=
+Candidate(e1) × Candidate(e2).
+
+Then, the log likelihood becomes:
+log p(n| pa(n), x)
+=
+�
+c
+Kc ∗ log p(n|(e1, e2) = c)
+To simplify this above expression, we notice that
+log p(n| pa(n), x) can be divided by the constant
+beam size K without impacting the inference. As
+a result, the log probability can be computed by
+simplify averaging the values of log p(v| pa(v) =
+ck) across the beam candidates:
+log p(n| pa(n), x)
+∝
+�
+c
+Kc
+K log p(n|(e1, e2) = c)
+= 1
+K
+K
+�
+k=1
+log p(n|(e1, e2) = ck)
+where ck is the value of (e1, e2) in kth beam candi-
+date.
+E
+Extensions and Practical
+Implementation
+E.1
+Infer Sparse Global Structure via
+Likelihood-based Pruning
+In practice, the meta graphG can contain spuri-
+ous elements v that have a high local likelihoods
+log p(v| pa(v), x) but very low global probabili-
+ties p(v|x).
+This happens when the element v
+only appears in a few low-probability beam se-
+quences. These spurious nodes and edges often
+adds redundancy to the generated graph (i.e., hurt-
+ing precision), and cannot be eliminated by the
+neural-symbolic inference procedure, due to their
+high local conditional probability p(v| pa(v), x).
+Consequently, we find it empirically effective to
+perform sparse structure inference forG based on
+global probabilities p(v|x) before diving into local
+neural-symbolic prediction for graph components.
+In this work, we carry out this global structure
+inference by considering a simple threshold-and-
+project procedure, i.e., pruning out all the graph
+elements whose global probability ||p(v|x)||∞ =
+maxs∈Candidate(v) p(v = s|x) is lower than a thresh-
+old t, but will keep v if its removal will lead to an
+invalid graph with disconnected subcomponents.
+Here ||p(v|x)||∞ is the total variation metric that
+returns the maximum probability.
+Algorithm 3 summarizes this procedure. From a
+theoretical perspective, this is equivalent to finding
+the most sparse solution with respect to threshold t
+within the space of valid (i.e., connected) subgraphs
+ofG.
+Algorithm 3 Likelihood-based Pruning
+Inputs:
+Meta Graph GM
+Marginal probabilities {p(s|x)}s∈G
+Threshold t
+Output:
+Pruned graph G′
+M
+Initialize:
+G′
+M = GM
+for v ∈ G′
+M do
+if ||p(v|x)||∞ < t AND G′
+M \ {v} is connected then
+Prune v : G′
+M = G′
+M \ {v}
+return G′
+M
+ARG
+unknown
+_abstract_n_1
+BV-of
+udef_q
+g1
+ARG
+unknown
+named "Abstract"
+BV-of
+proper_q
+g2
+ARG1-of
+unknown
+_abstract_a_1
+g3
+ARG1
+_abstract_a_1
+unknown
+g4
+ARG1
+_abstract_v_1
+pron
+g5
+BV-of
+pronoun_q
+Figure 4: Autoregressive Representation (i.e., beam se-
+quences) for the sentence “Abstract” from the Eric Ray-
+mond Essay dataset. Note that g3 and g4 are actually
+the same graph but with different linearization orders.
+E.2
+Handle Multi-modality via Mixture
+Modeling
+In some rare cases where the input sentence is frag-
+mented or ill-formed, the neural model may output
+multiple beam sequences with drastically differ-
+ent high-level structures, creating difficulty for the
+graph merging procedure (See Figure 4 for an ex-
+ample).
+We can handle this multi-modality in observed
+graph structure by extending p(G|x) to be a mix-
+ture of GAP distributions, so that the graphical
+model likelihood becomes:
+p(G|x) =
+�
+m∈M
+p(G|m, x)p(m|x)
+where p(m|x) is a categorical distribution over
+the mixture components m ∈ M.
+Here each
+component m induce a meta graph Gm for graph
+Gm = ⟨Nm, Em⟩, such that
+p(G|m, x) = p(Gm|x) =
+�
+v∈Gm
+p(v| pa(v), x)
+=
+�
+n∈Nm
+p(n| pa(n), x) ∗
+�
+e∈Em
+p(e| pa(e), x)
+
+Given beam sequences {gk}K
+k=1, the mixture com-
+ponents can be estimated using a standard cluster-
+ing algorithm based on an edit distance between
+beam candidate gk. Based on our experiments, hier-
+archical agglomerative clustering (HAC) combined
+with the longest common subsequence (LCS) dis-
+tance often leads to the best result. After clustering,
+p(m|x) is computed as the empirical probability of
+beam sequences belonging the mth cluster, and the
+meta graph Gm is computed by applying the graph
+merging procedure to the beam sequences in the
+mth cluster.
+To conduct neural symbolic inference, we also
+need to define the symbolic prior p0 for the mixture
+distribution:
+p0(G) =
+�
+m∈M
+p0(G|m) ∗ p0(m)
+=
+�
+m∈M
+[
+�
+v∈Gm
+p0(v) ∗ p0(m)]
+where p0(v
+=
+s)
+∝
+exp(I(s
+∈
+G0)) as
+define previously, and we define p0(m)
+=
+exp(−SMATCH(Gm, G0)) following the previous
+work (Lin et al., 2022).
+As a result, the decision criteria for neural-
+symbolic inference under the mixture model be-
+comes:
+R(Gm|x) = R(m|x) +
+�
+v∈Gm
+R(v|x)
+where �
+v∈Gm R(v|x) is the component-wise de-
+cision criteria as defined in the main text, and
+R(m|x) is the additional term for the mixture com-
+ponents:
+R(m|x) = α(m|x) ∗ log p(m|x)
++ (1 − α(m|x)) ∗ log p0(m)
+where α(m|x) = σ(− 1
+T H(m|x) + b) is the trade-
+off parameter driven by the average log likelihood
+of beam sequences in the mth cluster Cm, i.e.,
+H(m|x) =
+1
+|Cm|
+�
+gk∈Cm − log(gk|x).
+During inference, we can again proceed in a
+greedy fashion, first select the optimal ˆm based on
+R(m|x), and then perform compositional neural-
+symbolic inference with respect to G ˆm using
+�
+v∈G ˆ
+m R(v|x).
+As a result, the complete precedure with all op-
+tional extensions are shown in Algorithm 4.
+Algorithm 4 Complete Procedure with All Extensions
+Inputs:
+Beam candidates and associated token-level
+probabilities {p(gk|x)}K
+k=1
+Output:
+Neural-symbolic graph prediction G
+(Optional) Estimate:
+Mixture components {Gm}M
+m=1, {p(m|x)M
+m=1}
+from Cluster {p(gk|x)}K
+k=1
+Optimal mixture components G = G ˆ
+m, where
+ˆm = arg max R(m|x)
+Estimate:
+Marginal probability and graphical model likelihood
+(Algorithm 2):
+{p(v|x)}v∈G, log p(G|x) = GAP(G)
+Infer:
+Global graph structure via likelihood-based pruning
+(Algorithm 3)
+G′ = ThresholdAndProject(G, {p(v|x)}v∈G)
+Local node / edge prediction via compostitional
+neural-symbolic inference (Algorithm 1)
+G = NeuralSymbolicInference(G’)
+F
+Graph Matching Algorithm
+In general, finding the largest common subgraph
+is a well-known computationally intractable prob-
+lem in graph theory. However, for graph parsing
+problems where graphs have labels and a simple
+tree-like structure, some efficient heuristics are
+proposed to approximate the best match by a hill-
+climbing algorithm (Cai and Knight, 2013). The
+initial match is modified iteratively to optimize the
+total number of matches with a predefined number
+of iterations (default value set to 5). This algorithm
+is very efficient and effective, it was also used to
+calculate the SMATCH score in Cai and Knight
+(2013).
+G
+Details for OOD Datasets
+Wikipedia (Wiki)
+The DeepBank team con-
+structed a treebank for 100 Wikipedia articles on
+Computational Linguistics and closely related top-
+ics. The treebank of 11,558 sentences comprises 16
+sets of articles. The corpus contains mostly declar-
+ative, relatively long sentences, along with some
+fragments.
+The Brown Corpus (Brown)
+The Brown Cor-
+pus was a carefully compiled selection of current
+American English, totalling about a million words
+drawn from a wide variety of sources.
+
+The Eric Raymond Essay (Essay)
+The tree-
+bank is based on translations of the essay “The
+Cathedral and the Bazaar” by Eric Raymond. The
+average length and the linguistic complexity of
+these sentences is markedly higher than the other
+treebanked corpora.
+E-commerce
+While the ERG was being used in
+a commercial software product developed by the
+YY Software Corporation for automated response
+to customer emails, a corpus of training and test
+data was constructed and made freely available,
+consisting of email messages composed by people
+pretending to be customers of a fictional consumer
+products online store. The messages in the corpus
+fall into four roughly equal-sized categories: Prod-
+uct Availability, Order Status, Order Cancellation,
+and Product Return.
+Meeting/hotel scheduling (Verbmobil)
+This
+dataset is a collection of transcriptions of spoken
+dialogues, each of which reflected a negotiation
+either to schedule a meeting, or to plan a hotel stay.
+One dialogue usually consists of 20-30 turns, with
+most of the utterances relatively short, including
+greetings and closings, and not surprisingly with
+a high frequency of time and date expressions as
+well as questions and sentence fragments.
+Norwegian tourism (LOGON)
+The Norwe-
+gian/English machine translation research project
+LOGON acquired for its development and evalu-
+ation corpus a set of tourism brochures originally
+written in Norwegian and then professionally trans-
+lated into English. The corpus consists almost en-
+tirely of declarative sentences and many sentence
+fragments, where the average number of tokens
+per item is higher than in the Verbmobil and E-
+commerce data.
+The Tanaka Corpus (Tanaka)
+This treebank
+is based on parallel Japanese-English sentences,
+which was adopted to be used with in the
+WWWJDIC dictionary server as a set of example
+sentences associated within words in the dictio-
+nary.
+H
+Implementation and
+Hyperparameters
+T5 Model
+We use the open-sourced T5X 4,
+which is a new and improved implementation of
+T5 codebase in JAX and Flax. Specifically, we
+4https://github.com/google-research/t5x
+use the official pretrained T5-Large (770 million
+parameters), which is the same size as the one used
+in Lin et al. (2022), and finetuned it on DeepBank
+in-domain training set. Specifically, the total train-
+ing step is 1,750,000 including 1,000,000 pretrain
+steps. For fine-tuning the T5 model on ERG pars-
+ing, batch size is set to 128, the output and input
+sequence length is set to 512, and dropout rate is
+set to 0.1.
+Hyperparameters
+For the trade-off parameter
+α(v|x) = σ(− 1
+T H(v|x) + b), we set temperature
+T = 0.1 and bias b = 0.25.
+I
+In-domain Evaluation
+Model
+Node
+Edge
+SMATCH
+ACE
+89.30
+85.05
+87.14
+ACE*
+93.18
+88.76
+90.94
+Buys and Blunsom (2017)
+89.06
+84.96
+87.00
+Chen et al. (2018)
+94.51
+87.29
+90.86
+Chen et al. (2019)
+95.63
+91.43
+93.56
+Chen et al. (2019)
+97.28
+94.03
+95.67
+Cao et al. (2021)
+96.42
+93.73
+95.05
+ACE-T5 (following Shaw et al. (2021))
+93.46
+89.19
+91.30
+T5-based (Lin et al., 2022)
+97.34
+95.80
+96.56
++ Hoang et al. (2021)
+88.89
+87.67
+88.22
++ Lin et al. (2022)
+97.64
+96.41
+97.01
++ Ours
+97.50
+96.07
+96.77
+Table 4: F1 score for node and edge predictions and the
+SMATCH scores on the in-domain test set. ACE* refers
+to evaluation results only for valid parse.
+Table 4 shows the in-domain performance,
+where we compare our parser with the grammar-
+based ACE parser and other data-driven parsers.
+The baseline models also include a similar practice
+with (Shaw et al., 2021) and (Hoang et al., 2021).
+The former one takes T5 as a backup for grammar-
+based parser (ACE), and the latter gets ensembled
+graph via a voting strategy based on the candidates
+from the T5 parser and ACE parser.
+From the table we can see that our methods out-
+performs the base model (T5-based) and most of
+the previous work. Specifically, we achieves a
+SMATCH score of 96.77, which is a 6.11% error
+reduction compared to the base T5 parser.
+J
+Fine-grained Linguistic Phenomena
+Lexical construction
+ERG uses the abstract
+node compound to denote compound words. The
+edge labeled with ARG1 refers to the root of the
+compound word, and thus can help to further dis-
+tinguish the type of the compound into (1) nominal
+
+0.6191
+0.8290
+0.9394
+0.9795
+0.9917
+0.9982
+0.9996
+0.9999
+1.0000
+Probabilities
+0.4
+0.5
+0.6
+0.7
+0.8
+0.9
+1.0
+Accuracies
+wsj
+T5
+ACE
+0.6406
+0.8290
+0.9248
+0.9709
+0.9893
+0.9965
+0.9993
+0.9999
+1.0000
+Probabilities
+0.4
+0.5
+0.6
+0.7
+0.8
+0.9
+1.0
+Accuracies
+wiki
+T5
+ACE
+0.6900
+0.8290
+0.9181
+0.9692
+0.9845
+0.9961
+0.9995
+0.9999
+1.0000
+Probabilities
+0.4
+0.5
+0.6
+0.7
+0.8
+0.9
+1.0
+Accuracies
+essay
+T5
+ACE
+0.1835
+0.6806
+0.8825
+0.9593
+0.9890
+0.9981
+0.9996
+0.9999
+1.0000
+Probabilities
+0.4
+0.5
+0.6
+0.7
+0.8
+0.9
+1.0
+Accuracies
+ecommerce
+T5
+ACE
+0.1779
+0.7490
+0.9105
+0.9664
+0.9840
+0.9934
+0.9982
+0.9997
+0.9999
+Probabilities
+0.4
+0.5
+0.6
+0.7
+0.8
+0.9
+1.0
+Accuracies
+verbmobil
+T5
+ACE
+0.6127
+0.8138
+0.9177
+0.9716
+0.9907
+0.9972
+0.9994
+0.9999
+1.0000
+Probabilities
+0.4
+0.5
+0.6
+0.7
+0.8
+0.9
+1.0
+Accuracies
+logon
+T5
+ACE
+Figure 5: Diagrams for the T5 model’s probabilities verses the T5 model’s and ACE parser’s accuracies at subgraph
+level on the other datasets. Each bin contains the same number of examples. Since at most of the subgraphs, the
+model is pretty certain (log P > −1e − 5), we exclude these pretty certain predictions in the figures.
+with normalization, e.g., “flag burning”; (2) nomi-
+nal with noun, e.g., “pilot union”; (3) verbal, e.g.,
+“state-owned”; (4) named entities, e.g., “West Ger-
+many”.
+Argument structure
+In ERG, there are differ-
+ent types of core predicates in argument structures,
+specifically, verbs, nouns and adjectives. We also
+categorize verb in to basic verb (e.g., _look_v_1)
+and verb particle constructions (e.g., _look_v_up).
+The verb particle construction is handled semanti-
+cally by having the verb contribute a relation par-
+ticular to the combination.
+Coreference
+ERG resolves sentence-level coref-
+erence, i.e., if the sentence referring to the same
+entity, the entity will be an argument for all the
+nodes that it is an argument of, e.g., in the sen-
+tence, “What we want to do is take a more aggres-
+sive stance”, the predicates “want” (_want_v_1)
+and “take” (_take_v_1) share the same agent “we”
+(pron). Coreference can be presented as reentran-
+cies in the ERG graph, we notice that one important
+type of reentrancies is the passive construction, so
+we also report evaluation on passive construction
+in Table 2.
+K
+Calibration Performance on Other
+Datasets
+The correlations between the subgraph’s probabil-
+ity and performance on other datasets are shown in
+Figure 5. The conclusions drew from the figure is
+similar to the one discussed in Section 3.
+
diff --git a/P9FJT4oBgHgl3EQfJCz4/content/tmp_files/load_file.txt b/P9FJT4oBgHgl3EQfJCz4/content/tmp_files/load_file.txt
new file mode 100644
index 0000000000000000000000000000000000000000..0a2507572e671413d0118797d52b28659716d191
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+++ b/P9FJT4oBgHgl3EQfJCz4/content/tmp_files/load_file.txt
@@ -0,0 +1,1433 @@
+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf,len=1432
+page_content='Neural-Symbolic Inference for Robust Autoregressive Graph Parsing via  Compositional Uncertainty Quantification  Zi Lin  UC San Diego  lzi@ucsd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content='edu  Jeremiah Liu†‡  Google Research & Harvard University  jereliu@google.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content='com  Jingbo Shang†  UC San Diego  jshang@ucsd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content='edu  Abstract  Pre-trained seq2seq models excel at graph se-  mantic parsing with rich annotated data, but  generalize worse to out-of-distribution (OOD)  and long-tail examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content=' In comparison, sym-  bolic parsers under-perform on population-  level metrics, but exhibit unique strength in  OOD and tail generalization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content='  In this work,  we study compositionality-aware approach to  neural-symbolic inference informed by model  confidence, performing fine-grained neural-  symbolic reasoning at subgraph level (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content=',  nodes and edges) and precisely targeting sub-  graph components with high uncertainty in  the neural parser.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content='  As a result, the method  combines the distinct strength of the neural  and symbolic approaches in capturing differ-  ent aspects of the graph prediction, leading to  well-rounded generalization performance both  across domains and in the tail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content='  We empiri-  cally investigate the approach in the English  Resource Grammar (ERG) parsing problem  on a diverse suite of standard in-domain and  seven OOD corpora.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content=' Our approach leads to  35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content='26% and 35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content='60% error reduction in aggre-  gated SMATCH score over neural and sym-  bolic approaches respectively, and 14% abso-  lute accuracy gain in key tail linguistic cate-  gories over the neural model, outperforming  prior state-of-art methods that do not account  for compositionality or uncertainty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content='  1  Introduction  A structured account of compositional meaning has  become a longstanding goal for Natural Language  Processing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content=' To this end, a number of efforts have  focused on encoding semantic relationships and at-  tributes into graph-based meaning representations  (MRs, see Appendix A for details).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content=' In particular,  graph semantic parsing has been an important task  in almost every Semantic Evaluation (SemEval)  exercise since 2014.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content=' In recent years, we have wit-  nessed the burgeoning of applying neural networks  † Co-senior authors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content=' ‡ Work done at Google.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content='  to semantic parsing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content=' Pre-trained language model-  based approaches have led to significant improve-  ments across different MRs (Oepen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content=', 2019,  2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content=' However, these models often generalize  poorly to out-of-distribution (OOD) and tail ex-  amples (Cheng et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content=', 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content=' Shaw et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content=', 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content='  Kim, 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content=' Lin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content=', 2022), while grammar or  rule-based parser work relatively robustly across  different linguistic phenomena and language do-  mains (Cao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content=', 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content=' Lin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content=', 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content=' See  Section 6 for a review of related work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content='  In this paper, we propose a novel compositional  neural-symbolic inference for graph semantic pars-  ing, which takes advantage of both uncertainty  quantification from a seq2seq parser and prior  knowledge from a symbolic parser at the subgraph  level (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content=', nodes and edges).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content=' We take graph seman-  tic parsing for English Resource Grammar (ERG)  as our case study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content=' ERG is a compositional semantic  representation explicitly coupled with the syntactic  structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content=' Compared to other graph-based meaning  representations like Abstract Meaning Representa-  tion (AMR), ERG has high coverage of English text  and strong transferability across domains, render-  ing itself as an attractive target formalism for auto-  mated semantic parsing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content=' Furthermore, many years  of ERG research has led to well-established sym-  bolic parser and a rich set of carefully constructed  corpus across different application domains and  fine-grained linguistic phenomena, making it an  ideal candidate for studying cross-domain general-  ization of neural-symbolic methods (Oepen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content=',  2002;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content=' Crysmann and Packard, 2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content='  We start with a novel investigation of the uncer-  tainty calibration behaviour of a T5-based state-of-  the-art neural ERG parser (Lin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content=', 2022) on the  subgraph level (Section 3), where we make some  key observations: (1) the performance of the neu-  ral parser degrades when it becomes uncertain at  the subgraph level, while (2) the symbolic parser  works still robustly when the neural parser is un-  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content='11459v1  [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content='CL]  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content='26 Jan 2023  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content='_the_q<0>  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content='_want_v_1<2>  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content='_boy_n_1<1>  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content='_believe_v_1<6>  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content='_girl_n_1<0>  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content='pron<7>  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content='pronoun_q  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content='_the_q<3>  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content='BV  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content='BV  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content='BV  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content='ARG1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content='ARG2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content='ARG1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content='ARG2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content='The<0> boy<1> wants<2> the<3> girl<4> to<5> believe<6> him<7>  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content='Abstract Concepts  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content='(grammatical function)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content='Root  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content='Token-Node Alignments  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content='Surface Concepts  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content='(related to surface tokens)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content='<·>  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content='(a) EDS Representation  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content='( _want_v_1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content=':ARG1 ( _boy_n_1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content=':BV-of ( _the_q ) )  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content=':AGR2 ( _believe_v_1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content=':ARG1 ( _girl_n_1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content=':BV-of ( _the_q ) )  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content=':ARG2 ( pron  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content=':BV-of ( pronoun_q ) ) ) )  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content='(b) Variable-free PENMAN notation  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content='Figure 1: The EDS representation for ERG and the corresponding linearization of the example sentence “The boy  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content='wants the girl to believe him”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content='  certain at the subgraph level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content=' This motivates us to  develop a compositional neural-symbolic inference  process where the neural and symbolic parser col-  laborates at a more fine-grained level and guided  by model uncertainty, which is an aspect missing in  the previous neural-symbolic and ensemble parsing  literature (see Appendix 6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content='  We then propose a decision-theoretic criteria to  allow for neural-symbolic inference at subgraph  level (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content=', nodes and edges) and incorporates the  neural parser’s fine-grained uncertainty for each  graph component prediction (Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content=' The  key to this approach is a meta graph GM that enu-  merates possible candidates for each node/edge  prediction, and is constructed by merging multiple  beam predictions from the neural seq2seq model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content='  The core challenge here is how to properly quan-  tify compositional uncertainty using a seq2seq  model, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content=', assigning model probability for a node  or edge prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content=' For example, our interest is to  express the conditional probability of a graph node  v with respect to its parent p(v|pa(v), x), rather  than the likelihood of v conditioning on the previ-  ous tokens in the linearized string.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content=' As a result, it  cannot be achieved by relying on the naive token-  level autoregressive probabilities from the beam  search.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content=' To address this issue, we introduce a sim-  ple probabilistic formalism termed Graph Autore-  gressive Process (GAP) (Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content=' GAP adopts  a dual representation of an autoregressive process  and a probabilistic graphical model, and can serve  as a powerful medium for expressing compositional  uncertainty for seq2seq graph parsing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content='  We demonstrate the effectiveness of our ap-  proach in experiments across a diverse suite of  eight in-domain and OOD evaluation datasets en-  compassing domains including Wikipedia entries,  news articles, email communications, etc (Section  5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content='  We achieve the best results on the overall  performance across the eight domains, attaining  35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content='26% and 35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content='60% error reduction in the aggre-  gated SMATCH score over the neural and symbolic  parser, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content=' Our approach also exhibits sig-  nificantly stronger robustness in generalization to  OOD datasets and long-tail linguistic phenomena  than previous work, while maintaining the state-  of-the-art performance on in-domain test.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content=' Further  study also shows that the compositionality aspects  of neural-symbolic inference helps the model to as-  semble novel graph solution that the original infer-  ence process (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content=', beam search or symbolic parse)  fails to provide (Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content='  In summary, our contributions are four-fold:  We present a novel investigation of neural graph  parser’s uncertainty calibration performance at  subgraph level (Section 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content=' Our study confirms  the seq2seq uncertainty is effective for detecting  model error even out-of-distribution, establishing  the first empirical basis for the utility of compo-  sitional uncertainty in seq2seq graph parsing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content='  We propose a practical and principled framework  for neural-symbolic graph parsing that utilizes  model uncertainty and exploits compositionality  (Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content=' The method is fully compatible  with modern large pre-trained seq2seq network  using beam decoding, and is general-purpose and  applicable to any graph semantic parsing task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content='  We propose a simple probabilistic formalism  (GAP) to express a seq2seq model’s composi-  tional uncertainty (Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content=' GAP allows  us to go beyond the conventional autoregres-  sive sequence probability and express long-range  parent-child conditional probability on the graph,  serving as a useful medium of compositional un-  certainty quantification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content='  We conduct a comprehensive study to evalu-  ate the state-of-the-art graph parsing approaches  across a diverse suite of in-domain and out-of-  distribution datasets (Section 5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content=' Our study re-  veals surprising weakness of previous neural-  symbolic methods in OOD generalization, and  confirms the proposed method significantly im-  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
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+page_content="0000  T5 Model's Probabilities  0." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
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+page_content='0  Node/Edge Accuracies  tanaka  T5  ACE  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
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+page_content='9998  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content="0000  T5 Model's Probabilities  0." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
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+page_content='7  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content='8  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content='9  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content='0  Node/Edge Accuracies  brown  T5  ACE  Figure 2: Bar charts for the predictive accuracies of the T5 parser (blue) and ACE parser (orange) for all the  node / edge prediction across different uncertainty buckets based on T5 model’s probabilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content=' The performance  is evaluated on the Tanaka and Brown datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content=' Each bin represents a quantile bucket of the model probability  (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content=', they contain the same number of examples).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content=' Since at most of the subgraphs, the model is pretty certain  (log P > −1e − 5), we exclude these pretty certain predictions in the figures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content='  proves models OOD and tail performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content='  Reproducibility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content='  Our  code  is  available  on  Github:  https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content='com/google/  uncertainty-baselines/tree/main/  baselines/t5/data/deepbank.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content='  2  Background  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content='1  English Resource Grammar (ERG)  In this work, we take the representations from En-  glish Resource Grammar (ERG;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content=' Flickinger et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content=',  2014) as our target meaning representations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content=' ERG  is a broad-coverage computational grammar of En-  glish that derives underspecified logical-form rep-  resentations of meaning (Oepen and Flickinger,  2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content=' It is rooted in the general linguistic theory  of Head-driven Phrase Structure Grammar (HPSG;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content='  Pollard and Sag, 1994).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content='  ERG can be presented into different types of an-  notation formalism (Copestake et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content=', 2005).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content=' This  work focuses on the Elementary Dependency Struc-  ture (EDS;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content=' Oepen and Lønning, 2006) which is  a compact representation that can be expressed  as a directed acyclic graph (DAG) and is widely  adopted in the neural parsing approaches (Buys and  Blunsom, 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content=' Chen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content=', 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content=' An example is  shown in Figure 1(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content='2  Parsing Approaches  In this section, we review the state-of-the-art sym-  bolic and neural parsers utilized in our work, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content=',  the ACE parser (Crysmann and Packard, 2012) and  the T5 parser (Lin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content=', 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content=' Appendix B re-  views other ERG parsing techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content='  The symbolic parser: ACE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content=' The ACE parser  (Crysmann and Packard, 2012) is one of the state-  of-the-art symbolic parsers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content=' It first decomposes  sentences into ERG-consistent candidate derivation  trees, and the parser will rank candidates based on  the structural features in the nodes of the deriva-  tion trees via maximum entropy models (Oepen  and Lønning, 2006;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content=' Toutanova et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content=', 2005).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content=' This  approach fails to parse sentences for which no valid  derivation is found.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content='  The neural parser: T5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content=' Lin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content=' (2022) pro-  posed a T5-based ERG parser which achieves the  best known results on the in-domain DeepBank  benchmark.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content=' It is the first work that successfully  transfers the ERG parsing problem into a pure end-  to-end translation problem via compositionality-  aware tokenization and a variable-free top-down  graph linearization based on the PENMAN nota-  tion (Kasper, 1989).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content=' Figure 1(b) shows an example  of the linearized graph string from the original EDS  graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content='  3  Motivation: Subgraph-level  Uncertainty in Seq2seq Graph Parsing  We hypothesize that when the neural seq2seq  model is uncertain at the subgraph level, it is more  likely to make mistakes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content=' Assuming the symbolic  parser performs more robustly in these situations,  we can then design a procedure to ask the symbolic  parser for help when the model is uncertain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content=' To  validate this hypothesis, we conduct experiments  to empirically explore the following two questions:  (1) how does the model perform when it is uncer-  tain at the subgraph level?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content=' and (2) how does the  symbolic parser perform when the model is uncer-  tain?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content='  First, we compute model probabilities for each  graph element (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content=', node and edge) prediction (see  Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content='2 for how to compute these quanitities),  and identify the corresponding ACE parser pre-  diction using the graph matching algorithm from  SMATCH (Cai and Knight, 2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content=' We then evaluate  the accuracies of those graph element predictions  with respect to the gold labels, and compare it to  that of the ACE parser.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content='  In Figure 2, we plot the bar charts compare  the neural and symbolic performance in different  bucket of seq2seq model uncertainties on the two  largest datasets (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content=', Tanaka and Brown, see Ap-  pendix G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content=' Results on other datasets can be found in  the Appendix K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content=' As shown in the figure, low model  probability generally corresponds to low T5 per-  formance, while the corresponding ACE parser’s  accuracies spread relatively stably (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content=', it attains  > 90% accuracy in the lowest-confidence buck-  ets, while T5 accuracy is < 50%).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content=' This implies  that when the model is uncertain, the accuracy of  the neural model tend to be low, while the ACE  parser still performs well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content=' This has motivated us  to develop a compositional neural-symbolic infer-  ence procedure guided the model’s subgraph level  uncertainty, such that the T5 and ACE parser can  collaborate at a more fine-grained level via com-  postional uncertainty quantification (Section 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content='  4  Methods  Notation & Problem Statement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content=' For graph se-  mantic parsing, the input is a natural language ut-  terance x, and the output is a directed acyclic graph  (DAG) G = ⟨N, E⟩, where N is the set of nodes  and E ∈ N × N is the set of edges (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content=', Figure  1(a)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content=' In the case of seq2seq parsing, G is repre-  sented as a linearized graph string g = s1s2 · · · sL  which consists of symbols {sl}L  l=1 (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content=', Figure  1(b)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content=' As the graph prediction is probabilistic, each  of the graph element v ∈ N ∪ E is a random vari-  able whose values are the symbols si observed  from the beam outputs, leading to marginal prob-  abilities p(v = si|x) and conditional probabilities  p(v = si|v′ = sj, x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content='  To this end, our goal is to produce a principled  inference procedure for graph prediction account-  ing for model uncertainty on predicting graph ele-  ments v ∈ G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content=' In the sequel, Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content='1 presents  a decision-theoretic criterion that leverages the  graphical model likelihood p(G|x) to conduct com-  positional neural-symbolic inference for graph pre-  diction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content=' To properly express the graphic model  likelihood p(G|x) = �  v∈G p(v|pa(v), x) using  a learned seq2seq model, Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content='2 introduces  a simple probabilistic formalism termed Graph  Autoregressive Process (GAP) to translate the au-  toregressive sequence probability from the seq2seq  model to graphical model probability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content=' Appendix E  discusses some additional extensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content='1  Compositional Neural-Symbolic  Inference  Previously, an uncertainty-aware decision criteria  was proposed for neural-symbolic inference based  on the Hurwicz pessimism-optimism criteria  R(G|x) (Lin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content=', 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content=' Specifically, the criteria  is written as:  R(G|x) = α(x)∗Rp(G|x)+(1−α(x))∗R0(G),  where R(G|x) = − log p(G|x) is the neural model  likelihood, R0(G) = log p0(G) is the symbolic  prior likelihood, and α(x) is a the uncertainty-  driven trade-off coefficient to balance between  the optimistic MLE criteria Rp(G|x) and the pes-  simistic, prior-centered criteria R0(G|x) centered  around symbolic prediction G0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content='  A key drawback of this approach is the lack of  accounting for the compositionality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content=' This motivates  us to consider synthesizing the multiple graph pre-  dictions {Gk}K  k=1 from the neural parser to form  a meta graph G 1, where we can leverage the dis-  entangled uncertainty of p(G|x) to perform fine-  grained neural-symbolic inference for each graph  component v ∈ G (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content=', nodes or edges).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content=' Specifi-  cally, we leverage the factorized graphical model  likelihood p(G|x) = �  v∈G p(v| pa(v), x) to de-  compose the overall decision criteria R(G|x) into  that of individual components R(v|x):  R(v|x) = α(v|x) ∗ log p(v| pa(v), x)  + (1 − α(v|x)) ∗ log p0(v),  (1)  and the overall criteria is written as R(G|x) =  �  v∈G R(v|x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content=' Here pa(v) refers to the parents  of v in G, and α(v|x) = sigmoid(− 1  T H(v|x) +  b) is the component-specific trade-off param-  eter driven by model uncertainty H(v|x)  =  − log p(v| pa(v), x), and (T, b) are scalar calibra-  tion hyperparameters that can be tuned on the dev  set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content='  Following previous work (Lin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content=', 2022), the  symbolic prior p0 for each graph component v is  defined as a Boltzmann distribution based on the  graph output G0 from the symbolic parser, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content=',  p0(v = s) ∝ exp(I(s ∈ G0)), so that it is pro-  portional to the empirical probability of whether  a symbol s appears in G0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content=' Notice that we have  1Given a group of candidate graphs {Gk}K  k=1, well-  established algorithm exists to synthesize different graph pre-  dictions into a meta graph G (Cai and Knight, 2013;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content=' Hoang  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content=', 2021) (see Appendix F for a more detailed review).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content='  ignored the normalizing constants since they do  not impact optimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content='  Algorithm 1 summarizes the full algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content='  As shown, during inference, the method pro-  ceeds  by  starting  from  the  root  node  v0  and  selects  the  optimal  prediction  ˆv0  =  arg maxc0∈Candidate(v0) R(c0|x), where c0 are dif-  ferent candidates for v0 given by the meta graph G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content='  The algorithm then recursively performs the same  neural-symbolic inference procedure for the chil-  dren of v0 (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content=', ch(v)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content=' The algorithm terminates  when the optimal candidates for all graph variables  v ∈ G are determined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content='  As a result, the algorithm is able to adap-  tively combine subgraph predictions across mul-  tiple beam candidates thanks to the meta graph G,  and appropriately weight between the local neural  and symbolic information thanks to the uncertainty-  aware decision criteria R(v|x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content=' Empirically, this  also gives the algorithm the ability to synthesize  novel graph predictions that are distinct from its  base models (Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content='  Algorithm 1 Compositional Neural-Symbolic Inference  Inputs:  Meta graph G  Graphical model likelihood log p(G|x)  Symbolic prior p0  Output:  Neural-symbolic graph prediction G  Initialize:  v = root(GM);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content=' G = GM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content='  if G does not contain undecided candidates then return G  else  for cv ∈ Candidate(v) do  Compute decision criteria R(cv|x) (Equation 1)  Select optimal candidate ˆv = arg maxc R(c|x)  Remove non-optimal candidates of v from G  Recursively perform Algorithm 1 for all v′ ∈ ch(v)  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content='2  Compositional Uncertainty  Quantification with Graph Autogressive  Process (GAP)  To properly model the uncertainty p(G|x) from a  seq2seq model, we need an intermediate probabilis-  tic representation to translate the raw token-level  probability to the distribution over graph elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content='  To this end, we introduce a simple probabilistic  formalism termed Graph Autoregressive Process  (GAP), which is a probability distribution assigning  seq2seq learned probability to the graph elements  v ∈ G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content=' Specifically, as the seq2seq-predicted graph  adopts both a sequence-based representation g =  s1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content=', sL and a graph representation G = ⟨N, E⟩,  the GAP model adopts both an autoregressive repre-  sentation p(g|x) = �  i p(si|s<i, x) (Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content='1),  and also a probabilistic graphical model repre-  sentation p(G|x) = �  v∈G p(v| pa(v), x) (Section  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content=' Both representations share the same set of  underlying probability measures (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content=', the graphical-  model likelihood p(G|x) can be derived from the  autoregressive probabilities p(si|s<i, x)) (Figure  3), rendering itself a useful medium for princi-  pled compositional neural-symbolic inference us-  ing seq2seq probabilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content='1  Autoregressive Representation for  Linearized Sequence g  Given an input sequence x and output sequence  y = y1y2 · · · yN, the token-level autoregressive  distribution from a seq2seq model is p(y|x) =  �N  i=1 p(yi|y<i, x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content=' In the context of graph pars-  ing, the output sequence describes a linearized  graph g = s1s2 · · · sL, where each symbol si =  {yi1yi2 · · · yiNi} represents either a node n ∈  N or an edge e ∈ E of the graph and corre-  sponds to a collection of beam-decoded tokens  {yi1yi2 · · · yiNi}, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content=', the node _the_q in Figure  1(a) is represented by tokens {_, the, _q}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content=' This  process is illustrated in follows:  s1  s2  s3  sL  g:  y:  y1 y2 y3  y4 y5  y6 y7 y8  yN-1 yN  …  …  To  this  end,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content='  the  Graph  Autoregressive  Process  (GAP)  assigns  probability  to  each  linearized  graph  g  =  s1s2 · · · sL  autore-  greesviely  as  p(g|x)  =  �L  i=1 p(si|s<i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content=' x),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content='  and the conditional probability p(si|s<i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content=' x) is  computed by aggregating the token probability:  p(si|s<i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content=' x) = p({yi1 · · · yiNi }|s<i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content=' x) =  Ni  �  j=1  p(yij|yi<j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content=' s<i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content=' x)  Marginal and Conditional Probability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content=' Impor-  tantly, GAP allows us to compute the marginal  and (non-local) conditional probabilities for  graph elements si.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content='  Given the input x, the  marginal probability of si  is computed as  p(si|x) =  �  s<i  p(si|s<i, x)p(s<i|x)ds<i  by integrating over the space of all possible sub-  sequences s<i prior to the symbol si.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content=' Then, the  (non-local) conditional probability between two  graph elements (si, sj) with i < j is computed as  p(sj|si, x) =  �  si→j,s<i  p(si, si→j|si, s<i, x)p(si|s<i, x)p(s<i|x)dsi→jds<i  by integrating over the space of subsequences si→j  between (si, sj) and the subsequence s<i before  si.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content=' Higher order conditional (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content=', p(sj|(si, sl), x))  can be computed analogously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content=' Notice this gives us  the ability to reason about long-range dependencies  between non-adjacent symbols on the sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content='  Furthermore, the conditional probability on the  reverse direction can also be computed using the  Bayes’ rule: p(si|sj, x) = p(sj|si,x)p(si|x)  p(sj|x)  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content='  Efficient Estimation Using Beam Outputs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content=' In  practice, we can estimate p(si|x) and p(sj|si, x)  efficiently via importance sampling using the  output from the beam decoding {gk}K  k=1, where  K is the beam size (Malinin and Gales, 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content='  The marginal probability can be computed as  ˆp(si|x) =  K  �  k=1  πkp(si|sk,<i, x)  (2)  where πk =  exp( 1  t log p(gk|x))  �K  k=1 exp( 1  t log p(gk|x)) is the impor-  tance weight proportional to the beam candidate  gk’s log likelihoods, and t > 0 is the temperature  parameter fixed to a small constant (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content=', t = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content='1,  see Appendix C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content='1 further discussion) (Malinin and  Gales, 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content=' If the symbol si does not appear in  the kth beam, we set p(si|sk,<i, x) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content='  Then, for two symbols (si, sj) with i < j, we  can estimate the joint probability as  ˆp(sj|si, x) =  K  �  k=1  πi  kp(sj|si, sk,i→j, sk,<i, x)  (3)  where πi  k =  exp( 1  t log p(gk|x))∗I(si∈gk)  �K  k=1 exp( 1  t log p(gk|x))∗I(si∈gk) is the  importance weight among beam candidates that  contains si.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content=' Notice this is different from Equation  2 where πk is computed over all beam candidates  regardless of whether it contains si.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content='2  Probabilistic Graphical Model  Representation for G  So far, we have focused on probability computa-  tion based on the graph’s linearized representation  p(g|x) = �  i p(si|s<i, x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content=' To conduct the compo-  sitional neural-symbolic inference (Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content='1),  we also need to consider GAP’s graphical model  representation p(G|x) = �  v∈G p(v| pa(v), x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content='  GAP’s graphical model representation G de-  pends on the meta graph G constructed from K  candidate graphs {Gk}K  k=1 (Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content=' Figure 3  shows an example, where ni and ej are the candi-  dates for the node and edge predictions collected  from beam sequences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content=' Compared to the sequence-  based representation g, G provides two advantages:  it (1) explicitly enumerates different candidates for  each node and edge prediction (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content=', n2 v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content=' n3 for  predicting the third element), and (2) provides an  explicit account of the parent-child relationships  between variables on the graph (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content=', e2 is a child  node of n1 in the predicted graph, which is not re-  flected in the autoregressive representation).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content=' From  the probabilistic learning perspective, G describes  the space of possible graphs (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content=', the support) for  a graph distribution p(G|x) : G → [0, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content='  To this end, GAP assigns proper graph-level  probability p(G|x) to graphs G sampled from the  meta graph G via the graphical model likelihood:  p(G|x) =  �  v∈G  p(v| pa(v), x)  =  �  n∈N  p(n| pa(n), x) ∗  �  e∈E  p(e| pa(e), x)  where p(v| pa(v), x) is the conditional probability  for v with respect to their parents pa(v) in  G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content='  Given the candidates graphs {Gk}K  k=1, we  can express the likelihood for p(v| pa(v), x)  by  writing  down  a  multinomial  likelihood  enumerating over different values of pa(v)  (Murphy, 2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content=' This in fact leads to a simple  expression for the model likelihood as a sim-  ple averaging of the beam-sequence log likelihoods:  log p(n| pa(n), x) ∝ 1  K  K  �  k=1  log p(n| pa(n) = ck)  (4)  where ck is the value of pa(n) in kth beam se-  quence, and the conditional probabilities are  computed using Equation (3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content=' See Appendix D for  a detailed derivation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content='  Algorithm 2 Graph Autoregressive Process  Inputs:  Beam candidates with probabilities {p(gk|x)}K  k=1  Meta graph G  Output:  Marginal probabilities {p(s|x)}  Graph model likelihood log p(G|x)  for v ∈ G do  Compute marginal likelihood:  p(v = s|x) (Equation 2)  Compute graphical model likelihood:  log p(v = s| pa(v),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content=' x) (Equation 4)  return {p(v|x)},' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content=' log p(G|x)) = �  v∈G log p(v| pa(v),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content=' x)  In summary,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content=' for each graph element variable  v ∈ G,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content=' GAP allows us to compute the graphical-  model conditional likelihood p(v|pa(v),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content=' x) via its  graphical model representation,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content=' and also to com-  pute the marginal probability p(v|x) via its autore-  gressive presentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content=' The conditional likelihood is  e1: L-INDEX  n1: _and_c  n2: _cathedral_n_1  e2: R-INDEX  n4: named "Bazaar"  g1  e1: L-INDEX  n1: _and_c  n2: _cathedral_n_1  e2: R-INDEX  n5: named "bazar"  e1: L-INDEX  n1: _and_c  n3: named "Cathedral"  e2: R-INDEX  n6: _bazaar_n_1  g2  g3  Autoregressive Sequential Representation (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content=', Beam Sequences)  …  …  …  …  …  Graphical Model Representation (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content=', Meta Graph)  v  n1: _and_c  e1: L-INDEX  n2: _cathedral_n_1  n3: named "Cathedral"  e2: R-INDEX  n4: named "Bazaar"  n5: named "bazar"  n6: _bazaar_n_1  GM  Figure 3: Visual illustration of constructing graphical model representation GM from autorepressive representation  {gk}K  k=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content=' The example here represents the sentence “The Cathedral and the Bazaar” from the Eric Raymond  Essay dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content=' Note that here we have omitted the brackets in g for simplicity (see 1(b)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content='  crucial for neural-symbolic inference (Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content='1),  and the marginal probability is useful for sparsity  regularization in global graph structure inference  (Appendix E).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content=' Algorithm 2 summarizes the full  GAP computation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content='  5  Experiments  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content='1  Experiment Setup  Datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content=' Consistent with previous ERG works,  we train the neural model on DeepBank v1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content='1 an-  notation of the Wall Stree Journal (WSJ), sections  00-21 (the same text annotated in the Penn Tree  Bank) that correspond to ERG version 1214.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content='  For OOD evaluation, we select 7 diverse datasets  from the Redwoods Treebank corpus: Wikipedia  (Wiki), the Brown Corpus (Brown), the Eric  Raymond Essay (Essay), customer emails (E-  commerce), meeting/hotel scheduling (Verbmobil),  Norwegian tourism (LOGON) and the Tanaka Cor-  pus (Tanaka) (See Appendix G for more details).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content='  Model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content=' Following Lin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content=' (2022), We train a  T5large using the official T5X finetune pipeline2,  and use beam search with size K = 5 at inference  time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content=' Further details are collected in Appendix H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content='  Evaluation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content='  we use the standard eval metric  SMATCH (Cai and Knight, 2013), which computes  the maximum F1-score obtainable from an align-  ment between the predicted and gold graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content=' We  evaluate the models’ average-case performance on  all the 8 in-domain and OOD datasets, and also  conduct fine-grained evaluation of the models’ tail  generalization performance across 19 important  linguistic subcategories (Appendix J, Table 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content='  Baselines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content='  We compare with two recent state-  of-the-art approaches from the neural-symbolic  2https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content='com/google-research/t5x/blob/  main/t5x/train.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content='py  #  T5  ACE  Vote  Collab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content='  Ours  ACE*  WSJ (in-domain)  1,437  96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content='56  87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content='14  88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content='22  97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content='01  96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content='77  90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content='94  Wiki  1,307  90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content='12  80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content='25  80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content='55  90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content='58  90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content='04  90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content='42  Brown  2,182  92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content='05  91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content='74  85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content='46  93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content='58  93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content='11  93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content='20  Essay  591  92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content='19  92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content='64  83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content='72  93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content='57  93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content='76  93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content='52  E-commerce  1,114  93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content='15  97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content='25  87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content='38  95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content='44  97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content='37  98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content='36  Verbmobil  931  90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content='06  95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content='15  84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content='80  92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content='24  96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content='42  97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content='62  LOGON  1,895  87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content='13  93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
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+page_content='50  92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
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+page_content='05  Table 1:  SMATCH for T5, ACE, and collabora-  tive/compostional inference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content=' # refers to the number of  sentences in the dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content=' ACE* refers to the evaluation  results only for valid parse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content=' Collab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content=' refers to collabo-  rative inference from Lin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content=' (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content=' Vote refers to  voting strategy from Hoang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content=' (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content=' The bold and  underlined refer to the best and the second best results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content='  and ensemble graph parsing literature, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content='  (see Appendix 6 for a review) (1) Lin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content=' (2022)  is uncertainty-aware neural-symbolic framework  method attained state-of-the-art performance on the  in-domain DeepBank test set, and (2) Hoang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content='  (2021), a majority-voting-based graph ensemble  method that uses a voting strategy based on beam  sequences from the T5 model and predictions from  the ACE parser 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content=' It doesn’t exploit uncertainty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content='2  Results  The results are shown in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content=' Detailed in-  domain comparision with other previous work is in  Appendix I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content=' As shown, among the base models, the  T5 and ACE parser achieve similar overall perfor-  mance, with T5 strongly outperforms on in-domain  data but underperforms on the OOD data (see last  row in Table 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content=' Our approach achieves best re-  3We have tried several other variants for the voting can-  didates, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content=', top K predictions from the T5 parser and top 1  prediction + ACE prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content=' It turns out the best one is using  top K predictions from the T5 parser and ACE predictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content='  Essay  E-commerce  Verbmobil  Type  #  ACE  T5  Collab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
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+page_content='37  12  100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content='00  79.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content='10  100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content='00  100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content='00  Table 2: Comparing ACE, Collab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content=' (Lin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content=', 2022) and our parsers on fine-grained linguistic categories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content=' All  scores are reported in accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content=' The gray colored row means long-tail phenomenon (< 500 cases in the training  set).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content=' The bold indicates the best results among neural approaches (T5, Collab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content=' and Ours).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content=' * indicates the result is  better than ACE parser.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content='  sults on overall performance, which is ∼ 35% er-  ror reduction in aggregated SMATCH score over the  T5-based and symbolic approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content='  We now compare with the previous state-of-the-  art methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content=' Though in-domain performance is not  the focus of this work, our approach is still compa-  rable to Collab, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content=', the neural-symbolic method  from Lin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content=' (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content=' However, on the challenging  out-of-domain eval sets (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content=', E-commerce, Verb-  mobil whose topic and style are significantly differ-  ent from WSJ), the performance of Collab starts  to deteriorate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content=' In comparison, our neural-symbolic  approach remains robust out-of-domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content=' Its perfor-  mance stays competitive with and even sometimes  outperforms the ACE parser on difficult domains,  illustrating the advantage of compositionality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content='  We also notice that the voting-based ensemble  method Vote (Hoang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content=', 2021) performs poorly  in the neural-symbolic setting, despite based a mod-  erate number of beam sequences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content=' This is likely  because the majority-voting approach requires a  large number of diverse predictions from distinct  models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content=' When there are only two models, the abil-  ity of quantifying uncertainty becomes important.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content='3  Fine-grained Linguistic Evaluation  ERG provides different levels of linguistic informa-  tion that can benefit many NLP tasks, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content=', named  entity recognition and semantic role labeling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content=' This  rich linguistic annotation provides an oppurtunity  to evaluate model performance in meaningful pop-  ulation subgroups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content=' Detailed description of those  linguistic phenomena is in Appendix J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content='  Result is in Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content=' As shown, on OOD datasets,  the T5 parser underperforms the ACE parser on  most of the linguistic categories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content=' Our approach  outperforms both the neural model and the non-  compositional neural-symbolic method especially  on long-tail categories (the gray colored rows in the  table), attaining an > 14% average absolute gain  compared to the base model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content=' In some categories,  our method even outperforms the ACE parser while  all base model underperforms, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content=', ARG3 of basic  verb on Verbmobil and ARG3 of verb-particle on  E-commerce.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content='4  Case Study: Synthesizing Novel Graphs  To test if our methods can generate optimal graph  solution which the base models fail to obtain, we  further explore the percentage of novel graphs  (graphs that are not identical to any of the can-  didate predictions of the neural or symbolic model)  for each dataset, and compare the corresponding  SMATCH scores on those novel cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content=' The results  are shown in Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content=' We see that our method syn-  thesize novel graph parses that are in general of  higher quality than that of the base models, thanks  to the calibrated uncertainty (Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content=' This  indicates the compositional neural-symbolic infer-  ence can synthesize evidence across neural and  symbolic results and produce novel graphs that are  closer to ground truth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content='  %  Top 1  Top 2  Top 3  Top 4  Top 5  Collab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content='  ACE  Ours  In-domain  31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content='25  94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content='95  93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content='01  91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content='91  89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content='92  89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content='58  95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content='10  82.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content='80  98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content='44  Wiki  32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content='29  87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
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+page_content='00  83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content='90  88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content='77  82.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content='67  92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content='24  Brown  46.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content='84  90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content='54  89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
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+page_content='11  92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content='53  96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content='15  96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content='56  Essay  50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content='93  90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content='71  90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content='02  89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content='31  89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content='02  87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
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+page_content='73  96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content='08  E-commerce  34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content='65  90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content='03  88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content='34  86.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content='61  85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content='56  82.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content='91  92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content='82  98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content='96  97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content='54  Verbmobil  39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content='96  85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content='45  83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content='06  81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content='54  79.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content='30  78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content='27  88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content='42  97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content='78  96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content='70  LOGON  58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content='10  90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content='75  89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content='65  88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content='20  87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content='90  86.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content='95  92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content='50  96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content='70  97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content='06  Tanaka  24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content='89  89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content='35  87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content='46  85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content='60  83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content='55  83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content='16  92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content='30  98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content='23  98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content='27  All  38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content='76  90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content='57  89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content='18  88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content='01  87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content='24  86.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content='13  92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content='29  93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content='93  96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content='28  Table 3: SMATCH performance on novel graphs, where the results of our inference process are not identical to any  of the candidates from the base model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content='  6  Related Work  In this section we introduce related work for neural-  symbolic and ensemble learning for graph semantic  parsing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content=' For a broader context of graph semantic  parsing, please refer to Appendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content='  Neural-Symbolic  Graph  Semantic  Parsing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content='  Though neural models excel at semantic parsing,  they have been shown to struggle with out-of-  distribution compositional generalization, while  grammar or rule-based approaches work relatively  robustly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content=' This has motivated the work in neural-  symbolic parsing where symbolic approaches are  imported as inductive bias (Shaw et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content=', 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content=' Kim,  2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content=' Cheng et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content=', 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content=' Cole et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content=', 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content=' For  graph meaning representations, importing induc-  tive bias into neural model was somehow difficult  due to the much more complicated structure com-  pared to pure syntactic rules or logical formalism  (Peng et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content=', 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content=' Peng and Gildea, 2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content=' To  address this, Lin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content=' (2022) proposes a collabora-  tive framework by designing a decision criterion for  beam search that incorporates the prior knowledge  from a symbolic parser and accounts for model un-  certainty, which achieves the state-of-the-art results  on the in-domain test set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content='  Ensemble Learning for Graph Parsing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content=' Ensem-  ble learning is a popular machine learning approach  that combines predictions from multiple candidates  to create a new one that is more robust and ac-  curate than individual predictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content=' Previous stud-  ies have explored various ensemble learning ap-  proaches for graph parsing (Green and Žabokrtský,  2012;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content=' Barzdins and Gosko, 2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content=' Specifically, for  graph semantic parsing at subgraph level, Hoang  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content=' (2021) make use of checkpoints from models  of different architectures, and mining the largest  graph that is the most supported by a collection of  graph predictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content=' They then propose a heuristic  algorithm to approximate the optimal solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content='  Compare to the previous ensemble work, our  work differ in three ways: (1) Our decision rule is  based on neural model confidence, so the decision  is driven not by model consensus, but by model  confidence which indicates when the main (neural)  result is untrustworthy and needs to be comple-  mented by symbolic result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content=' Model consensus is  effective when there exists a large number of candi-  date models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content=' However, in the neural-symbolic set-  ting when there are only two models, the ability of  quantifying model uncertainty becomes important.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content='  (2) A secondary contribution of our work is to pro-  duce an parsing approach for the ERG community  that not only exhibits strong average-case perfor-  mance on in-domain and OOD environments, but  also generalizes robustly in important categories of  tail linguistic phenomena.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content=' Therefore, our investi-  gation goes beyond average-case performance and  evaluates in tail generalization as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content=' (3) We re-  veal a more nuance picture of neural models’ OOD  performance: a neural model’s top K parses in fact  often contains subgraphs that generalize well to  OOD scenarios, but the vanilla MLE-based infer-  ence fails to select them (see Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content='4 for more  details).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content='  7  Conclusions  We have shown how to perform accurate and ro-  bust semantic parsing across a diverse range of gen-  res and linguistic categories for English Resource  Grammar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content=' We achieve this by taking the advan-  tage of both the symbolic parser (ACE) and the  neural parser (T5) at a fine-grained subgraph level  using compositional uncertainty, an aspect miss-  ing in the previous neural-symbolic or ensemble  parsing work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content=' Our approach attains the best known  result on the aggregated SMATCH score across  eight evaluation corpus from Redwoods Treebank,  attaining 35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content='26% and 35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content='60% error reduction over  the neural and symbolic parser, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content='  Acknowledgement  Our work is sponsored in part by National Sci-  ence Foundation Convergence Accelerator under  award OIA-2040727 as well as generous gifts  from Google, Adobe, and Teradata.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content=' Any opinions,  findings, and conclusions or recommendations ex-  pressed herein are those of the authors and should  not be interpreted as necessarily representing the  views, either expressed or implied, of the U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content=' Gov-  ernment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content=' The U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content=' Government is authorized to  reproduce and distribute reprints for government  purposes not withstanding any copyright annota-  tion hereon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content=' We thank Du Phan, Panupong Pasupat,  Jie Ren, Balaji Lakshminarayanan and Deepak Ra-  machandran for helpful discussion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content='  Limitation  Here we discuss a potential limitations of the cur-  rent study:  Problem domain  In this work, we have selected  English Resource Grammar as the target formalism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content='  This is a deliberate choice based on the availabil-  ity of (1) realistic out-of-distribution evaluation  corpus, and (2) well-established, high-quality sym-  bolic parser.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content=' This is a common setting in indus-  trial applications, where an practitioner is tempted  to combine large pre-trained neural model with  expert-developed symbolic rules to improve perfor-  mance for a new domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content=' Unfortunately, we are not  aware of another popular meaning representation  for which both resources are available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content=' To over-  come this challenge, we may consider studying  collaborative inference between a standard seq2seq  model and some indirect symbolic supervision, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content=',  syntactic parser or CCG parser (Steedman, 2001),  which is an interesting direction for future work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content='  Uncertainty estimation techniques  The vanilla  seq2seq model is known to under-estimate the true  probability of the high-likelihood output sequences,  wasting a considerable amount of probability mass  towards the space of improbable outputs (Ott et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content=',  2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content=' LeBrun et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content=', 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content=' This systematic un-  derestimation of neural likelihood may lead to a  conservative neural-symbolic procedure that im-  plicitly favors the information from the symbolic  prior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content=' It may also negatively impact calibration  quality, leading the model to under-detect wrong  predictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content=' To this end, it is interesting to ask  if a more advanced seq2seq uncertainty method  (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content=', Monte Carlo dropout or Gaussian process  (Gal and Ghahramani, 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content=' Liu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content=', 2020)) can  provide systematically better uncertainty quantifi-  cation, and consequently improved downstream  performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content='  Graphical model specification  The GAP model  presented  in  this  work  considers  a  classi-  cal  graphical  model  likelihood  p(G|x)  =  �  v∈G p(v| pa(v), x) , which leads to a clean fac-  torization between graph elements v and fast prob-  ability computation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content=' However, it also assumes a  local Markov property that v is conditional inde-  pendent to its ancestors given the parent pa(v).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content=' In  theory, the probability learned by a seq2seq model  is capable of modeling higher order conditionals  between arbitrary elements on the graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content=' Therefore  it is interesting to ask if a more sophisticated graph-  ical model with higher-order dependency structure  can lead to better performance in practice while  maintaining reasonable computational complexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content='  Understanding different types of uncertainty  There exists many different types of uncertainties  occur in a machine learning system (Hüllermeier  and Waegeman, 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content=' This includes data uncer-  tainty (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content=', erroneously annotated training labels,  ill-formedness of the input sentence, or inherent  ambiguity in the example-to-label mapping), and  also model uncertainty which occurs the test ex-  ample not containing familiar patterns the model  learned from the training data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content=' In this work, we  quantifies uncertainty using mean log likelihood,  which broadly captures both types of uncertainty  and does not make a distinction between these dif-  ferent subtypes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content=' As different source of uncertainty  may lead to different strategy in neural-symbolic  parsing, the future work should look into more  fine-grained uncertainty signal that can decompose  these different sources of error and uncertainty, and  propose adaptive strategy to handle different sce-  narios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content='  Ethical Consideration  This paper focused on neural-symbolic semantic  parsing for the English Resource Grammar (ERG).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content='  Our architecture are built based on open-source  models and datasets (all available online).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content=' We do  not anticipate any major ethical concerns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content='  References  Omri Abend and Ari Rappoport.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
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+page_content=' 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content=' Robust incremen-  tal neural semantic graph parsing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content=' In Proceedings  of the 55th Annual Meeting of the Association for  Computational Linguistics (Volume 1: Long Papers),  pages 1215–1226, Vancouver, Canada.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content=' Association  for Computational Linguistics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content='  Shu Cai and Kevin Knight.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content=' 2013.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content=' Smatch: an evalua-  tion metric for semantic feature structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content=' In Pro-  ceedings of the 51st Annual Meeting of the Associa-  tion for Computational Linguistics (Volume 2: Short  Papers), pages 748–752, Sofia, Bulgaria.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content=' Associa-  tion for Computational Linguistics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content='  Ulrich Callmeier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content=' 2000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content='  Pet–a platform for experi-  mentation with efficient hpsg processing techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content='  Natural Language Engineering, 6(1):99–107.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content='  Junjie Cao, Zi Lin, Weiwei Sun, and Xiaojun Wan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content='  2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content='  Comparing knowledge-intensive and data-  intensive models for English resource semantic pars-  ing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
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+page_content=' In Proceed-  ings of the 56th Annual Meeting of the Association  for Computational Linguistics (Volume 1: Long Pa-  pers), pages 408–418, Melbourne, Australia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
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+page_content=' Peking  at MRP 2019:  Factorization- and composition-  based parsing for elementary dependency struc-  tures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content=' In Proceedings of the Shared Task on Cross-  Framework Meaning Representation Parsing at the  2019 Conference on Natural Language Learning,  pages 166–176, Hong Kong.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
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+page_content='  Jianpeng Cheng, Siva Reddy, Vijay Saraswat, and  Mirella Lapata.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
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+page_content='  In  Proceedings of the 12th Conference of the Euro-  pean Chapter of the ACL (EACL 2009), pages 1–9,  Athens, Greece.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content=' Association for Computational Lin-  guistics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content='  Ann Copestake, Dan Flickinger, Carl Pollard, and  Ivan A Sag.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
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+page_content='  Berthold Crysmann and Woodley Packard.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content=' 2012.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content=' To-  wards efficient HPSG generation for German, a non-  configurational language.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content=' In Proceedings of COL-  ING 2012, pages 695–710, Mumbai, India.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content=' The  COLING 2012 Organizing Committee.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content='  Dan Flickinger, Emily M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content=' Bender, and Stephan Oepen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
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+page_content=' In Proceedings of the Ninth In-  ternational Conference on Language Resources and  Evaluation (LREC’14), pages 875–881, Reykjavik,  Iceland.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content=' European Language Resources Association  (ELRA).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content='  Yarin Gal and Zoubin Ghahramani.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content=' 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content=' Dropout as a  bayesian approximation: Representing model uncer-  tainty in deep learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content=' In international conference  on machine learning, pages 1050–1059.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content=' PMLR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content='  Nathan Green and Zdenˇek Žabokrtský.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content=' 2012.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content='  Hy-  brid combination of constituency and dependency  trees into an ensemble dependency parser.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content=' In Pro-  ceedings of the Workshop on Innovative Hybrid Ap-  proaches to the Processing of Textual Data, pages  19–26, Avignon, France.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content=' Association for Computa-  tional Linguistics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content='  Thanh Lam Hoang, Gabriele Picco, Yufang Hou,  Young-Suk Lee, Lam Nguyen, Dzung Phan, Vanessa  López, and Ramon Fernandez Astudillo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
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+page_content=' En-  sembling graph predictions for amr parsing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content='  Ad-  vances in Neural Information Processing Systems,  34:8495–8505.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
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+page_content='  Head-driven  phrase structure grammar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content=' University of Chicago  Press.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content='  Colin Raffel, Noam Shazeer, Adam Roberts, Katherine  Lee, Sharan Narang, Michael Matena, Yanqi Zhou,  Wei Li, and Peter J Liu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content=' 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content=' Exploring the lim-  its of transfer learning with a unified text-to-text  transformer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content=' Journal of Machine Learning Research,  21:1–67.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content='  Peter Shaw, Ming-Wei Chang, Panupong Pasupat, and  Kristina Toutanova.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content=' 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content=' Compositional general-  ization and natural language variation: Can a se-  mantic parsing approach handle both?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content=' In Proceed-  ings of the 59th Annual Meeting of the Association  for Computational Linguistics and the 11th Interna-  tional Joint Conference on Natural Language Pro-  cessing (Volume 1: Long Papers), pages 922–938,  Online.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content=' Association for Computational Linguistics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content='  Mark Steedman.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content=' 2001.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content='  The syntactic process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content='  MIT  press.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content='  Kristina Toutanova, Christopher D Manning, Dan  Flickinger, and Stephan Oepen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content=' 2005.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content='  Stochas-  tic hpsg parse disambiguation using the redwoods  corpus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content='  Research on Language and Computation,  3(1):83–105.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content='  Hiroyasu Yamada and Yuji Matsumoto.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content=' 2003.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content=' Statis-  tical dependency analysis with support vector ma-  chines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content=' In Proceedings of the Eighth International  Conference on Parsing Technologies, pages 195–  206, Nancy, France.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content='  A  Graph-based Meaning Representation  Considerable NLP research has been devoted to  the transformation of natural language utterances  into a desired linguistically motivated semantic rep-  resentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content=' Such a representation can be under-  stood as a class of discrete structures that describe  lexical, syntactic, semantic, pragmatic, as well as  many other aspects of the phenomenon of human  language.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content=' In this domain, graph-based representa-  tions provide a light-weight yet effective way to en-  code rich semantic information of natural language  sentences and have been receiving heightened at-  tention in recent years.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content=' Popular frameworks un-  der this umbrella includes Bi-lexical Semantic De-  pendency Graphs (SDG;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content=' Bos et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content=', 2004;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content=' Ivanova  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content=', 2012;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content=' Oepen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content=', 2015), Abstract Mean-  ing Representation (AMR;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content=' Banarescu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content=', 2013),  Graph-based Representations for English Resource  Grammar (ERG;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content=' Oepen and Lønning, 2006;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content=' Copes-  take, 2009), and Universal Conceptual Cognitive  Annotation (UCCA;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content=' Abend and Rappoport, 2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content='  B  Literature Review on Graph Semantic  Parsing  In this section, we present a summary of different  parsing technologies for graph-based meaning rep-  resentations in addition to the ones discussed in  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content='2, with a focus on English Resource Grammar  (ERG).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content='  Grammar-based approach  In this type of ap-  proach, a semantic graph is derived according to  a set of lexical and syntactico-semantic rules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content=' For  ERG parsing, sentences are parsed to HPSG deriva-  tions consistent with ERG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content=' The nodes in the deriva-  tion trees are feature structures, from which MRS  is extracted through unification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content=' The parser has a  default parse ranking procedure trained on a tree-  bank, where maximum entropy models are used  to score the derivations in order to find the most  likely parse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content=' However, this approach fails to parse  sentences for which no valid derivation is found  (Toutanova et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content=', 2005).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content=' There are two main exist-  ing grammar-based parsers for ERG parsing: the  PET system (Callmeier, 2000) and the ACE system  (Crysmann and Packard, 2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content=' The core algo-  rithms implemented by both systems are the same,  but ACE is faster in certain common configurations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content='  We choose ACE as the symbolic parser in our work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content='  Factorization-based approach  This type of ap-  proach is inspired by graph-based dependency tree  parsing (McDonald, 2006).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content=' A factorization-based  parser explicitly models the target semantic struc-  tures by defining a score function that can eval-  uate the probability of any candidate graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content=' For  ERG parsing, Cao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content=' (2021) implemented a two-  step pipeline architecture that identifies the concept  nodes and dependencies by solving two optimiza-  tion problems, where prediction of the first step is  utilized as the input for the second step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content=' Chen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content='  (2019) presented a four-stage pipeline to incremen-  tally construct an ERG graph, whose core idea is  similar to previous work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content='  Transition-based approach  In these parsing  systems, the meaning representations graph is gen-  erated via a series of actions, in a process that is  very similar to dependency tree parsing (Yamada  and Matsumoto, 2003;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content=' Nivre, 2008), with the dif-  ference being that the actions for graph parsing  need to allow reentrancies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content=' For ERG parsing, Buys  and Blunsom (2017) proposed a neural encoder-  decoder transition-based parser, which uses stack-  based embedding features to predict graphs jointly  with unlexicalized predicates and their token align-  ments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content='  Composition-based approach  Following a prin-  ciple of compositionality, a semantic graph can  be viewed as the result of a derivation process, in  which a set of lexical and syntactico-semantic rules  are iteratively applied and evaluated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content=' For ERG pars-  ing, based on Chen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content=' (2018), Chen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content=' (2019)  proposed a composition-based parser whose core  engine is a graph rewriting system that explicitly  explores the syntactico-semantic recursive deriva-  tions that are governed by a synchronous SHRG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content='  Translation-based approach  This type of ap-  proach is inspired by the success of seq2seq mod-  els which are the heart of modern Neural Machine  Translation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content=' A translation-based parser encodes  and views a target semantic graph as a string from  another language.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content=' In a broader context of graph  semantic parsing, simply applying seq2seq models  is not successful, in part because effective lineariza-  tion (encoding graphs as linear sequences) and data  sparsity were thought to pose significant challenges  (Konstas et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content=', 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content=' Alternatively, some specifi-  cally designed preprocessing procedures for vocab-  ulary and entities can help to address these issues  (Konstas et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content=', 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content=' Peng et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content=', 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content=' These pre-  processing procedures are very specific to a certain  type of meaning representation and are difficult  to transfer to others.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content=' To address this, Lin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content='  (2022) propose a variable-free top-down lineariza-  tion and a compositionality-aware tokenization for  ERG graph preprocessing, and successfully trans-  fer the ERG parsing into a translation problem that  can be solved by a state-of-the-art seq2seq model  T5 (Raffel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content=', 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content=' The parser achieves the  best known results on the in-domain test set from  the DeepBank benchmark.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content='  C  Additional Methods Discussions  C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content='1  Efficient Probability Estimation Using  Beam Outputs  The marginalized probability ˆp(si|x) provides a  way to reason about the global importance of si by  integrating the probabilistic evidence p(si|sk,<i, x)  over the whole beam-sampled posterior space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content=' It is  able to capture the cases of spurious graph elements  si with high local probability p(si|sk,<i, x) but low  global likelihood (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content=', only appear in a few low-  probability beam candidates), which is useful for  inferring sparse global structures for the meta graph  (Appendix E).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content='  In the importance weight πk, the temperature pa-  rameter t controls how evidence for p(si|x) is ag-  gregated across beam samples {gk}K  k=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content=' When t →  0, the above is equivalent to selecting p(si|sk,<i, x)  from the most probable subsequence sk,<i;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content=' when  t → ∞, the above is equivalent to simple averaging  of p(si|sk,<i, x) from all beam candidates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content=' In the  experiments, we find that the value of t does not  have a significant impact on the final performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content='  In general, we recommend fixing it to a small value  (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content=', t = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content='1) to suitably downweighting the con-  tribution from improbable beam candidates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content='  D  Simplified Expression for Graphical  Model Likelihood  Given the candidates graphs {Gk}K  k=1, we can  express the likelihood for p(v| pa(v), x) by writing  down a multinomial likelihood enumerating over  different values of pa(v) (Murphy, 2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content=' For  example, say pa(n) = (e1, e2) which represents  a subgraph of two edges (e1, e2) pointing into  a node n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content='  Then the conditional probability  p(n| pa(n), x) can be computed by enumer-  ating over the observed values of (e1, e2) pair:  p(n| pa(n), x) = p(n|(e1, e2), x)  ∝  �  c∈Candidate(e1,e2)  p(n|(e1, e2) = c, x)Kc  where Candidate(e) is the collection of possi-  ble symbols s the variable e can take, and  Kc is the number of times (e1, e2) takes a  particular value c  ∈  Candidate(e1, e2)  =  Candidate(e1) × Candidate(e2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content='  Then, the log likelihood becomes:  log p(n| pa(n), x)  =  �  c  Kc ∗ log p(n|(e1, e2) = c)  To simplify this above expression, we notice that  log p(n| pa(n), x) can be divided by the constant  beam size K without impacting the inference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content=' As  a result, the log probability can be computed by  simplify averaging the values of log p(v| pa(v) =  ck) across the beam candidates:  log p(n| pa(n), x)  ∝  �  c  Kc  K log p(n|(e1, e2) = c)  = 1  K  K  �  k=1  log p(n|(e1, e2) = ck)  where ck is the value of (e1, e2) in kth beam candi-  date.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content='  E  Extensions and Practical  Implementation  E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content='1  Infer Sparse Global Structure via  Likelihood-based Pruning  In practice, the meta graphG can contain spuri-  ous elements v that have a high local likelihoods  log p(v| pa(v), x) but very low global probabili-  ties p(v|x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content='  This happens when the element v  only appears in a few low-probability beam se-  quences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content=' These spurious nodes and edges often  adds redundancy to the generated graph (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content=', hurt-  ing precision), and cannot be eliminated by the  neural-symbolic inference procedure, due to their  high local conditional probability p(v| pa(v), x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content='  Consequently, we find it empirically effective to  perform sparse structure inference forG based on  global probabilities p(v|x) before diving into local  neural-symbolic prediction for graph components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content='  In this work, we carry out this global structure  inference by considering a simple threshold-and-  project procedure, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content=', pruning out all the graph  elements whose global probability ||p(v|x)||∞ =  maxs∈Candidate(v) p(v = s|x) is lower than a thresh-  old t, but will keep v if its removal will lead to an  invalid graph with disconnected subcomponents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content='  Here ||p(v|x)||∞ is the total variation metric that  returns the maximum probability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content='  Algorithm 3 summarizes this procedure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content=' From a  theoretical perspective, this is equivalent to finding  the most sparse solution with respect to threshold t  within the space of valid (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content=', connected) subgraphs  ofG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content='  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content='Algorithm 3 Likelihood-based Pruning  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content='Inputs:  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content='Meta Graph GM  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content='Marginal probabilities {p(s|x)}s∈G  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content='Threshold t  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content='Output:  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content='Pruned graph G′  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content='M  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content='Initialize:  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content='G′  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content='M = GM  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content='for v ∈ G′  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content='M do  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content='if ||p(v|x)||∞ < t AND G′  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content='M \\ {v} is connected then  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content='Prune v : G′  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content='M = G′  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content='M \\ {v}  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content='return G′  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content='M  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content='ARG  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content='unknown  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content='_abstract_n_1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content='BV-of  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content='udef_q  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content='g1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content='ARG  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content='unknown  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content='named "Abstract"  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content='BV-of  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content='proper_q  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content='g2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content='ARG1-of  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content='unknown  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content='_abstract_a_1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content='g3  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content='ARG1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content='_abstract_a_1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content='unknown  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content='g4  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content='ARG1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content='_abstract_v_1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content='pron  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content='g5  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content='BV-of  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content='pronoun_q  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content='Figure 4: Autoregressive Representation (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content=', beam se-  quences) for the sentence “Abstract” from the Eric Ray-  mond Essay dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content=' Note that g3 and g4 are actually  the same graph but with different linearization orders.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content='  E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content='2  Handle Multi-modality via Mixture  Modeling  In some rare cases where the input sentence is frag-  mented or ill-formed, the neural model may output  multiple beam sequences with drastically differ-  ent high-level structures, creating difficulty for the  graph merging procedure (See Figure 4 for an ex-  ample).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content='  We can handle this multi-modality in observed  graph structure by extending p(G|x) to be a mix-  ture of GAP distributions, so that the graphical  model likelihood becomes:  p(G|x) =  �  m∈M  p(G|m, x)p(m|x)  where p(m|x) is a categorical distribution over  the mixture components m ∈ M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content='  Here each  component m induce a meta graph Gm for graph  Gm = ⟨Nm, Em⟩, such that  p(G|m, x) = p(Gm|x) =  �  v∈Gm  p(v| pa(v), x)  =  �  n∈Nm  p(n| pa(n), x) ∗  �  e∈Em  p(e| pa(e), x)  Given beam sequences {gk}K  k=1, the mixture com-  ponents can be estimated using a standard cluster-  ing algorithm based on an edit distance between  beam candidate gk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content=' Based on our experiments, hier-  archical agglomerative clustering (HAC) combined  with the longest common subsequence (LCS) dis-  tance often leads to the best result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content=' After clustering,  p(m|x) is computed as the empirical probability of  beam sequences belonging the mth cluster, and the  meta graph Gm is computed by applying the graph  merging procedure to the beam sequences in the  mth cluster.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content='  To conduct neural symbolic inference, we also  need to define the symbolic prior p0 for the mixture  distribution:  p0(G) =  �  m∈M  p0(G|m) ∗ p0(m)  =  �  m∈M  [  �  v∈Gm  p0(v) ∗ p0(m)]  where p0(v  =  s)  ∝  exp(I(s  ∈  G0)) as  define previously, and we define p0(m)  =  exp(−SMATCH(Gm, G0)) following the previous  work (Lin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content=', 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content='  As a result,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content=' the decision criteria for neural-  symbolic inference under the mixture model be-  comes:  R(Gm|x) = R(m|x) +  �  v∈Gm  R(v|x)  where �  v∈Gm R(v|x) is the component-wise de-  cision criteria as defined in the main text,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content=' and  R(m|x) is the additional term for the mixture com-  ponents:  R(m|x) = α(m|x) ∗ log p(m|x)  + (1 − α(m|x)) ∗ log p0(m)  where α(m|x) = σ(− 1  T H(m|x) + b) is the trade-  off parameter driven by the average log likelihood  of beam sequences in the mth cluster Cm,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content=' i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content=',  H(m|x) =  1  |Cm|  �  gk∈Cm − log(gk|x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content='  During inference, we can again proceed in a  greedy fashion, first select the optimal ˆm based on  R(m|x), and then perform compositional neural-  symbolic inference with respect to G ˆm using  �  v∈G ˆ  m R(v|x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content='  As a result, the complete precedure with all op-  tional extensions are shown in Algorithm 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content='  Algorithm 4 Complete Procedure with All Extensions  Inputs:  Beam candidates and associated token-level  probabilities {p(gk|x)}K  k=1  Output:  Neural-symbolic graph prediction G  (Optional) Estimate:  Mixture components {Gm}M  m=1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content=' {p(m|x)M  m=1}  from Cluster {p(gk|x)}K  k=1  Optimal mixture components G = G ˆ  m,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content=' where  ˆm = arg max R(m|x)  Estimate:  Marginal probability and graphical model likelihood  (Algorithm 2):  {p(v|x)}v∈G,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content=' log p(G|x) = GAP(G)  Infer:  Global graph structure via likelihood-based pruning  (Algorithm 3)  G′ = ThresholdAndProject(G,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content=' {p(v|x)}v∈G)  Local node / edge prediction via compostitional  neural-symbolic inference (Algorithm 1)  G = NeuralSymbolicInference(G’)  F  Graph Matching Algorithm  In general,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content=' finding the largest common subgraph  is a well-known computationally intractable prob-  lem in graph theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content=' However, for graph parsing  problems where graphs have labels and a simple  tree-like structure, some efficient heuristics are  proposed to approximate the best match by a hill-  climbing algorithm (Cai and Knight, 2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content=' The  initial match is modified iteratively to optimize the  total number of matches with a predefined number  of iterations (default value set to 5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content=' This algorithm  is very efficient and effective, it was also used to  calculate the SMATCH score in Cai and Knight  (2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content='  G  Details for OOD Datasets  Wikipedia (Wiki)  The DeepBank team con-  structed a treebank for 100 Wikipedia articles on  Computational Linguistics and closely related top-  ics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content=' The treebank of 11,558 sentences comprises 16  sets of articles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content=' The corpus contains mostly declar-  ative, relatively long sentences, along with some  fragments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content='  The Brown Corpus (Brown)  The Brown Cor-  pus was a carefully compiled selection of current  American English, totalling about a million words  drawn from a wide variety of sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content='  The Eric Raymond Essay (Essay)  The tree-  bank is based on translations of the essay “The  Cathedral and the Bazaar” by Eric Raymond.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content=' The  average length and the linguistic complexity of  these sentences is markedly higher than the other  treebanked corpora.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content='  E-commerce  While the ERG was being used in  a commercial software product developed by the  YY Software Corporation for automated response  to customer emails, a corpus of training and test  data was constructed and made freely available,  consisting of email messages composed by people  pretending to be customers of a fictional consumer  products online store.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content=' The messages in the corpus  fall into four roughly equal-sized categories: Prod-  uct Availability, Order Status, Order Cancellation,  and Product Return.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content='  Meeting/hotel scheduling (Verbmobil)  This  dataset is a collection of transcriptions of spoken  dialogues, each of which reflected a negotiation  either to schedule a meeting, or to plan a hotel stay.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content='  One dialogue usually consists of 20-30 turns, with  most of the utterances relatively short, including  greetings and closings, and not surprisingly with  a high frequency of time and date expressions as  well as questions and sentence fragments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content='  Norwegian tourism (LOGON)  The Norwe-  gian/English machine translation research project  LOGON acquired for its development and evalu-  ation corpus a set of tourism brochures originally  written in Norwegian and then professionally trans-  lated into English.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content=' The corpus consists almost en-  tirely of declarative sentences and many sentence  fragments, where the average number of tokens  per item is higher than in the Verbmobil and E-  commerce data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content='  The Tanaka Corpus (Tanaka)  This treebank  is based on parallel Japanese-English sentences,  which was adopted to be used with in the  WWWJDIC dictionary server as a set of example  sentences associated within words in the dictio-  nary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content='  H  Implementation and  Hyperparameters  T5 Model  We use the open-sourced T5X 4,  which is a new and improved implementation of  T5 codebase in JAX and Flax.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content=' Specifically, we  4https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content='com/google-research/t5x  use the official pretrained T5-Large (770 million  parameters), which is the same size as the one used  in Lin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content=' (2022), and finetuned it on DeepBank  in-domain training set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content=' Specifically, the total train-  ing step is 1,750,000 including 1,000,000 pretrain  steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content=' For fine-tuning the T5 model on ERG pars-  ing, batch size is set to 128, the output and input  sequence length is set to 512, and dropout rate is  set to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content='  Hyperparameters  For the trade-off parameter  α(v|x) = σ(− 1  T H(v|x) + b), we set temperature  T = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content='1 and bias b = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content='25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content='  I  In-domain Evaluation  Model  Node  Edge  SMATCH  ACE  89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content='30  85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content='05  87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content='14  ACE*  93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content='18  88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content='76  90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content='94  Buys and Blunsom (2017)  89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content='06  84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content='96  87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content='00  Chen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content=' (2018)  94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content='51  87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content='29  90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content='86  Chen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content=' (2019)  95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content='63  91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content='43  93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content='56  Chen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content=' (2019)  97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content='28  94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content='03  95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content='67  Cao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content=' (2021)  96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content='42  93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content='73  95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content='05  ACE-T5 (following Shaw et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content=' (2021))  93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content='46  89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content='19  91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content='30  T5-based (Lin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content=', 2022)  97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content='34  95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
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+page_content='56  + Hoang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content=' (2021)  88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content='89  87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content='67  88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content='22  + Lin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content=' (2022)  97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content='64  96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content='41  97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content='01  + Ours  97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content='50  96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content='07  96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content='77  Table 4: F1 score for node and edge predictions and the  SMATCH scores on the in-domain test set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content=' ACE* refers  to evaluation results only for valid parse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content='  Table 4 shows the in-domain performance,  where we compare our parser with the grammar-  based ACE parser and other data-driven parsers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content='  The baseline models also include a similar practice  with (Shaw et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content=', 2021) and (Hoang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content=', 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content='  The former one takes T5 as a backup for grammar-  based parser (ACE), and the latter gets ensembled  graph via a voting strategy based on the candidates  from the T5 parser and ACE parser.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content='  From the table we can see that our methods out-  performs the base model (T5-based) and most of  the previous work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content=' Specifically, we achieves a  SMATCH score of 96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
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+page_content='11% error  reduction compared to the base T5 parser.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
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+page_content='0  Accuracies  logon  T5  ACE  Figure 5: Diagrams for the T5 model’s probabilities verses the T5 model’s and ACE parser’s accuracies at subgraph  level on the other datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content=' Each bin contains the same number of examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content=' Since at most of the subgraphs, the  model is pretty certain (log P > −1e − 5), we exclude these pretty certain predictions in the figures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content='  with normalization, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content=', “flag burning”;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content=' (2) nomi-  nal with noun, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content=', “pilot union”;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content=' (3) verbal, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content=',  “state-owned”;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content=' (4) named entities, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content=', “West Ger-  many”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content='  Argument structure  In ERG, there are differ-  ent types of core predicates in argument structures,  specifically, verbs, nouns and adjectives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content=' We also  categorize verb in to basic verb (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content=', _look_v_1)  and verb particle constructions (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content=', _look_v_up).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content='  The verb particle construction is handled semanti-  cally by having the verb contribute a relation par-  ticular to the combination.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content='  Coreference  ERG resolves sentence-level coref-  erence, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content=', if the sentence referring to the same  entity, the entity will be an argument for all the  nodes that it is an argument of, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content=', in the sen-  tence, “What we want to do is take a more aggres-  sive stance”, the predicates “want” (_want_v_1)  and “take” (_take_v_1) share the same agent “we”  (pron).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content=' Coreference can be presented as reentran-  cies in the ERG graph, we notice that one important  type of reentrancies is the passive construction, so  we also report evaluation on passive construction  in Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content='  K  Calibration Performance on Other  Datasets  The correlations between the subgraph’s probabil-  ity and performance on other datasets are shown in  Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
+page_content=' The conclusions drew from the figure is  similar to the one discussed in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQfJCz4/content/2301.11459v1.pdf'}
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+arXiv:2301.08706v1  [math.DG]  20 Jan 2023
+NASH RESOLUTIONS OF SINGULAR FOLIATIONS WITH A VIEW
+TOWARDS AFFINE VARIETIES
+RUBEN LOUIS
+Abstract. We use the universal Lie ∞-algebroid of a singular foliation to construct several
+notions of resolution of singularities. For instance, we recover Nash modification or the monoidal
+transformation of an affine variety and a resolution method due to Mohsen [Moh21]. One of the
+important features is that any singular foliation becomes a Debord foliation after one blowup.
+Some examples are also given.
+Contents
+Introduction
+1
+1.
+Preliminaries
+2
+1.1.
+Grassmannian
+2
+1.1.1.
+Topological structure
+2
+1.1.2.
+Manifold structure
+2
+1.2.
+Singular foliations and universal Lie ∞-algebroids
+3
+1.2.1.
+Definitions
+4
+2.
+The blowup procedures
+7
+2.1.
+Blowup of vector bundle morphisms.
+7
+2.2.
+Main constructions and results
+8
+3.
+Proof of the main results
+10
+3.1.
+Proof of Theorem 2.5
+12
+3.2.
+Proof of Theorem 2.12 and 2.14
+14
+4.
+Examples
+16
+5.
+Conclusion
+20
+References
+21
+Introduction
+The results of this paper are taken from Chapter 7 of my PhD thesis. It is shown in [LLS20,
+LGL22] that behind any singular foliation there is a unique up to homotopy Lie ∞-algebroid,
+i.e., a Lie ∞-algebroid built over a geometric resolution of F. Now, in [Moh21], Omar Mohsen
+introduced a notion of blow-up of a smooth manifold along the singular leaves of a singular
+foliation. In this paper, we give an interpretation of the latter in terms of the universal Lie
+∞-algebroid of F, in fact in terms of an almost Lie algebroid over a geometric resolution of F.
+Indeed, we show that this construction is the 1st type of a series indexed by N0 of blowups of
+a singular foliation. For n = 0, we recover the Nash blowups for a singular foliation, which
+matches Nash blowups for an affine variety W when applied to the vector fields tangent to W.
+1
+
+2
+RUBEN LOUIS
+A consequence of our construction for n = 1 is that a resolution of any singular foliation can
+be constructed, which is given by an action of a Lie algebroid whose anchor map is injective on
+an open dense subset, i.e., a Debord foliation [Deb01].
+The paper is organized ad follows. In section 1, we recall the definition of Grassmann bundles
+and their properties. Then, we recall the concept of singular foliations and its universal Lie
+∞-algebroids. In section 2, we give the constructions of several notions of blowup of a singular
+foliation. Then, we state the main theorems. In Section 3 we write the proofs of the results.
+In section 4, we give some examples, where we recover the usual notions of blowups for affine
+varieties.
+1. Preliminaries
+1.1. Grassmannian. For E a finite dimension vector space over a field K ∈ {R, C}, we denote
+by Gr−r(E) the set of all vector subspaces of E of co-dimension r ∈ N. Let us recall a few facts
+on Gr−r(E).
+1.1.1. Topological structure. Gr−r(E) is metric space, the corresponding metric is defined by
+δ(V, V ′) = ∥pV − pV ′∥,
+(1)
+where pV stands for the orthogonal projection of E onto V ⊂ E. It is important to notice that:
+for all V, V ∈ Gr−r(E),
+δ(V, V ′) = δ(V ⊥, V ′⊥)
+here V ⊥ stands for the orthogonal space of V . It is proven (see e.g., [FGP94]) that Gr−r(E)
+equipped with the topology induced by the so-called “gap” metric (1), is equivalent to the
+Grassmann topology, i.e., the topology on Gr−r(E) whose open subsets W ⊆ Gr−r(E) are such
+that τ −1(W) is open in Str(d, K) := {A ∈ Md×r(K) | rk(A) = r}, with
+τ : Str(d, K) −→ Gr−r(E), A �−→ {vector space spanned by the columns of A}.
+Also, Gr−r(E) is a compact space.
+1.1.2. Manifold structure. Gr−r(E) is moreover a compact manifold of dimension r(d − r) and
+also, a projective variety.
+(1) Coordinates charts: One manner to define the standard affine coordinates on the
+Grassmannian Gr−r(E) is as follows. Fix a basis e1, . . . , ed=dim E for E. Let us describe
+the first chart. Consider
+ψ: Mr,d−r(K) −→ Md,d−r(K)
+A′ �−→
+�
+Id−r
+A′
+�
+.
+The vector space V = τ
+��
+Id−r
+A′
+��
+admits a basis of the form
+vj := ej +
+ℓ
+�
+k=1
+akjek,
+j = 1, . . . , d − r.
+(2)
+V is completely determined by the matrix A′. Hence, τ ◦ ψ is the first chart.
+For a permutation σ ∈ Sd, let P(σ) be the permutation matrix of lines associated to
+σ. We claim that the family τ ◦ P(σ) ◦ ψ(Mr,d−r(K)), indexed by σ ∈ Sd is an atlas of
+Grr(E). Its image consists of (2) up to permutation.
+
+NASH RESOLUTIONS OF SINGULAR FOLIATIONS WITH A VIEW TOWARDS AFFINE VARIETIES
+3
+(2) Grassmann bundle: For E → M a vector bundle of rank d over a manifold M (or a
+quasi-projective variety 1). Let r ≤ d. The disjoint union:
+Gr−r(E) :=
+�
+x∈M
+Gr−r(E|x)
+comes equipped with a natural manifold structure in the smooth or complex case and a
+quasi-projective variety structure when M is a quasi-projective variety. Also
+Π: Gr−r(E) −→ M
+(3)
+is a fibration. It is called (d − r)-th Grassmann bundle. To fix notations, for x ∈ M,
+elements of the fiber Π−1(x), i.e., points of Gr−r(E|x), are denoted by (V, x) with V ∈
+Gr−r(E|x).
+For every open subset U ⊂ M on which E is trivial, Π−1(U) ≃ Gr−r(Kd) × U. An
+adapted chart for Gr−r(E) −→ M around a point x ∈ M is a set of local coordinates
+of the form (Π∗x1, . . . Π∗xn, z1, . . . , zr(d−r)), where (x1, . . . , xn) are local coordinates on
+M and (z1, . . . , zr(d−r)) are functions which are standard affine coordinates on an open
+subset of each fiber of Π as in item (1).
+Convention 1.1. Let x ∈ M. Let e1, . . . , ed be local frame for E in a neighborhood U
+of x. For y ∈ U, let κy be the linear isomorphism defined by
+κy : Ex −→ Ey, κy(ei(x)) = ei(y),
+for all
+i ∈ {1, . . . , d}.
+(4)
+Let (xn) be a sequence of M that converges to x. We will say that a sequence of vector
+space Vxn ∈ Gr−r(E) with Vxn ⊂ Exn, converges to V ⊂ Ex and write Vxn
+−→
+n→+∞ V if
+κ−1(Vxn) −→
+n→+∞ V in Gr−r(Ex).
+(5)
+In the sequel we will not mention κ, since this notion of convergence does not depend
+on the chosen local frames of E.
+(3) Tautological subbundle: The Grassmann bundle Gr−r(E) comes equipped with two
+vector bundles τ E and AE, called tautological subbundle and tautological quotient bun-
+dle, that fit into the exact sequence
+0 −→ τ E −→ Π∗E −→ AE −→ 0.
+(6)
+Precisely, the fiber of τ E over the point (V, x) ∈ Gr−r(E|x) is the codimension r subvector
+space V of E|x = E|Π(V,x) = (Π∗E)|(V,x). By construction, τ E is a subbundle of the pull-
+back bundle Π∗E. Furthermore, AE ≃ Π∗E/τ E.
+This tautological quotient bundle is important for us to express some results of this
+paper.
+1.2. Singular foliations and universal Lie ∞-algebroids. We refer the reader to [AS09,
+AZ14, Cer79, Deb01, LLS20, LGLR22] for the topic of singular foliations, in particular to [LLS20,
+LGL22] for the notion of universal Lie ∞-algebroids. For Lie algebroids, see [Mac05].
+1the intersection inside some projective space of a Zariski-open and a Zariski-closed subset.
+
+4
+RUBEN LOUIS
+1.2.1. Definitions. We recall some basic definitions and properties on singular foliations.
+(1) A singular foliation on a smooth, real analytic, or complex manifold M, or a quasi-
+projective variety, is a subsheaf F ⊆ X(M) that fulfills the following conditions,
+(a) Stability under Lie bracket : [F, F] ⊆ F.
+(b) F is a module over its respective relevant sheaf of functions.
+(c) Locally finitely generateness : every m ∈ M admits an open neighborhood U to-
+gether with a finite number of vector fields X1, . . . , Xk ∈ X(U) such that for every
+open subset U ⊆ M the vector fields X1|V, . . . , Xk|V generates F on V as a mod-
+ule over functions on V. This condition is only necessary in the smooth case (see,
+[LGLR22]).
+There are two classes of singular foliations we are interested in.
+A locally polynomial singular foliation is a singular foliation over a smooth or complex
+manifold which admits, around each point, polynomial generators in some local chart.
+Proposition 1.2. A singular foliation F is projective as a module over functions if and
+only if there exists a Lie algebroid (A, [· , ·] , ρ) such that ρ(Γ(A)) = F whose anchor is
+injective on an open dense subset.
+These singular foliations are called Debord singular foliations.
+(2) Here are some important features of the above definition in the smooth/complex cases.
+- Singular foliation admits leaves : there exists a partition of M into submanifolds
+called leaves such that for all m ∈ M, the image of the evaluation map F → TmM
+is the tangent space of the leaf through m.
+- Singular foliations are self-preserving: the flow φX
+t of vector fields X ∈ F, whenever
+defined, preserves F [AS09, GY18], i.e.,
+∀ m ∈ M, ∃ ε > 0 such that ∀t ∈] − ε, ε[, (φX
+t )∗(F) = F.
+(3) We introduce here a notion of tangency and singular foliation for arbitrary closed subsets
+of a smooth or complex manifold. Let M be a smooth or complex manifold. Let S ⊆ M
+be a closed subset.
+(a) A vector field Z on M is said to be tangent to S if its flows φZ
+t preserves S, [MP82].
+We define X(S) to be the restriction to S of vector fields of M tangent to S, it is a
+OM/IS -module and a Lie algebra, with IS being the ideal of smooth functions on
+M that vanish on S .
+(b) A singular foliation over S is a OM/IS -submodule F of X(S) which is
+(i) finitely (locally) generated,
+(ii) stable under Lie bracket.
+It is said to be projective if it is projective as a module.
+(c) A Lie algebroid over S is a locally generated projective Lie-Rinehart algebra over
+OM/IS.
+Remark 1.3. The previous definitions match the usual notions of tangency and singular
+foliations, for affine or Stein subvarieties in the algebraic or holomorphic cases, and for
+S a smooth submanifold in the smooth case.
+
+NASH RESOLUTIONS OF SINGULAR FOLIATIONS WITH A VIEW TOWARDS AFFINE VARIETIES
+5
+(4) Let F ⊆ X(M) be a submodule. A complex of vector bundles (E, d, ρ)
+� E−i−1
+d(i+1) �
+�
+E−i
+d(i) �
+�
+E−i+1
+�
+�
+d(2)� E−1
+ρ
+�
+�
+TM
+�
+M
+M
+M
+M
+M
+is said to be a geometric resolution of F if the following complex is an exact sequence of
+sheaves:
+−→Γ(E−i−1) d(i+1)
+−→ Γ(E−i) d(i)
+−→ Γ(E−i+1)−→ · · · −→Γ(E−1)
+ρ
+−→ F.
+(7)
+A geometric resolution is said to be minimal at a point x ∈ M if, for all i ≥ 2, the
+linear maps d(i)|x : E−i|x −→ E−i+1|x vanish. Recall that it exists in many contexts,
+e.g., singular foliation with polynomial generators on Rn or Cn or locally real analytic
+singular foliation on a compact manifold (see, [LLS20]).
+(5) An almost graded Lie algebroid over M is the datum of a sequence (E, d = ℓ1, ρ) of
+vector bundles over M equipped with a graded symmetric degree +1 K-bilinear bracket
+ℓ2 : Γ(E) ⊙ Γ(E) → Γ(E)
+such that:
+(a) ℓ2 satisfies the Leibniz identity with respect to ρ: Γ(E−1) −→ X(M), i.e.,
+ℓ2(x, fy) = fℓ2(x, y) + ρ(x)[f]y
+(8)
+for all x ∈ Γ(E−1), y ∈ Γ(E) and f ∈ O.
+(b) ℓ1 is degree +1-derivation of ℓ2, i.e., for all x ∈ Γ(E−i), y ∈ Γ(E):
+ℓ1(ℓ2(x, y)) + ℓ2(ℓ1(x), y) + (−1)iℓ2(x, ℓ1(y)) = 0,
+(c) ρ is a morphism, i.e., for all x, y ∈ Γ(E−1)
+ρ(ℓ2(x, y)) = [ρ(x), ρ(y)].
+The O-linear map ρ is called the anchor map, and ℓ1 the differential.
+(6) A Lie ∞-algebroid over M is the datum of a sequence E = (E−i), 1 ≤ i < ∞ of vector
+bundles over M together with a structure of Lie ∞-algebra (ℓk)k≥1 on the sheaf of
+sections of E and a vector bundle morphism, ρ: E−1 → TM, called anchor map such
+that the k-ary brackets ℓk, k ̸= 2 are O-multilinear and such that
+ℓ2(e1, fe2) = ρ(e1)[f]e2 + fℓ2(e1, e2)
+(9)
+for all e1 ∈ Γ(E−1), e2 ∈ Γ(E•) and f ∈ O. The sequence
+· · ·
+ℓ1 � E−2
+ℓ1
+� E−1
+ρ
+� TM,
+(10)
+is a complex called the linear part of the Lie ∞-algebroid.
+(7) The following theorem is important, (see [Lav17, LLS20, LGL22] for more details) :
+Theorem 1.4. Let F be a singular foliation over M. Any geometric resolution of F
+· · ·
+d
+−→ E−3
+d
+−→ E−2
+d
+−→ E−1
+ρ
+−→ TM
+(11)
+comes equipped with a Lie ∞-algebroid structure whose unary bracket is d and whose
+anchor map is ρ (in particular ρ(Γ(E−1)) = F). Such a Lie ∞-algebroid structure is
+unique up to homotopy and is called a universal Lie ∞-algebroid of F.
+
+6
+RUBEN LOUIS
+In particular, this structure can be truncated to an almost graded Lie algebroid for F.
+(8) Let (E•, ℓ•, ρ) a universal Lie ∞-algebroid of a singular foliation F. For every point
+x ∈ M,
+(a) We let H•(F, x) = ⊕i≥1H−i(F, x) be the cohomology of the complex (11). The
+cohomology groups H•(F, x) do not depend on the choice of a geometric resolution
+of F. Notice that when the complex (11) is minimal at x, H−i(F, x) ≃ E−i|x for
+every i ≥ 1.
+(b) The 1, 2-ary brackets restrict to the graded vector space
+
+�
+i≥2
+E−i|x
+
+ ⊕ ker(ρx)
+and equip the latter with an almost graded Lie ∞-algebra structure as follows : for
+every k ∈ {1, 2},
+{x1, . . . , xk}k := ℓk(s1, . . . , sk)|x
+(12)
+for all x1, . . . , xk ∈ ev(E, x) and s1, . . . , sk ∈ Γ(E) sections of E such that si(x) = xi
+with i = 1, . . . , k.
+The bracket {· , · }2 induces a graded Lie algebra on H•(F, x). In particular, the 2-ary
+bracket {· , ·}2 satisfies the Jacobi identity on H−1(F, x) = ker(ρx)
+im(d(2)
+x ), and equips the latter
+with a Lie algebra structure.
+(9) Let (M, F) be a singular foliation, let Ix := {f ∈ C∞(M) | f(x) = 0} and F(x) :=
+{X ∈ F | X(x) = 0}. The quotient gx = F(x)
+IxF is a Lie algebra and is called the isotropy
+Lie algebra of F at x.
+Lemma 1.5. Let (E, ℓ•, ρ) be a universal Lie ∞-algebroid of F. Consider its underlying
+geometric resolution
+(E, d, ρ) :
+· · · ℓ1=d(4)
+−→ E−3
+ℓ1=d(3)
+−→ E−2
+ℓ1=d(2)
+−→ E−1
+ρ=d(1)
+−→ TM.
+(13)
+Then,
+(a) for all x ∈ M, we have H−1(F, x) ≃ gx as Lie algebras;
+(b) the subset of regular points of F in M satisfies
+Mreg,F = {x ∈ M | rk(d(2)
+x ) = dim(ker ρx)}
+= {x ∈ M | H−i(F, x) = 0, ∀i ≥ 1},
+Mreg,F is open and dense in M;
+(c) the restriction of the foliation F to Mreg,F is the set of sections of a subbundle of
+TM, i.e., is a regular foliation;
+(d) For every i ≥ 0, the dimension of im
+�
+d(i+1)�
+is locally constant on Mreg,F. More-
+over, if r the dimension of a regular leaf, then im(d(i+1)) is of codimension
+ri =
+i−1
+�
+j=1
+(−1)j+1rk(E−j) + (−1)i+1r, for i ≥ 1
+in E−i or r0 = dim M − r, with E0 := TM;
+(e) if (E, d, ρ) is of finite length, then (in the smooth case) all the regular leaves have
+the same dimension.
+
+NASH RESOLUTIONS OF SINGULAR FOLIATIONS WITH A VIEW TOWARDS AFFINE VARIETIES
+7
+In what follows, we assume that a geometric resolution of finite length exists.
+Under these
+assumptions, all the regular leaves have the same dimension. Denote by r the common dimension
+of the regular leaves.
+2. The blowup procedures
+In what follows M can be a connected complex or smooth manifold or an irreducible affine
+or quasi-projective. Firstly, let us explain a general construction on morphisms of vector bundles.
+2.1. Blowup of vector bundle morphisms. Let E, F be vector bundles over M and
+F
+d
+�
+�❆
+❆
+❆
+❆
+❆
+❆
+❆
+❆
+E
+�⑥⑥⑥⑥⑥⑥⑥⑥
+M
+a morphism of vector bundles. In the smooth case, we assume that d is of constant rank on
+an open dense subset Mreg,d ⊂ M, i.e., the dimensions of im(dx) or ker(dx) are constant for
+x ∈ Mreg,d, called the regular part (this is automatically true when M is complex or a quasi-
+projective variety). Let r be the codimension of im(dx) ⊆ Ex for a point x ∈ Mreg,d. Notice that
+for every x ∈ Mreg,d, im(dx) is a point of the Grassmannian Gr−r(Ex) and ker(dx) is a point of
+Grrk(F )−r(Fx). We consider the natural section of Gr−r(E) −→ M which is defined on Mreg,d
+by:
+σ: Mreg,d −→ Gr−r(E), x �−→ (im(dx), x) .
+(14)
+Then we define �
+M := σ(Mreg,d) to be the closure of the image of the section σ in Gr−r(E),
+together with the projection π: �
+M −→ M, where π denotes the restriction of Π: Gr−r(E) −→ M
+to �
+M.
+Remark 2.1. Intuitively, for x ∈ M, π−1(x) = �
+M ∩ Π−1(x) is the set of all possible limits of
+the images imdy when y ∈ Mreg,d converges to x. One can make a similar construction with the
+kernel of d.
+Here is an immediate property of that construction.
+Proposition 2.2. Let F
+d
+−→ E be a vector bundle morphism over M. The projection π: �
+M →
+M has the following property:
+(1) π is proper and surjective. In particular, for each point x ∈ M, the fiber π−1(x) is
+non-empty.
+(2) For every x ∈ M and V ∈ π−1(x), one has im(dx) ⊆ V .
+(3) For every x ∈ Mreg,d, π−1(x) = im(dx) is reduced to a point in Gr−r(E).
+Also,
+π−1(Mreg,d) is a manifold 2 and the restriction π: π−1(Mreg,d) −→ Mreg,d is invertible3
+in the smooth and holomorphic contexts.
+Proof. We prove it in the smooth and complex settings. Properness derives from the fact that
+the projection Π admits compact fibers. For any x ∈ M, choose U ⊂ M an open neighborhood
+of x that trivializes E −→ M over U. Then, Gr−r(E) ≃ Gr−r(Krk(E)) × U. Notice that,
+π−1(x) =
+�
+V ⊂ Ex
+���� ∃ (xn) ∈ MN
+reg,d, such that, im(dxn) −→
+n→+∞ V as xn
+−→
+n→+∞ x
+�
+.
+2Manifold is to be understood as quasi-projective when M is quasi-projective.
+3Invertible here means: diffeomorphism, in the smooth case; bi-holomorphism, in the complex case.
+
+8
+RUBEN LOUIS
+For any sequence (xn) in (Mreg,d ∩ U)N that converges to x, we can extract a sequence (xϕ(n))
+such that n �→ im(dxϕ(n)) ∈ Gr−r(Krk(E)) has a limit V , since the Grassmannian manifold
+Gr−r(Krk(E)) is compact. Hence, π−1(x) ̸= ∅ and π is onto. This proves item 1.
+Let us show item 2.
+Let V ∈ π−1(x) and (xn) ∈ (Mreg,d)N such that xn
+−→
+n→+∞ x and
+im(dxn) −→
+n→+∞ V . Let v ∈ im(dx). We have v = dxu for some u ∈ Fx. Choose a (local) section
+�u of F through u. By continuity, dxn �u(xn) −→
+n→+∞ dxu, hence dxu ∈ V . Thus, im(dx) ⊆ V .
+In particular, if x ∈ Mreg,d and V ∈ π−1(x) one has im(dx) = V since dim V = dim(im(dx)).
+Therefore, π−1(Mreg,d) is the image of the map σ on Mreg,d, it is isomorphic/biholomorphic to
+Mreg,d. This proves item 3.
+□
+We are now going to apply the constructions above to a sequence of vector bundle morphisms
+which are all of constant rank on an open dense subset.
+2.2. Main constructions and results. Let us work in the complex or holomorphic setting.
+Let F be a locally finitely generated O(M)-submodule of X(M), i.e., every point of M admits
+an open neighborhood U and a finite number of vector fields X1, . . . , Xn ∈ X(M) such that
+F|U = �n
+k=1 fkXk for fk ∈ O(M). We assume that there exists a geometric resolution, i.e.,
+complex of vector bundles (E•, d(•), ρ) of finite length
+0 · · ·
+� E−i−1
+d(i+1) �
+�
+E−i
+d(i) �
+�
+E−i+1
+�
+�
+d(2)� E−1
+ρ=d(1)
+�
+�
+TM
+�
+M · · ·
+M
+M
+M
+M
+M
+(15)
+such that ρ(Γ(E−1)) = F and exact as in (7), (see [LLS20] for conditions on existence of geometric
+resolutions at least locally). Denote by Mreg,F (as in Lemma 1.5) the open dense subset such
+that the fibers of im(d(i)) are of constant rank and im(d(i+1)) = ker(d(i)), for all i ≥ 1. We set
+E0 := TM by convention and proceed as the following:
+(a) For every i ≥ 0, let Πi : Gr−ri(E−i) −→ M be the Grassmann bundle of E−i with ri is
+as in Lemma 1.5 (d). Consider the natural section of Πi on Mreg,F defined by :
+σi : Mreg,F −→ Gr−ri(E−i), x �−→
+�
+im
+�
+d(i+1)
+x
+�
+, x
+�
+(16)
+(b) Let �
+Mi := σi(Mreg,F) be the closure of the image of σi in Gr−ri(E−i). Let πi : �
+Mi −→ M
+denote the restriction of Πi to �
+Mi.
+We also consider the section
+σ∞: Mreg,F −→
+�
+x∈M
+�
+i≥1
+Gr−ri(E−i|x), x �→ (σ1(x), σ2(x), . . . , σi(x), . . . )
+and define �
+M∞ := σ∞(Mreg,F). Here, �
+M∞ should be understood as the tuples made of elements
+V1 ∈ Gr−r1(E−1|x), . . . , Vi ∈ Gr−ri(E−i|x), . . . such that there exists (xn) ∈ MN
+reg,d such that
+im
+�
+d(i+1)
+xn
+�
+−→
+n→+∞ Vi as xn
+−→
+n→+∞ x for all i ∈ N. It is important to notice that the Vi’s are
+given by the same sequence (xn) ∈ MN
+reg,F.
+Remark 2.3. The construction also makes sense for M an affine variety in CN, upon replacing
+TM by TCN|M ≃ M × CN. Notice that in the definition of �
+M∞ we leave �
+M0 out.
+
+NASH RESOLUTIONS OF SINGULAR FOLIATIONS WITH A VIEW TOWARDS AFFINE VARIETIES
+9
+By Proposition 2.2, for each i ≥ 0, the projection πi : �
+Mi → M is invertible on the open dense
+subset Mreg,F, it is proper and surjective. Moreover, for each point x ∈ M and for every i ≥ 0,
+the fiber π−1
+i
+(x) is non-empty. Also, π−1
+∞ (x) is non-empty.
+From now on, most of the proofs are delayed to Section 3.
+As sets, �
+Mi, �
+M∞ do not need to be manifolds. They can be singular, see Section 4.
+Proposition 2.4. Let F be a singular foliation on a smooth or complex variety on M ∈
+�
+CN, RN�
+that admits polynomial generators, then it admits a geometric resolution of finite
+length and �
+Mi and �
+M∞ are quasi-projective varieties for all i ≥ 0.
+For F a singular foliation on an affine variety, �
+Mi is a quasi-projective variety.
+The following assertion follows from the existence of homotopy equivalence between any two
+geometric resolutions.
+Theorem 2.5. Let F be a singular foliation on M that admits geometric resolution. For each
+i ≥ 1, �
+Mi does not depend on the choice of a geometric resolution of F. The same is true for
+�
+M∞.
+To prove Theorem 2.5, we first establish the following results.
+Proposition 2.6. Let (15) be a geometric resolution for F. For every x ∈ M, for every i ≥ 1
+and V ∈ π−1
+i
+(x) one has,
+im(d(i+1)
+x
+) ⊆ V ⊆ ker(d(i)
+x ).
+(17)
+In particular, for all x ∈ Mreg,F and i ≥ 1, ker(d(i)
+x ) = im(d(i+1)
+x
+) = π−1
+i
+(x).
+Proposition 2.7. Fix a geometric resolution (E, d, ρ) of F and a universal Lie ∞-algebroid of
+F. The following are satisfied:
+(1) For every x ∈ M and V ∈ π−1
+1 (x), the 2-ary bracket {· , · }2 on ker ρx restricts to V and
+the image of V in H−1(F, x) ≃ gx, is a Lie subalgebra of codimension r−dim(Lx), where
+dim(Lx) is the dimension the leaf through x.
+(2) For all x ∈ M, and (V1 ⊂ E−1|x, . . . , Vk ⊂ E−k|x, . . .) ∈ π−1
+∞ (x) we have {Vi, Vj}2 ⊂
+Vi+j−1 for every i, j ∈ N0.
+The corollary below is a direct consequence of Proposition 2.6, and is another manner to state
+that Mi does not depend on the geometric resolution.
+Corollary 2.8. There are inclusions
+�
+Mi ֒→
+�
+x∈M
+Gr
+rk
+�
+d(i)
+x
+�
+−ri(H−i(F, x))
+and
+�
+M∞ ֒→
+�
+x∈M
+�
+i≥1
+Gr
+rk
+�
+d(i)
+x
+�
+−ri(H−i(F, x)).
+(18)
+Proof. Let x ∈ M and i ≥ 0. By Proposition 2.6, elements V ∈ π−1
+i
+(x) satisfy im(d(i+1)
+x
+) ⊆ V ⊆
+ker(d(i)
+x ), they correspond injectively to a (unique) sub-vector space of codimension ri−rk(d(i)) in
+H−i(F, x). In particular, this implies the existence of an inclusion π−1
+i
+(x) ֒→ Grri−rk(d(i))(H−i(F, x)).
+□
+Remark 2.9. In particular, �
+M1 ֒→ ⊔x∈MGrLiedim(Lx)−r(gx), where gx is the isotropy Lie algebra
+of the singular foliation F at x and Lx is the leaf that passes through x. Here, GrLier−dim(Lx)(gx)
+denotes the Grassmannian of Lie subalgebras of gx of codimension r − dim(Lx).
+
+10
+RUBEN LOUIS
+Assume now that F is a singular foliation and that Equation (15) is a geometric resolution
+for F.
+Definition 2.10. Let i ≥ 0. We say that X ∈ F lifts to �
+Mi ⊂ Gr−ri(E−i), or �
+M∞, if there
+exists a vector field �X ∈ X(Gr−ri(E−i)) or X
+��
+x∈M
+�
+i≥1 Gr−ri(E−i|x)
+�
+, projectable to X and
+tangent to Mi in the sense of Section 1.2.1 (3). We denote by �Xi or �X∞ the restriction of �X to
+Mi or �X∞ respectively.
+We say that a F lifts to �
+Mi if every vector field X ∈ F lifts to �
+Mi.
+Remark 2.11. �Xi on π−1
+i
+(Mreg,F) is tangent in the usual sense to the submanifold and projects
+to X through πi. In particular, if a lift exists, its restriction to π−1(Mreg,F)) is unique, because
+πi : π−1
+i
+(Mreg,F)
+∼
+−→ Mreg,F. Since the other points of �
+Mi are limits of elements of π−1
+i
+(Mreg,F),
+thus its restriction to �
+Mi is unique.
+Theorem 2.12. Let F be a singular foliation on M that admits geometric resolution. For every
+i ≥ 1, the following items hold:
+(1) Every vector field X ∈ F lifts to a vector field �Xi on �
+Mi,
+(2) the map X ∈ F −→ �X| �
+Mi does not depend on any choices. In particular, it is a Lie
+algebra morphism.
+(3) The module �Fi over functions of �
+Mi generated by the
+�X′
+is for X ∈ F is a singular
+foliation.
+The same holds for �
+M∞.
+Definition 2.13. For each i ≥ 1, �Fi is called the blowup of F on �
+Mi.
+Here is a remarkable fact.
+Theorem 2.14. Let F be a singular foliation on M that admits geometric resolution (E, d, ρ).
+�F1 is projective, i.e., it is the image of a Lie algebroid over �
+M1 whose anchor map is injective
+on an open dense subset.
+In Theorem 2.14, we do not need the existence of geometric resolutions of F. An almost Lie
+algebroid of F is enough, the latter always exists [LLS20].
+3. Proof of the main results
+In this section, we prove Proposition 2.4, Proposition2.6, Proposition 2.7 and Theorem 2.12.
+Proof (of Proposition 2.4). Since M = RN or CN and F is generated by polynomial vector fields,
+we can choose a polynomial geometric resolution (E, d, ρ) of F by trivial vector bundles, [LLS20].
+Here by polynomial we mean the d and ρ are given by polynomials.
+Let e1, . . . , ed resp. e′
+1, . . . , e′
+d′ be a basis by constant sections of E−i resp. E−i+1. One has,
+d(i)(el) =
+d′
+�
+k=1
+f k
+l e′
+k
+(19)
+for some polynomial functions f k
+l on KN. Without any lost of generality, let us describe �
+Mi using
+local coordinates on Gr−ri(E−i) consider for example the first standard coordinates chart U1 for
+the Grassmannian Gr−ri(E−i), (see Section 1.1.2). Denote by x = (x1, . . . , xN) the coordinates
+
+NASH RESOLUTIONS OF SINGULAR FOLIATIONS WITH A VIEW TOWARDS AFFINE VARIETIES
+11
+on M. Let m ∈ M. For W ∈ U1∩Gr−ri(E−i|m) and (apq(m)) ∈ Md,d−ri(K) be the homogeneous
+coordinates of W, let us define the sections
+�wq(x) := eq(x) +
+ri
+�
+k=1
+akq(x)ek(x),
+q = 1, . . . , d − ri.
+(20)
+By construction, the �wq(x)’s, evaluated at m, form a basis for W. Notice that we have,
+d(i)( �wq)(x) =
+d′
+�
+k=1
+�
+f k
+q (x) +
+ri
+�
+s=1
+asqf k
+s (x)
+�
+e′
+k(x).
+Therefore, W ⊆ ker d(i)
+m if and only if
+f k
+q (m) +
+ri
+�
+s=1
+asq(m)f k
+s (m) = 0,
+q = 1, . . . , d − ri
+k = 1, . . . , d′.
+(21)
+This defines an affine variety. As a result, �
+Mi is given in local coordinates U1 by elements that
+satisfy
+(1) Equation (21)
+(2) and that are limits of solutions of (21) in nearby regular points, elements which are
+unique.
+Hence, it is, on the affine variety, the irreducible components of (21) that
+projects onto M.
+This concludes the proof in the smooth or complex cases. For a quasi-projective variety, the
+proof is similar.
+□
+Remark 3.1. When F is generated by polynomial vector fields, π−1
+i
+(Mreg,F) is locally defined
+by polynomial equations.
+Proof (of Proposition 2.6). We know by Proposition 2.2(2) that, for every x ∈ M and V ∈
+π−1
+i
+(x), one has im(d(i+1)
+x
+) ⊆ V . Now, for any element v ∈ V , there exists a sequence vn ∈
+ker(d(i)
+xn) = im(d(i+1)
+xn
+), n ∈ N that converges to v. In particular, d(i)
+xn(vn) = 0 for all n. Hence,
+by continuity, one has v ∈ ker(d(i)
+x ). Hence, V ⊆ ker d(i)
+x . This completes the proof.
+□
+Proof. (of Proposition 2.7). For all i ≥ 1, choose a local frame e(i)
+1 , . . . , e(i)
+qi , . . . e(i)
+qi+ri of E−i on a
+neighborhood U of x such that e(i)
+1 (x), . . . , e(i)
+qi (x) is an orthogonal basis for Vi for an arbitrary
+Hermitian structure on E−i. For i, j ≥ 1, let (cij,s
+kl ) ∈ O(U) be a family of functions over U such
+that for all k ≤ qi and l ≤ qj,
+ℓ2
+�
+e(i)
+k , e(j)
+l
+�
+=
+�
+s≥1
+cij,s
+kl e(i+j−1)
+s
+∈ ΓU(E−i−j+1).
+In particular,
+�
+e(i)
+k (x), e(j)
+l (x)
+�
+2 =
+�
+s≥1
+cij,s
+kl (x)e(i+j−1)
+s
+(x).
+(22)
+The bracket in Equation 22 is well-defined even for i = 1 or j = 1, although only the 2-ary
+bracket of local sections is defined in such cases, because even if i or j = 1, we are taking the
+brackets of elements in ker ρx. Let u ∈ Vi, v ∈ Vj with u =
+qi
+�
+s=1
+αse(i)
+s (x), and v =
+qj
+�
+s=1
+βse(j)
+s (x).
+
+12
+RUBEN LOUIS
+Let (xn) ∈ MN
+reg,F be a sequence that converges to x such that im(d(i+1)
+xn
+)
+−→
+n→+∞ Vi and
+im(d(j+1)
+xn
+) −→
+n→+∞ Vj. There exist sequences
+un =
+qi+ri
+�
+k=1
+αk
+ne(i)
+k (xn) −→
+n→+∞ u;
+vn =
+qj+rj
+�
+l=1
+βl
+ne(j)
+l (xn) −→
+n→+∞ v
+with un ∈ im(d(i+1)
+xn
+) = ker d(i)
+xn and vn ∈ im(d(j+1)
+xn
+) = ker d(j)
+xn, for all n ∈ N. In particular,
+the sequences (αk
+n), (βl
+n) ∈ KN satisfy αk
+n
+−→
+n→+∞ αk;
+βl
+n
+−→
+n→+∞ βl with αk = βl = 0 for
+k ≥ qi + 1, l ≥ qj + 1. Therefore, for every n ∈ N we have
+�
+αk
+nβl
+ncij,s
+kl (xn)e(i+j−1)
+s
+(xn) = {un, vn}2 ∈ im(d(i+j)
+xn
+) = ker d(i+j−1)
+xn
+).
+(23)
+We have used in (23), the fact that {du1, du2}2 ∈ im(d), for all u1, u2 ∈ E≤−2. Since
+�
+αk
+nβs
+ncij,s
+kl (xn)e(i+j+1)
+s
+(xn) −→
+n→+∞
+�
+αkβlcij,s
+kl (x)e(i+j−1)
+s
+(x) ∈ E−i−j+1|x
+= {u, v}2.
+(24)
+As a result, {u, v}2 ∈ Vi+j−1 ∈ π−1
+i−j−1(x). Hence, for every point (V1, . . . , Vi, . . . , Vj, . . . ) ∈ π−1
+∞ (x)
+one has {Vi, Vj}2 ⊆ Vi+j−1. This proves item 2. By taking i = j = 1 and Vi = Vj = V ∈ π−1
+1 (x),
+Equation (24) means that {u, v}2 ∈ V . This proves item 1.
+□
+3.1. Proof of Theorem 2.5. In this section, we give a proof of Theorem 2.5. By Corollary
+2.8 (whose proof is independent of Theorem 2.5), for every i ≥ 1, we have an inclusion �
+Mi ֒→
+�
+x∈M Gr
+rk
+�
+d(i)
+x
+�
+−ri(H−i(F, x)), where ri is defined as in Lemma 1.5(d). We now need to show
+this inclusion is canonical, i.e., independent of the choice of a geometric resolution (E, d, ρ).
+Convention 3.2. For (E, d, ρ) a geometric resolution of F. Denote by �
+ME
+i
+:= �
+Mi constructed
+out of (E, d, ρ) and �
+ME′
+i
+:= �
+Mi constructed out of (E′, d′, ρ′) for i ≥ 1. Also, for x ∈ M and
+V ∈ π−1
+i
+(x), we denote by V the image of V in Gr
+rk
+�
+d(i)
+x
+�
+−ri(H−i(F, x)).
+Remark 3.3. Let x ∈ M. Consider a minimal geometric resolution (E′, d′, ρ′) of F at x (see
+Definition (4)). For (V, x) ∈ �
+ME
+1
+and (V ′, x) ∈ �
+ME′
+1
+one has that dim V ′ ≤ dim V , because
+rk(E′
+−1) ≤ rk(E−1) by minimality. Hence, V, V ′ do not necessarily belong to the same Grass-
+mannian. However, dim V = dim V ′. We prove the latter in the next Lemma.
+Lemma 3.4. Let (E, d, ρ) and (E′, d′, ρ′) be geometric resolutions of F. For all i ≥ 1, and for
+all (V, x) ∈ �
+ME
+i
+and (V ′, x) ∈ �
+ME′
+i , one has, dim V = dim V ′.
+Proof. If x ∈ M is a regular point, then V = V ′ = {0}. Thus, the equality holds. Let x ∈ M
+be a singular point. We prove it only for i = 1, 2, since i = 1 is a special case and for i ≥ 3 the
+proof uses a similar argument as for the one of i = 2. The key point in the latter is, for every
+x ∈ M, the restriction of the complexes (E, d, ρ) and (E′, d′, ρ′) at x are quasi-isomorphic. This
+implies that the codimension of im
+�
+d(i+1)
+x
+�
+inside ker d(i)
+x , resp. im
+�
+d′
+x
+(i+1)�
+inside ker d′
+x
+(i), is
+invariant.
+
+NASH RESOLUTIONS OF SINGULAR FOLIATIONS WITH A VIEW TOWARDS AFFINE VARIETIES
+13
+Let (V, x) ∈ �
+ME
+1 and (V ′, x) ∈ �
+ME′
+1 . We have
+dim V = dim V − dim(im (d(2)
+x ))
+= dim V − (dim ker ρx − dim ker ρ′
+x + dim(im (d′
+x
+(2)))
+= dim V − rk(E−1) + rk(E′
+−1) − dim(im (d′
+x
+(2)))
+= dim V ′ − dim(im (d′
+x
+(2)))
+= dim V ′.
+We have used the fact the cohomology groups at degree −1 of both complexes are isomorphic
+and the Rank–nullity theorem.
+For i = 2, let (V, x) ∈ �
+ME
+2 and (V ′, x) ∈ �
+ME′
+2 . Notice that dim V = rk(E−2) − rk(E−1) + r.
+We have a similar formula for dim V ′. By direct computation we find that
+dim V = dim V − dim(im d(3)
+x )
+= dim V − rk(E−2) + rk(E′
+−2) + dim(im (d(2)
+x )) − dim(im (d′
+x
+(2))) − dim(im (d′
+x
+(3))).
+(25)
+We have used the fact the cohomology groups at degree −2 of both complexes are isomorphic
+and the Rank–nullity theorem. But
+dim(im (d(2)
+x )) = rk(E−1) − dim(im(ρx)) − dim W,
+where W is such that dim(im (d(2)
+x )) ⊕ W = ker ρx. A similar formula holds for dim(im (d′
+x
+(2)))
+by adding ′ everywhere. Substituting them into the Equation (25) we obtain
+dim V = dim V ′ + dim W ′ − dim W = dim V ′,
+since dim W ′ = dim W.
+□
+Proof of (Theorem 2.5). For simplicity, we prove it for i = 1. For i ≥ 1, the same arguments
+hold.
+Let (E, d, ρ) and (E′, d′, ρ′) be geometric resolutions of F.
+There exists chain morphisms
+ϕ: E −→ E′ and ψ: E′ −→ E whose compositions are homotopic to identity. In particular, ϕ
+defines an isomorphism ϕ at the level of cohomology, the latter is canonical (see [LLS20], Lemma
+4.1). All we need to show is ϕ sends �
+ME
+1 to �
+ME′
+1 . The strategy is to show that for x ∈ M, the fiber
+�
+ME
+1 ∩Grr−dim(Lx)(E−1|x) over x is independent of any choices of minimal geometric resolutions at
+x, then deduce the result for every point y ∈ M, and for any two arbitrary geometric resolutions
+(E, d, ρ) and (E”, d”, ρ”) of F by introducing a minimal geometric resolution (E′, d′, ρ′) at y of
+F between them, then compose the underlying quasi-isomorphisms.
+Let x ∈ �
+ME
+1 . Like we said, let us assume that (E′, d′, ρ′) is minimal at x, i.e., d′
+x = 0. Let
+e1, . . . , ek be local sections around x of E−2 such that
+span
+�
+d(2)e1|x, . . . d(2)ek|x
+�
+= im(d(2)
+x ).
+There is a neighborhood Ux of x such that Fy = span
+�
+d(2)e1|y, . . . d(2)ek|y
+�
+⊆ im(d(2)
+y ) is of
+constant rank. These sections define a vector bundle F on Ux and Fx = im(d(2)
+x ). Likewise,
+
+14
+RUBEN LOUIS
+one consider the vector bundle F ′ ⊆ im(d′(2)) on a neighborhood of x such that ϕy(Fy) ⊆ F ′
+y.
+Therefore, ϕ induces a map
+im(d(2)
+y )
+Fy
+−→ im(d′(2)
+y
+)
+F ′y
+.
+The latter is injective, because d′
+x = 0.
+Since ϕy
+�
+im(d(2)
+y )
+Fy
+�
+֒→
+im(d′(2)
+y
+)
+F ′y
+, then ϕy
+�
+im(d(2)
+y )
+Fy
+�
+converges to W ⊆ im(d′(2)
+y
+)
+F ′y
+when y ∈ Mreg,F tends to x. Consequently, for (V, x), (V ′, x) such
+that im(d(2)
+y ) and im(d′(2)
+y
+) converges to V and V ′ respectively, one has, ϕx(V ) ⊆ V ′. Hence,
+ϕx(V ) = V ′. Also, also ψx(V ′) = V since ψx and ϕx is are the inverse of each other.
+□
+3.2. Proof of Theorem 2.12 and 2.14. Theorem 2.12 follows from Lemma 3.6 which itself
+requires Lemma 3.5. We prove those in the smooth context. We recall that for p: E −→ M a
+vector bundle over M, a linear vector field on E is a pair (Z, X) ∈ X(E) × X(M) such that
+E
+Z
+�
+p
+�
+TE
+dp
+�
+M
+X � TM
+is a morphism of vector bundles (see e.g [Mac05], p. 110). Equivalently,
+(1) Z[C∞
+lin(E)] ⊂ C∞
+lin(E) and Z[p∗C∞(M)] ⊂ p∗C∞(M).
+or
+(2) The flow of Z on E are (local) vector bundle morphisms E −→ E over the flow of X on
+M.
+where C∞
+lin(E) is the subalgebra of smooth functions on E which are fiberwise linear. The latter
+is canonically isomorphic to Γ(E∗) as C∞(M)-modules. Notice in particular that, a linear vector
+field is p-projectable to X.
+Lemma 3.5. A linear vector field on E −→ M induces a vector field on Π: Gr−r(E) −→ M
+that is Π-projectable on M.
+Proof. Let (Z, X) be a linear vector field on E −→ M. Its flow φZ
+t : E −→ E is a vector bundle
+isomorphism whenever it is defined, induces a diffeomorphism Gr−r(E) so that it induces a map
+Gr−r(E) −→ Gr−r(E), V �→ φZ
+t (V ) that we still denote by φZ
+t . Define �Z ∈ X(Gr−r(E)) for
+(V, x) ∈ Gr−r(E) by
+�Z(V ) := d
+dt |t=0
+c(t) ∈ T(V,x)Gr−r(E)
+(26)
+where c(t) =
+�
+φZ
+t |x(V ), φX
+t (x)
+�
+for t in some interval I. Also, �Z is Π-projectable to X, by
+construction.
+□
+Lemma 3.6. Let i ≥ 1. For every X ∈ F, there is a linear vector field (Zi, X) on the vector
+bundle pi : E−i −→ M or on p0 : E0 := TM −→ M, pi-projectable to X.
+Their flows are
+compatible with the complex of vector bundles,
+· · · ℓ1=d(4)
+−→ E−3
+ℓ1=d(3)
+−→ E−2
+ℓ1=d(2)
+−→ E−1
+ρ=d(1)
+−→ TM.
+(27)
+
+NASH RESOLUTIONS OF SINGULAR FOLIATIONS WITH A VIEW TOWARDS AFFINE VARIETIES
+15
+i.e., the diagram below commutes for all i ≥ 1,
+M
+φX
+t
+�
+�
+①①①①①①①①①
+M
+�
+①①①①①①①①①
+E−i
+φZi
+t
+�
+d(i+1)
+�
+E−i
+d(i+1)
+�
+M
+φX
+t
+�
+�
+①①①①①①①①①
+M
+�
+①①①①①①①①①
+E−i+1
+φZi−1
+t
+� E−i+1
+(28)
+and induces a vector field �Zi on Gr−ri(E−i) such that
+(1) �Zi is tangent to �
+Mi.
+(2) �Zi projects onto X.
+where φZi
+t
+or φX
+t
+denotes the flow of Zi or X, when defined.
+Proof. Let (E, ℓ1, ℓ2, ρ) be an almost graded Lie algebroid of F, see Section 1.2.1(5). Let X ∈ F
+and i ≥ 0. For i ̸= 0, there exists a section υ of the vector bundle pi : E−i → M such that
+ρ(υ) = X. Consider the linear vector field Zi ∈ X(E−i) defined as follows
+Zi[p∗
+i f] : = p∗
+i (X[f]), ∀ f ∈ C∞(M),
+(29)
+Zi
+e[α] : = X[⟨α, e⟩] − ⟨α, ℓ2(υ, e)⟩, ∀ α ∈ Γ(E∗
+−i), e ∈ Γ(E−i).
+(30)
+For i = 0, one replaces ℓ2(υ, e) in (30) by [X, Y ] with Y ∈ Γ(E0) = X(M). Notice that Zi
+depends on the choice of the almost graded Lie algebroid bracket ℓ2 and X. The items 1, 2, 3
+and 4 hold (see [LGLR22], Lemma 2.1.19. p. 61).
+By Lemma 3.5, the linear vector field (Zi, X) induces a vector field �Zi on the Grassmanian
+bundle Gr−ri(E−i). Let us show item 1, φZi
+t
+preserves �
+Mi: to see this take (V, x) ∈ �
+Mi, let
+xn
+−→
+n→+∞ x be such that im d(i+1)
+xn
+−→
+n→+∞ V with (xn) ⊂ Mreg,F. Since, d(i+1)◦φZi
+t
+= φZi−1
+t
+◦d(i+1)
+for i ≥ 0, one has
+φZi
+t |xn
+�
+im d(i+1)
+xn
+�
+= im d(i+1)
+φX
+t (xn),
+for every n ∈ N0.
+Thus,
+φZi
+t |x(V ) =
+lim
+n→+∞ φZi
+t |xn
+�
+im d(i+1)
+xn
+�
+=
+lim
+n→+∞
+�
+im d(i+1)
+φX
+t (xn)
+�
+∈ π−1 �
+φX
+t (x)
+�
+.
+By consequence, �Xi is tangent to �
+Mi, by Equation (26).
+□
+Proof (of Theorem 2.12). By Lemma 3.6, every vector field X ∈ F extends to a linear field
+Xi ∈ X(Gr−ri(E−i)) which is tangent to �
+Mi in the sense of Definition 3a. This proves item
+1. Furthermore, the restriction �Xi of Xi to �
+Mi is unique, since πi|π−1
+i
+(Mreg,F) : π−1
+i
+(Mreg,F) −→
+Mreg,F is invertible. In particular, the map X ∈ F −→ �X| �
+Mi does not depend on any choices
+and is a Lie algebra morphism. The module which is generated by the �Xi is closed under Lie
+bracket by item 2 of Theorem 2.12). This ends the proof.
+□
+Proof (of Theorem 2.14). Let (E, d, ρ) be a geometric resolution of F. Fix a universal Lie ∞-
+algebroid of F on (E, d, ρ). Let τ E−1 and AE−1 be the tautological subbundle and tautological
+quotient bundle on Gr−r(E−1), that fit into the exact sequence
+0 −→ τ E−1 −→ Π∗E−1 −→ AE−1 −→ 0.
+(31)
+
+16
+RUBEN LOUIS
+with AE−1 ≃ Π∗E−1/τ E−1. In particular, rk(AE−1) is the dimension of the regular leaves. One
+has
+(1) �F1 the image of an almost Lie algebroid on Π∗E−1| �
+M1 via the anchor map �ρ: Γ(Π∗E−1)| �
+M1 −→
+X( �
+M1) definded by π∗
+1e �−→
+�
+ρ(e) ∈ �F1.
+(2) The tautological subbundle τ E−1 lies in the kernel of the anchor map: indeed, the fiber
+of τ E−1 over a point (V, x) ∈ �
+M1 is equal to V by definition. By Proposition 2.6, the
+latter is included in ker ρx with equality if x ∈ Mreg,F.
+Therefore, the anchor map �ρ goes to quotient
+0
+� τ E−1
+� Π∗E−1
+�
+�ρ
+�
+AE−1
+�
+�✈ ✈ ✈ ✈ ✈
+0
+T �
+M1
+.
+(32)
+and makes �F1 the image of an almost Lie algebroid on AE−1| �
+M1 whose anchor is injective on
+the open dense subset Mreg,F.
+Thus, AE−1| �
+M1 is a Lie algebroid with anchor, injective on
+π−1
+1 (Mreg,F), whose image is �F1. This proves the result.
+□
+Remark 3.7. Notice that in the proof of Theorem 2.14 we do not need the existence of a
+geometric resolution, we only make use of the anchor map and the bracket of an almost Lie
+algebroid of F, i.e., we only need E−1 and ρ: E−1 −→ TM.
+4. Examples
+Let us start with some examples where our constructions give nothing new, i.e., �
+Mi ≃ M or
+�
+M∞ ≃ M.
+Example 4.1. If F is a projective singular foliation, then �
+Mi ≃ M, for all i ≥ 1 and i = +∞.
+This comes from the fact that there exists a vector bundle E−1 −→ M such that Γ(E−1) ≃ F by
+Serre-Swan theorem [Swa62, Mor13]. This isomorphism is given by a vector bundle morphism,
+E−1
+ρ→ TM which is injective on an open dense subset Mreg,F. As a consequence, E−1
+ρ→ TM
+is a geometric resolution of F. Therefore, �
+Mi ≃ M since E−i = 0 for i ≥ 2. Also, if r is the
+dimension of the regular leaves of F, then r = rk(E−1). Hence Gr−r(E−1) ≃ M. In particular,
+�
+M1 ≃ M.
+Example 4.2. If the regular leaves of F are open, then �
+M0 ≃ M, since Gr−0(TM) ≃ M.
+Example 4.3. If there exists a geometrical resolution (E, d, ρ) of length k, then �
+Mi ≃ M for all
+i ≥ k+1. Notice that one also has �
+Mk ≃ M since the last differential map d(k): E−k −→ E−k+1 is
+injective on an open dense subset so that the considered Grassmann bundle is Gr−rk(E−k)(E−k) ≃ M.
+In contrast with Examples 4.1, 4.2 and 4.3, we construct two related examples where our
+construction is not trivial.
+Example 4.4. Consider the projective singular foliation F generated by the Euler vector field
+−→
+E = �N
+i=1 xi ∂
+∂xi on M = CN. Here, Mreg,F = CN \ {0}. It is easily checked that �
+M0 is the
+closure of the graph {([x1 : · · · : xN], x) ∈ PN−1(C) × CN | x ̸= 0}. The latter is the blowup of
+CN at 0. This is an example where F is projective and �
+M0 ̸= M.
+Example 4.5. Let F be the singular foliation of all vector fields vanishing at the origin 0 ∈
+M = CN. Here, Mreg,F = CN \ {0}. Let us compute �
+M1. To do this, we only need E−1 the
+
+NASH RESOLUTIONS OF SINGULAR FOLIATIONS WITH A VIEW TOWARDS AFFINE VARIETIES
+17
+degree −1 of a geometric resolution (E, d, ρ) of F. See Example 3.34 of [LLS20] for a complete
+description of the latter. Here E−1 ≃ MN(C)×CN and the anchor map ρ is Eij �→ xi ∂
+∂xj , where
+MN(C) is the vector space of N ×N matrix with coefficient in C and (Eij)i,j=1,...,N its canonical
+basis.
+A direct computation for every x ̸= 0 tells that ker ρx is the subspace of matrices M ∈ MN(C)
+such that Mx = 0, where x = (x1, . . . , xN) is seen as a column vector. Equivalently, this kernel
+can be described as N copies of [x1 : · · · : xN]⊥. Hence, �
+M1 is the blowup of CN at the origin.
+This is an example of a singular foliation whose regular leaves are open, but such that �
+M1 ̸= M.
+Let us study some examples related to the notion of an affine variety in Cd.
+Let Ad be an affine space over K with a set of coordinates x1, . . . , xd. Recall that an affine
+variety W is a subset of the affine space Ad given by the zero locus Z(IW) of a radical ideal
+IW ⊆ K[x1, . . . , xd] and equipped with the induced Zariski topology of Ad. The coordinate ring
+of W is the quotient ring OW = K[x1, . . . , xd]/IW . The Lie algebra X(W) of vector fields on W
+are derivations of OW . We denote by Wreg the set of regular points of W. For every x ∈ Ad
+we denote by mx the maximal ideal of vanishing polynomials at x. See, for instance, [Har77] for
+more details on these notions.
+Example 4.6. Let M = Cd and ϕ ∈ C[x1, . . . , xd]. Consider the singular foliation Fϕ = {X ∈
+X(Cd) | X[ϕ] = 0}. In this case, Mreg,F = {x ∈ Cd, |, dxϕ ̸= 0}. For every y ∈ Cd, (TyFϕ)⊥ =
+⟨∇yϕ⟩.
+For a convergent sequence yn
+−→
+n→+∞ y with yn ∈ Wreg,F.
+The sequence im(ρyn) =
+TynF converges if and only if ∇ynϕ converges in Gr−(d−1)(Cd), that is,
+�
+∂ϕ
+∂x1 (yn): · · · :
+∂ϕ
+∂xd (yn)
+�
+converges in the projective space Pd−1(C). Therefore, �
+M0 is the closure of the image of the map,
+y �→ (y,
+� ∂ϕ
+∂x1(y): · · · :
+∂ϕ
+∂xd (y)
+�
+) which is the blow up of Cd along the singular locus of ϕ, i.e.,
+along dϕ = 0.
+Example 4.7. (Nash modification).
+Let M = W be an affine irreducible affine variety of
+dimension r embedded in Cd.
+Let Σ be its singular locus.
+Let F = Der(OW ) the singular
+foliation of vector fields on W tangent to Σ, where IΣ stands for the polynomial functions that
+vanish on Σ. Here, Wreg,F = Wreg = W \ Σ. Consider a geometric resolution (E•, d, ρ) of F by
+trivial vector bundles (which exists because OW is Noetherian, see Section 3.3 in [LLS20]).
+Let us show that for every x ∈ W \ Σ, im(ρx) = TxF = TxW. It is clear that im(ρx) ⊆ TxW.
+Conversely, it is a classical property that x ∈ W is a regular point if and only if there exists
+“local coordinates” y1, . . . , yd ∈ Ox such that W is of the form
+y1 = · · · = yk = 0,
+i.e., the localization of IW is generated by these variables, where Ox denotes the local ring at
+x. Hence, the tangent space of W at x is the vector space, span{ ∂
+∂yi |m, i ≥ k + 1}. Therefore,
+for v ∈ TxW the local vector field
+X =
+dim W
+�
+i=1
+vi
+∂
+∂yk+i
+maps Ox to Ox, in particular it maps O to Ox and we have X[IW ] ⊂ (IW )mx. Therefore, for
+every, i ∈ {1, . . . , d} there exists a polynomial function gi that does not vanish at x such that
+giY [xi] ∈ C[x1, . . . , xd]. By construction, the vector field ˆX =
+g1···gr
+g1(x)···gr(x)X is tangent to W, i.e.,
+ˆX[IW ] ⊂ IW, and satisfies ˆX(x) = v.
+
+18
+RUBEN LOUIS
+The map π0 : W \ Σ −→ Gr−(d−r)(Cd) x �−→ im(ρx) = TxW is the so-called Gauss map
+[LU81]. The Zariski closure �
+W0 of the image of such a map is by definition the classical Nash
+blow-up of W along its singular locus Σ.
+Example 4.8. (Monoidal transformation). Let W = Cd. Let C = Z(IC) ⊂ Cd be a subvariety
+of Cd defined by the ideal IC generated by f1, . . . , fk ∈ OW . Let F = ICX(W) the singular
+foliation of vector fields vanishing along C. By Hilbert’s Syzygy theorem [Eis04], there exists
+a free resolution of finite length for the ideal IC of polynomial functions vanishing on C of the
+form
+· · · −→ K−2
+∂
+−→ K−1
+∂
+−→ IC −→ 0
+(33)
+Since X(W) is a flat OW = C[x1, . . . , xd]-module (in fact X(W) ≃ Od
+W is a free module), the
+sequence
+· · ·
+∂⊗id
+� K−2 ⊗OW X(W)
+∂⊗id
+� K−1 ⊗OW X(W)
+ρ
+� F.
+(34)
+is a free resolution K[W] by finitely generated K[W]-modules of the singular foliation F =
+ICX(W), where for (µ1, . . . , µk) a set of generators of K−1 the map given by, ρ(µi⊗ ∂
+∂yj ) = fi ∂
+∂yj ,
+for i = 1, . . . , k and j = 1, . . . , d. By Theorem 2.1 in [LGL22], F admits a universal Lie ∞-
+algebroid structure over the complex (34) whose unary bracket is ℓ1 = ∂ ⊗ id and whose anchor
+is ρ.
+Here, Wreg,F = W \ C. A direct computation shows that, for every x ∈ W \ C, ker ρx is equal
+to d copies of [f1(x) : · · · : fk(x)]⊥, i.e.,
+ker ρx =
+�
+[f1(x) : · · · : fk(x)]⊥�d ,
+where [f1(x) : · · · : fk(x)] is a well-defined straight line of Kk generated by the vector (f1(x), . . . , fk(x)) ∈
+Kk seen as a point of the projective space Pk−1(K) = Gr−(k−1)(Ck).
+One has,
+π−1
+1 (x) =
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+�
+[f1(x) : · · · : fk(x)]⊥�d , for x ∈ W \ C,
+V d ∈
+�
+Gr−1(Ck)
+�d such that ∃ (xn) ∈ W N
+reg,F, [f1(xn) : · · · : fk(xn)]⊥
+−→
+n→+∞ V,
+with V ∈ Gr−1(Ck) as xn
+−→
+n→+∞ x, for x ∈ C.
+The d components converge if and only if one of them converges. Since [f1(xn) : · · · : fk(xn)]⊥
+converges in Gr−1(Kk) if and only if the straight line [f1(xn) : · · · : fk(xn)] converges in Pk−1(K),
+�
+W1 corresponds to the usual monoidal transformation of W with center C (see, for instance
+[Hau14]). In particular, �
+W1 does not depend, up to isomorphism over W, on the choice of the
+generators f1, . . . , fk.
+When f1, . . . , fk form a regular sequence, (33) can be chosen to be the Koszul complex. Then
+for each i ≥ 1, �
+Wi is again the blowup of Cd along C. In particular, �
+Wi = �
+W∞ is the blowup
+of W = Cd along C. Let us prove it. The proof is similar for every i ≥ 2. Let us show it for
+i = 2. Since (34) is given by E−i = �i M⊗X(Cd), where M is a C[x1, . . . , xd]-module generated
+by some degree −1 symbols µ1, . . . , µk and the differential map is defined by: for every i ≥ 1,
+j = 1, . . . , d and 1 ≤ j1 < · · · < ji ≤ k,
+d(i)(µj1 ∧ · · · ∧ µji ⊗
+∂
+∂xj
+) =
+i
+�
+l=1
+(−1)l+1fjlµj1 ∧ · · · ∧ �
+µjl ∧ · · · ∧ µji ⊗
+∂
+∂xj
+,
+
+NASH RESOLUTIONS OF SINGULAR FOLIATIONS WITH A VIEW TOWARDS AFFINE VARIETIES
+19
+here the hat �· means the term is omitted. The computations of �
+Mi are similar for all i ∈ N0,
+let us compute �
+M2. Let (gj
+pq) ∈ C[x1 . . . , xd] with j = 1, . . . , d and p, q = 1, . . . , k such that
+d(2)
+
+�
+p,q,j
+gj
+p,qµp ∧ µq ⊗
+∂
+∂xj
+
+ = 0. This implies that
+0 =
+�
+p,q,j
+gj
+p,q(fpµq − fqµp) ⊗
+∂
+∂xj
+=
+�
+q,j
+��
+p
+gj
+p,qfp
+�
+µq ⊗
+∂
+∂xj
+−
+�
+p,j
+��
+q
+gj
+p,qfq
+�
+µp ⊗
+∂
+∂xj
+.
+For x ∈ Wreg,F, one finds that for every (p, j) and (q, j) that (gj
+1,q(x), . . . , gj
+k,q(x)), (gj
+p,1(x), . . . , gj
+p,k(x)) ∈
+[f1(x) : · · · : fk(x)]⊥. This proves the result.
+Here is an example related to Poisson manifolds.
+Example 4.9. Let (M, P) a smooth or holomorphic Poisson manifold with P ∈ Γ(∧2TM).
+Consider the singular foliation generated by the Hamiltonian vector fields associated to P, i.e.,
+F = P ♯(Γ(T ∗M)), where P ♯ : T ∗M −→ TM, α �→ P(α, · ). Assume that a geometric resolution
+exists. By Lemma 3.6, every Hamiltonian vector field lifts to a vector field tangent to �
+Mi, i ≥ 1.
+It is natural to ask whether the bivector field P lifts to �
+Mi. Assume that �
+Mi is smooth. Since for
+every i ≥ 1, π−1
+i
+(Mreg,F) −→ Mreg,F is invertible, the restriction P|U lifts to a Poisson bivector
+field on π−1
+i
+(Mreg,F). However, it does not lift to �
+Mi in general. Indeed, consider the Poisson
+manifold M = so∗(3) ≃ R3 with
+P = x ∂
+∂y ∧ ∂
+∂z + y ∂
+∂z ∧ ∂
+∂x + z ∂
+∂x ∧ ∂
+∂y.
+(35)
+Here F is generated by the vector fields P ♯(dx) = z ∂
+∂y − y ∂
+∂z, P ♯(dy) = x ∂
+∂z − z ∂
+∂x, P ♯(dz) =
+y ∂
+∂x − x ∂
+∂y. Let us compute �
+M1. Given a point m ∈ Mreg,F = R3 \ {0}, we find that
+ker P ♯|m =
+�
+(a, b, c) ∈ R3 | (a, b, c) ∈ [x(m) : y(m) : z(m)] ∈ P2(R)
+�
+= [x(m) : y(m) : z(m)].
+Hence, �
+M1 is the usual blowup B0(R3) of R3 at the origin.
+The bivector field P does not lift to �
+M1. Recall that the blowup space of R3 at the origin
+B0(R3) ⊂ P2(R) × R3 is covered by three charts given by x ̸= 0, y ̸= 0 and z ̸= 0. Let us
+look at the x-chart where the projection π1 becomes (x, y, z) �→ (x, xy, xz). In this chart P pulls
+back to
+y ∂
+∂z ∧ ∂
+∂x + z ∂
+∂x ∧ ∂
+∂y + 1
+x(1 + y2 + z2) ∂
+∂y ∧ ∂
+∂x.
+(36)
+For x = 0 Equation (36) is not defined. In conclusion, the Hamiltonian vector fields of the
+Poisson structure P in (35) lift to �
+M1, but the bivector field P does not lift to �
+M1.
+Example 4.10. Let (E−1, [· , ·] , ρ) be a smooth or holomorphic Lie algebroid over a manifold
+M and denote by F = ρ(Γ(E−1)) the induced singular foliation. Assume there exists geometric
+resolutions. The Lie algebroid E−1 acts on the spaces �
+Mi for all i ∈ N0 also on �
+M∞, in a way
+that the following
+X( �
+M1)
+�
+Γ(E−1)
+ρ
+�✉
+✉
+✉
+✉
+✉
+ρ
+� X(M)
+(37)
+
+20
+RUBEN LOUIS
+is a commutative diagram of Lie algebra morphisms. In addition, for each i ∈ N0, �Fi is the image
+of a Lie algebroid on �
+Mi, namely the pull-back to �
+Mi of the Lie Algebroid E−1. In particular,
+if F is given by a Lie algebra action of a Lie algebra g on M, then �Fi is given by an action of g
+on �
+Mi.
+In the following example, we show that our �
+M1 is the blowup construction blup(F) given by
+Mohsen in Section 2.2 of [Moh21]
+Example 4.11. Let F be a singular foliation that admits a geometrical resolution (E, d, ρ).
+For every x ∈ M, blup(F)x is constructed out of minimal generators X1, . . . , Xd of F in a
+neighborhood of x as follows: for y ∈ Mreg,F, let φy be the surjective linear map defined by
+φy :
+F
+IxF −→ TyF, φy([Xi]x) = Xi(y),
+for all
+i ∈ {1, . . . , d},
+(38)
+where TyF is the image of the evaluation map ey : F −→ TyM at y. By definition, blup(F)x is
+made of subspaces V ⊆
+F
+IxF such that there exists a sequence xn ∈ Mreg,F such that
+xn −→ x, φ−1
+xn (0) −→ V ∈ Gr−r
+� F
+IxF
+�
+.
+(39)
+We claim that for every x ∈ M, blup(F)x ≃ π−1
+1 (x). Indeed, we can assume that (E, d, ρ) is
+a minimal geometric resolution at x such that ρ(ei) = Xi for i = 1, . . . , d, where (ei)i=1,...,d
+is a local frame of E−1. Since
+Γ(E−1)
+Ix′Γ(E−1) ≃ E−1|x′ for all x′ ∈ M, the anchor map defines an
+isomorphism ρx : E−1|x −→
+F
+IxF such that the diagram
+E−1|x
+≃
+ρx
+�
+≃
+κy
+�
+F
+IxF
+φy
+�
+E−1|y
+ρy
+� TyF
+(40)
+commutes. The claim follows.
+5. Conclusion
+Let us conclude this paper with an open question. Can we desingularize a singular affine
+variety W by applying the constructions above to the singular foliation F = X(W) of vector
+fields tangent to W? We should then understand �
+W0 as the Nash modification of W [LU81].
+The meaning of �
+W1 when F is the vector fields that vanish along a subvariety, is the monoidal
+transformation of W along of that subvariety. �F1 is projective, and is included into the vector
+fields tangent to �
+W1. It is logical to apply the construction again to �
+W1 or apply it to a sub-
+singular foliation of F.
+Notice that until now we have used the anchor map, the unary bracket and the 2-ary bracket
+of a universal Lie-∞-algebroid, we would like to go further to understand, e.g., the role of the
+3-ary bracket in this procedure.
+Acknowledgements.
+I would like to thank C. Laurent-Gengoux, my PhD supervisor, for
+supporting me in writing this article and directing me to such questions that are originated
+from Claire Debord and Georges Skandalis. Also, I thank the management of the University of
+Lorraine for having supported me financially through an ATER position.
+
+NASH RESOLUTIONS OF SINGULAR FOLIATIONS WITH A VIEW TOWARDS AFFINE VARIETIES
+21
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+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf,len=1060
+page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content='08706v1  [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content='DG]  20 Jan 2023  NASH RESOLUTIONS OF SINGULAR FOLIATIONS WITH A VIEW  TOWARDS AFFINE VARIETIES  RUBEN LOUIS  Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content=' We use the universal Lie ∞-algebroid of a singular foliation to construct several  notions of resolution of singularities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content=' For instance, we recover Nash modification or the monoidal  transformation of an affine variety and a resolution method due to Mohsen [Moh21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content=' One of the  important features is that any singular foliation becomes a Debord foliation after one blowup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content='  Some examples are also given.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content='  Contents  Introduction  1  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content='  Preliminaries  2  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content='  Grassmannian  2  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content='  Topological structure  2  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content='  Manifold structure  2  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content='  Singular foliations and universal Lie ∞-algebroids  3  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content='  Definitions  4  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content='  The blowup procedures  7  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content='  Blowup of vector bundle morphisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content='  7  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content='  Main constructions and results  8  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content='  Proof of the main results  10  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content='  Proof of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content='5  12  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content='  Proof of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content='12 and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content='14  14  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content='  Examples  16  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content='  Conclusion  20  References  21  Introduction  The results of this paper are taken from Chapter 7 of my PhD thesis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content=' It is shown in [LLS20,  LGL22] that behind any singular foliation there is a unique up to homotopy Lie ∞-algebroid,  i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content=', a Lie ∞-algebroid built over a geometric resolution of F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content=' Now, in [Moh21], Omar Mohsen  introduced a notion of blow-up of a smooth manifold along the singular leaves of a singular  foliation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content=' In this paper, we give an interpretation of the latter in terms of the universal Lie  ∞-algebroid of F, in fact in terms of an almost Lie algebroid over a geometric resolution of F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content='  Indeed, we show that this construction is the 1st type of a series indexed by N0 of blowups of  a singular foliation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content=' For n = 0, we recover the Nash blowups for a singular foliation, which  matches Nash blowups for an affine variety W when applied to the vector fields tangent to W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content='  1  2  RUBEN LOUIS  A consequence of our construction for n = 1 is that a resolution of any singular foliation can  be constructed, which is given by an action of a Lie algebroid whose anchor map is injective on  an open dense subset, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content=', a Debord foliation [Deb01].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content='  The paper is organized ad follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content=' In section 1, we recall the definition of Grassmann bundles  and their properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content=' Then, we recall the concept of singular foliations and its universal Lie  ∞-algebroids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content=' In section 2, we give the constructions of several notions of blowup of a singular  foliation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content=' Then, we state the main theorems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content=' In Section 3 we write the proofs of the results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content='  In section 4, we give some examples, where we recover the usual notions of blowups for affine  varieties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content=' Preliminaries  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content=' Grassmannian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content=' For E a finite dimension vector space over a field K ∈ {R, C}, we denote  by Gr−r(E) the set of all vector subspaces of E of co-dimension r ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content=' Let us recall a few facts  on Gr−r(E).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content=' Topological structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content=' Gr−r(E) is metric space, the corresponding metric is defined by  δ(V, V ′) = ∥pV − pV ′∥,  (1)  where pV stands for the orthogonal projection of E onto V ⊂ E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content=' It is important to notice that:  for all V, V ∈ Gr−r(E),  δ(V, V ′) = δ(V ⊥, V ′⊥)  here V ⊥ stands for the orthogonal space of V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content=' It is proven (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content=', [FGP94]) that Gr−r(E)  equipped with the topology induced by the so-called “gap” metric (1), is equivalent to the  Grassmann topology, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content=', the topology on Gr−r(E) whose open subsets W ⊆ Gr−r(E) are such  that τ −1(W) is open in Str(d, K) := {A ∈ Md×r(K) | rk(A) = r}, with  τ : Str(d, K) −→ Gr−r(E), A �−→ {vector space spanned by the columns of A}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content='  Also, Gr−r(E) is a compact space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content=' Manifold structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content=' Gr−r(E) is moreover a compact manifold of dimension r(d − r) and  also, a projective variety.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content='  (1) Coordinates charts: One manner to define the standard affine coordinates on the  Grassmannian Gr−r(E) is as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content=' Fix a basis e1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content=' , ed=dim E for E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content=' Let us describe  the first chart.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content=' Consider  ψ: Mr,d−r(K) −→ Md,d−r(K)  A′ �−→  �  Id−r  A′  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content='  The vector space V = τ  ��  Id−r  A′  ��  admits a basis of the form  vj := ej +  ℓ  �  k=1  akjek,  j = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content=' , d − r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content='  (2)  V is completely determined by the matrix A′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content=' Hence, τ ◦ ψ is the first chart.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content='  For a permutation σ ∈ Sd, let P(σ) be the permutation matrix of lines associated to  σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content=' We claim that the family τ ◦ P(σ) ◦ ψ(Mr,d−r(K)), indexed by σ ∈ Sd is an atlas of  Grr(E).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content=' Its image consists of (2) up to permutation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content='  NASH RESOLUTIONS OF SINGULAR FOLIATIONS WITH A VIEW TOWARDS AFFINE VARIETIES  3  (2) Grassmann bundle: For E → M a vector bundle of rank d over a manifold M (or a  quasi-projective variety 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content=' Let r ≤ d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content=' The disjoint union:  Gr−r(E) :=  �  x∈M  Gr−r(E|x)  comes equipped with a natural manifold structure in the smooth or complex case and a  quasi-projective variety structure when M is a quasi-projective variety.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content=' Also  Π: Gr−r(E) −→ M  (3)  is a fibration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content=' It is called (d − r)-th Grassmann bundle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content=' To fix notations, for x ∈ M,  elements of the fiber Π−1(x), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content=', points of Gr−r(E|x), are denoted by (V, x) with V ∈  Gr−r(E|x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content='  For every open subset U ⊂ M on which E is trivial, Π−1(U) ≃ Gr−r(Kd) × U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content=' An  adapted chart for Gr−r(E) −→ M around a point x ∈ M is a set of local coordinates  of the form (Π∗x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content=' Π∗xn, z1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content=' , zr(d−r)), where (x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content=' , xn) are local coordinates on  M and (z1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content=' , zr(d−r)) are functions which are standard affine coordinates on an open  subset of each fiber of Π as in item (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content='  Convention 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content=' Let x ∈ M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content=' Let e1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content=' , ed be local frame for E in a neighborhood U  of x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content=' For y ∈ U, let κy be the linear isomorphism defined by  κy : Ex −→ Ey, κy(ei(x)) = ei(y),  for all  i ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content=' , d}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content='  (4)  Let (xn) be a sequence of M that converges to x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content=' We will say that a sequence of vector  space Vxn ∈ Gr−r(E) with Vxn ⊂ Exn, converges to V ⊂ Ex and write Vxn  −→  n→+∞ V if  κ−1(Vxn) −→  n→+∞ V in Gr−r(Ex).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content='  (5)  In the sequel we will not mention κ, since this notion of convergence does not depend  on the chosen local frames of E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content='  (3) Tautological subbundle: The Grassmann bundle Gr−r(E) comes equipped with two  vector bundles τ E and AE, called tautological subbundle and tautological quotient bun-  dle, that fit into the exact sequence  0 −→ τ E −→ Π∗E −→ AE −→ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content='  (6)  Precisely, the fiber of τ E over the point (V, x) ∈ Gr−r(E|x) is the codimension r subvector  space V of E|x = E|Π(V,x) = (Π∗E)|(V,x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content=' By construction, τ E is a subbundle of the pull-  back bundle Π∗E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content=' Furthermore, AE ≃ Π∗E/τ E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content='  This tautological quotient bundle is important for us to express some results of this  paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content=' Singular foliations and universal Lie ∞-algebroids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content=' We refer the reader to [AS09,  AZ14, Cer79, Deb01, LLS20, LGLR22] for the topic of singular foliations, in particular to [LLS20,  LGL22] for the notion of universal Lie ∞-algebroids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content=' For Lie algebroids, see [Mac05].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content='  1the intersection inside some projective space of a Zariski-open and a Zariski-closed subset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content='  4  RUBEN LOUIS  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content=' Definitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content=' We recall some basic definitions and properties on singular foliations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content='  (1) A singular foliation on a smooth, real analytic, or complex manifold M, or a quasi-  projective variety, is a subsheaf F ⊆ X(M) that fulfills the following conditions,  (a) Stability under Lie bracket : [F, F] ⊆ F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content='  (b) F is a module over its respective relevant sheaf of functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content='  (c) Locally finitely generateness : every m ∈ M admits an open neighborhood U to-  gether with a finite number of vector fields X1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content=' , Xk ∈ X(U) such that for every  open subset U ⊆ M the vector fields X1|V, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content=' , Xk|V generates F on V as a mod-  ule over functions on V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content=' This condition is only necessary in the smooth case (see,  [LGLR22]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content='  There are two classes of singular foliations we are interested in.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content='  A locally polynomial singular foliation is a singular foliation over a smooth or complex  manifold which admits, around each point, polynomial generators in some local chart.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content='  Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content=' A singular foliation F is projective as a module over functions if and  only if there exists a Lie algebroid (A, [· , ·] , ρ) such that ρ(Γ(A)) = F whose anchor is  injective on an open dense subset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content='  These singular foliations are called Debord singular foliations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content='  (2) Here are some important features of the above definition in the smooth/complex cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content='  Singular foliation admits leaves : there exists a partition of M into submanifolds  called leaves such that for all m ∈ M, the image of the evaluation map F → TmM  is the tangent space of the leaf through m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content='  Singular foliations are self-preserving: the flow φX  t of vector fields X ∈ F, whenever  defined, preserves F [AS09, GY18], i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content=',  ∀ m ∈ M, ∃ ε > 0 such that ∀t ∈] − ε, ε[, (φX  t )∗(F) = F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content='  (3) We introduce here a notion of tangency and singular foliation for arbitrary closed subsets  of a smooth or complex manifold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content=' Let M be a smooth or complex manifold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content=' Let S ⊆ M  be a closed subset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content='  (a) A vector field Z on M is said to be tangent to S if its flows φZ  t preserves S, [MP82].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content='  We define X(S) to be the restriction to S of vector fields of M tangent to S, it is a  OM/IS -module and a Lie algebra, with IS being the ideal of smooth functions on  M that vanish on S .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content='  (b) A singular foliation over S is a OM/IS -submodule F of X(S) which is  (i) finitely (locally) generated,  (ii) stable under Lie bracket.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content='  It is said to be projective if it is projective as a module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content='  (c) A Lie algebroid over S is a locally generated projective Lie-Rinehart algebra over  OM/IS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content='  Remark 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content=' The previous definitions match the usual notions of tangency and singular  foliations, for affine or Stein subvarieties in the algebraic or holomorphic cases, and for  S a smooth submanifold in the smooth case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content='  NASH RESOLUTIONS OF SINGULAR FOLIATIONS WITH A VIEW TOWARDS AFFINE VARIETIES  5  (4) Let F ⊆ X(M) be a submodule.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content=' A complex of vector bundles (E, d, ρ)  � E−i−1  d(i+1) �  �  E−i  d(i) �  �  E−i+1  �  �  d(2)� E−1  ρ  �  �  TM  �  M  M  M  M  M  is said to be a geometric resolution of F if the following complex is an exact sequence of  sheaves:  −→Γ(E−i−1) d(i+1)  −→ Γ(E−i) d(i)  −→ Γ(E−i+1)−→ · · · −→Γ(E−1)  ρ  −→ F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content='  (7)  A geometric resolution is said to be minimal at a point x ∈ M if, for all i ≥ 2, the  linear maps d(i)|x : E−i|x −→ E−i+1|x vanish.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content=' Recall that it exists in many contexts,  e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content=', singular foliation with polynomial generators on Rn or Cn or locally real analytic  singular foliation on a compact manifold (see, [LLS20]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content='  (5) An almost graded Lie algebroid over M is the datum of a sequence (E, d = ℓ1, ρ) of  vector bundles over M equipped with a graded symmetric degree +1 K-bilinear bracket  ℓ2 : Γ(E) ⊙ Γ(E) → Γ(E)  such that:  (a) ℓ2 satisfies the Leibniz identity with respect to ρ: Γ(E−1) −→ X(M), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content=',  ℓ2(x, fy) = fℓ2(x, y) + ρ(x)[f]y  (8)  for all x ∈ Γ(E−1), y ∈ Γ(E) and f ∈ O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content='  (b) ℓ1 is degree +1-derivation of ℓ2, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content=', for all x ∈ Γ(E−i), y ∈ Γ(E):  ℓ1(ℓ2(x, y)) + ℓ2(ℓ1(x), y) + (−1)iℓ2(x, ℓ1(y)) = 0,  (c) ρ is a morphism, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content=', for all x, y ∈ Γ(E−1)  ρ(ℓ2(x, y)) = [ρ(x), ρ(y)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content='  The O-linear map ρ is called the anchor map, and ℓ1 the differential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content='  (6) A Lie ∞-algebroid over M is the datum of a sequence E = (E−i), 1 ≤ i < ∞ of vector  bundles over M together with a structure of Lie ∞-algebra (ℓk)k≥1 on the sheaf of  sections of E and a vector bundle morphism, ρ: E−1 → TM, called anchor map such  that the k-ary brackets ℓk, k ̸= 2 are O-multilinear and such that  ℓ2(e1, fe2) = ρ(e1)[f]e2 + fℓ2(e1, e2)  (9)  for all e1 ∈ Γ(E−1), e2 ∈ Γ(E•) and f ∈ O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content=' The sequence  · ·  ℓ1 � E−2  ℓ1  � E−1  ρ  � TM,  (10)  is a complex called the linear part of the Lie ∞-algebroid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content='  (7) The following theorem is important, (see [Lav17, LLS20, LGL22] for more details) :  Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content=' Let F be a singular foliation over M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content=' Any geometric resolution of F  · ·  d  −→ E−3  d  −→ E−2  d  −→ E−1  ρ  −→ TM  (11)  comes equipped with a Lie ∞-algebroid structure whose unary bracket is d and whose  anchor map is ρ (in particular ρ(Γ(E−1)) = F).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content=' Such a Lie ∞-algebroid structure is  unique up to homotopy and is called a universal Lie ∞-algebroid of F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content='  6  RUBEN LOUIS  In particular, this structure can be truncated to an almost graded Lie algebroid for F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content='  (8) Let (E•, ℓ•, ρ) a universal Lie ∞-algebroid of a singular foliation F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content=' For every point  x ∈ M,  (a) We let H•(F, x) = ⊕i≥1H−i(F, x) be the cohomology of the complex (11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content=' The  cohomology groups H•(F, x) do not depend on the choice of a geometric resolution  of F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content=' Notice that when the complex (11) is minimal at x, H−i(F, x) ≃ E−i|x for  every i ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content='  (b) The 1, 2-ary brackets restrict to the graded vector space  \uf8eb  \uf8ed�  i≥2  E−i|x  \uf8f6  \uf8f8 ⊕ ker(ρx)  and equip the latter with an almost graded Lie ∞-algebra structure as follows : for  every k ∈ {1, 2},  {x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content=' , xk}k := ℓk(s1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content=' , sk)|x  (12)  for all x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content=' , xk ∈ ev(E, x) and s1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content=' , sk ∈ Γ(E) sections of E such that si(x) = xi  with i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content=' , k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content='  The bracket {· , · }2 induces a graded Lie algebra on H•(F, x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content=' In particular, the 2-ary  bracket {· , ·}2 satisfies the Jacobi identity on H−1(F, x) = ker(ρx)  im(d(2)  x ), and equips the latter  with a Lie algebra structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content='  (9) Let (M, F) be a singular foliation, let Ix := {f ∈ C∞(M) | f(x) = 0} and F(x) :=  {X ∈ F | X(x) = 0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content=' The quotient gx = F(x)  IxF is a Lie algebra and is called the isotropy  Lie algebra of F at x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content='  Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content=' Let (E, ℓ•, ρ) be a universal Lie ∞-algebroid of F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content=' Consider its underlying  geometric resolution  (E, d, ρ) :  · · ℓ1=d(4)  −→ E−3  ℓ1=d(3)  −→ E−2  ℓ1=d(2)  −→ E−1  ρ=d(1)  −→ TM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content='  (13)  Then,  (a) for all x ∈ M, we have H−1(F, x) ≃ gx as Lie algebras;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content='  (b) the subset of regular points of F in M satisfies  Mreg,F = {x ∈ M | rk(d(2)  x ) = dim(ker ρx)}  = {x ∈ M | H−i(F, x) = 0, ∀i ≥ 1},  Mreg,F is open and dense in M;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content='  (c) the restriction of the foliation F to Mreg,F is the set of sections of a subbundle of  TM, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content=', is a regular foliation;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content='  (d) For every i ≥ 0, the dimension of im  �  d(i+1)�  is locally constant on Mreg,F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content=' More-  over, if r the dimension of a regular leaf, then im(d(i+1)) is of codimension  ri =  i−1  �  j=1  (−1)j+1rk(E−j) + (−1)i+1r, for i ≥ 1  in E−i or r0 = dim M − r, with E0 := TM;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content='  (e) if (E, d, ρ) is of finite length, then (in the smooth case) all the regular leaves have  the same dimension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content='  NASH RESOLUTIONS OF SINGULAR FOLIATIONS WITH A VIEW TOWARDS AFFINE VARIETIES  7  In what follows, we assume that a geometric resolution of finite length exists.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content='  Under these  assumptions, all the regular leaves have the same dimension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content=' Denote by r the common dimension  of the regular leaves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content=' The blowup procedures  In what follows M can be a connected complex or smooth manifold or an irreducible affine  or quasi-projective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content=' Firstly, let us explain a general construction on morphisms of vector bundles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content=' Blowup of vector bundle morphisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content=' Let E, F be vector bundles over M and  F  d  �  �❆  ❆  ❆  ❆  ❆  ❆  ❆  ❆  E  �⑥⑥⑥⑥⑥⑥⑥⑥  M  a morphism of vector bundles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content=' In the smooth case, we assume that d is of constant rank on  an open dense subset Mreg,d ⊂ M, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content=', the dimensions of im(dx) or ker(dx) are constant for  x ∈ Mreg,d, called the regular part (this is automatically true when M is complex or a quasi-  projective variety).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content=' Let r be the codimension of im(dx) ⊆ Ex for a point x ∈ Mreg,d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content=' Notice that  for every x ∈ Mreg,d, im(dx) is a point of the Grassmannian Gr−r(Ex) and ker(dx) is a point of  Grrk(F )−r(Fx).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content=' We consider the natural section of Gr−r(E) −→ M which is defined on Mreg,d  by:  σ: Mreg,d −→ Gr−r(E), x �−→ (im(dx), x) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content='  (14)  Then we define �  M := σ(Mreg,d) to be the closure of the image of the section σ in Gr−r(E),  together with the projection π: �  M −→ M, where π denotes the restriction of Π: Gr−r(E) −→ M  to �  M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content='  Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content=' Intuitively, for x ∈ M, π−1(x) = �  M ∩ Π−1(x) is the set of all possible limits of  the images imdy when y ∈ Mreg,d converges to x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content=' One can make a similar construction with the  kernel of d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content='  Here is an immediate property of that construction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content='  Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content=' Let F  d  −→ E be a vector bundle morphism over M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content=' The projection π: �  M →  M has the following property:  (1) π is proper and surjective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content=' In particular, for each point x ∈ M, the fiber π−1(x) is  non-empty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content='  (2) For every x ∈ M and V ∈ π−1(x), one has im(dx) ⊆ V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content='  (3) For every x ∈ Mreg,d, π−1(x) = im(dx) is reduced to a point in Gr−r(E).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content='  Also,  π−1(Mreg,d) is a manifold 2 and the restriction π: π−1(Mreg,d) −→ Mreg,d is invertible3  in the smooth and holomorphic contexts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content=' We prove it in the smooth and complex settings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content=' Properness derives from the fact that  the projection Π admits compact fibers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content=' For any x ∈ M, choose U ⊂ M an open neighborhood  of x that trivializes E −→ M over U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content=' Then, Gr−r(E) ≃ Gr−r(Krk(E)) × U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content=' Notice that,  π−1(x) =  �  V ⊂ Ex  ���� ∃ (xn) ∈ MN  reg,d, such that, im(dxn) −→  n→+∞ V as xn  −→  n→+∞ x  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content='  2Manifold is to be understood as quasi-projective when M is quasi-projective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content='  3Invertible here means: diffeomorphism, in the smooth case;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content=' bi-holomorphism, in the complex case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content='  8  RUBEN LOUIS  For any sequence (xn) in (Mreg,d ∩ U)N that converges to x, we can extract a sequence (xϕ(n))  such that n �→ im(dxϕ(n)) ∈ Gr−r(Krk(E)) has a limit V , since the Grassmannian manifold  Gr−r(Krk(E)) is compact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content=' Hence, π−1(x) ̸= ∅ and π is onto.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content=' This proves item 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content='  Let us show item 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content='  Let V ∈ π−1(x) and (xn) ∈ (Mreg,d)N such that xn  −→  n→+∞ x and  im(dxn) −→  n→+∞ V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content=' Let v ∈ im(dx).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content=' We have v = dxu for some u ∈ Fx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content=' Choose a (local) section  �u of F through u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content=' By continuity, dxn �u(xn) −→  n→+∞ dxu, hence dxu ∈ V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content=' Thus, im(dx) ⊆ V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content='  In particular, if x ∈ Mreg,d and V ∈ π−1(x) one has im(dx) = V since dim V = dim(im(dx)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content='  Therefore, π−1(Mreg,d) is the image of the map σ on Mreg,d, it is isomorphic/biholomorphic to  Mreg,d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content=' This proves item 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content='  □  We are now going to apply the constructions above to a sequence of vector bundle morphisms  which are all of constant rank on an open dense subset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content=' Main constructions and results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content=' Let us work in the complex or holomorphic setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content='  Let F be a locally finitely generated O(M)-submodule of X(M), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content=', every point of M admits  an open neighborhood U and a finite number of vector fields X1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content=' , Xn ∈ X(M) such that  F|U = �n  k=1 fkXk for fk ∈ O(M).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content=' We assume that there exists a geometric resolution, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content=',  complex of vector bundles (E•, d(•), ρ) of finite length  0 · · ·  � E−i−1  d(i+1) �  �  E−i  d(i) �  �  E−i+1  �  �  d(2)� E−1  ρ=d(1)  �  �  TM  �  M · · ·  M  M  M  M  M  (15)  such that ρ(Γ(E−1)) = F and exact as in (7), (see [LLS20] for conditions on existence of geometric  resolutions at least locally).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content=' Denote by Mreg,F (as in Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content='5) the open dense subset such  that the fibers of im(d(i)) are of constant rank and im(d(i+1)) = ker(d(i)), for all i ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content=' We set  E0 := TM by convention and proceed as the following:  (a) For every i ≥ 0, let Πi : Gr−ri(E−i) −→ M be the Grassmann bundle of E−i with ri is  as in Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content='5 (d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content=' Consider the natural section of Πi on Mreg,F defined by :  σi : Mreg,F −→ Gr−ri(E−i), x �−→  �  im  �  d(i+1)  x  �  , x  �  (16)  (b) Let �  Mi := σi(Mreg,F) be the closure of the image of σi in Gr−ri(E−i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content=' Let πi : �  Mi −→ M  denote the restriction of Πi to �  Mi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content='  We also consider the section  σ∞: Mreg,F −→  �  x∈M  �  i≥1  Gr−ri(E−i|x), x �→ (σ1(x), σ2(x), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content=' , σi(x), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content=' )  and define �  M∞ := σ∞(Mreg,F).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content=' Here, �  M∞ should be understood as the tuples made of elements  V1 ∈ Gr−r1(E−1|x), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content=' , Vi ∈ Gr−ri(E−i|x), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content=' such that there exists (xn) ∈ MN  reg,d such that  im  �  d(i+1)  xn  �  −→  n→+∞ Vi as xn  −→  n→+∞ x for all i ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content=' It is important to notice that the Vi’s are  given by the same sequence (xn) ∈ MN  reg,F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content='  Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content=' The construction also makes sense for M an affine variety in CN, upon replacing  TM by TCN|M ≃ M × CN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content=' Notice that in the definition of �  M∞ we leave �  M0 out.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content='  NASH RESOLUTIONS OF SINGULAR FOLIATIONS WITH A VIEW TOWARDS AFFINE VARIETIES  9  By Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content='2, for each i ≥ 0, the projection πi : �  Mi → M is invertible on the open dense  subset Mreg,F, it is proper and surjective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content=' Moreover, for each point x ∈ M and for every i ≥ 0,  the fiber π−1  i  (x) is non-empty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content=' Also, π−1  ∞ (x) is non-empty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content='  From now on, most of the proofs are delayed to Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content='  As sets, �  Mi, �  M∞ do not need to be manifolds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content=' They can be singular, see Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content='  Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content=' Let F be a singular foliation on a smooth or complex variety on M ∈  �  CN, RN�  that admits polynomial generators, then it admits a geometric resolution of finite  length and �  Mi and �  M∞ are quasi-projective varieties for all i ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content='  For F a singular foliation on an affine variety, �  Mi is a quasi-projective variety.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content='  The following assertion follows from the existence of homotopy equivalence between any two  geometric resolutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content='  Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content=' Let F be a singular foliation on M that admits geometric resolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content=' For each  i ≥ 1, �  Mi does not depend on the choice of a geometric resolution of F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content=' The same is true for  �  M∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content='  To prove Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content='5, we first establish the following results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content='  Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content=' Let (15) be a geometric resolution for F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content=' For every x ∈ M, for every i ≥ 1  and V ∈ π−1  i  (x) one has,  im(d(i+1)  x  ) ⊆ V ⊆ ker(d(i)  x ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content='  (17)  In particular, for all x ∈ Mreg,F and i ≥ 1, ker(d(i)  x ) = im(d(i+1)  x  ) = π−1  i  (x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content='  Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content=' Fix a geometric resolution (E, d, ρ) of F and a universal Lie ∞-algebroid of  F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content=' The following are satisfied:  (1) For every x ∈ M and V ∈ π−1  1 (x), the 2-ary bracket {· , · }2 on ker ρx restricts to V and  the image of V in H−1(F, x) ≃ gx, is a Lie subalgebra of codimension r−dim(Lx), where  dim(Lx) is the dimension the leaf through x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content='  (2) For all x ∈ M, and (V1 ⊂ E−1|x, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content=' , Vk ⊂ E−k|x, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content=') ∈ π−1  ∞ (x) we have {Vi, Vj}2 ⊂  Vi+j−1 for every i, j ∈ N0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content='  The corollary below is a direct consequence of Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content='6, and is another manner to state  that Mi does not depend on the geometric resolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content='  Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content=' There are inclusions  �  Mi ֒→  �  x∈M  Gr  rk  �  d(i)  x  �  −ri(H−i(F, x))  and  �  M∞ ֒→  �  x∈M  �  i≥1  Gr  rk  �  d(i)  x  �  −ri(H−i(F, x)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content='  (18)  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content=' Let x ∈ M and i ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content=' By Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content='6, elements V ∈ π−1  i  (x) satisfy im(d(i+1)  x  ) ⊆ V ⊆  ker(d(i)  x ), they correspond injectively to a (unique) sub-vector space of codimension ri−rk(d(i)) in  H−i(F, x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content=' In particular, this implies the existence of an inclusion π−1  i  (x) ֒→ Grri−rk(d(i))(H−i(F, x)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content='  □  Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content=' In particular, �  M1 ֒→ ⊔x∈MGrLiedim(Lx)−r(gx), where gx is the isotropy Lie algebra  of the singular foliation F at x and Lx is the leaf that passes through x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content=' Here, GrLier−dim(Lx)(gx)  denotes the Grassmannian of Lie subalgebras of gx of codimension r − dim(Lx).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content='  10  RUBEN LOUIS  Assume now that F is a singular foliation and that Equation (15) is a geometric resolution  for F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content='  Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content=' Let i ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content=' We say that X ∈ F lifts to �  Mi ⊂ Gr−ri(E−i), or �  M∞, if there  exists a vector field �X ∈ X(Gr−ri(E−i)) or X  ��  x∈M  �  i≥1 Gr−ri(E−i|x)  �  , projectable to X and  tangent to Mi in the sense of Section 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content='1 (3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content=' We denote by �Xi or �X∞ the restriction of �X to  Mi or �X∞ respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content='  We say that a F lifts to �  Mi if every vector field X ∈ F lifts to �  Mi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content='  Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content=' �Xi on π−1  i  (Mreg,F) is tangent in the usual sense to the submanifold and projects  to X through πi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content=' In particular, if a lift exists, its restriction to π−1(Mreg,F)) is unique, because  πi : π−1  i  (Mreg,F)  ∼  −→ Mreg,F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content=' Since the other points of �  Mi are limits of elements of π−1  i  (Mreg,F),  thus its restriction to �  Mi is unique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content='  Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content=' Let F be a singular foliation on M that admits geometric resolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content=' For every  i ≥ 1, the following items hold:  (1) Every vector field X ∈ F lifts to a vector field �Xi on �  Mi,  (2) the map X ∈ F −→ �X| �  Mi does not depend on any choices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content=' In particular, it is a Lie  algebra morphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content='  (3) The module �Fi over functions of �  Mi generated by the  �X′  is for X ∈ F is a singular  foliation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content='  The same holds for �  M∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content='  Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content='13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content=' For each i ≥ 1, �Fi is called the blowup of F on �  Mi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content='  Here is a remarkable fact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content='  Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content='14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content=' Let F be a singular foliation on M that admits geometric resolution (E, d, ρ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content='  �F1 is projective, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content=', it is the image of a Lie algebroid over �  M1 whose anchor map is injective  on an open dense subset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content='  In Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content='14, we do not need the existence of geometric resolutions of F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content=' An almost Lie  algebroid of F is enough, the latter always exists [LLS20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content=' Proof of the main results  In this section, we prove Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content='4, Proposition2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content='6, Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content='7 and Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content='  Proof (of Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content=' Since M = RN or CN and F is generated by polynomial vector fields,  we can choose a polynomial geometric resolution (E, d, ρ) of F by trivial vector bundles, [LLS20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content='  Here by polynomial we mean the d and ρ are given by polynomials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content='  Let e1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content=' , ed resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content=' e′  1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content=' , e′  d′ be a basis by constant sections of E−i resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content=' E−i+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content=' One has,  d(i)(el) =  d′  �  k=1  f k  l e′  k  (19)  for some polynomial functions f k  l on KN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content=' Without any lost of generality, let us describe �  Mi using  local coordinates on Gr−ri(E−i) consider for example the first standard coordinates chart U1 for  the Grassmannian Gr−ri(E−i), (see Section 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content=' Denote by x = (x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content=' , xN) the coordinates  NASH RESOLUTIONS OF SINGULAR FOLIATIONS WITH A VIEW TOWARDS AFFINE VARIETIES  11  on M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content=' Let m ∈ M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content=' For W ∈ U1∩Gr−ri(E−i|m) and (apq(m)) ∈ Md,d−ri(K) be the homogeneous  coordinates of W, let us define the sections  �wq(x) := eq(x) +  ri  �  k=1  akq(x)ek(x),  q = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content=' , d − ri.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content='  (20)  By construction, the �wq(x)’s, evaluated at m, form a basis for W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content=' Notice that we have,  d(i)( �wq)(x) =  d′  �  k=1  �  f k  q (x) +  ri  �  s=1  asqf k  s (x)  �  e′  k(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content='  Therefore, W ⊆ ker d(i)  m if and only if  f k  q (m) +  ri  �  s=1  asq(m)f k  s (m) = 0,  q = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content=' , d − ri  k = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content=' , d′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content='  (21)  This defines an affine variety.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content=' As a result, �  Mi is given in local coordinates U1 by elements that  satisfy  (1) Equation (21)  (2) and that are limits of solutions of (21) in nearby regular points, elements which are  unique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content='  Hence, it is, on the affine variety, the irreducible components of (21) that  projects onto M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content='  This concludes the proof in the smooth or complex cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content=' For a quasi-projective variety, the  proof is similar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content='  □  Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content=' When F is generated by polynomial vector fields, π−1  i  (Mreg,F) is locally defined  by polynomial equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content='  Proof (of Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content='6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content=' We know by Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content='2(2) that, for every x ∈ M and V ∈  π−1  i  (x), one has im(d(i+1)  x  ) ⊆ V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content=' Now, for any element v ∈ V , there exists a sequence vn ∈  ker(d(i)  xn) = im(d(i+1)  xn  ), n ∈ N that converges to v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content=' In particular, d(i)  xn(vn) = 0 for all n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content=' Hence,  by continuity, one has v ∈ ker(d(i)  x ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content=' Hence, V ⊆ ker d(i)  x .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content=' This completes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content='  □  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content=' (of Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content='7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content=' For all i ≥ 1, choose a local frame e(i)  1 , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content=' , e(i)  qi , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content=' e(i)  qi+ri of E−i on a  neighborhood U of x such that e(i)  1 (x), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content=' , e(i)  qi (x) is an orthogonal basis for Vi for an arbitrary  Hermitian structure on E−i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content=' For i, j ≥ 1, let (cij,s  kl ) ∈ O(U) be a family of functions over U such  that for all k ≤ qi and l ≤ qj,  ℓ2  �  e(i)  k , e(j)  l  �  =  �  s≥1  cij,s  kl e(i+j−1)  s  ∈ ΓU(E−i−j+1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content='  In particular,  �  e(i)  k (x), e(j)  l (x)  �  2 =  �  s≥1  cij,s  kl (x)e(i+j−1)  s  (x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content='  (22)  The bracket in Equation 22 is well-defined even for i = 1 or j = 1, although only the 2-ary  bracket of local sections is defined in such cases, because even if i or j = 1, we are taking the  brackets of elements in ker ρx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content=' Let u ∈ Vi, v ∈ Vj with u =  qi  �  s=1  αse(i)  s (x), and v =  qj  �  s=1  βse(j)  s (x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content='  12  RUBEN LOUIS  Let (xn) ∈ MN  reg,F be a sequence that converges to x such that im(d(i+1)  xn  )  −→  n→+∞ Vi and  im(d(j+1)  xn  ) −→  n→+∞ Vj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content=' There exist sequences  un =  qi+ri  �  k=1  αk  ne(i)  k (xn) −→  n→+∞ u;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content='  vn =  qj+rj  �  l=1  βl  ne(j)  l (xn) −→  n→+∞ v  with un ∈ im(d(i+1)  xn  ) = ker d(i)  xn and vn ∈ im(d(j+1)  xn  ) = ker d(j)  xn, for all n ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content=' In particular,  the sequences (αk  n), (βl  n) ∈ KN satisfy αk  n  −→  n→+∞ αk;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content='  βl  n  −→  n→+∞ βl with αk = βl = 0 for  k ≥ qi + 1, l ≥ qj + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content=' Therefore, for every n ∈ N we have  �  αk  nβl  ncij,s  kl (xn)e(i+j−1)  s  (xn) = {un, vn}2 ∈ im(d(i+j)  xn  ) = ker d(i+j−1)  xn  ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content='  (23)  We have used in (23), the fact that {du1, du2}2 ∈ im(d), for all u1, u2 ∈ E≤−2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content=' Since  �  αk  nβs  ncij,s  kl (xn)e(i+j+1)  s  (xn) −→  n→+∞  �  αkβlcij,s  kl (x)e(i+j−1)  s  (x) ∈ E−i−j+1|x  = {u, v}2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content='  (24)  As a result, {u, v}2 ∈ Vi+j−1 ∈ π−1  i−j−1(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content=' Hence, for every point (V1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content=' , Vi, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content=' , Vj, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content=' ) ∈ π−1  ∞ (x)  one has {Vi, Vj}2 ⊆ Vi+j−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content=' This proves item 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content=' By taking i = j = 1 and Vi = Vj = V ∈ π−1  1 (x),  Equation (24) means that {u, v}2 ∈ V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content=' This proves item 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content='  □  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content=' Proof of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content=' In this section, we give a proof of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content=' By Corollary  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content='8 (whose proof is independent of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content='5), for every i ≥ 1, we have an inclusion �  Mi ֒→  �  x∈M Gr  rk  �  d(i)  x  �  −ri(H−i(F, x)), where ri is defined as in Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content='5(d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content=' We now need to show  this inclusion is canonical, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content=', independent of the choice of a geometric resolution (E, d, ρ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content='  Convention 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content=' For (E, d, ρ) a geometric resolution of F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content=' Denote by �  ME  i  := �  Mi constructed  out of (E, d, ρ) and �  ME′  i  := �  Mi constructed out of (E′, d′, ρ′) for i ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content=' Also, for x ∈ M and  V ∈ π−1  i  (x), we denote by V the image of V in Gr  rk  �  d(i)  x  �  −ri(H−i(F, x)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content='  Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content=' Let x ∈ M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content=' Consider a minimal geometric resolution (E′, d′, ρ′) of F at x (see  Definition (4)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content=' For (V, x) ∈ �  ME  1  and (V ′, x) ∈ �  ME′  1  one has that dim V ′ ≤ dim V , because  rk(E′  −1) ≤ rk(E−1) by minimality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content=' Hence, V, V ′ do not necessarily belong to the same Grass-  mannian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content=' However, dim V = dim V ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content=' We prove the latter in the next Lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content='  Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content=' Let (E, d, ρ) and (E′, d′, ρ′) be geometric resolutions of F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content=' For all i ≥ 1, and for  all (V, x) ∈ �  ME  i  and (V ′, x) ∈ �  ME′  i , one has, dim V = dim V ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content=' If x ∈ M is a regular point, then V = V ′ = {0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content=' Thus, the equality holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content=' Let x ∈ M  be a singular point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content=' We prove it only for i = 1, 2, since i = 1 is a special case and for i ≥ 3 the  proof uses a similar argument as for the one of i = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content=' The key point in the latter is, for every  x ∈ M, the restriction of the complexes (E, d, ρ) and (E′, d′, ρ′) at x are quasi-isomorphic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content=' This  implies that the codimension of im  �  d(i+1)  x  �  inside ker d(i)  x , resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content=' im  �  d′  x  (i+1)�  inside ker d′  x  (i), is  invariant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content='  NASH RESOLUTIONS OF SINGULAR FOLIATIONS WITH A VIEW TOWARDS AFFINE VARIETIES  13  Let (V, x) ∈ �  ME  1 and (V ′, x) ∈ �  ME′  1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content=' We have  dim V = dim V − dim(im (d(2)  x ))  = dim V − (dim ker ρx − dim ker ρ′  x + dim(im (d′  x  (2)))  = dim V − rk(E−1) + rk(E′  −1) − dim(im (d′  x  (2)))  = dim V ′ − dim(im (d′  x  (2)))  = dim V ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content='  We have used the fact the cohomology groups at degree −1 of both complexes are isomorphic  and the Rank–nullity theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content='  For i = 2, let (V, x) ∈ �  ME  2 and (V ′, x) ∈ �  ME′  2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content=' Notice that dim V = rk(E−2) − rk(E−1) + r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content='  We have a similar formula for dim V ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content=' By direct computation we find that  dim V = dim V − dim(im d(3)  x )  = dim V − rk(E−2) + rk(E′  −2) + dim(im (d(2)  x )) − dim(im (d′  x  (2))) − dim(im (d′  x  (3))).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content='  (25)  We have used the fact the cohomology groups at degree −2 of both complexes are isomorphic  and the Rank–nullity theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content=' But  dim(im (d(2)  x )) = rk(E−1) − dim(im(ρx)) − dim W,  where W is such that dim(im (d(2)  x )) ⊕ W = ker ρx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content=' A similar formula holds for dim(im (d′  x  (2)))  by adding ′ everywhere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content=' Substituting them into the Equation (25) we obtain  dim V = dim V ′ + dim W ′ − dim W = dim V ′,  since dim W ′ = dim W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content='  □  Proof of (Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content=' For simplicity, we prove it for i = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content=' For i ≥ 1, the same arguments  hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content='  Let (E, d, ρ) and (E′, d′, ρ′) be geometric resolutions of F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content='  There exists chain morphisms  ϕ: E −→ E′ and ψ: E′ −→ E whose compositions are homotopic to identity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content=' In particular, ϕ  defines an isomorphism ϕ at the level of cohomology, the latter is canonical (see [LLS20], Lemma  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content=' All we need to show is ϕ sends �  ME  1 to �  ME′  1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content=' The strategy is to show that for x ∈ M, the fiber  �  ME  1 ∩Grr−dim(Lx)(E−1|x) over x is independent of any choices of minimal geometric resolutions at  x, then deduce the result for every point y ∈ M, and for any two arbitrary geometric resolutions  (E, d, ρ) and (E”, d”, ρ”) of F by introducing a minimal geometric resolution (E′, d′, ρ′) at y of  F between them, then compose the underlying quasi-isomorphisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content='  Let x ∈ �  ME  1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content=' Like we said, let us assume that (E′, d′, ρ′) is minimal at x, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content=', d′  x = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content=' Let  e1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content=' , ek be local sections around x of E−2 such that  span  �  d(2)e1|x, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content=' d(2)ek|x  �  = im(d(2)  x ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content='  There is a neighborhood Ux of x such that Fy = span  �  d(2)e1|y, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content=' d(2)ek|y  �  ⊆ im(d(2)  y ) is of  constant rank.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content=' These sections define a vector bundle F on Ux and Fx = im(d(2)  x ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content=' Likewise,  14  RUBEN LOUIS  one consider the vector bundle F ′ ⊆ im(d′(2)) on a neighborhood of x such that ϕy(Fy) ⊆ F ′  y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content='  Therefore, ϕ induces a map  im(d(2)  y )  Fy  −→ im(d′(2)  y  )  F ′y  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content='  The latter is injective, because d′  x = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content='  Since ϕy  �  im(d(2)  y )  Fy  �  ֒→  im(d′(2)  y  )  F ′y  , then ϕy  �  im(d(2)  y )  Fy  �  converges to W ⊆ im(d′(2)  y  )  F ′y  when y ∈ Mreg,F tends to x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content=' Consequently, for (V, x), (V ′, x) such  that im(d(2)  y ) and im(d′(2)  y  ) converges to V and V ′ respectively, one has, ϕx(V ) ⊆ V ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content=' Hence,  ϕx(V ) = V ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content=' Also, also ψx(V ′) = V since ψx and ϕx is are the inverse of each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content='  □  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content=' Proof of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content='12 and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content='14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content='12 follows from Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content='6 which itself  requires Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content=' We prove those in the smooth context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content=' We recall that for p: E −→ M a  vector bundle over M, a linear vector field on E is a pair (Z, X) ∈ X(E) × X(M) such that  E  Z  �  p  �  TE  dp  �  M  X � TM  is a morphism of vector bundles (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content='g [Mac05], p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content=' 110).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content=' Equivalently,  (1) Z[C∞  lin(E)] ⊂ C∞  lin(E) and Z[p∗C∞(M)] ⊂ p∗C∞(M).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content='  or  (2) The flow of Z on E are (local) vector bundle morphisms E −→ E over the flow of X on  M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content='  where C∞  lin(E) is the subalgebra of smooth functions on E which are fiberwise linear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content=' The latter  is canonically isomorphic to Γ(E∗) as C∞(M)-modules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content=' Notice in particular that, a linear vector  field is p-projectable to X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content='  Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content=' A linear vector field on E −→ M induces a vector field on Π: Gr−r(E) −→ M  that is Π-projectable on M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content=' Let (Z, X) be a linear vector field on E −→ M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content=' Its flow φZ  t : E −→ E is a vector bundle  isomorphism whenever it is defined, induces a diffeomorphism Gr−r(E) so that it induces a map  Gr−r(E) −→ Gr−r(E), V �→ φZ  t (V ) that we still denote by φZ  t .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content=' Define �Z ∈ X(Gr−r(E)) for  (V, x) ∈ Gr−r(E) by  �Z(V ) := d  dt |t=0  c(t) ∈ T(V,x)Gr−r(E)  (26)  where c(t) =  �  φZ  t |x(V ), φX  t (x)  �  for t in some interval I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content=' Also, �Z is Π-projectable to X, by  construction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content='  □  Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content=' Let i ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content=' For every X ∈ F, there is a linear vector field (Zi, X) on the vector  bundle pi : E−i −→ M or on p0 : E0 := TM −→ M, pi-projectable to X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content='  Their flows are  compatible with the complex of vector bundles,  · · ℓ1=d(4)  −→ E−3  ℓ1=d(3)  −→ E−2  ℓ1=d(2)  −→ E−1  ρ=d(1)  −→ TM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content='  (27)  NASH RESOLUTIONS OF SINGULAR FOLIATIONS WITH A VIEW TOWARDS AFFINE VARIETIES  15  i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content=', the diagram below commutes for all i ≥ 1,  M  φX  t  �  �  ①①①①①①①①①  M  �  ①①①①①①①①①  E−i  φZi  t  �  d(i+1)  �  E−i  d(i+1)  �  M  φX  t  �  �  ①①①①①①①①①  M  �  ①①①①①①①①①  E−i+1  φZi−1  t  � E−i+1  (28)  and induces a vector field �Zi on Gr−ri(E−i) such that  (1) �Zi is tangent to �  Mi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content='  (2) �Zi projects onto X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content='  where φZi  t  or φX  t  denotes the flow of Zi or X, when defined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content=' Let (E, ℓ1, ℓ2, ρ) be an almost graded Lie algebroid of F, see Section 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content='1(5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content=' Let X ∈ F  and i ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content=' For i ̸= 0, there exists a section υ of the vector bundle pi : E−i → M such that  ρ(υ) = X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content=' Consider the linear vector field Zi ∈ X(E−i) defined as follows  Zi[p∗  i f] : = p∗  i (X[f]), ∀ f ∈ C∞(M),  (29)  Zi  e[α] : = X[⟨α, e⟩] − ⟨α, ℓ2(υ, e)⟩, ∀ α ∈ Γ(E∗  −i), e ∈ Γ(E−i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content='  (30)  For i = 0, one replaces ℓ2(υ, e) in (30) by [X, Y ] with Y ∈ Γ(E0) = X(M).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content=' Notice that Zi  depends on the choice of the almost graded Lie algebroid bracket ℓ2 and X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content=' The items 1, 2, 3  and 4 hold (see [LGLR22], Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content='19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content=' p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content=' 61).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content='  By Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content='5, the linear vector field (Zi, X) induces a vector field �Zi on the Grassmanian  bundle Gr−ri(E−i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content=' Let us show item 1, φZi  t  preserves �  Mi: to see this take (V, x) ∈ �  Mi, let  xn  −→  n→+∞ x be such that im d(i+1)  xn  −→  n→+∞ V with (xn) ⊂ Mreg,F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content=' Since, d(i+1)◦φZi  t  = φZi−1  t  d(i+1)  for i ≥ 0, one has  φZi  t |xn  �  im d(i+1)  xn  �  = im d(i+1)  φX  t (xn),  for every n ∈ N0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content='  Thus,  φZi  t |x(V ) =  lim  n→+∞ φZi  t |xn  �  im d(i+1)  xn  �  =  lim  n→+∞  �  im d(i+1)  φX  t (xn)  �  ∈ π−1 �  φX  t (x)  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content='  By consequence, �Xi is tangent to �  Mi, by Equation (26).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content='  □  Proof (of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content='12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content=' By Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content='6, every vector field X ∈ F extends to a linear field  Xi ∈ X(Gr−ri(E−i)) which is tangent to �  Mi in the sense of Definition 3a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content=' This proves item  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content=' Furthermore, the restriction �Xi of Xi to �  Mi is unique, since πi|π−1  i  (Mreg,F) : π−1  i  (Mreg,F) −→  Mreg,F is invertible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content=' In particular, the map X ∈ F −→ �X| �  Mi does not depend on any choices  and is a Lie algebra morphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content=' The module which is generated by the �Xi is closed under Lie  bracket by item 2 of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content='12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content=' This ends the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content='  □  Proof (of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content='14).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content=' Let (E, d, ρ) be a geometric resolution of F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content=' Fix a universal Lie ∞-  algebroid of F on (E, d, ρ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content=' Let τ E−1 and AE−1 be the tautological subbundle and tautological  quotient bundle on Gr−r(E−1), that fit into the exact sequence  0 −→ τ E−1 −→ Π∗E−1 −→ AE−1 −→ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content='  (31)  16  RUBEN LOUIS  with AE−1 ≃ Π∗E−1/τ E−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content=' In particular, rk(AE−1) is the dimension of the regular leaves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content=' One  has  (1) �F1 the image of an almost Lie algebroid on Π∗E−1| �  M1 via the anchor map �ρ: Γ(Π∗E−1)| �  M1 −→  X( �  M1) definded by π∗  1e �−→  �  ρ(e) ∈ �F1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content='  (2) The tautological subbundle τ E−1 lies in the kernel of the anchor map: indeed, the fiber  of τ E−1 over a point (V, x) ∈ �  M1 is equal to V by definition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content=' By Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content='6, the  latter is included in ker ρx with equality if x ∈ Mreg,F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content='  Therefore, the anchor map �ρ goes to quotient  0  � τ E−1  � Π∗E−1  �  �ρ  �  AE−1  �  �✈ ✈ ✈ ✈ ✈  0  T �  M1  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content='  (32)  and makes �F1 the image of an almost Lie algebroid on AE−1| �  M1 whose anchor is injective on  the open dense subset Mreg,F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content='  Thus, AE−1| �  M1 is a Lie algebroid with anchor, injective on  π−1  1 (Mreg,F), whose image is �F1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content=' This proves the result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content='  □  Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content=' Notice that in the proof of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content='14 we do not need the existence of a  geometric resolution, we only make use of the anchor map and the bracket of an almost Lie  algebroid of F, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content=', we only need E−1 and ρ: E−1 −→ TM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content=' Examples  Let us start with some examples where our constructions give nothing new, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content=', �  Mi ≃ M or  �  M∞ ≃ M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content='  Example 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content=' If F is a projective singular foliation, then �  Mi ≃ M, for all i ≥ 1 and i = +∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content='  This comes from the fact that there exists a vector bundle E−1 −→ M such that Γ(E−1) ≃ F by  Serre-Swan theorem [Swa62, Mor13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content=' This isomorphism is given by a vector bundle morphism,  E−1  ρ→ TM which is injective on an open dense subset Mreg,F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content=' As a consequence, E−1  ρ→ TM  is a geometric resolution of F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content=' Therefore, �  Mi ≃ M since E−i = 0 for i ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content=' Also, if r is the  dimension of the regular leaves of F, then r = rk(E−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content=' Hence Gr−r(E−1) ≃ M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content=' In particular,  �  M1 ≃ M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content='  Example 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content=' If the regular leaves of F are open, then �  M0 ≃ M, since Gr−0(TM) ≃ M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content='  Example 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content=' If there exists a geometrical resolution (E, d, ρ) of length k, then �  Mi ≃ M for all  i ≥ k+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content=' Notice that one also has �  Mk ≃ M since the last differential map d(k): E−k −→ E−k+1 is  injective on an open dense subset so that the considered Grassmann bundle is Gr−rk(E−k)(E−k) ≃ M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content='  In contrast with Examples 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content='1, 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content='2 and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content='3, we construct two related examples where our  construction is not trivial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content='  Example 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content=' Consider the projective singular foliation F generated by the Euler vector field  −→  E = �N  i=1 xi ∂  ∂xi on M = CN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content=' Here, Mreg,F = CN \\ {0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content=' It is easily checked that �  M0 is the  closure of the graph {([x1 : · · · : xN], x) ∈ PN−1(C) × CN | x ̸= 0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content=' The latter is the blowup of  CN at 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content=' This is an example where F is projective and �  M0 ̸= M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content='  Example 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content=' Let F be the singular foliation of all vector fields vanishing at the origin 0 ∈  M = CN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content=' Here, Mreg,F = CN \\ {0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content=' Let us compute �  M1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content=' To do this, we only need E−1 the  NASH RESOLUTIONS OF SINGULAR FOLIATIONS WITH A VIEW TOWARDS AFFINE VARIETIES  17  degree −1 of a geometric resolution (E, d, ρ) of F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content=' See Example 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content='34 of [LLS20] for a complete  description of the latter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content=' Here E−1 ≃ MN(C)×CN and the anchor map ρ is Eij �→ xi ∂  ∂xj , where  MN(C) is the vector space of N ×N matrix with coefficient in C and (Eij)i,j=1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content=',N its canonical  basis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content='  A direct computation for every x ̸= 0 tells that ker ρx is the subspace of matrices M ∈ MN(C)  such that Mx = 0, where x = (x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content=' , xN) is seen as a column vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content=' Equivalently, this kernel  can be described as N copies of [x1 : · · · : xN]⊥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content=' Hence, �  M1 is the blowup of CN at the origin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content='  This is an example of a singular foliation whose regular leaves are open, but such that �  M1 ̸= M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content='  Let us study some examples related to the notion of an affine variety in Cd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content='  Let Ad be an affine space over K with a set of coordinates x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content=' , xd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content=' Recall that an affine  variety W is a subset of the affine space Ad given by the zero locus Z(IW) of a radical ideal  IW ⊆ K[x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content=' , xd] and equipped with the induced Zariski topology of Ad.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content=' The coordinate ring  of W is the quotient ring OW = K[x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content=' , xd]/IW .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content=' The Lie algebra X(W) of vector fields on W  are derivations of OW .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content=' We denote by Wreg the set of regular points of W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content=' For every x ∈ Ad  we denote by mx the maximal ideal of vanishing polynomials at x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content=' See, for instance, [Har77] for  more details on these notions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content='  Example 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content=' Let M = Cd and ϕ ∈ C[x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content=' , xd].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content=' Consider the singular foliation Fϕ = {X ∈  X(Cd) | X[ϕ] = 0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content=' In this case, Mreg,F = {x ∈ Cd, |, dxϕ ̸= 0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content=' For every y ∈ Cd, (TyFϕ)⊥ =  ⟨∇yϕ⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content='  For a convergent sequence yn  −→  n→+∞ y with yn ∈ Wreg,F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content='  The sequence im(ρyn) =  TynF converges if and only if ∇ynϕ converges in Gr−(d−1)(Cd), that is,  �  ∂ϕ  ∂x1 (yn): · · · :  ∂ϕ  ∂xd (yn)  �  converges in the projective space Pd−1(C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content=' Therefore, �  M0 is the closure of the image of the map,  y �→ (y,  � ∂ϕ  ∂x1(y): · · · :  ∂ϕ  ∂xd (y)  �  ) which is the blow up of Cd along the singular locus of ϕ, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content=',  along dϕ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content='  Example 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content=' (Nash modification).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content='  Let M = W be an affine irreducible affine variety of  dimension r embedded in Cd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content='  Let Σ be its singular locus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content='  Let F = Der(OW ) the singular  foliation of vector fields on W tangent to Σ, where IΣ stands for the polynomial functions that  vanish on Σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content=' Here, Wreg,F = Wreg = W \\ Σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content=' Consider a geometric resolution (E•, d, ρ) of F by  trivial vector bundles (which exists because OW is Noetherian, see Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content='3 in [LLS20]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content='  Let us show that for every x ∈ W \\ Σ, im(ρx) = TxF = TxW.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content=' It is clear that im(ρx) ⊆ TxW.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content='  Conversely, it is a classical property that x ∈ W is a regular point if and only if there exists  “local coordinates” y1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content=' , yd ∈ Ox such that W is of the form  y1 = · · · = yk = 0,  i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content=', the localization of IW is generated by these variables, where Ox denotes the local ring at  x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content=' Hence, the tangent space of W at x is the vector space, span{ ∂  ∂yi |m, i ≥ k + 1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content=' Therefore,  for v ∈ TxW the local vector field  X =  dim W  �  i=1  vi  ∂  ∂yk+i  maps Ox to Ox, in particular it maps O to Ox and we have X[IW ] ⊂ (IW )mx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content=' Therefore, for  every, i ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content=' , d} there exists a polynomial function gi that does not vanish at x such that  giY [xi] ∈ C[x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content=' , xd].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content=' By construction, the vector field ˆX =  g1···gr  g1(x)···gr(x)X is tangent to W, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content=',  ˆX[IW ] ⊂ IW, and satisfies ˆX(x) = v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content='  18  RUBEN LOUIS  The map π0 : W \\ Σ −→ Gr−(d−r)(Cd) x �−→ im(ρx) = TxW is the so-called Gauss map  [LU81].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content=' The Zariski closure �  W0 of the image of such a map is by definition the classical Nash  blow-up of W along its singular locus Σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content='  Example 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content=' (Monoidal transformation).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content=' Let W = Cd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content=' Let C = Z(IC) ⊂ Cd be a subvariety  of Cd defined by the ideal IC generated by f1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content=' , fk ∈ OW .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content=' Let F = ICX(W) the singular  foliation of vector fields vanishing along C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content=' By Hilbert’s Syzygy theorem [Eis04], there exists  a free resolution of finite length for the ideal IC of polynomial functions vanishing on C of the  form  · · −→ K−2  ∂  −→ K−1  ∂  −→ IC −→ 0  (33)  Since X(W) is a flat OW = C[x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content=' , xd]-module (in fact X(W) ≃ Od  W is a free module), the  sequence  · ·  ∂⊗id  � K−2 ⊗OW X(W)  ∂⊗id  � K−1 ⊗OW X(W)  ρ  � F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content='  (34)  is a free resolution K[W] by finitely generated K[W]-modules of the singular foliation F =  ICX(W), where for (µ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content=' , µk) a set of generators of K−1 the map given by, ρ(µi⊗ ∂  ∂yj ) = fi ∂  ∂yj ,  for i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content=' , k and j = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content=' , d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content=' By Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content='1 in [LGL22], F admits a universal Lie ∞-  algebroid structure over the complex (34) whose unary bracket is ℓ1 = ∂ ⊗ id and whose anchor  is ρ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content='  Here, Wreg,F = W \\ C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content=' A direct computation shows that, for every x ∈ W \\ C, ker ρx is equal  to d copies of [f1(x) : · · · : fk(x)]⊥, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content=',  ker ρx =  �  [f1(x) : · · · : fk(x)]⊥�d ,  where [f1(x) : · · · : fk(x)] is a well-defined straight line of Kk generated by the vector (f1(x), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content=' , fk(x)) ∈  Kk seen as a point of the projective space Pk−1(K) = Gr−(k−1)(Ck).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content='  One has,  π−1  1 (x) =  \uf8f1  \uf8f4  \uf8f4  \uf8f4  \uf8f4  \uf8f4  \uf8f4  \uf8f4  \uf8f4  \uf8f2  \uf8f4  \uf8f4  \uf8f4  \uf8f4  \uf8f4  \uf8f4  \uf8f4  \uf8f4  \uf8f3  �  [f1(x) : · · · : fk(x)]⊥�d , for x ∈ W \\ C,  V d ∈  �  Gr−1(Ck)  �d such that ∃ (xn) ∈ W N  reg,F, [f1(xn) : · · · : fk(xn)]⊥  −→  n→+∞ V,  with V ∈ Gr−1(Ck) as xn  −→  n→+∞ x, for x ∈ C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content='  The d components converge if and only if one of them converges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content=' Since [f1(xn) : · · · : fk(xn)]⊥  converges in Gr−1(Kk) if and only if the straight line [f1(xn) : · · · : fk(xn)] converges in Pk−1(K),  �  W1 corresponds to the usual monoidal transformation of W with center C (see, for instance  [Hau14]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content=' In particular, �  W1 does not depend, up to isomorphism over W, on the choice of the  generators f1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content=' , fk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content='  When f1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content=' , fk form a regular sequence, (33) can be chosen to be the Koszul complex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content=' Then  for each i ≥ 1, �  Wi is again the blowup of Cd along C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content=' In particular, �  Wi = �  W∞ is the blowup  of W = Cd along C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content=' Let us prove it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content=' The proof is similar for every i ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content=' Let us show it for  i = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content=' Since (34) is given by E−i = �i M⊗X(Cd), where M is a C[x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content=' , xd]-module generated  by some degree −1 symbols µ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content=' , µk and the differential map is defined by: for every i ≥ 1,  j = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content=' , d and 1 ≤ j1 < · · · < ji ≤ k,  d(i)(µj1 ∧ · · · ∧ µji ⊗  ∂  ∂xj  ) =  i  �  l=1  (−1)l+1fjlµj1 ∧ · · · ∧ �  µjl ∧ · · · ∧ µji ⊗  ∂  ∂xj  ,  NASH RESOLUTIONS OF SINGULAR FOLIATIONS WITH A VIEW TOWARDS AFFINE VARIETIES  19  here the hat �· means the term is omitted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content=' The computations of �  Mi are similar for all i ∈ N0,  let us compute �  M2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content=' Let (gj  pq) ∈ C[x1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content=' , xd] with j = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content=' , d and p, q = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content=' , k such that  d(2)  \uf8eb  \uf8ed�  p,q,j  gj  p,qµp ∧ µq ⊗  ∂  ∂xj  \uf8f6  \uf8f8 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content=' This implies that  0 =  �  p,q,j  gj  p,q(fpµq − fqµp) ⊗  ∂  ∂xj  =  �  q,j  ��  p  gj  p,qfp  �  µq ⊗  ∂  ∂xj  −  �  p,j  ��  q  gj  p,qfq  �  µp ⊗  ∂  ∂xj  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content='  For x ∈ Wreg,F, one finds that for every (p, j) and (q, j) that (gj  1,q(x), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content=' , gj  k,q(x)), (gj  p,1(x), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content=' , gj  p,k(x)) ∈  [f1(x) : · · · : fk(x)]⊥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content=' This proves the result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content='  Here is an example related to Poisson manifolds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content='  Example 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content=' Let (M, P) a smooth or holomorphic Poisson manifold with P ∈ Γ(∧2TM).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content='  Consider the singular foliation generated by the Hamiltonian vector fields associated to P, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content=',  F = P ♯(Γ(T ∗M)), where P ♯ : T ∗M −→ TM, α �→ P(α, · ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content=' Assume that a geometric resolution  exists.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content=' By Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content='6, every Hamiltonian vector field lifts to a vector field tangent to �  Mi, i ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content='  It is natural to ask whether the bivector field P lifts to �  Mi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content=' Assume that �  Mi is smooth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content=' Since for  every i ≥ 1, π−1  i  (Mreg,F) −→ Mreg,F is invertible, the restriction P|U lifts to a Poisson bivector  field on π−1  i  (Mreg,F).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content=' However, it does not lift to �  Mi in general.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content=' Indeed, consider the Poisson  manifold M = so∗(3) ≃ R3 with  P = x ∂  ∂y ∧ ∂  ∂z + y ∂  ∂z ∧ ∂  ∂x + z ∂  ∂x ∧ ∂  ∂y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content='  (35)  Here F is generated by the vector fields P ♯(dx) = z ∂  ∂y − y ∂  ∂z, P ♯(dy) = x ∂  ∂z − z ∂  ∂x, P ♯(dz) =  y ∂  ∂x − x ∂  ∂y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content=' Let us compute �  M1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content=' Given a point m ∈ Mreg,F = R3 \\ {0}, we find that  ker P ♯|m =  �  (a, b, c) ∈ R3 | (a, b, c) ∈ [x(m) : y(m) : z(m)] ∈ P2(R)  �  = [x(m) : y(m) : z(m)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content='  Hence, �  M1 is the usual blowup B0(R3) of R3 at the origin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content='  The bivector field P does not lift to �  M1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content=' Recall that the blowup space of R3 at the origin  B0(R3) ⊂ P2(R) × R3 is covered by three charts given by x ̸= 0, y ̸= 0 and z ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content=' Let us  look at the x-chart where the projection π1 becomes (x, y, z) �→ (x, xy, xz).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content=' In this chart P pulls  back to  y ∂  ∂z ∧ ∂  ∂x + z ∂  ∂x ∧ ∂  ∂y + 1  x(1 + y2 + z2) ∂  ∂y ∧ ∂  ∂x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content='  (36)  For x = 0 Equation (36) is not defined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content=' In conclusion, the Hamiltonian vector fields of the  Poisson structure P in (35) lift to �  M1, but the bivector field P does not lift to �  M1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content='  Example 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content=' Let (E−1, [· , ·] , ρ) be a smooth or holomorphic Lie algebroid over a manifold  M and denote by F = ρ(Γ(E−1)) the induced singular foliation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content=' Assume there exists geometric  resolutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content=' The Lie algebroid E−1 acts on the spaces �  Mi for all i ∈ N0 also on �  M∞, in a way  that the following  X( �  M1)  �  Γ(E−1)  ρ  �✉  ✉  ✉  ✉  ✉  ρ  � X(M)  (37)  20  RUBEN LOUIS  is a commutative diagram of Lie algebra morphisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content=' In addition, for each i ∈ N0, �Fi is the image  of a Lie algebroid on �  Mi, namely the pull-back to �  Mi of the Lie Algebroid E−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content=' In particular,  if F is given by a Lie algebra action of a Lie algebra g on M, then �Fi is given by an action of g  on �  Mi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content='  In the following example, we show that our �  M1 is the blowup construction blup(F) given by  Mohsen in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content='2 of [Moh21]  Example 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content=' Let F be a singular foliation that admits a geometrical resolution (E, d, ρ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content='  For every x ∈ M, blup(F)x is constructed out of minimal generators X1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content=' , Xd of F in a  neighborhood of x as follows: for y ∈ Mreg,F, let φy be the surjective linear map defined by  φy :  F  IxF −→ TyF, φy([Xi]x) = Xi(y),  for all  i ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content=' , d},  (38)  where TyF is the image of the evaluation map ey : F −→ TyM at y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content=' By definition, blup(F)x is  made of subspaces V ⊆  F  IxF such that there exists a sequence xn ∈ Mreg,F such that  xn −→ x, φ−1  xn (0) −→ V ∈ Gr−r  � F  IxF  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content='  (39)  We claim that for every x ∈ M, blup(F)x ≃ π−1  1 (x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content=' Indeed, we can assume that (E, d, ρ) is  a minimal geometric resolution at x such that ρ(ei) = Xi for i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content=' , d, where (ei)i=1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content=',d  is a local frame of E−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content=' Since  Γ(E−1)  Ix′Γ(E−1) ≃ E−1|x′ for all x′ ∈ M, the anchor map defines an  isomorphism ρx : E−1|x −→  F  IxF such that the diagram  E−1|x  ≃  ρx  �  ≃  κy  �  F  IxF  φy  �  E−1|y  ρy  � TyF  (40)  commutes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content=' The claim follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content=' Conclusion  Let us conclude this paper with an open question.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content=' Can we desingularize a singular affine  variety W by applying the constructions above to the singular foliation F = X(W) of vector  fields tangent to W?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content=' We should then understand �  W0 as the Nash modification of W [LU81].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content='  The meaning of �  W1 when F is the vector fields that vanish along a subvariety, is the monoidal  transformation of W along of that subvariety.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content=' �F1 is projective, and is included into the vector  fields tangent to �  W1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content=' It is logical to apply the construction again to �  W1 or apply it to a sub-  singular foliation of F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content='  Notice that until now we have used the anchor map, the unary bracket and the 2-ary bracket  of a universal Lie-∞-algebroid, we would like to go further to understand, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content=', the role of the  3-ary bracket in this procedure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content='  Acknowledgements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content='  I would like to thank C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content=' Laurent-Gengoux, my PhD supervisor, for  supporting me in writing this article and directing me to such questions that are originated  from Claire Debord and Georges Skandalis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content=' Also, I thank the management of the University of  Lorraine for having supported me financially through an ATER position.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content='  NASH RESOLUTIONS OF SINGULAR FOLIATIONS WITH A VIEW TOWARDS AFFINE VARIETIES  21  References  [AS09]  Iakovos  Androulidakis  and  Georges  Skandalis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content='  The  holonomy  groupoid  of  a  singu-  lar  foliation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content='  Journal  für  die  reine  und  angewandte  Mathematik,  2009(626):1–37,  2009.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content='  https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content='5167/uzh-23589.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content='  [AZ14]  Iakovos  Androulidakis  and  Marco  Zambon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content='  Holonomy  transformations  for  singu-  lar  foliations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content='  Advances  in  Mathematics  (New  York.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content='  1965),  256:348–397,  2014.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content='  https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content='1016/j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content='aim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content='2014.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content='02.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content='003.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
+page_content='  [Cer79]  Dominique Cerveau.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFAT4oBgHgl3EQf0x7h/content/2301.08706v1.pdf'}
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diff --git a/StE5T4oBgHgl3EQfAA5y/content/tmp_files/2301.05375v1.pdf.txt b/StE5T4oBgHgl3EQfAA5y/content/tmp_files/2301.05375v1.pdf.txt
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@@ -0,0 +1,1067 @@
+MAPPING CLASS GROUPS OF CIRCLE BUNDLES OVER A
+SURFACE
+LEI CHEN AND BENA TSHISHIKU
+Abstract. In this paper, we study the algebraic structure of mapping class group
+Mod(X) of 3-manifolds X that fiber as a circle bundle over a surface S1 → X → Sg.
+There is an exact sequence 1 → H1(Sg) → Mod(X) → Mod(Sg) → 1. We relate this to
+the Birman exact sequence and determine when this sequence splits.
+1. Introduction
+For g ≥ 1, let Sg denote the closed oriented surface of genus g, and for k ∈ Z, let Xk
+g
+denote the closed 3-manifold that fibers
+S1 → Xk
+g → Sg
+as an oriented circle-bundle with Euler number k. Assuming (g, k) ̸= (1, 0), the mapping
+class group Mod(Xk
+g ) := π0
+�
+Homeo+(Xk
+g )
+�
+fits into a short exact sequence
+(1)
+1 → H1(Sg; Z) → Mod(Xk
+g ) → Mod(Sg) → 1.
+This paper is motivated by the following question.
+Question 1.1. For which values of g, k is the extension in (1) split?
+Interestingly, the extension does split for k = 2 − 2g, in which case Xk
+g is unit tangent
+bundle USg. In fact, there is a natural action of Mod(Sg) on USg by homeomorphisms,
+which gives a splitting of (1) upon taking isotopy classes. For g ≥ 2, this action comes
+from the action of the punctured mapping class group Mod(Sg,1) on triples of points on
+the boundary of hyperbolic space H2. This construction dates back to the work of Nielsen.
+See [FM12, §5.5.4, §8.2.6] and [Sou10, §1].
+In general, Question 1.1 reduces to a question about group cohomology. The extension
+(1) splits if and only if its Euler class euk ∈ H2�
+Mod(Sg); H1(Sg; Z)
+�
+vanishes [Bro82,
+§IV.3]. Here the coefficients are twisted via the natural action of Mod(Sg) on H1(Sg; Z).
+However, a computation of H2�
+Mod(Sg); H1(Sg; Z)
+�
+does not appear to be in the litera-
+ture.
+The extension (1) is related to the Birman exact sequence
+1 → π1(Sg) → Mod(Sg,1) → Mod(Sg) → 1.
+By taking quotients by the commutator subgroup π′ ≡ [π1(Sg), π1(Sg)], we obtain the
+following extension
+(2)
+1 → H1(Sg) → Mod(Sg,1)/π′ → Mod(Sg) → 1.
+Our main result relates the sequences (1) and (2).
+Date: January 16, 2023.
+1
+arXiv:2301.05375v1  [math.GT]  13 Jan 2023
+
+2
+LEI CHEN AND BENA TSHISHIKU
+Theorem A. Fix g ≥ 1 and k ∈ Z. Assume (g, k) ̸= (1, 0). There is a map between the
+short exact sequences (1) and (2)
+1
+H1(Sg)
+Mod(Sg,1)/π′
+Mod(Sg)
+1
+1
+H1(Sg; Z)
+Mod(Xk
+g )
+Mod(Sg)
+1
+�
+�
+�
+�
+�
+�
+�
+�
+� kδ
+�
+The homomorphism kδ is the Poincar´e duality isomorphism δ composed with multiplication
+by k. In particular, when k = 1, the exact sequences (1) and (2) are isomorphic.
+Theorem A implies the Euler classes of the extensions (1) satisfy euk = k eu1 for fixed
+g. Next we determine the subgroup generated by eu1 in H2�
+Mod(Sg); H1(Sg; Z)
+�
+.
+Theorem 1.2. Fix g ≥ 1, and let eu1 be the Euler class of the extension (1). Then eu1
+has order 2g − 2 in H2�
+Mod(Sg); H1(Sg; Z)
+�
+. Furthermore, if g ≥ 8, then eu1 generates
+this group, i.e.
+H2�
+Mod(Sg); H1(Sg; Z)
+� ∼= Z/(2g − 2)Z.
+Combining Theorem A and Theorem 1.2 we obtain the following answer to Question
+1.1.
+Corollary 1.3. For g ≥ 2 and k ∈ Z, the extension (1) splits if and only if k is divisible
+2g − 2. For g = 1 the extension splits for each k.
+When a splitting exists, the different possible splittings (up to the action of H1(Sg; Z)
+on Mod(Xk
+g ) by conjugation) are parameterized by elements of H1�
+Mod(Sg); H1(Sg; Z)
+�
+[Bro82, Ch. IV, Prop. 2.3].
+This group vanishes for g ≥ 1 [Mor85, Prop. 4.1], so the
+splitting, when it exists, is unique.
+Connection to Nielsen realization. Instead of Question 1.1, one can ask whether there
+is a splitting of the composite surjection
+Homeo(Xk
+g ) → Mod(Xk
+g ) → Mod(Sg).
+This is an instance of a Nielsen realization problem. Of course, if Mod(Xk
+g ) → Mod(Sg)
+does not split, then neither does Homeo(Xk
+g ) → Mod(Sg), and Corollary 1.3 gives examples
+of this. Since Mod(Sg) has a natural action on USg, the surjection Homeo(Xk
+g ) → Mod(Sg)
+does split for k = ±(2g − 2). This is somewhat surprising since mapping class groups are
+rarely realized as groups of surface homeomorphisms [Mar07, Che19, CS22]. We wonder
+whether this splitting is unique, or if a splitting exists for other values k divisible by 2g −2
+(for example, k = 0). We plan to study this in a future paper.
+Previous work and proof techniques. Waldhausen [Wal68, §7] proved that the group
+π0
+�
+Homeo(Xk
+g )
+�
+is isomorphic to the outer automorphism group Out
+�
+π1(Xk
+g )
+�
+. From this,
+the short exact sequence (1) can be derived from work of Conner–Raymond [CR77] and
+the Dehn–Nielsen–Baer theorem; alternatively, see McCullough [McC91, §3]. The Dehn–
+Nielsen–Baer theorem also plays a central role in Theorem A, since it allows us to translate
+back and forth between topology and group theory. There is a mix of both in the proof of
+Theorem A in §3.
+
+MAPPING CLASS GROUPS OF CIRCLE BUNDLES OVER A SURFACE
+3
+To prove Theorem A, we consider a version of Question 1.1 where Xk
+g and Sg are
+punctured. For the punctured manifolds, similar to (1), there is a short exact sequence
+1 → H1(Sg; Z) → Mod(Xk
+g,1) → Mod(Sg,1) → 1,
+and we construct a splitting
+σ : Mod(Sg,1) → Mod(Xk
+g,1).
+See Corollary 3.1. A key part of our proof of Theorem A is to determine the image of
+the point-pushing subgroup π1(Sg) < Mod(Sg,1) under σ. For this we relate three natural
+surface group representations π1(Sg) → Mod(Xk
+g,1) that appear in the following diagram,
+where the diagonal map is point pushing on Xk
+g (not a commutative diagram).
+π1(Sg)
+Mod(Sg,1)
+H1(Sg; Z)
+Mod(Xk
+g,1)
+�
+point-pushing on Sg
+� σ
+�
+��
+transvections
+See Proposition 3.4 for a precise statement.
+In order to deduce Corollary 1.3, we use a spectral sequence argument to prove that eu1
+generates a subgroup of H2�
+Mod(Sg); H1(Sg; Z)
+�
+isomorphic to Z/(2g − 2)Z. A different
+spectral sequence computation proves that eu1 generates H2�
+Mod(Sg); H1(Sg; Z)
+�
+when
+g is large. These computations use several known computations, including work of Morita
+[Mor85].
+Section outline. In §2 we collect the results we need about the manifolds Xk
+g and their
+mapping class groups, including Waldhausen’s work.
+Theorem A is proved in §3; this
+section is the core of the paper. In §4, we do two spectral sequence computations to prove
+Theorem 1.2.
+Acknowledgement.
+Thanks to B. Farb for sharing the reference [McC91] and to D.
+Margalit for comments on a draft.
+The authors are supported by NSF grants DMS-
+2203178, DMS-2104346 and DMS-2005409.
+2. Circle bundles over surfaces
+Here we review some results about circle bundles over surfaces that we will need in
+future sections.
+2.1. Classification. By an oriented circle bundle we mean a fiber bundle
+S1 → E → B
+with structure group Homeo+(S1), the group of orientation preserving homeomorphisms
+of S1. The inclusion of the rotation group SO(2) in Homeo+(S1) is a homotopy equiva-
+lence, so circle bundles are in bijection with rank-2 real vector bundles. The classifying
+space B SO(2) is homotopy equivalent to CP ∞, which is an Eilenberg–Maclane space
+K(Z, 2). Thus each circle bundle is uniquely determined up to isomorphism by its Euler
+class eu(E) ∈ H2(B; Z), which is the primary obstruction to a section of the bundle.
+
+4
+LEI CHEN AND BENA TSHISHIKU
+When B = Sg is a closed, oriented surface, H2(Sg; Z) ∼= Z, we can speak of the Euler
+number. We use Xk
+g to denote the total space of the circle bundle
+S1 → Xk
+g → Sg
+with Euler number k. For example, for the unit tangent bundle eu(USg) = 2 − 2g (the
+Euler characteristic), so USg ∼= X2−2g
+g
+. We also note that Xk
+g and X−k
+g
+are homeomorphic
+3-manifolds, since the sign of the Euler number of a circle bundle over Sg depends on the
+choice of orientation on Sg.
+2.2. Fundamental group π1(Xk
+g ) and its automorphisms. From the long exact se-
+quence of a fibration, we have an exact sequence
+1 → Z → π1(Xk
+g ) → π1(Sg) → 1.
+The group π1(Xk
+g ) has a presentation
+(3)
+π1(Xk
+g ) =
+�
+A1, B1, . . . , Ag, Bg, z | z central, [A1, B1] · · · [Ag, Bg] = zk�
+.
+Using this, one finds ⟨z⟩ ∼= Z is the center of π1(Xk
+g ) as long as (g, k) ̸= (1, 0). When g ≥ 2
+follows from the fact that the group π1(Sg) has trivial center; the case g = 1 can be treated
+directly.
+Given this computation of the center, any automorphism of π1(Xk
+g ) induces an auto-
+morphism of ⟨z⟩ ∼= Z and descends to an automorphism of π1(Sg). The latter gives a
+homomorphism
+Aut
+�
+π1(Xk
+g )
+�
+→ Aut
+�
+π1(Sg)
+�
+that restricts to an isomorphism between the inner automorphism groups
+(4)
+Inn
+�
+π1(Xk
+g )
+� ∼= π1(Sg) ∼= Inn
+�
+π1(Sg)
+�
+and hence descends to a homomorphism
+(5)
+Out
+�
+π1(Xk
+g )
+�
+→ Out
+�
+π1(Sg)
+�
+.
+Orientations. It will be convenient to define
+AUT
+�
+π1(Sg)
+�
+< Aut
+�
+π1(Sg)
+�
+as the subgroup that acts trivially on H2(π1(Sg); Z) ∼= Z (the “orientation-preserving”
+subgroup). We define
+AUT
+�
+π1(Xk
+g )
+�
+< Aut
+�
+π1(Xk
+g )
+�
+as the group of automorphisms that project into AUT
+�
+π1(Sg)
+�
+and that act trivially on
+the center ⟨z⟩ ∼= Z. In particular, AUT
+�
+π1(Xk
+g )
+�
+has index 4 in Aut
+�
+π1(Xk
+g )
+�
+.
+These orientation-preserving subgroups contain the (respective) inner automorphism
+groups, and we denote the quotients OUT
+�
+π1(Xk
+g )
+�
+and OUT
+�
+π1(Sg)
+�
+.
+
+MAPPING CLASS GROUPS OF CIRCLE BUNDLES OVER A SURFACE
+5
+2.3. Mapping class group Mod(Xk
+g ). Fix g ≥ 1 and k ∈ Z, and assume (g, k) ̸= (1, 0).
+Let Homeo+(Xk
+g ) denote the group of homeomorphisms whose image in Out
+�
+π1(Xk
+g )
+�
+is
+contained in OUT
+�
+π1(Sg)
+�
+. Define
+Mod(Xk
+g ) := π0
+�
+Homeo+(Xk
+g )
+�
+.
+Waldhausen [Wal68, Cor. 7.5] proved that the natural homomorphism
+π0
+�
+Homeo(Xk
+g )
+�
+→ Out
+�
+π1(Xk
+g )
+�
+is an isomorphism. Then, by the definitions, this homomorphism restricts to an isomor-
+phism Mod(Xk
+g ) ∼= OUT
+�
+π1(Xk
+g )
+�
+. Waldhausen also proved that π0 Homeo(Xk
+g ) is iso-
+morphic to the group of fiber-preserving homeomorphisms modulo homeomorphisms that
+are isotopic to the identity through fiber-preserving isotopies; see [Wal68, Rmk. following
+Cor. 7.5]. Consequently, there is a homomorphism
+(6)
+Mod(Xk
+g ) → Mod(Sg).
+Altogether, we have the following commutative diagram relating the maps (5) and (6).
+(7)
+Mod(Xk
+g )
+Mod(Sg)
+OUT
+�
+π1(Xk
+g )
+�
+OUT
+�
+π1(Sg)
+�
+�
+�
+∼=
+�
+∼=
+�
+The right vertical map is an isomorphism by the Dehn–Nielsen–Baer theorem [FM12,
+Thm. 8.1]. Furthermore, by Conner–Raymond [CR77, Thm. 8] that there is a short exact
+sequence
+(8)
+1 → Hom(π1(Sg), Z) → OUT
+�
+π1(Xk
+g )
+�
+→ OUT
+�
+π1(Sg)
+�
+→ 1.
+This establishes the short exact sequence (1) in the introduction. We will give a concrete
+derivation of this exact sequence in Corollary 3.1 below.
+3. Relating Mod(Xk
+g ) to the Birman exact sequence
+In this section, we prove Theorem A. To construct the map of short exact sequences in
+Theorem A, our main task is to first define a homomorphism Mod(Sg,1) → Mod(Xk
+g ) and
+then to compute that its kernel is the commutator subgroup of π1(Sg) < Mod(Sg,1) (the
+point-pushing subgroup). We do this is §3.1 and §3.2.
+3.1. A homomorphism Ψ : Mod(Sg,1) → Mod(Xk
+g ). Fix a basepoint ∗ ∈ Sg, and set
+Sg,1 = Sg\{∗}. By the Dehn–Nielsen–Baer theorem, Mod(Sg,1) is isomorphic to Out∗(F2g),
+where F2g is the free group of rank 2g and Out∗(F2g) < Out(F2g) is the subgroup that
+preserves the conjugacy class corresponding to the free homotopy class of the curve around
+the puncture in Sg,1. We construct Ψ as a composition
+(9)
+Ψ : Mod(Sg,1) ∼= Out∗(F2g) σ−→ AUT
+�
+π1(Xk
+g )
+�
+→ OUT
+�
+π1(Xk
+g )
+� ∼= Mod(Xk
+g ).
+To define σ, fix a generating set α1, β1, . . . , αg, βg for F2g such that c = �g
+i=1[αi, βi]
+represents the conjugacy class of the curve around the puncture. Let
+(10)
+ι : F2g → π1(Xk
+g )
+be the homomorphism defined by αi �→ Ai and βi �→ Bi. Given f ∈ Out∗(F2g), fix an
+automorphism ˜f : F2g → F2g that represents f, and assume that ˜f(c) = c (this can always
+
+6
+LEI CHEN AND BENA TSHISHIKU
+be achieved by composing any lift with an inner automorphism of F2g). Next we define
+σ(f) on generators of π1(Xk
+g ) by
+(11)
+σ(f)(Ai) = ι ˜f(αi),
+σ(f)(Bi) = ι ˜f(βi),
+σ(f)(z) = z.
+To show that σ(f) extends to a homomorphism of π1(Xk
+g ), we check that the relation
+[A1, B1] · · · [Ag, Bg] = zk is preserved under σ(f):
+�
+i
+[σ(f)(Ai), σ(f)(Bi)] =
+�
+i
+[ι ˜f(αi), ι ˜f(βi)] = ι(c) = zk = σ(f)(zk).
+The second equality uses the the fact that ˜f(c) = c. The map σ(f) is independent of
+the choice of ˜f because different choices of ˜f differ by conjugation by powers of c (be-
+cause the centralizer of c in F2g is the cyclic subgroup ⟨c⟩)1 and ι(c) = zk is central in
+π1(Xk
+g ). The homomorphism σ(f) : π1(Xk
+g ) → π1(Xk
+g ) is an automorphism and belongs to
+AUT
+�
+π1(Xk
+g )
+�
+by definition. Furthermore, f �→ σ(f) is a homomorphism, which is easy
+to check using the observation that if w = ιw′, then σ(f)(w) = ι ˜f(w′).
+Composing σ with AUT → OUT gives the desired homomorphism Ψ. As a corollary of
+this construction, we have proved the following.
+Corollary 3.1. Fix g ≥ 1 and k ∈ Z, and assume (g, k) ̸= (1, 0).
+The natural map
+Φ : AUT
+�
+π1(Xk
+g )
+�
+→ AUT
+�
+π1(Sg)
+�
+(see §2.2) fits into an exact sequence
+(12)
+1 → Hom(π1(Sg), Z) → AUT
+�
+π1(Xk
+g )
+� Φ−→ AUT
+�
+π1(Sg)
+�
+→ 1,
+and this exact sequence splits.
+Proof. First we compute the kernel of Φ.
+Using the presentation for π1(Xk
+g ) in (3), if
+f ∈ ker(Φ), then
+f(Ai) = Aizmi
+and
+f(Bi) = Bizni
+for some m1, n1, . . . , mg, ng ∈ Z. The map ai �→ mi, bi �→ ni extends to a homomorphism
+τ(f) : π1(Sg) → Z. It is elementary to check that the map ker(Φ) → Hom(π1(Sg), Z)
+defined by f �→ τ(f) is an isomorphism.
+The homomorphism σ defined above shows that Φ is a split surjection. Note that the
+Mod(Sg, ∗) ∼= Mod(Sg \ {∗}) (basepoint vs. puncture), so by Dehn–Nielsen–Baer there is
+an isomorphism AUT
+�
+π1(Sg)
+� ∼= Out∗(F2g), and we use this isomorphism to view σ as a
+splitting of Φ.
+□
+Remark 3.2. We call elements of ker(Φ) ∼= H1(Sg; Z) transvections.
+Remark 3.3. The homomorphism Ψ can be constructed on the level of topology as follows.
+Fix a regular neighborhood D of the puncture on Sg,1 (so D is a once-punctured disk).
+Given a mapping class f ∈ Mod(Sg,1), choose a representing homeomorphism f. Without
+loss of generality, we can assume that f is the identity on D. The bundle Xk
+g → Sg can be
+trivialized over S \ D (because the classifying space B SO(2) is simply connected). Fixing
+a trivialization (S \ D) × S1 over S \ D, we lift f to the product homeomorphism f × idS1.
+This homeomorphisms is the identity on the boundary ∂(S \ D) × S1, so we can extend
+by the identity to obtain a homeomorphism ˜f of Xk
+g . The map sending f ∈ Mod(Sg,1) to
+1Note that the centralizer is isomorphic to Z and contains ⟨c⟩. It is only bigger if c = ui for some u ∈ F2g
+and i ≥ 2. By contradiction, if c = ui for i ≥ 2, then u is cyclically reduced because c is. This implies that
+u is a subword of c = �[αi, βi], which is absurd.
+
+MAPPING CLASS GROUPS OF CIRCLE BUNDLES OVER A SURFACE
+7
+the isotopy class
+�
+˜f
+�
+∈ Mod(Xk
+g ) is the topological version of the homomorphism Ψ. Note
+that the isotopy class [f] is only well-defined up to Dehn twists about ∂D which is a loop
+around the puncture. This is analogous to the ambiguity encountered in the definition of
+σ, which ultimately does not affect the definition of Ψ.
+Corollary 3.1 and equation (4) combine to give the short exact sequence of outer auto-
+morphism groups (8).
+Warning. The splitting of the short exact sequence (12) does not give a splitting of the
+short exact sequence (8). Indeed we will show the latter sequence does not always split
+(Corollary 1.3). The subtlety comes from the fact that the inner automorphism group
+Inn
+�
+π1(Xk
+g )
+� ∼= π1(Sg) does not coincide with the image of π1(Sg) ∼= Inn
+�
+π1(Sg)
+�
+<
+Aut
+�
+π1(Sg)
+�
+under the section σ. Proposition 3.4 below describes the precise relationship.
+3.2. Kernel of Ψ : Mod(Sg,1) → Mod(Xk
+g ). Observe that the kernel of Ψ is contained
+in the point-pushing subgroup π1(Sg) < Mod(Sg,1). This is because Ψ composed with
+the natural map Mod(Xk
+g ) → Mod(Sg) is the natural map Mod(Sg,1) → Mod(Sg), whose
+kernel is the point-pushing subgroup.
+Thus we want to understand the image of the
+point-pushing subgroup under the section σ used to define Ψ. What we find is a simple
+relationship between three surface group representations:
+π1(Sg)
+π1(Sg)
+π1(Sg)
+AUT
+�
+π1(Xk
+g )
+�
+�
+inner auts of π1(Sg), lifted
+�
+inner auts of π1(Xk
+g )
+�
+σ
+�
+transvections
+The main results are Proposition 3.4 and Corollary 3.5 below. In order to state Propo-
+sition 3.4, we need the following notation. Let
+δ : H1(Sg; Z) → H1(Sg; Z)
+be the Poincar´e duality map, given explicitly by γ �→ ⟨−, γ⟩, where
+⟨−, −⟩ : H1(Sg; Z) × H1(Sg; Z) → Z
+is the algebraic intersection form. We use ˆδ denote the composition
+ˆδ : H1(Sg; Z) δ−→ H1(Sg; Z) �→ AUT
+�
+π1(Xk
+g )
+�
+.
+This map is given explicitly by ˆδ(γ)(w) = w · z⟨[ ¯w],γ⟩, where ¯w is the image of w under
+π1(Xk
+g ) → π1(Sg) and [ ¯w] ∈ H1(Sg; Z) is the corresponding homology class.
+Fix a basepoint ⋆ ∈ Sg,1. Recall that we have fixed a standard generating set {αi, βi} of
+π1(Sg,1, ⋆) ∼= F2g so that c := �
+i[αi, βi] is a loop around the puncture ∗ of Sg,1 = Sg \ {∗}.
+Define
+(13)
+Π : π1(Sg,1, ⋆) → π1(Sg, ∗)
+by γ �→ ϵ.γ.¯ϵ, where ϵ is a fixed arc from ∗ to ⋆.
+
+8
+LEI CHEN AND BENA TSHISHIKU
+Proposition 3.4. Fix t ∈ π1(Sg, ∗), and let Push(t) ∈ Mod(Sg,1) ∼= Out∗(F2g) be the
+point-pushing mapping class. If ˜t ∈ π1(Sg,1, ⋆) is any lift of t (i.e. Π(˜t) = t), then
+(14)
+σ
+�
+Push(t)
+�
+= Cι˜t ◦ ˆδ([kt]),
+Here Cx denotes conjugation by x, and the maps ι : F2g → π1(Xk
+g ) and σ : Out∗(F2g) →
+AUT
+�
+π1(Xk
+g )
+�
+are defined in (10) and (11).
+As a sanity check, observe that Cι˜t does not depend on the choice of lift ˜t because any
+two lifts differ by an element of the normal closure of c in π1(Sg,1, ⋆) = F2g, and conjugation
+by any such element is trivial on π1(Xk
+g ).
+Proof of Proposition 3.4. It suffices to prove the lemma for t ∈ π1(Sg, ∗) that are repre-
+sented by a non-separating simple closed curve. To see this, first note that π1(Sg, ∗) is
+generated by these curves. Furthermore, the groups Inn
+�
+π1(Xk
+g )
+�
+and H1(Sg; Z) commute
+in Aut
+�
+π1(Xk
+g )
+�
+, so
+�
+Cι˜t1 ◦ ˆδ([t1])
+�
+◦
+�
+Cι˜t2 ◦ ˆδ([t2])
+�
+= Cι(˜t1∗˜t2) ◦ ˆδ([t1 ∗ t2]).
+Assume now that t ∈ π1(Sg, ∗) is represented by a non-separating simple closed curve.
+After an isotopy, we can assume that t contains ϵ as a sub-arc. Choose ˜t as pictured in
+Figure 1.
+∗
+⋆
+t
+˜t
+ϵ
+Figure 1. A small regular neighborhood of a loop representing t ∈
+π1(Sg, ∗) and a lift ˜t ∈ π1(Sg,1, ⋆).
+We want to show that
+σ
+�
+Push(t)
+�
+(w) =
+�
+Cι˜t ◦ ˆδ([t])
+�
+(w)
+for each w ∈ π1(Xk
+g ).
+Since this is obviously true for w = z, it suffices to show this
+equality for w = ι(s) for s ∈ π1(Sg,1, ⋆); furthermore, it suffices to show the equality on
+any generating set of π1(Sg,1, ⋆). We use the (infinite) generating set consisting of curves
+of one of the forms pictured in Figure 2 (the intersection of these curves with the annulus
+around t has one component).
+Note that Push(t) fixes the basepoint ⋆, so we can compute the action of Push(t) on
+s ∈ π1(Sg,1, ⋆). We compute the action of Push(t) on the elements in Figure 2 as follows.
+See Figure 3 for an illustration.
+s1 �→ (˜t)−1s1˜tc−1
+and
+s2 �→ (˜t)−1s2˜t
+and
+s3 �→ c(˜t)−1s3˜tc−1
+and
+c �→ c.
+This proves that, for example, that
+σ
+�
+Push(t)
+�
+(ιs1) = (ι˜t)−1(ιs1)(ι˜t)z−k =
+�
+Cι˜t ◦ ˆδ([kt])
+�
+(ιs1).
+We conclude similarly for the generators s2, s3.
+This proves the desired formula for
+σ
+�
+Push(t)
+�
+.
+□
+
+MAPPING CLASS GROUPS OF CIRCLE BUNDLES OVER A SURFACE
+9
+c
+s1
+s2
+s3
+Figure 2. The group π1(Sg,1, ⋆) is generated by ˜t and loops of the form
+pictured above.
+Push(t)(s1)
+Push(t)(s2)
+Push(t)(s3)
+Figure 3. Action of point-pushing about t on the loops in Figure 2. The
+curve c is fixed up to isotopy (up to isotopy Push(t) is the identity on a
+neighborhood of t that contains c).
+The following corollary is an immediate consequence of Proposition 3.4.
+Corollary 3.5. Consider the composition
+(15)
+Ψ : AUT
+�
+π1(Sg)
+� σ−→ AUT
+�
+π1(Xk
+g )
+�
+→ OUT
+�
+π1(Xk
+g )
+�
+.
+The restriction of Ψ to π1(Sg) ∼= Inn
+�
+π1(Sg)
+�
+factors as follows.
+π1(Sg)
+AUT
+�
+π1(Sg)
+�
+H1(Sg; Z)
+OUT
+�
+π1(Xk
+g )
+�
+�
+conjugation
+� Ψ
+�
+kδ ◦ ab
+�
+transvections
+Here ab denotes the abelianization map π1(Sg, ∗) → H1(Sg; Z).
+3.3. Proof of Theorem A. Using the isomorphisms between mapping class groups and
+automorphism groups, the desired diagram is equivalent to the following one.
+1
+π1(Sg)ab
+AUT
+�
+π1(Sg)
+�
+/π′
+OUT
+�
+π1(Sg)
+�
+1
+1
+Hom(π1(Sg), Z)
+OUT
+�
+π1(Xk
+g )
+�
+OUT
+�
+π1(Sg
+�
+)
+1
+�
+�
+�
+�
+�
+�
+�
+�
+�
+kδ
+�
+The map Ψ in (15) descends to the middle vertical map and restricts to the left vertical
+map by Corollary 3.5. The fact that σ is a section (Corollary 3.1) implies that the middle
+
+10
+LEI CHEN AND BENA TSHISHIKU
+vertical map descends to the identity map on OUT
+�
+π1(Sg)
+�
+. When k = 1, the middle
+vertical map is an isomorphism by the five lemma. This concludes the proof of Theorem
+A.
+4. Spectral sequence computation
+In this section we prove Theorem 1.2. This is achieved by two different computations
+using the Lyndon–Hochschild–Serre (LHS) spectral sequence.
+Recall that this spectral
+sequence takes input a short exact sequence of groups 1 → N → G → Q → 1 and a
+G-module A, has E2 page
+Ep,q
+2
+= Hp�
+Q; Hq(N; A)
+�
+,
+and converges to Hp+q(G; A). For both computations we use the Birman exact sequence,
+but with different choices of the module A.
+Notational note.
+To simplify the notation, we use the convention that cohomology
+groups have Z coefficients unless otherwise specified.
+4.1. Euler class computation. Our goal in this section is to prove Proposition 4.1 below,
+which implies Corollary 1.3.
+Proposition 4.1. Fix g ≥ 1.
+Let euk be the Euler class of the extension (1).
+Then
+euk = k eu1, and eu1 has order 2g − 2 in H2�
+Mod(Sg); H1(Sg)
+�
+.
+Proof. The relation euk = k eu1 already follows from Theorem A. Indeed, choosing a set-
+theoretic section for the sequence in the top row of the diagram in Theorem A gives a
+cocycle representative for euk that is k times the cocycle representative for e1.
+Now we prove that eu1 generates a cyclic subgroup isomorphic to Z/(2g − 2)Z in
+H2�
+Mod(Sg); H1(Sg)
+�
+. Our method is to apply the LHS spectral sequence to the Bir-
+man exact sequence with the module A = H1(Sg). Here
+Ep,q
+2
+∼= Hp�
+Mod(Sg); Hq(Sg; A)
+�
+.
+A portion of the associated 5-term exact sequence is as follows.
+0 → H1�
+Mod(Sg); H1(Sg)
+�
+→ H1�
+Mod(Sg,1); H1(Sg)
+�
+A
+−→ Hom
+�
+H1(Sg), H1(Sg)
+�Mod(Sg)
+d0,1
+2
+−−→ H2�
+Mod(Sg); H1(Sg)
+�
+This sequence has been studied by Morita. Morita [Mor85, Prop. 4.1] computes that the
+first term vanishes, so the map A is injective. The group Hom
+�
+H1(Sg), H1(Sg)
+�Mod(Sg) is
+isomorphic to Z and generated the Poincar´e duality isomorphism δ. Morita [Mor85, proof
+of Prop. 6.4] shows that the image of A is (2g − 2)Z. Consequently, the differential d0,1
+2
+descends to an injection Z/(2g − 2)Z → H2�
+Mod(Sg); H1(Sg)
+�
+.
+It remains to show that d0,1
+2
+sends a generator to eu1.
+The differential d0,1
+2
+is the
+transgression; see e.g. [NSW08, Prop. 1.6.6, Thm. 2.4.3]. By standard knowledge of the
+transgression applied to our situation, we find that d0,1
+2
+sends a generator to δ∗(eu), where
+eu is the Euler class of the extension (2), and
+δ∗ : H2�
+Mod(Sg); H1(Sg)
+�
+→ H2�
+Mod(Sg); H1(Sg)
+�
+is the isomorphism induced by the Poincar´e duality isomorphism δ. (For this property
+of the transgression, see [NSW08, §I.6, Exercise 1-2].
+While that reference is mainly
+concerned with finite or profinite groups, the analysis of the transgression contained given
+there applies more generally.) Finally, we observe that δ∗(eu) = eu1 by Theorem A.
+□
+
+MAPPING CLASS GROUPS OF CIRCLE BUNDLES OVER A SURFACE
+11
+4.2. Computation of H2�
+Mod(Sg); H1(Sg)
+�
+. Running the LHS spectral sequence with
+the trivial module A = Z, we prove that if g ≥ 8, then
+(16)
+H2�
+Mod(Sg); H1(Sg)
+� ∼= Z/(2g − 2)Z.
+Combining this with Proposition 4.1 proves Theorem 1.2. The relevant portion of the
+spectral sequence appears below.
+2 H0(Mod(Sg); H2(Sg))
+1
+0
+0 H2(Mod(Sg); H1(Sg))
+0
+Z
+0
+H2(Mod(Sg))
+H3(Mod(Sg)) H4(Mod(Sg))
+0
+1
+2
+3
+4
+d0,2
+2
+d2,1
+2
+The computations in the first column are easy. In the second column, Morita [Mor85,
+Prop. 4.1] computed H1�
+Mod(Sg); H1(Sg)
+�
+= 0 for g ≥ 1.
+The other computation
+H1�
+Mod(Sg)
+�
+= 0 holds for g ≥ 1 because the abelianization of Mod(Sg) is finite [FM12,
+§5.1.2-3].
+According to [BT01, Cor. 1.2],
+H∗
+�
+Mod(Sg,1)
+� ∼= H∗
+�
+Mod(Sg)
+�
+⊗ Z[x]
+in degrees g ≥ 2∗. Here x has degree 2. Applying this and using the universal coefficients
+theorem, we conclude that
+Hi�
+Mod(Sg)
+�
+→ Hi�
+Mod(Sg,1)
+�
+is an isomorphism if i = 3 and g ≥ 6, and it is injective if i = 4 and if g ≥ 8.
+Since the map H4�
+Mod(Sg)
+�
+→ H4�
+Mod(Sg,1)
+�
+is injective, the differential d2,1
+2
+is zero.
+Since the map H3�
+Mod(Sg)
+�
+→ H3�
+Mod(Sg,1)
+�
+is an isomorphism, the differential d0,2
+2
+is
+surjective.
+Thus, the filtration of H2�
+Mod(Sg,1)
+�
+coming from the E∞ page gives an exact sequence
+0 → H2�
+Mod(Sg)
+�
+→ H2�
+Mod(Sg,1)
+�
+F−→
+H0�
+Mod(Sg); H2(Sg)
+� ∼= Z
+d0,2
+2
+−−→
+H2�
+Mod(Sg); H1(Sg)
+�
+→ 0.
+For g ≥ 4,
+H2�
+Mod(Sg)
+� ∼= Z[e1]
+and
+H2�
+Mod(Sg,1)
+� ∼= Z[e, e1]
+and the map Z[e1] → Z[e, e1] is the obvious one e1 �→ e1. We claim that F(e) = 2 − 2g.
+From this we deduce the desired isomorphism (16). The claim follows from the fact that
+the extension that defines e, when restricted to the point-pushing subgroup π1(Sg) <
+Mod(Sg,1), gives the extension
+1 → Z → π1(USg) → π1(Sg) → 1
+where USg is the unit tangent bundle. See [FM12, §5.5.5]. This extension has Euler class
+2 − 2g, so the claim follows.
+
+12
+LEI CHEN AND BENA TSHISHIKU
+References
+[Bro82]
+K. S. Brown. Cohomology of groups, volume 87 of Graduate Texts in Mathematics. Springer-
+Verlag, New York-Berlin, 1982.
+[BT01]
+C.-F. B¨odigheimer and U. Tillmann. Stripping and splitting decorated mapping class groups.
+In Cohomological methods in homotopy theory (Bellaterra, 1998), volume 196 of Progr. Math.,
+pages 47–57. Birkh¨auser, Basel, 2001.
+[Che19]
+L. Chen. On the nonrealizability of braid groups by homeomorphisms. Geom. Topol., 23(7):3735–
+3749, 2019.
+[CR77]
+P. E. Conner and F. Raymond. Deforming homotopy equivalences to homeomorphisms in aspher-
+ical manifolds. Bull. Amer. Math. Soc., 83(1):36–85, 1977.
+[CS22]
+L. Chen and N. Salter. Global fixed points of mapping class group actions and a theorem of
+Markovic. J. Topol., 15(3):1311–1324, 2022.
+[FM12]
+B. Farb and D. Margalit. A primer on mapping class groups, volume 49 of Princeton Mathematical
+Series. Princeton University Press, Princeton, NJ, 2012.
+[Mar07]
+V. Markovic. Realization of the mapping class group by homeomorphisms. Invent. Math.,
+168(3):523–566, 2007.
+[McC91] D. McCullough. Virtually geometrically finite mapping class groups of 3-manifolds. J. Differential
+Geom., 33(1):1–65, 1991.
+[Mor85]
+S. Morita. Family of Jacobian manifolds and characteristic classes of surface bundles. II. Proc.
+Japan Acad. Ser. A Math. Sci., 61(4):112–115, 1985.
+[NSW08] J. Neukirch, A. Schmidt, and K. Wingberg. Cohomology of number fields, volume 323 of
+Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sci-
+ences]. Springer-Verlag, Berlin, second edition, 2008.
+[Sou10]
+J. Souto. A remark on the action of the mapping class group on the unit tangent bundle. Ann.
+Fac. Sci. Toulouse Math. (6), 19(3-4):589–601, 2010.
+[Wal68]
+F. Waldhausen. On irreducible 3-manifolds which are sufficiently large. Ann. of Math. (2), 87:56–
+88, 1968.
+Lei Chen, Department of Mathematics, University of Maryland, 4176 Campus Drive, Col-
+lege Park, MD 20742, USA, chenlei@umd.edu
+Bena Tshishiku, Department of Mathematics, Brown University, 151 Thayer St., Provi-
+dence, RI, 02912, USA, bena tshishiku@brown.edu.
+
diff --git a/StE5T4oBgHgl3EQfAA5y/content/tmp_files/load_file.txt b/StE5T4oBgHgl3EQfAA5y/content/tmp_files/load_file.txt
new file mode 100644
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@@ -0,0 +1,522 @@
+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfAA5y/content/2301.05375v1.pdf,len=521
+page_content='MAPPING CLASS GROUPS OF CIRCLE BUNDLES OVER A  SURFACE  LEI CHEN AND BENA TSHISHIKU  Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfAA5y/content/2301.05375v1.pdf'}
+page_content=' In this paper, we study the algebraic structure of mapping class group  Mod(X) of 3-manifolds X that fiber as a circle bundle over a surface S1 → X → Sg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfAA5y/content/2301.05375v1.pdf'}
+page_content='  There is an exact sequence 1 → H1(Sg) → Mod(X) → Mod(Sg) → 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfAA5y/content/2301.05375v1.pdf'}
+page_content=' We relate this to  the Birman exact sequence and determine when this sequence splits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfAA5y/content/2301.05375v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfAA5y/content/2301.05375v1.pdf'}
+page_content=' Introduction  For g ≥ 1, let Sg denote the closed oriented surface of genus g, and for k ∈ Z, let Xk  g  denote the closed 3-manifold that fibers  S1 → Xk  g → Sg  as an oriented circle-bundle with Euler number k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfAA5y/content/2301.05375v1.pdf'}
+page_content=' Assuming (g, k) ̸= (1, 0), the mapping  class group Mod(Xk  g ) := π0  �  Homeo+(Xk  g )  �  fits into a short exact sequence  (1)  1 → H1(Sg;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfAA5y/content/2301.05375v1.pdf'}
+page_content=' Z) → Mod(Xk  g ) → Mod(Sg) → 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfAA5y/content/2301.05375v1.pdf'}
+page_content='  This paper is motivated by the following question.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfAA5y/content/2301.05375v1.pdf'}
+page_content='  Question 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfAA5y/content/2301.05375v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfAA5y/content/2301.05375v1.pdf'}
+page_content=' For which values of g, k is the extension in (1) split?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfAA5y/content/2301.05375v1.pdf'}
+page_content='  Interestingly, the extension does split for k = 2 − 2g, in which case Xk  g is unit tangent  bundle USg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfAA5y/content/2301.05375v1.pdf'}
+page_content=' In fact, there is a natural action of Mod(Sg) on USg by homeomorphisms,  which gives a splitting of (1) upon taking isotopy classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfAA5y/content/2301.05375v1.pdf'}
+page_content=' For g ≥ 2, this action comes  from the action of the punctured mapping class group Mod(Sg,1) on triples of points on  the boundary of hyperbolic space H2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfAA5y/content/2301.05375v1.pdf'}
+page_content=' This construction dates back to the work of Nielsen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfAA5y/content/2301.05375v1.pdf'}
+page_content='  See [FM12, §5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfAA5y/content/2301.05375v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfAA5y/content/2301.05375v1.pdf'}
+page_content='4, §8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfAA5y/content/2301.05375v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfAA5y/content/2301.05375v1.pdf'}
+page_content='6] and [Sou10, §1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfAA5y/content/2301.05375v1.pdf'}
+page_content='  In general, Question 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfAA5y/content/2301.05375v1.pdf'}
+page_content='1 reduces to a question about group cohomology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfAA5y/content/2301.05375v1.pdf'}
+page_content=' The extension  (1) splits if and only if its Euler class euk ∈ H2�  Mod(Sg);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfAA5y/content/2301.05375v1.pdf'}
+page_content=' H1(Sg;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfAA5y/content/2301.05375v1.pdf'}
+page_content=' Z)  �  vanishes [Bro82,  §IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfAA5y/content/2301.05375v1.pdf'}
+page_content='3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfAA5y/content/2301.05375v1.pdf'}
+page_content=' Here the coefficients are twisted via the natural action of Mod(Sg) on H1(Sg;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfAA5y/content/2301.05375v1.pdf'}
+page_content=' Z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfAA5y/content/2301.05375v1.pdf'}
+page_content='  However, a computation of H2�  Mod(Sg);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfAA5y/content/2301.05375v1.pdf'}
+page_content=' H1(Sg;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfAA5y/content/2301.05375v1.pdf'}
+page_content=' Z)  �  does not appear to be in the litera-  ture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfAA5y/content/2301.05375v1.pdf'}
+page_content='  The extension (1) is related to the Birman exact sequence  1 → π1(Sg) → Mod(Sg,1) → Mod(Sg) → 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfAA5y/content/2301.05375v1.pdf'}
+page_content='  By taking quotients by the commutator subgroup π′ ≡ [π1(Sg), π1(Sg)], we obtain the  following extension  (2)  1 → H1(Sg) → Mod(Sg,1)/π′ → Mod(Sg) → 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfAA5y/content/2301.05375v1.pdf'}
+page_content='  Our main result relates the sequences (1) and (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfAA5y/content/2301.05375v1.pdf'}
+page_content='  Date: January 16, 2023.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfAA5y/content/2301.05375v1.pdf'}
+page_content='  1  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfAA5y/content/2301.05375v1.pdf'}
+page_content='05375v1  [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfAA5y/content/2301.05375v1.pdf'}
+page_content='GT]  13 Jan 2023  2  LEI CHEN AND BENA TSHISHIKU  Theorem A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfAA5y/content/2301.05375v1.pdf'}
+page_content=' Fix g ≥ 1 and k ∈ Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfAA5y/content/2301.05375v1.pdf'}
+page_content=' Assume (g, k) ̸= (1, 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfAA5y/content/2301.05375v1.pdf'}
+page_content=' There is a map between the  short exact sequences (1) and (2)  1  H1(Sg)  Mod(Sg,1)/π′  Mod(Sg)  1  1  H1(Sg;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfAA5y/content/2301.05375v1.pdf'}
+page_content=' Z)  Mod(Xk  g )  Mod(Sg)  1  �  �  �  �  �  �  �  �  � kδ  �  The homomorphism kδ is the Poincar´e duality isomorphism δ composed with multiplication  by k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfAA5y/content/2301.05375v1.pdf'}
+page_content=' In particular, when k = 1, the exact sequences (1) and (2) are isomorphic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfAA5y/content/2301.05375v1.pdf'}
+page_content='  Theorem A implies the Euler classes of the extensions (1) satisfy euk = k eu1 for fixed  g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfAA5y/content/2301.05375v1.pdf'}
+page_content=' Next we determine the subgroup generated by eu1 in H2�  Mod(Sg);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfAA5y/content/2301.05375v1.pdf'}
+page_content=' H1(Sg;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfAA5y/content/2301.05375v1.pdf'}
+page_content=' Z)  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfAA5y/content/2301.05375v1.pdf'}
+page_content='  Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfAA5y/content/2301.05375v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfAA5y/content/2301.05375v1.pdf'}
+page_content=' Fix g ≥ 1, and let eu1 be the Euler class of the extension (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfAA5y/content/2301.05375v1.pdf'}
+page_content=' Then eu1  has order 2g − 2 in H2�  Mod(Sg);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfAA5y/content/2301.05375v1.pdf'}
+page_content=' H1(Sg;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfAA5y/content/2301.05375v1.pdf'}
+page_content=' Z)  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfAA5y/content/2301.05375v1.pdf'}
+page_content=' Furthermore, if g ≥ 8, then eu1 generates  this group, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfAA5y/content/2301.05375v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfAA5y/content/2301.05375v1.pdf'}
+page_content='  H2�  Mod(Sg);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfAA5y/content/2301.05375v1.pdf'}
+page_content=' H1(Sg;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfAA5y/content/2301.05375v1.pdf'}
+page_content=' Z)  � ∼= Z/(2g − 2)Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfAA5y/content/2301.05375v1.pdf'}
+page_content='  Combining Theorem A and Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfAA5y/content/2301.05375v1.pdf'}
+page_content='2 we obtain the following answer to Question  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfAA5y/content/2301.05375v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfAA5y/content/2301.05375v1.pdf'}
+page_content='  Corollary 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfAA5y/content/2301.05375v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfAA5y/content/2301.05375v1.pdf'}
+page_content=' For g ≥ 2 and k ∈ Z, the extension (1) splits if and only if k is divisible  2g − 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfAA5y/content/2301.05375v1.pdf'}
+page_content=' For g = 1 the extension splits for each k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfAA5y/content/2301.05375v1.pdf'}
+page_content='  When a splitting exists, the different possible splittings (up to the action of H1(Sg;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfAA5y/content/2301.05375v1.pdf'}
+page_content=' Z)  on Mod(Xk  g ) by conjugation) are parameterized by elements of H1�  Mod(Sg);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfAA5y/content/2301.05375v1.pdf'}
+page_content=' H1(Sg;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfAA5y/content/2301.05375v1.pdf'}
+page_content=' Z)  �  [Bro82, Ch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfAA5y/content/2301.05375v1.pdf'}
+page_content=' IV, Prop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfAA5y/content/2301.05375v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfAA5y/content/2301.05375v1.pdf'}
+page_content='3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfAA5y/content/2301.05375v1.pdf'}
+page_content='  This group vanishes for g ≥ 1 [Mor85, Prop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfAA5y/content/2301.05375v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfAA5y/content/2301.05375v1.pdf'}
+page_content='1], so the  splitting, when it exists, is unique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfAA5y/content/2301.05375v1.pdf'}
+page_content='  Connection to Nielsen realization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfAA5y/content/2301.05375v1.pdf'}
+page_content=' Instead of Question 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfAA5y/content/2301.05375v1.pdf'}
+page_content='1, one can ask whether there  is a splitting of the composite surjection  Homeo(Xk  g ) → Mod(Xk  g ) → Mod(Sg).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfAA5y/content/2301.05375v1.pdf'}
+page_content='  This is an instance of a Nielsen realization problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfAA5y/content/2301.05375v1.pdf'}
+page_content=' Of course, if Mod(Xk  g ) → Mod(Sg)  does not split, then neither does Homeo(Xk  g ) → Mod(Sg), and Corollary 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfAA5y/content/2301.05375v1.pdf'}
+page_content='3 gives examples  of this.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfAA5y/content/2301.05375v1.pdf'}
+page_content=' Since Mod(Sg) has a natural action on USg, the surjection Homeo(Xk  g ) → Mod(Sg)  does split for k = ±(2g − 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfAA5y/content/2301.05375v1.pdf'}
+page_content=' This is somewhat surprising since mapping class groups are  rarely realized as groups of surface homeomorphisms [Mar07, Che19, CS22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfAA5y/content/2301.05375v1.pdf'}
+page_content=' We wonder  whether this splitting is unique, or if a splitting exists for other values k divisible by 2g −2  (for example, k = 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfAA5y/content/2301.05375v1.pdf'}
+page_content=' We plan to study this in a future paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfAA5y/content/2301.05375v1.pdf'}
+page_content='  Previous work and proof techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfAA5y/content/2301.05375v1.pdf'}
+page_content=' Waldhausen [Wal68, §7] proved that the group  π0  �  Homeo(Xk  g )  �  is isomorphic to the outer automorphism group Out  �  π1(Xk  g )  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfAA5y/content/2301.05375v1.pdf'}
+page_content=' From this,  the short exact sequence (1) can be derived from work of Conner–Raymond [CR77] and  the Dehn–Nielsen–Baer theorem;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfAA5y/content/2301.05375v1.pdf'}
+page_content=' alternatively, see McCullough [McC91, §3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfAA5y/content/2301.05375v1.pdf'}
+page_content=' The Dehn–  Nielsen–Baer theorem also plays a central role in Theorem A, since it allows us to translate  back and forth between topology and group theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfAA5y/content/2301.05375v1.pdf'}
+page_content=' There is a mix of both in the proof of  Theorem A in §3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfAA5y/content/2301.05375v1.pdf'}
+page_content='  MAPPING CLASS GROUPS OF CIRCLE BUNDLES OVER A SURFACE  3  To prove Theorem A, we consider a version of Question 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfAA5y/content/2301.05375v1.pdf'}
+page_content='1 where Xk  g and Sg are  punctured.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfAA5y/content/2301.05375v1.pdf'}
+page_content=' For the punctured manifolds, similar to (1), there is a short exact sequence  1 → H1(Sg;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfAA5y/content/2301.05375v1.pdf'}
+page_content=' Z) → Mod(Xk  g,1) → Mod(Sg,1) → 1,  and we construct a splitting  σ : Mod(Sg,1) → Mod(Xk  g,1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfAA5y/content/2301.05375v1.pdf'}
+page_content='  See Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfAA5y/content/2301.05375v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfAA5y/content/2301.05375v1.pdf'}
+page_content=' A key part of our proof of Theorem A is to determine the image of  the point-pushing subgroup π1(Sg) < Mod(Sg,1) under σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfAA5y/content/2301.05375v1.pdf'}
+page_content=' For this we relate three natural  surface group representations π1(Sg) → Mod(Xk  g,1) that appear in the following diagram,  where the diagonal map is point pushing on Xk  g (not a commutative diagram).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfAA5y/content/2301.05375v1.pdf'}
+page_content='  π1(Sg)  Mod(Sg,1)  H1(Sg;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfAA5y/content/2301.05375v1.pdf'}
+page_content=' Z)  Mod(Xk  g,1)  �  point-pushing on Sg  � σ  �  ��  transvections  See Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfAA5y/content/2301.05375v1.pdf'}
+page_content='4 for a precise statement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfAA5y/content/2301.05375v1.pdf'}
+page_content='  In order to deduce Corollary 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfAA5y/content/2301.05375v1.pdf'}
+page_content='3, we use a spectral sequence argument to prove that eu1  generates a subgroup of H2�  Mod(Sg);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfAA5y/content/2301.05375v1.pdf'}
+page_content=' H1(Sg;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfAA5y/content/2301.05375v1.pdf'}
+page_content=' Z)  �  isomorphic to Z/(2g − 2)Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfAA5y/content/2301.05375v1.pdf'}
+page_content=' A different  spectral sequence computation proves that eu1 generates H2�  Mod(Sg);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfAA5y/content/2301.05375v1.pdf'}
+page_content=' H1(Sg;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfAA5y/content/2301.05375v1.pdf'}
+page_content=' Z)  �  when  g is large.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfAA5y/content/2301.05375v1.pdf'}
+page_content=' These computations use several known computations, including work of Morita  [Mor85].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfAA5y/content/2301.05375v1.pdf'}
+page_content='  Section outline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfAA5y/content/2301.05375v1.pdf'}
+page_content=' In §2 we collect the results we need about the manifolds Xk  g and their  mapping class groups, including Waldhausen’s work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfAA5y/content/2301.05375v1.pdf'}
+page_content='  Theorem A is proved in §3;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfAA5y/content/2301.05375v1.pdf'}
+page_content=' this  section is the core of the paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfAA5y/content/2301.05375v1.pdf'}
+page_content=' In §4, we do two spectral sequence computations to prove  Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfAA5y/content/2301.05375v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfAA5y/content/2301.05375v1.pdf'}
+page_content='  Acknowledgement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfAA5y/content/2301.05375v1.pdf'}
+page_content='  Thanks to B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfAA5y/content/2301.05375v1.pdf'}
+page_content=' Farb for sharing the reference [McC91] and to D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfAA5y/content/2301.05375v1.pdf'}
+page_content='  Margalit for comments on a draft.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfAA5y/content/2301.05375v1.pdf'}
+page_content='  The authors are supported by NSF grants DMS-  2203178, DMS-2104346 and DMS-2005409.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfAA5y/content/2301.05375v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfAA5y/content/2301.05375v1.pdf'}
+page_content=' Circle bundles over surfaces  Here we review some results about circle bundles over surfaces that we will need in  future sections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfAA5y/content/2301.05375v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfAA5y/content/2301.05375v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfAA5y/content/2301.05375v1.pdf'}
+page_content=' Classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfAA5y/content/2301.05375v1.pdf'}
+page_content=' By an oriented circle bundle we mean a fiber bundle  S1 → E → B  with structure group Homeo+(S1), the group of orientation preserving homeomorphisms  of S1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfAA5y/content/2301.05375v1.pdf'}
+page_content=' The inclusion of the rotation group SO(2) in Homeo+(S1) is a homotopy equiva-  lence, so circle bundles are in bijection with rank-2 real vector bundles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfAA5y/content/2301.05375v1.pdf'}
+page_content=' The classifying  space B SO(2) is homotopy equivalent to CP ∞, which is an Eilenberg–Maclane space  K(Z, 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfAA5y/content/2301.05375v1.pdf'}
+page_content=' Thus each circle bundle is uniquely determined up to isomorphism by its Euler  class eu(E) ∈ H2(B;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfAA5y/content/2301.05375v1.pdf'}
+page_content=' Z), which is the primary obstruction to a section of the bundle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfAA5y/content/2301.05375v1.pdf'}
+page_content='  4  LEI CHEN AND BENA TSHISHIKU  When B = Sg is a closed, oriented surface, H2(Sg;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfAA5y/content/2301.05375v1.pdf'}
+page_content=' Z) ∼= Z, we can speak of the Euler  number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfAA5y/content/2301.05375v1.pdf'}
+page_content=' We use Xk  g to denote the total space of the circle bundle  S1 → Xk  g → Sg  with Euler number k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfAA5y/content/2301.05375v1.pdf'}
+page_content=' For example, for the unit tangent bundle eu(USg) = 2 − 2g (the  Euler characteristic), so USg ∼= X2−2g  g  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfAA5y/content/2301.05375v1.pdf'}
+page_content=' We also note that Xk  g and X−k  g  are homeomorphic  3-manifolds, since the sign of the Euler number of a circle bundle over Sg depends on the  choice of orientation on Sg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfAA5y/content/2301.05375v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfAA5y/content/2301.05375v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfAA5y/content/2301.05375v1.pdf'}
+page_content=' Fundamental group π1(Xk  g ) and its automorphisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfAA5y/content/2301.05375v1.pdf'}
+page_content=' From the long exact se-  quence of a fibration, we have an exact sequence  1 → Z → π1(Xk  g ) → π1(Sg) → 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfAA5y/content/2301.05375v1.pdf'}
+page_content='  The group π1(Xk  g ) has a presentation  (3)  π1(Xk  g ) =  �  A1, B1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfAA5y/content/2301.05375v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfAA5y/content/2301.05375v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfAA5y/content/2301.05375v1.pdf'}
+page_content=' , Ag, Bg, z | z central, [A1, B1] · · · [Ag, Bg] = zk�  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfAA5y/content/2301.05375v1.pdf'}
+page_content='  Using this, one finds ⟨z⟩ ∼= Z is the center of π1(Xk  g ) as long as (g, k) ̸= (1, 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfAA5y/content/2301.05375v1.pdf'}
+page_content=' When g ≥ 2  follows from the fact that the group π1(Sg) has trivial center;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfAA5y/content/2301.05375v1.pdf'}
+page_content=' the case g = 1 can be treated  directly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfAA5y/content/2301.05375v1.pdf'}
+page_content='  Given this computation of the center, any automorphism of π1(Xk  g ) induces an auto-  morphism of ⟨z⟩ ∼= Z and descends to an automorphism of π1(Sg).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfAA5y/content/2301.05375v1.pdf'}
+page_content=' The latter gives a  homomorphism  Aut  �  π1(Xk  g )  �  → Aut  �  π1(Sg)  �  that restricts to an isomorphism between the inner automorphism groups  (4)  Inn  �  π1(Xk  g )  � ∼= π1(Sg) ∼= Inn  �  π1(Sg)  �  and hence descends to a homomorphism  (5)  Out  �  π1(Xk  g )  �  → Out  �  π1(Sg)  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfAA5y/content/2301.05375v1.pdf'}
+page_content='  Orientations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfAA5y/content/2301.05375v1.pdf'}
+page_content=' It will be convenient to define  AUT  �  π1(Sg)  �  < Aut  �  π1(Sg)  �  as the subgroup that acts trivially on H2(π1(Sg);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfAA5y/content/2301.05375v1.pdf'}
+page_content=' Z) ∼= Z (the “orientation-preserving”  subgroup).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfAA5y/content/2301.05375v1.pdf'}
+page_content=' We define  AUT  �  π1(Xk  g )  �  < Aut  �  π1(Xk  g )  �  as the group of automorphisms that project into AUT  �  π1(Sg)  �  and that act trivially on  the center ⟨z⟩ ∼= Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfAA5y/content/2301.05375v1.pdf'}
+page_content=' In particular, AUT  �  π1(Xk  g )  �  has index 4 in Aut  �  π1(Xk  g )  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfAA5y/content/2301.05375v1.pdf'}
+page_content='  These orientation-preserving subgroups contain the (respective) inner automorphism  groups, and we denote the quotients OUT  �  π1(Xk  g )  �  and OUT  �  π1(Sg)  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfAA5y/content/2301.05375v1.pdf'}
+page_content='  MAPPING CLASS GROUPS OF CIRCLE BUNDLES OVER A SURFACE  5  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfAA5y/content/2301.05375v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfAA5y/content/2301.05375v1.pdf'}
+page_content=' Mapping class group Mod(Xk  g ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfAA5y/content/2301.05375v1.pdf'}
+page_content=' Fix g ≥ 1 and k ∈ Z, and assume (g, k) ̸= (1, 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfAA5y/content/2301.05375v1.pdf'}
+page_content='  Let Homeo+(Xk  g ) denote the group of homeomorphisms whose image in Out  �  π1(Xk  g )  �  is  contained in OUT  �  π1(Sg)  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfAA5y/content/2301.05375v1.pdf'}
+page_content=' Define  Mod(Xk  g ) := π0  �  Homeo+(Xk  g )  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfAA5y/content/2301.05375v1.pdf'}
+page_content='  Waldhausen [Wal68, Cor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfAA5y/content/2301.05375v1.pdf'}
+page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfAA5y/content/2301.05375v1.pdf'}
+page_content='5] proved that the natural homomorphism  π0  �  Homeo(Xk  g )  �  → Out  �  π1(Xk  g )  �  is an isomorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfAA5y/content/2301.05375v1.pdf'}
+page_content=' Then, by the definitions, this homomorphism restricts to an isomor-  phism Mod(Xk  g ) ∼= OUT  �  π1(Xk  g )  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfAA5y/content/2301.05375v1.pdf'}
+page_content=' Waldhausen also proved that π0 Homeo(Xk  g ) is iso-  morphic to the group of fiber-preserving homeomorphisms modulo homeomorphisms that  are isotopic to the identity through fiber-preserving isotopies;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfAA5y/content/2301.05375v1.pdf'}
+page_content=' see [Wal68, Rmk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfAA5y/content/2301.05375v1.pdf'}
+page_content=' following  Cor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfAA5y/content/2301.05375v1.pdf'}
+page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfAA5y/content/2301.05375v1.pdf'}
+page_content='5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfAA5y/content/2301.05375v1.pdf'}
+page_content=' Consequently, there is a homomorphism  (6)  Mod(Xk  g ) → Mod(Sg).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfAA5y/content/2301.05375v1.pdf'}
+page_content='  Altogether, we have the following commutative diagram relating the maps (5) and (6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfAA5y/content/2301.05375v1.pdf'}
+page_content='  (7)  Mod(Xk  g )  Mod(Sg)  OUT  �  π1(Xk  g )  �  OUT  �  π1(Sg)  �  �  �  ∼=  �  ∼=  �  The right vertical map is an isomorphism by the Dehn–Nielsen–Baer theorem [FM12,  Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfAA5y/content/2301.05375v1.pdf'}
+page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfAA5y/content/2301.05375v1.pdf'}
+page_content='1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfAA5y/content/2301.05375v1.pdf'}
+page_content=' Furthermore, by Conner–Raymond [CR77, Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfAA5y/content/2301.05375v1.pdf'}
+page_content=' 8] that there is a short exact  sequence  (8)  1 → Hom(π1(Sg), Z) → OUT  �  π1(Xk  g )  �  → OUT  �  π1(Sg)  �  → 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfAA5y/content/2301.05375v1.pdf'}
+page_content='  This establishes the short exact sequence (1) in the introduction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfAA5y/content/2301.05375v1.pdf'}
+page_content=' We will give a concrete  derivation of this exact sequence in Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfAA5y/content/2301.05375v1.pdf'}
+page_content='1 below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfAA5y/content/2301.05375v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfAA5y/content/2301.05375v1.pdf'}
+page_content=' Relating Mod(Xk  g ) to the Birman exact sequence  In this section, we prove Theorem A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfAA5y/content/2301.05375v1.pdf'}
+page_content=' To construct the map of short exact sequences in  Theorem A, our main task is to first define a homomorphism Mod(Sg,1) → Mod(Xk  g ) and  then to compute that its kernel is the commutator subgroup of π1(Sg) < Mod(Sg,1) (the  point-pushing subgroup).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfAA5y/content/2301.05375v1.pdf'}
+page_content=' We do this is §3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfAA5y/content/2301.05375v1.pdf'}
+page_content='1 and §3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfAA5y/content/2301.05375v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfAA5y/content/2301.05375v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfAA5y/content/2301.05375v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfAA5y/content/2301.05375v1.pdf'}
+page_content=' A homomorphism Ψ : Mod(Sg,1) → Mod(Xk  g ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfAA5y/content/2301.05375v1.pdf'}
+page_content=' Fix a basepoint ∗ ∈ Sg, and set  Sg,1 = Sg\\{∗}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfAA5y/content/2301.05375v1.pdf'}
+page_content=' By the Dehn–Nielsen–Baer theorem, Mod(Sg,1) is isomorphic to Out∗(F2g),  where F2g is the free group of rank 2g and Out∗(F2g) < Out(F2g) is the subgroup that  preserves the conjugacy class corresponding to the free homotopy class of the curve around  the puncture in Sg,1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfAA5y/content/2301.05375v1.pdf'}
+page_content=' We construct Ψ as a composition  (9)  Ψ : Mod(Sg,1) ∼= Out∗(F2g) σ−→ AUT  �  π1(Xk  g )  �  → OUT  �  π1(Xk  g )  � ∼= Mod(Xk  g ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfAA5y/content/2301.05375v1.pdf'}
+page_content='  To define σ, fix a generating set α1, β1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfAA5y/content/2301.05375v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfAA5y/content/2301.05375v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfAA5y/content/2301.05375v1.pdf'}
+page_content=' , αg, βg for F2g such that c = �g  i=1[αi, βi]  represents the conjugacy class of the curve around the puncture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfAA5y/content/2301.05375v1.pdf'}
+page_content=' Let  (10)  ι : F2g → π1(Xk  g )  be the homomorphism defined by αi �→ Ai and βi �→ Bi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfAA5y/content/2301.05375v1.pdf'}
+page_content=' Given f ∈ Out∗(F2g), fix an  automorphism ˜f : F2g → F2g that represents f, and assume that ˜f(c) = c (this can always  6  LEI CHEN AND BENA TSHISHIKU  be achieved by composing any lift with an inner automorphism of F2g).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfAA5y/content/2301.05375v1.pdf'}
+page_content=' Next we define  σ(f) on generators of π1(Xk  g ) by  (11)  σ(f)(Ai) = ι ˜f(αi),  σ(f)(Bi) = ι ˜f(βi),  σ(f)(z) = z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfAA5y/content/2301.05375v1.pdf'}
+page_content='  To show that σ(f) extends to a homomorphism of π1(Xk  g ), we check that the relation  [A1, B1] · · · [Ag, Bg] = zk is preserved under σ(f):  �  i  [σ(f)(Ai), σ(f)(Bi)] =  �  i  [ι ˜f(αi), ι ˜f(βi)] = ι(c) = zk = σ(f)(zk).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfAA5y/content/2301.05375v1.pdf'}
+page_content='  The second equality uses the the fact that ˜f(c) = c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfAA5y/content/2301.05375v1.pdf'}
+page_content=' The map σ(f) is independent of  the choice of ˜f because different choices of ˜f differ by conjugation by powers of c (be-  cause the centralizer of c in F2g is the cyclic subgroup ⟨c⟩)1 and ι(c) = zk is central in  π1(Xk  g ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfAA5y/content/2301.05375v1.pdf'}
+page_content=' The homomorphism σ(f) : π1(Xk  g ) → π1(Xk  g ) is an automorphism and belongs to  AUT  �  π1(Xk  g )  �  by definition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfAA5y/content/2301.05375v1.pdf'}
+page_content=' Furthermore, f �→ σ(f) is a homomorphism, which is easy  to check using the observation that if w = ιw′, then σ(f)(w) = ι ˜f(w′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfAA5y/content/2301.05375v1.pdf'}
+page_content='  Composing σ with AUT → OUT gives the desired homomorphism Ψ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfAA5y/content/2301.05375v1.pdf'}
+page_content=' As a corollary of  this construction, we have proved the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfAA5y/content/2301.05375v1.pdf'}
+page_content='  Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfAA5y/content/2301.05375v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfAA5y/content/2301.05375v1.pdf'}
+page_content=' Fix g ≥ 1 and k ∈ Z, and assume (g, k) ̸= (1, 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfAA5y/content/2301.05375v1.pdf'}
+page_content='  The natural map  Φ : AUT  �  π1(Xk  g )  �  → AUT  �  π1(Sg)  �  (see §2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfAA5y/content/2301.05375v1.pdf'}
+page_content='2) fits into an exact sequence  (12)  1 → Hom(π1(Sg), Z) → AUT  �  π1(Xk  g )  � Φ−→ AUT  �  π1(Sg)  �  → 1,  and this exact sequence splits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfAA5y/content/2301.05375v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfAA5y/content/2301.05375v1.pdf'}
+page_content=' First we compute the kernel of Φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfAA5y/content/2301.05375v1.pdf'}
+page_content='  Using the presentation for π1(Xk  g ) in (3), if  f ∈ ker(Φ), then  f(Ai) = Aizmi  and  f(Bi) = Bizni  for some m1, n1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfAA5y/content/2301.05375v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfAA5y/content/2301.05375v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfAA5y/content/2301.05375v1.pdf'}
+page_content=' , mg, ng ∈ Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfAA5y/content/2301.05375v1.pdf'}
+page_content=' The map ai �→ mi, bi �→ ni extends to a homomorphism  τ(f) : π1(Sg) → Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfAA5y/content/2301.05375v1.pdf'}
+page_content=' It is elementary to check that the map ker(Φ) → Hom(π1(Sg), Z)  defined by f �→ τ(f) is an isomorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfAA5y/content/2301.05375v1.pdf'}
+page_content='  The homomorphism σ defined above shows that Φ is a split surjection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfAA5y/content/2301.05375v1.pdf'}
+page_content=' Note that the  Mod(Sg, ∗) ∼= Mod(Sg \\ {∗}) (basepoint vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfAA5y/content/2301.05375v1.pdf'}
+page_content=' puncture), so by Dehn–Nielsen–Baer there is  an isomorphism AUT  �  π1(Sg)  � ∼= Out∗(F2g), and we use this isomorphism to view σ as a  splitting of Φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfAA5y/content/2301.05375v1.pdf'}
+page_content='  □  Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfAA5y/content/2301.05375v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfAA5y/content/2301.05375v1.pdf'}
+page_content=' We call elements of ker(Φ) ∼= H1(Sg;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfAA5y/content/2301.05375v1.pdf'}
+page_content=' Z) transvections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfAA5y/content/2301.05375v1.pdf'}
+page_content='  Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfAA5y/content/2301.05375v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfAA5y/content/2301.05375v1.pdf'}
+page_content=' The homomorphism Ψ can be constructed on the level of topology as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfAA5y/content/2301.05375v1.pdf'}
+page_content='  Fix a regular neighborhood D of the puncture on Sg,1 (so D is a once-punctured disk).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfAA5y/content/2301.05375v1.pdf'}
+page_content='  Given a mapping class f ∈ Mod(Sg,1), choose a representing homeomorphism f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfAA5y/content/2301.05375v1.pdf'}
+page_content=' Without  loss of generality, we can assume that f is the identity on D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfAA5y/content/2301.05375v1.pdf'}
+page_content=' The bundle Xk  g → Sg can be  trivialized over S \\ D (because the classifying space B SO(2) is simply connected).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfAA5y/content/2301.05375v1.pdf'}
+page_content=' Fixing  a trivialization (S \\ D) × S1 over S \\ D, we lift f to the product homeomorphism f × idS1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfAA5y/content/2301.05375v1.pdf'}
+page_content='  This homeomorphisms is the identity on the boundary ∂(S \\ D) × S1, so we can extend  by the identity to obtain a homeomorphism ˜f of Xk  g .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfAA5y/content/2301.05375v1.pdf'}
+page_content=' The map sending f ∈ Mod(Sg,1) to  1Note that the centralizer is isomorphic to Z and contains ⟨c⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfAA5y/content/2301.05375v1.pdf'}
+page_content=' It is only bigger if c = ui for some u ∈ F2g  and i ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfAA5y/content/2301.05375v1.pdf'}
+page_content=' By contradiction, if c = ui for i ≥ 2, then u is cyclically reduced because c is.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfAA5y/content/2301.05375v1.pdf'}
+page_content=' This implies that  u is a subword of c = �[αi, βi], which is absurd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfAA5y/content/2301.05375v1.pdf'}
+page_content='  MAPPING CLASS GROUPS OF CIRCLE BUNDLES OVER A SURFACE  7  the isotopy class  �  ˜f  �  ∈ Mod(Xk  g ) is the topological version of the homomorphism Ψ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfAA5y/content/2301.05375v1.pdf'}
+page_content=' Note  that the isotopy class [f] is only well-defined up to Dehn twists about ∂D which is a loop  around the puncture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfAA5y/content/2301.05375v1.pdf'}
+page_content=' This is analogous to the ambiguity encountered in the definition of  σ, which ultimately does not affect the definition of Ψ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfAA5y/content/2301.05375v1.pdf'}
+page_content='  Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfAA5y/content/2301.05375v1.pdf'}
+page_content='1 and equation (4) combine to give the short exact sequence of outer auto-  morphism groups (8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfAA5y/content/2301.05375v1.pdf'}
+page_content='  Warning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfAA5y/content/2301.05375v1.pdf'}
+page_content=' The splitting of the short exact sequence (12) does not give a splitting of the  short exact sequence (8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfAA5y/content/2301.05375v1.pdf'}
+page_content=' Indeed we will show the latter sequence does not always split  (Corollary 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfAA5y/content/2301.05375v1.pdf'}
+page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfAA5y/content/2301.05375v1.pdf'}
+page_content=' The subtlety comes from the fact that the inner automorphism group  Inn  �  π1(Xk  g )  � ∼= π1(Sg) does not coincide with the image of π1(Sg) ∼= Inn  �  π1(Sg)  �  <  Aut  �  π1(Sg)  �  under the section σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfAA5y/content/2301.05375v1.pdf'}
+page_content=' Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfAA5y/content/2301.05375v1.pdf'}
+page_content='4 below describes the precise relationship.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfAA5y/content/2301.05375v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfAA5y/content/2301.05375v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfAA5y/content/2301.05375v1.pdf'}
+page_content=' Kernel of Ψ : Mod(Sg,1) → Mod(Xk  g ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfAA5y/content/2301.05375v1.pdf'}
+page_content=' Observe that the kernel of Ψ is contained  in the point-pushing subgroup π1(Sg) < Mod(Sg,1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfAA5y/content/2301.05375v1.pdf'}
+page_content=' This is because Ψ composed with  the natural map Mod(Xk  g ) → Mod(Sg) is the natural map Mod(Sg,1) → Mod(Sg), whose  kernel is the point-pushing subgroup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfAA5y/content/2301.05375v1.pdf'}
+page_content='  Thus we want to understand the image of the  point-pushing subgroup under the section σ used to define Ψ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfAA5y/content/2301.05375v1.pdf'}
+page_content=' What we find is a simple  relationship between three surface group representations:  π1(Sg)  π1(Sg)  π1(Sg)  AUT  �  π1(Xk  g )  �  �  inner auts of π1(Sg), lifted  �  inner auts of π1(Xk  g )  �  σ  �  transvections  The main results are Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfAA5y/content/2301.05375v1.pdf'}
+page_content='4 and Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfAA5y/content/2301.05375v1.pdf'}
+page_content='5 below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfAA5y/content/2301.05375v1.pdf'}
+page_content=' In order to state Propo-  sition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfAA5y/content/2301.05375v1.pdf'}
+page_content='4, we need the following notation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfAA5y/content/2301.05375v1.pdf'}
+page_content=' Let  δ : H1(Sg;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfAA5y/content/2301.05375v1.pdf'}
+page_content=' Z) → H1(Sg;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfAA5y/content/2301.05375v1.pdf'}
+page_content=' Z)  be the Poincar´e duality map, given explicitly by γ �→ ⟨−, γ⟩, where  ⟨−, −⟩ : H1(Sg;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfAA5y/content/2301.05375v1.pdf'}
+page_content=' Z) × H1(Sg;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfAA5y/content/2301.05375v1.pdf'}
+page_content=' Z) → Z  is the algebraic intersection form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfAA5y/content/2301.05375v1.pdf'}
+page_content=' We use ˆδ denote the composition  ˆδ : H1(Sg;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfAA5y/content/2301.05375v1.pdf'}
+page_content=' Z) δ−→ H1(Sg;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfAA5y/content/2301.05375v1.pdf'}
+page_content=' Z) �→ AUT  �  π1(Xk  g )  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfAA5y/content/2301.05375v1.pdf'}
+page_content='  This map is given explicitly by ˆδ(γ)(w) = w · z⟨[ ¯w],γ⟩, where ¯w is the image of w under  π1(Xk  g ) → π1(Sg) and [ ¯w] ∈ H1(Sg;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfAA5y/content/2301.05375v1.pdf'}
+page_content=' Z) is the corresponding homology class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfAA5y/content/2301.05375v1.pdf'}
+page_content='  Fix a basepoint ⋆ ∈ Sg,1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfAA5y/content/2301.05375v1.pdf'}
+page_content=' Recall that we have fixed a standard generating set {αi, βi} of  π1(Sg,1, ⋆) ∼= F2g so that c := �  i[αi, βi] is a loop around the puncture ∗ of Sg,1 = Sg \\ {∗}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfAA5y/content/2301.05375v1.pdf'}
+page_content='  Define  (13)  Π : π1(Sg,1, ⋆) → π1(Sg, ∗)  by γ �→ ϵ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfAA5y/content/2301.05375v1.pdf'}
+page_content='γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfAA5y/content/2301.05375v1.pdf'}
+page_content='¯ϵ, where ϵ is a fixed arc from ∗ to ⋆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfAA5y/content/2301.05375v1.pdf'}
+page_content='  8  LEI CHEN AND BENA TSHISHIKU  Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfAA5y/content/2301.05375v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfAA5y/content/2301.05375v1.pdf'}
+page_content=' Fix t ∈ π1(Sg, ∗), and let Push(t) ∈ Mod(Sg,1) ∼= Out∗(F2g) be the  point-pushing mapping class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfAA5y/content/2301.05375v1.pdf'}
+page_content=' If ˜t ∈ π1(Sg,1, ⋆) is any lift of t (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfAA5y/content/2301.05375v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfAA5y/content/2301.05375v1.pdf'}
+page_content=' Π(˜t) = t), then  (14)  σ  �  Push(t)  �  = Cι˜t ◦ ˆδ([kt]),  Here Cx denotes conjugation by x, and the maps ι : F2g → π1(Xk  g ) and σ : Out∗(F2g) →  AUT  �  π1(Xk  g )  �  are defined in (10) and (11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfAA5y/content/2301.05375v1.pdf'}
+page_content='  As a sanity check, observe that Cι˜t does not depend on the choice of lift ˜t because any  two lifts differ by an element of the normal closure of c in π1(Sg,1, ⋆) = F2g, and conjugation  by any such element is trivial on π1(Xk  g ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfAA5y/content/2301.05375v1.pdf'}
+page_content='  Proof of Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfAA5y/content/2301.05375v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfAA5y/content/2301.05375v1.pdf'}
+page_content=' It suffices to prove the lemma for t ∈ π1(Sg, ∗) that are repre-  sented by a non-separating simple closed curve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfAA5y/content/2301.05375v1.pdf'}
+page_content=' To see this, first note that π1(Sg, ∗) is  generated by these curves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfAA5y/content/2301.05375v1.pdf'}
+page_content=' Furthermore, the groups Inn  �  π1(Xk  g )  �  and H1(Sg;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfAA5y/content/2301.05375v1.pdf'}
+page_content=' Z) commute  in Aut  �  π1(Xk  g )  �  , so  �  Cι˜t1 ◦ ˆδ([t1])  � �  Cι˜t2 ◦ ˆδ([t2])  �  = Cι(˜t1∗˜t2) ◦ ˆδ([t1 ∗ t2]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfAA5y/content/2301.05375v1.pdf'}
+page_content='  Assume now that t ∈ π1(Sg, ∗) is represented by a non-separating simple closed curve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfAA5y/content/2301.05375v1.pdf'}
+page_content='  After an isotopy, we can assume that t contains ϵ as a sub-arc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfAA5y/content/2301.05375v1.pdf'}
+page_content=' Choose ˜t as pictured in  Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfAA5y/content/2301.05375v1.pdf'}
+page_content='  ∗  ⋆  t  ˜t  ϵ  Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfAA5y/content/2301.05375v1.pdf'}
+page_content=' A small regular neighborhood of a loop representing t ∈  π1(Sg, ∗) and a lift ˜t ∈ π1(Sg,1, ⋆).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfAA5y/content/2301.05375v1.pdf'}
+page_content='  We want to show that  σ  �  Push(t)  �  (w) =  �  Cι˜t ◦ ˆδ([t])  �  (w)  for each w ∈ π1(Xk  g ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfAA5y/content/2301.05375v1.pdf'}
+page_content='  Since this is obviously true for w = z, it suffices to show this  equality for w = ι(s) for s ∈ π1(Sg,1, ⋆);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfAA5y/content/2301.05375v1.pdf'}
+page_content=' furthermore, it suffices to show the equality on  any generating set of π1(Sg,1, ⋆).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfAA5y/content/2301.05375v1.pdf'}
+page_content=' We use the (infinite) generating set consisting of curves  of one of the forms pictured in Figure 2 (the intersection of these curves with the annulus  around t has one component).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfAA5y/content/2301.05375v1.pdf'}
+page_content='  Note that Push(t) fixes the basepoint ⋆, so we can compute the action of Push(t) on  s ∈ π1(Sg,1, ⋆).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfAA5y/content/2301.05375v1.pdf'}
+page_content=' We compute the action of Push(t) on the elements in Figure 2 as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfAA5y/content/2301.05375v1.pdf'}
+page_content='  See Figure 3 for an illustration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfAA5y/content/2301.05375v1.pdf'}
+page_content='  s1 �→ (˜t)−1s1˜tc−1  and  s2 �→ (˜t)−1s2˜t  and  s3 �→ c(˜t)−1s3˜tc−1  and  c �→ c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfAA5y/content/2301.05375v1.pdf'}
+page_content='  This proves that, for example, that  σ  �  Push(t)  �  (ιs1) = (ι˜t)−1(ιs1)(ι˜t)z−k =  �  Cι˜t ◦ ˆδ([kt])  �  (ιs1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfAA5y/content/2301.05375v1.pdf'}
+page_content='  We conclude similarly for the generators s2, s3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfAA5y/content/2301.05375v1.pdf'}
+page_content='  This proves the desired formula for  σ  �  Push(t)  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfAA5y/content/2301.05375v1.pdf'}
+page_content='  □  MAPPING CLASS GROUPS OF CIRCLE BUNDLES OVER A SURFACE  9  c  s1  s2  s3  Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfAA5y/content/2301.05375v1.pdf'}
+page_content=' The group π1(Sg,1, ⋆) is generated by ˜t and loops of the form  pictured above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfAA5y/content/2301.05375v1.pdf'}
+page_content='  Push(t)(s1)  Push(t)(s2)  Push(t)(s3)  Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfAA5y/content/2301.05375v1.pdf'}
+page_content=' Action of point-pushing about t on the loops in Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfAA5y/content/2301.05375v1.pdf'}
+page_content=' The  curve c is fixed up to isotopy (up to isotopy Push(t) is the identity on a  neighborhood of t that contains c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfAA5y/content/2301.05375v1.pdf'}
+page_content='  The following corollary is an immediate consequence of Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfAA5y/content/2301.05375v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfAA5y/content/2301.05375v1.pdf'}
+page_content='  Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfAA5y/content/2301.05375v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfAA5y/content/2301.05375v1.pdf'}
+page_content=' Consider the composition  (15)  Ψ : AUT  �  π1(Sg)  � σ−→ AUT  �  π1(Xk  g )  �  → OUT  �  π1(Xk  g )  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfAA5y/content/2301.05375v1.pdf'}
+page_content='  The restriction of Ψ to π1(Sg) ∼= Inn  �  π1(Sg)  �  factors as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfAA5y/content/2301.05375v1.pdf'}
+page_content='  π1(Sg)  AUT  �  π1(Sg)  �  H1(Sg;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfAA5y/content/2301.05375v1.pdf'}
+page_content=' Z)  OUT  �  π1(Xk  g )  �  �  conjugation  � Ψ  �  kδ ◦ ab  �  transvections  Here ab denotes the abelianization map π1(Sg, ∗) → H1(Sg;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfAA5y/content/2301.05375v1.pdf'}
+page_content=' Z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfAA5y/content/2301.05375v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfAA5y/content/2301.05375v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfAA5y/content/2301.05375v1.pdf'}
+page_content=' Proof of Theorem A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfAA5y/content/2301.05375v1.pdf'}
+page_content=' Using the isomorphisms between mapping class groups and  automorphism groups, the desired diagram is equivalent to the following one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfAA5y/content/2301.05375v1.pdf'}
+page_content='  1  π1(Sg)ab  AUT  �  π1(Sg)  �  /π′  OUT  �  π1(Sg)  �  1  1  Hom(π1(Sg), Z)  OUT  �  π1(Xk  g )  �  OUT  �  π1(Sg  �  )  1  �  �  �  �  �  �  �  �  �  kδ  �  The map Ψ in (15) descends to the middle vertical map and restricts to the left vertical  map by Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfAA5y/content/2301.05375v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfAA5y/content/2301.05375v1.pdf'}
+page_content=' The fact that σ is a section (Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfAA5y/content/2301.05375v1.pdf'}
+page_content='1) implies that the middle  10  LEI CHEN AND BENA TSHISHIKU  vertical map descends to the identity map on OUT  �  π1(Sg)  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfAA5y/content/2301.05375v1.pdf'}
+page_content=' When k = 1, the middle  vertical map is an isomorphism by the five lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfAA5y/content/2301.05375v1.pdf'}
+page_content=' This concludes the proof of Theorem  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfAA5y/content/2301.05375v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfAA5y/content/2301.05375v1.pdf'}
+page_content=' Spectral sequence computation  In this section we prove Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfAA5y/content/2301.05375v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfAA5y/content/2301.05375v1.pdf'}
+page_content=' This is achieved by two different computations  using the Lyndon–Hochschild–Serre (LHS) spectral sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfAA5y/content/2301.05375v1.pdf'}
+page_content='  Recall that this spectral  sequence takes input a short exact sequence of groups 1 → N → G → Q → 1 and a  G-module A, has E2 page  Ep,q  2  = Hp�  Q;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfAA5y/content/2301.05375v1.pdf'}
+page_content=' Hq(N;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfAA5y/content/2301.05375v1.pdf'}
+page_content=' A)  �  ,  and converges to Hp+q(G;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfAA5y/content/2301.05375v1.pdf'}
+page_content=' A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfAA5y/content/2301.05375v1.pdf'}
+page_content=' For both computations we use the Birman exact sequence,  but with different choices of the module A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfAA5y/content/2301.05375v1.pdf'}
+page_content='  Notational note.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfAA5y/content/2301.05375v1.pdf'}
+page_content='  To simplify the notation, we use the convention that cohomology  groups have Z coefficients unless otherwise specified.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfAA5y/content/2301.05375v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfAA5y/content/2301.05375v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfAA5y/content/2301.05375v1.pdf'}
+page_content=' Euler class computation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfAA5y/content/2301.05375v1.pdf'}
+page_content=' Our goal in this section is to prove Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfAA5y/content/2301.05375v1.pdf'}
+page_content='1 below,  which implies Corollary 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfAA5y/content/2301.05375v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfAA5y/content/2301.05375v1.pdf'}
+page_content='  Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfAA5y/content/2301.05375v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfAA5y/content/2301.05375v1.pdf'}
+page_content=' Fix g ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfAA5y/content/2301.05375v1.pdf'}
+page_content='  Let euk be the Euler class of the extension (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfAA5y/content/2301.05375v1.pdf'}
+page_content='  Then  euk = k eu1, and eu1 has order 2g − 2 in H2�  Mod(Sg);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfAA5y/content/2301.05375v1.pdf'}
+page_content=' H1(Sg)  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfAA5y/content/2301.05375v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfAA5y/content/2301.05375v1.pdf'}
+page_content=' The relation euk = k eu1 already follows from Theorem A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfAA5y/content/2301.05375v1.pdf'}
+page_content=' Indeed, choosing a set-  theoretic section for the sequence in the top row of the diagram in Theorem A gives a  cocycle representative for euk that is k times the cocycle representative for e1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfAA5y/content/2301.05375v1.pdf'}
+page_content='  Now we prove that eu1 generates a cyclic subgroup isomorphic to Z/(2g − 2)Z in  H2�  Mod(Sg);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfAA5y/content/2301.05375v1.pdf'}
+page_content=' H1(Sg)  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfAA5y/content/2301.05375v1.pdf'}
+page_content=' Our method is to apply the LHS spectral sequence to the Bir-  man exact sequence with the module A = H1(Sg).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfAA5y/content/2301.05375v1.pdf'}
+page_content=' Here  Ep,q  2  ∼= Hp�  Mod(Sg);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfAA5y/content/2301.05375v1.pdf'}
+page_content=' Hq(Sg;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfAA5y/content/2301.05375v1.pdf'}
+page_content=' A)  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfAA5y/content/2301.05375v1.pdf'}
+page_content='  A portion of the associated 5-term exact sequence is as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfAA5y/content/2301.05375v1.pdf'}
+page_content='  0 → H1�  Mod(Sg);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfAA5y/content/2301.05375v1.pdf'}
+page_content=' H1(Sg)  �  → H1�  Mod(Sg,1);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfAA5y/content/2301.05375v1.pdf'}
+page_content=' H1(Sg)  �  A  −→ Hom  �  H1(Sg), H1(Sg)  �Mod(Sg)  d0,1  2  −−→ H2�  Mod(Sg);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfAA5y/content/2301.05375v1.pdf'}
+page_content=' H1(Sg)  �  This sequence has been studied by Morita.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfAA5y/content/2301.05375v1.pdf'}
+page_content=' Morita [Mor85, Prop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfAA5y/content/2301.05375v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfAA5y/content/2301.05375v1.pdf'}
+page_content='1] computes that the  first term vanishes, so the map A is injective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfAA5y/content/2301.05375v1.pdf'}
+page_content=' The group Hom  �  H1(Sg), H1(Sg)  �Mod(Sg) is  isomorphic to Z and generated the Poincar´e duality isomorphism δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfAA5y/content/2301.05375v1.pdf'}
+page_content=' Morita [Mor85, proof  of Prop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfAA5y/content/2301.05375v1.pdf'}
+page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfAA5y/content/2301.05375v1.pdf'}
+page_content='4] shows that the image of A is (2g − 2)Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfAA5y/content/2301.05375v1.pdf'}
+page_content=' Consequently, the differential d0,1  2  descends to an injection Z/(2g − 2)Z → H2�  Mod(Sg);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfAA5y/content/2301.05375v1.pdf'}
+page_content=' H1(Sg)  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfAA5y/content/2301.05375v1.pdf'}
+page_content='  It remains to show that d0,1  2  sends a generator to eu1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfAA5y/content/2301.05375v1.pdf'}
+page_content='  The differential d0,1  2  is the  transgression;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfAA5y/content/2301.05375v1.pdf'}
+page_content=' see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfAA5y/content/2301.05375v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfAA5y/content/2301.05375v1.pdf'}
+page_content=' [NSW08, Prop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfAA5y/content/2301.05375v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfAA5y/content/2301.05375v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfAA5y/content/2301.05375v1.pdf'}
+page_content='6, Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfAA5y/content/2301.05375v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfAA5y/content/2301.05375v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfAA5y/content/2301.05375v1.pdf'}
+page_content='3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfAA5y/content/2301.05375v1.pdf'}
+page_content=' By standard knowledge of the  transgression applied to our situation, we find that d0,1  2  sends a generator to δ∗(eu), where  eu is the Euler class of the extension (2), and  δ∗ : H2�  Mod(Sg);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfAA5y/content/2301.05375v1.pdf'}
+page_content=' H1(Sg)  �  → H2�  Mod(Sg);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfAA5y/content/2301.05375v1.pdf'}
+page_content=' H1(Sg)  �  is the isomorphism induced by the Poincar´e duality isomorphism δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfAA5y/content/2301.05375v1.pdf'}
+page_content=' (For this property  of the transgression, see [NSW08, §I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfAA5y/content/2301.05375v1.pdf'}
+page_content='6, Exercise 1-2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfAA5y/content/2301.05375v1.pdf'}
+page_content='  While that reference is mainly  concerned with finite or profinite groups, the analysis of the transgression contained given  there applies more generally.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfAA5y/content/2301.05375v1.pdf'}
+page_content=') Finally, we observe that δ∗(eu) = eu1 by Theorem A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfAA5y/content/2301.05375v1.pdf'}
+page_content='  □  MAPPING CLASS GROUPS OF CIRCLE BUNDLES OVER A SURFACE  11  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfAA5y/content/2301.05375v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfAA5y/content/2301.05375v1.pdf'}
+page_content=' Computation of H2�  Mod(Sg);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfAA5y/content/2301.05375v1.pdf'}
+page_content=' H1(Sg)  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfAA5y/content/2301.05375v1.pdf'}
+page_content=' Running the LHS spectral sequence with  the trivial module A = Z, we prove that if g ≥ 8, then  (16)  H2�  Mod(Sg);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfAA5y/content/2301.05375v1.pdf'}
+page_content=' H1(Sg)  � ∼= Z/(2g − 2)Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfAA5y/content/2301.05375v1.pdf'}
+page_content='  Combining this with Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfAA5y/content/2301.05375v1.pdf'}
+page_content='1 proves Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfAA5y/content/2301.05375v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfAA5y/content/2301.05375v1.pdf'}
+page_content=' The relevant portion of the  spectral sequence appears below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfAA5y/content/2301.05375v1.pdf'}
+page_content='  2 H0(Mod(Sg);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfAA5y/content/2301.05375v1.pdf'}
+page_content=' H2(Sg))  1  0  0 H2(Mod(Sg);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfAA5y/content/2301.05375v1.pdf'}
+page_content=' H1(Sg))  0  Z  0  H2(Mod(Sg))  H3(Mod(Sg)) H4(Mod(Sg))  0  1  2  3  4  d0,2  2  d2,1  2  The computations in the first column are easy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfAA5y/content/2301.05375v1.pdf'}
+page_content=' In the second column, Morita [Mor85,  Prop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfAA5y/content/2301.05375v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfAA5y/content/2301.05375v1.pdf'}
+page_content='1] computed H1�  Mod(Sg);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfAA5y/content/2301.05375v1.pdf'}
+page_content=' H1(Sg)  �  = 0 for g ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfAA5y/content/2301.05375v1.pdf'}
+page_content='  The other computation  H1�  Mod(Sg)  �  = 0 holds for g ≥ 1 because the abelianization of Mod(Sg) is finite [FM12,  §5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfAA5y/content/2301.05375v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfAA5y/content/2301.05375v1.pdf'}
+page_content='2-3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfAA5y/content/2301.05375v1.pdf'}
+page_content='  According to [BT01, Cor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfAA5y/content/2301.05375v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfAA5y/content/2301.05375v1.pdf'}
+page_content='2],  H∗  �  Mod(Sg,1)  � ∼= H∗  �  Mod(Sg)  �  ⊗ Z[x]  in degrees g ≥ 2∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfAA5y/content/2301.05375v1.pdf'}
+page_content=' Here x has degree 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfAA5y/content/2301.05375v1.pdf'}
+page_content=' Applying this and using the universal coefficients  theorem, we conclude that  Hi�  Mod(Sg)  �  → Hi�  Mod(Sg,1)  �  is an isomorphism if i = 3 and g ≥ 6, and it is injective if i = 4 and if g ≥ 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfAA5y/content/2301.05375v1.pdf'}
+page_content='  Since the map H4�  Mod(Sg)  �  → H4�  Mod(Sg,1)  �  is injective, the differential d2,1  2  is zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfAA5y/content/2301.05375v1.pdf'}
+page_content='  Since the map H3�  Mod(Sg)  �  → H3�  Mod(Sg,1)  �  is an isomorphism, the differential d0,2  2  is  surjective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfAA5y/content/2301.05375v1.pdf'}
+page_content='  Thus, the filtration of H2�  Mod(Sg,1)  �  coming from the E∞ page gives an exact sequence  0 → H2�  Mod(Sg)  �  → H2�  Mod(Sg,1)  �  F−→  H0�  Mod(Sg);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfAA5y/content/2301.05375v1.pdf'}
+page_content=' H2(Sg)  � ∼= Z  d0,2  2  −−→  H2�  Mod(Sg);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfAA5y/content/2301.05375v1.pdf'}
+page_content=' H1(Sg)  �  → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfAA5y/content/2301.05375v1.pdf'}
+page_content='  For g ≥ 4,  H2�  Mod(Sg)  � ∼= Z[e1]  and  H2�  Mod(Sg,1)  � ∼= Z[e, e1]  and the map Z[e1] → Z[e, e1] is the obvious one e1 �→ e1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfAA5y/content/2301.05375v1.pdf'}
+page_content=' We claim that F(e) = 2 − 2g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfAA5y/content/2301.05375v1.pdf'}
+page_content='  From this we deduce the desired isomorphism (16).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfAA5y/content/2301.05375v1.pdf'}
+page_content=' The claim follows from the fact that  the extension that defines e, when restricted to the point-pushing subgroup π1(Sg) <  Mod(Sg,1), gives the extension  1 → Z → π1(USg) → π1(Sg) → 1  where USg is the unit tangent bundle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfAA5y/content/2301.05375v1.pdf'}
+page_content=' See [FM12, §5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfAA5y/content/2301.05375v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfAA5y/content/2301.05375v1.pdf'}
+page_content='5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfAA5y/content/2301.05375v1.pdf'}
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+page_content='  Lei Chen, Department of Mathematics, University of Maryland, 4176 Campus Drive, Col-  lege Park, MD 20742, USA, chenlei@umd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfAA5y/content/2301.05375v1.pdf'}
+page_content='edu  Bena Tshishiku, Department of Mathematics, Brown University, 151 Thayer St.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE5T4oBgHgl3EQfAA5y/content/2301.05375v1.pdf'}
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+arXiv:2301.02941v1  [math.AG]  7 Jan 2023
+DUAL EXCEPTIONAL COLLECTIONS ON LAGRANGIAN GRASSMANNIANS
+ANTON FONAREV
+Abstract. We construct graded left dual exceptional collections to the exceptional collections generating
+the blocks of Kuznetsov and Polishchuk on Lagrangian Grassmannians. As an application, we find explicit
+resolutions for some natural irreducible equivariant vector bundles.
+1. Introduction
+Derived categories of varieties are among the central objects in modern algebraic geometry. A skeptical
+reader might complain that the very notion of the derived category is too abstract, and they might have
+a point. However, since the pioneering work of Beilinson [1], derived categories have become a great
+computational tool. Nowadays one would say that Beilinson constructed a full exceptional collection in
+the bounded derived categories of coherent sheaves on projective spaces.
+The following analogy is commonly used to explain the computational power of exceptional collections.
+Given a finite dimensional real vector space V with a positive definite symmetric bilinear form ⟨−, −⟩, one
+can find an orthonormal basis. That is, a basis consisting of vectors (v1, v2, . . . , vn) such that (i) ⟨vi, vi⟩ = 1
+for all i = 1, . . . , n and (ii) ⟨vi, vj⟩ = 0 whenever i ̸= j. The first condition says that the vectors are
+unit vectors, while the second condition is the orthogonality condition. Given such a basis, every vector
+v ∈ V can be easily decomposed with respect to our basis:
+(1)
+v = ⟨v, v1⟩v1 + ⟨v, v2⟩v2 + · · · + ⟨v, vn⟩vn.
+Let us now relax some of the conditions; for instance, symmetry. We still want a basis which consists of
+unit vectors, but we have to modify the orthogonality condition (ii). Let us say that a basis (v1, v2, . . . , vn)
+is semiorthonormal if (i) ⟨vi, vi⟩ = 1 for all i = 1, . . . , n and (ii) ⟨vi, vj⟩ = 0 whenever i > j.
+One
+obviously needs and adjustment to the formula (1). Indeed, in (1) we explicitly used the fact that under
+the isomorphism V ∗ ∼
+−→ V given by the form ⟨−, −⟩ the dual basis (v1, . . . , vn) maps back to (v1, . . . , vn).
+Since we actually have two isomorphisms this time, V → V ∗, v �→ ⟨v, −⟩ and V → V ∗, v �→ ⟨−, v⟩, let us
+consider the second one and assume that its inverse maps vi to ui ∈ V . That is, we have
+⟨vi, uj⟩ = δij
+for all 1 ≤ i, j ≤ n.
+The vectors u1, . . . , un obviously form a basis, and for any v ∈ V we have the desired formula
+(2)
+v = ⟨v, u1⟩v1 + ⟨v, u2⟩v2 + · · · + ⟨v, un⟩vn.
+What is less immediate, the vectors (un, un−1, . . . , u1) (remark the reverse order) form an orthonormal
+basis called the left dual to (v1, v2, . . . , vn).
+One rather indirect way to see that the latter holds is via mutations. Consider a semiorthonormal pair
+(u, v): that is, ⟨u, u⟩ = ⟨v, v⟩ = 1 and ⟨v, u⟩ = 0. Let us define a new vector Luv = v − ⟨u, v⟩u, which is
+called the left mutation of v through u. A simple calculation shows that (Luv, u) is a semiorthonormal
+pair. As the name suggests, there is a sibling to the left mutation procedure: if one puts Rvu = u−⟨u, v⟩v,
+which is called the right mutation of u through v, then the pair (v, Rvu) is semiorthonormal. It is a very
+nice exercise in linear algebra to check that left and right mutations of adjacent elements define an action
+This work was supported by the Russian Science Foundation under grant no. 19-11-00164, https://rscf.ru/en/project/19-
+11-00164/.
+1
+
+of the braid group on n strands on the set of all semiorthonormal bases. Moreover, the left dual basis to
+a semiorthonormal basis (v1, v2, . . . , vn) is given by
+(3)
+Lv1Lv2 · · · Lvn−1vn, Lv1Lv2 · · · Lvn−2vn−1, . . . , Lv2v1, v1.
+We refer the reader to [2] for further insights and some interesting properties of this action.
+The linear-algebraic picture translates to triangulated categories in the following way. Instead of a
+vector space we consider a k-linear triangulated category T , while we treat Ext•
+T (−, −) as a kind of
+bilinear form. Then one says that an object E ∈ T is exceptional if Hom(E, E) = k and Exti(E, E) = 0
+for all i ̸= 0. A collection of objects (E1, E2, . . . , En) is called exceptional if every Ei is an exceptional
+object and Ext•(Ei, Ej) = 0 for all i > j. Finally, a collection is called full if it generates the category in
+a sense that no proper strictly full triangulated subcategory of T contains all Ei. If (E1, E2, . . . , En) is
+a full exceptional collection in T , we will write T = ⟨E1, E2, . . . , En⟩.
+The analogy with semiorthonormal bases should be clear by now. Assume that T has a full exceptional
+collection (E1, E2, . . . , En) (which is rarely the case). What is most interesting is that not only every
+object in T can be obtained from the finite set of Ei’s by iteratively taking shifts and cones, but this
+procedure can be made rather explicit.
+Recall that the decomposition of every vector in terms of a
+given semiorthonormal basis could be done with the help of a left dual semiorthonormal basis by (2).
+Let us mimic its definition in the categorical case.
+Namely, let us say that a collection of objects
+(∨En, ∨En−1, . . . , ∨E1) is left dual to (E1, E2, . . . , En) if Ext•(Ei, ∨Ej) = 0 for i ̸= j and Ext•(Ei, ∨Ei) = k
+for all i = 1, . . . , n. It should not surprise the reader at this point that (∨En, ∨En−1, . . . , ∨E1) is again an
+exceptional collection, and it is full whenever the original one is. Moreover, there is an action of the braid
+group on n strands on the set of all exceptional collections of length n in T , and a formula similar to (3)
+determines the left dual collection. Finally, instead of the decomposition (2) one has a spectral sequence
+which computes cohomological functors applied to objects in T , which we will talk about in Section 2.1.
+The bridge between exceptional collections and semiorthonormal bases is actually rather simple. Given
+a sufficiently nice T (say, we assume that Ext•(E, F) is finite-dimensional for all E, F ∈ T ) with a full
+exceptional collection (E1, E2, . . . , En), one checks that the classes of Ei form a basis in the Grothendieck
+group K0(T ). In particular, the length of any full exceptional collection equals the rank of the latter,
+which should a posteriori be a free finitely generated abelian group. If one takes �
+i(−1)i dimk Exti(−, −)
+as the bilinear form, exceptional collections become “categorifications” of the corresponding semiorthonor-
+mal bases.
+The first example of a full exceptional collection was given in [1], in which Beilinson showed that
+Db(Pn) = ⟨O, O(1), . . . , O(n)⟩, where Db stands for the bounded derived category of coherent sheaves.
+Interestingly enough, he also showed that the left dual is given by the collection ⟨Ωn(n), Ωn−1(n −
+1), . . . , Ω1(1), O⟩.
+A long-standing conjecture states that the bounded derived category of coherent
+sheaves on a rational homogeneous variety admits a full exceptional collection. Though a lot of work
+has been done over the years, the conjecture has been established in a very limited number of cases.
+Say, for classical groups of type ABCD and Picard rank 1 the problem was fully resolved only for
+Grassmannians [6], quadrics [7], symplectic and orthogonal Grassmannians of planes [8, 10], Lagrangian
+Grassmannians [4], and in some sporadic cases. We refer the reader to [9] for further details.
+Since many explicit constructions realize varieties as subvarieties in Grassmannians, exceptional collec-
+tions on the latter become an important computational tool. One of our favorite recent examples can be
+found in [11], where the authors use exceptional collections to study moduli of Ulrich bundles. In order
+to use the tool’s maximum power, it is important to know the dual collection, and finding one is a task of
+its own. In the present paper we find exceptional collections of Lagrangian Grassmannians dual to those
+constructed in [4]. We further use them to provide explicit resolutions of some very natural irreducible
+vector bundles.
+2
+
+From now on let we will be interested in the bounded derived category of coherent sheaves on LGr(n, V ),
+the Lagrangian Grassmannian of isotropic subspaces of dimension n in a fixed 2n-dimensional vector
+space V over an algebraically closed field k equipped with a non-degenerate symplectic form. Excep-
+tional collections of maximal length (equal to the rank of the Grothendieck group) were constructed on
+all symplectic and orthogonal Grassmannians by Kuznetsov and Polishchuk in [9]. All these collections
+are conjecturally full; however, the latter was checked only for Lagrangian Grassmannians in [4]. The
+construction of Kuznetsov and Polishchuk is rather indirect: they start with certain collections of ex-
+ceptional irreducible vector bundles, called blocks, which naturally form an exceptional collection in the
+equivariant derived category, then pass to dual collections within each block (again, in the equivariant
+derived category). Each block must satisfy certain homological conditions which guarantee that the dual
+collections become exceptional in the non-equivariant derived category. Most of the hard work in [9]
+is related to checking that certain collections of irreducible equivariant vector bundles satisfy the block
+condition, which is a rather difficult problem in representation theory.
+While the block condition will be discussed in Section 2.2, let us explain why isotropic Grassmannians
+are much harder than the classical ones. Full exceptional collections in the bounded derived categories of
+Grassmannians were constructed by Kapranov [7], who naturally extended Beilinson’s method. Consider
+the Grassmannian Gr(k, V ) of k-dimensional subspaces in a fixed N-dimensional vector space V over a
+field k of characteristic zero. Denote by U the tautological rank k subbundle of the trivial bundle V ⊗ O.
+Kapranov showed that
+(4)
+Db(Gr(k, V )) =
+�
+ΣλU∗ | λ ∈ Yk,N−k
+�
+,
+where λ runs over the set of Young diagrams Yk,N−k, Σλ is the corresponding Schur functor, and the
+order in this collection can be taken to be any linear order refining the partial inclusion order on the
+diagrams. Moreover, he simultaneously constructed the (graded) left dual to this collection. The latter
+is given by
+(5)
+Db(Gr(k, V )) =
+�
+ΣλT U⊥ | λ ∈ Yk,N−k
+�
+,
+where U⊥ = (V/U)∗ and λT denotes the transposed diagram.1 The duality relation might seem a little
+different from the one we described earlier:
+(6)
+Ext•(ΣλU∗, ΣµT U⊥) =
+�
+k[−|λ|],
+if λ = µ,
+0,
+otherwise,
+but this grading difference does not change much since the two definitions are equivalent up to shifts
+in the derived category. Actually, the grading choice in (6) is favorable since the dual collection then
+consists of vector bundles.
+Since LGr(n, V ) is naturally embedded as a closed subvariety in Gr(n, V ), one can ask whether any
+of elements of Kapranov’s collection restrict to exceptional vector bundles on LGr(n, V ). This is where
+surprising things happen. It turns out that the only Young diagrams for which ΣλU∗ are exceptional are
+λ ∈ Yn,1. These vector bundles are nothing but the exterior powers ΛiU∗, and there are only n of those,
+while rk K0(LGr(n, V )) = 2n. Remark also that U⊥ restricts to U on LGr(n, V ).
+The construction of Kuznetsov and Polishchuk produces for any λ ∈ Yh,n−h, 0 ≤ h ≤ n, an equivariant
+non-irreducible (in general) vector bundle Eλ on LGr(n, V ) with the following properties.
+First, Eλ
+belongs to the subcategory of the derived category generated by ΣµU∗ for µ ⊆ λ. Second, the G = Sp2n-
+equivariant groups Ext•
+G(Eλ, ΣµU∗) = 0 for all µ ⊊ λ, while Ext•
+G(Eλ, ΣλU∗) = k. A given diagram may
+1A careful reader might wonder why we choose this extra transposition and not index the collection directly by YN−k,k,
+as Kapranov does. After all, {λT | λ ∈ Yh,w} = Yw,h. Our choice becomes clear in Section 2.1, where we introduce our
+grading convention for dual exceptional collections.
+3
+
+belong to various sets Yh,n−h, but the resulting bundle does not depend on the choice of h since the
+previous conditions fully characterize it. The first nontrivial example of such a bundle is the universal
+extension
+0 → O → E2 → S2U∗ → 0.
+Kuznetsov and Polishchuk showed that for any 0 ≤ h ≤ n the bundles
+(7)
+�
+Eλ | λ ∈ Yh,n−h
+�
+form an exceptional collection in Db(LGr(n, V )). As in the case of Grassmannians, one can linearly order
+them in any way compatible with the partial inclusion order on the corresponding Young diagrams. Let
+us denote by Fλ the bundle dual (in the usual sense) to Eλ.
+Theorem A (Theorem 3.1). The bundles
+�
+FλT | λ ∈ Yh,n−h
+�
+form a graded left dual exceptional col-
+lection to (7).
+We formulated Theorem A in terms of transposed diagrams in order to show the parallel between
+our Lagrangian situation and the case of classical Grassmannians: compare the latter theorem with (4)
+and (5) keeping in mind that U⊥ is isomorphic to U = (U∗)∗ on LGr(n, V ).
+It is easy to see that the bundle ΣλU∗ belongs to the subcategory (7) whenever λ ∈ Yh,n−h. A nice
+geometric description (as a matter of fact, two of them) was given for the bundles Fµ in [4]. Using this
+description, one can compute the corresponding spectral sequences and get the following result as an
+application of Theorem A (see Section 3.2 for the details).
+Theorem B (Theorem 3.12). Let λ ∈ Yh,n−h for some 0 ≤ h ≤ n. Then there is an exact sequence of
+vector bundles on LGr(n, V ) of the form
+0 →
+�
+µ∈Bh(h+1)
+Eλ/µ → · · · →
+�
+µt∈B2t
+Eλ/µt → · · · →
+�
+µ2∈B4
+Eλ/µ2 →
+�
+µ1∈B2
+Eλ/µ1 → Eλ → ΣλU∗ → 0,
+where B2t denotes the set of balanced diagrams with 2t boxes.
+The paper is organized as follows. In Section 2 we collect all the preliminaries. It contains no new
+material except, maybe, our convention for dual exceptional collections for exceptional collections indexed
+by a graded partially ordered set (see Lemma 2.5). Section 3 contains our main results. That is, we
+prove Theorems A and B, which are Theorems 3.1 and 3.12 respectively.
+2. Preliminaries
+Throughout the paper we work over a field k of characteristic zero.
+2.1. Dual exceptional collections. In the present section we collect some preliminaries related to
+(dual) exceptional collections. The material is well known to specialists; however, since we naturally work
+with exceptional collections indexed by graded partially ordered sets, we introduce a certain convention
+in the definitions. This convention seems rather natural, as we show in various examples.
+2.1.1. Partially ordered sets. Recall that a partially ordered set (poset) P is a set equipped with a binary
+relation ⪯, called a partial order, satisfying the following three properties:
+Reflexivity: x ⪯ x for all x ∈ P.
+Antisymmetry: if x ⪯ y and y ⪯ x, then x = y for all x, y ∈ P.
+Transitivity: is x ⪯ y and y ⪯ z, then x ⪯ z for all x, y, z ∈ P.
+4
+
+Elements x and y are called comparable if either x ⪯ y or y ⪯ x.
+If every two elements in P are
+comparable, one calls P linearly ordered. If x ⪯ y and x ̸= y, one usually writes x ≺ y. If P is partially
+ordered, its dual P◦ is the set underlying P equipped with the converse relation: x ⪯ y in P◦ if and only
+if y ⪯ x in P.
+Let x and y be elements of a poset P. One says that y covers x, written x ⋖ y, if x ≺ y and there
+is no element z such that x ≺ z ≺ y. A grading function on P is a map ρ : P → Z with the following
+properties:
+• if x ≺ y then ρ(x) < ρ(y),
+• if x ⋖ y then ρ(y) = ρ(x) + 1.
+A poset equipped with a grading function is called a graded poset.
+Of course, not all posets can be
+turned into graded ones. We will be mainly interested in finite posets. If P is finite and admits a grading
+function, there is a rather natural choice for such a function: there exists a unique grading function | − |
+with the property that |x| = 0 whenever x is a minimal element (that is, there is no y such that y ≺ x).
+In the following by a graded poset we mean a finite poset equipped with this grading function.
+Example 2.1. Let Yh,w denote the set of Young diagrams of height at most h and width at most w. This set
+can be identified with the set of integer sequences (λ1, λ2, . . . , λh) such that w ≥ λ1 ≥ λ2 ≥ · · · ≥ λh ≥ 0.
+There is a natural partial order on Yh,w given by inclusion of diagrams: λ ⪯ µ if λi ≤ µi for all i = 1, . . . , h.
+With this partial order the poset Yh,w is graded, and |λ| = λ1 + λ2 + · · · + λh equals the number of boxes
+in the diagram λ.
+2.1.2. Exceptional collections. Let T be a k-linear triangulated category.
+An object E ∈ T is called
+exceptional if Hom(E, E) = k and Extt(E, E) = Hom(E, E[t]) = 0 for all t ̸= 0. Let P be a poset. An ex-
+ceptional collection indexed by P is a collection of exceptional objects {Ex}x∈P such that Ext•(Ex, Ey) = 0
+unless x ⪯ y. We denote by ⟨Ex|x ∈ P⟩ the smallest strictly full triangulated subcategory in T con-
+taining all Ex. If P is finite, one can always refine the order so that it becomes isomorphic to the poset
+{1, 2, . . . , l}. Under this isomorphism we get the usual definition of an exceptional collection: that is,
+a collection of exceptional objects (E1, E2, . . . , El) such that Ext•(Ej, Ei) = 0 for all l ≥ j > i ≥ 1.
+Example 2.2. Let V be an n-dimensional vector space over k. Consider the Grassmannian Gr(k, V ), and
+let U denote the tautological rank k subbundle in the trivial bundle V . Kapranov showed in [6] that the
+bounded derived category of coherent sheaves Db(Gr(k, V )) admits a full exceptional collection indexed
+by Yk,n−k:
+Db(Gr(k, V )) =
+�
+ΣλU∗ | λ ∈ Yk,n−k
+�
+,
+where Σλ denotes the Schur functor associated with λ.2 The fact that this collection is indexed by a poset
+gives more information about orthogonality of different objects: if λ and µ are incomparable (that is,
+neither is contained in the other), then both Ext•(ΣλU∗, ΣµU∗) = 0 and Ext•(ΣµU∗, ΣλU∗) = 0.
+2.1.3. Mutations and duality. Let (E, F) be an exceptional pair in T .
+The left mutation LEF of F
+through E is defined by the distinguished triangle
+LEF → Ext•(E, F) ⊗ E ev
+−→ F → LEF[1],
+where ev is the evaluation morphism. Similarly, define the right mutation RFE via the distinguished
+triangle
+RF E[−1] → E coev
+−−−→ Ext•(E, F)∗ ⊗ F → RFE,
+where coev is the coevaluation morphism.
+One easily checks that both (LEF, E) and (F, RF E) are
+exceptional pairs. Moreover, ⟨E, F⟩ = ⟨LEF, E⟩ = ⟨F, RF E⟩.
+2Our convention for Schur functors is such that Σ(p) is isomorphic to the p-th symmetric power Sp, so Σ(1,1) ≃ Λ2.
+5
+
+Example 2.3. Consider the Grassmannian Gr(k, V ). Then the structure sheaf and the dual tautological
+bundle form an exceptional pair (O, U∗). One quickly checks that LOU∗ ≃ U⊥, where U⊥ = (V/U)∗.
+Given an exceptional collection (E1, E2, . . . , El), mutations of adjacent elements define an action of
+the braid group Brl on l strands on the set of exceptional collections in ⟨E1, E2, . . . , El⟩, see [3]. For a
+fixed collection there are two important elements in the orbit. Namely, the dual collections. The left dual
+collection to (E1, E2, . . . , El) is defined as
+(LE1LE2 · · · LEl−1El, LE1LE2 · · · LEl−2El−1, . . . , LE1E2, E1).
+We denote it by (E∨
+l , E∨
+l−1, . . . , E∨
+1 ). The left dual exceptional collection can be fully characterized by
+the following three properties:
+(1) E∨
+i ∈ ⟨E1, E2, . . . , El⟩ for all 1 ≤ i ≤ l,
+(2) Ext•(Ei, E∨
+j ) = 0 for all i ̸= j,
+(3) Ext•(Ei, E∨
+i ) = k[−i + 1] for all 1 ≤ i ≤ l.
+Example 2.4. The bounded derived category of the projective space P(V ) has a full exceptional collection
+consisting of the line bundles ⟨O, O(1), . . . , O(n)⟩, where n is the dimension of V . Its left dual is given
+by ⟨Ωn(n), Ωn−1(n − 1), . . . , Ω1(1), O⟩, where Ωi = ΛiΩ1
+P(V ).
+Similarly, the right dual collection is defined as
+(En, REnEn−1, REnREn−1En−2, . . . , REnREn−1 · · · RE2E1),
+and will be denoted by (∨El, ∨El−1, . . . , ∨E1). It can be fully characterized by the following conditions:
+(1) ∨Ei ∈ ⟨E1, E2, . . . , El⟩ for all 1 ≤ i ≤ l,
+(2) Ext•(∨Ei, Ej) = 0 for all i ̸= j,
+(3) Ext•(∨Ei, Ei) = k[−n + i] for all 1 ≤ i ≤ l.
+Remark that in a given exceptional collection (E1, E2, . . . , El) one can replace any object Ei with its
+shift Ei[t] for any integer t. For instance, one can introduce an extra shift in the definitions of the right
+and left mutations. The first two defining conditions for the left and right dual collections will not change,
+while the third condition will become slightly nicer:
+Ext•(∨Ei, Ei) = Ext•(Ei, E∨
+i ) = k
+for any 1 ≤ i ≤ l.
+This convention is often reasonable, yet even in the case of the projective space, Example 2.4, the dual
+collection will not consist of vector bundles while the original collection does.
+Meanwhile, the definitions we have just given also have a downside. Imagine that (E, F) is a fully
+orthogonal pair in T . Then LEF ≃ F[−1], while RFE ≃ E[1]. This is often inconvenient as well. We
+propose the following lemma-definition, which is tailored to the case of an exceptional collection indexed
+by a finite graded poset P. The reader will immediately check that once the collection is linearly ordered,
+the left dual differs from the graded left dual by shifts of objects.
+Lemma 2.5. Let ⟨Ex | x ∈ P⟩ be an exceptional collection indexed by a finite graded poset P. For any
+y ∈ P there exists a unique (up to isomorphism) object E◦
+y ∈ ⟨Ex | x ∈ P⟩ such that
+(1) Ext•(Ex, E◦
+y) = 0 for all x ̸= y,
+(2) Ext•(Ey, E◦
+y) = k[−|y|].
+Moreover, the objects E◦
+y form an exceptional collection with respect to the opposite poset P◦, called the
+graded left dual, and ⟨E◦
+y | x ∈ P◦⟩ = ⟨Ex | x ∈ P⟩.
+Remark 2.6. If the poset P is linearly ordered, then the definition of the graded left dual agrees with the
+definition of the left dual.
+6
+
+Example 2.7. In Example 2.2 we have seen that Db(Gr(k, V )) admits a full exceptional collection indexed
+by the poset Yk,n−k:
+Db(Gr(k, V )) =
+�
+ΣλU∗ | λ ∈ Yk,n−k
+�
+,
+where U is the tautological rank k subbundle in V . In the same paper Kapranov showed that its graded
+left dual is given by
+Db(Gr(k, V )) =
+�
+ΣλT U⊥ | λ ∈ Yk,n−k
+�
+,
+where U⊥ = (V/U)∗, and λT denotes the transpose diagram.
+We leave the definition of the graded right dual to the reader, indicating that for the right dual the
+grading function should be taken for the opposite poset.
+2.1.4. Graded spectral sequence. As stated in the introduction, dual collections provide a particularly nice
+computational tool. Let (E1, E2, . . . , El) be an exceptional collection in T . Recall that a cohomological
+functor from T to an abelian category A is a functor F : T → A which takes distinguished triangles to
+exact sequences. As usual, we denote by F i the composition F ◦ [i].
+Proposition 2.8 ([5, Section 2.7.3]). Let G ∈ ⟨E1, E2, . . . , En⟩. There is a spectral sequence with the
+first page given by
+(8)
+Ep,q
+1
+=
+�
+i+j=q
+Ext−i(G, E∨
+p+1)∗ ⊗ F j(Ep+1)
+converging to F p+q(G).
+We will be interested in the case when T = Db(A) is the bounded derived category of an abelian
+category A (for instance, the bounded derived category of coherent sheaves on a smooth projective
+variety), and F = H0 is the usual 0-th cohomology functor. Assume that the exceptional collection
+(E1, E2, . . . , El) consists of pure objects (for instance, of coherent sheaves). Then the spectral sequence (8)
+simplifies to
+(9)
+Ep,q
+1
+= Ext−q(G, E∨
+p+1)∗ ⊗ Ep+1 ⇒ Hp+q(G).
+In the case of an exceptional collection indexed by a graded poset P the spectral sequence (9) becomes
+(10)
+Ep,q
+1
+=
+�
+x∈P, |x|=p
+Ext−q(G, E◦
+x)∗ ⊗ Ex ⇒ Hp+q(G).
+2.2. Exceptional collections on Lagrangian Grassmannians. Let V be a 2n-dimensional vector
+space over k equipped with a non-degenerate skew-symmetric bilinear form. We denote by LGr(n, V ) the
+Lagrangian Grassmannian of maximal isotropic subspaces in V , and by U the tautological rank n bundle
+on LGr(n, V ). The Lagrangian Grassmannian comes with an action of the symplectic group G = Sp2n.
+We denote by P the set of weakly decreasing integer sequences of length n:
+P = {λ ∈ Zn | λ1 ≥ λ2 ≥ · · · ≥ λn}.
+Given λ ∈ P, we denote by Σλ the corresponding Schur functor. We follow the convention under which
+Σ(p,0,...,0) = Sp.
+7
+
+2.2.1. Exceptional blocks. It is well known that every equivariant irreducible vector bundle on LGr(n, V )
+is isomorphic to ΣλU∗ for some λ ∈ P. Moreover, these form an infinite full exceptional collection in the
+equivariant derived category:
+Db
+G(LGr(n, V )) =
+�
+ΣλU∗ | λ ∈ P◦�
+,
+where P is treated as an infinite poset with the partial order given by
+λ ⪯ µ
+if and only if
+λi ≤ µi for all i = 1, . . . , n.
+Any subset S ⊆ P with the induced partial order produces an exceptional collection ⟨ΣλU∗ | λ ∈ S◦⟩.
+If S is finite and graded, we denote by
+⟨Eλ | λ ∈ S⟩
+and
+⟨Fλ | λ ∈ S⟩
+the graded right and left duals to ⟨ΣλU∗ | λ ∈ S◦⟩ respectively.
+Kuznetsov and Polishchuk came up with a very simple3 condition under which the objects Eλ form an
+exceptional collection in the non-equivariant category.
+Definition 2.9 (See [9, Definition 3.1]). A subset S ⊂ P is called an exceptional block if for all λ, µ ∈ S
+the canonical map
+�
+ν∈S
+Ext•
+G(ΣλU∗, ΣνU∗) ⊗ Hom(ΣνU∗, ΣµU∗) → Ext•(ΣλU∗, ΣµU∗)
+is an isomorphism.
+In plain words the block condition says that every extension between a pair of objects can be de-
+composed as a sum of equivariant extensions followed by homomorphisms. What is rather surprising is
+that even though the original objects ΣλU∗ for λ ∈ S almost never form an exceptional collection in the
+non-equivariant category, the right dual do form an exceptional collection in the non-equivariant category
+as long as S is a block.
+Proposition 2.10 (See [9, Proposition 3.9]). If S ⊂ P is an exceptional block, then the corresponding
+right dual objects form an exceptional collection in Db(LGr(n, V )),
+�
+Eλ | λ ∈ S
+�
+⊂ Db(LGr(n, V )).
+It turns out that it is natural to call exceptional blocks right exceptional blocks, and that it is useful
+to consider left exceptional blocks as well.
+Definition 2.11 (See [4, Remark 2.12]). A subset S ⊂ P is called an left exceptional block if for all
+λ, µ ∈ S the canonical map
+Hom(ΣλU∗, ΣνU∗) ⊗
+�
+ν∈S
+Ext•
+G(ΣνU∗, ΣµU∗) → Ext•(ΣλU∗, ΣµU∗)
+is an isomorphism.
+Recall that both the equivariant and the non-equivariant categories have the duality functor, which
+is an anti-autoequivalence. Since passing to duals takes exceptional collections to exceptional collections
+(with respect to the opposite order) and left and right (graded) dual collections to right and left dual
+collections respectively, we immediately see that S is a right exceptional block if and only if −S is a left
+exceptional block, where −S = {−λ | λ ∈ S} and −λ = (−λn, −λn−1, . . . , −λ1). The latter follows from
+the isomorphism (ΣλU∗)∗ ≃ Σ−λU∗. The dual statement to Proposition 2.10 is the following.
+3Simple, yet hard to check.
+8
+
+Proposition 2.12 (See [9, Proposition 3.9]). If S ⊂ P is a left exceptional block, then the corresponding
+left dual objects form an exceptional collection in Db(LGr(n, V )),
+�
+Fλ | λ ∈ S
+�
+⊂ Db(LGr(n, V )).
+The following theorem uses the term semiorthogonal decomposition.
+Recall that full triangulated
+subcategories T1, T2 ⊆ T are called semiorthogonal, one writes T1 ⊆ T ⊥
+2 , if HomT (X2, X1) = 0 for all
+X1 ∈ T1 and X2 ∈ T2. A semiorthogonal decomposition of a category T is a collection of full triangulated
+subcategories T1, T2, . . . , Tr such that Ti ⊆ T ⊥
+j
+for all 1 ≤ i < j ≤ r and T is the smallest strictly
+full triangulated subcategory containing all Ti.
+The notation for a semiorthogonal decomposition is
+T = ⟨T1, T2, . . . , Tr⟩. Finally, for a full triangulated subcategory B ∈ Db(LGr(n, V )) we denote by B(i)
+its image under the autoequivalence given by tensor product with O(i).
+Theorem 2.13 (See [9, Theorem 9.2] and [4, Theorem 4]). The set
+Bh = {λ ∈ P | n − h ≥ λ1 ≥ λ2 ≥ · · · ≥ λh ≥ λh+1 = · · · = λn = 0}
+is a right exceptional block for all h = 0, . . . , n. Moreover, there is semiorthogonal decomposition
+(11)
+Db(LGr(n, V )) = ⟨B0, B1(1), B2(2), . . . , Bn(n)⟩ ,
+where Bh = ⟨Eλ | λ ∈ Bh⟩.
+Let us make a few remarks. First, Bh = Yh,n−h, where Yh,w denotes the set of Young diagrams of
+height at most h and width at most w. Second, the object Eλ for λ ∈ Yh,n−h has the following homological
+description:
+Eλ ∈ ⟨ΣµU∗ | µ ⊆ λ⟩ ⊂ Db
+G(LGr(n, V ))
+and
+Ext•
+G(Eλ, ΣµU∗) =
+�
+k[−|λ|]
+if µ = λ,
+0
+if µ ⊊ λ.
+In particular, it depends only on λ, one only needs to know that λ ∈ Yh,n−h for some 0 ≤ h ≤ n. Finally,
+by using the duality anti-autoequivalence we get a semiorthogonal decomposition
+Db(LGr(n, V )) = ⟨Cn(−n), Cn−1(−n + 1), . . . , C1(−1), C0⟩ ,
+where Ch = ⟨Fλ | λ ∈ −Bh⟩ and Fλ for λ ∈ Yh,n−h can be characterized by the following properties:
+Fλ ∈ ⟨ΣµU | µ ⊆ λ⟩ ⊂ Db
+G(LGr(n, V ))
+and
+Ext•
+G(ΣµU, Fλ) =
+�
+k[−|λ|]
+if µ = λ,
+0
+if µ ⊊ λ.
+2.2.2. Geometric constructions. We will need one of the two geometric constructions for the objects Fλ
+obtained in [4]. Since the cases h = 0 and h = n are trivial (B0 and Bn consist of a single weight
+λ = (0, 0, . . . , 0), and Fλ = Eλ = O), let us fix 0 < h < n. Consider the diagram
+(12)
+IFl(n − h, n; V )
+LGr(n, V )
+IGr(n − h, V )
+p
+q
+and denote by W the universal rank n−h bundle on IGr(n−h, V ) as well as its pullback on IFl(n−h, n; V ).
+Lemma 2.14 ([4, Lemma 3.4]). The bundles ⟨ΣµT W∗ | µ ∈ Yh,w−h⟩ form an exceptional collection in
+the non-equivariant derived category Db(IGr(n − h, V )).
+9
+
+The exceptional collection from the previous lemma is indexed by a graded poset, so we can pass in
+the non-equivariant category to its graded left dual, denoted by ⟨Gλ | λ ∈ Y◦
+h,w−h⟩, see [4, Definition 3.5].
+The graded dual exceptional collection conditions are
+(13)
+Gλ ∈ ⟨ΣµW∗ | µ ∈ Yn−h,h⟩
+and
+Ext•(ΣµW∗, Gλ) =
+�
+k[−|λ|]
+if µ = λT ,
+0
+if µ ∈ Yn−h,h and µ ̸= λT .
+Proposition 2.15 ([4, Proposition 3.6]). The object Fλ is isomorphic to p∗q∗Gλ.
+3. Main results
+We continue to use the notation introduced in Section 2.2.
+Our first goal is to prove the duality
+statement.
+3.1. Dual exceptional collections on Lagrangian Grassmannians. The first main result of the
+paper is the following theorem, where we identify the posets Yh,n−h and Yn−h,h via transposition of
+diagrams.
+Theorem 3.1. Let 0 ≤ h ≤ n. Then the exceptional collection ⟨Fµ | µ ∈ Y◦
+n−h,h⟩ is the graded left dual
+to ⟨Eλ | λ ∈ Yh,n−h⟩. That is, for µ ∈ Yn−h,h one has
+Fµ ∈ ⟨Eλ | λ ∈ Yh,n−h⟩
+and
+Ext•(Eλ, Fµ) =
+�
+k[−|λ|]
+if µ = λT ,
+0
+if µ ∈ Yn−h,h and µ ̸= λT .
+The proof of the preceding theorem will take the rest of this section. Our strategy is to compare the
+objects Eλ with the graded right dual exceptional collection to ⟨Fµ | µ ∈ Y◦
+n−h,h⟩. Since the cases h = 0
+and h = n are trivial (all the objects considered are isomorphic to O), we assume that 0 < h < n.
+Lemma 3.2. Let ν ∈ Yh,n−h. The following categories coincide:
+⟨Fµ | µ ⊆ νT ⟩ = ⟨Eλ | λ ⊆ ν⟩ = ⟨ΣλU∗ | λ ⊆ ν⟩,
+where the latter is the smallest strictly full triangulated subcategory in Db(LGr(n, V )) containing the cor-
+responding objects.
+Proof. Since, the objects Eλ form a graded right dual exceptional collection to ⟨ΣλU∗ | λ ⊆ ν⟩ in
+Db
+G(LGr(n, V )), the second equality holds in Db
+G(LGr(n, V )). Once we apply the forgetful functor from
+Db
+G(LGr(n, V )) to Db(LGr(n, V )), we see that the same equality holds in the non-equivariant derived
+category.4 In a similar fashion one proves that
+⟨Fµ | µ ⊆ νT ⟩ = ⟨ΣµU | µ ⊆ νT ⟩.
+It remains to show that ⟨ΣµU | µ ⊆ νT⟩ = ⟨ΣλU∗ | λ ⊆ ν⟩, which follows from the structure of Kapranov’s
+exceptional collection on Gr(n, V ) and its dual, see [4, Lemma 3.7].
+□
+Next, we need another very simple purely categorical statement.
+Let T ′ = ⟨Wx | x ∈ P⟩ be a
+triangulated category generated by a graded full exceptional collection. Denote by ⟨Gx | x ∈ P◦⟩ its
+graded left dual. Assume F : T ′ → T is an exact functor into another triangulated category T . Put
+Fx = F(Gx) for all x ∈ P◦ and assume that ⟨Fx | x ∈ P◦⟩ form a graded exceptional collection in T .
+Denote by ⟨ ˜Ex | x ∈ P⟩ its graded right dual.
+Lemma 3.3. For all x, y ∈ P one has
+Ext•
+T ( ˜Ex, F(Wy)) ≃ Ext•
+T ′(Wx, Wy).
+4See also the discussion preceding Corollary 3.8 in [9].
+10
+
+Proof. Since the category T ′ is saturated, the functor F has a left adjoint F ∗ (see [3]). In particular,
+Ext•
+T ( ˜Ex, F(Wy)) ≃ Ext•
+T ′(F ∗( ˜Ex), Wy).
+We claim that F ∗( ˜Ex) ≃ Wx. Indeed, for any y ∈ P one has
+Ext•
+T ′(F ∗( ˜Ex), Gy) ≃ Ext•
+T ( ˜Ex, Fy) =
+�
+k[−|x|]
+if x = y,
+0
+otherwise.
+The latter is precisely the defining condition of the graded right dual to ⟨Gx | x ∈ P◦⟩, which is ⟨Wx |
+x ∈ P⟩.
+□
+We apply the previous lemma in our situation. Consider the isotropic Grassmannian IGr(h, V ) with
+its tautological bundle W. Fix ν ∈ Yh,n−h and consider the poset P = {λ | λ ⊆ ν}. By [4, Lemma 3.4]
+the bundles ⟨ΣλW∗ | λ ∈ P⟩ form an exceptional collection in Db(IGr(h, V )). Since the pullback functor
+q∗ : Db(IGr(h, V )) → Db(IFl(h, n; V )) is fully faithful, the bundles ⟨ΣλW∗ | λ ∈ P⟩ form an exceptional
+collection in Db(IFl(h, n; V )), and its graded left dual coincides with the pullback of the graded left dual.
+Put T ′ = ⟨ΣλW∗ | λ ∈ P⟩ ⊆ Db(IFl(h, n; V )), Wλ = ΣλW∗, Gλ = q∗Gλ, T = Db(LGr(n, V )), F = p∗,
+where p∗ is the restriction of the pushforward functor under the projection p : IFl(h, n; V )) → LGr(n, V ).
+Denote by ⟨ ˜Eλ | λ ∈ P⟩ the graded right dual exceptional collection to ⟨Fµ | µ ⊆ νT ⟩.
+Since
+Fλ = F(Gλ), see Proposition 2.15, we can apply Lemma 3.3. The conclusion is that Ext•( ˜Eλ, p∗ΣµW∗) ≃
+Ext•(ΣλW∗, ΣµW∗). A simple Borel–Bott–Weil computation shows that p∗ΣµW∗ ≃ ΣµU∗ for µ ⊆ ν
+(see [4, Lemma A.4]). Finally, for µ ⊆ ν we get
+(14)
+Ext•( ˜Eν, ΣµU∗) =
+�
+k
+if µ = ν,
+0
+if µ ⊊ ν.
+Let us recall that our goal is to prove that ˜Eν ≃ Eν. It was shown in [9, Corollary 3.8] that for all
+λ, µ ∈ P there is an isomorphism
+Ext•(Eλ, ΣµU∗) ≃ Hom(ΣλU∗, ΣµU∗).
+It follows from the Lagrangian Borel–Bott–Weil theorem (see Section 3.2.2) that Hom(ΣλU∗, ΣλU∗) ≃ k
+and that Hom(ΣλU∗, ΣµU∗) = 0 if λ ̸⊆ µ. We conclude that
+(15)
+Ext•(Eν, ΣµU∗) =
+�
+k
+if µ = ν,
+0
+if µ ⊊ ν.
+Proof of Theorem 3.1. Let ν ∈ Yh,n−h. By Lemma 3.2 we know that ˜Eν, Eν ∈ ⟨ΣµU∗ | µ ⊆ ν⟩. Choose a
+nontrivial element φ ∈ Ext•(Eν, ΣνU∗) ≃ k. It follows from the discussion preceding Corollary 3.8 in [9]
+that one has an exact triangle in Db(LGr(n, V )) of the form
+Eν
+φ−→ ΣνU∗ → Cφ → Eν[1],
+where the cone Cφ is in ⟨ΣµU∗ | µ ⊊ ν⟩. Applying the functor Hom( ˜Eν, −) to the previous triangle,
+from (14) we get a morphism ψ : ˜Eν → T , which lifts a nontrivial ξ ∈ Hom( ˜Eν, ΣνU∗) ≃ k. Consider the
+cone of ψ:
+˜Eν
+ψ−→ Eν → Cψ → ˜Eν[1],
+On the one hand, Cψ ∈ ⟨ΣµU∗ | µ ⊆ ν⟩ since both ˜Eν, Eν belong to this subcategory. On the other hand,
+it follows from the construction of ψ and formulas (14) and (15) that Ext•(Cψ, ΣµE∗) = 0 for all µ ⊆ ν.
+Thus, Cψ = 0, and ψ is an isomorphism.
+□
+11
+
+Remark 3.4. Theorem 3.1 shows that the semiorthogonal decomposition (11) has a very nice symmetry.
+Namely, if one applies the duality anti-autoequivalence followed by the twist by O(n), the decomposition
+will map to itself, and the h-th block of the resulting decomposition will be generated by the objects
+⟨(Eλ)∗ | λ ∈ Yn−h,h⟩. Since (Eλ)∗ ≃ Fλ, we see that the exceptional collection generating block n − h
+maps to the graded left dual of the exceptional collection generating block h.
+Since we have a graded left dual exceptional collection, there is a spectral sequence associated to it;
+namely, spectral sequence (10) takes the following form.
+Corollary 3.5. For any object G ∈
+�
+Eλ | λ ∈ Yh,n−h
+�
+there is a spectral sequence of the form
+(16)
+Ep,q
+1
+=
+�
+λ∈Yh,n−h, |λ|=p
+Ext−q �
+G, FλT �∗
+⊗ Eλ ⇒ Hp+q(G).
+In the following section we will use this spectral sequence to produce certain resolutions of natural
+equivariant non-exceptional vector bundles.
+3.2. Resolutions of irreducible equivariant bundles. In the final section we produce nice resolutions
+for irreducible equivariant bundles on LGr(n, V ) of the from ΣλU∗ for λ ∈ Yh,w for some h + w = n in
+terms of bundles of the form Eµ.
+3.2.1. Balanced diagrams. Let λ be a Young diagram. Most commonly it is represented by a sequence of
+integers λ = (λ1, . . . , λk) such that λ1 ≥ λ2 ≥ · · · ≥ λk ≥ 0, where λi is the length of the i-th row of λ.
+There is an alternative description via hook length. Let us say that λ has rank s5 if λs ≥ s and λs+1 ≤ s.
+Graphically, s is the size of the largest square that fits into λ. There are exactly s boxes on the diagonal
+of λ. Let ai and bi denote the number of boxes to the right of the i-th diagonal box (including itself) and
+below it (including itself) respectively. Classically, ai and bi are called the arm and the leg length. One
+could alternatively say that ai = λi − (i − 1) and bi = max{1 ≤ j ≤ k | λj ≥ i} − (i − 1). We will write
+λ = (a1, a2, . . . , as|b1, b2, . . . , bs) if λ has rank s and its arm and leg length are ai and bj respectively.
+Definition 3.6. A diagram λ = (a1, a2, . . . , as|b1, b2, . . . , bs) is called balanced if ai = bi + 1 for all
+1 ≤ i ≤ s. The set of balanced diagrams with 2t boxes is denoted by B2t.6
+Proposition 3.7 ([12, Proposition 2.3.9]). Let E be a locally free module over a commutative ring of
+characteristic zero. Then
+Λt �
+S2E
+�
+≃
+�
+λ∈B2t
+ΣλE.
+Remark that (a1, a2, . . . , as|b1, b2, . . . , bs)T = (b1, b2, . . . , bs|a1, a2, . . . , as). In particular λ is symmetric,
+λ = λT , if and only if ai = bi for all 1 ≤ i ≤ s. It now follows from Definition 3.6 that λ is balanced if
+and only if µ = (λ1 − 1, λ2 − 1, . . . , λs − 1, λs+1, . . . , λk−1, λk) is symmetric. In such case the rank of µ
+equals that of λ.
+Let us now assume that µ = (µ1, µ2, . . . , µk) is of rank s. One can separate µ into three parts: a
+square of size s, everything to the right of it, and everything below it. The latter two are nothing by the
+diagrams µr = (µ1 − s, µ2 − s, . . . , µs − s) and µb = (µs+1, µs+2, . . . , µk). Under transposition the square
+maps to itself, while µr and µb get interchanged and transposed: (µT )r = (µb)T and (µT )b = (µr)T . One
+immediately concludes that µ is symmetric if and only if (µr)T = µb.
+Combining what is written in the two preceding paragraphs, we obtain the following characterization
+of balanced diagrams.
+5One also says that λ has a Durfee square of size s.
+6The number of boxes in a balanced diagram is always even.
+12
+
+Lemma 3.8. A diagram λ = (λ1, . . . , λk) of rank s is balanced if and only if
+(λ1 − (s + 1), λ2 − (s + 1), . . . , λs − (s + 1))T = (λs+1, λs+2, . . . , λk),
+where, in particular, we assume that λs ≥ s + 1.
+3.2.2. Lagrangian Borel–Bott–Weil. The celebrated Borel–Bott–Weil theorem fully describes the coho-
+mology of irreducible equivariant line bundles on rational homogeneous varieties. Since we only use it in
+one place, we formulate a much simplified version of it and refer the interested reader to [12, Chapter 4].
+Given a weakly decreasing sequence λ ∈ Zn, we denote by −λ the sequence (−λn, −λn−1, . . . , λ1),
+which is again weakly decreasing. The sum of weakly decreasing sequences is defined termwise. A weakly
+decreasing sequence is called non-singular if the absolute values of all of its terms are positive and distinct.
+If λ is non-singular, we denote by ∥λ∥ the sequence obtained by taking all the absolute values of the
+terms of λ and writing them in decreasing order. If λ is non-singular, then ∥λ∥−ρ is a weakly decreasing
+sequence with non-negative terms, where ρ = (n, n − 1, . . . , 1). The set of weakly decreasing sequences
+µ ∈ Zn with non-negative terms is identified with the set of dominant weights of Sp(V ). We denote by
+V ⟨µ⟩ the corresponding irreducible representation. For instance, if µ = (0, 0, . . . , 0), then V ⟨µ⟩ = k.
+Theorem 3.9 (Lagrangian Borel–Bott–Weil). Let λ ∈ Zn be a weakly decreasing sequence. If −λ + ρ is
+non-singular, then
+H•(LGr(n, V ), ΣλU) = V ⟨∥−λ+ρ∥−ρ⟩[−ℓ],
+where ℓ equals the number of negative terms in −λ + ρ plus the number of pairs 1 ≤ i < j ≤ n such that
+(−λ + ρ)i + (−λ + ρ)j < 0. Otherwise, H•(LGr(n, V ), ΣλU) = 0.
+3.2.3. Vanishing lemma. We will need the following result, which is definitely well known to experts.
+A proof is included for the sake of completeness.
+Lemma 3.10. Let λ ∈ Yn,n+1 be a Young diagram.
+(1) If λ is not balanced, then H•(LGr(n, V ), ΣλU) = 0.
+(2) If λ is balanced and |λ| = 2t, then H•(LGr(n, V ), ΣλU) = k[−t].
+Proof. The proof is a direct application of the Borel–Bott–Weil theorem stated in the previous Section.
+In order to show (1), we need to show that the weight −λ + ρ is non-singular if and only if λ is balanced.
+Consider the strictly decreasing sequence
+(17)
+− λ + ρ = (n − λn, (n − 1) − λn−1, . . . , 2 − λ2, 1 − λ1).
+Since 0 ≤ λi ≤ n + 1 for all i, all the absolute values of the terms of −λ + ρ are between 0 and n. Recall
+that −λ + ρ is non-singular if and only if all the absolute values of its terms are distinct and positive.
+Assume the latter. Let s denote the rank of λ. Then the first n − s terms of (17) are positive, while the
+last s terms are non-positive. Since the absolute values of the latter can not be zero (the weight being
+non-singular), we conclude that λi ≥ s + 1 for i = 1, . . . , s. Put µi = λi − (s + 1). Then the sequence
+(18)
+(n − λn, (n − 1) − λn−1, . . . , (s + 1) − λs+1, s + µ1, (s − 1) + µ2, . . . , 1 + µs)
+differs from (17) only in the last s terms: their signs have been changed, and their order has been reversed.
+Remark that (λs+1, λs+2, . . . , λn) = λb ∈ Yn−s,s, while µ ∈ Ys,n−s. The sequence (18) is very well known
+to anyone who has ever studied Kapranov’s work. It follows from [6, Section 2.7] that all the terms in (18)
+are distinct if and only if λb = µT . The latter is equivalent, by Lemma 3.8, to saying that λ is balanced.
+Let us turn to (2). Assume that λ is balanced. Since the absolute values of the terms of −λ + ρ take
+all the values between 1 and n, by the Borel–Bott–Weil theorem the only nonzero cohomology will be
+equal to k. We only need to determine the degree in which it sits, and the latter equals the number of
+negative terms in −λ+ρ plus the number of pairs 1 ≤ i < j ≤ n such that (−λ+ρ)i +(−λ+ρ)j < 0. The
+13
+
+first number equals s, and we need to compute the second number. Remark that if 1 ≤ j ≤ n − s, then
+both (−λ + ρ)i and (−λ + ρ)j are positive. Thus, we may assume that n − s < j ≤ n. If n − s < i ≤ n,
+then any i < j ≤ n contributes to the count since both terms are negative. There are s(s−1)
+2
+such pairs.
+Finally, assume that 1 ≤ i ≤ n − s and n − s < j ≤ n. Since (−λ + ρ)j < 0, we need to determine when
+(−λ + ρ)i < −(−λ + ρ)j. Such pairs correspond precisely to inversions in the sequence (18). It is easy
+to show that the number of those equals |λb| (see [6, Section 2.7]). We conclude that for a balanced λ
+the only nontrivial cohomology sits in degree s + s(s−1)
+2
++ |λb| = s(s+1)
+2
++ |λb|. It remains to recall that λ
+consists of λb, µ = (λb)T , and a square of size s×(s+1), so |λ| = |λb|+|µ|+s(s+1) = 2|λb|+s(s+1).
+□
+The following is a simple relative version of the previous lemma. As usual, all the functors are derived.
+Corollary 3.11. Let X be smooth projective variety, and let V be a rank 2k symplectic bundle on X.
+Consider the relative Lagrangian Grassmannian π : LGrX(n, V) → X, denote by U ⊂ π∗V the universal
+bundle. Let ν ∈ Yk,k+1.
+(1) If ν is not balanced, then π∗ΣλU = 0.
+(2) If ν is balanced and |λ| = 2t, then π∗ΣλU = OX[−t].
+3.2.4. Skew Schur functors. Before we formulate and prove the second main result of the paper, we need
+to say a few words about skew diagrams. Let λ and µ be two Young diagrams such that µ ⊆ λ. Then
+one can define the skew Schur functor Σλ/µ, see [12, Section 2.1].
+If E is a free module over a commutative ring of characteristic zero and α and β are two Young
+diagrams, one has a direct sum decomposition
+ΣαE ⊗ ΣβE ≃
+�
+|κ|=|α|+|β|
+(ΣκE)⊕c(α,β;κ) ,
+where c(α, β; κ) are the celebrated Littlewood–Richardson coefficients. It turns out that one has a direct
+sum decomposition for skew Schur functors controlled by the same coefficients. Namely,
+(19)
+Σλ/µE ≃
+�
+|ν|+|µ|=|λ|
+(ΣνE)⊕c(ν,µ;λ) ,
+see [12, Theorem 2.3.6].
+If c(ν, µ; λ) > 0, we will write ν ⊆ λ/µ. Remark that ν ⊆ λ/µ implies that ν ⊆ λ. In particular,
+if λ ∈ Yh,w, then ν ⊆ λ/µ implies ν ∈ Yh,w. Inspired by decomposition (19), for a pair of diagrams
+λ, µ ∈ Yh,w such that h + w = n and µ ⊆ λ, we put
+(20)
+Eλ/µ =
+�
+ν⊆λ/µ
+(Eν)⊕c(ν,µ;λ) .
+Extend the definition, as usual, by putting Eλ/µ = 0 for µ ⊈ λ.
+3.2.5. Resolutions. We are ready to present the main application of Theorem 3.1.
+Theorem 3.12. Let λ ∈ Yh,n−h for some 0 ≤ h ≤ n. There is an exact sequence of vector bundles on
+LGr(n, V ) of the form
+(21)
+0 →
+�
+µ∈Bh(h+1)
+Eλ/µ → · · · →
+�
+µt∈B2t
+Eλ/µt → · · · →
+�
+µ2∈B4
+Eλ/µ2 →
+�
+µ1∈B2
+Eλ/µ1 → Eλ → ΣλU∗ → 0,
+where B2t denotes the set of balanced diagrams with 2t boxes.
+14
+
+Proof. Since the vector bundle ΣλU∗ lies in the subcategory ⟨Eλ | λ ∈ Yh,n−h⟩, spectral sequence (16) is
+applicable. Precisely, one has
+(22)
+Ep,q
+1
+=
+�
+µ∈Yh,n−h, |α|=p
+Ext−q �
+ΣλU∗, FαT �∗
+⊗ Eα ⇒ Hp+q(ΣλU∗).
+Let us compute Ext•(ΣλU∗, Fµ) for µ = αT ∈ Yn−h,h using the isomorphism Fµ ≃ p∗q∗Gµ given in
+Proposition 2.15, where p and q come from the diagram
+IFl(h, n; V )
+LGr(n, V )
+IGr(h, V )
+p
+q
+(This is diagram (12) with h and n − h interchanged since µ ∈ Yn−h,h.) One has a sequence of isomor-
+phisms
+Ext•(ΣλU∗, Fµ) ≃ Ext•(ΣλU∗, p∗q∗Gµ)
+≃ Ext•(ΣλU∗, q∗Gµ)
+≃ H•(IFl(h, n; V ), ΣλU ⊗ q∗Gµ)
+≃ H•(IGr(h; V ), q∗(ΣλU) ⊗ Gµ),
+where the second isomorphism is given by adjunction, and the last one follows from the projection formula.
+As usual, all the functors are derived.
+Consider the spectral sequence associated with the composition of derived functors. Its second page
+is given by
+(23)
+Hi(IGr(n − h; V ), Rjq∗(ΣλU) ⊗ Gµ)
+⇒
+Hi+j(IGr(n − h; V ), q∗ΣλU ⊗ Gµ).
+Let us compute Rjq∗(ΣλU). Recall that we denoted by W ⊂ U the universal flag on IFl(h, n; V ). There
+is a filtration on ΣλU associated with the short exact sequence 0 → W → U → U/W whose associated
+quotients are isomorphic to
+(24)
+�
+ν⊆λ,|ν|=k
+Σλ/νW ⊗ Σν(U/W).
+Since for any ν ⊆ λ the bundle Σλ/νW is pulled back from IGr(h, V ), we conclude by the projection
+formula that Rjq∗(Σλ/νW ⊗ Σν(U/W)) ≃ Σλ/νW ⊗ Rjq∗Σν(U/W).
+In order to compute Σλ/νW ⊗ Rjq∗Σν(U/W), recall that IFl(h, n; V ) is the relative Lagrangian Grass-
+mannian LGr(n−h, W⊥/W) over IGr(h, V ). Since ν ⊆ λ, one has ν ∈ Yh,n−h. If the height of ν is greater
+than n − h, then Σν(U/W) = 0. Thus, Σλ/νW ⊗ Rjq∗Σν(U/W) = 0 unless ν ∈ Yn−h,n−h. Finally, if
+ν ∈ Yn−h,n−h, then we are in the situation when we can apply Corollary 3.11. We conclude that if ν ⊆ λ,
+then
+(25)
+Σλ/νW ⊗ R•q∗Σν(U/W) ≃
+�
+Σλ/νW[−t]
+if ν ∈ B2t,
+0
+otherwise.
+Using (25), we can compute Rj(ΣλU). Indeed, the spectral sequence associated with the filtration
+with the quotients (24) degenerates in the first page, and
+(26)
+Rjq∗(ΣλU) ≃
+�
+ν⊆λ,ν∈B2j
+Σλ/νW.
+15
+
+We are ready to get back to the spectral sequence (23).
+Hi(IGr(h; V ), Rjq∗(ΣλU) ⊗ Gµ) ≃
+�
+ν⊆λ,ν∈B2j
+Hi(IGr(h; V ), Σλ/νW ⊗ Gµ)
+≃
+�
+ν⊆λ,ν∈B2j
+Exti(Σλ/νW∗, Gµ)
+≃
+�
+ν⊆λ,ν∈B2j
+�
+κ⊆λ/ν
+Exti((ΣκW∗)⊕c(κ,ν;λ), Gµ),
+where the last isomorphism comes from (19). Since κ ⊆ λ/ν implies that κ ⊆ λ, using (13) we see that
+the last term in (26) is zero unless κ = µT and i = |µ|, and is isomorphic to k otherwise. Finally,
+H|µ|(IGr(h; V ), Rjq∗(ΣλU) ⊗ Gµ) ≃
+�
+ν∈B2j
+k⊕c(ν,µ;λ),
+where 2j = |ν| = |λ| − |µ|, are the only potentially nontrivial cohomology groups, and the spectral
+sequence (23) degenerates. We conclude that
+H|µ|+j(IGr(h; V ), q∗ΣλU ⊗ Gµ) =
+�
+ν∈B2j
+k⊕c(ν,µ;λ).
+Let us return to spectral sequence (22). From the computations above we know that the only nontrivial
+terms in it are
+(27)
+E−|λ|+t,|λ|−2t
+1
+≃
+�
+ν∈B2t
+�
+µ⊆λ/ν
+|µ|=|λ|−2t
+(Eµ)⊕c(ν,µ;λ) =
+�
+ν∈B2t
+Eλ/ν,
+where the last equality comes from the definition in (20). We conclude that the spectral sequence contains
+at most one non-trivial term in each diagonal.
+As the spectral sequence converges to H•(Σλ), one must have a long exact sequence of the form
+· · · → E−|λ|+t,|λ|−2t
+1
+→ · · · → E−|λ|+2,|λ|−4
+1
+→ E−|λ|+1,|λ|−2
+1
+→ E−|λ|,|λ|
+1
+→ ΣλU∗ → 0.
+Since any balanced diagram contained in λ has height at most h, its width is at most h + 1 and
+E−|λ|+t,|λ|−2t
+1
+= 0 for t > h(h+1)/2. It remains to use (27) to get the desired long exact sequence (21).
+□
+Example 3.13. We present some examples of resolutions introduced in Theorem 3.12.
+h = 1: In this case λ consists of just a single row of length at most n − 1. If λ = (0), then Eλ ≃ O. If
+λ = (1), then Eλ ≃ ΣλU∗ = U∗. The interesting case is λ = (p) for some 2 ≤ p ≤ n − 1. Since (2) is the
+only nontrivial balanced diagram of height one, and (p)/(2) = (p − 2), resolution (21) takes form
+0 → E(p−2) → E(p) → SpU∗ → 0.
+As a consequence, we see that E(p) has a filtration with the associated graded quotients of the form
+Sp−2tU∗ for all 0 ≤ p ≤ p/2.
+w = 1 : In this case λ consists of a column of length at most n − 1. Since there are no nontrivial
+balanced diagrams of width 1, we conclude that for λ = (t)T there is an isomorphism Eλ ≃ ΛtU∗.
+λ = (3, 1): This is the first case when resolution (21) has length greater than 1. Remark that (2) ∈ B2
+and (3, 1) ∈ B4 are the only balanced diagrams contained in λ. Moreover, the only nontrivial Littlewood–
+Richardson coefficients for (3, 1)/(2) are c ((2), (2); λ) = c ((1, 1), (2); λ) = 1. Thus, one has a resolution
+of the form
+0 → O → E(2) ⊕ E(1,1) → E(3,1) → Σ(3,1)U∗ → 0.
+Since E(1,1) ≃ Λ2U∗ and E(2) fits into a short exact sequence 0 → O → E(2) → S2U∗ → 0, with a little
+bit of work one can show that Eλ is an extension of Σ(3,1)U∗ by S2U∗ and Λ2U∗.
+16
+
+References
+[1]
+A. A. Beilinson. “Coherent sheaves on P n and problems of linear algebra”. In: Functional Analysis
+and Its Applications 12.3 (July 1978), pp. 214–216. doi: 10.1007/bf01681436.
+[2]
+A. I. Bondal. “A symplectic groupoid of triangular bilinear forms and the braid group”. In: Izvestiya
+Rossiiskoi Akademii Nauk. Seriya Matematicheskaya 68.4 (2004), pp. 19–74. issn: 1607-0046. doi:
+10.1070/IM2004v068n04ABEH000495.
+[3]
+A. I. Bondal and M. M. Kapranov. “Representable functors, Serre functors, and reconstructions”.
+In: Izvestiya Akademii Nauk SSSR. Seriya Matematicheskaya 53.6 (1989), pp. 1183–1205, 1337.
+issn: 0373-2436. doi: 10.1070/IM1990v035n03ABEH000716.
+[4]
+Anton Fonarev. “Full exceptional collections on Lagrangian Grassmannians”. In: International
+Mathematics Research Notices. IMRN 2 (2022), pp. 1081–1122. issn: 1073-7928. doi: 10.1093/imrn/rnaa098.
+[5]
+A. L. Gorodentsev and S. A. Kuleshov. “Helix theory”. In: Moscow Mathematical Journal 4.2
+(2004), pp. 377–440, 535. issn: 1609-3321. doi: 10.17323/1609-4514-2004-4-2-377-440.
+[6]
+M. M. Kapranov. “Derived category of coherent sheaves on Grassmann manifolds”. In: Izvestiya
+Akademii Nauk SSSR. Seriya Matematicheskaya 48.1 (1984), pp. 192–202. issn: 0373-2436.
+[7]
+M. M. Kapranov. “On the derived categories of coherent sheaves on some homogeneous spaces”. In:
+Inventiones Mathematicae 92.3 (1988), pp. 479–508. issn: 0020-9910. doi: 10.1007/BF01393744.
+[8]
+Alexander Kuznetsov. “Exceptional collections for Grassmannians of isotropic lines”. In: Proceedings
+of the London Mathematical Society. Third Series 97.1 (2008), pp. 155–182. issn: 0024-6115. doi:
+10.1112/plms/pdm056.
+[9]
+Alexander Kuznetsov and Alexander Polishchuk. “Exceptional collections on isotropic Grassman-
+nians”. In: Journal of the European Mathematical Society (JEMS) 18.3 (2016), pp. 507–574. issn:
+1435-9855. doi: 10.4171/JEMS/596.
+[10]
+Alexander Kuznetsov and Maxim Smirnov. “Residual categories for (co)adjoint Grassmannians in
+classical types”. In: Compositio Mathematica 157.6 (2021), pp. 1172–1206. issn: 0010-437X. doi:
+10.1112/s0010437x21007090.
+[11]
+Kyoung-Seog Lee and Kyeong-Dong Park. “Moduli spaces of Ulrich bundles on the Fano 3-fold V5”.
+In: Journal of Algebra 574 (2021), pp. 262–277. issn: 0021-8693. doi: 10.1016/j.jalgebra.2021.01.015.
+[12]
+Jerzy Weyman. Cohomology of vector bundles and syzygies. Vol. 149. Cambridge Tracts in Math-
+ematics. Cambridge University Press, Cambridge, 2003, pp. xiv+371. isbn: 0-521-62197-6. doi:
+10.1017/CBO9780511546556.
+Algebraic Geometry Section, Steklov Mathematical Institute of Russian Academy of Sciences, 8
+Gubkin str., Moscow 119991 Russia
+Email address: avfonarev@mi-ras.ru
+17
+
diff --git a/U9E1T4oBgHgl3EQfIgPL/content/tmp_files/load_file.txt b/U9E1T4oBgHgl3EQfIgPL/content/tmp_files/load_file.txt
new file mode 100644
index 0000000000000000000000000000000000000000..13a31e053a88836f196bd8780674fe64ccae526f
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@@ -0,0 +1,976 @@
+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf,len=975
+page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content='02941v1  [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content='AG]  7 Jan 2023  DUAL EXCEPTIONAL COLLECTIONS ON LAGRANGIAN GRASSMANNIANS  ANTON FONAREV  Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content=' We construct graded left dual exceptional collections to the exceptional collections generating  the blocks of Kuznetsov and Polishchuk on Lagrangian Grassmannians.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content=' As an application, we find explicit  resolutions for some natural irreducible equivariant vector bundles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content=' Introduction  Derived categories of varieties are among the central objects in modern algebraic geometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content=' A skeptical  reader might complain that the very notion of the derived category is too abstract, and they might have  a point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content=' However, since the pioneering work of Beilinson [1], derived categories have become a great  computational tool.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content=' Nowadays one would say that Beilinson constructed a full exceptional collection in  the bounded derived categories of coherent sheaves on projective spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content='  The following analogy is commonly used to explain the computational power of exceptional collections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content='  Given a finite dimensional real vector space V with a positive definite symmetric bilinear form ⟨−, −⟩, one  can find an orthonormal basis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content=' That is, a basis consisting of vectors (v1, v2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content=' , vn) such that (i) ⟨vi, vi⟩ = 1  for all i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content=' , n and (ii) ⟨vi, vj⟩ = 0 whenever i ̸= j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content=' The first condition says that the vectors are  unit vectors, while the second condition is the orthogonality condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content=' Given such a basis, every vector  v ∈ V can be easily decomposed with respect to our basis:  (1)  v = ⟨v, v1⟩v1 + ⟨v, v2⟩v2 + · · · + ⟨v, vn⟩vn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content='  Let us now relax some of the conditions;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content=' for instance, symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content=' We still want a basis which consists of  unit vectors, but we have to modify the orthogonality condition (ii).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content=' Let us say that a basis (v1, v2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content=' , vn)  is semiorthonormal if (i) ⟨vi, vi⟩ = 1 for all i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content=' , n and (ii) ⟨vi, vj⟩ = 0 whenever i > j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content='  One  obviously needs and adjustment to the formula (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content=' Indeed, in (1) we explicitly used the fact that under  the isomorphism V ∗ ∼  −→ V given by the form ⟨−, −⟩ the dual basis (v1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content=' , vn) maps back to (v1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content=' , vn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content='  Since we actually have two isomorphisms this time, V → V ∗, v �→ ⟨v, −⟩ and V → V ∗, v �→ ⟨−, v⟩, let us  consider the second one and assume that its inverse maps vi to ui ∈ V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content=' That is, we have  ⟨vi, uj⟩ = δij  for all 1 ≤ i, j ≤ n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content='  The vectors u1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content=' , un obviously form a basis, and for any v ∈ V we have the desired formula  (2)  v = ⟨v, u1⟩v1 + ⟨v, u2⟩v2 + · · · + ⟨v, un⟩vn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content='  What is less immediate, the vectors (un, un−1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content=' , u1) (remark the reverse order) form an orthonormal  basis called the left dual to (v1, v2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content=' , vn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content='  One rather indirect way to see that the latter holds is via mutations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content=' Consider a semiorthonormal pair  (u, v): that is, ⟨u, u⟩ = ⟨v, v⟩ = 1 and ⟨v, u⟩ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content=' Let us define a new vector Luv = v − ⟨u, v⟩u, which is  called the left mutation of v through u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content=' A simple calculation shows that (Luv, u) is a semiorthonormal  pair.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content=' As the name suggests, there is a sibling to the left mutation procedure: if one puts Rvu = u−⟨u, v⟩v,  which is called the right mutation of u through v, then the pair (v, Rvu) is semiorthonormal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content=' It is a very  nice exercise in linear algebra to check that left and right mutations of adjacent elements define an action  This work was supported by the Russian Science Foundation under grant no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content=' 19-11-00164, https://rscf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content='ru/en/project/19-  11-00164/.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content='  1  of the braid group on n strands on the set of all semiorthonormal bases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content=' Moreover, the left dual basis to  a semiorthonormal basis (v1, v2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content=' , vn) is given by  (3)  Lv1Lv2 · · · Lvn−1vn, Lv1Lv2 · · · Lvn−2vn−1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content=' , Lv2v1, v1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content='  We refer the reader to [2] for further insights and some interesting properties of this action.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content='  The linear-algebraic picture translates to triangulated categories in the following way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content=' Instead of a  vector space we consider a k-linear triangulated category T , while we treat Ext•  T (−, −) as a kind of  bilinear form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content=' Then one says that an object E ∈ T is exceptional if Hom(E, E) = k and Exti(E, E) = 0  for all i ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content=' A collection of objects (E1, E2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content=' , En) is called exceptional if every Ei is an exceptional  object and Ext•(Ei, Ej) = 0 for all i > j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content=' Finally, a collection is called full if it generates the category in  a sense that no proper strictly full triangulated subcategory of T contains all Ei.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content=' If (E1, E2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content=' , En) is  a full exceptional collection in T , we will write T = ⟨E1, E2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content=' , En⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content='  The analogy with semiorthonormal bases should be clear by now.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content=' Assume that T has a full exceptional  collection (E1, E2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content=' , En) (which is rarely the case).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content=' What is most interesting is that not only every  object in T can be obtained from the finite set of Ei’s by iteratively taking shifts and cones, but this  procedure can be made rather explicit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content='  Recall that the decomposition of every vector in terms of a  given semiorthonormal basis could be done with the help of a left dual semiorthonormal basis by (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content='  Let us mimic its definition in the categorical case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content='  Namely, let us say that a collection of objects  (∨En, ∨En−1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content=' , ∨E1) is left dual to (E1, E2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content=' , En) if Ext•(Ei, ∨Ej) = 0 for i ̸= j and Ext•(Ei, ∨Ei) = k  for all i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content=' , n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content=' It should not surprise the reader at this point that (∨En, ∨En−1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content=' , ∨E1) is again an  exceptional collection, and it is full whenever the original one is.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content=' Moreover, there is an action of the braid  group on n strands on the set of all exceptional collections of length n in T , and a formula similar to (3)  determines the left dual collection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content=' Finally, instead of the decomposition (2) one has a spectral sequence  which computes cohomological functors applied to objects in T , which we will talk about in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content='  The bridge between exceptional collections and semiorthonormal bases is actually rather simple.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content=' Given  a sufficiently nice T (say, we assume that Ext•(E, F) is finite-dimensional for all E, F ∈ T ) with a full  exceptional collection (E1, E2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content=' , En), one checks that the classes of Ei form a basis in the Grothendieck  group K0(T ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content=' In particular, the length of any full exceptional collection equals the rank of the latter,  which should a posteriori be a free finitely generated abelian group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content=' If one takes �  i(−1)i dimk Exti(−, −)  as the bilinear form, exceptional collections become “categorifications” of the corresponding semiorthonor-  mal bases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content='  The first example of a full exceptional collection was given in [1], in which Beilinson showed that  Db(Pn) = ⟨O, O(1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content=' , O(n)⟩, where Db stands for the bounded derived category of coherent sheaves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content='  Interestingly enough, he also showed that the left dual is given by the collection ⟨Ωn(n), Ωn−1(n −  1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content=' , Ω1(1), O⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content='  A long-standing conjecture states that the bounded derived category of coherent  sheaves on a rational homogeneous variety admits a full exceptional collection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content=' Though a lot of work  has been done over the years, the conjecture has been established in a very limited number of cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content='  Say, for classical groups of type ABCD and Picard rank 1 the problem was fully resolved only for  Grassmannians [6], quadrics [7], symplectic and orthogonal Grassmannians of planes [8, 10], Lagrangian  Grassmannians [4], and in some sporadic cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content=' We refer the reader to [9] for further details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content='  Since many explicit constructions realize varieties as subvarieties in Grassmannians, exceptional collec-  tions on the latter become an important computational tool.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content=' One of our favorite recent examples can be  found in [11], where the authors use exceptional collections to study moduli of Ulrich bundles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content=' In order  to use the tool’s maximum power, it is important to know the dual collection, and finding one is a task of  its own.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content=' In the present paper we find exceptional collections of Lagrangian Grassmannians dual to those  constructed in [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content=' We further use them to provide explicit resolutions of some very natural irreducible  vector bundles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content='  2  From now on let we will be interested in the bounded derived category of coherent sheaves on LGr(n, V ),  the Lagrangian Grassmannian of isotropic subspaces of dimension n in a fixed 2n-dimensional vector  space V over an algebraically closed field k equipped with a non-degenerate symplectic form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content=' Excep-  tional collections of maximal length (equal to the rank of the Grothendieck group) were constructed on  all symplectic and orthogonal Grassmannians by Kuznetsov and Polishchuk in [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content=' All these collections  are conjecturally full;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content=' however, the latter was checked only for Lagrangian Grassmannians in [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content=' The  construction of Kuznetsov and Polishchuk is rather indirect: they start with certain collections of ex-  ceptional irreducible vector bundles, called blocks, which naturally form an exceptional collection in the  equivariant derived category, then pass to dual collections within each block (again, in the equivariant  derived category).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content=' Each block must satisfy certain homological conditions which guarantee that the dual  collections become exceptional in the non-equivariant derived category.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content=' Most of the hard work in [9]  is related to checking that certain collections of irreducible equivariant vector bundles satisfy the block  condition, which is a rather difficult problem in representation theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content='  While the block condition will be discussed in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content='2, let us explain why isotropic Grassmannians  are much harder than the classical ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content=' Full exceptional collections in the bounded derived categories of  Grassmannians were constructed by Kapranov [7], who naturally extended Beilinson’s method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content=' Consider  the Grassmannian Gr(k, V ) of k-dimensional subspaces in a fixed N-dimensional vector space V over a  field k of characteristic zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content=' Denote by U the tautological rank k subbundle of the trivial bundle V ⊗ O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content='  Kapranov showed that  (4)  Db(Gr(k, V )) =  �  ΣλU∗ | λ ∈ Yk,N−k  �  ,  where λ runs over the set of Young diagrams Yk,N−k, Σλ is the corresponding Schur functor, and the  order in this collection can be taken to be any linear order refining the partial inclusion order on the  diagrams.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content=' Moreover, he simultaneously constructed the (graded) left dual to this collection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content=' The latter  is given by  (5)  Db(Gr(k, V )) =  �  ΣλT U⊥ | λ ∈ Yk,N−k  �  ,  where U⊥ = (V/U)∗ and λT denotes the transposed diagram.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content='1 The duality relation might seem a little  different from the one we described earlier:  (6)  Ext•(ΣλU∗, ΣµT U⊥) =  �  k[−|λ|],  if λ = µ,  0,  otherwise,  but this grading difference does not change much since the two definitions are equivalent up to shifts  in the derived category.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content=' Actually, the grading choice in (6) is favorable since the dual collection then  consists of vector bundles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content='  Since LGr(n, V ) is naturally embedded as a closed subvariety in Gr(n, V ), one can ask whether any  of elements of Kapranov’s collection restrict to exceptional vector bundles on LGr(n, V ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content=' This is where  surprising things happen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content=' It turns out that the only Young diagrams for which ΣλU∗ are exceptional are  λ ∈ Yn,1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content=' These vector bundles are nothing but the exterior powers ΛiU∗, and there are only n of those,  while rk K0(LGr(n, V )) = 2n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content=' Remark also that U⊥ restricts to U on LGr(n, V ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content='  The construction of Kuznetsov and Polishchuk produces for any λ ∈ Yh,n−h, 0 ≤ h ≤ n, an equivariant  non-irreducible (in general) vector bundle Eλ on LGr(n, V ) with the following properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content='  First, Eλ  belongs to the subcategory of the derived category generated by ΣµU∗ for µ ⊆ λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content=' Second, the G = Sp2n-  equivariant groups Ext•  G(Eλ, ΣµU∗) = 0 for all µ ⊊ λ, while Ext•  G(Eλ, ΣλU∗) = k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content=' A given diagram may  1A careful reader might wonder why we choose this extra transposition and not index the collection directly by YN−k,k,  as Kapranov does.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content=' After all, {λT | λ ∈ Yh,w} = Yw,h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content=' Our choice becomes clear in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content='1, where we introduce our  grading convention for dual exceptional collections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content='  3  belong to various sets Yh,n−h, but the resulting bundle does not depend on the choice of h since the  previous conditions fully characterize it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content=' The first nontrivial example of such a bundle is the universal  extension  0 → O → E2 → S2U∗ → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content='  Kuznetsov and Polishchuk showed that for any 0 ≤ h ≤ n the bundles  (7)  �  Eλ | λ ∈ Yh,n−h  �  form an exceptional collection in Db(LGr(n, V )).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content=' As in the case of Grassmannians, one can linearly order  them in any way compatible with the partial inclusion order on the corresponding Young diagrams.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content=' Let  us denote by Fλ the bundle dual (in the usual sense) to Eλ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content='  Theorem A (Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content=' The bundles  �  FλT | λ ∈ Yh,n−h  �  form a graded left dual exceptional col-  lection to (7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content='  We formulated Theorem A in terms of transposed diagrams in order to show the parallel between  our Lagrangian situation and the case of classical Grassmannians: compare the latter theorem with (4)  and (5) keeping in mind that U⊥ is isomorphic to U = (U∗)∗ on LGr(n, V ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content='  It is easy to see that the bundle ΣλU∗ belongs to the subcategory (7) whenever λ ∈ Yh,n−h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content=' A nice  geometric description (as a matter of fact, two of them) was given for the bundles Fµ in [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content=' Using this  description, one can compute the corresponding spectral sequences and get the following result as an  application of Theorem A (see Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content='2 for the details).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content='  Theorem B (Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content='12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content=' Let λ ∈ Yh,n−h for some 0 ≤ h ≤ n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content=' Then there is an exact sequence of  vector bundles on LGr(n, V ) of the form  0 →  �  µ∈Bh(h+1)  Eλ/µ → · · · →  �  µt∈B2t  Eλ/µt → · · · →  �  µ2∈B4  Eλ/µ2 →  �  µ1∈B2  Eλ/µ1 → Eλ → ΣλU∗ → 0,  where B2t denotes the set of balanced diagrams with 2t boxes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content='  The paper is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content=' In Section 2 we collect all the preliminaries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content=' It contains no new  material except, maybe, our convention for dual exceptional collections for exceptional collections indexed  by a graded partially ordered set (see Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content=' Section 3 contains our main results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content=' That is, we  prove Theorems A and B, which are Theorems 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content='1 and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content='12 respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content=' Preliminaries  Throughout the paper we work over a field k of characteristic zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content=' Dual exceptional collections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content=' In the present section we collect some preliminaries related to  (dual) exceptional collections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content=' The material is well known to specialists;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content=' however, since we naturally work  with exceptional collections indexed by graded partially ordered sets, we introduce a certain convention  in the definitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content=' This convention seems rather natural, as we show in various examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content=' Partially ordered sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content=' Recall that a partially ordered set (poset) P is a set equipped with a binary  relation ⪯, called a partial order, satisfying the following three properties:  Reflexivity: x ⪯ x for all x ∈ P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content='  Antisymmetry: if x ⪯ y and y ⪯ x, then x = y for all x, y ∈ P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content='  Transitivity: is x ⪯ y and y ⪯ z, then x ⪯ z for all x, y, z ∈ P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content='  4  Elements x and y are called comparable if either x ⪯ y or y ⪯ x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content='  If every two elements in P are  comparable, one calls P linearly ordered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content=' If x ⪯ y and x ̸= y, one usually writes x ≺ y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content=' If P is partially  ordered, its dual P◦ is the set underlying P equipped with the converse relation: x ⪯ y in P◦ if and only  if y ⪯ x in P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content='  Let x and y be elements of a poset P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content=' One says that y covers x, written x ⋖ y, if x ≺ y and there  is no element z such that x ≺ z ≺ y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content=' A grading function on P is a map ρ : P → Z with the following  properties:  if x ≺ y then ρ(x) < ρ(y),  if x ⋖ y then ρ(y) = ρ(x) + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content='  A poset equipped with a grading function is called a graded poset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content='  Of course, not all posets can be  turned into graded ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content=' We will be mainly interested in finite posets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content=' If P is finite and admits a grading  function, there is a rather natural choice for such a function: there exists a unique grading function | − |  with the property that |x| = 0 whenever x is a minimal element (that is, there is no y such that y ≺ x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content='  In the following by a graded poset we mean a finite poset equipped with this grading function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content='  Example 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content=' Let Yh,w denote the set of Young diagrams of height at most h and width at most w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content=' This set  can be identified with the set of integer sequences (λ1, λ2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content=' , λh) such that w ≥ λ1 ≥ λ2 ≥ · · · ≥ λh ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content='  There is a natural partial order on Yh,w given by inclusion of diagrams: λ ⪯ µ if λi ≤ µi for all i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content=' , h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content='  With this partial order the poset Yh,w is graded, and |λ| = λ1 + λ2 + · · · + λh equals the number of boxes  in the diagram λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content=' Exceptional collections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content=' Let T be a k-linear triangulated category.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content='  An object E ∈ T is called  exceptional if Hom(E, E) = k and Extt(E, E) = Hom(E, E[t]) = 0 for all t ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content=' Let P be a poset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content=' An ex-  ceptional collection indexed by P is a collection of exceptional objects {Ex}x∈P such that Ext•(Ex, Ey) = 0  unless x ⪯ y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content=' We denote by ⟨Ex|x ∈ P⟩ the smallest strictly full triangulated subcategory in T con-  taining all Ex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content=' If P is finite, one can always refine the order so that it becomes isomorphic to the poset  {1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content=' , l}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content=' Under this isomorphism we get the usual definition of an exceptional collection: that is,  a collection of exceptional objects (E1, E2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content=' , El) such that Ext•(Ej, Ei) = 0 for all l ≥ j > i ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content='  Example 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content=' Let V be an n-dimensional vector space over k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content=' Consider the Grassmannian Gr(k, V ), and  let U denote the tautological rank k subbundle in the trivial bundle V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content=' Kapranov showed in [6] that the  bounded derived category of coherent sheaves Db(Gr(k, V )) admits a full exceptional collection indexed  by Yk,n−k:  Db(Gr(k, V )) =  �  ΣλU∗ | λ ∈ Yk,n−k  �  ,  where Σλ denotes the Schur functor associated with λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content='2 The fact that this collection is indexed by a poset  gives more information about orthogonality of different objects: if λ and µ are incomparable (that is,  neither is contained in the other), then both Ext•(ΣλU∗, ΣµU∗) = 0 and Ext•(ΣµU∗, ΣλU∗) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content=' Mutations and duality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content=' Let (E, F) be an exceptional pair in T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content='  The left mutation LEF of F  through E is defined by the distinguished triangle  LEF → Ext•(E, F) ⊗ E ev  −→ F → LEF[1],  where ev is the evaluation morphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content=' Similarly, define the right mutation RFE via the distinguished  triangle  RF E[−1] → E coev  −−−→ Ext•(E, F)∗ ⊗ F → RFE,  where coev is the coevaluation morphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content='  One easily checks that both (LEF, E) and (F, RF E) are  exceptional pairs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content=' Moreover, ⟨E, F⟩ = ⟨LEF, E⟩ = ⟨F, RF E⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content='  2Our convention for Schur functors is such that Σ(p) is isomorphic to the p-th symmetric power Sp, so Σ(1,1) ≃ Λ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content='  5  Example 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content=' Consider the Grassmannian Gr(k, V ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content=' Then the structure sheaf and the dual tautological  bundle form an exceptional pair (O, U∗).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content=' One quickly checks that LOU∗ ≃ U⊥, where U⊥ = (V/U)∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content='  Given an exceptional collection (E1, E2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content=' , El), mutations of adjacent elements define an action of  the braid group Brl on l strands on the set of exceptional collections in ⟨E1, E2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content=' , El⟩, see [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content=' For a  fixed collection there are two important elements in the orbit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content=' Namely, the dual collections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content=' The left dual  collection to (E1, E2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content=' , El) is defined as  (LE1LE2 · · · LEl−1El, LE1LE2 · · · LEl−2El−1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content=' , LE1E2, E1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content='  We denote it by (E∨  l , E∨  l−1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content=' , E∨  1 ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content=' The left dual exceptional collection can be fully characterized by  the following three properties:  (1) E∨  i ∈ ⟨E1, E2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content=' , El⟩ for all 1 ≤ i ≤ l,  (2) Ext•(Ei, E∨  j ) = 0 for all i ̸= j,  (3) Ext•(Ei, E∨  i ) = k[−i + 1] for all 1 ≤ i ≤ l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content='  Example 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content=' The bounded derived category of the projective space P(V ) has a full exceptional collection  consisting of the line bundles ⟨O, O(1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content=' , O(n)⟩, where n is the dimension of V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content=' Its left dual is given  by ⟨Ωn(n), Ωn−1(n − 1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content=' , Ω1(1), O⟩, where Ωi = ΛiΩ1  P(V ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content='  Similarly, the right dual collection is defined as  (En, REnEn−1, REnREn−1En−2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content=' , REnREn−1 · · · RE2E1),  and will be denoted by (∨El, ∨El−1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content=' , ∨E1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content=' It can be fully characterized by the following conditions:  (1) ∨Ei ∈ ⟨E1, E2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content=' , El⟩ for all 1 ≤ i ≤ l,  (2) Ext•(∨Ei, Ej) = 0 for all i ̸= j,  (3) Ext•(∨Ei, Ei) = k[−n + i] for all 1 ≤ i ≤ l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content='  Remark that in a given exceptional collection (E1, E2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content=' , El) one can replace any object Ei with its  shift Ei[t] for any integer t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content=' For instance, one can introduce an extra shift in the definitions of the right  and left mutations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content=' The first two defining conditions for the left and right dual collections will not change,  while the third condition will become slightly nicer:  Ext•(∨Ei, Ei) = Ext•(Ei, E∨  i ) = k  for any 1 ≤ i ≤ l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content='  This convention is often reasonable, yet even in the case of the projective space, Example 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content='4, the dual  collection will not consist of vector bundles while the original collection does.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content='  Meanwhile, the definitions we have just given also have a downside.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content=' Imagine that (E, F) is a fully  orthogonal pair in T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content=' Then LEF ≃ F[−1], while RFE ≃ E[1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content=' This is often inconvenient as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content=' We  propose the following lemma-definition, which is tailored to the case of an exceptional collection indexed  by a finite graded poset P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content=' The reader will immediately check that once the collection is linearly ordered,  the left dual differs from the graded left dual by shifts of objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content='  Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content=' Let ⟨Ex | x ∈ P⟩ be an exceptional collection indexed by a finite graded poset P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content=' For any  y ∈ P there exists a unique (up to isomorphism) object E◦  y ∈ ⟨Ex | x ∈ P⟩ such that  (1) Ext•(Ex, E◦  y) = 0 for all x ̸= y,  (2) Ext•(Ey, E◦  y) = k[−|y|].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content='  Moreover, the objects E◦  y form an exceptional collection with respect to the opposite poset P◦, called the  graded left dual, and ⟨E◦  y | x ∈ P◦⟩ = ⟨Ex | x ∈ P⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content='  Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content=' If the poset P is linearly ordered, then the definition of the graded left dual agrees with the  definition of the left dual.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content='  6  Example 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content=' In Example 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content='2 we have seen that Db(Gr(k, V )) admits a full exceptional collection indexed  by the poset Yk,n−k:  Db(Gr(k, V )) =  �  ΣλU∗ | λ ∈ Yk,n−k  �  ,  where U is the tautological rank k subbundle in V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content=' In the same paper Kapranov showed that its graded  left dual is given by  Db(Gr(k, V )) =  �  ΣλT U⊥ | λ ∈ Yk,n−k  �  ,  where U⊥ = (V/U)∗, and λT denotes the transpose diagram.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content='  We leave the definition of the graded right dual to the reader, indicating that for the right dual the  grading function should be taken for the opposite poset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content=' Graded spectral sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content=' As stated in the introduction, dual collections provide a particularly nice  computational tool.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content=' Let (E1, E2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content=' , El) be an exceptional collection in T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content=' Recall that a cohomological  functor from T to an abelian category A is a functor F : T → A which takes distinguished triangles to  exact sequences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content=' As usual, we denote by F i the composition F ◦ [i].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content='  Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content='8 ([5, Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content='3]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content=' Let G ∈ ⟨E1, E2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content=' , En⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content=' There is a spectral sequence with the  first page given by  (8)  Ep,q  1  =  �  i+j=q  Ext−i(G, E∨  p+1)∗ ⊗ F j(Ep+1)  converging to F p+q(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content='  We will be interested in the case when T = Db(A) is the bounded derived category of an abelian  category A (for instance, the bounded derived category of coherent sheaves on a smooth projective  variety), and F = H0 is the usual 0-th cohomology functor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content=' Assume that the exceptional collection  (E1, E2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content=' , El) consists of pure objects (for instance, of coherent sheaves).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content=' Then the spectral sequence (8)  simplifies to  (9)  Ep,q  1  = Ext−q(G, E∨  p+1)∗ ⊗ Ep+1 ⇒ Hp+q(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content='  In the case of an exceptional collection indexed by a graded poset P the spectral sequence (9) becomes  (10)  Ep,q  1  =  �  x∈P, |x|=p  Ext−q(G, E◦  x)∗ ⊗ Ex ⇒ Hp+q(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content=' Exceptional collections on Lagrangian Grassmannians.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content=' Let V be a 2n-dimensional vector  space over k equipped with a non-degenerate skew-symmetric bilinear form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content=' We denote by LGr(n, V ) the  Lagrangian Grassmannian of maximal isotropic subspaces in V , and by U the tautological rank n bundle  on LGr(n, V ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content=' The Lagrangian Grassmannian comes with an action of the symplectic group G = Sp2n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content='  We denote by P the set of weakly decreasing integer sequences of length n:  P = {λ ∈ Zn | λ1 ≥ λ2 ≥ · · · ≥ λn}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content='  Given λ ∈ P, we denote by Σλ the corresponding Schur functor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content=' We follow the convention under which  Σ(p,0,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content=',0) = Sp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content='  7  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content=' Exceptional blocks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content=' It is well known that every equivariant irreducible vector bundle on LGr(n, V )  is isomorphic to ΣλU∗ for some λ ∈ P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content=' Moreover, these form an infinite full exceptional collection in the  equivariant derived category:  Db  G(LGr(n, V )) =  �  ΣλU∗ | λ ∈ P◦�  ,  where P is treated as an infinite poset with the partial order given by  λ ⪯ µ  if and only if  λi ≤ µi for all i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content=' , n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content='  Any subset S ⊆ P with the induced partial order produces an exceptional collection ⟨ΣλU∗ | λ ∈ S◦⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content='  If S is finite and graded, we denote by  ⟨Eλ | λ ∈ S⟩  and  ⟨Fλ | λ ∈ S⟩  the graded right and left duals to ⟨ΣλU∗ | λ ∈ S◦⟩ respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content='  Kuznetsov and Polishchuk came up with a very simple3 condition under which the objects Eλ form an  exceptional collection in the non-equivariant category.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content='  Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content='9 (See [9, Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content='1]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content=' A subset S ⊂ P is called an exceptional block if for all λ, µ ∈ S  the canonical map  �  ν∈S  Ext•  G(ΣλU∗, ΣνU∗) ⊗ Hom(ΣνU∗, ΣµU∗) → Ext•(ΣλU∗, ΣµU∗)  is an isomorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content='  In plain words the block condition says that every extension between a pair of objects can be de-  composed as a sum of equivariant extensions followed by homomorphisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content=' What is rather surprising is  that even though the original objects ΣλU∗ for λ ∈ S almost never form an exceptional collection in the  non-equivariant category, the right dual do form an exceptional collection in the non-equivariant category  as long as S is a block.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content='  Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content='10 (See [9, Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content='9]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content=' If S ⊂ P is an exceptional block, then the corresponding  right dual objects form an exceptional collection in Db(LGr(n, V )),  �  Eλ | λ ∈ S  �  ⊂ Db(LGr(n, V )).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content='  It turns out that it is natural to call exceptional blocks right exceptional blocks, and that it is useful  to consider left exceptional blocks as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content='  Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content='11 (See [4, Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content='12]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content=' A subset S ⊂ P is called an left exceptional block if for all  λ, µ ∈ S the canonical map  Hom(ΣλU∗, ΣνU∗) ⊗  �  ν∈S  Ext•  G(ΣνU∗, ΣµU∗) → Ext•(ΣλU∗, ΣµU∗)  is an isomorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content='  Recall that both the equivariant and the non-equivariant categories have the duality functor, which  is an anti-autoequivalence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content=' Since passing to duals takes exceptional collections to exceptional collections  (with respect to the opposite order) and left and right (graded) dual collections to right and left dual  collections respectively, we immediately see that S is a right exceptional block if and only if −S is a left  exceptional block, where −S = {−λ | λ ∈ S} and −λ = (−λn, −λn−1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content=' , −λ1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content=' The latter follows from  the isomorphism (ΣλU∗)∗ ≃ Σ−λU∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content=' The dual statement to Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content='10 is the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content='  3Simple, yet hard to check.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content='  8  Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content='12 (See [9, Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content='9]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content=' If S ⊂ P is a left exceptional block, then the corresponding  left dual objects form an exceptional collection in Db(LGr(n, V )),  �  Fλ | λ ∈ S  �  ⊂ Db(LGr(n, V )).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content='  The following theorem uses the term semiorthogonal decomposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content='  Recall that full triangulated  subcategories T1, T2 ⊆ T are called semiorthogonal, one writes T1 ⊆ T ⊥  2 , if HomT (X2, X1) = 0 for all  X1 ∈ T1 and X2 ∈ T2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content=' A semiorthogonal decomposition of a category T is a collection of full triangulated  subcategories T1, T2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content=' , Tr such that Ti ⊆ T ⊥  j  for all 1 ≤ i < j ≤ r and T is the smallest strictly  full triangulated subcategory containing all Ti.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content='  The notation for a semiorthogonal decomposition is  T = ⟨T1, T2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content=' , Tr⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content=' Finally, for a full triangulated subcategory B ∈ Db(LGr(n, V )) we denote by B(i)  its image under the autoequivalence given by tensor product with O(i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content='  Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content='13 (See [9, Theorem 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content='2] and [4, Theorem 4]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content=' The set  Bh = {λ ∈ P | n − h ≥ λ1 ≥ λ2 ≥ · · · ≥ λh ≥ λh+1 = · · · = λn = 0}  is a right exceptional block for all h = 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content=' , n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content=' Moreover, there is semiorthogonal decomposition  (11)  Db(LGr(n, V )) = ⟨B0, B1(1), B2(2), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content=' , Bn(n)⟩ ,  where Bh = ⟨Eλ | λ ∈ Bh⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content='  Let us make a few remarks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content=' First, Bh = Yh,n−h, where Yh,w denotes the set of Young diagrams of  height at most h and width at most w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content=' Second, the object Eλ for λ ∈ Yh,n−h has the following homological  description:  Eλ ∈ ⟨ΣµU∗ | µ ⊆ λ⟩ ⊂ Db  G(LGr(n, V ))  and  Ext•  G(Eλ, ΣµU∗) =  �  k[−|λ|]  if µ = λ,  0  if µ ⊊ λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content='  In particular, it depends only on λ, one only needs to know that λ ∈ Yh,n−h for some 0 ≤ h ≤ n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content=' Finally,  by using the duality anti-autoequivalence we get a semiorthogonal decomposition  Db(LGr(n, V )) = ⟨Cn(−n), Cn−1(−n + 1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content=' , C1(−1), C0⟩ ,  where Ch = ⟨Fλ | λ ∈ −Bh⟩ and Fλ for λ ∈ Yh,n−h can be characterized by the following properties:  Fλ ∈ ⟨ΣµU | µ ⊆ λ⟩ ⊂ Db  G(LGr(n, V ))  and  Ext•  G(ΣµU, Fλ) =  �  k[−|λ|]  if µ = λ,  0  if µ ⊊ λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content=' Geometric constructions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content=' We will need one of the two geometric constructions for the objects Fλ  obtained in [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content=' Since the cases h = 0 and h = n are trivial (B0 and Bn consist of a single weight  λ = (0, 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content=' , 0), and Fλ = Eλ = O), let us fix 0 < h < n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content=' Consider the diagram  (12)  IFl(n − h, n;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content=' V )  LGr(n, V )  IGr(n − h, V )  p  q  and denote by W the universal rank n−h bundle on IGr(n−h, V ) as well as its pullback on IFl(n−h, n;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content=' V ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content='  Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content='14 ([4, Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content='4]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content=' The bundles ⟨ΣµT W∗ | µ ∈ Yh,w−h⟩ form an exceptional collection in  the non-equivariant derived category Db(IGr(n − h, V )).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content='  9  The exceptional collection from the previous lemma is indexed by a graded poset, so we can pass in  the non-equivariant category to its graded left dual, denoted by ⟨Gλ | λ ∈ Y◦  h,w−h⟩, see [4, Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content='5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content='  The graded dual exceptional collection conditions are  (13)  Gλ ∈ ⟨ΣµW∗ | µ ∈ Yn−h,h⟩  and  Ext•(ΣµW∗, Gλ) =  �  k[−|λ|]  if µ = λT ,  0  if µ ∈ Yn−h,h and µ ̸= λT .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content='  Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content='15 ([4, Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content='6]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content=' The object Fλ is isomorphic to p∗q∗Gλ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content=' Main results  We continue to use the notation introduced in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content='  Our first goal is to prove the duality  statement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content=' Dual exceptional collections on Lagrangian Grassmannians.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content=' The first main result of the  paper is the following theorem, where we identify the posets Yh,n−h and Yn−h,h via transposition of  diagrams.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content='  Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content=' Let 0 ≤ h ≤ n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content=' Then the exceptional collection ⟨Fµ | µ ∈ Y◦  n−h,h⟩ is the graded left dual  to ⟨Eλ | λ ∈ Yh,n−h⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content=' That is, for µ ∈ Yn−h,h one has  Fµ ∈ ⟨Eλ | λ ∈ Yh,n−h⟩  and  Ext•(Eλ, Fµ) =  �  k[−|λ|]  if µ = λT ,  0  if µ ∈ Yn−h,h and µ ̸= λT .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content='  The proof of the preceding theorem will take the rest of this section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content=' Our strategy is to compare the  objects Eλ with the graded right dual exceptional collection to ⟨Fµ | µ ∈ Y◦  n−h,h⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content=' Since the cases h = 0  and h = n are trivial (all the objects considered are isomorphic to O), we assume that 0 < h < n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content='  Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content=' Let ν ∈ Yh,n−h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content=' The following categories coincide:  ⟨Fµ | µ ⊆ νT ⟩ = ⟨Eλ | λ ⊆ ν⟩ = ⟨ΣλU∗ | λ ⊆ ν⟩,  where the latter is the smallest strictly full triangulated subcategory in Db(LGr(n, V )) containing the cor-  responding objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content=' Since, the objects Eλ form a graded right dual exceptional collection to ⟨ΣλU∗ | λ ⊆ ν⟩ in  Db  G(LGr(n, V )), the second equality holds in Db  G(LGr(n, V )).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content=' Once we apply the forgetful functor from  Db  G(LGr(n, V )) to Db(LGr(n, V )), we see that the same equality holds in the non-equivariant derived  category.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content='4 In a similar fashion one proves that  ⟨Fµ | µ ⊆ νT ⟩ = ⟨ΣµU | µ ⊆ νT ⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content='  It remains to show that ⟨ΣµU | µ ⊆ νT⟩ = ⟨ΣλU∗ | λ ⊆ ν⟩, which follows from the structure of Kapranov’s  exceptional collection on Gr(n, V ) and its dual, see [4, Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content='7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content='  □  Next, we need another very simple purely categorical statement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content='  Let T ′ = ⟨Wx | x ∈ P⟩ be a  triangulated category generated by a graded full exceptional collection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content=' Denote by ⟨Gx | x ∈ P◦⟩ its  graded left dual.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content=' Assume F : T ′ → T is an exact functor into another triangulated category T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content=' Put  Fx = F(Gx) for all x ∈ P◦ and assume that ⟨Fx | x ∈ P◦⟩ form a graded exceptional collection in T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content='  Denote by ⟨ ˜Ex | x ∈ P⟩ its graded right dual.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content='  Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content=' For all x, y ∈ P one has  Ext•  T ( ˜Ex, F(Wy)) ≃ Ext•  T ′(Wx, Wy).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content='  4See also the discussion preceding Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content='8 in [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content='  10  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content=' Since the category T ′ is saturated, the functor F has a left adjoint F ∗ (see [3]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content=' In particular,  Ext•  T ( ˜Ex, F(Wy)) ≃ Ext•  T ′(F ∗( ˜Ex), Wy).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content='  We claim that F ∗( ˜Ex) ≃ Wx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content=' Indeed, for any y ∈ P one has  Ext•  T ′(F ∗( ˜Ex), Gy) ≃ Ext•  T ( ˜Ex, Fy) =  �  k[−|x|]  if x = y,  0  otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content='  The latter is precisely the defining condition of the graded right dual to ⟨Gx | x ∈ P◦⟩, which is ⟨Wx |  x ∈ P⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content='  □  We apply the previous lemma in our situation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content=' Consider the isotropic Grassmannian IGr(h, V ) with  its tautological bundle W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content=' Fix ν ∈ Yh,n−h and consider the poset P = {λ | λ ⊆ ν}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content=' By [4, Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content='4]  the bundles ⟨ΣλW∗ | λ ∈ P⟩ form an exceptional collection in Db(IGr(h, V )).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content=' Since the pullback functor  q∗ : Db(IGr(h, V )) → Db(IFl(h, n;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content=' V )) is fully faithful, the bundles ⟨ΣλW∗ | λ ∈ P⟩ form an exceptional  collection in Db(IFl(h, n;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content=' V )), and its graded left dual coincides with the pullback of the graded left dual.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content='  Put T ′ = ⟨ΣλW∗ | λ ∈ P⟩ ⊆ Db(IFl(h, n;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content=' V )), Wλ = ΣλW∗, Gλ = q∗Gλ, T = Db(LGr(n, V )), F = p∗,  where p∗ is the restriction of the pushforward functor under the projection p : IFl(h, n;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content=' V )) → LGr(n, V ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content='  Denote by ⟨ ˜Eλ | λ ∈ P⟩ the graded right dual exceptional collection to ⟨Fµ | µ ⊆ νT ⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content='  Since  Fλ = F(Gλ), see Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content='15, we can apply Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content=' The conclusion is that Ext•( ˜Eλ, p∗ΣµW∗) ≃  Ext•(ΣλW∗, ΣµW∗).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content=' A simple Borel–Bott–Weil computation shows that p∗ΣµW∗ ≃ ΣµU∗ for µ ⊆ ν  (see [4, Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content='4]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content=' Finally, for µ ⊆ ν we get  (14)  Ext•( ˜Eν, ΣµU∗) =  �  k  if µ = ν,  0  if µ ⊊ ν.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content='  Let us recall that our goal is to prove that ˜Eν ≃ Eν.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content=' It was shown in [9, Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content='8] that for all  λ, µ ∈ P there is an isomorphism  Ext•(Eλ, ΣµU∗) ≃ Hom(ΣλU∗, ΣµU∗).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content='  It follows from the Lagrangian Borel–Bott–Weil theorem (see Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content='2) that Hom(ΣλU∗, ΣλU∗) ≃ k  and that Hom(ΣλU∗, ΣµU∗) = 0 if λ ̸⊆ µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content=' We conclude that  (15)  Ext•(Eν, ΣµU∗) =  �  k  if µ = ν,  0  if µ ⊊ ν.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content='  Proof of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content=' Let ν ∈ Yh,n−h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content=' By Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content='2 we know that ˜Eν, Eν ∈ ⟨ΣµU∗ | µ ⊆ ν⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content=' Choose a  nontrivial element φ ∈ Ext•(Eν, ΣνU∗) ≃ k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content=' It follows from the discussion preceding Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content='8 in [9]  that one has an exact triangle in Db(LGr(n, V )) of the form  Eν  φ−→ ΣνU∗ → Cφ → Eν[1],  where the cone Cφ is in ⟨ΣµU∗ | µ ⊊ ν⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content=' Applying the functor Hom( ˜Eν, −) to the previous triangle,  from (14) we get a morphism ψ : ˜Eν → T , which lifts a nontrivial ξ ∈ Hom( ˜Eν, ΣνU∗) ≃ k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content=' Consider the  cone of ψ:  ˜Eν  ψ−→ Eν → Cψ → ˜Eν[1],  On the one hand, Cψ ∈ ⟨ΣµU∗ | µ ⊆ ν⟩ since both ˜Eν, Eν belong to this subcategory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content=' On the other hand,  it follows from the construction of ψ and formulas (14) and (15) that Ext•(Cψ, ΣµE∗) = 0 for all µ ⊆ ν.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content='  Thus, Cψ = 0, and ψ is an isomorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content='  □  11  Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content='1 shows that the semiorthogonal decomposition (11) has a very nice symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content='  Namely, if one applies the duality anti-autoequivalence followed by the twist by O(n), the decomposition  will map to itself, and the h-th block of the resulting decomposition will be generated by the objects  ⟨(Eλ)∗ | λ ∈ Yn−h,h⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content=' Since (Eλ)∗ ≃ Fλ, we see that the exceptional collection generating block n − h  maps to the graded left dual of the exceptional collection generating block h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content='  Since we have a graded left dual exceptional collection, there is a spectral sequence associated to it;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content='  namely, spectral sequence (10) takes the following form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content='  Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content=' For any object G ∈  �  Eλ | λ ∈ Yh,n−h  �  there is a spectral sequence of the form  (16)  Ep,q  1  =  �  λ∈Yh,n−h, |λ|=p  Ext−q �  G, FλT �∗  ⊗ Eλ ⇒ Hp+q(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content='  In the following section we will use this spectral sequence to produce certain resolutions of natural  equivariant non-exceptional vector bundles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content=' Resolutions of irreducible equivariant bundles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content=' In the final section we produce nice resolutions  for irreducible equivariant bundles on LGr(n, V ) of the from ΣλU∗ for λ ∈ Yh,w for some h + w = n in  terms of bundles of the form Eµ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content=' Balanced diagrams.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content=' Let λ be a Young diagram.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content=' Most commonly it is represented by a sequence of  integers λ = (λ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content=' , λk) such that λ1 ≥ λ2 ≥ · · · ≥ λk ≥ 0, where λi is the length of the i-th row of λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content='  There is an alternative description via hook length.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content=' Let us say that λ has rank s5 if λs ≥ s and λs+1 ≤ s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content='  Graphically, s is the size of the largest square that fits into λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content=' There are exactly s boxes on the diagonal  of λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content=' Let ai and bi denote the number of boxes to the right of the i-th diagonal box (including itself) and  below it (including itself) respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content=' Classically, ai and bi are called the arm and the leg length.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content=' One  could alternatively say that ai = λi − (i − 1) and bi = max{1 ≤ j ≤ k | λj ≥ i} − (i − 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content=' We will write  λ = (a1, a2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content=' , as|b1, b2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content=' , bs) if λ has rank s and its arm and leg length are ai and bj respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content='  Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content=' A diagram λ = (a1, a2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content=' , as|b1, b2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content=' , bs) is called balanced if ai = bi + 1 for all  1 ≤ i ≤ s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content=' The set of balanced diagrams with 2t boxes is denoted by B2t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content='6  Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content='7 ([12, Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content='9]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content=' Let E be a locally free module over a commutative ring of  characteristic zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content=' Then  Λt �  S2E  �  ≃  �  λ∈B2t  ΣλE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content='  Remark that (a1, a2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content=' , as|b1, b2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content=' , bs)T = (b1, b2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content=' , bs|a1, a2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content=' , as).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content=' In particular λ is symmetric,  λ = λT , if and only if ai = bi for all 1 ≤ i ≤ s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content=' It now follows from Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content='6 that λ is balanced if  and only if µ = (λ1 − 1, λ2 − 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content=' , λs − 1, λs+1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content=' , λk−1, λk) is symmetric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content=' In such case the rank of µ  equals that of λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content='  Let us now assume that µ = (µ1, µ2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content=' , µk) is of rank s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content=' One can separate µ into three parts: a  square of size s, everything to the right of it, and everything below it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content=' The latter two are nothing by the  diagrams µr = (µ1 − s, µ2 − s, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content=' , µs − s) and µb = (µs+1, µs+2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content=' , µk).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content=' Under transposition the square  maps to itself, while µr and µb get interchanged and transposed: (µT )r = (µb)T and (µT )b = (µr)T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content=' One  immediately concludes that µ is symmetric if and only if (µr)T = µb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content='  Combining what is written in the two preceding paragraphs, we obtain the following characterization  of balanced diagrams.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content='  5One also says that λ has a Durfee square of size s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content='  6The number of boxes in a balanced diagram is always even.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content='  12  Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content=' A diagram λ = (λ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content=' , λk) of rank s is balanced if and only if  (λ1 − (s + 1), λ2 − (s + 1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content=' , λs − (s + 1))T = (λs+1, λs+2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content=' , λk),  where, in particular, we assume that λs ≥ s + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content=' Lagrangian Borel–Bott–Weil.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content=' The celebrated Borel–Bott–Weil theorem fully describes the coho-  mology of irreducible equivariant line bundles on rational homogeneous varieties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content=' Since we only use it in  one place, we formulate a much simplified version of it and refer the interested reader to [12, Chapter 4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content='  Given a weakly decreasing sequence λ ∈ Zn, we denote by −λ the sequence (−λn, −λn−1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content=' , λ1),  which is again weakly decreasing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content=' The sum of weakly decreasing sequences is defined termwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content=' A weakly  decreasing sequence is called non-singular if the absolute values of all of its terms are positive and distinct.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content='  If λ is non-singular, we denote by ∥λ∥ the sequence obtained by taking all the absolute values of the  terms of λ and writing them in decreasing order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content=' If λ is non-singular, then ∥λ∥−ρ is a weakly decreasing  sequence with non-negative terms, where ρ = (n, n − 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content=' , 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content=' The set of weakly decreasing sequences  µ ∈ Zn with non-negative terms is identified with the set of dominant weights of Sp(V ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content=' We denote by  V ⟨µ⟩ the corresponding irreducible representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content=' For instance, if µ = (0, 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content=' , 0), then V ⟨µ⟩ = k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content='  Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content='9 (Lagrangian Borel–Bott–Weil).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content=' Let λ ∈ Zn be a weakly decreasing sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content=' If −λ + ρ is  non-singular, then  H•(LGr(n, V ), ΣλU) = V ⟨∥−λ+ρ∥−ρ⟩[−ℓ],  where ℓ equals the number of negative terms in −λ + ρ plus the number of pairs 1 ≤ i < j ≤ n such that  (−λ + ρ)i + (−λ + ρ)j < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content=' Otherwise, H•(LGr(n, V ), ΣλU) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content=' Vanishing lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content=' We will need the following result, which is definitely well known to experts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content='  A proof is included for the sake of completeness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content='  Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content=' Let λ ∈ Yn,n+1 be a Young diagram.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content='  (1) If λ is not balanced, then H•(LGr(n, V ), ΣλU) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content='  (2) If λ is balanced and |λ| = 2t, then H•(LGr(n, V ), ΣλU) = k[−t].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content=' The proof is a direct application of the Borel–Bott–Weil theorem stated in the previous Section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content='  In order to show (1), we need to show that the weight −λ + ρ is non-singular if and only if λ is balanced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content='  Consider the strictly decreasing sequence  (17)  − λ + ρ = (n − λn, (n − 1) − λn−1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content=' , 2 − λ2, 1 − λ1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content='  Since 0 ≤ λi ≤ n + 1 for all i, all the absolute values of the terms of −λ + ρ are between 0 and n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content=' Recall  that −λ + ρ is non-singular if and only if all the absolute values of its terms are distinct and positive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content='  Assume the latter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content=' Let s denote the rank of λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content=' Then the first n − s terms of (17) are positive, while the  last s terms are non-positive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content=' Since the absolute values of the latter can not be zero (the weight being  non-singular), we conclude that λi ≥ s + 1 for i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content=' , s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content=' Put µi = λi − (s + 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content=' Then the sequence  (18)  (n − λn, (n − 1) − λn−1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content=' , (s + 1) − λs+1, s + µ1, (s − 1) + µ2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content=' , 1 + µs)  differs from (17) only in the last s terms: their signs have been changed, and their order has been reversed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content='  Remark that (λs+1, λs+2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content=' , λn) = λb ∈ Yn−s,s, while µ ∈ Ys,n−s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content=' The sequence (18) is very well known  to anyone who has ever studied Kapranov’s work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content=' It follows from [6, Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content='7] that all the terms in (18)  are distinct if and only if λb = µT .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content=' The latter is equivalent, by Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content='8, to saying that λ is balanced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content='  Let us turn to (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content=' Assume that λ is balanced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content=' Since the absolute values of the terms of −λ + ρ take  all the values between 1 and n, by the Borel–Bott–Weil theorem the only nonzero cohomology will be  equal to k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content=' We only need to determine the degree in which it sits, and the latter equals the number of  negative terms in −λ+ρ plus the number of pairs 1 ≤ i < j ≤ n such that (−λ+ρ)i +(−λ+ρ)j < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content=' The  13  first number equals s, and we need to compute the second number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content=' Remark that if 1 ≤ j ≤ n − s, then  both (−λ + ρ)i and (−λ + ρ)j are positive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content=' Thus, we may assume that n − s < j ≤ n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content=' If n − s < i ≤ n,  then any i < j ≤ n contributes to the count since both terms are negative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content=' There are s(s−1)  2  such pairs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content='  Finally, assume that 1 ≤ i ≤ n − s and n − s < j ≤ n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content=' Since (−λ + ρ)j < 0, we need to determine when  (−λ + ρ)i < −(−λ + ρ)j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content=' Such pairs correspond precisely to inversions in the sequence (18).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content=' It is easy  to show that the number of those equals |λb| (see [6, Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content='7]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content=' We conclude that for a balanced λ  the only nontrivial cohomology sits in degree s + s(s−1)  2  + |λb| = s(s+1)  2  + |λb|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content=' It remains to recall that λ  consists of λb, µ = (λb)T , and a square of size s×(s+1), so |λ| = |λb|+|µ|+s(s+1) = 2|λb|+s(s+1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content='  □  The following is a simple relative version of the previous lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content=' As usual, all the functors are derived.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content='  Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content=' Let X be smooth projective variety, and let V be a rank 2k symplectic bundle on X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content='  Consider the relative Lagrangian Grassmannian π : LGrX(n, V) → X, denote by U ⊂ π∗V the universal  bundle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content=' Let ν ∈ Yk,k+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content='  (1) If ν is not balanced, then π∗ΣλU = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content='  (2) If ν is balanced and |λ| = 2t, then π∗ΣλU = OX[−t].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content=' Skew Schur functors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content=' Before we formulate and prove the second main result of the paper, we need  to say a few words about skew diagrams.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content=' Let λ and µ be two Young diagrams such that µ ⊆ λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content=' Then  one can define the skew Schur functor Σλ/µ, see [12, Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content='1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content='  If E is a free module over a commutative ring of characteristic zero and α and β are two Young  diagrams, one has a direct sum decomposition  ΣαE ⊗ ΣβE ≃  �  |κ|=|α|+|β|  (ΣκE)⊕c(α,β;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content='κ) ,  where c(α, β;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content=' κ) are the celebrated Littlewood–Richardson coefficients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content=' It turns out that one has a direct  sum decomposition for skew Schur functors controlled by the same coefficients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content=' Namely,  (19)  Σλ/µE ≃  �  |ν|+|µ|=|λ|  (ΣνE)⊕c(ν,µ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content='λ) ,  see [12, Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content='6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content='  If c(ν, µ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content=' λ) > 0, we will write ν ⊆ λ/µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content=' Remark that ν ⊆ λ/µ implies that ν ⊆ λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content=' In particular,  if λ ∈ Yh,w, then ν ⊆ λ/µ implies ν ∈ Yh,w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content=' Inspired by decomposition (19), for a pair of diagrams  λ, µ ∈ Yh,w such that h + w = n and µ ⊆ λ, we put  (20)  Eλ/µ =  �  ν⊆λ/µ  (Eν)⊕c(ν,µ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content='λ) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content='  Extend the definition, as usual, by putting Eλ/µ = 0 for µ ⊈ λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content=' Resolutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content=' We are ready to present the main application of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content='  Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content=' Let λ ∈ Yh,n−h for some 0 ≤ h ≤ n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content=' There is an exact sequence of vector bundles on  LGr(n, V ) of the form  (21)  0 →  �  µ∈Bh(h+1)  Eλ/µ → · · · →  �  µt∈B2t  Eλ/µt → · · · →  �  µ2∈B4  Eλ/µ2 →  �  µ1∈B2  Eλ/µ1 → Eλ → ΣλU∗ → 0,  where B2t denotes the set of balanced diagrams with 2t boxes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content='  14  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content=' Since the vector bundle ΣλU∗ lies in the subcategory ⟨Eλ | λ ∈ Yh,n−h⟩, spectral sequence (16) is  applicable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content=' Precisely, one has  (22)  Ep,q  1  =  �  µ∈Yh,n−h, |α|=p  Ext−q �  ΣλU∗, FαT �∗  ⊗ Eα ⇒ Hp+q(ΣλU∗).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content='  Let us compute Ext•(ΣλU∗, Fµ) for µ = αT ∈ Yn−h,h using the isomorphism Fµ ≃ p∗q∗Gµ given in  Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content='15, where p and q come from the diagram  IFl(h, n;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content=' V )  LGr(n, V )  IGr(h, V )  p  q  (This is diagram (12) with h and n − h interchanged since µ ∈ Yn−h,h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content=') One has a sequence of isomor-  phisms  Ext•(ΣλU∗, Fµ) ≃ Ext•(ΣλU∗, p∗q∗Gµ)  ≃ Ext•(ΣλU∗, q∗Gµ)  ≃ H•(IFl(h, n;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content=' V ), ΣλU ⊗ q∗Gµ)  ≃ H•(IGr(h;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content=' V ), q∗(ΣλU) ⊗ Gµ),  where the second isomorphism is given by adjunction, and the last one follows from the projection formula.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content='  As usual, all the functors are derived.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content='  Consider the spectral sequence associated with the composition of derived functors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content=' Its second page  is given by  (23)  Hi(IGr(n − h;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content=' V ), Rjq∗(ΣλU) ⊗ Gµ)  ⇒  Hi+j(IGr(n − h;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content=' V ), q∗ΣλU ⊗ Gµ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content='  Let us compute Rjq∗(ΣλU).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content=' Recall that we denoted by W ⊂ U the universal flag on IFl(h, n;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content=' V ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content=' There  is a filtration on ΣλU associated with the short exact sequence 0 → W → U → U/W whose associated  quotients are isomorphic to  (24)  �  ν⊆λ,|ν|=k  Σλ/νW ⊗ Σν(U/W).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content='  Since for any ν ⊆ λ the bundle Σλ/νW is pulled back from IGr(h, V ), we conclude by the projection  formula that Rjq∗(Σλ/νW ⊗ Σν(U/W)) ≃ Σλ/νW ⊗ Rjq∗Σν(U/W).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content='  In order to compute Σλ/νW ⊗ Rjq∗Σν(U/W), recall that IFl(h, n;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content=' V ) is the relative Lagrangian Grass-  mannian LGr(n−h, W⊥/W) over IGr(h, V ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content=' Since ν ⊆ λ, one has ν ∈ Yh,n−h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content=' If the height of ν is greater  than n − h, then Σν(U/W) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content=' Thus, Σλ/νW ⊗ Rjq∗Σν(U/W) = 0 unless ν ∈ Yn−h,n−h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content=' Finally, if  ν ∈ Yn−h,n−h, then we are in the situation when we can apply Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content=' We conclude that if ν ⊆ λ,  then  (25)  Σλ/νW ⊗ R•q∗Σν(U/W) ≃  �  Σλ/νW[−t]  if ν ∈ B2t,  0  otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content='  Using (25), we can compute Rj(ΣλU).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content=' Indeed, the spectral sequence associated with the filtration  with the quotients (24) degenerates in the first page, and  (26)  Rjq∗(ΣλU) ≃  �  ν⊆λ,ν∈B2j  Σλ/νW.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content='  15  We are ready to get back to the spectral sequence (23).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content='  Hi(IGr(h;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content=' V ), Rjq∗(ΣλU) ⊗ Gµ) ≃  �  ν⊆λ,ν∈B2j  Hi(IGr(h;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content=' V ), Σλ/νW ⊗ Gµ)  ≃  �  ν⊆λ,ν∈B2j  Exti(Σλ/νW∗, Gµ)  ≃  �  ν⊆λ,ν∈B2j  �  κ⊆λ/ν  Exti((ΣκW∗)⊕c(κ,ν;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content='λ), Gµ),  where the last isomorphism comes from (19).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content=' Since κ ⊆ λ/ν implies that κ ⊆ λ, using (13) we see that  the last term in (26) is zero unless κ = µT and i = |µ|, and is isomorphic to k otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content=' Finally,  H|µ|(IGr(h;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content=' V ), Rjq∗(ΣλU) ⊗ Gµ) ≃  �  ν∈B2j  k⊕c(ν,µ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content='λ),  where 2j = |ν| = |λ| − |µ|, are the only potentially nontrivial cohomology groups, and the spectral  sequence (23) degenerates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content=' We conclude that  H|µ|+j(IGr(h;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content=' V ), q∗ΣλU ⊗ Gµ) =  �  ν∈B2j  k⊕c(ν,µ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content='λ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content='  Let us return to spectral sequence (22).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content=' From the computations above we know that the only nontrivial  terms in it are  (27)  E−|λ|+t,|λ|−2t  1  ≃  �  ν∈B2t  �  µ⊆λ/ν  |µ|=|λ|−2t  (Eµ)⊕c(ν,µ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content='λ) =  �  ν∈B2t  Eλ/ν,  where the last equality comes from the definition in (20).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content=' We conclude that the spectral sequence contains  at most one non-trivial term in each diagonal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content='  As the spectral sequence converges to H•(Σλ), one must have a long exact sequence of the form  · · → E−|λ|+t,|λ|−2t  1  → · · · → E−|λ|+2,|λ|−4  1  → E−|λ|+1,|λ|−2  1  → E−|λ|,|λ|  1  → ΣλU∗ → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content='  Since any balanced diagram contained in λ has height at most h, its width is at most h + 1 and  E−|λ|+t,|λ|−2t  1  = 0 for t > h(h+1)/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content=' It remains to use (27) to get the desired long exact sequence (21).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content='  □  Example 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content='13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content=' We present some examples of resolutions introduced in Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content='  h = 1: In this case λ consists of just a single row of length at most n − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content=' If λ = (0), then Eλ ≃ O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content=' If  λ = (1), then Eλ ≃ ΣλU∗ = U∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content=' The interesting case is λ = (p) for some 2 ≤ p ≤ n − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content=' Since (2) is the  only nontrivial balanced diagram of height one, and (p)/(2) = (p − 2), resolution (21) takes form  0 → E(p−2) → E(p) → SpU∗ → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content='  As a consequence, we see that E(p) has a filtration with the associated graded quotients of the form  Sp−2tU∗ for all 0 ≤ p ≤ p/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content='  w = 1 : In this case λ consists of a column of length at most n − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content=' Since there are no nontrivial  balanced diagrams of width 1, we conclude that for λ = (t)T there is an isomorphism Eλ ≃ ΛtU∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content='  λ = (3, 1): This is the first case when resolution (21) has length greater than 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content=' Remark that (2) ∈ B2  and (3, 1) ∈ B4 are the only balanced diagrams contained in λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content=' Moreover, the only nontrivial Littlewood–  Richardson coefficients for (3, 1)/(2) are c ((2), (2);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content=' λ) = c ((1, 1), (2);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content=' λ) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content=' Thus, one has a resolution  of the form  0 → O → E(2) ⊕ E(1,1) → E(3,1) → Σ(3,1)U∗ → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content='  Since E(1,1) ≃ Λ2U∗ and E(2) fits into a short exact sequence 0 → O → E(2) → S2U∗ → 0, with a little  bit of work one can show that Eλ is an extension of Σ(3,1)U∗ by S2U∗ and Λ2U∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content='  16  References  [1]  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content=' Beilinson.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content=' “Coherent sheaves on P n and problems of linear algebra”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content=' In: Functional Analysis  and Its Applications 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content='3 (July 1978), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content=' 214–216.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
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+page_content=' In: Izvestiya  Rossiiskoi Akademii Nauk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
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+page_content=' doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
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+page_content=' “Helix theory”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
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+page_content=' Seriya Matematicheskaya 48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
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+page_content=' “On the derived categories of coherent sheaves on some homogeneous spaces”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content=' In:  Inventiones Mathematicae 92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
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+page_content=' issn: 0020-9910.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content=' doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content='1007/BF01393744.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content='  [8]  Alexander Kuznetsov.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content=' “Exceptional collections for Grassmannians of isotropic lines”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content=' In: Proceedings  of the London Mathematical Society.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content=' Third Series 97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
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+page_content=' issn: 0024-6115.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content=' doi:  10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content='1112/plms/pdm056.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content='  [9]  Alexander Kuznetsov and Alexander Polishchuk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content=' “Exceptional collections on isotropic Grassman-  nians”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content=' In: Journal of the European Mathematical Society (JEMS) 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content='3 (2016), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content=' 507–574.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content=' issn:  1435-9855.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content=' doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content='4171/JEMS/596.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content='  [10]  Alexander Kuznetsov and Maxim Smirnov.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content=' “Residual categories for (co)adjoint Grassmannians in  classical types”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content=' In: Compositio Mathematica 157.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content='6 (2021), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content=' 1172–1206.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content=' issn: 0010-437X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content=' doi:  10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content='1112/s0010437x21007090.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content='  [11]  Kyoung-Seog Lee and Kyeong-Dong Park.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content=' “Moduli spaces of Ulrich bundles on the Fano 3-fold V5”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content='  In: Journal of Algebra 574 (2021), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content=' 262–277.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content=' issn: 0021-8693.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content=' doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content='1016/j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content='jalgebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content='2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content='01.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content='015.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content='  [12]  Jerzy Weyman.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content=' Cohomology of vector bundles and syzygies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content=' Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
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+page_content=' Cambridge Tracts in Math-  ematics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content=' Cambridge University Press, Cambridge, 2003, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content=' xiv+371.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content=' isbn: 0-521-62197-6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content=' doi:  10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content='1017/CBO9780511546556.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content='  Algebraic Geometry Section, Steklov Mathematical Institute of Russian Academy of Sciences, 8  Gubkin str.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content=', Moscow 119991 Russia  Email address: avfonarev@mi-ras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
+page_content='ru  17' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E1T4oBgHgl3EQfIgPL/content/2301.02941v1.pdf'}
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+Heavy-flavour production at the LHC
+Jaime Norman 𝑎,1,∗
+𝑎University of Liverpool,
+Oliver Lodge Laboratory, Oxford St, Liverpool, L69 7ZE, UK
+E-mail: jaime.norman@cern.ch
+Heavy flavour production measurements in pp collisions are a crucial test of QCD. The LHC
+experiments ALICE, ATLAS, CMS and LHCb, provide complementary abilities to measure many
+aspects of heavy-flavour production. This contribution summarises recent LHC measurements
+within this topic.
+The Tenth Annual Conference on Large Hadron Collider Physics - LHCP2022
+16-20 May 2022
+online
+1on behalf of the ALICE, ATLAS, CMS and LHCb collaborations
+∗Speaker
+© Copyright owned by the author(s) under the terms of the Creative Commons
+Attribution-NonCommercial-NoDerivatives 4.0 International License (CC BY-NC-ND 4.0).
+https://pos.sissa.it/
+arXiv:2301.00859v1  [hep-ex]  2 Jan 2023
+
+Heavy-flavour production at the LHC
+Jaime Norman
+Heavy-flavour (charm or beauty) hadron production in proton-proton collisions can be de-
+scribed with the factorisation approach as a convolution of the heavy quark partonic cross section,
+the parton distribution functions (PDFs) of the proton, and the fragmentation function (FF) of
+the heavy-flavour quark into a given hadron. The hard parton cross sections are calculated with
+perturbative QCD techniques, where the large mass of the heavy quark sets the hard scale meaning
+production can be calculated over all 𝑝T. The PDFs and FFs on the other hand must be determined
+through measurements. At the LHC, the quark and gluon densities in the proton means multiple
+hard-parton scattering becomes more relevant, and the underlying event activity becomes much
+larger, which may affect the fragmentation and hadronisation of quarks into hadrons. Measure-
+ments of heavy-flavour hadrons in general constitute some of the most important tests of QCD over
+many length scales at hadron colliders.
+Heavy-flavour meson and quarkonia production has been studied extensively at the LHC.
+Measurements of the prompt [1, 2] and non-prompt (from beauty decays) [3] production of D
+mesons agree well with QCD calculations [4, 5], indicating that heavy-flavour meson production
+is well understood. The measurement of strange to non-strange beauty meson production has also
+been measured from non-prompt D mesons with ALICE [3], and a combined analysis of B meson
+decays from LHCb [6]. The LHCb measurement hints at a slight increase in the production ratio
+of strange to non-strange beauty meson with collision energy. This is also a crucial measurement
+for reducing the uncertainty in many B0
+s branching fractions, which are a dominant source of
+uncertainty in many searches for new physics. Prompt 𝐽/𝜓 production has been measured by
+ALICE [7], ATLAS [8], CMS [9] and LHCb [10], and non-prompt 𝐽/𝜓 production is also measured
+by ALICE [7], ATLAS [8] and CMS [9].
+Comprehensive studies of heavy-flavour baryon production have recently been performed
+at the LHC, and currently heavy-flavour baryon production is less well understood than heavy-
+flavour meson production. Λ+
+c production has been measured in pp collisions by ALICE [11–13]
+and CMS [14], and is significantly underestimated by QCD calculations. The baryon-to-meson
+production ratio Λ+
+c/D0 is significantly larger than the same measurements in e+e− and ep collisions,
+by up to a factor of 5 at low 𝑝T, and exhibits a strong 𝑝T dependence. Recently the baryon-to-meson
+ratios Ξ0,+
+c /D0 [15, 16], Σ++,+
+c
+/D0 [11] and Ω0
+c/D0 [17] have also been measured by ALICE to
+be significantly underestimated by predictions utilising fragmentation parameterisations based on
+measurements in e+e− and ep collisions. The fragmentation fractions of charm quarks into charmed
+hadrons have been measured for the first time in pp collisions [18], which are shown in figure 1.
+The charmed hadron fragmentation fractions differ significantly from measurements in e+e− and
+ep collisions, indicating the assumption of universal, independent parton-to-hadron fragmentation
+across collision systems is not sufficient to describe charmed hadron production in pp collisions at
+the LHC. Possible explanations of this difference include colour reconnection between independent
+partons [19], quark coalescence [20, 21], or enhanced baryon production originating from the decay
+of as-yet-undiscovered charm baryon states [22], which describe baryon-to-meson ratios better,
+though still underpredict the Ω0
+c/D0 and Ξ0,+
+c /D0 ratios. The relative production of beauty baryons
+and mesons has also been measured by LHCb [23], where a similar enhancement of the ratio
+Λ0
+b/(B0 + B+) is measured at low 𝑝T. A measurement of the ratio of non-prompt Λ+
+c baryons and
+D0 mesons was made as a function of 𝑝T by ALICE which aims to indirectly probe beauty quark
+fragmentation. This measurement was compared to predictions utilising quark production with
+2
+
+Heavy-flavour production at the LHC
+Jaime Norman
+FONLL, and hadronisation/hadron decays with PYTHIA - the predictions require the fragmentation
+fractions measured by LHCb to describe the data, and the measurement is underestimated if
+utilising fragmentation fractions measured in e+e− collisions. These measurements also suggest
+that fragmentation fractions of beauty baryons are not universal in different collision systems.
+The production cross section measurements of different charmed hadrons has allowed for precise
+measurement of the total 𝑐 ¯𝑐 [18] and 𝑏 ¯𝑏 [7] production cross sections.
+Heavy-flavour hadron production has been studied as a function of the event multiplicity. The
+strange to non-strange production ratio of beauty mesons B0
+s/B0 measured by LHCb in pp col-
+lisions [24] is shown in figure 1 (right). When measuring the multiplicity in the same rapidity
+region as the beauty hadron (with the VELO tracker) there is evidence at a level of 3.4𝜎 that the
+ratio B0
+s/B0 increases with multiplicity. When measuring the multiplicity in the opposite rapidity
+direction as the beauty hadron, the ratio is instead independent of this multiplicity, indicating the
+enhancement is due to the local event multiplicity. The charmed baryon-to-meson ratio measured
+by ALICE [25] also displays a significant enhancement in the region 2 < 𝑝T < 12 GeV/𝑐 at
+high multiplicity compared to low multiplicity. This multiplicity dependence is described well by
+PYTHIA when including colour reconnection mechanisms beyond the leading colour approxima-
+tion. The charmed strange to non-strange ration D+
+s /D0 is instead independent of multiplicity within
+the current experimental uncertainties [25].
+0
+D
++
+D
+s
++
+D
+c
++
+Λ
+c
+0
+Ξ
++
+D*
+0.2
+0.4
+0.6
+0.8
+1.0
+)
+c
+ H
+→
+(c 
+f
+ = 5.02 TeV
+s
+ALICE, pp, 
+ = 10.5 GeV
+s
+, 
+−
+e
++
+B factories, e
+Z
+m
+ = 
+s
+, 
+−
+e
++
+LEP, e
+HERA, ep, DIS
+HERA, ep, PHP
+ALI-PUB-500750
+Figure 1: Left: Charm quark fragmentation fractions into charm hadrons in pp collisions at √𝑠 = 5.02 TeV,
+compared with measurements at LEP and B factories, and ep collisions [18]. Right: The production ratios
+of strange to non-strange B mesons B0
+s/B0 as a function of the event multiplicity measured either in the
+direction of the beauty meson (left) or in the opposite direction (right) [24].
+Fragmentation dynamics of heavy quarks are probed with measurements of heavy flavour
+hadrons within jets. Figure 2 (left) shows a measurement of B± mesons within jets by ATLAS [26]
+- in particular, the relative momentum of the jet that is carried by the hadron in the direction of the
+jet, 𝑧 = �𝑝𝐵,𝐷 · �𝑝𝑗
+| �𝑝𝑗 |
+. The transverse momentum profile is also reported in [26]. ALICE measured the
+𝑧 distribution of D mesons [27] at √𝑠 = 5.02 TeV and √𝑠 = 13 TeV. Comparisons to Monte Carlo
+event generators were made in all cases, which helps constrain different approaches to fragmentation
+in these generators.
+The first observation of 𝑏-hadron production asymmetry (i.e., asymmetrical production of a
+3
+
+0.5
+0.5
+LHCb
+pp Vs = 13 TeV
+LHCb
+pp Vs = 13 TeV-
+0 < p_<20 GeV/c
+5.4 fb-1
+0 <p_<20 GeV/c
+5.4 fb-1
+B
+B
+b
+0.4
+b
+0.4
+0.3
+0.3
+0.2
+1 Pp→BB+X
+0.2
+1 Pp→BB+X
+Il +e-→Z°→BB
+I e+e-→Z°→BB
+a)
+ie+e-→Y(5S)→BB
+b)
+{ete-→Y(5S)→BB
+0.1
+0.1
+0
+2
+4
+6
+0
+2
+4
+6
+tback
+tracks
+tracks
+NoBias
+tracks
+NoBiasHeavy-flavour production at the LHC
+Jaime Norman
+beauty hadron compared to its anti-particle, measured using 𝐴𝑝𝑟𝑜𝑑 =
+𝜎( 𝑝𝑝→Λ0
+𝑏𝑌 )−𝜎( 𝑝𝑝→ ¯
+Λ0
+𝑏𝑌 )
+𝜎( 𝑝𝑝→Λ0
+𝑏𝑌 )+𝜎( 𝑝𝑝→ ¯
+Λ0
+𝑏𝑌 ) )
+was reported by LHCb [28]. Figure 2 shows 𝐴𝑝𝑟𝑜𝑑 as a function of Λ0
+b rapidity at √𝑠 = 7 TeV.
+Combining the measurement at √𝑠 = 7 TeV and 8 TeV, the results are incompatible with symmetric
+production with a significance of 5.8 standard deviations, assuming no CP violation in the decay.
+There is also evidence at the level of 4𝜎 of an increase in production asymmetry with rapidity.
+Comparisons from MC generators show that the results agree with PYTHIA when an improved
+colour reconnection model is included.
+Figure 2: Left: The distribution of the longitudinal profile 𝑧 of B± mesons in jets [26]. Right: the production
+asymmetry of Λ0
+b baryons [28].
+CMS reported the observation of simultaneous production of three 𝐽/𝜓 meson particles [29].
+In this analysis, five events are found to be consistent with triple-𝐽/𝜓 production, with a statistical
+significance relative to the background-only expectation of 5 standard deviations. The measured
+cross section is 𝜎(pp → 𝐽/𝜓𝐽/𝜓𝐽/𝜓𝑋) = 272+141
+−104(stat.) ± 17(syst.) fb. This cross section is
+consistent with theoretical expectation of production via primarily double-parton scattering and
+triple-parton scattering (under the simplest assumption of factorization of multiple hard-scattering
+probabilities in terms of SPS cross sections). This measurement is the first observation of the
+simultaneous production of three heavy particles, and represents an important step in the quest to
+constrain the proton PDF.
+In summary, heavy-flavour production at the LHC constitutes a stringent test of QCD, and Run
+2 of the LHC has allowed for a diverse range of heavy-flavour production measurements. Run 3
+has just begun after significant detector upgrade programmes for all 4 LHC experiments during
+Long Shutdown 2, which will provide unprecedented accuracy for future heavy-flavour production
+measurements.
+4
+
+(1/o) do / dz
+ATLAS/s=13TeV,139fb1
+B→ J/u Kt
+70 GeV<p
+<100GeV
+anti-k, jets R = 0.4
+4
+Data (stat.  sys.)
+P8 (Monash + Peterson)
+P8 (Monash + L-B)
+△ P8 (A14 + L-B)
+3
+ P8 (A14-rb + L-B)
+Herwig 7 (ang. ord.)
+ Herwig 7 (dipole)
+2
+Sherpa (Lund)
+Sherpa (cluster)
+MC/Data
+Total uncertainty
+0.5
+MC /Data
+0.3
+0.4
+0.5
+0.6
+0.7
+0.8
+0.9
+z16
+LHCb
+prod
+vs= 7 TeV
+14
+- Data 1fb-1
+A
+12
+Pythia8 (CR1)
+10
+Pythia8 (CR2)
+8
+Pythia8 (Monash)
+6
+4
+2
+0
+2
+2.5
+3
+3.5
+4Heavy-flavour production at the LHC
+Jaime Norman
+References
+[1] LHCb collaboration, Measurements of prompt charm production cross-sections in 𝑝𝑝
+collisions at √𝑠 = 13 TeV, JHEP 03 (2016) 159 [1510.01707].
+[2] CMS collaboration, Measurement of prompt open-charm production cross sections in
+proton-proton collisions at √𝑠 = 13 TeV, JHEP 11 (2021) 225 [2107.01476].
+[3] ALICE collaboration, Measurement of beauty and charm production in pp collisions at √𝑠 =
+5.02 TeV via non-prompt and prompt D mesons, JHEP 05 (2021) 220 [2102.13601].
+[4] B.A. Kniehl, G. Kramer, I. Schienbein and H. Spiesberger, Inclusive Charmed-Meson
+Production at the CERN LHC, Eur. Phys. J. C 72 (2012) 2082 [1202.0439].
+[5] M. Cacciari, M. Greco and P. Nason, The P(T) spectrum in heavy flavor hadroproduction,
+JHEP 05 (1998) 007 [hep-ph/9803400].
+[6] LHCb collaboration, Precise measurement of the 𝑓𝑠/ 𝑓𝑑 ratio of fragmentation fractions and
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+[7] ALICE collaboration, Prompt and non-prompt J/𝜓 production cross sections at midrapidity
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+[8] ATLAS collaboration, Measurement of the production cross-section of 𝐽/𝜓 and 𝜓(2S)
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+detector, Tech. Rep. ATLAS-CONF-2019-047, CERN, Geneva (2019).
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+√𝑠 = 5 TeV, JHEP 11 (2021) 181 [2109.00220].
+[11] ALICE collaboration, Measurement of Prompt D0, Λ+
+𝑐, and Σ0,++
+𝑐
+(2455) Production in
+Proton–Proton Collisions at √𝑠 = 13 TeV, Phys. Rev. Lett. 128 (2022) 012001
+[2106.08278].
+[12] ALICE collaboration, Λ+
+𝑐 Production and Baryon-to-Meson Ratios in pp and p-Pb Collisions
+at √𝑠𝑁 𝑁 =5.02 TeV at the LHC, Phys. Rev. Lett. 127 (2021) 202301 [2011.06078].
+[13] ALICE collaboration, Λ+
+𝑐 production in 𝑝𝑝 and in 𝑝-Pb collisions at √𝑠𝑁 𝑁 =5.02 TeV, Phys.
+Rev. C 104 (2021) 054905 [2011.06079].
+[14] CMS collaboration, Production of Λ+
+c baryons in proton-proton and lead-lead collisions at
+√𝑠NN = 5.02 TeV, Phys. Lett. B 803 (2020) 135328 [1906.03322].
+[15] ALICE collaboration, Measurement of the production cross section of prompt Ξ0
+c baryons at
+midrapidity in pp collisions at √𝑠 = 5.02 TeV, JHEP 10 (2021) 159 [2105.05616].
+5
+
+Heavy-flavour production at the LHC
+Jaime Norman
+[16] ALICE collaboration, Measurement of the Cross Sections of Ξ0
+𝑐 and Ξ+
+𝑐 Baryons and of the
+Branching-Fraction Ratio BR(Ξ0
+𝑐 → Ξ−𝑒+𝜈𝑒)/BR(Ξ0
+𝑐 → Ξ−𝜋+) in pp collisions at 13 TeV,
+Phys. Rev. Lett. 127 (2021) 272001 [2105.05187].
+[17] ALICE collaboration, First measurement of Ω0
+c production in pp collisions at √𝑠 = 13 TeV,
+Submitted to Phys. Lett. B [2205.13993].
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+midrapidity in pp collisions at the LHC, Phys. Rev. D 105 (2022) L011103 [2105.06335].
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+(2015) 003 [1505.01681].
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+a coalescence plus fragmentation approach, Phys. Lett. B 821 (2021) 136622 [2012.12001].
+[21] J. Song, H.-h. Li and F.-l. Shao, New feature of low 𝑝𝑇 charm quark hadronization in 𝑝𝑝
+collisions at √𝑠 = 7 TeV, Eur. Phys. J. C 78 (2018) 344 [1801.09402].
+[22] M. He and R. Rapp, Charm-Baryon Production in Proton-Proton Collisions, Phys. Lett. B
+795 (2019) 117 [1902.08889].
+[23] LHCb collaboration, Measurement of 𝑏 hadron fractions in 13 TeV 𝑝𝑝 collisions, Phys. Rev.
+D 100 (2019) 031102 [1902.06794].
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+𝑝𝑝 collisions at √𝑠 = 13 TeV, Submitted to Phys. Rev. Lett. [2204.13042].
+[25] ALICE collaboration, Observation of a multiplicity dependence in the 𝑝T-differential charm
+baryon-to-meson ratios in proton-proton collisions at √𝑠 = 13 TeV, Phys. Lett. B 829 (2022)
+137065 [2111.11948].
+[26] ATLAS collaboration, Measurement of 𝑏-quark fragmentation properties in jets using the
+decay 𝐵± → 𝐽/𝜓𝐾± in 𝑝𝑝 collisions at √𝑠 = 13 TeV with the ATLAS detector, JHEP 12
+(2021) 131 [2108.11650].
+[27] ALICE collaboration, Measurement of the production of charm jets tagged with D0 mesons
+in pp collisions at √𝑠 = 5.02 and 13 TeV, Submitted to JHEP [2204.10167].
+[28] LHCb collaboration, Observation of a Λ0
+𝑏 − Λ
+0
+𝑏 production asymmetry in proton-proton
+collisions at √𝑠 = 7 and 8 TeV, JHEP 10 (2021) 060 [2107.09593].
+[29] CMS collaboration, Observation of triple J/𝜓 meson production in proton-proton collisions
+at √𝑠 = 13 TeV, Submitted to Nature Physics [2111.05370].
+6
+
diff --git a/UdAyT4oBgHgl3EQf8fre/content/tmp_files/load_file.txt b/UdAyT4oBgHgl3EQf8fre/content/tmp_files/load_file.txt
new file mode 100644
index 0000000000000000000000000000000000000000..0ea3b72b61d803285772e615956f11ef55643d33
--- /dev/null
+++ b/UdAyT4oBgHgl3EQf8fre/content/tmp_files/load_file.txt
@@ -0,0 +1,233 @@
+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQf8fre/content/2301.00859v1.pdf,len=232
+page_content='Heavy-flavour production at the LHC  Jaime Norman 𝑎,1,∗  𝑎University of Liverpool,  Oliver Lodge Laboratory, Oxford St, Liverpool, L69 7ZE, UK  E-mail: jaime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQf8fre/content/2301.00859v1.pdf'}
+page_content='norman@cern.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQf8fre/content/2301.00859v1.pdf'}
+page_content='ch  Heavy flavour production measurements in pp collisions are a crucial test of QCD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQf8fre/content/2301.00859v1.pdf'}
+page_content=' The LHC  experiments ALICE, ATLAS, CMS and LHCb, provide complementary abilities to measure many  aspects of heavy-flavour production.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQf8fre/content/2301.00859v1.pdf'}
+page_content=' This contribution summarises recent LHC measurements  within this topic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQf8fre/content/2301.00859v1.pdf'}
+page_content='  The Tenth Annual Conference on Large Hadron Collider Physics - LHCP2022  16-20 May 2022  online  1on behalf of the ALICE, ATLAS, CMS and LHCb collaborations  ∗Speaker  © Copyright owned by the author(s) under the terms of the Creative Commons  Attribution-NonCommercial-NoDerivatives 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQf8fre/content/2301.00859v1.pdf'}
+page_content='0 International License (CC BY-NC-ND 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQf8fre/content/2301.00859v1.pdf'}
+page_content='0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQf8fre/content/2301.00859v1.pdf'}
+page_content='  https://pos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQf8fre/content/2301.00859v1.pdf'}
+page_content='sissa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQf8fre/content/2301.00859v1.pdf'}
+page_content='it/  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQf8fre/content/2301.00859v1.pdf'}
+page_content='00859v1  [hep-ex]  2 Jan 2023  Heavy-flavour production at the LHC  Jaime Norman  Heavy-flavour (charm or beauty) hadron production in proton-proton collisions can be de-  scribed with the factorisation approach as a convolution of the heavy quark partonic cross section,  the parton distribution functions (PDFs) of the proton, and the fragmentation function (FF) of  the heavy-flavour quark into a given hadron.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQf8fre/content/2301.00859v1.pdf'}
+page_content=' The hard parton cross sections are calculated with  perturbative QCD techniques, where the large mass of the heavy quark sets the hard scale meaning  production can be calculated over all 𝑝T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQf8fre/content/2301.00859v1.pdf'}
+page_content=' The PDFs and FFs on the other hand must be determined  through measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQf8fre/content/2301.00859v1.pdf'}
+page_content=' At the LHC, the quark and gluon densities in the proton means multiple  hard-parton scattering becomes more relevant, and the underlying event activity becomes much  larger, which may affect the fragmentation and hadronisation of quarks into hadrons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQf8fre/content/2301.00859v1.pdf'}
+page_content=' Measure-  ments of heavy-flavour hadrons in general constitute some of the most important tests of QCD over  many length scales at hadron colliders.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQf8fre/content/2301.00859v1.pdf'}
+page_content='  Heavy-flavour meson and quarkonia production has been studied extensively at the LHC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQf8fre/content/2301.00859v1.pdf'}
+page_content='  Measurements of the prompt [1, 2] and non-prompt (from beauty decays) [3] production of D  mesons agree well with QCD calculations [4, 5], indicating that heavy-flavour meson production  is well understood.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQf8fre/content/2301.00859v1.pdf'}
+page_content=' The measurement of strange to non-strange beauty meson production has also  been measured from non-prompt D mesons with ALICE [3], and a combined analysis of B meson  decays from LHCb [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQf8fre/content/2301.00859v1.pdf'}
+page_content=' The LHCb measurement hints at a slight increase in the production ratio  of strange to non-strange beauty meson with collision energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQf8fre/content/2301.00859v1.pdf'}
+page_content=' This is also a crucial measurement  for reducing the uncertainty in many B0  s branching fractions, which are a dominant source of  uncertainty in many searches for new physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQf8fre/content/2301.00859v1.pdf'}
+page_content=' Prompt 𝐽/𝜓 production has been measured by  ALICE [7], ATLAS [8], CMS [9] and LHCb [10], and non-prompt 𝐽/𝜓 production is also measured  by ALICE [7], ATLAS [8] and CMS [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQf8fre/content/2301.00859v1.pdf'}
+page_content='  Comprehensive studies of heavy-flavour baryon production have recently been performed  at the LHC, and currently heavy-flavour baryon production is less well understood than heavy-  flavour meson production.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQf8fre/content/2301.00859v1.pdf'}
+page_content=' Λ+  c production has been measured in pp collisions by ALICE [11–13]  and CMS [14], and is significantly underestimated by QCD calculations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQf8fre/content/2301.00859v1.pdf'}
+page_content=' The baryon-to-meson  production ratio Λ+  c/D0 is significantly larger than the same measurements in e+e− and ep collisions,  by up to a factor of 5 at low 𝑝T, and exhibits a strong 𝑝T dependence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQf8fre/content/2301.00859v1.pdf'}
+page_content=' Recently the baryon-to-meson  ratios Ξ0,+  c /D0 [15, 16], Σ++,+  c  /D0 [11] and Ω0  c/D0 [17] have also been measured by ALICE to  be significantly underestimated by predictions utilising fragmentation parameterisations based on  measurements in e+e− and ep collisions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQf8fre/content/2301.00859v1.pdf'}
+page_content=' The fragmentation fractions of charm quarks into charmed  hadrons have been measured for the first time in pp collisions [18], which are shown in figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQf8fre/content/2301.00859v1.pdf'}
+page_content='  The charmed hadron fragmentation fractions differ significantly from measurements in e+e− and  ep collisions, indicating the assumption of universal, independent parton-to-hadron fragmentation  across collision systems is not sufficient to describe charmed hadron production in pp collisions at  the LHC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQf8fre/content/2301.00859v1.pdf'}
+page_content=' Possible explanations of this difference include colour reconnection between independent  partons [19], quark coalescence [20, 21], or enhanced baryon production originating from the decay  of as-yet-undiscovered charm baryon states [22], which describe baryon-to-meson ratios better,  though still underpredict the Ω0  c/D0 and Ξ0,+  c /D0 ratios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQf8fre/content/2301.00859v1.pdf'}
+page_content=' The relative production of beauty baryons  and mesons has also been measured by LHCb [23], where a similar enhancement of the ratio  Λ0  b/(B0 + B+) is measured at low 𝑝T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQf8fre/content/2301.00859v1.pdf'}
+page_content=' A measurement of the ratio of non-prompt Λ+  c baryons and  D0 mesons was made as a function of 𝑝T by ALICE which aims to indirectly probe beauty quark  fragmentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQf8fre/content/2301.00859v1.pdf'}
+page_content=' This measurement was compared to predictions utilising quark production with  2  Heavy-flavour production at the LHC  Jaime Norman  FONLL, and hadronisation/hadron decays with PYTHIA - the predictions require the fragmentation  fractions measured by LHCb to describe the data, and the measurement is underestimated if  utilising fragmentation fractions measured in e+e− collisions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQf8fre/content/2301.00859v1.pdf'}
+page_content=' These measurements also suggest  that fragmentation fractions of beauty baryons are not universal in different collision systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQf8fre/content/2301.00859v1.pdf'}
+page_content='  The production cross section measurements of different charmed hadrons has allowed for precise  measurement of the total 𝑐 ¯𝑐 [18] and 𝑏 ¯𝑏 [7] production cross sections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQf8fre/content/2301.00859v1.pdf'}
+page_content='  Heavy-flavour hadron production has been studied as a function of the event multiplicity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQf8fre/content/2301.00859v1.pdf'}
+page_content=' The  strange to non-strange production ratio of beauty mesons B0  s/B0 measured by LHCb in pp col-  lisions [24] is shown in figure 1 (right).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQf8fre/content/2301.00859v1.pdf'}
+page_content=' When measuring the multiplicity in the same rapidity  region as the beauty hadron (with the VELO tracker) there is evidence at a level of 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQf8fre/content/2301.00859v1.pdf'}
+page_content='4𝜎 that the  ratio B0  s/B0 increases with multiplicity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQf8fre/content/2301.00859v1.pdf'}
+page_content=' When measuring the multiplicity in the opposite rapidity  direction as the beauty hadron, the ratio is instead independent of this multiplicity, indicating the  enhancement is due to the local event multiplicity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQf8fre/content/2301.00859v1.pdf'}
+page_content=' The charmed baryon-to-meson ratio measured  by ALICE [25] also displays a significant enhancement in the region 2 < 𝑝T < 12 GeV/𝑐 at  high multiplicity compared to low multiplicity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQf8fre/content/2301.00859v1.pdf'}
+page_content=' This multiplicity dependence is described well by  PYTHIA when including colour reconnection mechanisms beyond the leading colour approxima-  tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQf8fre/content/2301.00859v1.pdf'}
+page_content=' The charmed strange to non-strange ration D+  s /D0 is instead independent of multiplicity within  the current experimental uncertainties [25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQf8fre/content/2301.00859v1.pdf'}
+page_content='  0  D  +  D  s  +  D  c  +  Λ  c  0  Ξ  +  D*  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQf8fre/content/2301.00859v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQf8fre/content/2301.00859v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQf8fre/content/2301.00859v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQf8fre/content/2301.00859v1.pdf'}
+page_content='8  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQf8fre/content/2301.00859v1.pdf'}
+page_content='0  )  c  H  →  (c  f  = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQf8fre/content/2301.00859v1.pdf'}
+page_content='02 TeV  s  ALICE, pp,  = 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQf8fre/content/2301.00859v1.pdf'}
+page_content='5 GeV  s  ,  −  e  +  B factories, e  Z  m  =  s  ,  −  e  +  LEP, e  HERA, ep, DIS  HERA, ep, PHP  ALI-PUB-500750  Figure 1: Left: Charm quark fragmentation fractions into charm hadrons in pp collisions at √𝑠 = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQf8fre/content/2301.00859v1.pdf'}
+page_content='02 TeV,  compared with measurements at LEP and B factories, and ep collisions [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQf8fre/content/2301.00859v1.pdf'}
+page_content=' Right: The production ratios  of strange to non-strange B mesons B0  s/B0 as a function of the event multiplicity measured either in the  direction of the beauty meson (left) or in the opposite direction (right) [24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQf8fre/content/2301.00859v1.pdf'}
+page_content='  Fragmentation dynamics of heavy quarks are probed with measurements of heavy flavour  hadrons within jets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQf8fre/content/2301.00859v1.pdf'}
+page_content=' Figure 2 (left) shows a measurement of B± mesons within jets by ATLAS [26]  in particular, the relative momentum of the jet that is carried by the hadron in the direction of the  jet, 𝑧 = �𝑝𝐵,𝐷 · �𝑝𝑗  | �𝑝𝑗 |  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQf8fre/content/2301.00859v1.pdf'}
+page_content=' The transverse momentum profile is also reported in [26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQf8fre/content/2301.00859v1.pdf'}
+page_content=' ALICE measured the  𝑧 distribution of D mesons [27] at √𝑠 = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQf8fre/content/2301.00859v1.pdf'}
+page_content='02 TeV and √𝑠 = 13 TeV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQf8fre/content/2301.00859v1.pdf'}
+page_content=' Comparisons to Monte Carlo  event generators were made in all cases, which helps constrain different approaches to fragmentation  in these generators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQf8fre/content/2301.00859v1.pdf'}
+page_content='  The first observation of 𝑏-hadron production asymmetry (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQf8fre/content/2301.00859v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQf8fre/content/2301.00859v1.pdf'}
+page_content=', asymmetrical production of a  3  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQf8fre/content/2301.00859v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQf8fre/content/2301.00859v1.pdf'}
+page_content='5  LHCb  pp Vs = 13 TeV  LHCb  pp Vs = 13 TeV-  0 < p_<20 GeV/c  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQf8fre/content/2301.00859v1.pdf'}
+page_content='4 fb-1  0 <p_<20 GeV/c  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQf8fre/content/2301.00859v1.pdf'}
+page_content='4 fb-1  B  B  b  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQf8fre/content/2301.00859v1.pdf'}
+page_content='4  b  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQf8fre/content/2301.00859v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQf8fre/content/2301.00859v1.pdf'}
+page_content='3  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQf8fre/content/2301.00859v1.pdf'}
+page_content='3  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQf8fre/content/2301.00859v1.pdf'}
+page_content='2  1 Pp→BB+X  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQf8fre/content/2301.00859v1.pdf'}
+page_content='2  1 Pp→BB+X  Il +e-→Z°→BB  I e+e-→Z°→BB  a)  ie+e-→Y(5S)→BB  b)  {ete-→Y(5S)→BB  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQf8fre/content/2301.00859v1.pdf'}
+page_content='1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQf8fre/content/2301.00859v1.pdf'}
+page_content='1  0  2  4  6  0  2  4  6  tback  tracks  tracks  NoBias  tracks  NoBiasHeavy-flavour production at the LHC  Jaime Norman  beauty hadron compared to its anti-particle, measured using 𝐴𝑝𝑟𝑜𝑑 =  𝜎( 𝑝𝑝→Λ0  𝑏𝑌 )−𝜎( 𝑝𝑝→ ¯  Λ0  𝑏𝑌 )  𝜎( 𝑝𝑝→Λ0  𝑏𝑌 )+𝜎( 𝑝𝑝→ ¯  Λ0  𝑏𝑌 ) )  was reported by LHCb [28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQf8fre/content/2301.00859v1.pdf'}
+page_content=' Figure 2 shows 𝐴𝑝𝑟𝑜𝑑 as a function of Λ0  b rapidity at √𝑠 = 7 TeV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQf8fre/content/2301.00859v1.pdf'}
+page_content='  Combining the measurement at √𝑠 = 7 TeV and 8 TeV, the results are incompatible with symmetric  production with a significance of 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQf8fre/content/2301.00859v1.pdf'}
+page_content='8 standard deviations, assuming no CP violation in the decay.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQf8fre/content/2301.00859v1.pdf'}
+page_content='  There is also evidence at the level of 4𝜎 of an increase in production asymmetry with rapidity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQf8fre/content/2301.00859v1.pdf'}
+page_content='  Comparisons from MC generators show that the results agree with PYTHIA when an improved  colour reconnection model is included.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQf8fre/content/2301.00859v1.pdf'}
+page_content='  Figure 2: Left: The distribution of the longitudinal profile 𝑧 of B± mesons in jets [26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQf8fre/content/2301.00859v1.pdf'}
+page_content=' Right: the production  asymmetry of Λ0  b baryons [28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQf8fre/content/2301.00859v1.pdf'}
+page_content='  CMS reported the observation of simultaneous production of three 𝐽/𝜓 meson particles [29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQf8fre/content/2301.00859v1.pdf'}
+page_content='  In this analysis, five events are found to be consistent with triple-𝐽/𝜓 production, with a statistical  significance relative to the background-only expectation of 5 standard deviations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQf8fre/content/2301.00859v1.pdf'}
+page_content=' The measured  cross section is 𝜎(pp → 𝐽/𝜓𝐽/𝜓𝐽/𝜓𝑋) = 272+141  −104(stat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQf8fre/content/2301.00859v1.pdf'}
+page_content=') ± 17(syst.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQf8fre/content/2301.00859v1.pdf'}
+page_content=') fb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQf8fre/content/2301.00859v1.pdf'}
+page_content=' This cross section is  consistent with theoretical expectation of production via primarily double-parton scattering and  triple-parton scattering (under the simplest assumption of factorization of multiple hard-scattering  probabilities in terms of SPS cross sections).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQf8fre/content/2301.00859v1.pdf'}
+page_content=' This measurement is the first observation of the  simultaneous production of three heavy particles, and represents an important step in the quest to  constrain the proton PDF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQf8fre/content/2301.00859v1.pdf'}
+page_content='  In summary, heavy-flavour production at the LHC constitutes a stringent test of QCD, and Run  2 of the LHC has allowed for a diverse range of heavy-flavour production measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQf8fre/content/2301.00859v1.pdf'}
+page_content=' Run 3  has just begun after significant detector upgrade programmes for all 4 LHC experiments during  Long Shutdown 2, which will provide unprecedented accuracy for future heavy-flavour production  measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQf8fre/content/2301.00859v1.pdf'}
+page_content='  4  (1/o) do / dz  ATLAS/s=13TeV,139fb1  B→ J/u Kt  70 GeV<p  <100GeV  anti-k, jets R = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQf8fre/content/2301.00859v1.pdf'}
+page_content='4  4  Data (stat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQf8fre/content/2301.00859v1.pdf'}
+page_content=' sys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQf8fre/content/2301.00859v1.pdf'}
+page_content=')  P8 (Monash + Peterson)  P8 (Monash + L-B)  △ P8 (A14 + L-B)  3  P8 (A14-rb + L-B)  Herwig 7 (ang.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQf8fre/content/2301.00859v1.pdf'}
+page_content=' ord.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQf8fre/content/2301.00859v1.pdf'}
+page_content=')  Herwig 7 (dipole)  2  Sherpa (Lund)  Sherpa (cluster)  MC/Data  Total uncertainty  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQf8fre/content/2301.00859v1.pdf'}
+page_content='5  MC /Data  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQf8fre/content/2301.00859v1.pdf'}
+page_content='3  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQf8fre/content/2301.00859v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQf8fre/content/2301.00859v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQf8fre/content/2301.00859v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQf8fre/content/2301.00859v1.pdf'}
+page_content='7  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQf8fre/content/2301.00859v1.pdf'}
+page_content='8  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQf8fre/content/2301.00859v1.pdf'}
+page_content='9  z16  LHCb  prod  vs= 7 TeV  14  Data 1fb-1  A  12  Pythia8 (CR1)  10  Pythia8 (CR2)  8  Pythia8 (Monash)  6  4  2  0  2  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQf8fre/content/2301.00859v1.pdf'}
+page_content='5  3  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQf8fre/content/2301.00859v1.pdf'}
+page_content='5  4Heavy-flavour production at the LHC  Jaime Norman  References  [1] LHCb collaboration, Measurements of prompt charm production cross-sections in 𝑝𝑝  collisions at √𝑠 = 13 TeV, JHEP 03 (2016) 159 [1510.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQf8fre/content/2301.00859v1.pdf'}
+page_content='01707].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQf8fre/content/2301.00859v1.pdf'}
+page_content='  [2] CMS collaboration, Measurement of prompt open-charm production cross sections in  proton-proton collisions at √𝑠 = 13 TeV, JHEP 11 (2021) 225 [2107.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQf8fre/content/2301.00859v1.pdf'}
+page_content='01476].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQf8fre/content/2301.00859v1.pdf'}
+page_content='  [3] ALICE collaboration, Measurement of beauty and charm production in pp collisions at √𝑠 =  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQf8fre/content/2301.00859v1.pdf'}
+page_content='02 TeV via non-prompt and prompt D mesons, JHEP 05 (2021) 220 [2102.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQf8fre/content/2301.00859v1.pdf'}
+page_content='13601].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQf8fre/content/2301.00859v1.pdf'}
+page_content='  [4] B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQf8fre/content/2301.00859v1.pdf'}
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+page_content='  [26] ATLAS collaboration, Measurement of 𝑏-quark fragmentation properties in jets using the  decay 𝐵± → 𝐽/𝜓𝐾± in 𝑝𝑝 collisions at √𝑠 = 13 TeV with the ATLAS detector, JHEP 12  (2021) 131 [2108.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQf8fre/content/2301.00859v1.pdf'}
+page_content='11650].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQf8fre/content/2301.00859v1.pdf'}
+page_content='  [27] ALICE collaboration, Measurement of the production of charm jets tagged with D0 mesons  in pp collisions at √𝑠 = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQf8fre/content/2301.00859v1.pdf'}
+page_content='02 and 13 TeV, Submitted to JHEP [2204.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQf8fre/content/2301.00859v1.pdf'}
+page_content='10167].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQf8fre/content/2301.00859v1.pdf'}
+page_content='  [28] LHCb collaboration, Observation of a Λ0  𝑏 − Λ  0  𝑏 production asymmetry in proton-proton  collisions at √𝑠 = 7 and 8 TeV, JHEP 10 (2021) 060 [2107.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQf8fre/content/2301.00859v1.pdf'}
+page_content='09593].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQf8fre/content/2301.00859v1.pdf'}
+page_content='  [29] CMS collaboration, Observation of triple J/𝜓 meson production in proton-proton collisions  at √𝑠 = 13 TeV, Submitted to Nature Physics [2111.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQf8fre/content/2301.00859v1.pdf'}
+page_content='05370].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQf8fre/content/2301.00859v1.pdf'}
+page_content='  6' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQf8fre/content/2301.00859v1.pdf'}
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+Realistic pattern formations on rough surfaces∗
+Siqing LI†
+Leevan LING‡
+Steven J. RUUTH§
+Xuemeng Wang¶
+January 31, 2023
+Abstract
+We are interested in simulating patterns on rough surfaces.
+First,
+we consider periodic rough surfaces with analytic parametric equations,
+which are defined by some superposition of wave functions with random
+frequencies and angles of propagation. The amplitude of such surfaces
+is also an important variable in the provided eigenvalue analysis for the
+Laplace-Beltrami operator and in our numerical studies. Simulations show
+that the patterns become irregular as the amplitude and frequency of the
+rough surface increase. Next, for the sake of easy generalization to closed
+manifolds, we propose another construction method of rough surfaces by
+using random nodal values and discretized heat filters. We provide numer-
+ical evidence that both surface constructions yield comparable patterns
+to those found in real-life animals.
+keywords Laplace-Beltrami operator, reaction-diffusion system, random
+surfaces, Turing pattern
+AMS subject classifications 65M06, 35K57
+1
+Introduction
+Pattern formation by reaction-diffusion systems has been an intensively studied
+field for decades. In 1952, Turing proposed the idea of diffusion-driven instabil-
+ity [47], in which simple mechanisms evolve from a homogeneous state into spa-
+tial heterogeneous patterns. In recent years, a variety of application areas have
+∗This work was funded by the Hong Kong Research Grant Council GRF Grants
+(12303818,12301419,12301520), a National Youth Science Foundation of China (12201449),
+and the financial support of NSERC Canada (RGPIN-2022-03302).
+†College
+of
+Data
+Science,
+Taiyuan
+University
+of
+Technology,
+Shanxi,
+China
+(lisiqing@tyut.edu.cn).
+‡Department of Mathematics, Hong Kong Baptist University, Kowloon Tong, Hong Kong
+(lling@hkbu.edu.hk).
+§Department of Mathematics, Simon Fraser University, Burnaby, British Columbia,
+Canada V5A1S6 (sruuth@sfu.ca).
+¶Department of Mathematics, University of British Columbia, Vancouver, Canada, and
+Department of Mathematics, Hong Kong Baptist University, Kowloon Tong, Hong Kong
+(17251109@life.hkbu.edu.hk).
+1
+arXiv:2301.13148v1  [math.NA]  13 Jan 2023
+
+(a)
+(b)
+(c)
+(d)
+Figure 1: Skin patterns in real-life: (a) emperor angelfish [26], (b) genet [1], (c)
+pecostomus [12], (d) cheetah [46].
+been actively developed, including vegetation pattern, plant root hair initiation,
+flock simulation and boundary drop configurations [4,7,13,32]. Mechanisms of
+pattern generation under different types of transport or domain size have been
+considered as well; see [6,25]. In theoretical biology, reaction-diffusion systems
+provide a relatively generic and concise approach for simulating animal skin pat-
+terns [18,35], as well as regenerative processes of organisms [21]. In particular,
+reaction-diffusion systems are a well-accepted class of models for multiple pig-
+mentation processes. Examples include marine [27] and emperor [26] angelfish,
+genets [1], plecostomi [12], cheetah [46], zebra fish [2], and many other mam-
+mal skin patterns [34]; see Figure 1. Although animal skin pattern simulation
+through reaction-diffusion systems has been explored in many researches, see
+the second row of Figure 5 for typical examples, very few of them have taken
+the skin texture into account. In reality, the skin surfaces are often rugged, and
+consequently the pattern will actually not be regular spots or stripes. There-
+fore, combining surface properties with reaction-diffusion systems can provide
+an opportunity for improved simulation of real pattern. In [11, 30], parameter
+functions instead of constants were used in PDEs to generate nonuniform and
+complex, i.e., more real-life, patterns.
+Characterizing random rough surfaces can be carried out by a variety of
+methods. These include the reference parameter method, motif method, fractal
+method, watershed method, and wavelet method, etc [42]. For three-dimensional
+surface topography, there are a number of parameters involved [20]. To generate
+random rough surfaces, both the Fast Fourier Transform (FFT) [37] and digital
+filter [23] methods can be applied to approximate the auto-correlation function
+for surfaces with wavelengths. In [24], the authors proposed an approach that
+applies the covariance function and Karhunen-Lo`eve expansion to generate ran-
+dom surfaces. Rough surfaces may also be generated from spatial frequencies
+via the FFT method; in this case, the rough surface is built by summing up
+2
+
+trigonometric functions [43].
+In Section 2, the rough surface M with parametric equations will be for-
+mulated in detail. We consider periodic rough surfaces M characterized with
+spatial frequency, which come with analytic parametric equations. In Section 5,
+we consider surfaces S constructed by random nodal values and discretized heat
+filters. The latter surface construction technique can be easily extended to add
+“roughness” to more general manifolds. Along with the construction of rough
+surfaces, we will review the concept of the heat equation which is a surface PDE
+involving the Laplace operator on manifolds.
+Moving from a flat geometry to rough surfaces, generally speaking, we have
+to deal with surface dependent differential operators. Surface PDEs have been
+extensively studied in recent years, and there are a variety of techniques for their
+analysis and computation. The approaches for surface PDEs can be separated
+into intrinsic methods and embedding methods. Intrinsic methods solve PDEs
+on the manifold directly.
+However, embedding methods formulate and solve
+PDEs on a band around the manifold. Intrinsic and embedding methods that
+are dependent on mesh construction have been explored by many researchers.
+Examples of such methods include finite difference [40], finite element [5,16,36],
+and finite volume [14,15] methods. There are also numerical methods that do
+not require meshes, namely meshfree methods. Meshfree methods for surface
+PDEs, such as radial basis function (RBF) methods [8–10,19] and the meshfree
+generalized finite difference method [44,45], have the advantage of avoiding mesh
+construction.
+In Section 3, we apply the finite difference method to solve pattern forma-
+tion PDEs on rough surfaces M; the FDM is chosen because of its simplicity
+and effectiveness on the class of problem domains considered. The rough sur-
+face pattern formation models here are not yet extendable to add roughness to
+other manifolds, e.g., we cannot yet create red-blood cells with rough surfaces.
+Our aim is to study the effect of roughness on pattern formation using the sim-
+plest setup. Numerical simulation results on rough surfaces M with parametric
+equations are demonstrated in Section 4.
+In Section 5, we put forward a different construction method of rough sur-
+faces S by random discrete data. After updating our numerical schemes for M
+to work on S, we see that surface types M and S yield comparable patterns.
+In Section 4 and Section 5, we present some simulated animal coat patterns on
+both types of surfaces, M and S. Results for PDEs with constant parameters
+on rough surfaces with various roughness are similar to the real-life patterns
+displayed in Figure 1.
+2
+Rough surfaces M with analytic parametric
+equations
+Consider Ck–smooth (k ≥ 2), codimension 1, and periodic Riemannian surfaces
+M =
+�
+(x, y, z) ∈ R3 : z = z(x, y) for (x, y) ∈ V ⊂ R2�
+,
+(1)
+3
+
+for some Ck function z(·, ·) defined on a global parameter space V ⊂ R2. A
+corresponding parametric representation of M is given by
+⃗r(x, y) =
+�
+x, y, z(x, y)
+�T ∈ M,
+(x, y) ∈ V.
+For convenience, let (x, y) := (x1, x2); we shall use both notations interchange-
+ably. Recall that the first fundamental form G : V → R2×2 of M is defined
+by
+G(x, y) =
+�
+g11
+g12
+g21
+g22
+�
+(x, y)
+where
+gij(x, y) = ∂⃗r(x, y)
+∂xi
+· ∂⃗r(x, y)
+∂xj
+,
+see [16] for a review. In the case of a surface function M as in (1), we have
+G(x, y) =
+�g11
+g12
+g21
+g22
+�
+(x, y) =
+�1 + z2
+x
+zxzy
+zxzy
+1 + z2
+y
+�
+,
+(2)
+whose determinant and inverse are
+g(x, y) := det(G)(x, y) = 1 + z2
+x + z2
+y,
+(3)
+and
+G−1(x, y) =
+�g11
+g12
+g21
+g22
+�
+(x, y) =
+1
+1 + z2x + z2y
+� 1 + z2
+y
+−zxzy
+−zxzy
+1 + z2
+x
+�
+.
+(4)
+Using (3)–(4), the Laplace-Beltrami operator on M is given by
+∆Mf =
+1
+√g
+�
+1≤i,j≤2
+∂
+∂i
+�√ggij ∂
+∂j
+f
+�
+=:
+1
+√g ∇ ·
+�
+A∇f
+�
+,
+(5)
+for any C2 function f : M → R.
+Now, we focus on random rough surfaces M ⊆ R3 in the form of (1), whose
+surface function z : I2 = [−L, L]2 → R is a superposition of elementary waves
+[17,43]. The function z is stochastically determined by
+z(x, y) =
+M
+�
+m=−M
+N
+�
+n=−N
+am,n cos
+�
+2π(mx + ny) + φm,n
+�
+,
+(6)
+for some random variables am,n and φm,n. Similar to a Fourier series expansion,
+z(x, y) in (6) is constructed by trigonometric functions in which m, n correspond
+to spatial frequencies on the x and y axes respectively. As in [43], the spatial
+frequencies m and n allow values taken up to maximum integers M and N
+respectively; this corresponds to a high frequency cut off.
+To determine (6), we first introduce ˜am,n ∼ N(0, 1) and φm,n ∼ U(0, π).
+This specifies the pre-surface function ˜z = ˜z(x, y). Next, we control the ampli-
+tude of the rough surface M to be within a specific range [−δM, δM] by scaling
+according to
+z :=
+δM
+∥˜z∥∞
+˜z,
+i.e., am,n :=
+δM
+∥˜z∥∞
+˜am,n.
+(7)
+4
+
+M
+(a) M = 5 = N
+Lower roughness
+S
+(d)
+(b) M = 5, N = 15
+Increasing M, N
+←−−−−−−−−−−−−−−
+(e)
+(c) M = 15 = N
+Higher roughness
+(f)
+Figure 2: Bird’s-eye view of some random rough surfaces M from Section 2
+(left) and S from Section 5 (right).
+Figures 2 (a)–(c), first column, show some rough surfaces M with amplitude
+δM = 10−3 and various (M, N).
+The construction of the rough surfaces S
+appearing in the second column will be discussed in Section 5.
+Remark: One can also add a decay condition with respect to frequencies
+m, n and a frequency attenuation parameter β to the pre-coefficient ˜am,n via
+˜am,n ∼
+1
+(m2 + n2)β/2 N(µ, σ).
+2.1
+Surface roughness and Laplace-Beltrami operator
+Applying the definition of the Laplace-Beltrami operator (5) to the rough sur-
+face function in (6)–(7), the relationship between the eigenvalues of the diffusion
+tensor and the geometry of the surface can be made explicit. Firstly, the diffu-
+5
+
+sion tensor A in (5) is
+A(x, y) =
+�A1
+A2
+A2
+A4
+�
+(x, y) := √gG−1 =
+1
+√g
+�1 + z2
+y
+−zxzy
+−zxzy
+1 + z2
+x
+�
+,
+(8)
+with the partial derivatives zx and zy computing from (6). We are interested in
+obtaining the eigenvalues and eigenvectors of A. Rather than computing these
+quantities directly, it turns out to be easier to first compute the eigenvalues and
+eigenvectors of the Riemannian metric tensor G in (2). From the characteristic
+equation of G
+|G − λI| = (1 + z2
+x − λ)(1 + z2
+y − λ) − z2
+xz2
+y = 0,
+we find that the eigenvalues λG of G are 1 + z2
+x + z2
+y = g and 1. Since
+A⃗v = √gG−1⃗v =
+√g
+λG ⃗v,
+(9)
+any eigenvector ⃗v of G is also an eigenvector of A and the eigenvalues λA of A
+are
+λA =
+�√g =
+�
+1 + z2x + z2y,
+1
+√g
+�
+.
+(10)
+Note that, by (7), the matrix function [G−λGI](zx, zy) = C(δM)[G−λGI](˜zx, ˜zy)
+for some δM-dependent constant C(δM). This implies that all eigendirections
+of G (and A) are independent of the amplitude δM. Simple calculations show
+that λA = 1 + O(M 2N 2δ2
+M) varies with δM nonlinearly by a bijection. Thus,
+the contour lines (but not the height) of λA are independent of δM.
+Figure 3 illustrates the close relationship between the eigenvalues of A and
+the geometric properties of a rough surface.
+In particular, we can see from
+subfigures (b), (c), and (e) that the maximum (and minimum) eigenvalues are
+larger (and smaller) at regions of the surface with the steepest gradients. In
+subfigures (d) and (f), eigendirections are plotted on top of subfigure (b). We can
+clearly see that the eigenvectors corresponding to the maximum (and minimum)
+eigenvalues are tangent (and orthogonal) to the contour plots of eigenvalues.
+3
+Solving heat equations on rough surfaces M
+We begin by constructing a finite difference scheme for solving heat equations on
+rough surfaces M ⊂ R3 defined by (6)–(7). We work intrinsically by transform-
+ing the PDEs on rough surfaces to the parameter space I2 = [−L, L]2 ⊂ R2.
+Subsequently, we move on to solving reaction-diffusion systems for pattern for-
+mation.
+We consider the heat equation on a rough surface
+∂u
+∂t (ξ, t) − ∆Mu(ξ, t) = h(ξ, t)
+for ξ ∈ M, t ∈ (0, T],
+(11a)
+6
+
+(a) M = 1 = N
+(b) Contour of a rough surface M
+(c) maximum eigenvalue λA
+max
+(d) eigendirection of λA
+max
+(e) minimum eigenvalue λA
+min
+(f) eigendirection of λA
+min
+Figure 3:
+(a, b): 3D and contour plots of a rough surface in the form of (7)
+with M = N = 1 and amplitude δM = 1E −1, (c, d): maximum eigenvalue and
+eigendirection of A, (e, f): minimum eigenvalue and eigendirection of A.
+7
+
+0.1
+0.5
+0.05
+0
+0
+-0.5
+1
+-0.05
+1
+0
+0
+.1
+-10.05
+0.5
+0
+0
+-0.5
+-0.05
+-0.5
+0
+0.5
+11.15
+0.5
+1.1
+0
+-0.5
+1.05
+-0.5
+0
+0.5
+10.05
+0.5
+0
+0
+-0.5
+-0.05
+1
+-0.5
+0
+0.5
+10.98
+0.96
+0.5
+0.94
+0
+0.92
+0.9
+-0.5
+0.88
+0.86
+-0.5
+0
+0.5
+1
+10.05
+0.5
+0
+0
+-0.5
+-0.05
+1
+-0.5
+0
+0.5
+1where u : M × (0, T] → R, subject to periodic boundary conditions on ∂M :=
+∂I2 × z(∂I2) with z(∂I2) being the surface function in (6), i.e.,
+u([−L, y, z(−L, y)]T , t) = u([L, y, z(L, y)]T , t),
+for y ∈ I, t ∈ (0, T],
+u([x, −L, z(x, −L)]T , t) = u([x, L, z(x, L)]T , t),
+for x ∈ I, t ∈ (0, T].
+(11b)
+Since all surface points are in the form ξ = [x, y, z(x, y)]T ∈ M for [x, y]T ∈
+I2 and z in (6), we discretize the parameter space I2 by some set of nXnY
+tensor-product grid points [X, Y ] ∈ RnXnY ×2 ⊂ I2. Let uj(x, y) ≈ u([x, y, z(x, y)]T , tj)
+for [x, y]T ∈ I2. We begin by temporal discretization of (11a) by the θ-method
+uj+1 − uj
+τ
+= θ
+�
+∆Muj+1 + hj+1�
++ (1 − θ)
+�
+∆Muj + hj�
+,
+(12)
+for an equispaced partition {tj}M
+j=0 of [0, T] with step-size τ. By some user-
+selected finite difference scheme for the first derivative, we construct differenti-
+ation matrices Dk ∈ RnXnY ×nXnY such that, for any C1-function w : I2 → R
+satisfying periodic boundary condition (11b), we have
+Dkw(X, Y ) ≈ ∂w
+∂xk
+(X, Y ),
+for k ∈ {1, 2},
+where the nXnY × 1 vector w(X, Y ) (and
+∂w
+∂xk (X, Y )) contains nodal values of
+w (and
+∂w
+∂xk ) at grid points in [X, Y ]. Working directly on definitions (5)–(8),
+we can approximate the Laplace-Beltrami term ∆Mw at grids [X, Y ] by nodal
+function values W := w(X, Y ) according to
+∆Mw(X, Y ) ≈
+1
+√g (X, Y )⊛
+�
+D1
+�
+A1(X, Y ) ⊛ (D1W)
+�
++ D2
+�
+A4(X, Y ) ⊛ (D2W)
+�
++ D1
+�
+A2(X, Y ) ⊛ (D2W)
+�
++ D2
+�
+A2(X, Y ) ⊛ (D1W)
+��
+=
+1
+√g (X, Y )⊚
+�
+D1
+�
+A1(X, Y ) ⊚ D1
+�
++ D2
+�
+A4(X, Y ) ⊚ D2
+�
++ D1
+�
+A2(X, Y ) ⊚ D2
+�
++ D2
+�
+A2(X, Y ) ⊚ D1
+��
+W
+=: △M,hW,
+(13)
+where △M,h ∈ RnXnY ×nXnY , ⊛ is the element-wise Hadamard product of ma-
+trices and ⊚ is a vector-matrix product defined by ⃗a ⊚ [⃗b1, · · · ,⃗bn] := [⃗a ⊛
+⃗b1, ...,⃗a ⊛ ⃗bn].
+See Appendix for the detailed construction of differentiation
+matrices D1, D2. Combining (12) and (13) yields a fully discretized scheme for
+the update in time.
+3.1
+Visualizing heat flow on a rough surface M
+We show the heat flow under different amplitudes of rough surface M in (3) by
+solving the heat equations (11a) with zero flux h(ξ, t) = 0 and periodic boundary
+8
+
+conditions (11b). The rough surfaces M are defined over the parameter space
+I2 = [−1, 1]2 ⊂ R2. The compatible initial condition is given by
+u(x, y, z, 0) = cos(πx/2) cos(πy/2).
+(14)
+Figure 4 shows the numerical solutions on the surface M in Figure 3 (b) with
+δM ∈ {0.1, 0.5, 1} in the respective rows. In all cases, we use backward Euler
+method, and set τ = 1E − 3, T = 1 and nX = 41 = nY . Figure 4 shows
+that the range of the numerical solutions becomes larger as the amplitude of M
+increases from δM = 0.1 to δM = 1. From the rough surface in Figure 3 (b) and
+solutions in Figure 4, it can be concluded that the heat flow when projected to
+the plane is greatest in flat regions.
+4
+Pattern formation on a rough surface M
+In this section, we consider reaction-diffusion systems (RDS) on a rough surface
+M
+�
+∂tu = δu∆Mu + fu(u, v),
+∂tv = δv∆Mv + fv(u, v),
+(15)
+for some (concentration) functions u, v : M × (0, T] → R, and reaction terms
+�
+�
+�
+fu(u, v) = αu(1 − ξ1v2) + v(1 − ξ2u),
+fv(u, v) = βv
+�
+1 + αξ1
+β uv
+�
++ u(γ + ξ2v),
+(16)
+with parameters δu, δv, α, β, ξ1, ξ2. Pattern formation on very smooth (and usu-
+ally closed) surfaces with little variation is discussed in [10, 29, 33, 41]; pattern
+formation on rough surfaces has not been studied as far as the authors know. In
+this section, we aim to analyze the effect of rough surfaces on pattern formation
+generated by the reaction-diffusion system (15)-(16). Numerical methods are
+not our focus and we simply extend the finite difference scheme in the previous
+section to work here. Other options for the discretization of reaction-diffusion
+systems include the finite element method [22,28,38], and various types of mesh-
+free methods [30,31].
+As in Section 3, we work on an equispaced temporal partition {tj}NT
+j=0 with
+some time step-size τ > 0 and tensor-product grid points [X, Y ] ∈ RnXnY ×2 ⊂
+I2. Let
+U j ≈ u(X, Y, z(X, Y ), tj)
+and
+V j ≈ v(X, Y, z(X, Y ), tj),
+1 ≤ j ≤ NT ,
+be the unknown nodal values we seek based on initial conditions U 0 and V 0.
+Using the second order backward differentiation formula (BDF2) [3,39] and (13)
+to discretize the RDS (15) with periodic boundary condition (11b), we obtain
+9
+
+(a)
+δM = 0.1
+(b)
+(c)
+δM = 0.5
+(d)
+(e)
+δM = 1
+(f)
+Figure 4:
+Numerical solution of heat equations under different amplitudes
+δM.
+(a, b): δM = 0.1, (c, d): δM = 0.5, (e, f): δM = 1.
+In all cases,
+τ = 1E − 3, T = 1, nX = 41 = nY .
+10
+
+0.5
+0.5
+0.45
+0
+0.4
+-0.5
+0.35
+-0.5
+0
+0.5
+10.5
+0.5
+0.45
+0
+0.4
+-0.5
+0.35
+-0.5
+0
+0.5
+10.5
+0.5
+0.45
+0
+0.4
+-0.5
+0.35
+-0.5
+0
+0.5
+10.5
+0.5
+0.45
+0
+0.4
+-0.5
+0.35
+-0.5
+0
+0.5
+10.5
+0.5
+0.45
+0
+0.4
+-0.5
+0.35
+-0.5
+0
+0.5
+10.5
+0.5
+0.45
+0
+0.4
+-0.5
+0.35
+-0.5
+0
+0.5
+1Table 1:
+Parameters of the reaction-diffusion system (15)-(16) for generating
+spots and stripes patterns on rough surfaces.
+Pattern
+δv
+δu
+α
+β
+γ
+ξ1
+ξ2
+Spots
+10−3
+0.516δv
+0.899
+−0.91
+−0.899
+0.02
+0.2
+Stripes
+10−3
+0.516δv
+0.899
+−0.91
+−0.899
+3.5
+0
+the following fully-discrete system of equations on M
+�
+�
+�
+�
+�
+3U j+1 − 2τδu△M,hU j+1 = 4τfu
+�
+U j, V j�
+− 2τfu
+�
+U j−1, V j−1�
++ 4U j − U j−1,
+3V j+1 − 2τδv△M,hV j+1 = 4τfv
+�
+U j, V j�
+− 2τfv
+�
+U j−1, V j−1�
++ 4V j − V j−1,
+(17)
+for 1 ≤ j ≤ NT , subject to some yet-to-be specified (usually random) initial
+condition and some first order approximations to the solutions at the first time
+step, U 1 and V 1. Note that the two equations in (17) are not coupled; the com-
+putational cost is of the same order as that of solving two scalar heat equations.
+4.1
+The pattern generation process
+In this subsection, we show the formation of irregular patterns as we go from
+a flat two dimensional domain to rough surfaces with different amplitudes. We
+start with patterns on the flat domain [−1, 1]2 as in [30], i.e., the rough sur-
+face with zero amplitude (δM = 0). Model and surface parameters are chosen
+according to the values in Table 1. To discretize, we select grid parameters
+nX = 90, nY = nX, and a time step-size τ = 0.5. The rough surfaces M with
+M = N = 5 are used in this part.
+The first row of Figure 5 plots the initial conditions used to compute the
+spots and stripes patterns, respectively. These are random values generated
+within the interval [−0.5, 0.5]. The second row of Figure 5 shows the steady
+state patterns on the surface with zero amplitude. Perfect spots and stripes
+are obtained, which is similar to our previous results in [30]. Next, we set the
+initial conditions for the next amplitude to be the steady solutions from the zero
+amplitude rough surface, i.e, we use the solution of spots with δM = 0, T = 800
+and the solution of stripes with δM = 0, T = 4000. By increasing the amplitude
+of the rough surface from δM = 0 to δM = 0.1 with increments of 0.01, and
+setting initial conditions using the previous steady state, we achieve the final
+patterns for δM = 0.1. These are shown in the third row of Figure 5. For rough
+surfaces under small amplitude δM = 0.05, the spots and stripes are similar
+to those with δM = 0. However, for larger amplitude δM = 0.1, both spots
+and stripes become irregular. We can conclude that the steady state patterns
+become irregular as the amplitude of the rough surface M increases.
+11
+
+Spots
+δM = 0, t = 0
+Stripes
+δM = 0, t = 0
+δM = 0, t = 800
+δM = 0, t = 4000
+Increase δM by 0.01 at a time
+−−−−−→
+δM = 0.1, t = 800 × 11
+δM = 0.1, t = 4000 × 11
+Figure 5: Pattern generation on rough surfaces M with M = N = 5 and
+amplitude δM increasing from 0 to 0.1. Parameters for spots and stripes are set
+according to Table 1. Discretization parameters are nX = 90 = nY , τ = 0.5.
+12
+
+4.2
+Patterns on surfaces with different spatial frequencies
+and amplitudes
+Properties of rough surfaces are influenced by the amplitude δM and the spatial
+frequencies M, N in (7). To better understand patterns on rough surfaces with
+different properties, we compute steady state patterns for
+δM ∈ {0.05, 0.1},
+(M, N) ∈ {(5, 15), (15, 15)},
+on the parameter space I2 = [−0.5, 0.5]2 ⊂ R2. Periodic boundary conditions
+on ∂M := ∂I2 × z(∂I2) are prescribed.
+These patterns in both 2-D view and zoom in 3-D view are presented in
+Figures 6-7 for spots, and in Figures 8-9 for stripes.
+As before, the initial
+conditions are assigned to be random data in [−0.5, 0.5]. In order to capture
+details of the spot and stripe patterns, we increase the spatial resolution from
+nX = 90 to nX = 170, again keeping nY = nX to ensure the irregular patterns
+were due to the roughness rather than low resolution.
+Firstly, from Figures 6 and 7, we can see that the number of spots increases
+as the frequencies M, N and the amplitude δM increase. For fixed amplitude
+δM, patterns are deformed along the x−axis as the frequency N increases. This
+conclusion is clearly illustrated in the case δM = 0.1.
+On the other hand,
+for fixed values of M, N, the patterns maintain their shape as spots when
+δM ≤ 0.05, and start to deform when δM ≥ 0.05. For δ = 0.1, all patterns
+become deformed spots.
+Secondly, for fixed δM = 0.05, zoom in 3-D profiles of region [−0.3, 0.1] ×
+[−0.3, 0] for [M, N] = [5, 15] and region [0.2, 0.5]×[−0.3, 0] for [M, N] = [15, 15]
+reveal that irregular spots appear when part of the pattern is located in a
+valley or ridge of the rough surface. When δM increases to 0.1, as in Figure 7,
+the deformation of the patterns is more severe.
+This makes sense since the
+amplitude δM is twice that of Figure 6. From zoom in 3-D figures of region
+[−0.1, 0.5] × [0.1, 0.4] for [M, N] = [5, 15] and region [−0.4, 0.1] × [−0.2, 0.1]
+for [M, N] = [15, 15], the deformed patterns again appear when their locations
+cover the local ridge/valley/mountain of the rough surfaces.
+As frequencies and amplitudes are varied, similar behavior (to the case of
+spots) can be observed in stripe formation; see Figures 8 and 9. High amplitudes
+and frequencies yield a particularly strong effect in the zoom in 3-D plots in
+Figure 9, where the stripes break into small separate components. Further, we
+observe that the largest concentration values appear at the local peaks of the
+rough surface M.
+4.3
+Animal coat simulation results
+In simulation results on rough surfaces M which are characterized by spatial
+frequencies, we observe interesting similarities between some steady state pat-
+terns and actual animal coat patterns. In Figure 10, we show the simulations
+of the animal coats in Figure 1 by patterns on rough surfaces M. The specific
+13
+
+2-D view
+M = 5, N = 15
+Zoom in 3-D view
+M = 15 = N
+Figure 6: Patterns on rough surfaces with amplitude δM = 0.05. Discretization
+parameters are nX = 170 = nY , τ = 0.5, T = 800, I2 := [−0.5, 0.5]2.
+parameter values for generating each animal coat pattern are listed in Tables 1
+and 2.
+Table 2: Parameters for animal coats simulation on M in Figure 10 with nX = nY = 90
+M
+N
+τ
+T
+δM
+Emperor angelfish [26]
+Figure 10 (a)
+5
+5
+0.5
+4000
+0.05
+Genet [1]
+Figure 10 (b)
+15
+15
+0.5
+800
+0.1
+Plecostomus [12]
+Figure 10 (c)
+15
+5
+0.5
+4000
+0.1
+Cheetah [46]
+Figure 10 (d)
+15
+5
+0.5
+800
+0.1
+5
+Random rough surfaces S by discrete data
+While the parametric equation (6) allows us to work analytically on rough sur-
+faces, i.e., via the evaluation of metric tensor (2), its generalization to mani-
+folds [9,10,40] is not trivial.
+In [24], the authors used the covariance function of random deformation fields
+and the surface Karhunen-Lo`eve expansion to generate random surfaces. In [23],
+the authors applied the 2D digital filter and Fourier analysis to generate random
+rough surfaces. In this section, we introduce a new approach for constructing
+14
+
+0.5
+10
+5
+0
+5
+-0.5
+-0.5
+0
+0.5-0.3
+-0.2
+0.05
+-0.1
+0
+0
+-0.05
+X
+-0.3
+-0.2
+-0.1
+00.5
+10
+5
+0
+0
+5
+-0.5
+-0.5
+0.5
+00.05
+0
+0.4
+-0.05
+0
+0.3
+-0.1
+-0.2
+0.2
+X
+-0.3
+y2-D view
+M = 5, N = 15
+Zoom in 3-D view
+M = 15 = N
+Figure 7: Patterns on rough surfaces with amplitude δM = 0.1. Discretization
+parameters are nX = 170 = nY , τ = 0.5, T = 800, I2 := [−0.5, 0.5]2.
+rough surfaces S based on random data and heat filters. The desired reaction-
+diffusion systems are then solved on S.
+5.1
+Construction of S by heat filters
+We want to generate some random rough surfaces S, which have similar rough-
+ness as rough surfaces M in (7) with different M and N.
+Let [X, Y ] ∈ RnXnY ×2 ⊂ I2 as in Section 3.
+We assign uniform ran-
+dom numbers to each node to obtain the initial random surface values ˜Z0 ∼
+�
+U[−1, 1]
+�nXnY at nodes [X, Y ]. We define the discretized heat filter according
+to the method of Section 3 but with some filter-diffusion tensor F (instead of
+the diffusion tensor A for surface M) in the Laplace-Beltrami operator △F,h.
+For an isotropic filter, we take F = I2×2. This can generate an M-like surface
+with M = N. For the anisotropic case with M ̸= N, we use F = diag(2, 1). We
+can now smooth the surface data J-times via
+˜Zj+1 = (Q + κh△F,h) ˜Zj,
+for j = 0, . . . , J,
+where Q is an nXnY by nXnY matrix of ones, κ > 0 is the parameter to control
+the weights of △F,h, and h is the fill distance of the discrete set. This completes
+the definition of the pre-surface that is the counterpart to ˜Z in (7) of the surface
+15
+
+0.5
+10
+5
+0
+-5
+-0.5
+-0.5
+0
+0.50.4
+0.1
+0
+0.2
+-0.1
+0.3
+0
+X
+0.2
+0.1
+y0.5
+10
+5
+0
+0
+5
+-0.5
+-0.5
+0
+0.50.1
+0
+-0.1
+0.1
+0
+0
+-0.1
+-0.2
+y
+-0.2
+X
+-0.42-D view
+M = 5, N = 15
+Zoom in 3-D view
+M = 15 = N
+Figure 8:
+Patterns on rough surfaces with amplitude δM ∈ {0.05} and
+(M, N) ∈ {(5, 15), (15, 15)}. The model parameters for stripes are set according
+to Table 1. Discretization parameters are nX = 170 = nY , τ = 0.5, T = 4000.
+16
+
+0.5
+0.1
+0.05
+0
+0
+-0.05
+-0.1
+-0.5
+0.15
+-0.5
+0
+0.50.05
+-0.05
+-0.2
+0.5
+0
+0
+0.2
+-0.5
+X
+y0.5
+0.1
+0.05
+0
+0
+-0.05
+-0.1
+-0.5
+-0.15
+-0.5
+0
+0.50.05
+0
+-0.05
+0
+-0.2
+-0.4
+-0.1
+-0.2
+-0.3
+y
+-0.4
+-0.5
+X2-D view
+M = 5, N = 15
+Zoom in 3-D view
+M = 15 = N
+Figure 9:
+Patterns on rough surfaces with amplitude δM = 0.1 and (M, N) ∈
+{(5, 15), (15, 15)}. The model parameters for stripes are set according to Table 1.
+Discretization parameters are nX = 170 = nY , τ = 0.5, T = 4000.
+17
+
+0.5
+0.1
+0.05
+0
+0
+-0.05
+-0.1
+-0.5
+0.15
+-0.5
+0
+0.50.1
+0
+-0.1
+-0.2
+-0.3
+0.4
+-0.4
+0.2
+X
+-0.5
+y
+00.5
+0.1
+0.05
+0
+0
+-0.05
+-0.1
+-0.5
+0.15
+-0.5
+0.5
+0-0.1
+0.1
+-0.2
+0
+-0.1
+-0.3
+0.4
+-0.4
+0.2
+X
+-0.5
+0
+y(a)
+(b)
+(c)
+(d)
+Figure 10: On random rough surface M with parameters in Table 1 and Table 2,
+actual animal coat patterns simulation results: (a) emperor angelfish in [26]; (b)
+genet in [1]; (c) plecostomus in [12]; (d) cheetah in [46].
+18
+
+(a)
+(b)
+(c)
+Figure 11:
+Rough surfaces S with κ = 2 and the number of filter steps J =
+15, under various filter-diffusion tensors F: (a) F = diag(0.01, 1), (b) F =
+diag(1, 1), (c) F = diag(5, 1).
+type M. Similar to the scaling in (7), we define S only by nodal values
+�
+[X, Y, Z] : Z =
+δS
+∥ ˜ZJ∥∞
+˜ZJ
+�
+⊂ S,
+for some amplitude δS > 0.
+Figure 11(a, b, c) shows the rough surfaces S derived for fixed nX =
+90, nY = nX, κ = 2 and J = 15 filter steps under filter-diffusion tensors
+F ∈ {diag(0.01, 1), diag(1, 1), diag(5, 1)},
+respectively.
+Taking F(1, 1) = 1 yields a rough surface S which is similar
+to a rough surface M with equal frequencies M, N (cf.
+Figure 2).
+Taking
+F(1, 1) = 0.01 scales down space in the x-direction, yielding a rough surface
+S which is similar to a rough surface M with a larger frequency M, M > N.
+Conversely, taking F(1, 1) = 5, a rough surface S is obtained which is similar
+to M with a smaller frequency M, M < N.
+To reproduce the properties of surface M, the required value of filter-diffusion
+tensors F and number J of filtering steps will depend on the density of the given
+data points [X, Y ].
+For fixed amplitude δS = 1E − 3 and nX = 90, nY = nX,
+Table 3 gives values of κ, F and the number of filtering steps J to generate
+surfaces that give good qualitative agreement with the rough surfaces M in (7)
+for various M, N. Figure 2 gives a comparison showing that rough surfaces S
+by heat filters (column 2) are qualitatively similar to our earlier surfaces M by
+(7) (see column 1).
+While not the focus of the current work, our construction methods for S
+can be extended to generate rough closed manifolds which can be used as the
+surfaces for the numerical approximation of PDEs. See Figure 12 for a graph-
+ical illustration of rough closed manifolds generated by type-M and type-S
+approaches respectively. Figure 12(a) was obtained by applying the type-M
+procedure to the parameter space (θ, φ) ∈ [0, 2π] × [0, π] with roughness added
+to the constant function r = 1. Figure 12(b) was obtained by adding noise to
+r = 1 for points on the unit sphere. In contrast to that in (13), the heat filter
+here makes use of the discrete Laplace-Beltrami operator for the rough sphere.
+19
+
+Table 3: Coefficients for constructing the surfaces S that appear in the second column of Figure 2.
+Parameters are chosen to give qualitative agreement with the rough surfaces M (7) with δS =
+1E − 3, nX = 90 = nY
+M in (7)
+S by heat filter
+κ
+F
+filter number
+[M, N] = [5, 5]
+5
+diag(1, 1)
+15
+[M, N] = [5, 15]
+8
+diag(1, 0.01)
+10
+[M, N] = [15, 15]
+0.2
+diag(20, 20)
+2
+(a)
+(b)
+Figure 12: Graphical illustration of rough closed manifolds by(a) type-M, and
+(b) type-S approaches on parameter size and manifold respectively.
+20
+
+0
+0
+1
+0
+-10
+0
+1
+1
+0Figure 13: For δS = 0.1, nX = 90, nY = nX, spot and stripe patterns on
+a rough surface S with number of filter steps J = 15, F = diag(1, 1), and
+κ = 5.
+These values from Table 3 give a rough surface similar to M with
+M = 5, N = 15. Parameters for spots and stripes are set according to Table 1.
+The FDM in Section 3 can be used to solve PDEs on rough surfaces S.
+The only difference in the method is that we no longer have the parametric
+equation to calculate the metric tensor G in (2) and hence the diffusion tensor
+A in (13). Instead of computing metric tensor G analytically as in Section 3
+and 4, centered finite difference formulas are applied to approximate the metric
+tensor G. For solving reaction-diffusion systems on rough surfaces S, we set
+initial conditions to be steady state solutions on a zero amplitude rough surface
+M (7). Figure 13 plots the spot and stripe patterns on a rough surface S using
+the reaction-diffusion parameters provided in Table 1. As shown in the second
+line of Table 3, rough surface S takes κ = 5, F = diag(1, 1) with J = 15 filter
+steps to approximate rough surface M with M = 5, N = 5. It can be seen that
+the spot and stripe patterns generated on S are similar to those on M under
+parameters M = 5, N = 5, δM = 0.1 (see the third row of Figure 5).
+We conclude by simulating the same set of animal coats displayed in Fig-
+ure 1 using Turing models on rough surfaces S with parameters set according
+to Table 1 and Table 4.
+Figure 14 demonstrates again that adding surface
+roughness into the process of pattern generation can indeed provide more var-
+ied simulations for animal coats. Moreover, the patterns on surface S provide
+valid simulations just like M.
+Table 4: Parameters for animal coats simulation on S with nX = nY = 90
+κ
+F
+J
+τ
+T
+δS
+Emperor angelfish [26]
+Figure 14 (a)
+5
+diag(1, 1)
+15
+0.5
+400
+0.05
+Genet [1]
+Figure 14 (b)
+8
+diag(1, 0.01)
+10
+0.5
+800
+0.1
+Plecostomus [12]
+Figure 14 (c)
+8
+diag(1, 0.01)
+10
+0.5
+3000
+0.1
+Cheetah [46]
+Figure 14 (d)
+0.2
+diag(20, 20)
+2
+0.5
+400
+0.05
+21
+
+(a)
+(b)
+(c)
+(d)
+Figure 14: On random rough surface S with parameters in Table 1 and Table 4,
+actual animal coat patterns simulation results: (a) emperor angelfish in [26]; (b)
+genet in [1]; (c) plecostomus in [12]; (d) cheetah in [46].
+22
+
+6
+Conclusion
+In this paper, we discussed pattern formation on rough surfaces, with the surface
+characterized by two different methods: spatial frequency and discretized heat
+filters. The patterns generated on random rough surfaces of different ampli-
+tudes and spatial frequencies are illustrated, and the simulation results indicate
+that the patterns became relatively more irregular as amplitude increases. The
+change of spatial frequencies on the x and y axes will also lead to pattern defor-
+mation along the x and y directions. Numerical results indicate that combining
+reaction-diffusion systems with rough surfaces can provide better simulations
+for animal coat patterns in the real world. What’s more, the method for gen-
+erating rough surfaces by heat filters can be further applied to obtain closed
+rough manifolds. We plan to explore this generalization in future work.
+Appendix: Finite difference algorithm for the heat
+equation
+The heat equation (11) on a rough surface defined over the parameter space
+I2 = [−1, 1]2 is considered. The set of discrete data points on I2 are defined as:
+[X, Y ] :=
+��
+(xi
+1, xj
+2)
+�nX
+i=1
+�nY
+j=1 ∈ RnXnY ×2,
+with mesh size hx1, hx2 on each axis. To fully discretize (12), we require dis-
+cretization of the Laplacian-Beltrami operator. For any twice differentiable func-
+tion u : I2 → R, we introduce differentiation matrices Dk ∈ RnXnY ×nXnY , k ∈
+{1, 2} satisfying periodic boundary condition (11b) as follows. Using a second-
+order centered finite differences, we have for each point (xi
+1, xj
+2) that
+∂u
+∂x1
+(xi
+1, xj
+2) =
+�
+−
+1
+2hx1
+1
+2hx1
+� �u(xi−1
+1
+, xj
+2)
+u(xi+1
+1
+, xj
+2)
+�
+,
+∂u
+∂x2
+(xi
+1, xj
+2) =
+�
+−
+1
+2hx2
+1
+2hx2
+� �u(xi
+1, xj−1
+2
+)
+u(xi
+1, xj+1
+2
+)
+�
+.
+The following fictitious node approach is applied to deal with the periodic
+boundary conditions:
+u(x0
+1, xj
+2) = u(xnX−1
+1
+, xj
+2), u(xnX+1
+1
+, xj
+2) = u(x2
+1, xj
+2), u(xi
+1, x0
+2) = u(xi
+1, xnY −1
+2
+),
+u(xi
+1, xnY +1
+2
+) = u(xi
+1, x2
+2), u(x1
+1, xj
+2) = u(xnX
+1 , xj
+2),
+u(xi
+1, x1
+2) = u(xi
+1, xnY
+2 ).
+By assembling all points in [X, Y ], the nodal values of
+∂u
+∂xk (k = 1, 2) at [X, Y ]
+can be obtained by
+∂u
+∂xk
+(X, Y ) ≈ Dku(X, Y ),
+for k ∈ {1, 2},
+23
+
+where u(X, Y ) := [u(x1
+1, x1
+2), · · · , u(xnX
+1 , x1
+2), · · · , u(x1
+1, xnY
+2 ), · · · , u(xnX
+1 , xnY
+2 )]T
+is the vector of nodal function values and differential matrices Dk ∈ RnXnY ×nXnY
+are in the form of
+D1 =
+�
+��
+DB
+1
+· · ·
+0
+...
+...
+...
+0
+· · ·
+DB
+1
+�
+�� , DB
+1 =
+1
+2hx1
+�
+������
+0
+1
+0
+· · ·
+0
+−1
+0
+−1
+0
+1
+· · ·
+0
+0
+0
+...
+...
+...
+...
+...
+...
+0
+0
+0
+· · ·
+−1
+0
+1
+1
+0
+0
+· · ·
+0
+0
+−1
+�
+������
+,
+and
+D2 =
+�
+������
+0
+DB
+2
+0
+· · ·
+0
+−DB
+2
+0
+−DB
+2
+0
+DB
+2
+· · ·
+0
+0
+0
+...
+...
+...
+...
+...
+...
+...
+0
+0
+0
+· · ·
+−DB
+2
+0
+DB
+2
+DB
+2
+0
+0
+· · ·
+0
+0
+−DB
+2
+�
+������
+, DB
+2 =
+1
+2hx2
+�
+��
+1
+· · ·
+0
+...
+...
+...
+0
+· · ·
+1
+�
+�� ,
+with DB
+k ∈ RnX×nX, k ∈ {1, 2}.
+References
+[1] https://commons.wikimedia.org/wiki/Category:Genetta%
+20tigrina?uselang=it.
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+ment strategy for reaction-diffusion systems on evolving domains, Discrete
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+droplets in a self-organizing inhibitory system, SIAM Journal on Applied
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+[35] J. Murray, A pre-pattern formation mechanism for animal coat markings,
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+of random surface topography modeling, Tribology International, (2020),
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+ing partial differential equations on surfaces, Journal of Computational
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+ity formulation for stable RBF-FD discretizations of advection-diffusion-
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+42 (2020), pp. A2371–A2401.
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+Letters, 122 (2021), p. 107534.
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+Lond., 237 (1952), pp. 37–72.
+27
+
diff --git a/XdFPT4oBgHgl3EQfsTWZ/content/tmp_files/load_file.txt b/XdFPT4oBgHgl3EQfsTWZ/content/tmp_files/load_file.txt
new file mode 100644
index 0000000000000000000000000000000000000000..ffedddd153dca00be4d5941c1d9d8eaf99dc97f3
--- /dev/null
+++ b/XdFPT4oBgHgl3EQfsTWZ/content/tmp_files/load_file.txt
@@ -0,0 +1,872 @@
+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFPT4oBgHgl3EQfsTWZ/content/2301.13148v1.pdf,len=871
+page_content='Realistic pattern formations on rough surfaces∗  Siqing LI†  Leevan LING‡  Steven J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFPT4oBgHgl3EQfsTWZ/content/2301.13148v1.pdf'}
+page_content=' RUUTH§  Xuemeng Wang¶  January 31, 2023  Abstract  We are interested in simulating patterns on rough surfaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFPT4oBgHgl3EQfsTWZ/content/2301.13148v1.pdf'}
+page_content='  First,  we consider periodic rough surfaces with analytic parametric equations,  which are defined by some superposition of wave functions with random  frequencies and angles of propagation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFPT4oBgHgl3EQfsTWZ/content/2301.13148v1.pdf'}
+page_content=' The amplitude of such surfaces  is also an important variable in the provided eigenvalue analysis for the  Laplace-Beltrami operator and in our numerical studies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFPT4oBgHgl3EQfsTWZ/content/2301.13148v1.pdf'}
+page_content=' Simulations show  that the patterns become irregular as the amplitude and frequency of the  rough surface increase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFPT4oBgHgl3EQfsTWZ/content/2301.13148v1.pdf'}
+page_content=' Next, for the sake of easy generalization to closed  manifolds, we propose another construction method of rough surfaces by  using random nodal values and discretized heat filters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFPT4oBgHgl3EQfsTWZ/content/2301.13148v1.pdf'}
+page_content=' We provide numer-  ical evidence that both surface constructions yield comparable patterns  to those found in real-life animals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFPT4oBgHgl3EQfsTWZ/content/2301.13148v1.pdf'}
+page_content='  keywords Laplace-Beltrami operator, reaction-diffusion system, random  surfaces, Turing pattern  AMS subject classifications 65M06, 35K57  1  Introduction  Pattern formation by reaction-diffusion systems has been an intensively studied  field for decades.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFPT4oBgHgl3EQfsTWZ/content/2301.13148v1.pdf'}
+page_content=' In 1952, Turing proposed the idea of diffusion-driven instabil-  ity [47], in which simple mechanisms evolve from a homogeneous state into spa-  tial heterogeneous patterns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFPT4oBgHgl3EQfsTWZ/content/2301.13148v1.pdf'}
+page_content=' In recent years, a variety of application areas have  ∗This work was funded by the Hong Kong Research Grant Council GRF Grants  (12303818,12301419,12301520), a National Youth Science Foundation of China (12201449),  and the financial support of NSERC Canada (RGPIN-2022-03302).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFPT4oBgHgl3EQfsTWZ/content/2301.13148v1.pdf'}
+page_content='  †College  of  Data  Science,  Taiyuan  University  of  Technology,  Shanxi,  China  (lisiqing@tyut.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFPT4oBgHgl3EQfsTWZ/content/2301.13148v1.pdf'}
+page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFPT4oBgHgl3EQfsTWZ/content/2301.13148v1.pdf'}
+page_content='cn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFPT4oBgHgl3EQfsTWZ/content/2301.13148v1.pdf'}
+page_content='  ‡Department of Mathematics, Hong Kong Baptist University, Kowloon Tong, Hong Kong  (lling@hkbu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFPT4oBgHgl3EQfsTWZ/content/2301.13148v1.pdf'}
+page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFPT4oBgHgl3EQfsTWZ/content/2301.13148v1.pdf'}
+page_content='hk).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFPT4oBgHgl3EQfsTWZ/content/2301.13148v1.pdf'}
+page_content='  §Department of Mathematics, Simon Fraser University, Burnaby, British Columbia,  Canada V5A1S6 (sruuth@sfu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFPT4oBgHgl3EQfsTWZ/content/2301.13148v1.pdf'}
+page_content='ca).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFPT4oBgHgl3EQfsTWZ/content/2301.13148v1.pdf'}
+page_content='  ¶Department of Mathematics, University of British Columbia, Vancouver, Canada, and  Department of Mathematics, Hong Kong Baptist University, Kowloon Tong, Hong Kong  (17251109@life.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFPT4oBgHgl3EQfsTWZ/content/2301.13148v1.pdf'}
+page_content='hkbu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFPT4oBgHgl3EQfsTWZ/content/2301.13148v1.pdf'}
+page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFPT4oBgHgl3EQfsTWZ/content/2301.13148v1.pdf'}
+page_content='hk).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFPT4oBgHgl3EQfsTWZ/content/2301.13148v1.pdf'}
+page_content='  1  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFPT4oBgHgl3EQfsTWZ/content/2301.13148v1.pdf'}
+page_content='13148v1  [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFPT4oBgHgl3EQfsTWZ/content/2301.13148v1.pdf'}
+page_content='NA]  13 Jan 2023  (a)  (b)  (c)  (d)  Figure 1: Skin patterns in real-life: (a) emperor angelfish [26], (b) genet [1], (c)  pecostomus [12], (d) cheetah [46].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFPT4oBgHgl3EQfsTWZ/content/2301.13148v1.pdf'}
+page_content='  been actively developed, including vegetation pattern, plant root hair initiation,  flock simulation and boundary drop configurations [4,7,13,32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFPT4oBgHgl3EQfsTWZ/content/2301.13148v1.pdf'}
+page_content=' Mechanisms of  pattern generation under different types of transport or domain size have been  considered as well;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFPT4oBgHgl3EQfsTWZ/content/2301.13148v1.pdf'}
+page_content=' see [6,25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFPT4oBgHgl3EQfsTWZ/content/2301.13148v1.pdf'}
+page_content=' In theoretical biology, reaction-diffusion systems  provide a relatively generic and concise approach for simulating animal skin pat-  terns [18,35], as well as regenerative processes of organisms [21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFPT4oBgHgl3EQfsTWZ/content/2301.13148v1.pdf'}
+page_content=' In particular,  reaction-diffusion systems are a well-accepted class of models for multiple pig-  mentation processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFPT4oBgHgl3EQfsTWZ/content/2301.13148v1.pdf'}
+page_content=' Examples include marine [27] and emperor [26] angelfish,  genets [1], plecostomi [12], cheetah [46], zebra fish [2], and many other mam-  mal skin patterns [34];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFPT4oBgHgl3EQfsTWZ/content/2301.13148v1.pdf'}
+page_content=' see Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFPT4oBgHgl3EQfsTWZ/content/2301.13148v1.pdf'}
+page_content=' Although animal skin pattern simulation  through reaction-diffusion systems has been explored in many researches, see  the second row of Figure 5 for typical examples, very few of them have taken  the skin texture into account.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFPT4oBgHgl3EQfsTWZ/content/2301.13148v1.pdf'}
+page_content=' In reality, the skin surfaces are often rugged, and  consequently the pattern will actually not be regular spots or stripes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFPT4oBgHgl3EQfsTWZ/content/2301.13148v1.pdf'}
+page_content=' There-  fore, combining surface properties with reaction-diffusion systems can provide  an opportunity for improved simulation of real pattern.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFPT4oBgHgl3EQfsTWZ/content/2301.13148v1.pdf'}
+page_content=' In [11, 30], parameter  functions instead of constants were used in PDEs to generate nonuniform and  complex, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFPT4oBgHgl3EQfsTWZ/content/2301.13148v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFPT4oBgHgl3EQfsTWZ/content/2301.13148v1.pdf'}
+page_content=', more real-life, patterns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFPT4oBgHgl3EQfsTWZ/content/2301.13148v1.pdf'}
+page_content='  Characterizing random rough surfaces can be carried out by a variety of  methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFPT4oBgHgl3EQfsTWZ/content/2301.13148v1.pdf'}
+page_content=' These include the reference parameter method, motif method, fractal  method, watershed method, and wavelet method, etc [42].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFPT4oBgHgl3EQfsTWZ/content/2301.13148v1.pdf'}
+page_content=' For three-dimensional  surface topography, there are a number of parameters involved [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFPT4oBgHgl3EQfsTWZ/content/2301.13148v1.pdf'}
+page_content=' To generate  random rough surfaces, both the Fast Fourier Transform (FFT) [37] and digital  filter [23] methods can be applied to approximate the auto-correlation function  for surfaces with wavelengths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFPT4oBgHgl3EQfsTWZ/content/2301.13148v1.pdf'}
+page_content=' In [24], the authors proposed an approach that  applies the covariance function and Karhunen-Lo`eve expansion to generate ran-  dom surfaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFPT4oBgHgl3EQfsTWZ/content/2301.13148v1.pdf'}
+page_content=' Rough surfaces may also be generated from spatial frequencies  via the FFT method;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFPT4oBgHgl3EQfsTWZ/content/2301.13148v1.pdf'}
+page_content=' in this case, the rough surface is built by summing up  2  trigonometric functions [43].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFPT4oBgHgl3EQfsTWZ/content/2301.13148v1.pdf'}
+page_content='  In Section 2, the rough surface M with parametric equations will be for-  mulated in detail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFPT4oBgHgl3EQfsTWZ/content/2301.13148v1.pdf'}
+page_content=' We consider periodic rough surfaces M characterized with  spatial frequency, which come with analytic parametric equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFPT4oBgHgl3EQfsTWZ/content/2301.13148v1.pdf'}
+page_content=' In Section 5,  we consider surfaces S constructed by random nodal values and discretized heat  filters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFPT4oBgHgl3EQfsTWZ/content/2301.13148v1.pdf'}
+page_content=' The latter surface construction technique can be easily extended to add  “roughness” to more general manifolds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFPT4oBgHgl3EQfsTWZ/content/2301.13148v1.pdf'}
+page_content=' Along with the construction of rough  surfaces, we will review the concept of the heat equation which is a surface PDE  involving the Laplace operator on manifolds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFPT4oBgHgl3EQfsTWZ/content/2301.13148v1.pdf'}
+page_content='  Moving from a flat geometry to rough surfaces, generally speaking, we have  to deal with surface dependent differential operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFPT4oBgHgl3EQfsTWZ/content/2301.13148v1.pdf'}
+page_content=' Surface PDEs have been  extensively studied in recent years, and there are a variety of techniques for their  analysis and computation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFPT4oBgHgl3EQfsTWZ/content/2301.13148v1.pdf'}
+page_content=' The approaches for surface PDEs can be separated  into intrinsic methods and embedding methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFPT4oBgHgl3EQfsTWZ/content/2301.13148v1.pdf'}
+page_content=' Intrinsic methods solve PDEs  on the manifold directly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFPT4oBgHgl3EQfsTWZ/content/2301.13148v1.pdf'}
+page_content='  However, embedding methods formulate and solve  PDEs on a band around the manifold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFPT4oBgHgl3EQfsTWZ/content/2301.13148v1.pdf'}
+page_content=' Intrinsic and embedding methods that  are dependent on mesh construction have been explored by many researchers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFPT4oBgHgl3EQfsTWZ/content/2301.13148v1.pdf'}
+page_content='  Examples of such methods include finite difference [40], finite element [5,16,36],  and finite volume [14,15] methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFPT4oBgHgl3EQfsTWZ/content/2301.13148v1.pdf'}
+page_content=' There are also numerical methods that do  not require meshes, namely meshfree methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFPT4oBgHgl3EQfsTWZ/content/2301.13148v1.pdf'}
+page_content=' Meshfree methods for surface  PDEs, such as radial basis function (RBF) methods [8–10,19] and the meshfree  generalized finite difference method [44,45], have the advantage of avoiding mesh  construction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFPT4oBgHgl3EQfsTWZ/content/2301.13148v1.pdf'}
+page_content='  In Section 3, we apply the finite difference method to solve pattern forma-  tion PDEs on rough surfaces M;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFPT4oBgHgl3EQfsTWZ/content/2301.13148v1.pdf'}
+page_content=' the FDM is chosen because of its simplicity  and effectiveness on the class of problem domains considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFPT4oBgHgl3EQfsTWZ/content/2301.13148v1.pdf'}
+page_content=' The rough sur-  face pattern formation models here are not yet extendable to add roughness to  other manifolds, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFPT4oBgHgl3EQfsTWZ/content/2301.13148v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFPT4oBgHgl3EQfsTWZ/content/2301.13148v1.pdf'}
+page_content=', we cannot yet create red-blood cells with rough surfaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFPT4oBgHgl3EQfsTWZ/content/2301.13148v1.pdf'}
+page_content='  Our aim is to study the effect of roughness on pattern formation using the sim-  plest setup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFPT4oBgHgl3EQfsTWZ/content/2301.13148v1.pdf'}
+page_content=' Numerical simulation results on rough surfaces M with parametric  equations are demonstrated in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFPT4oBgHgl3EQfsTWZ/content/2301.13148v1.pdf'}
+page_content='  In Section 5, we put forward a different construction method of rough sur-  faces S by random discrete data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFPT4oBgHgl3EQfsTWZ/content/2301.13148v1.pdf'}
+page_content=' After updating our numerical schemes for M  to work on S, we see that surface types M and S yield comparable patterns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFPT4oBgHgl3EQfsTWZ/content/2301.13148v1.pdf'}
+page_content='  In Section 4 and Section 5, we present some simulated animal coat patterns on  both types of surfaces, M and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFPT4oBgHgl3EQfsTWZ/content/2301.13148v1.pdf'}
+page_content=' Results for PDEs with constant parameters  on rough surfaces with various roughness are similar to the real-life patterns  displayed in Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFPT4oBgHgl3EQfsTWZ/content/2301.13148v1.pdf'}
+page_content='  2  Rough surfaces M with analytic parametric  equations  Consider Ck–smooth (k ≥ 2), codimension 1, and periodic Riemannian surfaces  M =  �  (x, y, z) ∈ R3 : z = z(x, y) for (x, y) ∈ V ⊂ R2�  ,  (1)  3  for some Ck function z(·, ·) defined on a global parameter space V ⊂ R2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFPT4oBgHgl3EQfsTWZ/content/2301.13148v1.pdf'}
+page_content=' A  corresponding parametric representation of M is given by  ⃗r(x, y) =  �  x, y, z(x, y)  �T ∈ M,  (x, y) ∈ V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFPT4oBgHgl3EQfsTWZ/content/2301.13148v1.pdf'}
+page_content='  For convenience, let (x, y) := (x1, x2);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFPT4oBgHgl3EQfsTWZ/content/2301.13148v1.pdf'}
+page_content=' we shall use both notations interchange-  ably.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFPT4oBgHgl3EQfsTWZ/content/2301.13148v1.pdf'}
+page_content=' Recall that the first fundamental form G : V → R2×2 of M is defined  by  G(x, y) =  �  g11  g12  g21  g22  �  (x, y)  where  gij(x, y) = ∂⃗r(x, y)  ∂xi  ∂⃗r(x, y)  ∂xj  ,  see [16] for a review.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFPT4oBgHgl3EQfsTWZ/content/2301.13148v1.pdf'}
+page_content=' In the case of a surface function M as in (1), we have  G(x, y) =  �g11  g12  g21  g22  �  (x, y) =  �1 + z2  x  zxzy  zxzy  1 + z2  y  �  ,  (2)  whose determinant and inverse are  g(x, y) := det(G)(x, y) = 1 + z2  x + z2  y,  (3)  and  G−1(x, y) =  �g11  g12  g21  g22  �  (x, y) =  1  1 + z2x + z2y  � 1 + z2  y  −zxzy  −zxzy  1 + z2  x  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFPT4oBgHgl3EQfsTWZ/content/2301.13148v1.pdf'}
+page_content='  (4)  Using (3)–(4), the Laplace-Beltrami operator on M is given by  ∆Mf =  1  √g  �  1≤i,j≤2  ∂  ∂i  �√ggij ∂  ∂j  f  �  =:  1  √g ∇ ·  �  A∇f  �  ,  (5)  for any C2 function f : M → R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFPT4oBgHgl3EQfsTWZ/content/2301.13148v1.pdf'}
+page_content='  Now, we focus on random rough surfaces M ⊆ R3 in the form of (1), whose  surface function z : I2 = [−L, L]2 → R is a superposition of elementary waves  [17,43].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFPT4oBgHgl3EQfsTWZ/content/2301.13148v1.pdf'}
+page_content=' The function z is stochastically determined by  z(x, y) =  M  �  m=−M  N  �  n=−N  am,n cos  �  2π(mx + ny) + φm,n  �  ,  (6)  for some random variables am,n and φm,n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFPT4oBgHgl3EQfsTWZ/content/2301.13148v1.pdf'}
+page_content=' Similar to a Fourier series expansion,  z(x, y) in (6) is constructed by trigonometric functions in which m, n correspond  to spatial frequencies on the x and y axes respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFPT4oBgHgl3EQfsTWZ/content/2301.13148v1.pdf'}
+page_content=' As in [43], the spatial  frequencies m and n allow values taken up to maximum integers M and N  respectively;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFPT4oBgHgl3EQfsTWZ/content/2301.13148v1.pdf'}
+page_content=' this corresponds to a high frequency cut off.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFPT4oBgHgl3EQfsTWZ/content/2301.13148v1.pdf'}
+page_content='  To determine (6), we first introduce ˜am,n ∼ N(0, 1) and φm,n ∼ U(0, π).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFPT4oBgHgl3EQfsTWZ/content/2301.13148v1.pdf'}
+page_content='  This specifies the pre-surface function ˜z = ˜z(x, y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFPT4oBgHgl3EQfsTWZ/content/2301.13148v1.pdf'}
+page_content=' Next, we control the ampli-  tude of the rough surface M to be within a specific range [−δM, δM] by scaling  according to  z :=  δM  ∥˜z∥∞  ˜z,  i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFPT4oBgHgl3EQfsTWZ/content/2301.13148v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFPT4oBgHgl3EQfsTWZ/content/2301.13148v1.pdf'}
+page_content=', am,n :=  δM  ∥˜z∥∞  ˜am,n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFPT4oBgHgl3EQfsTWZ/content/2301.13148v1.pdf'}
+page_content='  (7)  4  M  (a) M = 5 = N  Lower roughness  S  (d)  (b) M = 5, N = 15  Increasing M, N  ←−−−−−−−−−−−−−−  (e)  (c) M = 15 = N  Higher roughness  (f)  Figure 2: Bird’s-eye view of some random rough surfaces M from Section 2  (left) and S from Section 5 (right).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFPT4oBgHgl3EQfsTWZ/content/2301.13148v1.pdf'}
+page_content='  Figures 2 (a)–(c), first column, show some rough surfaces M with amplitude  δM = 10−3 and various (M, N).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFPT4oBgHgl3EQfsTWZ/content/2301.13148v1.pdf'}
+page_content='  The construction of the rough surfaces S  appearing in the second column will be discussed in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFPT4oBgHgl3EQfsTWZ/content/2301.13148v1.pdf'}
+page_content='  Remark: One can also add a decay condition with respect to frequencies  m, n and a frequency attenuation parameter β to the pre-coefficient ˜am,n via  ˜am,n ∼  1  (m2 + n2)β/2 N(µ, σ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFPT4oBgHgl3EQfsTWZ/content/2301.13148v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFPT4oBgHgl3EQfsTWZ/content/2301.13148v1.pdf'}
+page_content='1  Surface roughness and Laplace-Beltrami operator  Applying the definition of the Laplace-Beltrami operator (5) to the rough sur-  face function in (6)–(7), the relationship between the eigenvalues of the diffusion  tensor and the geometry of the surface can be made explicit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFPT4oBgHgl3EQfsTWZ/content/2301.13148v1.pdf'}
+page_content=' Firstly, the diffu-  5  sion tensor A in (5) is  A(x, y) =  �A1  A2  A2  A4  �  (x, y) := √gG−1 =  1  √g  �1 + z2  y  −zxzy  −zxzy  1 + z2  x  �  ,  (8)  with the partial derivatives zx and zy computing from (6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFPT4oBgHgl3EQfsTWZ/content/2301.13148v1.pdf'}
+page_content=' We are interested in  obtaining the eigenvalues and eigenvectors of A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFPT4oBgHgl3EQfsTWZ/content/2301.13148v1.pdf'}
+page_content=' Rather than computing these  quantities directly, it turns out to be easier to first compute the eigenvalues and  eigenvectors of the Riemannian metric tensor G in (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFPT4oBgHgl3EQfsTWZ/content/2301.13148v1.pdf'}
+page_content=' From the characteristic  equation of G  |G − λI| = (1 + z2  x − λ)(1 + z2  y − λ) − z2  xz2  y = 0,  we find that the eigenvalues λG of G are 1 + z2  x + z2  y = g and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFPT4oBgHgl3EQfsTWZ/content/2301.13148v1.pdf'}
+page_content=' Since  A⃗v = √gG−1⃗v =  √g  λG ⃗v,  (9)  any eigenvector ⃗v of G is also an eigenvector of A and the eigenvalues λA of A  are  λA =  �√g =  �  1 + z2x + z2y,  1  √g  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFPT4oBgHgl3EQfsTWZ/content/2301.13148v1.pdf'}
+page_content='  (10)  Note that, by (7), the matrix function [G−λGI](zx, zy) = C(δM)[G−λGI](˜zx, ˜zy)  for some δM-dependent constant C(δM).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFPT4oBgHgl3EQfsTWZ/content/2301.13148v1.pdf'}
+page_content=' This implies that all eigendirections  of G (and A) are independent of the amplitude δM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFPT4oBgHgl3EQfsTWZ/content/2301.13148v1.pdf'}
+page_content=' Simple calculations show  that λA = 1 + O(M 2N 2δ2  M) varies with δM nonlinearly by a bijection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFPT4oBgHgl3EQfsTWZ/content/2301.13148v1.pdf'}
+page_content=' Thus,  the contour lines (but not the height) of λA are independent of δM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFPT4oBgHgl3EQfsTWZ/content/2301.13148v1.pdf'}
+page_content='  Figure 3 illustrates the close relationship between the eigenvalues of A and  the geometric properties of a rough surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFPT4oBgHgl3EQfsTWZ/content/2301.13148v1.pdf'}
+page_content='  In particular, we can see from  subfigures (b), (c), and (e) that the maximum (and minimum) eigenvalues are  larger (and smaller) at regions of the surface with the steepest gradients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFPT4oBgHgl3EQfsTWZ/content/2301.13148v1.pdf'}
+page_content=' In  subfigures (d) and (f), eigendirections are plotted on top of subfigure (b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFPT4oBgHgl3EQfsTWZ/content/2301.13148v1.pdf'}
+page_content=' We can  clearly see that the eigenvectors corresponding to the maximum (and minimum)  eigenvalues are tangent (and orthogonal) to the contour plots of eigenvalues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFPT4oBgHgl3EQfsTWZ/content/2301.13148v1.pdf'}
+page_content='  3  Solving heat equations on rough surfaces M  We begin by constructing a finite difference scheme for solving heat equations on  rough surfaces M ⊂ R3 defined by (6)–(7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFPT4oBgHgl3EQfsTWZ/content/2301.13148v1.pdf'}
+page_content=' We work intrinsically by transform-  ing the PDEs on rough surfaces to the parameter space I2 = [−L, L]2 ⊂ R2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFPT4oBgHgl3EQfsTWZ/content/2301.13148v1.pdf'}
+page_content='  Subsequently, we move on to solving reaction-diffusion systems for pattern for-  mation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFPT4oBgHgl3EQfsTWZ/content/2301.13148v1.pdf'}
+page_content='  We consider the heat equation on a rough surface  ∂u  ∂t (ξ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFPT4oBgHgl3EQfsTWZ/content/2301.13148v1.pdf'}
+page_content=' t) − ∆Mu(ξ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFPT4oBgHgl3EQfsTWZ/content/2301.13148v1.pdf'}
+page_content=' t) = h(ξ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFPT4oBgHgl3EQfsTWZ/content/2301.13148v1.pdf'}
+page_content=' t)  for ξ ∈ M,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFPT4oBgHgl3EQfsTWZ/content/2301.13148v1.pdf'}
+page_content=' t ∈ (0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFPT4oBgHgl3EQfsTWZ/content/2301.13148v1.pdf'}
+page_content=' T],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFPT4oBgHgl3EQfsTWZ/content/2301.13148v1.pdf'}
+page_content='  (11a)  6  (a) M = 1 = N  (b) Contour of a rough surface M  (c) maximum eigenvalue λA  max  (d) eigendirection of λA  max  (e) minimum eigenvalue λA  min  (f) eigendirection of λA  min  Figure 3:  (a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFPT4oBgHgl3EQfsTWZ/content/2301.13148v1.pdf'}
+page_content=' b): 3D and contour plots of a rough surface in the form of (7)  with M = N = 1 and amplitude δM = 1E −1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFPT4oBgHgl3EQfsTWZ/content/2301.13148v1.pdf'}
+page_content=' (c,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFPT4oBgHgl3EQfsTWZ/content/2301.13148v1.pdf'}
+page_content=' d): maximum eigenvalue and  eigendirection of A,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFPT4oBgHgl3EQfsTWZ/content/2301.13148v1.pdf'}
+page_content=' (e,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFPT4oBgHgl3EQfsTWZ/content/2301.13148v1.pdf'}
+page_content=' f): minimum eigenvalue and eigendirection of A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFPT4oBgHgl3EQfsTWZ/content/2301.13148v1.pdf'}
+page_content='  7  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFPT4oBgHgl3EQfsTWZ/content/2301.13148v1.pdf'}
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+page_content='5  1where u : M × (0, T] → R, subject to periodic boundary conditions on ∂M :=  ∂I2 × z(∂I2) with z(∂I2) being the surface function in (6), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFPT4oBgHgl3EQfsTWZ/content/2301.13148v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFPT4oBgHgl3EQfsTWZ/content/2301.13148v1.pdf'}
+page_content=',  u([−L, y, z(−L, y)]T , t) = u([L, y, z(L, y)]T , t),  for y ∈ I, t ∈ (0, T],  u([x, −L, z(x, −L)]T , t) = u([x, L, z(x, L)]T , t),  for x ∈ I, t ∈ (0, T].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFPT4oBgHgl3EQfsTWZ/content/2301.13148v1.pdf'}
+page_content='  (11b)  Since all surface points are in the form ξ = [x, y, z(x, y)]T ∈ M for [x, y]T ∈  I2 and z in (6), we discretize the parameter space I2 by some set of nXnY  tensor-product grid points [X, Y ] ∈ RnXnY ×2 ⊂ I2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFPT4oBgHgl3EQfsTWZ/content/2301.13148v1.pdf'}
+page_content=' Let uj(x, y) ≈ u([x, y, z(x, y)]T , tj)  for [x, y]T ∈ I2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFPT4oBgHgl3EQfsTWZ/content/2301.13148v1.pdf'}
+page_content=' We begin by temporal discretization of (11a) by the θ-method  uj+1 − uj  τ  = θ  �  ∆Muj+1 + hj+1�  + (1 − θ)  �  ∆Muj + hj�  ,  (12)  for an equispaced partition {tj}M  j=0 of [0, T] with step-size τ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFPT4oBgHgl3EQfsTWZ/content/2301.13148v1.pdf'}
+page_content=' By some user-  selected finite difference scheme for the first derivative, we construct differenti-  ation matrices Dk ∈ RnXnY ×nXnY such that, for any C1-function w : I2 → R  satisfying periodic boundary condition (11b), we have  Dkw(X, Y ) ≈ ∂w  ∂xk  (X, Y ),  for k ∈ {1, 2},  where the nXnY × 1 vector w(X, Y ) (and  ∂w  ∂xk (X, Y )) contains nodal values of  w (and  ∂w  ∂xk ) at grid points in [X, Y ].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFPT4oBgHgl3EQfsTWZ/content/2301.13148v1.pdf'}
+page_content=' Working directly on definitions (5)–(8),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFPT4oBgHgl3EQfsTWZ/content/2301.13148v1.pdf'}
+page_content='  we can approximate the Laplace-Beltrami term ∆Mw at grids [X,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFPT4oBgHgl3EQfsTWZ/content/2301.13148v1.pdf'}
+page_content=' Y ] by nodal  function values W := w(X,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFPT4oBgHgl3EQfsTWZ/content/2301.13148v1.pdf'}
+page_content=' Y ) according to  ∆Mw(X,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFPT4oBgHgl3EQfsTWZ/content/2301.13148v1.pdf'}
+page_content=' Y ) ≈  1  √g (X,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFPT4oBgHgl3EQfsTWZ/content/2301.13148v1.pdf'}
+page_content=' Y )⊛  �  D1  �  A1(X,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFPT4oBgHgl3EQfsTWZ/content/2301.13148v1.pdf'}
+page_content=' Y ) ⊛ (D1W)  �  + D2  �  A4(X,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFPT4oBgHgl3EQfsTWZ/content/2301.13148v1.pdf'}
+page_content=' Y ) ⊛ (D2W)  �  + D1  �  A2(X,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFPT4oBgHgl3EQfsTWZ/content/2301.13148v1.pdf'}
+page_content=' Y ) ⊛ (D2W)  �  + D2  �  A2(X,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFPT4oBgHgl3EQfsTWZ/content/2301.13148v1.pdf'}
+page_content=' Y ) ⊛ (D1W)  ��  =  1  √g (X,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFPT4oBgHgl3EQfsTWZ/content/2301.13148v1.pdf'}
+page_content=' Y )⊚  �  D1  �  A1(X,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFPT4oBgHgl3EQfsTWZ/content/2301.13148v1.pdf'}
+page_content=' Y ) ⊚ D1  �  + D2  �  A4(X,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFPT4oBgHgl3EQfsTWZ/content/2301.13148v1.pdf'}
+page_content=' Y ) ⊚ D2  �  + D1  �  A2(X,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFPT4oBgHgl3EQfsTWZ/content/2301.13148v1.pdf'}
+page_content=' Y ) ⊚ D2  �  + D2  �  A2(X,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFPT4oBgHgl3EQfsTWZ/content/2301.13148v1.pdf'}
+page_content=' Y ) ⊚ D1  ��  W  =: △M,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFPT4oBgHgl3EQfsTWZ/content/2301.13148v1.pdf'}
+page_content='hW,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFPT4oBgHgl3EQfsTWZ/content/2301.13148v1.pdf'}
+page_content='  (13)  where △M,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFPT4oBgHgl3EQfsTWZ/content/2301.13148v1.pdf'}
+page_content='h ∈ RnXnY ×nXnY ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFPT4oBgHgl3EQfsTWZ/content/2301.13148v1.pdf'}
+page_content=' ⊛ is the element-wise Hadamard product of ma-  trices and ⊚ is a vector-matrix product defined by ⃗a ⊚ [⃗b1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFPT4oBgHgl3EQfsTWZ/content/2301.13148v1.pdf'}
+page_content=' · · · ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFPT4oBgHgl3EQfsTWZ/content/2301.13148v1.pdf'}
+page_content='⃗bn] := [⃗a ⊛  ⃗b1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFPT4oBgHgl3EQfsTWZ/content/2301.13148v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFPT4oBgHgl3EQfsTWZ/content/2301.13148v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFPT4oBgHgl3EQfsTWZ/content/2301.13148v1.pdf'}
+page_content=',⃗a ⊛ ⃗bn].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFPT4oBgHgl3EQfsTWZ/content/2301.13148v1.pdf'}
+page_content='  See Appendix for the detailed construction of differentiation  matrices D1, D2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFPT4oBgHgl3EQfsTWZ/content/2301.13148v1.pdf'}
+page_content=' Combining (12) and (13) yields a fully discretized scheme for  the update in time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFPT4oBgHgl3EQfsTWZ/content/2301.13148v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFPT4oBgHgl3EQfsTWZ/content/2301.13148v1.pdf'}
+page_content='1  Visualizing heat flow on a rough surface M  We show the heat flow under different amplitudes of rough surface M in (3) by  solving the heat equations (11a) with zero flux h(ξ, t) = 0 and periodic boundary  8  conditions (11b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFPT4oBgHgl3EQfsTWZ/content/2301.13148v1.pdf'}
+page_content=' The rough surfaces M are defined over the parameter space  I2 = [−1, 1]2 ⊂ R2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFPT4oBgHgl3EQfsTWZ/content/2301.13148v1.pdf'}
+page_content=' The compatible initial condition is given by  u(x, y, z, 0) = cos(πx/2) cos(πy/2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFPT4oBgHgl3EQfsTWZ/content/2301.13148v1.pdf'}
+page_content='  (14)  Figure 4 shows the numerical solutions on the surface M in Figure 3 (b) with  δM ∈ {0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFPT4oBgHgl3EQfsTWZ/content/2301.13148v1.pdf'}
+page_content='1, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFPT4oBgHgl3EQfsTWZ/content/2301.13148v1.pdf'}
+page_content='5, 1} in the respective rows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFPT4oBgHgl3EQfsTWZ/content/2301.13148v1.pdf'}
+page_content=' In all cases, we use backward Euler  method, and set τ = 1E − 3, T = 1 and nX = 41 = nY .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFPT4oBgHgl3EQfsTWZ/content/2301.13148v1.pdf'}
+page_content=' Figure 4 shows  that the range of the numerical solutions becomes larger as the amplitude of M  increases from δM = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFPT4oBgHgl3EQfsTWZ/content/2301.13148v1.pdf'}
+page_content='1 to δM = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFPT4oBgHgl3EQfsTWZ/content/2301.13148v1.pdf'}
+page_content=' From the rough surface in Figure 3 (b) and  solutions in Figure 4, it can be concluded that the heat flow when projected to  the plane is greatest in flat regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFPT4oBgHgl3EQfsTWZ/content/2301.13148v1.pdf'}
+page_content='  4  Pattern formation on a rough surface M  In this section, we consider reaction-diffusion systems (RDS) on a rough surface  M  �  ∂tu = δu∆Mu + fu(u, v),  ∂tv = δv∆Mv + fv(u, v),  (15)  for some (concentration) functions u, v : M × (0, T] → R, and reaction terms  �  �  �  fu(u, v) = αu(1 − ξ1v2) + v(1 − ξ2u),  fv(u, v) = βv  �  1 + αξ1  β uv  �  + u(γ + ξ2v),  (16)  with parameters δu, δv, α, β, ξ1, ξ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFPT4oBgHgl3EQfsTWZ/content/2301.13148v1.pdf'}
+page_content=' Pattern formation on very smooth (and usu-  ally closed) surfaces with little variation is discussed in [10, 29, 33, 41];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFPT4oBgHgl3EQfsTWZ/content/2301.13148v1.pdf'}
+page_content=' pattern  formation on rough surfaces has not been studied as far as the authors know.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFPT4oBgHgl3EQfsTWZ/content/2301.13148v1.pdf'}
+page_content=' In  this section, we aim to analyze the effect of rough surfaces on pattern formation  generated by the reaction-diffusion system (15)-(16).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFPT4oBgHgl3EQfsTWZ/content/2301.13148v1.pdf'}
+page_content=' Numerical methods are  not our focus and we simply extend the finite difference scheme in the previous  section to work here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFPT4oBgHgl3EQfsTWZ/content/2301.13148v1.pdf'}
+page_content=' Other options for the discretization of reaction-diffusion  systems include the finite element method [22,28,38], and various types of mesh-  free methods [30,31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFPT4oBgHgl3EQfsTWZ/content/2301.13148v1.pdf'}
+page_content='  As in Section 3, we work on an equispaced temporal partition {tj}NT  j=0 with  some time step-size τ > 0 and tensor-product grid points [X, Y ] ∈ RnXnY ×2 ⊂  I2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFPT4oBgHgl3EQfsTWZ/content/2301.13148v1.pdf'}
+page_content=' Let  U j ≈ u(X, Y, z(X, Y ), tj)  and  V j ≈ v(X, Y, z(X, Y ), tj),  1 ≤ j ≤ NT ,  be the unknown nodal values we seek based on initial conditions U 0 and V 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFPT4oBgHgl3EQfsTWZ/content/2301.13148v1.pdf'}
+page_content='  Using the second order backward differentiation formula (BDF2) [3,39] and (13)  to discretize the RDS (15) with periodic boundary condition (11b), we obtain  9  (a)  δM = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFPT4oBgHgl3EQfsTWZ/content/2301.13148v1.pdf'}
+page_content='1  (b)  (c)  δM = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFPT4oBgHgl3EQfsTWZ/content/2301.13148v1.pdf'}
+page_content='5  (d)  (e)  δM = 1  (f)  Figure 4:  Numerical solution of heat equations under different amplitudes  δM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFPT4oBgHgl3EQfsTWZ/content/2301.13148v1.pdf'}
+page_content='  (a, b): δM = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFPT4oBgHgl3EQfsTWZ/content/2301.13148v1.pdf'}
+page_content='1, (c, d): δM = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFPT4oBgHgl3EQfsTWZ/content/2301.13148v1.pdf'}
+page_content='5, (e, f): δM = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFPT4oBgHgl3EQfsTWZ/content/2301.13148v1.pdf'}
+page_content='  In all cases,  τ = 1E − 3, T = 1, nX = 41 = nY .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFPT4oBgHgl3EQfsTWZ/content/2301.13148v1.pdf'}
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+page_content='45  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFPT4oBgHgl3EQfsTWZ/content/2301.13148v1.pdf'}
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+page_content='5  1Table 1:  Parameters of the reaction-diffusion system (15)-(16) for generating  spots and stripes patterns on rough surfaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFPT4oBgHgl3EQfsTWZ/content/2301.13148v1.pdf'}
+page_content='  Pattern  δv  δu  α  β  γ  ξ1  ξ2  Spots  10−3  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFPT4oBgHgl3EQfsTWZ/content/2301.13148v1.pdf'}
+page_content='516δv  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFPT4oBgHgl3EQfsTWZ/content/2301.13148v1.pdf'}
+page_content='899  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFPT4oBgHgl3EQfsTWZ/content/2301.13148v1.pdf'}
+page_content='91  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFPT4oBgHgl3EQfsTWZ/content/2301.13148v1.pdf'}
+page_content='899  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFPT4oBgHgl3EQfsTWZ/content/2301.13148v1.pdf'}
+page_content='02  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFPT4oBgHgl3EQfsTWZ/content/2301.13148v1.pdf'}
+page_content='2  Stripes  10−3  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFPT4oBgHgl3EQfsTWZ/content/2301.13148v1.pdf'}
+page_content='516δv  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFPT4oBgHgl3EQfsTWZ/content/2301.13148v1.pdf'}
+page_content='899  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFPT4oBgHgl3EQfsTWZ/content/2301.13148v1.pdf'}
+page_content='91  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFPT4oBgHgl3EQfsTWZ/content/2301.13148v1.pdf'}
+page_content='899  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFPT4oBgHgl3EQfsTWZ/content/2301.13148v1.pdf'}
+page_content='5  0  the following fully-discrete system of equations on M  �  �  �  �  �  3U j+1 − 2τδu△M,hU j+1 = 4τfu  �  U j, V j�  − 2τfu  �  U j−1, V j−1�  + 4U j − U j−1,  3V j+1 − 2τδv△M,hV j+1 = 4τfv  �  U j, V j�  − 2τfv  �  U j−1, V j−1�  + 4V j − V j−1,  (17)  for 1 ≤ j ≤ NT , subject to some yet-to-be specified (usually random) initial  condition and some first order approximations to the solutions at the first time  step, U 1 and V 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFPT4oBgHgl3EQfsTWZ/content/2301.13148v1.pdf'}
+page_content=' Note that the two equations in (17) are not coupled;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFPT4oBgHgl3EQfsTWZ/content/2301.13148v1.pdf'}
+page_content=' the com-  putational cost is of the same order as that of solving two scalar heat equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFPT4oBgHgl3EQfsTWZ/content/2301.13148v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFPT4oBgHgl3EQfsTWZ/content/2301.13148v1.pdf'}
+page_content='1  The pattern generation process  In this subsection, we show the formation of irregular patterns as we go from  a flat two dimensional domain to rough surfaces with different amplitudes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFPT4oBgHgl3EQfsTWZ/content/2301.13148v1.pdf'}
+page_content=' We  start with patterns on the flat domain [−1, 1]2 as in [30], i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFPT4oBgHgl3EQfsTWZ/content/2301.13148v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFPT4oBgHgl3EQfsTWZ/content/2301.13148v1.pdf'}
+page_content=', the rough sur-  face with zero amplitude (δM = 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFPT4oBgHgl3EQfsTWZ/content/2301.13148v1.pdf'}
+page_content=' Model and surface parameters are chosen  according to the values in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFPT4oBgHgl3EQfsTWZ/content/2301.13148v1.pdf'}
+page_content=' To discretize, we select grid parameters  nX = 90, nY = nX, and a time step-size τ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFPT4oBgHgl3EQfsTWZ/content/2301.13148v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFPT4oBgHgl3EQfsTWZ/content/2301.13148v1.pdf'}
+page_content=' The rough surfaces M with  M = N = 5 are used in this part.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFPT4oBgHgl3EQfsTWZ/content/2301.13148v1.pdf'}
+page_content='  The first row of Figure 5 plots the initial conditions used to compute the  spots and stripes patterns, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFPT4oBgHgl3EQfsTWZ/content/2301.13148v1.pdf'}
+page_content=' These are random values generated  within the interval [−0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFPT4oBgHgl3EQfsTWZ/content/2301.13148v1.pdf'}
+page_content='5, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFPT4oBgHgl3EQfsTWZ/content/2301.13148v1.pdf'}
+page_content='5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFPT4oBgHgl3EQfsTWZ/content/2301.13148v1.pdf'}
+page_content=' The second row of Figure 5 shows the steady  state patterns on the surface with zero amplitude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFPT4oBgHgl3EQfsTWZ/content/2301.13148v1.pdf'}
+page_content=' Perfect spots and stripes  are obtained, which is similar to our previous results in [30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFPT4oBgHgl3EQfsTWZ/content/2301.13148v1.pdf'}
+page_content=' Next, we set the  initial conditions for the next amplitude to be the steady solutions from the zero  amplitude rough surface, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFPT4oBgHgl3EQfsTWZ/content/2301.13148v1.pdf'}
+page_content='e, we use the solution of spots with δM = 0, T = 800  and the solution of stripes with δM = 0, T = 4000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFPT4oBgHgl3EQfsTWZ/content/2301.13148v1.pdf'}
+page_content=' By increasing the amplitude  of the rough surface from δM = 0 to δM = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFPT4oBgHgl3EQfsTWZ/content/2301.13148v1.pdf'}
+page_content='1 with increments of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFPT4oBgHgl3EQfsTWZ/content/2301.13148v1.pdf'}
+page_content='01, and  setting initial conditions using the previous steady state, we achieve the final  patterns for δM = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFPT4oBgHgl3EQfsTWZ/content/2301.13148v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFPT4oBgHgl3EQfsTWZ/content/2301.13148v1.pdf'}
+page_content=' These are shown in the third row of Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFPT4oBgHgl3EQfsTWZ/content/2301.13148v1.pdf'}
+page_content=' For rough  surfaces under small amplitude δM = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFPT4oBgHgl3EQfsTWZ/content/2301.13148v1.pdf'}
+page_content='05, the spots and stripes are similar  to those with δM = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFPT4oBgHgl3EQfsTWZ/content/2301.13148v1.pdf'}
+page_content=' However, for larger amplitude δM = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFPT4oBgHgl3EQfsTWZ/content/2301.13148v1.pdf'}
+page_content='1, both spots  and stripes become irregular.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFPT4oBgHgl3EQfsTWZ/content/2301.13148v1.pdf'}
+page_content=' We can conclude that the steady state patterns  become irregular as the amplitude of the rough surface M increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFPT4oBgHgl3EQfsTWZ/content/2301.13148v1.pdf'}
+page_content='  11  Spots  δM = 0, t = 0  Stripes  δM = 0, t = 0  δM = 0, t = 800  δM = 0, t = 4000  Increase δM by 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFPT4oBgHgl3EQfsTWZ/content/2301.13148v1.pdf'}
+page_content='01 at a time  −−−−−→  δM = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFPT4oBgHgl3EQfsTWZ/content/2301.13148v1.pdf'}
+page_content='1, t = 800 × 11  δM = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFPT4oBgHgl3EQfsTWZ/content/2301.13148v1.pdf'}
+page_content='1, t = 4000 × 11  Figure 5: Pattern generation on rough surfaces M with M = N = 5 and  amplitude δM increasing from 0 to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFPT4oBgHgl3EQfsTWZ/content/2301.13148v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFPT4oBgHgl3EQfsTWZ/content/2301.13148v1.pdf'}
+page_content=' Parameters for spots and stripes are set  according to Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFPT4oBgHgl3EQfsTWZ/content/2301.13148v1.pdf'}
+page_content=' Discretization parameters are nX = 90 = nY , τ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFPT4oBgHgl3EQfsTWZ/content/2301.13148v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFPT4oBgHgl3EQfsTWZ/content/2301.13148v1.pdf'}
+page_content='  12  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFPT4oBgHgl3EQfsTWZ/content/2301.13148v1.pdf'}
+page_content='2  Patterns on surfaces with different spatial frequencies  and amplitudes  Properties of rough surfaces are influenced by the amplitude δM and the spatial  frequencies M, N in (7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFPT4oBgHgl3EQfsTWZ/content/2301.13148v1.pdf'}
+page_content=' To better understand patterns on rough surfaces with  different properties, we compute steady state patterns for  δM ∈ {0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFPT4oBgHgl3EQfsTWZ/content/2301.13148v1.pdf'}
+page_content='05, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFPT4oBgHgl3EQfsTWZ/content/2301.13148v1.pdf'}
+page_content='1},  (M, N) ∈ {(5, 15), (15, 15)},  on the parameter space I2 = [−0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFPT4oBgHgl3EQfsTWZ/content/2301.13148v1.pdf'}
+page_content='5, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFPT4oBgHgl3EQfsTWZ/content/2301.13148v1.pdf'}
+page_content='5]2 ⊂ R2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFPT4oBgHgl3EQfsTWZ/content/2301.13148v1.pdf'}
+page_content=' Periodic boundary conditions  on ∂M := ∂I2 × z(∂I2) are prescribed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFPT4oBgHgl3EQfsTWZ/content/2301.13148v1.pdf'}
+page_content='  These patterns in both 2-D view and zoom in 3-D view are presented in  Figures 6-7 for spots, and in Figures 8-9 for stripes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFPT4oBgHgl3EQfsTWZ/content/2301.13148v1.pdf'}
+page_content='  As before, the initial  conditions are assigned to be random data in [−0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFPT4oBgHgl3EQfsTWZ/content/2301.13148v1.pdf'}
+page_content='5, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFPT4oBgHgl3EQfsTWZ/content/2301.13148v1.pdf'}
+page_content='5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFPT4oBgHgl3EQfsTWZ/content/2301.13148v1.pdf'}
+page_content=' In order to capture  details of the spot and stripe patterns, we increase the spatial resolution from  nX = 90 to nX = 170, again keeping nY = nX to ensure the irregular patterns  were due to the roughness rather than low resolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFPT4oBgHgl3EQfsTWZ/content/2301.13148v1.pdf'}
+page_content='  Firstly, from Figures 6 and 7, we can see that the number of spots increases  as the frequencies M, N and the amplitude δM increase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFPT4oBgHgl3EQfsTWZ/content/2301.13148v1.pdf'}
+page_content=' For fixed amplitude  δM, patterns are deformed along the x−axis as the frequency N increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFPT4oBgHgl3EQfsTWZ/content/2301.13148v1.pdf'}
+page_content=' This  conclusion is clearly illustrated in the case δM = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFPT4oBgHgl3EQfsTWZ/content/2301.13148v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFPT4oBgHgl3EQfsTWZ/content/2301.13148v1.pdf'}
+page_content='  On the other hand,  for fixed values of M, N, the patterns maintain their shape as spots when  δM ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFPT4oBgHgl3EQfsTWZ/content/2301.13148v1.pdf'}
+page_content='05, and start to deform when δM ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFPT4oBgHgl3EQfsTWZ/content/2301.13148v1.pdf'}
+page_content='05.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFPT4oBgHgl3EQfsTWZ/content/2301.13148v1.pdf'}
+page_content=' For δ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFPT4oBgHgl3EQfsTWZ/content/2301.13148v1.pdf'}
+page_content='1, all patterns  become deformed spots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFPT4oBgHgl3EQfsTWZ/content/2301.13148v1.pdf'}
+page_content='  Secondly, for fixed δM = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFPT4oBgHgl3EQfsTWZ/content/2301.13148v1.pdf'}
+page_content='05, zoom in 3-D profiles of region [−0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFPT4oBgHgl3EQfsTWZ/content/2301.13148v1.pdf'}
+page_content='3, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFPT4oBgHgl3EQfsTWZ/content/2301.13148v1.pdf'}
+page_content='1] ×  [−0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFPT4oBgHgl3EQfsTWZ/content/2301.13148v1.pdf'}
+page_content='3, 0] for [M, N] = [5, 15] and region [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFPT4oBgHgl3EQfsTWZ/content/2301.13148v1.pdf'}
+page_content='2, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFPT4oBgHgl3EQfsTWZ/content/2301.13148v1.pdf'}
+page_content='5]×[−0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFPT4oBgHgl3EQfsTWZ/content/2301.13148v1.pdf'}
+page_content='3, 0] for [M, N] = [15, 15]  reveal that irregular spots appear when part of the pattern is located in a  valley or ridge of the rough surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFPT4oBgHgl3EQfsTWZ/content/2301.13148v1.pdf'}
+page_content=' When δM increases to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFPT4oBgHgl3EQfsTWZ/content/2301.13148v1.pdf'}
+page_content='1, as in Figure 7,  the deformation of the patterns is more severe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFPT4oBgHgl3EQfsTWZ/content/2301.13148v1.pdf'}
+page_content='  This makes sense since the  amplitude δM is twice that of Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFPT4oBgHgl3EQfsTWZ/content/2301.13148v1.pdf'}
+page_content=' From zoom in 3-D figures of region  [−0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFPT4oBgHgl3EQfsTWZ/content/2301.13148v1.pdf'}
+page_content='1, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFPT4oBgHgl3EQfsTWZ/content/2301.13148v1.pdf'}
+page_content='5] × [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFPT4oBgHgl3EQfsTWZ/content/2301.13148v1.pdf'}
+page_content='1, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFPT4oBgHgl3EQfsTWZ/content/2301.13148v1.pdf'}
+page_content='4] for [M, N] = [5, 15] and region [−0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFPT4oBgHgl3EQfsTWZ/content/2301.13148v1.pdf'}
+page_content='4, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFPT4oBgHgl3EQfsTWZ/content/2301.13148v1.pdf'}
+page_content='1] × [−0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFPT4oBgHgl3EQfsTWZ/content/2301.13148v1.pdf'}
+page_content='2, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFPT4oBgHgl3EQfsTWZ/content/2301.13148v1.pdf'}
+page_content='1]  for [M, N] = [15, 15], the deformed patterns again appear when their locations  cover the local ridge/valley/mountain of the rough surfaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFPT4oBgHgl3EQfsTWZ/content/2301.13148v1.pdf'}
+page_content='  As frequencies and amplitudes are varied, similar behavior (to the case of  spots) can be observed in stripe formation;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFPT4oBgHgl3EQfsTWZ/content/2301.13148v1.pdf'}
+page_content=' see Figures 8 and 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFPT4oBgHgl3EQfsTWZ/content/2301.13148v1.pdf'}
+page_content=' High amplitudes  and frequencies yield a particularly strong effect in the zoom in 3-D plots in  Figure 9, where the stripes break into small separate components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFPT4oBgHgl3EQfsTWZ/content/2301.13148v1.pdf'}
+page_content=' Further, we  observe that the largest concentration values appear at the local peaks of the  rough surface M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFPT4oBgHgl3EQfsTWZ/content/2301.13148v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFPT4oBgHgl3EQfsTWZ/content/2301.13148v1.pdf'}
+page_content='3  Animal coat simulation results  In simulation results on rough surfaces M which are characterized by spatial  frequencies, we observe interesting similarities between some steady state pat-  terns and actual animal coat patterns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFPT4oBgHgl3EQfsTWZ/content/2301.13148v1.pdf'}
+page_content=' In Figure 10, we show the simulations  of the animal coats in Figure 1 by patterns on rough surfaces M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFPT4oBgHgl3EQfsTWZ/content/2301.13148v1.pdf'}
+page_content=' The specific  13  2-D view  M = 5, N = 15  Zoom in 3-D view  M = 15 = N  Figure 6: Patterns on rough surfaces with amplitude δM = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFPT4oBgHgl3EQfsTWZ/content/2301.13148v1.pdf'}
+page_content='05.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFPT4oBgHgl3EQfsTWZ/content/2301.13148v1.pdf'}
+page_content=' Discretization  parameters are nX = 170 = nY , τ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFPT4oBgHgl3EQfsTWZ/content/2301.13148v1.pdf'}
+page_content='5, T = 800, I2 := [−0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFPT4oBgHgl3EQfsTWZ/content/2301.13148v1.pdf'}
+page_content='5, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFPT4oBgHgl3EQfsTWZ/content/2301.13148v1.pdf'}
+page_content='5]2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFPT4oBgHgl3EQfsTWZ/content/2301.13148v1.pdf'}
+page_content='  parameter values for generating each animal coat pattern are listed in Tables 1  and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFPT4oBgHgl3EQfsTWZ/content/2301.13148v1.pdf'}
+page_content='  Table 2: Parameters for animal coats simulation on M in Figure 10 with nX = nY = 90  M  N  τ  T  δM  Emperor angelfish [26]  Figure 10 (a)  5  5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFPT4oBgHgl3EQfsTWZ/content/2301.13148v1.pdf'}
+page_content='5  4000  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFPT4oBgHgl3EQfsTWZ/content/2301.13148v1.pdf'}
+page_content='05  Genet [1]  Figure 10 (b)  15  15  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFPT4oBgHgl3EQfsTWZ/content/2301.13148v1.pdf'}
+page_content='5  800  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFPT4oBgHgl3EQfsTWZ/content/2301.13148v1.pdf'}
+page_content='1  Plecostomus [12]  Figure 10 (c)  15  5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFPT4oBgHgl3EQfsTWZ/content/2301.13148v1.pdf'}
+page_content='5  4000  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFPT4oBgHgl3EQfsTWZ/content/2301.13148v1.pdf'}
+page_content='1  Cheetah [46]  Figure 10 (d)  15  5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFPT4oBgHgl3EQfsTWZ/content/2301.13148v1.pdf'}
+page_content='5  800  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFPT4oBgHgl3EQfsTWZ/content/2301.13148v1.pdf'}
+page_content='1  5  Random rough surfaces S by discrete data  While the parametric equation (6) allows us to work analytically on rough sur-  faces, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFPT4oBgHgl3EQfsTWZ/content/2301.13148v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFPT4oBgHgl3EQfsTWZ/content/2301.13148v1.pdf'}
+page_content=', via the evaluation of metric tensor (2), its generalization to mani-  folds [9,10,40] is not trivial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFPT4oBgHgl3EQfsTWZ/content/2301.13148v1.pdf'}
+page_content='  In [24], the authors used the covariance function of random deformation fields  and the surface Karhunen-Lo`eve expansion to generate random surfaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFPT4oBgHgl3EQfsTWZ/content/2301.13148v1.pdf'}
+page_content=' In [23],  the authors applied the 2D digital filter and Fourier analysis to generate random  rough surfaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFPT4oBgHgl3EQfsTWZ/content/2301.13148v1.pdf'}
+page_content=' In this section, we introduce a new approach for constructing  14  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFPT4oBgHgl3EQfsTWZ/content/2301.13148v1.pdf'}
+page_content='5  10  5  0  5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFPT4oBgHgl3EQfsTWZ/content/2301.13148v1.pdf'}
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+page_content='5  10  5  0  0  5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFPT4oBgHgl3EQfsTWZ/content/2301.13148v1.pdf'}
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+page_content='1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFPT4oBgHgl3EQfsTWZ/content/2301.13148v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFPT4oBgHgl3EQfsTWZ/content/2301.13148v1.pdf'}
+page_content='2  X  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFPT4oBgHgl3EQfsTWZ/content/2301.13148v1.pdf'}
+page_content='3  y2-D view  M = 5, N = 15  Zoom in 3-D view  M = 15 = N  Figure 7: Patterns on rough surfaces with amplitude δM = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFPT4oBgHgl3EQfsTWZ/content/2301.13148v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFPT4oBgHgl3EQfsTWZ/content/2301.13148v1.pdf'}
+page_content=' Discretization  parameters are nX = 170 = nY , τ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFPT4oBgHgl3EQfsTWZ/content/2301.13148v1.pdf'}
+page_content='5, T = 800, I2 := [−0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFPT4oBgHgl3EQfsTWZ/content/2301.13148v1.pdf'}
+page_content='5, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFPT4oBgHgl3EQfsTWZ/content/2301.13148v1.pdf'}
+page_content='5]2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFPT4oBgHgl3EQfsTWZ/content/2301.13148v1.pdf'}
+page_content='  rough surfaces S based on random data and heat filters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFPT4oBgHgl3EQfsTWZ/content/2301.13148v1.pdf'}
+page_content=' The desired reaction-  diffusion systems are then solved on S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFPT4oBgHgl3EQfsTWZ/content/2301.13148v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFPT4oBgHgl3EQfsTWZ/content/2301.13148v1.pdf'}
+page_content='1  Construction of S by heat filters  We want to generate some random rough surfaces S, which have similar rough-  ness as rough surfaces M in (7) with different M and N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFPT4oBgHgl3EQfsTWZ/content/2301.13148v1.pdf'}
+page_content='  Let [X, Y ] ∈ RnXnY ×2 ⊂ I2 as in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFPT4oBgHgl3EQfsTWZ/content/2301.13148v1.pdf'}
+page_content='  We assign uniform ran-  dom numbers to each node to obtain the initial random surface values ˜Z0 ∼  �  U[−1, 1]  �nXnY at nodes [X, Y ].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFPT4oBgHgl3EQfsTWZ/content/2301.13148v1.pdf'}
+page_content=' We define the discretized heat filter according  to the method of Section 3 but with some filter-diffusion tensor F (instead of  the diffusion tensor A for surface M) in the Laplace-Beltrami operator △F,h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFPT4oBgHgl3EQfsTWZ/content/2301.13148v1.pdf'}
+page_content='  For an isotropic filter, we take F = I2×2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFPT4oBgHgl3EQfsTWZ/content/2301.13148v1.pdf'}
+page_content=' This can generate an M-like surface  with M = N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFPT4oBgHgl3EQfsTWZ/content/2301.13148v1.pdf'}
+page_content=' For the anisotropic case with M ̸= N, we use F = diag(2, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFPT4oBgHgl3EQfsTWZ/content/2301.13148v1.pdf'}
+page_content=' We  can now smooth the surface data J-times via  ˜Zj+1 = (Q + κh△F,h) ˜Zj,  for j = 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFPT4oBgHgl3EQfsTWZ/content/2301.13148v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFPT4oBgHgl3EQfsTWZ/content/2301.13148v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFPT4oBgHgl3EQfsTWZ/content/2301.13148v1.pdf'}
+page_content=' , J,  where Q is an nXnY by nXnY matrix of ones, κ > 0 is the parameter to control  the weights of △F,h, and h is the fill distance of the discrete set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFPT4oBgHgl3EQfsTWZ/content/2301.13148v1.pdf'}
+page_content=' This completes  the definition of the pre-surface that is the counterpart to ˜Z in (7) of the surface  15  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFPT4oBgHgl3EQfsTWZ/content/2301.13148v1.pdf'}
+page_content='5  10  5  0  5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFPT4oBgHgl3EQfsTWZ/content/2301.13148v1.pdf'}
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+page_content='3  0  X  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFPT4oBgHgl3EQfsTWZ/content/2301.13148v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFPT4oBgHgl3EQfsTWZ/content/2301.13148v1.pdf'}
+page_content='1  y0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFPT4oBgHgl3EQfsTWZ/content/2301.13148v1.pdf'}
+page_content='5  10  5  0  0  5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFPT4oBgHgl3EQfsTWZ/content/2301.13148v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFPT4oBgHgl3EQfsTWZ/content/2301.13148v1.pdf'}
+page_content='5  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFPT4oBgHgl3EQfsTWZ/content/2301.13148v1.pdf'}
+page_content='50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFPT4oBgHgl3EQfsTWZ/content/2301.13148v1.pdf'}
+page_content='1  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFPT4oBgHgl3EQfsTWZ/content/2301.13148v1.pdf'}
+page_content='1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFPT4oBgHgl3EQfsTWZ/content/2301.13148v1.pdf'}
+page_content='1  0  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFPT4oBgHgl3EQfsTWZ/content/2301.13148v1.pdf'}
+page_content='1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFPT4oBgHgl3EQfsTWZ/content/2301.13148v1.pdf'}
+page_content='2  y  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFPT4oBgHgl3EQfsTWZ/content/2301.13148v1.pdf'}
+page_content='2  X  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFPT4oBgHgl3EQfsTWZ/content/2301.13148v1.pdf'}
+page_content='42-D view  M = 5, N = 15  Zoom in 3-D view  M = 15 = N  Figure 8:  Patterns on rough surfaces with amplitude δM ∈ {0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFPT4oBgHgl3EQfsTWZ/content/2301.13148v1.pdf'}
+page_content='05} and  (M, N) ∈ {(5, 15), (15, 15)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFPT4oBgHgl3EQfsTWZ/content/2301.13148v1.pdf'}
+page_content=' The model parameters for stripes are set according  to Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFPT4oBgHgl3EQfsTWZ/content/2301.13148v1.pdf'}
+page_content=' Discretization parameters are nX = 170 = nY , τ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFPT4oBgHgl3EQfsTWZ/content/2301.13148v1.pdf'}
+page_content='5, T = 4000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFPT4oBgHgl3EQfsTWZ/content/2301.13148v1.pdf'}
+page_content='  16  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFPT4oBgHgl3EQfsTWZ/content/2301.13148v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFPT4oBgHgl3EQfsTWZ/content/2301.13148v1.pdf'}
+page_content='1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFPT4oBgHgl3EQfsTWZ/content/2301.13148v1.pdf'}
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+page_content='5  X2-D view  M = 5, N = 15  Zoom in 3-D view  M = 15 = N  Figure 9:  Patterns on rough surfaces with amplitude δM = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFPT4oBgHgl3EQfsTWZ/content/2301.13148v1.pdf'}
+page_content='1 and (M, N) ∈  {(5, 15), (15, 15)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFPT4oBgHgl3EQfsTWZ/content/2301.13148v1.pdf'}
+page_content=' The model parameters for stripes are set according to Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFPT4oBgHgl3EQfsTWZ/content/2301.13148v1.pdf'}
+page_content='  Discretization parameters are nX = 170 = nY , τ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFPT4oBgHgl3EQfsTWZ/content/2301.13148v1.pdf'}
+page_content='5, T = 4000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFPT4oBgHgl3EQfsTWZ/content/2301.13148v1.pdf'}
+page_content='  17  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFPT4oBgHgl3EQfsTWZ/content/2301.13148v1.pdf'}
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+page_content='5  0  y(a)  (b)  (c)  (d)  Figure 10: On random rough surface M with parameters in Table 1 and Table 2,  actual animal coat patterns simulation results: (a) emperor angelfish in [26];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFPT4oBgHgl3EQfsTWZ/content/2301.13148v1.pdf'}
+page_content=' (b)  genet in [1];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFPT4oBgHgl3EQfsTWZ/content/2301.13148v1.pdf'}
+page_content=' (c) plecostomus in [12];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFPT4oBgHgl3EQfsTWZ/content/2301.13148v1.pdf'}
+page_content=' (d) cheetah in [46].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFPT4oBgHgl3EQfsTWZ/content/2301.13148v1.pdf'}
+page_content='  18  (a)  (b)  (c)  Figure 11:  Rough surfaces S with κ = 2 and the number of filter steps J =  15, under various filter-diffusion tensors F: (a) F = diag(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFPT4oBgHgl3EQfsTWZ/content/2301.13148v1.pdf'}
+page_content='01, 1), (b) F =  diag(1, 1), (c) F = diag(5, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFPT4oBgHgl3EQfsTWZ/content/2301.13148v1.pdf'}
+page_content='  type M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFPT4oBgHgl3EQfsTWZ/content/2301.13148v1.pdf'}
+page_content=' Similar to the scaling in (7), we define S only by nodal values  �  [X, Y, Z] : Z =  δS  ∥ ˜ZJ∥∞  ˜ZJ  �  ⊂ S,  for some amplitude δS > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFPT4oBgHgl3EQfsTWZ/content/2301.13148v1.pdf'}
+page_content='  Figure 11(a, b, c) shows the rough surfaces S derived for fixed nX =  90, nY = nX, κ = 2 and J = 15 filter steps under filter-diffusion tensors  F ∈ {diag(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFPT4oBgHgl3EQfsTWZ/content/2301.13148v1.pdf'}
+page_content='01, 1), diag(1, 1), diag(5, 1)},  respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFPT4oBgHgl3EQfsTWZ/content/2301.13148v1.pdf'}
+page_content='  Taking F(1, 1) = 1 yields a rough surface S which is similar  to a rough surface M with equal frequencies M, N (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFPT4oBgHgl3EQfsTWZ/content/2301.13148v1.pdf'}
+page_content='  Figure 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFPT4oBgHgl3EQfsTWZ/content/2301.13148v1.pdf'}
+page_content='  Taking  F(1, 1) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFPT4oBgHgl3EQfsTWZ/content/2301.13148v1.pdf'}
+page_content='01 scales down space in the x-direction, yielding a rough surface  S which is similar to a rough surface M with a larger frequency M, M > N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFPT4oBgHgl3EQfsTWZ/content/2301.13148v1.pdf'}
+page_content='  Conversely, taking F(1, 1) = 5, a rough surface S is obtained which is similar  to M with a smaller frequency M, M < N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFPT4oBgHgl3EQfsTWZ/content/2301.13148v1.pdf'}
+page_content='  To reproduce the properties of surface M, the required value of filter-diffusion  tensors F and number J of filtering steps will depend on the density of the given  data points [X, Y ].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFPT4oBgHgl3EQfsTWZ/content/2301.13148v1.pdf'}
+page_content='  For fixed amplitude δS = 1E − 3 and nX = 90, nY = nX,  Table 3 gives values of κ, F and the number of filtering steps J to generate  surfaces that give good qualitative agreement with the rough surfaces M in (7)  for various M, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFPT4oBgHgl3EQfsTWZ/content/2301.13148v1.pdf'}
+page_content=' Figure 2 gives a comparison showing that rough surfaces S  by heat filters (column 2) are qualitatively similar to our earlier surfaces M by  (7) (see column 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFPT4oBgHgl3EQfsTWZ/content/2301.13148v1.pdf'}
+page_content='  While not the focus of the current work, our construction methods for S  can be extended to generate rough closed manifolds which can be used as the  surfaces for the numerical approximation of PDEs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFPT4oBgHgl3EQfsTWZ/content/2301.13148v1.pdf'}
+page_content=' See Figure 12 for a graph-  ical illustration of rough closed manifolds generated by type-M and type-S  approaches respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFPT4oBgHgl3EQfsTWZ/content/2301.13148v1.pdf'}
+page_content=' Figure 12(a) was obtained by applying the type-M  procedure to the parameter space (θ, φ) ∈ [0, 2π] × [0, π] with roughness added  to the constant function r = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFPT4oBgHgl3EQfsTWZ/content/2301.13148v1.pdf'}
+page_content=' Figure 12(b) was obtained by adding noise to  r = 1 for points on the unit sphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFPT4oBgHgl3EQfsTWZ/content/2301.13148v1.pdf'}
+page_content=' In contrast to that in (13), the heat filter  here makes use of the discrete Laplace-Beltrami operator for the rough sphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFPT4oBgHgl3EQfsTWZ/content/2301.13148v1.pdf'}
+page_content='  19  Table 3: Coefficients for constructing the surfaces S that appear in the second column of Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFPT4oBgHgl3EQfsTWZ/content/2301.13148v1.pdf'}
+page_content='  Parameters are chosen to give qualitative agreement with the rough surfaces M (7) with δS =  1E − 3, nX = 90 = nY  M in (7)  S by heat filter  κ  F  filter number  [M, N] = [5, 5]  5  diag(1, 1)  15  [M, N] = [5, 15]  8  diag(1, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFPT4oBgHgl3EQfsTWZ/content/2301.13148v1.pdf'}
+page_content='01)  10  [M, N] = [15, 15]  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFPT4oBgHgl3EQfsTWZ/content/2301.13148v1.pdf'}
+page_content='2  diag(20, 20)  2  (a)  (b)  Figure 12: Graphical illustration of rough closed manifolds by(a) type-M, and  (b) type-S approaches on parameter size and manifold respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFPT4oBgHgl3EQfsTWZ/content/2301.13148v1.pdf'}
+page_content='  20  0  0  1  0  10  0  1  1  0Figure 13: For δS = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFPT4oBgHgl3EQfsTWZ/content/2301.13148v1.pdf'}
+page_content='1, nX = 90, nY = nX, spot and stripe patterns on  a rough surface S with number of filter steps J = 15, F = diag(1, 1), and  κ = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFPT4oBgHgl3EQfsTWZ/content/2301.13148v1.pdf'}
+page_content='  These values from Table 3 give a rough surface similar to M with  M = 5, N = 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFPT4oBgHgl3EQfsTWZ/content/2301.13148v1.pdf'}
+page_content=' Parameters for spots and stripes are set according to Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFPT4oBgHgl3EQfsTWZ/content/2301.13148v1.pdf'}
+page_content='  The FDM in Section 3 can be used to solve PDEs on rough surfaces S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFPT4oBgHgl3EQfsTWZ/content/2301.13148v1.pdf'}
+page_content='  The only difference in the method is that we no longer have the parametric  equation to calculate the metric tensor G in (2) and hence the diffusion tensor  A in (13).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFPT4oBgHgl3EQfsTWZ/content/2301.13148v1.pdf'}
+page_content=' Instead of computing metric tensor G analytically as in Section 3  and 4, centered finite difference formulas are applied to approximate the metric  tensor G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFPT4oBgHgl3EQfsTWZ/content/2301.13148v1.pdf'}
+page_content=' For solving reaction-diffusion systems on rough surfaces S, we set  initial conditions to be steady state solutions on a zero amplitude rough surface  M (7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFPT4oBgHgl3EQfsTWZ/content/2301.13148v1.pdf'}
+page_content=' Figure 13 plots the spot and stripe patterns on a rough surface S using  the reaction-diffusion parameters provided in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFPT4oBgHgl3EQfsTWZ/content/2301.13148v1.pdf'}
+page_content=' As shown in the second  line of Table 3, rough surface S takes κ = 5, F = diag(1, 1) with J = 15 filter  steps to approximate rough surface M with M = 5, N = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFPT4oBgHgl3EQfsTWZ/content/2301.13148v1.pdf'}
+page_content=' It can be seen that  the spot and stripe patterns generated on S are similar to those on M under  parameters M = 5, N = 5, δM = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFPT4oBgHgl3EQfsTWZ/content/2301.13148v1.pdf'}
+page_content='1 (see the third row of Figure 5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFPT4oBgHgl3EQfsTWZ/content/2301.13148v1.pdf'}
+page_content='  We conclude by simulating the same set of animal coats displayed in Fig-  ure 1 using Turing models on rough surfaces S with parameters set according  to Table 1 and Table 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFPT4oBgHgl3EQfsTWZ/content/2301.13148v1.pdf'}
+page_content='  Figure 14 demonstrates again that adding surface  roughness into the process of pattern generation can indeed provide more var-  ied simulations for animal coats.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFPT4oBgHgl3EQfsTWZ/content/2301.13148v1.pdf'}
+page_content=' Moreover, the patterns on surface S provide  valid simulations just like M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFPT4oBgHgl3EQfsTWZ/content/2301.13148v1.pdf'}
+page_content='  Table 4: Parameters for animal coats simulation on S with nX = nY = 90  κ  F  J  τ  T  δS  Emperor angelfish [26]  Figure 14 (a)  5  diag(1, 1)  15  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFPT4oBgHgl3EQfsTWZ/content/2301.13148v1.pdf'}
+page_content='5  400  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFPT4oBgHgl3EQfsTWZ/content/2301.13148v1.pdf'}
+page_content='05  Genet [1]  Figure 14 (b)  8  diag(1, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFPT4oBgHgl3EQfsTWZ/content/2301.13148v1.pdf'}
+page_content='01)  10  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFPT4oBgHgl3EQfsTWZ/content/2301.13148v1.pdf'}
+page_content='5  800  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFPT4oBgHgl3EQfsTWZ/content/2301.13148v1.pdf'}
+page_content='1  Plecostomus [12]  Figure 14 (c)  8  diag(1, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFPT4oBgHgl3EQfsTWZ/content/2301.13148v1.pdf'}
+page_content='01)  10  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFPT4oBgHgl3EQfsTWZ/content/2301.13148v1.pdf'}
+page_content='5  3000  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFPT4oBgHgl3EQfsTWZ/content/2301.13148v1.pdf'}
+page_content='1  Cheetah [46]  Figure 14 (d)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFPT4oBgHgl3EQfsTWZ/content/2301.13148v1.pdf'}
+page_content='2  diag(20, 20)  2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFPT4oBgHgl3EQfsTWZ/content/2301.13148v1.pdf'}
+page_content='5  400  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFPT4oBgHgl3EQfsTWZ/content/2301.13148v1.pdf'}
+page_content='05  21  (a)  (b)  (c)  (d)  Figure 14: On random rough surface S with parameters in Table 1 and Table 4,  actual animal coat patterns simulation results: (a) emperor angelfish in [26];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFPT4oBgHgl3EQfsTWZ/content/2301.13148v1.pdf'}
+page_content=' (b)  genet in [1];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFPT4oBgHgl3EQfsTWZ/content/2301.13148v1.pdf'}
+page_content=' (c) plecostomus in [12];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFPT4oBgHgl3EQfsTWZ/content/2301.13148v1.pdf'}
+page_content=' (d) cheetah in [46].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFPT4oBgHgl3EQfsTWZ/content/2301.13148v1.pdf'}
+page_content='  22  6  Conclusion  In this paper, we discussed pattern formation on rough surfaces, with the surface  characterized by two different methods: spatial frequency and discretized heat  filters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFPT4oBgHgl3EQfsTWZ/content/2301.13148v1.pdf'}
+page_content=' The patterns generated on random rough surfaces of different ampli-  tudes and spatial frequencies are illustrated, and the simulation results indicate  that the patterns became relatively more irregular as amplitude increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFPT4oBgHgl3EQfsTWZ/content/2301.13148v1.pdf'}
+page_content=' The  change of spatial frequencies on the x and y axes will also lead to pattern defor-  mation along the x and y directions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFPT4oBgHgl3EQfsTWZ/content/2301.13148v1.pdf'}
+page_content=' Numerical results indicate that combining  reaction-diffusion systems with rough surfaces can provide better simulations  for animal coat patterns in the real world.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFPT4oBgHgl3EQfsTWZ/content/2301.13148v1.pdf'}
+page_content=' What’s more, the method for gen-  erating rough surfaces by heat filters can be further applied to obtain closed  rough manifolds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFPT4oBgHgl3EQfsTWZ/content/2301.13148v1.pdf'}
+page_content=' We plan to explore this generalization in future work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFPT4oBgHgl3EQfsTWZ/content/2301.13148v1.pdf'}
+page_content='  Appendix: Finite difference algorithm for the heat  equation  The heat equation (11) on a rough surface defined over the parameter space  I2 = [−1, 1]2 is considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFPT4oBgHgl3EQfsTWZ/content/2301.13148v1.pdf'}
+page_content=' The set of discrete data points on I2 are defined as:  [X, Y ] :=  ��  (xi  1, xj  2)  �nX  i=1  �nY  j=1 ∈ RnXnY ×2,  with mesh size hx1, hx2 on each axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFPT4oBgHgl3EQfsTWZ/content/2301.13148v1.pdf'}
+page_content=' To fully discretize (12), we require dis-  cretization of the Laplacian-Beltrami operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFPT4oBgHgl3EQfsTWZ/content/2301.13148v1.pdf'}
+page_content=' For any twice differentiable func-  tion u : I2 → R, we introduce differentiation matrices Dk ∈ RnXnY ×nXnY , k ∈  {1, 2} satisfying periodic boundary condition (11b) as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFPT4oBgHgl3EQfsTWZ/content/2301.13148v1.pdf'}
+page_content=' Using a second-  order centered finite differences, we have for each point (xi  1, xj  2) that  ∂u  ∂x1  (xi  1, xj  2) =  �  −  1  2hx1  1  2hx1  � �u(xi−1  1  , xj  2)  u(xi+1  1  , xj  2)  �  ,  ∂u  ∂x2  (xi  1, xj  2) =  �  −  1  2hx2  1  2hx2  � �u(xi  1, xj−1  2  )  u(xi  1, xj+1  2  )  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFPT4oBgHgl3EQfsTWZ/content/2301.13148v1.pdf'}
+page_content='  The following fictitious node approach is applied to deal with the periodic  boundary conditions:  u(x0  1, xj  2) = u(xnX−1  1  , xj  2), u(xnX+1  1  , xj  2) = u(x2  1, xj  2), u(xi  1, x0  2) = u(xi  1, xnY −1  2  ),  u(xi  1, xnY +1  2  ) = u(xi  1, x2  2), u(x1  1, xj  2) = u(xnX  1 , xj  2),  u(xi  1, x1  2) = u(xi  1, xnY  2 ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFPT4oBgHgl3EQfsTWZ/content/2301.13148v1.pdf'}
+page_content='  By assembling all points in [X, Y ], the nodal values of  ∂u  ∂xk (k = 1, 2) at [X, Y ]  can be obtained by  ∂u  ∂xk  (X, Y ) ≈ Dku(X, Y ),  for k ∈ {1, 2},  23  where u(X, Y ) := [u(x1  1, x1  2), · · · , u(xnX  1 , x1  2), · · · , u(x1  1, xnY  2 ), · · · , u(xnX  1 , xnY  2 )]T  is the vector of nodal function values and differential matrices Dk ∈ RnXnY ×nXnY  are in the form of  D1 =  �  ��  DB  1  · ·  0  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFPT4oBgHgl3EQfsTWZ/content/2301.13148v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFPT4oBgHgl3EQfsTWZ/content/2301.13148v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFPT4oBgHgl3EQfsTWZ/content/2301.13148v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFPT4oBgHgl3EQfsTWZ/content/2301.13148v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFPT4oBgHgl3EQfsTWZ/content/2301.13148v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFPT4oBgHgl3EQfsTWZ/content/2301.13148v1.pdf'}
+page_content='  0  · ·  DB  1  �  �� , DB  1 =  1  2hx1  �  ������  0  1  0  · ·  0  −1  0  −1  0  1  · ·  0  0  0  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFPT4oBgHgl3EQfsTWZ/content/2301.13148v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFPT4oBgHgl3EQfsTWZ/content/2301.13148v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFPT4oBgHgl3EQfsTWZ/content/2301.13148v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFPT4oBgHgl3EQfsTWZ/content/2301.13148v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFPT4oBgHgl3EQfsTWZ/content/2301.13148v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFPT4oBgHgl3EQfsTWZ/content/2301.13148v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFPT4oBgHgl3EQfsTWZ/content/2301.13148v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFPT4oBgHgl3EQfsTWZ/content/2301.13148v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFPT4oBgHgl3EQfsTWZ/content/2301.13148v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFPT4oBgHgl3EQfsTWZ/content/2301.13148v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFPT4oBgHgl3EQfsTWZ/content/2301.13148v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFPT4oBgHgl3EQfsTWZ/content/2301.13148v1.pdf'}
+page_content='  0  0  0  · ·  −1  0  1  1  0  0  · ·  0  0  −1  �  ������  ,  and  D2 =  �  ������  0  DB  2  0  · ·  0  −DB  2  0  −DB  2  0  DB  2  · ·  0  0  0  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFPT4oBgHgl3EQfsTWZ/content/2301.13148v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFPT4oBgHgl3EQfsTWZ/content/2301.13148v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFPT4oBgHgl3EQfsTWZ/content/2301.13148v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFPT4oBgHgl3EQfsTWZ/content/2301.13148v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFPT4oBgHgl3EQfsTWZ/content/2301.13148v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFPT4oBgHgl3EQfsTWZ/content/2301.13148v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFPT4oBgHgl3EQfsTWZ/content/2301.13148v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFPT4oBgHgl3EQfsTWZ/content/2301.13148v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFPT4oBgHgl3EQfsTWZ/content/2301.13148v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFPT4oBgHgl3EQfsTWZ/content/2301.13148v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFPT4oBgHgl3EQfsTWZ/content/2301.13148v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFPT4oBgHgl3EQfsTWZ/content/2301.13148v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFPT4oBgHgl3EQfsTWZ/content/2301.13148v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFPT4oBgHgl3EQfsTWZ/content/2301.13148v1.pdf'}
+page_content='  0  0  0  · ·  −DB  2  0  DB  2  DB  2  0  0  · ·  0  0  −DB  2  �  ������  , DB  2 =  1  2hx2  �  ��  1  · ·  0  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFPT4oBgHgl3EQfsTWZ/content/2301.13148v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFPT4oBgHgl3EQfsTWZ/content/2301.13148v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFPT4oBgHgl3EQfsTWZ/content/2301.13148v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFPT4oBgHgl3EQfsTWZ/content/2301.13148v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFPT4oBgHgl3EQfsTWZ/content/2301.13148v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFPT4oBgHgl3EQfsTWZ/content/2301.13148v1.pdf'}
+page_content='  0  · ·  1  �  �� ,  with DB  k ∈ RnX×nX, k ∈ {1, 2}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFPT4oBgHgl3EQfsTWZ/content/2301.13148v1.pdf'}
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+Manipulation of the magnetic state of a porphyrin-based molecule on gold : From Kondo to
+quantum nanomagnet via the charge fluctuation regime
+Yingzheng Gao,1 Sergio Vlaic,1 Tommaso Gorni,1 Luca de’ Medici,1
+Sylvain Clair,2 Dimitri Roditchev,1, 3 and Stéphane Pons1, ∗
+1Laboratoire de Physique et d’Étude des Matériaux (LPEM), ESPCI Paris,
+Université PSL, CNRS UMR8213, Sorbonne Université, 75005 Paris, France
+2Aix Marseille Univ, CNRS, IM2NP, Marseille, France
+3Institut des Nanosciences de Paris, Sorbonne Université, CNRS UMR7588, 75005 Paris, France
+(Dated: 4 janvier 2023)
+By moving individual Fe-Porphyrin-based molecules with the tip of Scanning Tunneling Microscope
+in the vicinity of a Br-atom containing elbow of the herringbone-reconstructed Au(111), we reversibly
+and continuously control their magnetic state. Several regimes are obtained experimentally and explo-
+red theoretically : from the integer spin limit, through intermediate magnetic states with renormalized
+magnetic anisotropy, until the Kondo-screened regime, corresponding to a progressive increase of charge
+fluctuations and mixed valency due to an increase in the interaction of the molecular Fe states with the
+substrate Fermi sea. Our results open a route for the realization, tuning and experimental studies of novel
+quantum magnetic states in molecule-surface hybrids.
+KEYWORDS
+spin-flip excitation, spin states, magnetic anisotropy,
+Kondo, charge fluctuation, mixed-valence states, scanning
+tunneling microscopy, transiton metal complexes
+INTRODUCTION
+Addressing and manipulating the spin state of molecular
+species at interfaces is a challenge that could greatly bene-
+fit spintronics [1], nanoelectronics [2, 3] and quantum electro-
+nics [4] in the near future. When the valence of a magnetic
+molecule deposited on a surface is integer, the description of
+spin-polarized molecular orbitals can be done in the frame-
+work of the atomic limit in terms of crystal field and spin-orbit
+coupling : the spin state is simply interpreted as a quantum
+magnet for which the Hund’s rule determines the fundamental
+magnetic state. However, the atomic limit is no longer valid in
+the mixed-valence regime which is the most general behavior
+of an interacting magnetic impurity with an electron bath.
+Among the most studied classes of magnetic molecules are
+the transition metal phthalocyanine (Pc) and porphyrin (P) fa-
+milies which are easily deposited at surface in vacuum and
+studied by various local probe microscopies and spectrosco-
+pies. In these molecules, the spin state is mainly given by
+the spin polarization of the central transition metal ion d-
+states [5, 6]. Several works have focused on the influence of
+external parameters on the magnetic ground state such as the
+influence of charge transfer to the orbitals of the molecule [7–
+14], the effect of surface spin-orbit coupling and magnetic ani-
+sotropy [15–17], the coupling to the substrate, [15, 18], the in-
+teraction with attached and neighboring molecules [19, 20],
+the structural deformation [21, 22] or the chemical substitu-
+tion of ligands [23]. For all these studies, a systematic unders-
+tanding of the effect of mixed valence and charge fluctuations
+is still missing, although they have a strong influence on the
+effect of magnetic anistropy.
+Here
+we
+show
+that
+charge
+fluctuations
+in
+Fe
+5,15-di-4-pyridyl-10,20-di-4-bromophenyl
+porphyrin
+(Fe−DPyDBrPP) molecules adsorbed on the Br de-
+corated Au(111) surface allow the magnetic state of the
+molecule to be driven between high-spin (S=1) and Kondo-
+screened states in a reversible and continuous manner through
+the intermediate valence regimes.
+RESULTS AND DISCUSSION
+Structural and spectroscopic properties of monomers
+Fe−DPyDBrPP were deposited in ultrahigh vacuum on
+Au(111) and studied in situ by scanning tunneling microscopy
+(STM) and spectroscopy (STS) at 1.3K (see Methods sec-
+tion for more details). The molecules are randomly located on
+hcp domains, fcc domains or stacking faults lines of the her-
+ringbone reconstruction [24] of Au(111) and appear as bright
+spots in constant-current topographic STM images in figures 1
+(a-f). The Fe−DPyDBrPP molecules do not exhibit a pla-
+nar configuration on the Au(111) surface because two pyrrole
+rings bend toward the vacuum and exhibit a higher height in
+STM images; the other two pyrrole rings bend toward the sub-
+strate, resulting in a lower height in images. Such deforma-
+tion corresponds to the saddling distortion of porphyrin-based
+molecules [25] that is quite commonly observed [17, 26–29].
+Thus, the shape of the molecules in STM images allows us
+to determine the central position of the Fe-atom, the porphine
+macrocycle and the bromophenyl and pyridyl ligands attached
+to the macrocycle, as well as their location relative to the sur-
+face reconstruction. Importantly, during deposition and annea-
+ling, some Br-atoms detach from the molecules [30], migrate
+at the surface, and get trapped by highly reactive elbows of the
+Au(111) herringbone reconstruction [31, 32]. The intersection
+arXiv:2301.01101v1  [cond-mat.mes-hall]  3 Jan 2023
+
+2
+Bias Voltage (mV)
+dI/dV (arb. units)
+0.25
+0.50
+0.75
+1.00
+0
+-20
+-10
+10
+20
+Bias Voltage (mV)
+dI/dV (arb. units)
+0.6
+1.2
+1.8
+2.4
+3.0
+0
+-20
+-10
+10
+20
+(g)
+(h)
+(d)
+(e)
+(f)
+(a)
+(b)
+(c)
+FIGURE 1: Scanning tunneling study of Fe−DPyDBrPP mono-
+mers. (a-f) Topography image of Fe−DPyDBrPP molecules at
+various location of the reconstructed surface of gold. The intersec-
+tion of the dotted lines indicates the location of the Br adatom which
+decorates the elbow of the herringbone reconstruction. (g) Norma-
+lized scanning tunneling spectra taken on upper pyrroles in images
+(a-c) showing spin inelastic excitations depending on the location of
+the molecule on the surface. (h) Normalized scanning tunneling spec-
+tra taken on upper pyrroles in images (d-f), when the molecule is on
+top of the Br-decorated elbow site of the reconstruction. These spec-
+tra show an Abrikosov-Suhl resonance witnessing a Kondo mecha-
+nism which does not depend on the in-plane rotation of the molecule.
+The solid lines correspond to the fits of the data (see Methods sec-
+tion). Experimental parameters : topography images : V = 125 mV,
+Istab = 20 pA, size 10 × 10 nm2 ; spectroscopy : Vstab = 30 mV,
+Istab = 200 pA, lock-in parameters : Vm = 0.2 mV, f = 750 Hz.
+of the dotted lines in figures 1 (a-c) points to the position of
+these elbows decorated by Br-adatoms.
+Depending on their position on the surface, the molecules
+exhibit different spectral signatures. When a molecule is loca-
+ted on the hcp and fcc domains (figures 1 (a-c)), the tunneling
+spectra show conductance steps characteristic of a spin = 1
+quantum magnet affected by the presence of magnetic anis-
+tropy [16, 17, 29, 33, 34], figure 1 (g). The steps of conduc-
+tance are due to the opening of additional spin-flip tunne-
+ling channels trough inelastic excitations. This is in agree-
+ment with the known behavior of Fe-Pc and Fe-P which be-
+have as S=1 nanomagnets once deposited on the Au(111) sur-
+face [15, 17, 20, 22, 29, 35]. The step-like spin-flip signatures
+are recorded at both upper pyrrole and Fe-atom locations, pro-
+viding evidence for hybridization of the molecular states of
+the pyrrole with the Fe magnetic d-states [17, 29].
+When the molecule is located on the elbow of the recons-
+truction, it behaves differently and exhibits a spectral reso-
+nance at the Fermi level. The molecule rotation induced by
+the microscope tip over the elbow site barely affects the shape
+and amplitude of the spectral resonance, figures 1 (d-f, h).
+In the following we show that the two distinct spectral si-
+gnatures are fully controlled by the adsorption site of the mo-
+lecule. To this end we have prepared chains of 3 covalently
+bonded Fe−DPyDBrPP molecules by Ulmann’s coupling
+(see Methods section). In figure 2, a trimer chain is moved by
+the tip of the microscope to various positions of the recons-
+tructed surface. The targeted locations are the hcp and fcc do-
+mains and the Br-decorated elbow of the reconstruction. In the
+first panel (I) of figure 2, the 3 molecules are located inside a
+fcc domain. In panels (II-IV), the molecular chain is sequen-
+tially repositioned with the microscope tip to move the iron
+center of each molecule over the Br-decorated site. When the
+molecules are located inside fcc and hcp domains, the spec-
+tra exhibit inelastic excitations of independent spin 1 nano-
+magnets in presence of magnetic anisotropy, similar to which
+was already measured for monomers in figure 1. This observa-
+tion means that, here, neither the nature of the covalent bonds
+nor the substrate mediated interaction are efficient enough for
+coupling the molecules together. The spectra usually display a
+symmetric double-step structure. Following the standard ana-
+lysis [34], the characteristic voltages of the steps is interpreted
+to be related to the out of plane and in plane magnetic aniso-
+tropy energies. Depending on adsorption sites, the out of plane
+and in plane magnetic anisotropy energies can vary from 6.8
+to 10.0 meV and from 0 to 1.5 meV respectively. These aniso-
+tropy parameters correspond to typical values of Fe porphy-
+rin and phthalocyanine based magnetic molecules adsorbed
+on gold [15–17, 22, 29]. In panels (II-IV), the molecules are
+moved one by one above a Br-decorated elbow. In each case,
+the molecule exhibits a spectral resonance at 0 bias similarly
+to the monomer in figure 1. In figure 2, the resonance is found
+to be reversible once the molecule is removed from the elbow
+site and independent of the selected molecule. We identify the
+spectral peak as an Abrikosov-Suhl resonance [36–38] (also
+named Kondo peak) due to the many body Kondo interactions
+of the Fe d states with the substrate electron bath. The charac-
+teristic Kondo temperature, TK ≈ 11 K, evaluated by fitting
+the lineshape with a Frota function [39, 40], is found to be
+independent of the molecular orientation with respect to the
+surface atomic lattice. We therefore expect that magnetic ani-
+sotropy plays no role.
+The Kondo effect originating from degenerate triplet
+ground state is extremely sensitive to magnetic anisotropy
+which tends to lift the degeneracy [21]. In the present case,
+KBTK ≈ 0.9 meV is much smaller than the measured aniso-
+tropy. Three possible phenomena can explain the robustness
+of the Kondo effect on the magnetic anisotropy occurring at
+the Br-decorated elbow site : 1. The charge fluctuations due to
+the coupling of the Fe states with the substrate states overw-
+helm the magnetic anisotropy energy at this location, effecti-
+vely restoring the degeneracy and screening the magnetic S=1
+moment; 2. A sizable charge transfer from the substrate or
+the ligands onto the Fe atom induces a spin reduction from
+S = 1 to S = 1/2, resulting in a spin 1/2 Kondo effect which
+is naturally immune against magnetic anisotropy [41]; 3. The
+deformation of the molecule at this position induces the sup-
+pression of the magnetic anistropy energy [21]. This last ex-
+planation is not in agreement with our tunneling topography
+
+3
+FIGURE 2: Scanning tunneling study of Fe−DPyDBrPP chains
+made by Ullman’s coupling. (a) topography images (I-IV) of a tri-
+mer chain of molecules made by Ullman’s coupling which is ma-
+nipulated by the tip of the microscope in order to position sequen-
+tially the monomers above the Br-site. The intersection of the dotted
+lines indicates the Br-site. The molecules are labeled as a function
+of their spectroscopic signature measured in (b) : "SF" stands for
+spin-flip and "K" for Kondo. Experimental parameters of images (I-
+IV) : size 15×15 nm2, V = 125 mV, I = 20 pA. (b) related scanning
+tunneling spectra taken above the left upper-pyrrole of each mole-
+cule of the chain, showing the presence of the Kondo peak for the
+molecules located above the Br-site which otherwise shows a spin-
+flip signature. Vstab = 30 mV, Istab = 200 pA. Lock-in parameters :
+Vm = 0.2 mV, f = 750 Hz. The solid lines correspond to the fits
+of the data (see Methods section). (c) topographic zoom on a mole-
+cule of the chain exhibiting a spin-flip spectroscopic signature. (d-
+e) differential conductance image recorded simultaneously to image
+(c) at −100 mV and 100 mV respectively showing the symmetry of
+the frontier orbitals below and above the Fermi level. The red dotted
+line delimits the molecule. Experimental parameters of (c-e), size
+3 × 3 nm2, Vstab = 800 mV, Istab = 500 pA. Lock-in parameters :
+Vm = 5 mV, f = 900 Hz.
+studies which did not reveal a significant deformation of the
+molecule whose shape is preserved on the Br-site. It has also
+been shown that the anisotropic energy is not significantly af-
+fected when a similar molecule is pressed onto the surface
+with the tip of a microscope while staying away from the tip-
+molecule contact [14, 15, 29]. As for scenarios 1 and 2, they
+are both possible, but the DFT calculations and the theoreti-
+cal interpretation that follow lean in the direction of the first
+explanation.
+Density Functional Theory analysis
+DFT was used to study the electronic properties as a func-
+tion of the distance of the molecule from the substrate. In or-
+der to rationalize the effect of the Br-adatom on the electronic
+properties of Fe−DPyDBrPP, two sets of situations were
+simulated : 1. as a function of the distance to the genuine
+Au surface 2. as a function of the distance to a Br adatom
+on Au slabs positioned below the Fe atom of the molecule
+Fe−DPyDBrPP.
+The Fe 3d states are dominant in the coupling with the sub-
+strate through the Br states. Fe dxz, dyz states are mainly hy-
+bridized with the pyrroles molecular orbitals. Therefore, fur-
+ther analysis will be discussed on the basis of the Fe 3d hy-
+brids with molecular orbital. In particular the Fe dxz, dyz and
+dz2 hybrid states were proven to be at the origin of the obser-
+ved phenomena.
+The DFT simulation shows that the magnetic polarization
+(contrary to the total charge) of the Fe atom, figure 3 (b), de-
+pends strongly on the molecule-surface distance when the sur-
+face is decorated with a Br (or Au) adatom, whereas it is much
+less sensitive when approaching the clean surface. When the
+molecule is away from the Br site, the total occupancy and
+magnetic moment are about 6.4 e− and 2µB which corres-
+ponds to a configuration close to the spin 1 state (high spin
+state) and the hybridization of the Fe d states with the sub-
+strate is small. This is in good agreement with the measured
+spectroscopic signature of a spin 1 quantum magnet when the
+molecule is above fcc or hcp domains. Indeed, the lack of a
+Kondo signature may be the result of weak hopping integrals
+from Fe orbitals to surface orbitals when the molecule is far
+away. Hybridization with the substrate cannot compete with
+the magnetic anisotropy and the Kondo effect is prevented.
+The magnetic anisotropy energy dominates the physics and
+the experimental spectroscopic signature is that of a spin 1
+subjected to an anisotropy of a few meV.
+Moving the molecule of 1 Å towards the Br adatom, from
+5 to 4 Å, leaves the occupancy of the Fe states roughly un-
+changed (only a minor reshuffling of the charge distribution
+between the orbitals is observed (figure 3 (c)) - but induces a
+strong reduction of the magnetic polarization from about 2µB
+to about 1µB. The approximately constant charge suggests
+that charge fluctuations are the cause of this reduction, which
+is confirmed by the fact that the hybridization of the Fe and Br
+states increases exponentially as the molecule is moved closer
+
+(a) (l)
+()
+SF
+SF
+SF
+SF
+SF
+K
+(I)
+(IV)
+SF
+K
+K
+SF
+SF
+SF
+(b)
+(c)
+(1)
+(d)
+(II)
+(e)
+(IV)4
+FIGURE 3: Theory analysis by DFT calculations. (a) Simulated adsorption geometry for the Iron(II) 5,15-(di-4-bromophenyl)-10,20-(di-4-
+pyridyl) porphyrin molecule on a Au(111) surface. The (Au,Br) impurities are adsorbed on the hollow-hcp site. (b) polarizations as a function
+of the distance from the substrate, computed for three different substrates, clean Au(111), Au(111) + a Au adatom and Au(111) + a Br adatom,
+obtained via the same Mulliken analysis as in (c). (c) Orbitally-resolved Fe(3d) charge as a function of the distance from the substrate. The
+charge and magnetization values have been obtained via a Mulliken analysis projecting the Kohn-Sham orbitals over the atomic states of the Fe
+pseudopotential. (d) Calculated non-zero squared hoppings between the Fe(3d) and the Br(4s,4p) states in the low-spin (Fe-Au(111) distance
+= 4.0 Å, in blue) and high spin (Fe-Au(111) distance = 5.0 Å, in red) case. Only the squared average of the spin-up and spin-down hopping is
+reported, since no significant difference has been found in the two channels.
+to the surface (the squared hoppings that enters in this form in
+the hybridization function between the impurity and the sub-
+strate are plotted in the figure 3 (d)). The squared hopping in-
+tegrals of the Fe states to the surface are indeed one order
+of magnitude smaller when the molecule is far away. The Fe
+3dz2 hybrids and Br 4s states, which have a almost perfect ro-
+tational symmetry, are supposed to dominate the physics when
+the molecule is closer (figure 3 (d)). This configuration is favo-
+rable to the emergence of a charge transport that is insensitive
+to magnetic anistropy and independent of the relative orienta-
+tion between the molecule and the surface once the iron atom
+is located just above the adsorbed Br atom, in perfect agree-
+ment with observations.
+The cases presented above are perfectly consistent with the
+scenario in which the molecule evolves from a quasi-atomic
+moment (spin 1) under the influence of magnetic anisotropy
+to an Fe atom carrying a screened magnetic moment resul-
+ting from a strong hybridization with the substrate. Finally,
+the DFT calculations clearly indicate that the strength of the
+charge fluctuations should vary with the distance to the Br
+atom. Indeed the transition from spin 1 nanomagnet to Kondo
+screening, through the mixed-valence regime has been expe-
+rimentally probed in a reversible manner and summarized in
+figure 4.
+Crossover from spin 1 nanomagnet to Kondo-screened ground
+state through the mixed-valence regime
+It was possible to finely tune the electronic configuration of
+the iron atom by using a chain of molecules in order to sta-
+bilize positions for which a molecule deviates slightly from
+the Br site. The studied position of the molecules are presen-
+ted in topography images (a-d) of figure 4 and the density of
+states close to the Fermi energy are measured by STS spectra
+figure 4 (g)). The spin 1 signature is observed when the mole-
+cule is far from the elbow site as already mentioned above, fi-
+gure 4 (a). When the molecule is positioned in such a way that
+the Br site is located below the upper pyrrole ring, image 4
+(b), the LDOS shows a broadened and asymmetric step like
+shape originating from spin flip inelastic excitations but with
+a lower characteristic energy, indicating a renormalization of
+the anisotropy energy. At intermediate positions where the Br
+adsorption-site is located in between the Fe site and the upper
+pyrrole location, 4 (c), the spectroscopy shows a small anti-
+resonance at the Fermi level with a Fano-like shape which
+we believe is the extrapolation of the inelastic spin flip si-
+gnature but for stronger charge fluctuations. When the mo-
+lecule is strictly coinciding with the Br-adsorption site, fi-
+gure 4 (d), it is exhibiting a Kondo peak. Therefore, the si-
+tuation of images 4 (b) & (c) corresponds to two interme-
+diate mixed-valence regimes between spin 1 nanomagnet and
+Kondo-screened ground states. Indeed as shown by D. Jacob
+in Ref. [41] the gradual increase of charge fluctuations alters
+the spectroscopic signatures of the spin 1 nanomagnet by re-
+normalizing the magnetic excitations to lower energy. This
+gradually closes the gap and finally restores a Kondo peak
+as in the case of image 4 (d). In this process, the absence of
+particle-hole symmetry causes a typical Fano resonance to ap-
+pear. The renormalization of the magnetic excitations is evi-
+denced in pannels 4 (h-k) where differential conductance is
+used to map the inelastic spin-flip probability density in the
+real space in the same configuration as in (b). This situation
+corresponds to the occurrence of one molecule in the inter-
+mediate state and 2 molecules in the regime of the spin 1 na-
+nomagnet. At ±9.3 mV, spin-flip channels are open in all 3
+molecules and the probability to induce a spin flip follows a
+two-lobes spatial pattern in all three molecules. At ±3 mV,
+panels (i) & (j) the spin-flip excitations are hindered by the
+magnetic anisotropy in the molecules far from the Br-site. On
+the contrary, the molecule in the intermediate regime shows
+spin-flip processes already being developed at ±3 mV, indica-
+ting the renormalization of the magnetic excitation energy to
+
+(a)5
+dI/dV (arb. units)
+Bias Voltage (mV)
+Upper-pyrrole
+(e)
+Bias Voltage (mV)
+(g)
+dI/dV (arb. units)
+(f)
+0
+10
+20
+-10
+-20
+(h)
+-9.3mV
+(i)
+-3mV
+(j)
+3mV
+(k)
+9.3mV
+(a)
+(b)
+(c)
+(d)
+-500
+-250
+500
+-250
+0
+dI/dV (arb. units)
+Bias Voltage (mV)
+Iron center
+-500
+-250
+500
+-250
+0
+FIGURE 4: Crossover from spin 1 nanomagnet to Kondo screen ground state through the mixed-valence regime. (a-d) topography images
+showing the position of the first molecule of a chain with respect to the Br-decorated elbow site of the gold reconstruction. (a) the molecule
+is far from the Br site. (b) the upper pyrrole is centered on the Br-site. (c) the Br-site is located in between the center of the upper pyrole and
+the Fe ion position. (d) the Fe atom is located just above the Br-site. (e) conductance spectra taken above the center of the upper pyrole. (f and
+g) conductance spectra taken above the iron atom. red, green, blue and yellow colors correspond to the configurations of (a), (b), (c) and (d)
+respectively. (h-k) Background subtracted (see Methods section) differential conductance images at V = −9.3 mV, V = −3.0 mV,V = 3 mV
+and V = 9.3 mV respectively recorded in the configuration of image (b) where, the left molecule is in the mixed-valence intermediate state
+(blue dashed line) and the two other molecules behave as spin 1 nanomagnets (e.g. red dashed line). Experimental parameters : (a) & (d)
+V = 125 mV, I = 20 pA, scalebar = 2 nm; (b) & (c) V = 200 mV, I = 100 pA, scalebar = 2 nm; (e) & (f) Vstab = 800 mV, Istab = 500 pA,
+lock-in parameters : Vm = 5 mV, f = 900 Hz; (g) Vstab = 30 mV, Istab = 200 pA, lock-in parameters : Vm = 0.2 mV, f = 750 Hz; (h-k)
+Vstab = 30 mV, Istab = 200 pA, size 12×12 nm2, lock-in parameters : Vm = 0.2 mV, f = 750 Hz. All spectra are normalized.
+a smaller scale.
+This transition through a mixed-valence regime might be
+due to a net increase of Fe charge while remaining still in a
+regime of weak coupling with the substrate (our previously
+outlined scenario 2). Our DFT calculations rather suggest that
+the predominant effect is the drastic increase of the hopping
+from/to the substrate (scenario 1), instead. Accordingly, the
+spin crossover was found to be accompanied with an energy
+shift of the frontier orbitals which are localized at the up-
+per pyrrole rings (frontier orbital A, FOA) and at the iron
+atom position (frontier orbital B, FOB). A carefully analysis
+of figures 4 (e) & (f) allows to follow the evolution of FOA
+and FOB energies when moving the molecule. FOA shows a
+continuous evolution of density of states from the unoccupied
+states towards occupied states when approaching the molecule
+to the Br-site, indicating an increase of the orbital occupation.
+FOB shows a reverse behavior with a shift from below the
+Fermi level towards the unoccupied states. The gap between
+the frontier orbitals is reducing when approaching the Br site
+so the d level splitting tends to be weakened and could be
+favoring a decrease of orbital momentum and a renormaliza-
+tion of the magnetic anisotropy in agreement with the closing
+of the steps of conductance due to spin-flip excitations in the
+measured data of panel 4 (g) and figures 4 (h-k).
+The conductance images presented figure 2 (c-e) support
+the hybrid Fe 3dxz, dyz character of FOA and the strong Fe
+3dz2 character of FOB in agreement with the orbital occupa-
+tions in the DFT calculations (figure 3 (c)). It is interesting
+to note that according to conductance images of figure 4 (h-
+k), the spin-flip probability density is following the 3dxz, dyz
+symmetry at energies above and below the Fermi energy (two-
+lobes shape). On the contrary, the 3dz2 character of the exci-
+tations (bright localized spot at the Fe site) is only visible at
+negative bias, so for occupied states. This difference is explai-
+ning the breaking of the electron-hole symmetry observed in
+inelastic spectra.
+The fano-like shape of the blue spectrum in figure 4 (g) oc-
+curs when the frontier orbitals are overlapping at the Fermi
+level, as visible in figure 4 (e) & (f) and when the charge fluc-
+tuations between orbitals are expected to be strong. The hypo-
+thesis of intermediate regimes driven by charge fluctuations is
+well described by prediction in ref. [41] as well as the shape
+of the spectrum. However in the present case the frontier or-
+bitals, could also recover their degeneracy when overlapping
+in the intermediate regimes, implying a possible Kondo SU(4)
+effect [35, 42], although this is less likely in a molecular sys-
+tem [43] and cannot be easily distinguished from the simple
+effect of strong charge fluctuations [44]. Finally, when the iron
+atom is located above the Br site, FOA migrates below the
+Fermi level and FOB is washed out towards incoherent exci-
+tation and only the Kondo resonance survives at EF.
+CONCLUSION
+We have revealed in experiments and DFT calculations that
+the spin state switching is caused by the strong hybridiza-
+tion potential at Br-decorated elbow sites which induce charge
+
+6
+fluctuation and a renormalization of the magnetic anisotropy
+when molecules are adsorbed on these sites.
+The direct consequence of charge fluctuations is to tune the
+energies of two molecular frontier orbitals which are respec-
+tively closest to the Fermi level in positive and negative ener-
+gies. The energetic overlapping of these two frontier orbitals
+gives rise to low-energy spin excitation channels near Fermi
+level. Under the condition of relative small charge fluctua-
+tions, the two frontier orbitals are far from Fermi level and
+the overlapping is small, hence the molecule will have high
+spin (S = 1) state, while under strong charge fluctuation these
+two orbitals can cross Fermi level, resulting a renormalization
+of the magnetic anisotropy and ultimately the Kondo screened
+ground state.
+Our work provides a new approach for tuning not only the
+spin state but also the intermediate valence character/charge
+fluctuation in hybrids made of molecules on reconstructed sur-
+faces.
+METHODS
+Fe 5,15-di-4-pyridyl-10,20-di-4-bromophenyl porphyrin
+molecules were ordered from Alpha Aesar company. Samples
+were prepared under ultrahigh vacuum (base pressure 1 ×
+10−10 mbar) just before their scanning tunneling studies. The
+gold substrate was prepared by repeated cycles of Ar ion sput-
+tering followed by annealing at 900 K. The molecules were
+deposited by organic molecular beam epitaxy (OMBE) when
+the sample was kept at Room Temperature (Tcrucible=575K).
+It was observed that the molecule source is also depositing Br
+atoms on the surface of gold. We have attributed this conco-
+mitant deposition of Br with Fe−DPyDBrPP to the stochas-
+tic Br detaching from Fe−DPyDBrPP molecules inside the
+crucible when heated at evaporation temperature. In-crucible
+Br detaching form similar molecules was already reported in
+ref. [30] at 590K. In our case, the low amount of deposited
+Br atoms was seen to be enough for occupying all the elbow
+sites of the herringbone reconstruction. Once the molecules
+deposited, the sample temperature was quenched at liquid ni-
+trogen temperature then at helium liquid temperature in less
+than 20 minutes. For low coverage this procedure produces
+single monomers scattered on the surface. A post annealing
+at 400 K for 10 minutes was used for producing molecular
+chains by Ullman’s coupling at the surface following recipes
+in ref. [30, 45–47].
+STM/STS measurements were achieved in-situ in ultrahigh
+vacuum by means of a modified "Tyto SPM" from SPECSTM
+at a base temperature of 1.3 K; the tunneling bias was applied
+to the sample. Mechanically cut PtIr tips were used. Soft la-
+teral manipulation as follows was used for moving the mole-
+cules 1) positioning the tip above the molecule to move 2) Tur-
+ning off the microscope feedback loop and approaching the
+tip towards molecule in order to increase tip-molecule inter-
+action 2) Moving the tip to the destination in constant height
+mode 3) Retracting the tip and engaging the feedback loop.
+The analyses of the microscopy and spectroscopy data were
+done with WSxM software [48] and homemade python proce-
+dures respectively [49].
+The fit of spectra from figure 1 & 2 involved a large Gaus-
+sian background mimicking the frontier orbital DOS plus a
+specific function associated to spin-flips (step functions intro-
+duced in [33]) or Kondo ground states (Frota function). Mar-
+ginally, a convolution with a Gaussian function was used to
+clean up the incoherent noise of all conductance spectra pre-
+sented here.
+The spin-polarized Density-Functional Theory (DFT) cal-
+culations have been carried out with v.∼6.8 of the QUAN-
+TUM ESPRESSO package [50, 51] within the Projector
+Augmented-Wave (PAW) scheme [52] and using the Perdew-
+Burke-Ernzerhof (PBE) functional [53]. PAW pseudopoten-
+tials have been taken from the PSlibrary [54–56].
+The Au(111) surface has been modeled by a finite slab
+consisting of 3 gold layers with 15 Åof vacuum between
+periodic replicas along the z direction. The Au-Au nearest-
+neighbor distance has been set to 2.93 Å, the PBE equili-
+brium distance of the corresponding fcc bulk structure, which
+is ≲ 2% larger than the experimental room-temperature va-
+lue [57].
+Given the sizeable extension of the molecule, a 10×10 su-
+percell of Au atoms on the xy plane has been used in order
+to avoid interaction between its periodic replicas, for a total
+of 300 Au atoms in the simulation cell. The molecule geo-
+metry has been relaxed in the vacuum until all components
+of the forces were smaller than 10−3 Ry/aB, with aB being
+the Bohr radius. The projected density of states (pDoS) and
+the Mulliken orbital occupations and polarizations have been
+evaluated by projecting the Kohn-Sham orbitals on the atomic
+states of the pseudopotentials. They have been computed by
+considering different adsorption geometries between the mo-
+lecule and three different substrates : the clean Au(111) sur-
+face, the Au(111) surface with a Au impurity and the Au(111)
+surface with a Br impurity. The impurities have been placed
+at the preferred PBE adsorption site at the equilibrium dis-
+tance, which has resulted in the hollow-hcp site for both the
+Au and the Br impurity, respectively at 2.16 Åand 2.23 Åfrom
+the Au(111) surface. All calculations have been performed
+with and a plane-wave cutoff of ϵcut = 40 Ry on the wave-
+functions and 320 Ry on the density. Brillouin zone integrals
+have been evaluated at the Γ point using a gaussian smearing
+of σ = 0.01 Ry. The hopping parameters have been computed
+via the projection method as implemented in the WANNIER90
+package [58].
+ACKNOWLEDGEMENT
+Cheryl Feuillet-Palma is acknowledged for her construc-
+tive discussions. TG thanks Paolo Restuccia for fruitful dis-
+cussions. LdM acknowledges Giorgio Sangiovanni for useful
+discussions. SV, DR and SP acknowledge the French National
+Research Agency for the support of the SHOGUN project, Re.
+
+7
+ANR-22-CE09-0028-01. TG and LdM are supported by the
+European Commission through the ERC-CoG2016, Strong-
+CoPhy4Energy, GA No724177.
+∗ Electronic address: stephane.pons@espci.fr
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+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAzT4oBgHgl3EQfKvs6/content/2301.01101v1.pdf,len=919
+page_content='Manipulation of the magnetic state of a porphyrin-based molecule on gold : From Kondo to  quantum nanomagnet via the charge fluctuation regime  Yingzheng Gao,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAzT4oBgHgl3EQfKvs6/content/2301.01101v1.pdf'}
+page_content='1 Sergio Vlaic,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAzT4oBgHgl3EQfKvs6/content/2301.01101v1.pdf'}
+page_content='1 Tommaso Gorni,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAzT4oBgHgl3EQfKvs6/content/2301.01101v1.pdf'}
+page_content='1 Luca de’ Medici,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAzT4oBgHgl3EQfKvs6/content/2301.01101v1.pdf'}
+page_content='1  Sylvain Clair,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAzT4oBgHgl3EQfKvs6/content/2301.01101v1.pdf'}
+page_content='2 Dimitri Roditchev,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAzT4oBgHgl3EQfKvs6/content/2301.01101v1.pdf'}
+page_content='1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAzT4oBgHgl3EQfKvs6/content/2301.01101v1.pdf'}
+page_content=' 3 and Stéphane Pons1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAzT4oBgHgl3EQfKvs6/content/2301.01101v1.pdf'}
+page_content=' ∗  1Laboratoire de Physique et d’Étude des Matériaux (LPEM),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAzT4oBgHgl3EQfKvs6/content/2301.01101v1.pdf'}
+page_content=' ESPCI Paris,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAzT4oBgHgl3EQfKvs6/content/2301.01101v1.pdf'}
+page_content='  Université PSL,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAzT4oBgHgl3EQfKvs6/content/2301.01101v1.pdf'}
+page_content=' CNRS UMR8213,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAzT4oBgHgl3EQfKvs6/content/2301.01101v1.pdf'}
+page_content=' Sorbonne Université,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAzT4oBgHgl3EQfKvs6/content/2301.01101v1.pdf'}
+page_content=' 75005 Paris,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAzT4oBgHgl3EQfKvs6/content/2301.01101v1.pdf'}
+page_content=' France  2Aix Marseille Univ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAzT4oBgHgl3EQfKvs6/content/2301.01101v1.pdf'}
+page_content=' CNRS,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAzT4oBgHgl3EQfKvs6/content/2301.01101v1.pdf'}
+page_content=' IM2NP,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAzT4oBgHgl3EQfKvs6/content/2301.01101v1.pdf'}
+page_content=' Marseille,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAzT4oBgHgl3EQfKvs6/content/2301.01101v1.pdf'}
+page_content=' France  3Institut des Nanosciences de Paris,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAzT4oBgHgl3EQfKvs6/content/2301.01101v1.pdf'}
+page_content=' Sorbonne Université,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAzT4oBgHgl3EQfKvs6/content/2301.01101v1.pdf'}
+page_content=' CNRS UMR7588,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAzT4oBgHgl3EQfKvs6/content/2301.01101v1.pdf'}
+page_content=' 75005 Paris,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAzT4oBgHgl3EQfKvs6/content/2301.01101v1.pdf'}
+page_content=' France  (Dated: 4 janvier 2023)  By moving individual Fe-Porphyrin-based molecules with the tip of Scanning Tunneling Microscope  in the vicinity of a Br-atom containing elbow of the herringbone-reconstructed Au(111),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAzT4oBgHgl3EQfKvs6/content/2301.01101v1.pdf'}
+page_content=' we reversibly  and continuously control their magnetic state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAzT4oBgHgl3EQfKvs6/content/2301.01101v1.pdf'}
+page_content=' Several regimes are obtained experimentally and explo-  red theoretically : from the integer spin limit, through intermediate magnetic states with renormalized  magnetic anisotropy, until the Kondo-screened regime, corresponding to a progressive increase of charge  fluctuations and mixed valency due to an increase in the interaction of the molecular Fe states with the  substrate Fermi sea.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAzT4oBgHgl3EQfKvs6/content/2301.01101v1.pdf'}
+page_content=' Our results open a route for the realization, tuning and experimental studies of novel  quantum magnetic states in molecule-surface hybrids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAzT4oBgHgl3EQfKvs6/content/2301.01101v1.pdf'}
+page_content='  KEYWORDS  spin-flip excitation, spin states, magnetic anisotropy,  Kondo, charge fluctuation, mixed-valence states, scanning  tunneling microscopy, transiton metal complexes  INTRODUCTION  Addressing and manipulating the spin state of molecular  species at interfaces is a challenge that could greatly bene-  fit spintronics [1], nanoelectronics [2, 3] and quantum electro-  nics [4] in the near future.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAzT4oBgHgl3EQfKvs6/content/2301.01101v1.pdf'}
+page_content=' When the valence of a magnetic  molecule deposited on a surface is integer, the description of  spin-polarized molecular orbitals can be done in the frame-  work of the atomic limit in terms of crystal field and spin-orbit  coupling : the spin state is simply interpreted as a quantum  magnet for which the Hund’s rule determines the fundamental  magnetic state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAzT4oBgHgl3EQfKvs6/content/2301.01101v1.pdf'}
+page_content=' However, the atomic limit is no longer valid in  the mixed-valence regime which is the most general behavior  of an interacting magnetic impurity with an electron bath.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAzT4oBgHgl3EQfKvs6/content/2301.01101v1.pdf'}
+page_content='  Among the most studied classes of magnetic molecules are  the transition metal phthalocyanine (Pc) and porphyrin (P) fa-  milies which are easily deposited at surface in vacuum and  studied by various local probe microscopies and spectrosco-  pies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAzT4oBgHgl3EQfKvs6/content/2301.01101v1.pdf'}
+page_content=' In these molecules, the spin state is mainly given by  the spin polarization of the central transition metal ion d-  states [5, 6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAzT4oBgHgl3EQfKvs6/content/2301.01101v1.pdf'}
+page_content=' Several works have focused on the influence of  external parameters on the magnetic ground state such as the  influence of charge transfer to the orbitals of the molecule [7–  14], the effect of surface spin-orbit coupling and magnetic ani-  sotropy [15–17], the coupling to the substrate, [15, 18], the in-  teraction with attached and neighboring molecules [19, 20],  the structural deformation [21, 22] or the chemical substitu-  tion of ligands [23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAzT4oBgHgl3EQfKvs6/content/2301.01101v1.pdf'}
+page_content=' For all these studies, a systematic unders-  tanding of the effect of mixed valence and charge fluctuations  is still missing, although they have a strong influence on the  effect of magnetic anistropy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAzT4oBgHgl3EQfKvs6/content/2301.01101v1.pdf'}
+page_content='  Here  we  show  that  charge  fluctuations  in  Fe  5,15-di-4-pyridyl-10,20-di-4-bromophenyl  porphyrin  (Fe−DPyDBrPP) molecules adsorbed on the Br de-  corated Au(111) surface allow the magnetic state of the  molecule to be driven between high-spin (S=1) and Kondo-  screened states in a reversible and continuous manner through  the intermediate valence regimes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAzT4oBgHgl3EQfKvs6/content/2301.01101v1.pdf'}
+page_content='  RESULTS AND DISCUSSION  Structural and spectroscopic properties of monomers  Fe−DPyDBrPP were deposited in ultrahigh vacuum on  Au(111) and studied in situ by scanning tunneling microscopy  (STM) and spectroscopy (STS) at 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAzT4oBgHgl3EQfKvs6/content/2301.01101v1.pdf'}
+page_content='3K (see Methods sec-  tion for more details).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAzT4oBgHgl3EQfKvs6/content/2301.01101v1.pdf'}
+page_content=' The molecules are randomly located on  hcp domains, fcc domains or stacking faults lines of the her-  ringbone reconstruction [24] of Au(111) and appear as bright  spots in constant-current topographic STM images in figures 1  (a-f).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAzT4oBgHgl3EQfKvs6/content/2301.01101v1.pdf'}
+page_content=' The Fe−DPyDBrPP molecules do not exhibit a pla-  nar configuration on the Au(111) surface because two pyrrole  rings bend toward the vacuum and exhibit a higher height in  STM images;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAzT4oBgHgl3EQfKvs6/content/2301.01101v1.pdf'}
+page_content=' the other two pyrrole rings bend toward the sub-  strate, resulting in a lower height in images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAzT4oBgHgl3EQfKvs6/content/2301.01101v1.pdf'}
+page_content=' Such deforma-  tion corresponds to the saddling distortion of porphyrin-based  molecules [25] that is quite commonly observed [17, 26–29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAzT4oBgHgl3EQfKvs6/content/2301.01101v1.pdf'}
+page_content='  Thus, the shape of the molecules in STM images allows us  to determine the central position of the Fe-atom, the porphine  macrocycle and the bromophenyl and pyridyl ligands attached  to the macrocycle, as well as their location relative to the sur-  face reconstruction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAzT4oBgHgl3EQfKvs6/content/2301.01101v1.pdf'}
+page_content=' Importantly, during deposition and annea-  ling, some Br-atoms detach from the molecules [30], migrate  at the surface, and get trapped by highly reactive elbows of the  Au(111) herringbone reconstruction [31, 32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAzT4oBgHgl3EQfKvs6/content/2301.01101v1.pdf'}
+page_content=' The intersection  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAzT4oBgHgl3EQfKvs6/content/2301.01101v1.pdf'}
+page_content='01101v1  [cond-mat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAzT4oBgHgl3EQfKvs6/content/2301.01101v1.pdf'}
+page_content='mes-hall]  3 Jan 2023  2  Bias Voltage (mV)  dI/dV (arb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAzT4oBgHgl3EQfKvs6/content/2301.01101v1.pdf'}
+page_content=' units)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAzT4oBgHgl3EQfKvs6/content/2301.01101v1.pdf'}
+page_content='25  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAzT4oBgHgl3EQfKvs6/content/2301.01101v1.pdf'}
+page_content='50  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAzT4oBgHgl3EQfKvs6/content/2301.01101v1.pdf'}
+page_content='75  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAzT4oBgHgl3EQfKvs6/content/2301.01101v1.pdf'}
+page_content='00  0  20  10  10  20  Bias Voltage (mV)  dI/dV (arb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAzT4oBgHgl3EQfKvs6/content/2301.01101v1.pdf'}
+page_content=' units)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAzT4oBgHgl3EQfKvs6/content/2301.01101v1.pdf'}
+page_content='6  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAzT4oBgHgl3EQfKvs6/content/2301.01101v1.pdf'}
+page_content='2  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAzT4oBgHgl3EQfKvs6/content/2301.01101v1.pdf'}
+page_content='8  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAzT4oBgHgl3EQfKvs6/content/2301.01101v1.pdf'}
+page_content='4  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAzT4oBgHgl3EQfKvs6/content/2301.01101v1.pdf'}
+page_content='0  0  20  10  10  20  (g)  (h)  (d)  (e)  (f)  (a)  (b)  (c)  FIGURE 1: Scanning tunneling study of Fe−DPyDBrPP mono-  mers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAzT4oBgHgl3EQfKvs6/content/2301.01101v1.pdf'}
+page_content=' (a-f) Topography image of Fe−DPyDBrPP molecules at  various location of the reconstructed surface of gold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAzT4oBgHgl3EQfKvs6/content/2301.01101v1.pdf'}
+page_content=' The intersec-  tion of the dotted lines indicates the location of the Br adatom which  decorates the elbow of the herringbone reconstruction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAzT4oBgHgl3EQfKvs6/content/2301.01101v1.pdf'}
+page_content=' (g) Norma-  lized scanning tunneling spectra taken on upper pyrroles in images  (a-c) showing spin inelastic excitations depending on the location of  the molecule on the surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAzT4oBgHgl3EQfKvs6/content/2301.01101v1.pdf'}
+page_content=' (h) Normalized scanning tunneling spec-  tra taken on upper pyrroles in images (d-f), when the molecule is on  top of the Br-decorated elbow site of the reconstruction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAzT4oBgHgl3EQfKvs6/content/2301.01101v1.pdf'}
+page_content=' These spec-  tra show an Abrikosov-Suhl resonance witnessing a Kondo mecha-  nism which does not depend on the in-plane rotation of the molecule.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAzT4oBgHgl3EQfKvs6/content/2301.01101v1.pdf'}
+page_content='  The solid lines correspond to the fits of the data (see Methods sec-  tion).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAzT4oBgHgl3EQfKvs6/content/2301.01101v1.pdf'}
+page_content=' Experimental parameters : topography images : V = 125 mV,  Istab = 20 pA, size 10 × 10 nm2 ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAzT4oBgHgl3EQfKvs6/content/2301.01101v1.pdf'}
+page_content=' spectroscopy : Vstab = 30 mV,  Istab = 200 pA, lock-in parameters : Vm = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAzT4oBgHgl3EQfKvs6/content/2301.01101v1.pdf'}
+page_content='2 mV, f = 750 Hz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAzT4oBgHgl3EQfKvs6/content/2301.01101v1.pdf'}
+page_content='  of the dotted lines in figures 1 (a-c) points to the position of  these elbows decorated by Br-adatoms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAzT4oBgHgl3EQfKvs6/content/2301.01101v1.pdf'}
+page_content='  Depending on their position on the surface, the molecules  exhibit different spectral signatures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAzT4oBgHgl3EQfKvs6/content/2301.01101v1.pdf'}
+page_content=' When a molecule is loca-  ted on the hcp and fcc domains (figures 1 (a-c)), the tunneling  spectra show conductance steps characteristic of a spin = 1  quantum magnet affected by the presence of magnetic anis-  tropy [16, 17, 29, 33, 34], figure 1 (g).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAzT4oBgHgl3EQfKvs6/content/2301.01101v1.pdf'}
+page_content=' The steps of conduc-  tance are due to the opening of additional spin-flip tunne-  ling channels trough inelastic excitations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAzT4oBgHgl3EQfKvs6/content/2301.01101v1.pdf'}
+page_content=' This is in agree-  ment with the known behavior of Fe-Pc and Fe-P which be-  have as S=1 nanomagnets once deposited on the Au(111) sur-  face [15, 17, 20, 22, 29, 35].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAzT4oBgHgl3EQfKvs6/content/2301.01101v1.pdf'}
+page_content=' The step-like spin-flip signatures  are recorded at both upper pyrrole and Fe-atom locations, pro-  viding evidence for hybridization of the molecular states of  the pyrrole with the Fe magnetic d-states [17, 29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAzT4oBgHgl3EQfKvs6/content/2301.01101v1.pdf'}
+page_content='  When the molecule is located on the elbow of the recons-  truction, it behaves differently and exhibits a spectral reso-  nance at the Fermi level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAzT4oBgHgl3EQfKvs6/content/2301.01101v1.pdf'}
+page_content=' The molecule rotation induced by  the microscope tip over the elbow site barely affects the shape  and amplitude of the spectral resonance, figures 1 (d-f, h).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAzT4oBgHgl3EQfKvs6/content/2301.01101v1.pdf'}
+page_content='  In the following we show that the two distinct spectral si-  gnatures are fully controlled by the adsorption site of the mo-  lecule.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAzT4oBgHgl3EQfKvs6/content/2301.01101v1.pdf'}
+page_content=' To this end we have prepared chains of 3 covalently  bonded Fe−DPyDBrPP molecules by Ulmann’s coupling  (see Methods section).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAzT4oBgHgl3EQfKvs6/content/2301.01101v1.pdf'}
+page_content=' In figure 2, a trimer chain is moved by  the tip of the microscope to various positions of the recons-  tructed surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAzT4oBgHgl3EQfKvs6/content/2301.01101v1.pdf'}
+page_content=' The targeted locations are the hcp and fcc do-  mains and the Br-decorated elbow of the reconstruction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAzT4oBgHgl3EQfKvs6/content/2301.01101v1.pdf'}
+page_content=' In the  first panel (I) of figure 2, the 3 molecules are located inside a  fcc domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAzT4oBgHgl3EQfKvs6/content/2301.01101v1.pdf'}
+page_content=' In panels (II-IV), the molecular chain is sequen-  tially repositioned with the microscope tip to move the iron  center of each molecule over the Br-decorated site.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAzT4oBgHgl3EQfKvs6/content/2301.01101v1.pdf'}
+page_content=' When the  molecules are located inside fcc and hcp domains, the spec-  tra exhibit inelastic excitations of independent spin 1 nano-  magnets in presence of magnetic anisotropy, similar to which  was already measured for monomers in figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAzT4oBgHgl3EQfKvs6/content/2301.01101v1.pdf'}
+page_content=' This observa-  tion means that, here, neither the nature of the covalent bonds  nor the substrate mediated interaction are efficient enough for  coupling the molecules together.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAzT4oBgHgl3EQfKvs6/content/2301.01101v1.pdf'}
+page_content=' The spectra usually display a  symmetric double-step structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAzT4oBgHgl3EQfKvs6/content/2301.01101v1.pdf'}
+page_content=' Following the standard ana-  lysis [34], the characteristic voltages of the steps is interpreted  to be related to the out of plane and in plane magnetic aniso-  tropy energies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAzT4oBgHgl3EQfKvs6/content/2301.01101v1.pdf'}
+page_content=' Depending on adsorption sites, the out of plane  and in plane magnetic anisotropy energies can vary from 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAzT4oBgHgl3EQfKvs6/content/2301.01101v1.pdf'}
+page_content='8  to 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAzT4oBgHgl3EQfKvs6/content/2301.01101v1.pdf'}
+page_content='0 meV and from 0 to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAzT4oBgHgl3EQfKvs6/content/2301.01101v1.pdf'}
+page_content='5 meV respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAzT4oBgHgl3EQfKvs6/content/2301.01101v1.pdf'}
+page_content=' These aniso-  tropy parameters correspond to typical values of Fe porphy-  rin and phthalocyanine based magnetic molecules adsorbed  on gold [15–17, 22, 29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAzT4oBgHgl3EQfKvs6/content/2301.01101v1.pdf'}
+page_content=' In panels (II-IV), the molecules are  moved one by one above a Br-decorated elbow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAzT4oBgHgl3EQfKvs6/content/2301.01101v1.pdf'}
+page_content=' In each case,  the molecule exhibits a spectral resonance at 0 bias similarly  to the monomer in figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAzT4oBgHgl3EQfKvs6/content/2301.01101v1.pdf'}
+page_content=' In figure 2, the resonance is found  to be reversible once the molecule is removed from the elbow  site and independent of the selected molecule.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAzT4oBgHgl3EQfKvs6/content/2301.01101v1.pdf'}
+page_content=' We identify the  spectral peak as an Abrikosov-Suhl resonance [36–38] (also  named Kondo peak) due to the many body Kondo interactions  of the Fe d states with the substrate electron bath.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAzT4oBgHgl3EQfKvs6/content/2301.01101v1.pdf'}
+page_content=' The charac-  teristic Kondo temperature, TK ≈ 11 K, evaluated by fitting  the lineshape with a Frota function [39, 40], is found to be  independent of the molecular orientation with respect to the  surface atomic lattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAzT4oBgHgl3EQfKvs6/content/2301.01101v1.pdf'}
+page_content=' We therefore expect that magnetic ani-  sotropy plays no role.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAzT4oBgHgl3EQfKvs6/content/2301.01101v1.pdf'}
+page_content='  The Kondo effect originating from degenerate triplet  ground state is extremely sensitive to magnetic anisotropy  which tends to lift the degeneracy [21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAzT4oBgHgl3EQfKvs6/content/2301.01101v1.pdf'}
+page_content=' In the present case,  KBTK ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAzT4oBgHgl3EQfKvs6/content/2301.01101v1.pdf'}
+page_content='9 meV is much smaller than the measured aniso-  tropy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAzT4oBgHgl3EQfKvs6/content/2301.01101v1.pdf'}
+page_content=' Three possible phenomena can explain the robustness  of the Kondo effect on the magnetic anisotropy occurring at  the Br-decorated elbow site : 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAzT4oBgHgl3EQfKvs6/content/2301.01101v1.pdf'}
+page_content=' The charge fluctuations due to  the coupling of the Fe states with the substrate states overw-  helm the magnetic anisotropy energy at this location, effecti-  vely restoring the degeneracy and screening the magnetic S=1  moment;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAzT4oBgHgl3EQfKvs6/content/2301.01101v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAzT4oBgHgl3EQfKvs6/content/2301.01101v1.pdf'}
+page_content=' A sizable charge transfer from the substrate or  the ligands onto the Fe atom induces a spin reduction from  S = 1 to S = 1/2, resulting in a spin 1/2 Kondo effect which  is naturally immune against magnetic anisotropy [41];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAzT4oBgHgl3EQfKvs6/content/2301.01101v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAzT4oBgHgl3EQfKvs6/content/2301.01101v1.pdf'}
+page_content=' The  deformation of the molecule at this position induces the sup-  pression of the magnetic anistropy energy [21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAzT4oBgHgl3EQfKvs6/content/2301.01101v1.pdf'}
+page_content=' This last ex-  planation is not in agreement with our tunneling topography  3  FIGURE 2: Scanning tunneling study of Fe−DPyDBrPP chains  made by Ullman’s coupling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAzT4oBgHgl3EQfKvs6/content/2301.01101v1.pdf'}
+page_content=' (a) topography images (I-IV) of a tri-  mer chain of molecules made by Ullman’s coupling which is ma-  nipulated by the tip of the microscope in order to position sequen-  tially the monomers above the Br-site.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAzT4oBgHgl3EQfKvs6/content/2301.01101v1.pdf'}
+page_content=' The intersection of the dotted  lines indicates the Br-site.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAzT4oBgHgl3EQfKvs6/content/2301.01101v1.pdf'}
+page_content=' The molecules are labeled as a function  of their spectroscopic signature measured in (b) : "SF" stands for  spin-flip and "K" for Kondo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAzT4oBgHgl3EQfKvs6/content/2301.01101v1.pdf'}
+page_content=' Experimental parameters of images (I-  IV) : size 15×15 nm2, V = 125 mV, I = 20 pA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAzT4oBgHgl3EQfKvs6/content/2301.01101v1.pdf'}
+page_content=' (b) related scanning  tunneling spectra taken above the left upper-pyrrole of each mole-  cule of the chain, showing the presence of the Kondo peak for the  molecules located above the Br-site which otherwise shows a spin-  flip signature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAzT4oBgHgl3EQfKvs6/content/2301.01101v1.pdf'}
+page_content=' Vstab = 30 mV, Istab = 200 pA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAzT4oBgHgl3EQfKvs6/content/2301.01101v1.pdf'}
+page_content=' Lock-in parameters :  Vm = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAzT4oBgHgl3EQfKvs6/content/2301.01101v1.pdf'}
+page_content='2 mV, f = 750 Hz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAzT4oBgHgl3EQfKvs6/content/2301.01101v1.pdf'}
+page_content=' The solid lines correspond to the fits  of the data (see Methods section).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAzT4oBgHgl3EQfKvs6/content/2301.01101v1.pdf'}
+page_content=' (c) topographic zoom on a mole-  cule of the chain exhibiting a spin-flip spectroscopic signature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAzT4oBgHgl3EQfKvs6/content/2301.01101v1.pdf'}
+page_content=' (d-  e) differential conductance image recorded simultaneously to image  (c) at −100 mV and 100 mV respectively showing the symmetry of  the frontier orbitals below and above the Fermi level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAzT4oBgHgl3EQfKvs6/content/2301.01101v1.pdf'}
+page_content=' The red dotted  line delimits the molecule.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAzT4oBgHgl3EQfKvs6/content/2301.01101v1.pdf'}
+page_content=' Experimental parameters of (c-e), size  3 × 3 nm2, Vstab = 800 mV, Istab = 500 pA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAzT4oBgHgl3EQfKvs6/content/2301.01101v1.pdf'}
+page_content=' Lock-in parameters :  Vm = 5 mV, f = 900 Hz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAzT4oBgHgl3EQfKvs6/content/2301.01101v1.pdf'}
+page_content='  studies which did not reveal a significant deformation of the  molecule whose shape is preserved on the Br-site.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAzT4oBgHgl3EQfKvs6/content/2301.01101v1.pdf'}
+page_content=' It has also  been shown that the anisotropic energy is not significantly af-  fected when a similar molecule is pressed onto the surface  with the tip of a microscope while staying away from the tip-  molecule contact [14, 15, 29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAzT4oBgHgl3EQfKvs6/content/2301.01101v1.pdf'}
+page_content=' As for scenarios 1 and 2, they  are both possible, but the DFT calculations and the theoreti-  cal interpretation that follow lean in the direction of the first  explanation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAzT4oBgHgl3EQfKvs6/content/2301.01101v1.pdf'}
+page_content='  Density Functional Theory analysis  DFT was used to study the electronic properties as a func-  tion of the distance of the molecule from the substrate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAzT4oBgHgl3EQfKvs6/content/2301.01101v1.pdf'}
+page_content=' In or-  der to rationalize the effect of the Br-adatom on the electronic  properties of Fe−DPyDBrPP, two sets of situations were  simulated : 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAzT4oBgHgl3EQfKvs6/content/2301.01101v1.pdf'}
+page_content=' as a function of the distance to the genuine  Au surface 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAzT4oBgHgl3EQfKvs6/content/2301.01101v1.pdf'}
+page_content=' as a function of the distance to a Br adatom  on Au slabs positioned below the Fe atom of the molecule  Fe−DPyDBrPP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAzT4oBgHgl3EQfKvs6/content/2301.01101v1.pdf'}
+page_content='  The Fe 3d states are dominant in the coupling with the sub-  strate through the Br states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAzT4oBgHgl3EQfKvs6/content/2301.01101v1.pdf'}
+page_content=' Fe dxz, dyz states are mainly hy-  bridized with the pyrroles molecular orbitals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAzT4oBgHgl3EQfKvs6/content/2301.01101v1.pdf'}
+page_content=' Therefore, fur-  ther analysis will be discussed on the basis of the Fe 3d hy-  brids with molecular orbital.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAzT4oBgHgl3EQfKvs6/content/2301.01101v1.pdf'}
+page_content=' In particular the Fe dxz, dyz and  dz2 hybrid states were proven to be at the origin of the obser-  ved phenomena.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAzT4oBgHgl3EQfKvs6/content/2301.01101v1.pdf'}
+page_content='  The DFT simulation shows that the magnetic polarization  (contrary to the total charge) of the Fe atom, figure 3 (b), de-  pends strongly on the molecule-surface distance when the sur-  face is decorated with a Br (or Au) adatom, whereas it is much  less sensitive when approaching the clean surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAzT4oBgHgl3EQfKvs6/content/2301.01101v1.pdf'}
+page_content=' When the  molecule is away from the Br site, the total occupancy and  magnetic moment are about 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAzT4oBgHgl3EQfKvs6/content/2301.01101v1.pdf'}
+page_content='4 e− and 2µB which corres-  ponds to a configuration close to the spin 1 state (high spin  state) and the hybridization of the Fe d states with the sub-  strate is small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAzT4oBgHgl3EQfKvs6/content/2301.01101v1.pdf'}
+page_content=' This is in good agreement with the measured  spectroscopic signature of a spin 1 quantum magnet when the  molecule is above fcc or hcp domains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAzT4oBgHgl3EQfKvs6/content/2301.01101v1.pdf'}
+page_content=' Indeed, the lack of a  Kondo signature may be the result of weak hopping integrals  from Fe orbitals to surface orbitals when the molecule is far  away.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAzT4oBgHgl3EQfKvs6/content/2301.01101v1.pdf'}
+page_content=' Hybridization with the substrate cannot compete with  the magnetic anisotropy and the Kondo effect is prevented.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAzT4oBgHgl3EQfKvs6/content/2301.01101v1.pdf'}
+page_content='  The magnetic anisotropy energy dominates the physics and  the experimental spectroscopic signature is that of a spin 1  subjected to an anisotropy of a few meV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAzT4oBgHgl3EQfKvs6/content/2301.01101v1.pdf'}
+page_content='  Moving the molecule of 1 Å towards the Br adatom, from  5 to 4 Å, leaves the occupancy of the Fe states roughly un-  changed (only a minor reshuffling of the charge distribution  between the orbitals is observed (figure 3 (c)) - but induces a  strong reduction of the magnetic polarization from about 2µB  to about 1µB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAzT4oBgHgl3EQfKvs6/content/2301.01101v1.pdf'}
+page_content=' The approximately constant charge suggests  that charge fluctuations are the cause of this reduction, which  is confirmed by the fact that the hybridization of the Fe and Br  states increases exponentially as the molecule is moved closer  (a) (l)  ()  SF  SF  SF  SF  SF  K  (I)  (IV)  SF  K  K  SF  SF  SF  (b)  (c)  (1)  (d)  (II)  (e)  (IV)4  FIGURE 3: Theory analysis by DFT calculations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAzT4oBgHgl3EQfKvs6/content/2301.01101v1.pdf'}
+page_content=' (a) Simulated adsorption geometry for the Iron(II) 5,15-(di-4-bromophenyl)-10,20-(di-4-  pyridyl) porphyrin molecule on a Au(111) surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAzT4oBgHgl3EQfKvs6/content/2301.01101v1.pdf'}
+page_content=' The (Au,Br) impurities are adsorbed on the hollow-hcp site.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAzT4oBgHgl3EQfKvs6/content/2301.01101v1.pdf'}
+page_content=' (b) polarizations as a function  of the distance from the substrate, computed for three different substrates, clean Au(111), Au(111) + a Au adatom and Au(111) + a Br adatom,  obtained via the same Mulliken analysis as in (c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAzT4oBgHgl3EQfKvs6/content/2301.01101v1.pdf'}
+page_content=' (c) Orbitally-resolved Fe(3d) charge as a function of the distance from the substrate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAzT4oBgHgl3EQfKvs6/content/2301.01101v1.pdf'}
+page_content=' The  charge and magnetization values have been obtained via a Mulliken analysis projecting the Kohn-Sham orbitals over the atomic states of the Fe  pseudopotential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAzT4oBgHgl3EQfKvs6/content/2301.01101v1.pdf'}
+page_content=' (d) Calculated non-zero squared hoppings between the Fe(3d) and the Br(4s,4p) states in the low-spin (Fe-Au(111) distance  = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAzT4oBgHgl3EQfKvs6/content/2301.01101v1.pdf'}
+page_content='0 Å, in blue) and high spin (Fe-Au(111) distance = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAzT4oBgHgl3EQfKvs6/content/2301.01101v1.pdf'}
+page_content='0 Å, in red) case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAzT4oBgHgl3EQfKvs6/content/2301.01101v1.pdf'}
+page_content=' Only the squared average of the spin-up and spin-down hopping is  reported, since no significant difference has been found in the two channels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAzT4oBgHgl3EQfKvs6/content/2301.01101v1.pdf'}
+page_content='  to the surface (the squared hoppings that enters in this form in  the hybridization function between the impurity and the sub-  strate are plotted in the figure 3 (d)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAzT4oBgHgl3EQfKvs6/content/2301.01101v1.pdf'}
+page_content=' The squared hopping in-  tegrals of the Fe states to the surface are indeed one order  of magnitude smaller when the molecule is far away.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAzT4oBgHgl3EQfKvs6/content/2301.01101v1.pdf'}
+page_content=' The Fe  3dz2 hybrids and Br 4s states, which have a almost perfect ro-  tational symmetry, are supposed to dominate the physics when  the molecule is closer (figure 3 (d)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAzT4oBgHgl3EQfKvs6/content/2301.01101v1.pdf'}
+page_content=' This configuration is favo-  rable to the emergence of a charge transport that is insensitive  to magnetic anistropy and independent of the relative orienta-  tion between the molecule and the surface once the iron atom  is located just above the adsorbed Br atom, in perfect agree-  ment with observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAzT4oBgHgl3EQfKvs6/content/2301.01101v1.pdf'}
+page_content='  The cases presented above are perfectly consistent with the  scenario in which the molecule evolves from a quasi-atomic  moment (spin 1) under the influence of magnetic anisotropy  to an Fe atom carrying a screened magnetic moment resul-  ting from a strong hybridization with the substrate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAzT4oBgHgl3EQfKvs6/content/2301.01101v1.pdf'}
+page_content=' Finally,  the DFT calculations clearly indicate that the strength of the  charge fluctuations should vary with the distance to the Br  atom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAzT4oBgHgl3EQfKvs6/content/2301.01101v1.pdf'}
+page_content=' Indeed the transition from spin 1 nanomagnet to Kondo  screening, through the mixed-valence regime has been expe-  rimentally probed in a reversible manner and summarized in  figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAzT4oBgHgl3EQfKvs6/content/2301.01101v1.pdf'}
+page_content='  Crossover from spin 1 nanomagnet to Kondo-screened ground  state through the mixed-valence regime  It was possible to finely tune the electronic configuration of  the iron atom by using a chain of molecules in order to sta-  bilize positions for which a molecule deviates slightly from  the Br site.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAzT4oBgHgl3EQfKvs6/content/2301.01101v1.pdf'}
+page_content=' The studied position of the molecules are presen-  ted in topography images (a-d) of figure 4 and the density of  states close to the Fermi energy are measured by STS spectra  figure 4 (g)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAzT4oBgHgl3EQfKvs6/content/2301.01101v1.pdf'}
+page_content=' The spin 1 signature is observed when the mole-  cule is far from the elbow site as already mentioned above, fi-  gure 4 (a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAzT4oBgHgl3EQfKvs6/content/2301.01101v1.pdf'}
+page_content=' When the molecule is positioned in such a way that  the Br site is located below the upper pyrrole ring, image 4  (b), the LDOS shows a broadened and asymmetric step like  shape originating from spin flip inelastic excitations but with  a lower characteristic energy, indicating a renormalization of  the anisotropy energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAzT4oBgHgl3EQfKvs6/content/2301.01101v1.pdf'}
+page_content=' At intermediate positions where the Br  adsorption-site is located in between the Fe site and the upper  pyrrole location, 4 (c), the spectroscopy shows a small anti-  resonance at the Fermi level with a Fano-like shape which  we believe is the extrapolation of the inelastic spin flip si-  gnature but for stronger charge fluctuations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAzT4oBgHgl3EQfKvs6/content/2301.01101v1.pdf'}
+page_content=' When the mo-  lecule is strictly coinciding with the Br-adsorption site, fi-  gure 4 (d), it is exhibiting a Kondo peak.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAzT4oBgHgl3EQfKvs6/content/2301.01101v1.pdf'}
+page_content=' Therefore, the si-  tuation of images 4 (b) & (c) corresponds to two interme-  diate mixed-valence regimes between spin 1 nanomagnet and  Kondo-screened ground states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAzT4oBgHgl3EQfKvs6/content/2301.01101v1.pdf'}
+page_content=' Indeed as shown by D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAzT4oBgHgl3EQfKvs6/content/2301.01101v1.pdf'}
+page_content=' Jacob  in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAzT4oBgHgl3EQfKvs6/content/2301.01101v1.pdf'}
+page_content=' [41] the gradual increase of charge fluctuations alters  the spectroscopic signatures of the spin 1 nanomagnet by re-  normalizing the magnetic excitations to lower energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAzT4oBgHgl3EQfKvs6/content/2301.01101v1.pdf'}
+page_content=' This  gradually closes the gap and finally restores a Kondo peak  as in the case of image 4 (d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAzT4oBgHgl3EQfKvs6/content/2301.01101v1.pdf'}
+page_content=' In this process, the absence of  particle-hole symmetry causes a typical Fano resonance to ap-  pear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAzT4oBgHgl3EQfKvs6/content/2301.01101v1.pdf'}
+page_content=' The renormalization of the magnetic excitations is evi-  denced in pannels 4 (h-k) where differential conductance is  used to map the inelastic spin-flip probability density in the  real space in the same configuration as in (b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAzT4oBgHgl3EQfKvs6/content/2301.01101v1.pdf'}
+page_content=' This situation  corresponds to the occurrence of one molecule in the inter-  mediate state and 2 molecules in the regime of the spin 1 na-  nomagnet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAzT4oBgHgl3EQfKvs6/content/2301.01101v1.pdf'}
+page_content=' At ±9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAzT4oBgHgl3EQfKvs6/content/2301.01101v1.pdf'}
+page_content='3 mV, spin-flip channels are open in all 3  molecules and the probability to induce a spin flip follows a  two-lobes spatial pattern in all three molecules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAzT4oBgHgl3EQfKvs6/content/2301.01101v1.pdf'}
+page_content=' At ±3 mV,  panels (i) & (j) the spin-flip excitations are hindered by the  magnetic anisotropy in the molecules far from the Br-site.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAzT4oBgHgl3EQfKvs6/content/2301.01101v1.pdf'}
+page_content=' On  the contrary, the molecule in the intermediate regime shows  spin-flip processes already being developed at ±3 mV, indica-  ting the renormalization of the magnetic excitation energy to  (a)5  dI/dV (arb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAzT4oBgHgl3EQfKvs6/content/2301.01101v1.pdf'}
+page_content=' units)  Bias Voltage (mV)  Upper-pyrrole  (e)  Bias Voltage (mV)  (g)  dI/dV (arb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAzT4oBgHgl3EQfKvs6/content/2301.01101v1.pdf'}
+page_content=' units)  (f)  0  10  20  10  20  (h)  9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAzT4oBgHgl3EQfKvs6/content/2301.01101v1.pdf'}
+page_content='3mV  (i)  3mV  (j)  3mV  (k)  9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAzT4oBgHgl3EQfKvs6/content/2301.01101v1.pdf'}
+page_content='3mV  (a)  (b)  (c)  (d)  500  250  500  250  0  dI/dV (arb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAzT4oBgHgl3EQfKvs6/content/2301.01101v1.pdf'}
+page_content=' units)  Bias Voltage (mV)  Iron center  500  250  500  250  0  FIGURE 4: Crossover from spin 1 nanomagnet to Kondo screen ground state through the mixed-valence regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAzT4oBgHgl3EQfKvs6/content/2301.01101v1.pdf'}
+page_content=' (a-d) topography images  showing the position of the first molecule of a chain with respect to the Br-decorated elbow site of the gold reconstruction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAzT4oBgHgl3EQfKvs6/content/2301.01101v1.pdf'}
+page_content=' (a) the molecule  is far from the Br site.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAzT4oBgHgl3EQfKvs6/content/2301.01101v1.pdf'}
+page_content=' (b) the upper pyrrole is centered on the Br-site.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAzT4oBgHgl3EQfKvs6/content/2301.01101v1.pdf'}
+page_content=' (c) the Br-site is located in between the center of the upper pyrole and  the Fe ion position.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAzT4oBgHgl3EQfKvs6/content/2301.01101v1.pdf'}
+page_content=' (d) the Fe atom is located just above the Br-site.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAzT4oBgHgl3EQfKvs6/content/2301.01101v1.pdf'}
+page_content=' (e) conductance spectra taken above the center of the upper pyrole.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAzT4oBgHgl3EQfKvs6/content/2301.01101v1.pdf'}
+page_content=' (f and  g) conductance spectra taken above the iron atom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAzT4oBgHgl3EQfKvs6/content/2301.01101v1.pdf'}
+page_content=' red, green, blue and yellow colors correspond to the configurations of (a), (b), (c) and (d)  respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAzT4oBgHgl3EQfKvs6/content/2301.01101v1.pdf'}
+page_content=' (h-k) Background subtracted (see Methods section) differential conductance images at V = −9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAzT4oBgHgl3EQfKvs6/content/2301.01101v1.pdf'}
+page_content='3 mV, V = −3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAzT4oBgHgl3EQfKvs6/content/2301.01101v1.pdf'}
+page_content='0 mV,V = 3 mV  and V = 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAzT4oBgHgl3EQfKvs6/content/2301.01101v1.pdf'}
+page_content='3 mV respectively recorded in the configuration of image (b) where, the left molecule is in the mixed-valence intermediate state  (blue dashed line) and the two other molecules behave as spin 1 nanomagnets (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAzT4oBgHgl3EQfKvs6/content/2301.01101v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAzT4oBgHgl3EQfKvs6/content/2301.01101v1.pdf'}
+page_content=' red dashed line).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAzT4oBgHgl3EQfKvs6/content/2301.01101v1.pdf'}
+page_content=' Experimental parameters : (a) & (d)  V = 125 mV, I = 20 pA, scalebar = 2 nm;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAzT4oBgHgl3EQfKvs6/content/2301.01101v1.pdf'}
+page_content=' (b) & (c) V = 200 mV, I = 100 pA, scalebar = 2 nm;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAzT4oBgHgl3EQfKvs6/content/2301.01101v1.pdf'}
+page_content=' (e) & (f) Vstab = 800 mV, Istab = 500 pA,  lock-in parameters : Vm = 5 mV, f = 900 Hz;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAzT4oBgHgl3EQfKvs6/content/2301.01101v1.pdf'}
+page_content=' (g) Vstab = 30 mV, Istab = 200 pA, lock-in parameters : Vm = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAzT4oBgHgl3EQfKvs6/content/2301.01101v1.pdf'}
+page_content='2 mV, f = 750 Hz;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAzT4oBgHgl3EQfKvs6/content/2301.01101v1.pdf'}
+page_content=' (h-k)  Vstab = 30 mV, Istab = 200 pA, size 12×12 nm2, lock-in parameters : Vm = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAzT4oBgHgl3EQfKvs6/content/2301.01101v1.pdf'}
+page_content='2 mV, f = 750 Hz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAzT4oBgHgl3EQfKvs6/content/2301.01101v1.pdf'}
+page_content=' All spectra are normalized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAzT4oBgHgl3EQfKvs6/content/2301.01101v1.pdf'}
+page_content='  a smaller scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAzT4oBgHgl3EQfKvs6/content/2301.01101v1.pdf'}
+page_content='  This transition through a mixed-valence regime might be  due to a net increase of Fe charge while remaining still in a  regime of weak coupling with the substrate (our previously  outlined scenario 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAzT4oBgHgl3EQfKvs6/content/2301.01101v1.pdf'}
+page_content=' Our DFT calculations rather suggest that  the predominant effect is the drastic increase of the hopping  from/to the substrate (scenario 1), instead.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAzT4oBgHgl3EQfKvs6/content/2301.01101v1.pdf'}
+page_content=' Accordingly, the  spin crossover was found to be accompanied with an energy  shift of the frontier orbitals which are localized at the up-  per pyrrole rings (frontier orbital A, FOA) and at the iron  atom position (frontier orbital B, FOB).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAzT4oBgHgl3EQfKvs6/content/2301.01101v1.pdf'}
+page_content=' A carefully analysis  of figures 4 (e) & (f) allows to follow the evolution of FOA  and FOB energies when moving the molecule.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAzT4oBgHgl3EQfKvs6/content/2301.01101v1.pdf'}
+page_content=' FOA shows a  continuous evolution of density of states from the unoccupied  states towards occupied states when approaching the molecule  to the Br-site, indicating an increase of the orbital occupation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAzT4oBgHgl3EQfKvs6/content/2301.01101v1.pdf'}
+page_content='  FOB shows a reverse behavior with a shift from below the  Fermi level towards the unoccupied states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAzT4oBgHgl3EQfKvs6/content/2301.01101v1.pdf'}
+page_content=' The gap between  the frontier orbitals is reducing when approaching the Br site  so the d level splitting tends to be weakened and could be  favoring a decrease of orbital momentum and a renormaliza-  tion of the magnetic anisotropy in agreement with the closing  of the steps of conductance due to spin-flip excitations in the  measured data of panel 4 (g) and figures 4 (h-k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAzT4oBgHgl3EQfKvs6/content/2301.01101v1.pdf'}
+page_content='  The conductance images presented figure 2 (c-e) support  the hybrid Fe 3dxz, dyz character of FOA and the strong Fe  3dz2 character of FOB in agreement with the orbital occupa-  tions in the DFT calculations (figure 3 (c)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAzT4oBgHgl3EQfKvs6/content/2301.01101v1.pdf'}
+page_content=' It is interesting  to note that according to conductance images of figure 4 (h-  k), the spin-flip probability density is following the 3dxz, dyz  symmetry at energies above and below the Fermi energy (two-  lobes shape).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAzT4oBgHgl3EQfKvs6/content/2301.01101v1.pdf'}
+page_content=' On the contrary, the 3dz2 character of the exci-  tations (bright localized spot at the Fe site) is only visible at  negative bias, so for occupied states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAzT4oBgHgl3EQfKvs6/content/2301.01101v1.pdf'}
+page_content=' This difference is explai-  ning the breaking of the electron-hole symmetry observed in  inelastic spectra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAzT4oBgHgl3EQfKvs6/content/2301.01101v1.pdf'}
+page_content='  The fano-like shape of the blue spectrum in figure 4 (g) oc-  curs when the frontier orbitals are overlapping at the Fermi  level, as visible in figure 4 (e) & (f) and when the charge fluc-  tuations between orbitals are expected to be strong.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAzT4oBgHgl3EQfKvs6/content/2301.01101v1.pdf'}
+page_content=' The hypo-  thesis of intermediate regimes driven by charge fluctuations is  well described by prediction in ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAzT4oBgHgl3EQfKvs6/content/2301.01101v1.pdf'}
+page_content=' [41] as well as the shape  of the spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAzT4oBgHgl3EQfKvs6/content/2301.01101v1.pdf'}
+page_content=' However in the present case the frontier or-  bitals, could also recover their degeneracy when overlapping  in the intermediate regimes, implying a possible Kondo SU(4)  effect [35, 42], although this is less likely in a molecular sys-  tem [43] and cannot be easily distinguished from the simple  effect of strong charge fluctuations [44].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAzT4oBgHgl3EQfKvs6/content/2301.01101v1.pdf'}
+page_content=' Finally, when the iron  atom is located above the Br site, FOA migrates below the  Fermi level and FOB is washed out towards incoherent exci-  tation and only the Kondo resonance survives at EF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAzT4oBgHgl3EQfKvs6/content/2301.01101v1.pdf'}
+page_content='  CONCLUSION  We have revealed in experiments and DFT calculations that  the spin state switching is caused by the strong hybridiza-  tion potential at Br-decorated elbow sites which induce charge  6  fluctuation and a renormalization of the magnetic anisotropy  when molecules are adsorbed on these sites.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAzT4oBgHgl3EQfKvs6/content/2301.01101v1.pdf'}
+page_content='  The direct consequence of charge fluctuations is to tune the  energies of two molecular frontier orbitals which are respec-  tively closest to the Fermi level in positive and negative ener-  gies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAzT4oBgHgl3EQfKvs6/content/2301.01101v1.pdf'}
+page_content=' The energetic overlapping of these two frontier orbitals  gives rise to low-energy spin excitation channels near Fermi  level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAzT4oBgHgl3EQfKvs6/content/2301.01101v1.pdf'}
+page_content=' Under the condition of relative small charge fluctua-  tions, the two frontier orbitals are far from Fermi level and  the overlapping is small, hence the molecule will have high  spin (S = 1) state, while under strong charge fluctuation these  two orbitals can cross Fermi level, resulting a renormalization  of the magnetic anisotropy and ultimately the Kondo screened  ground state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAzT4oBgHgl3EQfKvs6/content/2301.01101v1.pdf'}
+page_content='  Our work provides a new approach for tuning not only the  spin state but also the intermediate valence character/charge  fluctuation in hybrids made of molecules on reconstructed sur-  faces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAzT4oBgHgl3EQfKvs6/content/2301.01101v1.pdf'}
+page_content='  METHODS  Fe 5,15-di-4-pyridyl-10,20-di-4-bromophenyl porphyrin  molecules were ordered from Alpha Aesar company.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAzT4oBgHgl3EQfKvs6/content/2301.01101v1.pdf'}
+page_content=' Samples  were prepared under ultrahigh vacuum (base pressure 1 ×  10−10 mbar) just before their scanning tunneling studies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAzT4oBgHgl3EQfKvs6/content/2301.01101v1.pdf'}
+page_content=' The  gold substrate was prepared by repeated cycles of Ar ion sput-  tering followed by annealing at 900 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAzT4oBgHgl3EQfKvs6/content/2301.01101v1.pdf'}
+page_content=' The molecules were  deposited by organic molecular beam epitaxy (OMBE) when  the sample was kept at Room Temperature (Tcrucible=575K).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAzT4oBgHgl3EQfKvs6/content/2301.01101v1.pdf'}
+page_content='  It was observed that the molecule source is also depositing Br  atoms on the surface of gold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAzT4oBgHgl3EQfKvs6/content/2301.01101v1.pdf'}
+page_content=' We have attributed this conco-  mitant deposition of Br with Fe−DPyDBrPP to the stochas-  tic Br detaching from Fe−DPyDBrPP molecules inside the  crucible when heated at evaporation temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAzT4oBgHgl3EQfKvs6/content/2301.01101v1.pdf'}
+page_content=' In-crucible  Br detaching form similar molecules was already reported in  ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAzT4oBgHgl3EQfKvs6/content/2301.01101v1.pdf'}
+page_content=' [30] at 590K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAzT4oBgHgl3EQfKvs6/content/2301.01101v1.pdf'}
+page_content=' In our case, the low amount of deposited  Br atoms was seen to be enough for occupying all the elbow  sites of the herringbone reconstruction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAzT4oBgHgl3EQfKvs6/content/2301.01101v1.pdf'}
+page_content=' Once the molecules  deposited, the sample temperature was quenched at liquid ni-  trogen temperature then at helium liquid temperature in less  than 20 minutes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAzT4oBgHgl3EQfKvs6/content/2301.01101v1.pdf'}
+page_content=' For low coverage this procedure produces  single monomers scattered on the surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAzT4oBgHgl3EQfKvs6/content/2301.01101v1.pdf'}
+page_content=' A post annealing  at 400 K for 10 minutes was used for producing molecular  chains by Ullman’s coupling at the surface following recipes  in ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAzT4oBgHgl3EQfKvs6/content/2301.01101v1.pdf'}
+page_content=' [30, 45–47].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAzT4oBgHgl3EQfKvs6/content/2301.01101v1.pdf'}
+page_content='  STM/STS measurements were achieved in-situ in ultrahigh  vacuum by means of a modified "Tyto SPM" from SPECSTM  at a base temperature of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAzT4oBgHgl3EQfKvs6/content/2301.01101v1.pdf'}
+page_content='3 K;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAzT4oBgHgl3EQfKvs6/content/2301.01101v1.pdf'}
+page_content=' the tunneling bias was applied  to the sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAzT4oBgHgl3EQfKvs6/content/2301.01101v1.pdf'}
+page_content=' Mechanically cut PtIr tips were used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAzT4oBgHgl3EQfKvs6/content/2301.01101v1.pdf'}
+page_content=' Soft la-  teral manipulation as follows was used for moving the mole-  cules 1) positioning the tip above the molecule to move 2) Tur-  ning off the microscope feedback loop and approaching the  tip towards molecule in order to increase tip-molecule inter-  action 2) Moving the tip to the destination in constant height  mode 3) Retracting the tip and engaging the feedback loop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAzT4oBgHgl3EQfKvs6/content/2301.01101v1.pdf'}
+page_content='  The analyses of the microscopy and spectroscopy data were  done with WSxM software [48] and homemade python proce-  dures respectively [49].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAzT4oBgHgl3EQfKvs6/content/2301.01101v1.pdf'}
+page_content='  The fit of spectra from figure 1 & 2 involved a large Gaus-  sian background mimicking the frontier orbital DOS plus a  specific function associated to spin-flips (step functions intro-  duced in [33]) or Kondo ground states (Frota function).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAzT4oBgHgl3EQfKvs6/content/2301.01101v1.pdf'}
+page_content=' Mar-  ginally, a convolution with a Gaussian function was used to  clean up the incoherent noise of all conductance spectra pre-  sented here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAzT4oBgHgl3EQfKvs6/content/2301.01101v1.pdf'}
+page_content='  The spin-polarized Density-Functional Theory (DFT) cal-  culations have been carried out with v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAzT4oBgHgl3EQfKvs6/content/2301.01101v1.pdf'}
+page_content='∼6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAzT4oBgHgl3EQfKvs6/content/2301.01101v1.pdf'}
+page_content='8 of the QUAN-  TUM ESPRESSO package [50, 51] within the Projector  Augmented-Wave (PAW) scheme [52] and using the Perdew-  Burke-Ernzerhof (PBE) functional [53].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAzT4oBgHgl3EQfKvs6/content/2301.01101v1.pdf'}
+page_content=' PAW pseudopoten-  tials have been taken from the PSlibrary [54–56].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAzT4oBgHgl3EQfKvs6/content/2301.01101v1.pdf'}
+page_content='  The Au(111) surface has been modeled by a finite slab  consisting of 3 gold layers with 15 Åof vacuum between  periodic replicas along the z direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAzT4oBgHgl3EQfKvs6/content/2301.01101v1.pdf'}
+page_content=' The Au-Au nearest-  neighbor distance has been set to 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAzT4oBgHgl3EQfKvs6/content/2301.01101v1.pdf'}
+page_content='93 Å, the PBE equili-  brium distance of the corresponding fcc bulk structure, which  is ≲ 2% larger than the experimental room-temperature va-  lue [57].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAzT4oBgHgl3EQfKvs6/content/2301.01101v1.pdf'}
+page_content='  Given the sizeable extension of the molecule, a 10×10 su-  percell of Au atoms on the xy plane has been used in order  to avoid interaction between its periodic replicas, for a total  of 300 Au atoms in the simulation cell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAzT4oBgHgl3EQfKvs6/content/2301.01101v1.pdf'}
+page_content=' The molecule geo-  metry has been relaxed in the vacuum until all components  of the forces were smaller than 10−3 Ry/aB, with aB being  the Bohr radius.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAzT4oBgHgl3EQfKvs6/content/2301.01101v1.pdf'}
+page_content=' The projected density of states (pDoS) and  the Mulliken orbital occupations and polarizations have been  evaluated by projecting the Kohn-Sham orbitals on the atomic  states of the pseudopotentials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAzT4oBgHgl3EQfKvs6/content/2301.01101v1.pdf'}
+page_content=' They have been computed by  considering different adsorption geometries between the mo-  lecule and three different substrates : the clean Au(111) sur-  face, the Au(111) surface with a Au impurity and the Au(111)  surface with a Br impurity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAzT4oBgHgl3EQfKvs6/content/2301.01101v1.pdf'}
+page_content=' The impurities have been placed  at the preferred PBE adsorption site at the equilibrium dis-  tance, which has resulted in the hollow-hcp site for both the  Au and the Br impurity, respectively at 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAzT4oBgHgl3EQfKvs6/content/2301.01101v1.pdf'}
+page_content='16 Åand 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAzT4oBgHgl3EQfKvs6/content/2301.01101v1.pdf'}
+page_content='23 Åfrom  the Au(111) surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAzT4oBgHgl3EQfKvs6/content/2301.01101v1.pdf'}
+page_content=' All calculations have been performed  with and a plane-wave cutoff of ϵcut = 40 Ry on the wave-  functions and 320 Ry on the density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAzT4oBgHgl3EQfKvs6/content/2301.01101v1.pdf'}
+page_content=' Brillouin zone integrals  have been evaluated at the Γ point using a gaussian smearing  of σ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAzT4oBgHgl3EQfKvs6/content/2301.01101v1.pdf'}
+page_content='01 Ry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAzT4oBgHgl3EQfKvs6/content/2301.01101v1.pdf'}
+page_content=' The hopping parameters have been computed  via the projection method as implemented in the WANNIER90  package [58].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAzT4oBgHgl3EQfKvs6/content/2301.01101v1.pdf'}
+page_content='  ACKNOWLEDGEMENT  Cheryl Feuillet-Palma is acknowledged for her construc-  tive discussions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAzT4oBgHgl3EQfKvs6/content/2301.01101v1.pdf'}
+page_content=' TG thanks Paolo Restuccia for fruitful dis-  cussions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAzT4oBgHgl3EQfKvs6/content/2301.01101v1.pdf'}
+page_content=' LdM acknowledges Giorgio Sangiovanni for useful  discussions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAzT4oBgHgl3EQfKvs6/content/2301.01101v1.pdf'}
+page_content=' SV, DR and SP acknowledge the French National  Research Agency for the support of the SHOGUN project, Re.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAzT4oBgHgl3EQfKvs6/content/2301.01101v1.pdf'}
+page_content='  7  ANR-22-CE09-0028-01.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAzT4oBgHgl3EQfKvs6/content/2301.01101v1.pdf'}
+page_content=' TG and LdM are supported by the  European Commission through the ERC-CoG2016, Strong-  CoPhy4Energy, GA No724177.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAzT4oBgHgl3EQfKvs6/content/2301.01101v1.pdf'}
+page_content='  ∗ Electronic address: stephane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAzT4oBgHgl3EQfKvs6/content/2301.01101v1.pdf'}
+page_content='pons@espci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAzT4oBgHgl3EQfKvs6/content/2301.01101v1.pdf'}
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+page_content='  Valentyuk,  M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAzT4oBgHgl3EQfKvs6/content/2301.01101v1.pdf'}
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+page_content=' Pons, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAzT4oBgHgl3EQfKvs6/content/2301.01101v1.pdf'}
+page_content=' Fabris, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAzT4oBgHgl3EQfKvs6/content/2301.01101v1.pdf'}
+page_content=' Baroni, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAzT4oBgHgl3EQfKvs6/content/2301.01101v1.pdf'}
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+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAzT4oBgHgl3EQfKvs6/content/2301.01101v1.pdf'}
+page_content=' Chem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAzT4oBgHgl3EQfKvs6/content/2301.01101v1.pdf'}
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+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAzT4oBgHgl3EQfKvs6/content/2301.01101v1.pdf'}
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+page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAzT4oBgHgl3EQfKvs6/content/2301.01101v1.pdf'}
+page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAzT4oBgHgl3EQfKvs6/content/2301.01101v1.pdf'}
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+page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAzT4oBgHgl3EQfKvs6/content/2301.01101v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAzT4oBgHgl3EQfKvs6/content/2301.01101v1.pdf'}
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+page_content=' Kim, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAzT4oBgHgl3EQfKvs6/content/2301.01101v1.pdf'}
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+page_content='086602.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAzT4oBgHgl3EQfKvs6/content/2301.01101v1.pdf'}
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+page_content='aps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAzT4oBgHgl3EQfKvs6/content/2301.01101v1.pdf'}
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+page_content='aps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAzT4oBgHgl3EQfKvs6/content/2301.01101v1.pdf'}
+page_content='org/doi/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAzT4oBgHgl3EQfKvs6/content/2301.01101v1.pdf'}
+page_content='1103/PhysRev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAzT4oBgHgl3EQfKvs6/content/2301.01101v1.pdf'}
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+page_content='A515.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAzT4oBgHgl3EQfKvs6/content/2301.01101v1.pdf'}
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+page_content='aps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAzT4oBgHgl3EQfKvs6/content/2301.01101v1.pdf'}
+page_content='org/doi/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAzT4oBgHgl3EQfKvs6/content/2301.01101v1.pdf'}
+page_content='1103/PhysRev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAzT4oBgHgl3EQfKvs6/content/2301.01101v1.pdf'}
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diff --git a/ZtE1T4oBgHgl3EQfwgWm/content/tmp_files/2301.03412v1.pdf.txt b/ZtE1T4oBgHgl3EQfwgWm/content/tmp_files/2301.03412v1.pdf.txt
new file mode 100644
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@@ -0,0 +1,2119 @@
+Neighbor Auto-Grouping Graph Neural Networks for Handover Parameter
+Configuration in Cellular Network
+Mehrtash Mehrabi,1,2 Walid Masoudimansour,2 Yingxue Zhang,2 Jie Chuai,2 Zhitang Chen,2
+Mark Coates,3 Jianye Hao,1, 4 Yanhui Geng2
+1 Huawei Noah’s Ark Lab, 2 University of Alberta, 3 McGill University, 4 Tianjin University
+{mehrtash.mehrabi, walid.masoudimansour, yingxue.zhang, chuaijie, chenzhitang2,
+haojianye, geng.yanhui}@huawei.com, mark.coates@mcgill.ca
+Abstract
+The mobile communication enabled by cellular networks is
+the one of the main foundations of our modern society. Op-
+timizing the performance of cellular networks and providing
+massive connectivity with improved coverage and user ex-
+perience has a considerable social and economic impact on
+our daily life. This performance relies heavily on the config-
+uration of the network parameters. However, with the mas-
+sive increase in both the size and complexity of cellular net-
+works, network management, especially parameter configu-
+ration, is becoming complicated. The current practice, which
+relies largely on experts’ prior knowledge, is not adequate
+and will require lots of domain experts and high mainte-
+nance costs. In this work, we propose a learning-based frame-
+work for handover parameter configuration. The key chal-
+lenge, in this case, is to tackle the complicated dependencies
+between neighboring cells and jointly optimize the whole net-
+work. Our framework addresses this challenge in two ways.
+First, we introduce a novel approach to imitate how the net-
+work responds to different network states and parameter val-
+ues, called auto-grouping graph convolutional network (AG-
+GCN). During the parameter configuration stage, instead of
+solving the global optimization problem, we design a local
+multi-objective optimization strategy where each cell con-
+siders several local performance metrics to balance its own
+performance and its neighbors. We evaluate our proposed al-
+gorithm via a simulator constructed using real network data.
+We demonstrate that the handover parameters our model can
+find, achieve better average network throughput compared to
+those recommended by experts as well as alternative base-
+lines, which can bring better network quality and stability. It
+has the potential to massively reduce costs arising from hu-
+man expert intervention and maintenance.
+1
+Introduction
+The rapid growth in the number of devices that need real
+time, high quality connection to the internet (e.g., internet of
+things (IoT) devices, health monitoring equipment, devices
+used for online education and remote working, autonomous
+vehicles, etc.) makes it essential to improve cellular network
+performance. Unsatisfactory user experience and network
+interruption have negative impacts in our modern society.
+Thus, improving the cellular network has both economic and
+Copyright © 2023, Association for the Advancement of Artificial
+Intelligence (www.aaai.org). All rights reserved.
+social impact towards achieving United Nations Sustainable
+Development Goals (UNSDGs) (Weisenborn 2018; World
+Economic Forum 2020). Moreover, it can highly contribute
+to enhancing infrastructure, promoting sustainable indus-
+trialization, fostering innovation, responsible consumption,
+enabling sustainable cities and communities, and promot-
+ing decent work and economic growth (Gohar and Nencioni
+2021; Rao and Prasad 2018; Siriwardhana et al. 2021).
+The performance of a cellular network relies heavily on
+its parameter configurations and it is becoming more cru-
+cial, as the number of mobile users continues to grow rapidly
+(statista 2022). These parameters govern access control,
+handover, and resource management (Dahlman et al. 2013;
+Bhat et al. 2012). One of the factors that has a significant
+impact on the quality of service (QoS) in such networks is
+the handover parameter (Tekinay et al. 1991). We provide
+more details concerning this parameter and its effects in the
+supplementary materials, Sec. A.1.
+Optimizing handover parameters is one of the most com-
+mon approaches to guarantee minimum service delay or in-
+terruption and improve coverage and throughput (Mu et al.
+2014). However, with the massive increase in both the size
+and complexity of cellular networks, parameter configura-
+tion is becoming complicated. The current practice, which
+relies largely on experts’ prior knowledge, is inadequate, re-
+quiring many domain experts and leading to high mainte-
+nance costs.
+One of the key challenges in the network parameter op-
+timization problem is the complex spatial and temporal de-
+pendencies in the cellular network. Any employed algorithm
+should be capable of tracking the non-stationary changes in
+the environment, i.e., the fluctuations of user number, net-
+work load, etc. (Agiwal et al. 2016). Also, due to the diverse
+characteristics of cells across the network, the best parame-
+ter configuration for one cell may not be optimal for another
+and parameter configuration of one cell not only affects its
+own performance, but also affects its neighbors’ (Dahlman
+et al. 2013). Therefore, there are strong interactions between
+neighboring cells which become extremely complicated in
+heterogeneous network. Consequently, developing an algo-
+rithm that can adapt to the temporal dynamics and cell diver-
+sity in real networks is essential for parameter configuration
+(Jiang et al. 2016).
+The current cellular network deployments are highly de-
+arXiv:2301.03412v1  [cs.NI]  29 Dec 2022
+
+pendent on human designed rules or analytical models based
+on domain knowledge and assumptions about the network
+dynamics which is far from optimal. They only consider
+a limited number of network states (e.g., user distribution,
+channel quality, etc.) and parameters, and cannot capture the
+complex relationships between network states, parameter
+configurations and network performance. Also, the assump-
+tions of the network dynamics, based on which the rules/-
+models are developed, are often simplified without consider-
+ing the non-stationary changes in real environments, which
+degrades their performance. Finally, these rules/models may
+not be able to deal with the cell diversities in the network
+which makes them sub-optimal (Imran et al. 2014).
+Recently, data-driven approaches based on machine learn-
+ing (ML) have been extensively used for parameter con-
+figuration and network management in cellular networks
+(Yu et al. 2017; Tabrizi et al. 2012; Riihijarvi et al. 2018;
+Chuai et al. 2019; Ye et al. 2013). It has been shown that
+the multi-layer perceptron (MLP) can be considered as a
+universal function approximator (Goodfellow et al. 2016).
+Thus, in environments such as cellular networks where there
+is lack of an accurate analytical model and the network is
+highly dynamic, neural-network-based methods can be used
+to achieve high-accuracy prediction. ML models can uti-
+lize high dimensional information and approximate complex
+functions to fully describe the relationship between network
+states, parameter configurations and network performance
+metrics, which cannot be achieved by human experience.
+In order to address the above challenges, we investigate
+two important questions: 1) Modeling: how to model the
+spatial and temporal dependencies of the cellular network?
+2) Decision-making: how to choose the parameter values to
+jointly optimize the overall performance of interconnected
+and interacting cells? We first, propose a ML-based model to
+precisely imitate the cellular network environment and then,
+use it to configure the parameters.
+We demonstrate that the handover parameters recom-
+mended by our model can achieve better average network
+throughput compared to the existing methods and our ap-
+proach can massively reduce costs from human expert in-
+tervention and maintenance. It opens up the potential for
+high-quality internet access to geographical areas that are
+currently under-served by the cellular network. Besides, this
+framework can bring new possibilities for important appli-
+cations to under-developed regions including online educa-
+tion, health monitoring devices by improving their real time
+connection (Attaran 2021).
+Our main contributions are summarized as follows.
+• We propose a novel method to model the impact from the
+neighbors of each cell in a distinguishable way to capture
+the complex spatial dependencies of the network.
+• We consider the changing dynamics of the network in our
+reward model to better reflect the temporal dependencies.
+• We introduce a multi-objective optimization strategy
+based on the model to consider several performance met-
+rics and improve the overall network throughput, which
+has the potential for high social impact applications.
+2
+Background and Related Work
+The adjustment of handover parameters helps to balance the
+traffic load in the network and it can dramatically affect the
+network throughput. During the handover process in cellular
+networks, in order to guarantee an acceptable service qual-
+ity, a user equipment (UE) must monitor the reference signal
+received power (RSRP) of the serving cell (3GPP TS36.331
+2016). As soon as the RSRP drops below a pre-defined
+threshold (called A2-threshold), the UE starts to report mea-
+surements to its serving cell and prepares for handover. In-
+creasing the value of A2-threshold decreases the number of
+UEs in the serving cell in which the handover is triggered,
+and this spreads the serving cell’s load to its neighbors, re-
+sulting in a significant change of throughput for the serving
+cell and its neighbors. While improving the load balance of
+the network, this can have adverse effects on the network
+performance since it forces frequent handovers which re-
+quires a considerable amount of bandwidth for measurement
+reporting and causes a drop in network throughput. Decreas-
+ing the value of A2-threshold, on the other hand, may cause
+a poor experience for edge UEs and lead to repeated connec-
+tion loss due to weak signal. In attempt to solve the problem
+of optimization of the parameters of a wireless network, dif-
+ferent techniques such as fuzzy systems, deep reinforcement
+learning (DRL), and contextual bandit have been used in the
+literature. (see Sec. A.2 for some details).
+The use of graph convolutoinal networks (GCNs) (Hamil-
+ton et al. 2017; Kipf et al. 2017; Fan et al. 2019) has also
+yielded significantly well-designed models to predict the
+network traffic and optimize the corresponding parameters.
+For example, in (Zhang et al. 2020), the authors introduce a
+novel handover strategy based on GCNs. The handover pro-
+cess is modeled as a directed graph by which the user tries to
+predict its future signal strength. Other works such as (Zhao
+et al. 2020) introduce novel methods of network traffic pre-
+diction combined with a greedy search or action configu-
+ration method to optimize handover parameters. However,
+these works fail to consider the heterogeneous aspect of the
+cellular networks.
+Despite being effective, none of the above-mentioned
+methods uses the capacity of the neighbors’ information to
+fully tailor the model to adapt to the spatial characteristics
+of a cellular network, where the interaction is complex and
+the network is heterogeneous. Also, despite the fact that
+these techniques consider some important measures of op-
+timization, none of them approaches the problem at hand
+by considering two of the most important measures simul-
+taneously (especially from the users’ perspective): load bal-
+ancing and throughput. In this article, we propose an effec-
+tive and efficient framework that models the network as a
+heterogeneous graph where we learn an implicit interaction
+type for each neighboring cell. Then, it incorporates the im-
+pact of neighboring cells from each interaction group in a
+unique way. Moreover, in contrast to the available methods
+in the literature, we exploit two important measures in the
+network simultaneously, to configure the parameters effec-
+tively: throughput and load balancing, which are directly re-
+lated to the user experience in the network.
+
+3
+Problem Formulation
+Let us consider a network with N cells, and form N clusters
+each composed of one of the network cells as its center cell
+along with its neighboring cells. As an example, we choose
+the optimization of the A2-threshold to investigate the per-
+formance of our algorithm. According to the 3GPP standard
+(3GPP TS36.331 2016), an A2 event is triggered when the
+received power at user u from cell n, P u,n, satisfies
+P u,n + Hys < Thresh,
+(1)
+where Hys is the hysteresis parameter to avoid frequent han-
+dovers and Thresh is the A2-threshold we are optimizing.
+We consider an online optimization process. In real prac-
+tice, network operators are often conservative and only allow
+a limited number of experiments. During the optimization
+period of L days, and the A2-threshold can be adjusted once
+for each cell at the beginning of each day. For day t, let Dt
+be the total bits transmitted by all the cells, and Tt be the to-
+tal transmission time. We would like to maximize the accu-
+mulated network throughput of the optimization period, i.e.,
+max �L
+t=1
+Dt
+Tt . Maximizing the overall network throughput
+by jointly optimizing the A2-threshold of all cells is diffi-
+cult. The problem becomes even more complicated as the
+network size increases, which makes a centralized solution
+not scalable. The adjustment of the A2-threshold of one cell
+only affects its local neighborhood and thus, we convert the
+centralized problem into a local decision problem. That is,
+each cell only examines its local performance metrics and
+chooses its own parameter configuration value.
+The adjustment of the A2-threshold affects the network
+throughput via two means: better resource utilization by load
+balancing, and improved cell throughput with less connec-
+tion loss and measurement reporting. Consequently, in order
+to configure it, these two metrics must be considered in the
+local decision problem. The throughput of cell i on day t is
+highly dependent on its A2-threshold, formulated as ai
+t, de-
+noted as αi
+t(ai
+t). The load balancing factor in the i-th cluster
+with center cell i on day t with ai
+t is defined as the ratio
+of the center cell throughput to the average throughput of
+its neighboring cells, denoted by βi
+t(ai
+t) and formulated as
+βi
+t(ai
+t) = αi
+t(ai
+t)/¯αi
+t, where ¯αi
+t is the average throughput
+of the neighbors of cell i with action ai
+t and, denoting by
+Nt(i) the set of all neighbors of cell i on day t, it can be
+formulated as ¯αi
+t =
+1
+|Nt(i)|
+�
+j∈Nt(i) αj
+t(aj
+t). The through-
+put ratio (rather than traffic/user ratio) is used since different
+cells have different capacities. This value approaches 1 when
+loads of different cells match their capacities.
+Our goal is to maximize the overall network throughput
+by optimizing the two important network performance met-
+rics, namely, throughput ratio βi
+t(ai
+t) and cell throughput
+αi
+t(ai
+t) for each cell i ∈ [1, Nt], where Nt is the total number
+of cells on day t, at the same time. Therefore, we propose the
+following optimization problem for tuning the A2-threshold
+for cell i:
+arg max
+ai
+t∈A
+�
+−
+���1 − βi
+t(ai
+t)
+��, αi
+t(ai
+t)
+�
+,
+(2)
+where A is the set of all possible values for the A2-threshold
+in the cellular network.
+The challenge of solving the above problem lies in sev-
+eral folds. First, since the network performance function is
+complex, dynamic and unknown, obtaining accurate βi
+t(ai
+t)
+and αi
+t(ai
+t) is difficult. Instead, in this work, we adopt a
+data-driven approach to learn reward models and estimate
+the performance metrics. Second, in real-world cases, only
+a limited experimental budget is allowed by network opera-
+tors leading to insufficient diverse historical data (state, ac-
+tion pairs) to train a data-driven learning model. In our de-
+sign, we use a data augmentation technique in the form of
+neighbor cell augmentation to enrich the features from each
+cell. Third, the handover parameter configuration is affected
+by adjacent cells. Thus, it is essential to model the informa-
+tion coming from the adjacent cells to achieve accurate re-
+ward modeling. Lastly, optimizing one performance metric
+greedily might hinder another, thus, how to jointly optimize
+different performance metrics needs careful consideration.
+4
+Temporal Auto-Grouping GCN for
+Reward Modeling
+In order to better capture the dependency between each
+cell and its neighboring cells, we first introduce our novel
+method for neighboring cell feature aggregation. Second, we
+propose a temporal feature aggregation step with recurrent
+neural networks (RNN) to model the temporal correlation
+from the historical sequence of the network states. Third, we
+elaborate the overall training process, considering the im-
+pact from the neighboring cells, the temporal correlation in
+the network and the action we aim to optimize.
+4.1
+Spatial Feature Modeling
+The handover parameters heavily impact the learning prob-
+lem on the graph of the center cells as well as the neighbor-
+ing cells, hence, we aim to capture the neighboring cells in-
+formation during our modeling process. Recently, message-
+passing neural networks (MPNNs) in the form of graph neu-
+ral networks (GNNs) have been introduced and showed to
+be effective in modeling real world applications with struc-
+tural information. The dependencies in the dataset are mod-
+eled using a graph (Hamilton et al. 2017; Ying et al. 2018;
+Wang et al. 2019). In each layer of a GNN, each node’s rep-
+resentation includes the features from itself as well as the
+features from its neighboring nodes (messages sent from the
+neighborhood). We believe the GNN framework is suitable
+for handling the dependencies between the center cell and
+the neighboring cells in cellular networks. We present more
+details on GNN and recent works on homogeneous and het-
+erogeneous graphs in the Sec. A.3.
+Graph-Based Cellular Network Modeling
+We construct
+a graph Gt=(Vt, Et, Xt) for day t, where each node v∈Vt
+represents one cell and is associated with a feature vector
+xv
+t ∈ Rd (v-th column of Xt ∈ Rd×|Vt|), including the sta-
+tistical properties of node v measured on day t. The sta-
+tistical properties could include several features such as the
+antenna transmission power, physical resource block (PRB)
+usage ratio, the amount of data traffic, and the transmission
+bandwidth. These features serve as the node attributes. The
+edge set Et encodes the interactions between cells based on
+
+the handover events between pairs of cells. Based on his-
+torical data, if any pair of cells has an average number of
+handover events above a threshold τ, we assume an edge
+between those two cells. The neighboring set for node v is
+denoted as N g
+t (v)={u|u ∈ Vt, (u, v) ∈ Et}.
+Due to the heterogeneous nature of the cellular network,
+the relationships between the neighboring cells can be com-
+plex. Concretely, there might be an implicit M latent rela-
+tionship types R = {r1, r2, · · · , rM} that can be learned
+to better handle the complex interactions in the cellular net-
+works. Assuming each cell is represented by its states such
+as PRB usage, traffic, etc. in the network graph, we aim at
+dividing the neighboring cells into different groups, each of
+which will provide some information that is shared between
+the neighbors in that group and help to better capture the
+rich information from neighboring cells in a distinguishable
+way. Thus, inspired by the above motivation and a recent
+work (Pei et al. 2020), we propose a novel GCN approach
+called auto-grouping GCN (AG-GCN) to characterize this
+special property of cellular networks when handling the in-
+teractions between neighboring cells. In the following, we
+elaborate upon the detailed steps to realize our design.
+Neighborhood Augmentation
+In cellular network mod-
+eling, since the experiment budget is limited, the historical
+data (state-action pairs) is not diverse enough to train our
+data-driven model. Besides, since we construct the graph
+based on the handover events, there are cells that have a very
+limited number of neighboring cells. Thus, in our design, we
+use a data augmentation technique in the form of neighbor
+cell augmentation based on the similarity between cells in a
+latent space, to enrich the features of each cell.
+We define a feature transformation function f(·) : Rd →
+Rl which maps the input node feature xv
+t ∈ Rd to a la-
+tent space yv
+t = f(xv
+t ) ∈ Rl. In order to capture the long-
+range dependencies and similarity in the cellular network,
+we design an additional neighborhood in the latent repre-
+sentation space based on Euclidean distance. For each node
+v ∈ Vt, we form the augmented neighborhood Nt(v) =
+N g
+t (v) ∪ N s
+t (v), where N g
+t (v) and N s
+t (v) are the neigh-
+bors of node v in the original graph and in the latent space,
+respectively. The neighbors in the latent space are selected
+based on their Euclidean distance to the center cell. The n
+nearest nodes in the latent space are selected to create N s
+t (v)
+for cell v, where the number of nodes we select based on the
+feature similarity is equal to the neighborhood size in the
+original graph |N g
+t (v)|=|N s
+t (v)|=n. The neighbor augmen-
+tation module in Fig. 1 illustrates this process.
+Neighborhood Auto-Grouping
+Once we have obtained
+the augmented neighborhood set, the neighbors in the
+augmented neighborhood Nt(v) are divided into different
+groups by a geometric operator γ. Consider node v and
+its neighbor node u ∈ Nt(v). The relation between them
+on day t is denoted as γ(yv
+t , yu
+t ) : (Rl, Rl) → R =
+{r1, r2, · · · , rM}. This grouping aims at combining neigh-
+bors’ information in groups with similar inter-group fea-
+tures. For each group ri ∈ R, the neighborhood feature set
+on day t is defined as N ri
+t (v) = {u|u ∈ Nt(v), γ(yv
+t , yu
+t ) =
+ri}. The auto-grouping module in Fig. 1 demonstrates this
+process. Note that yellow neighbors (marked with *) are the
+projected counterparts of the neighbors in the graph space,
+while the green neighbors (marked with +) correspond to the
+augmented neighbors from the latent space.
+Conditional Message Passing
+Since the order within
+each neighbor group should not impact the output of the
+representation, we apply a permutation invariant function
+π(·) on the neighbors within each group (mean pooling
+across each feature dimension) and aggregate them sepa-
+rately. Fig. 1 shows an example of the AG-GCN, where
+l = 2 and |R| = 4 and the representation after the per-
+mutation invariant function π(·) is shown by black dashed
+arrows ended to nodes 1, 2, 3, and 4. Then for each group
+ri ∈ R, a non-linear transform is further applied as:
+zv,ri
+t
+= σ
+�
+Wv,ri
+t
+· π
+�
+{xu
+t |u ∈ N ri
+t (v)}
+��
+,
+(3)
+where Wv,ri
+t
+is a learnable weight matrix for the neighbors
+in group ri of node v on day t, and σ(·) is a non-linear
+function, e.g., tanh. Then for each node we aim to aggre-
+gate the transformed neighborhood features from their dif-
+ferent groups of neighbors in a distinguishable way. The
+vectors zv,ri
+t
+for ri ∈ R are further aggregated as hv
+t =
+[zv,r1
+t
+; · · · ; zv,rM
+t
+], where [ ; ] represents concatenation.
+4.2
+Temporal Feature Modeling
+Capturing the trend in the evolution of the states of each cell
+within a day properly can benefit the prediction of the per-
+formance metric for the following day. We propose to use
+additional temporal features for each center cell to extract
+the changing dynamic pattern of its states within each day to
+further improve the reward model performance. We assume
+the samples of the center cell v on day t can be divided into
+K groups by their temporal order. For all the samples in each
+group k, we take the average network state for each group of
+samples and denote it as xv
+t,k. We use an RNN layer to cap-
+ture this temporal dependency of the features from different
+groups by feeding all the network states as an input sequence
+Pv
+t =
+�
+xv
+t,1
+T ; xv
+t,2
+T ; · · · ; xv
+t,K
+T �T ∈ RK×d, to obtain
+cv
+t = RNN(Pv
+t , δ) ∈ Rd′,
+(4)
+where δ and d′ are the set of trainable parameters and the
+output dimension of the RNN layer, respectively.
+4.3
+Overall Training Pipeline
+The main purpose of the model is to estimate the real
+network‘s response, and predict the throughput ratio and
+throughput of the center cell for the next day based on the
+observed network states in the current day. These perfor-
+mance metrics are not only affected by the current day’s
+states, but also highly correlated with the action we choose
+to configure for the next day. Thus, we also consider the ac-
+tions of the next day. Furthermore, the throughput ratio and
+throughput of the next day are highly dependent on the pre-
+vious performance metrics. Hence, we consider the current
+throughput ratio, i.e., βv
+t , in the prediction process.
+To make the final prediction, the learned representation of
+the neighborhood by the AG-GCN aggregation, the tempo-
+ral features of the center cell, and the throughput ratio of the
+
+RNN
+Predict 
+Throughout 
+or 
+Throughput 
+Ratio
+Neighbor 
+Augmentation Module
+Auto-Grouping 
+Module
+Daily Average
+Graph Space
+Euclidean Space
+Embedding function 
+Group 1
+Group 2
+Group 3
+Group 4
+1
+4
+3
+2
+*
++
+Neighbors from graph space
+Neighbors from Euclidean space
+TAG-GCN
+Neighbor Augmentation Module
+Auto-Grouping Module
+Capturing Spatial Information
+Capturing Temporal Information
+*
++
++
++
++
+*
+Temporal Feature 
+Modeling
+Neighbor 
+Information 
+Modeling
+Concatenation
+Concatenation
+Concatenation
+Figure 1: The flow of information from graph structure to final prediction, used to form the training pipeline of two models for
+predicting the throughput ˆα and the throughput ratio ˆβ for Tin = {1, 2, · · · , t − 1}. In this demo example, the auto-grouping
+module constructs M = 4 groups of neighbors, where l = 2. Empty groups are filled with the average of other groups.
+current day, i.e. βv
+t , are concatenated to form the state vec-
+tor of cell v as sv
+t = Ψ(Wv
+t ·
+�
+βv
+t ; cv
+t ; hv
+t
+�
+), where Wv
+t is
+a learnable weight matrix for node v on day t, and Ψ(·) is
+a non-linear function, e.g., tanh. Since the final representa-
+tion should be sensitive to the chosen input action (of which
+the decision making process will be elaborated in Sec. 5),
+the throughput ratio and throughput of the next day for cell
+v are formulated as the output of a non-linear transformation
+Λ(·) function of state and action:
+ˆβv
+t+1 = Λ
+�
+Wv
+β.([sv
+t ; av
+t ])
+�
+,
+(5)
+ˆαv
+t+1 = Λ
+�
+Wv
+α.([sv
+t ; av
+t ])
+�
+,
+(6)
+where Wv
+β and Wv
+α are trainable matrices of node v for
+throughput ratio and throughput models, respectively. The
+overall flow of data from the graph structure to the final pre-
+diction is represented in Fig. 1. Note that we train two sepa-
+rate models for predicting the throughput and the throughput
+ratio simultaneously.
+In order to properly use the A2-threshold for the predic-
+tion, we use the change in this parameter compared to the
+previous day as the action av
+t = A2v
+t+1 −A2v
+t , where A2v
+t+1
+and A2v
+t are the A2-thresholds for cell v on day t + 1 and t,
+respectively. The reason for this design choice has twofold.
+First, the original action space of A2 is large, but the range
+of the change of action can be smaller by controlling the
+adjustment steps, making it easier for the model to learn
+and conduct the decision making step. Besides, the delta ac-
+tion directly reflects the change in the cell coverage/loads, so
+they are more sensitive to the performance metrics. To form
+the training objective, we consider data of T + 1 consecu-
+tive days and form the pairs (t, t + 1), t ∈ {1, 2, · · · , T},
+to predict the throughput ratio and throughput of the center
+cell in day T + 1, trained by minimizing the following loss
+functions respectively:
+1
+T
+T
+�
+t=1
+1
+Nt
+Nt
+�
+v=1
+(ˆβv
+t+1 − βv
+t+1)2 + λ1||Θ1||2,
+(7)
+1
+T
+T
+�
+t=1
+1
+Nt
+Nt
+�
+v=1
+(ˆαv
+t+1 − αv
+t+1)2 + λ2||Θ2||2,
+(8)
+where λ1 and λ2 are the hyperparameters chosen for reg-
+ularization. Θ1 and Θ2 represent all the trainable parame-
+ters in the models. The trained reward model is now able to
+mimic the real network and predict both throughput ratio and
+throughput of each center cell for the coming day and can be
+used to check the impact of actions towards the performance
+metrics we are considering.
+5
+Action Configuration
+As discussed in the earlier sections the main objectives to
+consider in the action configuration process are load balanc-
+ing, identified by the throughput ratio, and the cell through-
+put. Hence, the best action for cell v on day t, i.e. av
+t ∈ A, is
+the one that optimizes the problem in (2). In general, when
+dealing with a multi-objective problem, different objectives
+are often conflicting, and we may not be able to optimize
+them simultaneously. One common way to tackle this prob-
+lem is to give different objectives weights and optimize the
+weighted objective value. However, in our scenario, it is dif-
+ficult to determine the weights and different clusters may
+require cluster-specific weights. Here we break the problem
+in (2) into two sub-problems, and solve them sequentially.
+We first optimize the action with respect to the predicted
+throughput ratio, i.e., ˆβv
+t+1(av
+t ) for cell v on day t, where
+av
+t ∈ A, and then optimize the throughput ˆαv
+t+1(av
+t ).
+Specifically, the throughput ratio is optimized and we find
+the set of best c values for av
+t , denoted Av
+c, such that
+min
+av
+t ∈Av
+c
+−
+���1 − ˆβv
+t+1(av
+t )
+�� ≥ max
+av
+t ∈A−Avc
+−
+���1 − ˆβv
+t+1(av
+t )
+��. (9)
+Then, our goal is to achieve the maximum possible through-
+put for cell v on day t and this is through
+ˆav
+t = arg max
+av
+t ∈Av
+c
+ˆαv
+t+1(av
+t ).
+(10)
+ˆav
+t is then the final recommended action for cell v on day t.
+This procedure for all the Nt cells of the network on day t is
+presented in Algorithm 1.
+6
+Experimental Results
+The experiments are conducted on a large-scale cellular net-
+work simulator constructed from real-world data which pre-
+sented in Sec. B.1. We use principal component analysis
+
+γ(yv
+t , yu
+t )
+yv
+t [0] > yu
+t [0]
+yv
+t [0] ≤ yu
+t [0]
+yv
+t [1] ≤ yu
+t [1]
+2
+1
+yv
+t [1] > yu
+t [1]
+3
+4
+Table 1: The relationship operator γ
+Algorithm 1: TAG-GCN for Action Configuration
+Input: Pv
+t and xu
+t where, u ∈ Nt(v) ∀v ∈ [1, Nt]
+Output: av
+t for ∀v ∈ [1, Nt]
+1: Let v = 1 and ∀j ∈ [1, Nt] set Aj
+c = ∅.
+2: while v ≤ Nt do
+3:
+Feed the TAG-GCN model with Pv
+t and xu
+t , for all
+neighbor u ∈ Nt(v).
+4:
+Freeze all inputs of TAG-GCN except actions and
+predict the performance metrics ˆαv
+t+1(·) and ˆβv
+t+1(·)
+for the input actions.
+5:
+for |Av
+c| ≤ ν do
+6:
+x = arg maxa∈{A−Avc} −
+���1 − ˆβv
+t+1(a)
+��
+7:
+Av
+c = Av
+c ∪ {x}
+8:
+end for
+9:
+av
+t = arg maxa∈Avc ˆαv
+t+1(a)
+10:
+v = v + 1
+11: end while
+12: return av
+t for ∀v ∈ [1, Nt]
+(PCA) as the mapping function f(·), as defined in Sec. 4, for
+obtaining the latent representation in the AG-GCN step. It
+transforms the original feature into a 2-dimensional space to
+perform the neighborhood augmentation and the neighbors
+group assignment process. After this transformation, the re-
+lationship operator γ for the auto-grouping assigns a group
+to each subset of points in each quadrant of this two dimen-
+sional space presented in Table 1. The permutation-invariant
+function π applied on each group of neighbors is average in
+our experiments.
+6.1
+Datasets
+To perform our experiments and evaluate the proposed
+model, two datasets are used in this study (see Sec. B.2 for
+more details):
+Dataset-A:
+A real metropolitan cellular network contain-
+ing around 1500 cells sampled hourly and collected from
+Oct. 17 to Oct. 31, 2019. Each data sample contains infor-
+mation such as the cell ID, sample time, configuration of cell
+parameters, and measurements of the cell states.
+Dataset-B:
+Also a real metropolitan cellular network. The
+network contains 1459 cells, and the data is collected from
+Sep. 1 to Sep. 29, 2021. Each data sample contains similar
+information as above.
+6.2
+Reward Model Accuracy Evaluation
+Dataset Generation
+In order to evaluate the prediction ac-
+curacy of our model, we use a simulator to modify Dataset-A
+with a random policy to diversify our network configuration.
+On each day, the A2-threshold for each cell is randomly se-
+lected around the default action -100 dBm within the range
+[−105, −95]. This approach provides us a fix data buffer
+with diverse action dataset to train all models and have a fair
+2019-10-22
+2019-10-23
+2019-10-24
+2019-10-25
+2019-10-28
+2019-10-29
+Time
+6
+7
+8
+9
+10
+11
+Throughput MSE
+MLP
+GCN
+AG-GCN
+TAG-GCN
+(a)
+2021-09-05 2021-09-06 2021-09-07 2021-09-08 2021-09-09 2021-09-10 2021-09-11 2021-09-12
+Time
+6
+7
+8
+9
+10
+11
+Throughput MSE
+MLP
+GCN
+AG-GCN
+TAG-GCN
+(b)
+Figure 2: Achieved MSE of the throughput for test data of
+(a) Dataset-A and (b) Dataset-B for different methods
+comparison of their accuracy. For Dataset-B, there exists a
+reasonable amount of the diversity in the handover parame-
+ter configuration, thus we directly use the raw dataset from
+the live network to perform the training and evaluation.
+Training Process and Metrics
+As samples generated
+hourly, we aggregate them within each day as described in
+Sec. 4. To evaluate the model accuracy in predicting cell
+throughput and throughput ratio, we train the model with
+the generated pairs {(1, 2), · · · , (t − 1, t)} for t = 9 and 12
+days for Dataset-A and B, respectively. At each day t > 2,
+data pairs {(1, 2), · · · , (t−2, t−1)} are used as training and
+validation sets, and (t − 1, t) serves as testing set for eval-
+uation across different models. We report the mean square
+error (MSE) to measure the reward model performance.
+Comparison with Benchmark Models
+It is important to
+note that, the social impact of this work has not been ad-
+dressed by ML approaches the same way as we propose.
+Due to the uniqueness of our problem, existing solutions
+for optimizing handover parameters are either not appropri-
+ate to solve it or there is no apparent way to directly adapt
+them to our problem. For instance, traditional handover op-
+timization methods rely on designing fuzzy rules based on
+different measures of QoS in the network (Vasu et al. 2012),
+however, designing proper rules is complex and cannot han-
+dle the change in highly dynamic systems well. Instead, we
+hope to use a data-driven approach, among which the (deep)
+RL method gains the most attention (Cao et al. 2018; Wang
+et al. 2018). However, in this type of problems, the network
+provider only allows limited exploration of the parameter
+values (e.g., allows changing the A2 value once a day) to en-
+sure the stability of the network. Thus, we only have limited
+days for exploring the best action. RL models, nevertheless,
+usually need longer episodes to optimize the accumulated
+return. Despite this fact, we made some preliminary attempt,
+presented in Sec. B.3, to adapt the RL paradigm from the lit-
+erature into our problem which did not show any advantage
+over our simpler design.
+In order to show the effectiveness of our proposed re-
+ward model, we compare it with alternative designs for the
+prediction model. It should be mentioned that all of these
+models are our contribution. In the Sec. B.4, we summarize
+our benchmarks and the properties of each model. The first
+model is MLP, where we only use the features of the cen-
+ter cells and ignore the neighboring cells’ features. In GCN
+model, we follow the typical GCN formulation (Hamilton
+et al. 2017) and process the network as a homogeneous
+
+Model
+Action 
+Configuration
+Simulator
+𝐱𝑡
+𝑣, {𝐱𝑡
+𝑢|𝑢 ∈ 𝒩𝑡(𝑣)}
+𝐱𝑡+1
+𝑣
+, {𝐱𝑡+1
+𝑢
+|𝑢 ∈ 𝒩𝑡+1(𝑣)}
+𝑎𝑡
+𝑣
+ 𝛽𝑡+1
+𝑣
+,  𝛼𝑡+1
+𝑣
+Figure 3: The process of action recommendation by the
+trained model and the simulator
+graph where the neighbor information is aggregated jointly
+without distinction. The AG-GCN model ignores the tem-
+poral dependencies of the data which we consider in TAG-
+GCN model. In Fig. 2, we compare the prediction accuracy
+of these models for throughput in Dataset-A and B. We ob-
+serve on average the best accuracy for the test set is achieved
+by AG-GCN and TAG-GCN, with TAG-GCN performing
+marginally better on the average rank metric across the eval-
+uation days, indicating that our neighbor aggregation and
+temporal features extraction have a considerable impact on
+the reward modeling for cellular networks. The same results
+also achieved for the throughput ratio model. The same test
+is performed for throughput ratio and included in the sup-
+plementary materials in Fig. 10.
+6.3
+Overall Parameter Optimization
+Performance
+The Action Recommendation Process
+In the following
+experiments we use the presented models to recommend the
+actions for Dataset-A. The actions in day 1, i.e., Oct. 17,
+has been set to the default action which is -100 dBm. Unless
+otherwise stated, the action for the second day, i.e., Oct. 18,
+is initialized by a set of random actions around the default
+action in the range of [−105, −95]. The model is trained it-
+eratively on each day and used to recommend actions for the
+next day. The process is depicted in Fig. 3, where states of
+the cells on day t are given to the trained model to predict
+performance metrics of the network on day t + 1 and the
+action av
+t is adjusted for each cell based on the predictions.
+Finally, the network states and performance measurements
+for day t + 1 are computed according to the new selected
+action by the cellular network simulator and used for model
+training and action recommendation in the following day.
+Baseline Performance Bounds
+In addition to the result
+achieved by the actions recommended by the models, we use
+three baseline performance bounds achieved by the default
+A2-threshold, the expert rule, and the optimal actions of the
+simulator. As stated before the default A2-threshold value
+is −100 dBm and this is used as the lower bound in the fol-
+lowing experiments. The optimal actions in the simulator are
+obtained by brute-force search and it introduces the upper
+performance bound. The expert rule-based method provided
+by experienced network operator is a simple rule presented
+in the Sec. B.6. The performance achieved by the expert rule
+is better than the default action. We hope to use our proposed
+learning based framework to further fill the gap with the re-
+ward achieved by optimal actions.
+Results
+In the following experiments, we compare the per-
+formance of the network under the actions recommended
+2019-10-17 2019-10-18 2019-10-21 2019-10-22 2019-10-23 2019-10-24 2019-10-25 2019-10-28 2019-10-29 2019-10-30 2019-10-31
+Time
+1.00
+0.75
+0.50
+0.25
+0.00
+0.25
+0.50
+0.75
+1.00
+1.25
+Reward
+MLP
+GCN
+AG-GCN
+TAG-GCN
+Best Action
+Experts Actions
+Default Action
+Figure 4: Performance comparison of different models along
+with optimal action curve initialized with random actions
+by different models in terms of the network throughput as
+defined in Sec. 3. We plot the trajectory of the throughput
+difference to the default A2-threshold baseline (dash black
+line) in Fig. 4. We repeat all the experiments 20 times for
+all models, where each run uses the same set of random ac-
+tions on the first action exploration day (Oct. 18) for all the
+models. We also show the performance achieved through the
+expert rule action recommendation, default action, and the
+optimal actions of the simulator (random actions are also
+used on Oct. 18 for the curve of the optimal action). The
+quantitative results for Fig. 4 are summarized in Table 6
+in the supplementary materials. TAG-GCN can achieve bet-
+ter average throughput in the final days which indicates the
+importance of our auto-grouping GCN design to tailor the
+heterogeneous property of the cellular networks. Besides, as
+expected, all the learning-based models can beat the expert
+rule algorithm which is highly dependent on human experi-
+ence and is unable to recover from the performance degra-
+dation due to bad random initialization on the first day. Fur-
+thermore, to show the effectiveness of our proposed model
+in terms of load balancing and enhancing cluster through-
+put ratio, we illustrate the progress of this ratio achieved by
+TAG-GCN for some selected severely unbalanced cells in
+Fig.5. As it can be seen, the throughput ratio of the clusters
+form a trajectory that converges to 1 which is the ideal target
+value. More ablation studies are presented in Sec. B.7.
+Based on the above experiments, our proposed ML-based
+solutions can improve the network performance and opti-
+mize the handover process compared to the conventional
+methods such as using the default action or human experts
+rule-based methods. Moreover, the automation of the param-
+eter optimization process achieved by our ML-based solu-
+tions reduces the domain expert’s intervention and, hence,
+the management cost of network operators and improves the
+maintenance efficiency of cellular networks. Consequently,
+the proposed solutions open up the possibilities to provide
+reliable and high-quality network access even to geograph-
+ical areas that are currently underserved by the cellular net-
+work. This can bring exciting new opportunities to these
+regions such as remote education, remote working, health
+monitoring, video streaming, etc.
+7
+Conclusion
+In this paper, we study the handover parameter configura-
+tion problem in cellular networks. We propose a reward pre-
+diction model to accurately imitate the cellular network and
+estimate the performance metrics. Our proposed model, i.e.,
+
+2019-10-17 2019-10-18 2019-10-21 2019-10-22 2019-10-23 2019-10-24 2019-10-25 2019-10-28 2019-10-29 2019-10-30 2019-10-31
+Time
+0.6
+0.8
+1.0
+1.2
+1.4
+1.6
+1.8
+Throughput Ratio
+CELL_CELLNAME
+cell132
+cell2575
+cell1304
+cell3141
+cell2955
+cell2790
+cell2250
+cell3872
+cell1015
+cell3140
+Figure 5: Trend of the throughput ratio for sample clusters
+TAG-GCN, investigates the impact of the adjacent cells and
+differentiate their impact on the center cell of each clus-
+ter. We also consider the network changing dynamics in our
+model to learn the temporal dependencies in the data. Based
+on the reward model, a novel multi-objective parameter con-
+figuration strategy is proposed to perform the optimization
+for each cluster and balance the performance metrics in each
+neighborhood. The conducted simulations shows the superi-
+ority of TAG-GCN which has a huge potential social impact
+by improving the cellular network parameters and providing
+massive connectivity and high coverage with a balanced traf-
+fic across the network. Hence, this can help the widespread
+adaptation of new technologies to benefit many sectors such
+as health and education.
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+A
+Extended Background and Related Work
+A.1
+Handover in Cellular Networks
+A cellular network is formed by many cells distributed over
+land areas, each served by at least one fixed-location base
+station (BS) providing wireless links with a wide geographic
+area coverage to support many users. A cluster in a cellular
+network (consisting of a cell and its neighboring cells) is
+shown in Fig. 6 where, user u is crossing one cell’s cov-
+erage area and monitors the received power from cell n to
+check the handover criteria. The cell BS is responsible for
+monitoring the strength of the signals received by served
+users, which can degrade when a user travels from one cell
+to another. The BS should trigger a handover process at the
+proper moment to avoid any service interruption and trans-
+fer the user to another cell BS that is receiving the strongest
+signals (Bhat et al. 2012; Tekinay et al. 1991).
+There are multiple parameters that impact handover in a
+network. These parameters generally are grouped in three
+categories. RSRP and RSRQ parameters are indicators of the
+signal strength and quality of the serving station, respec-
+tively, Hysteresis parameters act as a tolerance margin to
+avoid the Ping Pong (PP) effect, and Time To Trigger (TTT)
+parameters are set such that short term violations of han-
+dover conditions are ignored to avoid the PP effect. Also,
+different measures such as handover failure (HOF) rate, han-
+dover frequency, Ping-Pong (PP) rate, network throughput,
+and load balancing have been used in the literature to assess
+the effectiveness of the proposed algorithms for optimizing
+handover parameters. When performing optimization in a
+real network, some of these measures may conflict. There-
+fore, it is crucial to consider a combination of these quanti-
+ties to find a solution that satisfies different requirements of
+the network.
+A.2
+Related Work
+Different methods have been used in the literature for solv-
+ing the problem of optimization of handover parameters in
+a wireless network. Fuzzy system handover algorithms, for
+example, have been used in (Vasu et al. 2012). Such tech-
+niques, however, are not scalable despite being accurate and
+stable. Deep reinforcement learning (DRL) is another tech-
+nique that has been used to solve the handover optimization
+problem. In (Cao et al. 2018), the authors propose a frame-
+work based on DRL where actions are flexible and can be
+chosen by the user, and the objective of the optimization
+is the throughput and the handover count. In (Wang et al.
+2018), the authors propose a framework in which the UEs
+first are clustered based on their usage pattern and then the
+handover process is optimized in each cluster by a DRL
+method. Another newly introduced approach to the problem
+of cellular network parameter optimization is the Contex-
+tual Bandit model proposed in (Chuai et al. 2019)in which
+the throughput of the cells is used as a performance metric
+for optimizing some of the cell parameters.
+The above mentioned methods, despite being effective, do
+not exploit the information from the neighboring cells which
+
+<A2-threshold
+<A2-threshold
+<A2-threshold
+<A2-threshold
+<A2-threshold
+<A2-threshold
+<A2-threshold
+Center Cell 𝑛
+User 𝑢
+𝑃𝑢,𝑛
+Figure 6: A typical structure of a cluster in the cellular net-
+work serving different types of users.
+Model each sub-network as a graph where 
+nodes are cells and each edge represents heavy 
+handover events between this pair of cells 
+𝒢𝑡 = (𝒱𝑡, ℰ𝑡)
+Graph Construction
+Typical GCN Aggregation
+All neighbors are using the same 
+transformation function to perform 
+the feature mapping
+Figure 7: Commonly used GNN design.
+can be used to develop a more effective model of the net-
+work. Moreover, two of the most important measures from
+the user’s standpoint are throughput and load balancing,
+none of which are considered in the above models simulta-
+neously. Thus, in the current work, we propose a technique
+which remedies these shortcomings by utilizing the neigh-
+bors’ information, and optimizing the parameters of the net-
+work using two significantly important objectives, namely,
+load balancing and throughput.
+A.3
+Graph Neural Networks for Cellular
+Network
+Successful applications of GNNs mostly rely on a homoge-
+neous graph structure. For example, in social networks or
+recommender systems, each edge represents a positive cor-
+relation between the adjacent nodes (Kipf et al. 2017; Fan
+et al. 2019). The MPNN framework (Gilmer et al. 2017) ap-
+plies the same transformation function to every neighbor and
+aggregates across them to obtain the neighbor representa-
+tion, shown in Fig. 7. There is a rich literature that studies
+how to process a heterogeneous graph with multiple relation
+types (Schlichtkrull et al. 2018; Zhang et al. 2019). They
+are established on the prior knowledge of the given relation
+type between each pair of nodes and then use a meta-path
+design to distinguish the information coming from different
+types of neighbors. Such an approach is not applicable to our
+problem.
+B
+Experimental Details, Results, and
+Ablations
+B.1
+Simulator Construction
+The experiments are run through a proprietary cellular net-
+work simulator which uses statistical models to simulate the
+Figure 8: The internal workflow of the simulator.
+network response to any change in the parameter values.
+This simulator is designed based on the network correspond-
+ing to Dataset-A and uses the states of the network cells
+along with their action to simulate the throughput and the
+traffic of the cells.
+Dataset-A is collected under the default network configu-
+ration and the value of A2-threshold is not changed during
+the collection period. To evaluate our model and the action
+configuration strategy, the simulator needs to simulate the
+network performance given arbitrary A2-threshold values of
+the cells. More precisely, let xi
+t represents cell i’s states at
+day t under the default A2-threshold configuration, and A2i
+t
+be the configured A2-threshold value for cell i at day t, the
+simulator will output
+˜x0
+t, ..., ˜xNt
+t , r0
+t , ..., rNt
+t
+= g(x0
+t, ..., xNt
+t , A20
+t, ..., A2Nt
+t ),
+(11)
+where ri
+t is cell i’s throughput, and ˜xi
+t is the observed states
+of cell i under the new configurations at t.
+Fig. 8 gives a simple sketch on the internal workflow of
+the simulator. During simulation, when A20
+t, ..., A2Nt
+t
+are
+configured at day t, the cell states x0
+t, ..., xNt
+t
+are read from
+the collected data, and inputted to the simulator. The cell
+coverage Ci for each i is first computed by a built-in func-
+tion that considers cell i’s frequency, bandwidth and A2i
+t.
+Then cell i’s traffic load (including the number of users and
+the amount of data bits for transmission) at day t is redis-
+tributed among itself and its neighbors based on the change
+of Ci and Cj with j ∈ Nt(i). After each cell’s traffic load is
+updated, the throughput ri
+t is predicted based on cell i’s traf-
+fic load. The load-throughput prediction model for each cell
+is pre-trained from the collected historical data and stored
+when the simulator is initialized. The value ri
+t is further ad-
+justed by a factor that considers the throughput loss due to
+measurement reporting or connection loss. Finally, the sim-
+ulator outputs r0
+t , ..., rNt
+t . As each cell’s traffic load (part of
+the cell states) is modified, the updated states ˜x0
+t, ..., ˜xNt
+t
+are
+also outputted.
+B.2
+Dataset Samples
+Dataset-A and B contain wide range of cell states (e.g., the
+number of total users within the cell, the number of active
+users, the cell average CQI, the cell traffic load, etc.) and
+performance indicators (e.g., the average cell throughput,
+the edge user throughput, etc.). The neighbor relation infor-
+mation between cells in both datasets are collected and the
+hourly average handover counts between neighbor cells are
+recorded. The weekend dates are excluded from Dataset-A
+since these days are not considered as rush hour usage for
+this region of cellular networks.
+We present the values of four different features of three
+
+0
+Nt
+N
+Nt0
+N
+Estimate each cell
+Update each cell
+Predict each cell
+i's coverage ci
+i's traffic load
+i's throughput口
+A2-threshold
+Center Cell n
+A2.threshold
+-thres
+A2-threshold
+<A2-threshold
+A2-threshold
+UseruTime
+Cell Throughput
+AVG Traffic
+Cell Bandwidth
+AVG CQI
+00 - 01
+17.18
+86.14
+5.0
+11.84
+01 - 02
+12.97
+55.06
+5.0
+12.36
+Cell 999
+02 - 03
+19.25
+50.09
+5.0
+12.48
+(Dataset-A)
+...
+...
+...
+...
+...
+23 - 00
+7.17
+128.63
+5.0
+11.84
+Daily AVG
+17.78
+105.52
+5.0
+11.82
+00 - 01
+17.45
+16.31
+5.0
+11.46
+01 - 02
+24.15
+11.59
+5.0
+10.51
+Cell 1720
+02 - 03
+32.26
+7.36
+5.0
+9.49
+(Dataset-A)
+...
+...
+...
+...
+...
+23 - 00
+27.62
+18.38
+5.0
+11.19
+Daily AVG
+24.42
+18.08
+5.0
+10.24
+00 - 01
+26.56
+15.65
+5.0
+11.51
+01 - 02
+37.82
+7.52
+5.0
+10.97
+Cell 997
+02 - 03
+35.80
+4.62
+5.0
+11.53
+(Dataset-A)
+...
+...
+...
+...
+...
+23 - 00
+31.41
+22.94
+5.0
+11.67
+Daily AVG
+31.05
+26.72
+5.0
+11.48
+00 - 01
+16.83
+4.35
+5.0
+10.10
+01 - 02
+22.67
+7.27
+5.0
+9.90
+Cell 1284
+02 - 03
+22.94
+4.99
+5.0
+9.75
+(Dataset-B)
+...
+...
+...
+...
+...
+23 - 00
+28.62
+5.76
+5.0
+10.45
+Daily AVG
+17.92
+7.18
+5.0
+10.16
+00 - 01
+8.12
+6.28
+3.0
+8.95
+01 - 02
+18.22
+5.73
+3.0
+9.12
+Cell 1127
+02 - 03
+18.66
+6.00
+3.0
+9.24
+(Dataset-B)
+...
+...
+...
+...
+...
+23 - 00
+14.91
+7.61
+3.0
+8.96
+Daily AVG
+11.09
+10.38
+3.0
+8.98
+00 - 01
+32.50
+29.14
+5.0
+10.35
+01 - 02
+30.39
+22.81
+5.0
+10.58
+Cell 986
+02 - 03
+33.88
+22.67
+5.0
+10.99
+(Dataset-B)
+...
+...
+...
+...
+...
+23 - 00
+29.25
+37.42
+5.0
+10.21
+Daily AVG
+22.45
+43.41
+5.0
+10.32
+Table 2: Samples from Dataset-A and B.
+random cells in Dataset-A in Tables 2. Note that the repre-
+sented values are collected in the first day. As seen the range
+of variation across different cells and different hours for each
+single cell is too high and that is the reason we consider the
+modules to handle spatial and temporal dependencies in our
+model to better imitate the real-world environment. We have
+almost the same pattern in Dataset-B as well for these pre-
+sented features.
+B.3
+Reinforcement Learning Modeling
+As we mentioned in Sec. 6, for problems similar to ours
+which require following a policy to choose an action to inter-
+act with an environment, reinforcement learning (RL) mod-
+eling is a good candidate. Among all the RL methods, the
+actor-critic (AC) method is a potential match to our prob-
+lem due to its continuous action space and modeling flexibil-
+ity. AC methods have been widely used in cellular network
+problems and for different tasks such as power allocation
+and traffic forecasting (Li et al. 2014; Koushik et al. 2017;
+Jiang et al. 2020; Wei et al. 2018). Hence, as an additional
+benchmark we compare our proposed algorithm with a ba-
+sic AC algorithm we implemented to adapt to our problem
+where the actor model recommends actions in order to max-
+imize the reward predicted by the critic model.
+We use TAG-GCN as the critic model to predict the re-
+ward which can be either throughput ratio or throughput to
+train the actor model. For the actor model, we use the same
+modules presented for TAG-GCN to do the neighbor auto-
+grouping and capture the temporal dependencies in the data
+2019-10-17 2019-10-18 2019-10-21 2019-10-22 2019-10-23 2019-10-24 2019-10-25 2019-10-28 2019-10-29 2019-10-30 2019-10-31
+Time
+1.00
+0.75
+0.50
+0.25
+0.00
+0.25
+0.50
+0.75
+1.00
+1.25
+Reward
+TAG-GCN
+Actor-Critic
+Figure 9: Performance comparison between our proposed
+design and actor-critic design for action exploration.
+Model List
+Model Name
+Graph
+Heterogeneous
+Graph
+Daily Temporal
+Features
+MLP
+
+
+
+GCN
+
+
+
+AG-GCN
+
+
+
+TAG-GCN
+
+
+
+Table 3: Properties of Different Models
+and train it with the following loss function
+Loss(actor) = 1
+T
+T
+�
+t=1
+1
+Nt
+Nt
+�
+v=1
+���1 − ˆβv
+t (ˆav
+t )
+�� + λ3||Θ3||2,
+(12)
+where ˆav
+t is the recommended action by the actor model and
+ˆβv
+t (ˆav
+t ) is the predicted reward by the critic model using the
+recommended action. λ3 is the hyperparameter chosen for
+regularization and Θ3 represents all the trainable parameters
+in the actor model.
+Fig. 9 shows that TAG-GCN outperforms this basic RL
+model. The proposed AC model is a very preliminary one
+and obviously it can be improved even more which is out of
+the scope of this paper. However, this potential improvement
+comes with the cost of increasing complexity and requires
+a highly diverse data and more interactions with the envi-
+ronment. The RL approach generally requires frequent and
+plenty of interactions with the environment. However, in our
+case, the action is configured once a day, and the total explo-
+ration duration only consists of small number of days, and
+is far less than what is generally required by the RL mod-
+els. Furthermore, RL models usually consider accumulated
+returns, i.e., impact of the current action on the future per-
+formance. However, in our scenario, the current action only
+affects the loads distribution of the current hour, and does
+not affect the future performance. Therefore the standard RL
+approach does not apply here.
+B.4
+Alternative reward model comparison
+We summarize the alternative designs for the reward model
+we consider in our paper in Table 3.
+B.5
+Additional Model Evaluation Metrics (avg.
+Rank)
+We present the average rank of different models across dif-
+ferent evaluation days in terms of the achieved test MSE for
+both throughput ratio and cell throughput models in Dataset-
+A and B, in Table 4 and 5, respectively. As seen, considering
+
+Model Name
+Ratio Model
+Throughput Model
+Average
+MLP
+3.83
+3.83
+3.83
+GCN
+3.17
+3.00
+3.08
+AG-GCN
+1.67
+1.67
+1.67
+TAG-GCN
+1.33
+1.50
+1.42
+Table 4: Average rank of the learning models in terms of the
+achieved test MSE across the evaluation days for Dataset-A
+Model Name
+Ratio Model
+Throughput Model
+Average
+MLP
+3.50
+3.83
+3.67
+GCN
+3.00
+3.00
+3.00
+AG-GCN
+2.00
+1.50
+1.75
+TAG-GCN
+1.50
+1.67
+1.58
+Table 5: Average rank of the learning models in terms of the
+achieved test MSE across the evaluation days for Dataset-B
+2019-10-22
+2019-10-23
+2019-10-24
+2019-10-25
+2019-10-28
+2019-10-29
+Time
+0.008
+0.010
+0.012
+0.014
+0.016
+0.018
+0.020
+0.022
+Ratio MSE
+MLP
+GCN
+AG-GCN
+TAG-GCN
+(a)
+2021-09-05 2021-09-06 2021-09-07 2021-09-08 2021-09-09 2021-09-10 2021-09-11 2021-09-12
+Time
+0.008
+0.010
+0.012
+0.014
+0.016
+0.018
+Ratio MSE
+MLP
+GCN
+AG-GCN
+TAG-GCN
+(b)
+Figure 10: Mean squared error of the throughput ratio for
+test data of (a) Dataset-A and (b) Dataset-B for TAG-GCN.
+the results of both datasets, TAG-GCN can achieve better
+performance than other proposed model which is due to its
+capacity to learn the spatial and temporal dependencies in
+the network.
+B.6
+Expert Rule-Based Parameter Optimization
+Strategy
+The experts rule-based method can set the actions for each
+cell v ∈ Nt for day t + 1 based on the load balancing as:
+av
+t =
+�rv
+1(βv
+t − 1)
+βv
+t ≥ 1
+rv
+2(βv
+t − 1)
+βv
+t < 1 ,
+(13)
+where rv
+1, rv
+2 < 0 are the weights set by human experts based
+on domain knowledge and assumptions about the network
+dynamics. The intuition is to increase the A2-threshold to
+release traffic when a cell is overloaded and decrease the
+value in the opposite case.
+B.7
+Additional Experimental Results
+Reward Modeling Performance for Throughput Ratio
+We repeat the same experiment as Fig. 2 in the paper but
+for the throughput ratio reward and compare the perfor-
+mance measured in the achieved MSE by different models
+for Dataset-A and B in Fig. 10.
+Ablation Study on the Multi-Objective Action Recom-
+mendation Strategy
+In Fig. 11, we show an ablation study
+on the effectiveness of our proposed multi-objective action
+recommendation strategy. For each setting, we repeat our ex-
+periment five times with TAG-GCN as the reward model but
+follow different action configuration strategies. Other than
+the multi-object formulation, we consider two alternatives
+2019-10-17 2019-10-18 2019-10-21 2019-10-22 2019-10-23 2019-10-24 2019-10-25 2019-10-28 2019-10-29 2019-10-30 2019-10-31
+Time
+1.00
+0.75
+0.50
+0.25
+0.00
+0.25
+0.50
+0.75
+1.00
+Reward
+TAG-GCN (multi-objective)
+TAG-GCN (single objective by throughput ratio)
+TAG-GCN (single objective by cell throughput)
+Figure 11: The impact of different optimization objectives
+for the parameter configuration in our proposed TAG-GCN
+model.
+for action configuration, one by considering only the cell
+throughput in the optimization problem, and one by consid-
+ering only the load-balancing objective. As shown in Fig. 11,
+our proposed TAG-GCN model with both load-balancing
+and throughput as optimization objectives can outperform
+others, achieving the best average throughput in the final
+days across random trials.
+The Quantitative Results
+The quantitative results for
+Fig. 4 and Fig. 11 are summarized in Table 6 and Table 7, re-
+spectively. We present the achieved network throughput by
+different models divided by the performance achieved by the
+best actions of the simulator for the last 4 days of the exper-
+iments. It should be noted that in all these experiments the
+model is trained up to Oct. 29 and after that the model is
+not updated and only performs the inference step. As seen
+in Table 6, the proposed method is outperforming the rest of
+the methods and yields better average throughput. Further-
+more, our method also yields lower variance with respect to
+change in the initial values of the actions, which is a signif-
+icant advantage. In general, our method proves to be very
+robust in this regard. The second table demonstrates the ef-
+fectiveness of the multi-objective optimization strategy. In
+this table, we have denoted three different variants of TAG-
+GCN which represent three optimization strategies, namely,
+multi-objective strategy using both throughput and through-
+put ratio models (TAG-GCN1), single objective strategies
+using throughput ratio (TAG-GCN2), and throughput mod-
+els (TAG-GCN3). As the results in the table show, the multi-
+objective strategy outperforms the single objective ones and
+therefore is superior.
+The Average Rank of Different Models (Fig. 4)
+As
+stated earlier the models for the experiment depicted in
+Fig. 4 are trained up to Oct. 29 and for the rest of days they
+are not updated. In Table 8, we compare the performance of
+the proposed learning-based models in terms of the achieved
+average rank. We separate the results for before and after
+Oct. 29 to show the generalization of different methods. As
+seen TAG-GCN has the best performance both before and
+after freezing the learning model.
+Different action initialization strategies
+In Fig. 12, we
+present an ablation study on the different initialization
+schemes on the first action exploration day. In addition to the
+random initialization, we use two more action initialization
+schemes including the expert rule and the negative slope.
+
+Model Name
+10-28
+10-29
+10-30
+10-31
+Experts Rule-Based
+.47 ± .0026
+.37 ± .0173
+.62 ± .0007
+.41 ± .0090
+MLP
+.68 ± .0073
+.76 ± .0080
+.87 ± .0102
+.66 ± .0110
+GCN
+.66 ± .0148
+.76 ± .0077
+.86 ± .0002
+.65 ± .0118
+AG-GCN
+.68 ± .0345
+.77 ± .0055
+.86 ± .0019
+.67 ± .0068
+TAG-GCN
+.71 ± .0136
+.78 ± .0005
+.93 ± .0074
+.73 ± .0069
+Table 6: The Average throughput divided by the throughput
+achieved from the best action designed by the simulator (in
+last four days) for different reward models.
+Model Name
+10-28
+10-29
+10-30
+10-31
+TAG-GCN1
+.71 ± .0136
+.78 ± .0005
+.93 ± .0074
+.73 ± .0069
+TAG-GCN2
+.66 ± .0259
+.75 ± .0212
+.85 ± .0131
+.65 ± .0060
+TAG-GCN3
+.63 ± .0199
+.71 ± .0094
+.74 ± .0089
+.61 ± .0170
+Table 7: The 95% confidence interval for the average
+throughput divided by the throughput achieved from the best
+action designed by the simulator (in last four days) for dif-
+ferent optimization methods using TAG-GCN model.
+Model Name
+Before Oct. 29
+After Oct. 29
+All Days
+MLP
+2.80
+3.67
+3.13
+GCN
+2.80
+3.33
+3.00
+AG-GCN
+3.20
+2.00
+2.75
+TAG-GCN
+1.20
+1.00
+1.13
+Table 8: Average rank of different models in terms of the
+achieved reward before and after freezing the learning mod-
+els
+2019-10-17 2019-10-18 2019-10-21 2019-10-22 2019-10-23 2019-10-24 2019-10-25 2019-10-28 2019-10-29 2019-10-30 2019-10-31
+Time
+1.00
+0.75
+0.50
+0.25
+0.00
+0.25
+0.50
+0.75
+1.00
+Reward
+TAG-GCN (random initialization)
+TAG-GCN (negative slope initialization)
+TAG-GCN (experts rule initialization)
+Figure 12: Performance comparison of the TAG-GCN model
+under different types of action initialization for Oct. 18.
+The expert rule initialization is given in (13). The negative
+slope procedure for the action initialization is defined based
+on the throughput ratio of the first day of each cell v, i.e., βv
+t1
+as
+av
+t1 =
+�2(βv
+t1 − ρ)φ
+ρ − τ
++ φ
+�
+,
+(14)
+where ρ = minv∈Nt1 (βv
+t1), and τ = maxv∈Nt1 (βv
+t1) and
+[·] is the rounding operation. For controlling the variation
+of action, the maximum range of change is defined as φ.
+We can observe that regardless of the type of initialization,
+the model can converge to almost the same performance in
+the final days, which indicates that even without strong prior
+information from the network, our design can still quickly
+converge to the desired model performance.
+
diff --git a/ZtE1T4oBgHgl3EQfwgWm/content/tmp_files/load_file.txt b/ZtE1T4oBgHgl3EQfwgWm/content/tmp_files/load_file.txt
new file mode 100644
index 0000000000000000000000000000000000000000..d1ce17b7414f837a9ecc2fc79cf49b97044f7be8
--- /dev/null
+++ b/ZtE1T4oBgHgl3EQfwgWm/content/tmp_files/load_file.txt
@@ -0,0 +1,1273 @@
+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf,len=1272
+page_content='Neighbor Auto-Grouping Graph Neural Networks for Handover Parameter  Configuration in Cellular Network  Mehrtash Mehrabi,1,2 Walid Masoudimansour,2 Yingxue Zhang,2 Jie Chuai,2 Zhitang Chen,2  Mark Coates,3 Jianye Hao,1, 4 Yanhui Geng2  1 Huawei Noah’s Ark Lab, 2 University of Alberta, 3 McGill University, 4 Tianjin University  {mehrtash.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content='mehrabi, walid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content='masoudimansour, yingxue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content='zhang, chuaijie, chenzhitang2,  haojianye, geng.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content='yanhui}@huawei.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content='com, mark.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content='coates@mcgill.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content='ca  Abstract  The mobile communication enabled by cellular networks is  the one of the main foundations of our modern society.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content=' Op-  timizing the performance of cellular networks and providing  massive connectivity with improved coverage and user ex-  perience has a considerable social and economic impact on  our daily life.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content=' This performance relies heavily on the config-  uration of the network parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content=' However, with the mas-  sive increase in both the size and complexity of cellular net-  works, network management, especially parameter configu-  ration, is becoming complicated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content=' The current practice, which  relies largely on experts’ prior knowledge, is not adequate  and will require lots of domain experts and high mainte-  nance costs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content=' In this work, we propose a learning-based frame-  work for handover parameter configuration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content=' The key chal-  lenge, in this case, is to tackle the complicated dependencies  between neighboring cells and jointly optimize the whole net-  work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content=' Our framework addresses this challenge in two ways.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content='  First, we introduce a novel approach to imitate how the net-  work responds to different network states and parameter val-  ues, called auto-grouping graph convolutional network (AG-  GCN).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content=' During the parameter configuration stage, instead of  solving the global optimization problem, we design a local  multi-objective optimization strategy where each cell con-  siders several local performance metrics to balance its own  performance and its neighbors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content=' We evaluate our proposed al-  gorithm via a simulator constructed using real network data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content='  We demonstrate that the handover parameters our model can  find, achieve better average network throughput compared to  those recommended by experts as well as alternative base-  lines, which can bring better network quality and stability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content=' It  has the potential to massively reduce costs arising from hu-  man expert intervention and maintenance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content='  1  Introduction  The rapid growth in the number of devices that need real  time, high quality connection to the internet (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content=', internet of  things (IoT) devices, health monitoring equipment, devices  used for online education and remote working, autonomous  vehicles, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content=') makes it essential to improve cellular network  performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content=' Unsatisfactory user experience and network  interruption have negative impacts in our modern society.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content='  Thus, improving the cellular network has both economic and  Copyright © 2023, Association for the Advancement of Artificial  Intelligence (www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content='aaai.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content='org).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content=' All rights reserved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content='  social impact towards achieving United Nations Sustainable  Development Goals (UNSDGs) (Weisenborn 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content=' World  Economic Forum 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content=' Moreover, it can highly contribute  to enhancing infrastructure, promoting sustainable indus-  trialization, fostering innovation, responsible consumption,  enabling sustainable cities and communities, and promot-  ing decent work and economic growth (Gohar and Nencioni  2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content=' Rao and Prasad 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content=' Siriwardhana et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content='  The performance of a cellular network relies heavily on  its parameter configurations and it is becoming more cru-  cial, as the number of mobile users continues to grow rapidly  (statista 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content=' These parameters govern access control,  handover, and resource management (Dahlman et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content=' 2013;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content='  Bhat et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content=' 2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content=' One of the factors that has a significant  impact on the quality of service (QoS) in such networks is  the handover parameter (Tekinay et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content=' 1991).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content=' We provide  more details concerning this parameter and its effects in the  supplementary materials, Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content='  Optimizing handover parameters is one of the most com-  mon approaches to guarantee minimum service delay or in-  terruption and improve coverage and throughput (Mu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content='  2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content=' However, with the massive increase in both the size  and complexity of cellular networks, parameter configura-  tion is becoming complicated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content=' The current practice, which  relies largely on experts’ prior knowledge, is inadequate, re-  quiring many domain experts and leading to high mainte-  nance costs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content='  One of the key challenges in the network parameter op-  timization problem is the complex spatial and temporal de-  pendencies in the cellular network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content=' Any employed algorithm  should be capable of tracking the non-stationary changes in  the environment, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content=', the fluctuations of user number, net-  work load, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content=' (Agiwal et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content=' 2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content=' Also, due to the diverse  characteristics of cells across the network, the best parame-  ter configuration for one cell may not be optimal for another  and parameter configuration of one cell not only affects its  own performance, but also affects its neighbors’ (Dahlman  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content=' 2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content=' Therefore, there are strong interactions between  neighboring cells which become extremely complicated in  heterogeneous network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content=' Consequently, developing an algo-  rithm that can adapt to the temporal dynamics and cell diver-  sity in real networks is essential for parameter configuration  (Jiang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content=' 2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content='  The current cellular network deployments are highly de-  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content='03412v1  [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content='NI]  29 Dec 2022  pendent on human designed rules or analytical models based  on domain knowledge and assumptions about the network  dynamics which is far from optimal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content=' They only consider  a limited number of network states (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content=', user distribution,  channel quality, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content=') and parameters, and cannot capture the  complex relationships between network states, parameter  configurations and network performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content=' Also, the assump-  tions of the network dynamics, based on which the rules/-  models are developed, are often simplified without consider-  ing the non-stationary changes in real environments, which  degrades their performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content=' Finally, these rules/models may  not be able to deal with the cell diversities in the network  which makes them sub-optimal (Imran et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content=' 2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content='  Recently, data-driven approaches based on machine learn-  ing (ML) have been extensively used for parameter con-  figuration and network management in cellular networks  (Yu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content=' 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content=' Tabrizi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content=' 2012;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content=' Riihijarvi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content=' 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content='  Chuai et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content=' 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content=' Ye et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content=' 2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content=' It has been shown that  the multi-layer perceptron (MLP) can be considered as a  universal function approximator (Goodfellow et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content=' 2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content='  Thus, in environments such as cellular networks where there  is lack of an accurate analytical model and the network is  highly dynamic, neural-network-based methods can be used  to achieve high-accuracy prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content=' ML models can uti-  lize high dimensional information and approximate complex  functions to fully describe the relationship between network  states, parameter configurations and network performance  metrics, which cannot be achieved by human experience.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content='  In order to address the above challenges, we investigate  two important questions: 1) Modeling: how to model the  spatial and temporal dependencies of the cellular network?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content='  2) Decision-making: how to choose the parameter values to  jointly optimize the overall performance of interconnected  and interacting cells?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content=' We first, propose a ML-based model to  precisely imitate the cellular network environment and then,  use it to configure the parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content='  We demonstrate that the handover parameters recom-  mended by our model can achieve better average network  throughput compared to the existing methods and our ap-  proach can massively reduce costs from human expert in-  tervention and maintenance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content=' It opens up the potential for  high-quality internet access to geographical areas that are  currently under-served by the cellular network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content=' Besides, this  framework can bring new possibilities for important appli-  cations to under-developed regions including online educa-  tion, health monitoring devices by improving their real time  connection (Attaran 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content='  Our main contributions are summarized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content='  We propose a novel method to model the impact from the  neighbors of each cell in a distinguishable way to capture  the complex spatial dependencies of the network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content='  We consider the changing dynamics of the network in our  reward model to better reflect the temporal dependencies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content='  We introduce a multi-objective optimization strategy  based on the model to consider several performance met-  rics and improve the overall network throughput, which  has the potential for high social impact applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content='  2  Background and Related Work  The adjustment of handover parameters helps to balance the  traffic load in the network and it can dramatically affect the  network throughput.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content=' During the handover process in cellular  networks, in order to guarantee an acceptable service qual-  ity, a user equipment (UE) must monitor the reference signal  received power (RSRP) of the serving cell (3GPP TS36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content='331  2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content=' As soon as the RSRP drops below a pre-defined  threshold (called A2-threshold), the UE starts to report mea-  surements to its serving cell and prepares for handover.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content=' In-  creasing the value of A2-threshold decreases the number of  UEs in the serving cell in which the handover is triggered,  and this spreads the serving cell’s load to its neighbors, re-  sulting in a significant change of throughput for the serving  cell and its neighbors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content=' While improving the load balance of  the network, this can have adverse effects on the network  performance since it forces frequent handovers which re-  quires a considerable amount of bandwidth for measurement  reporting and causes a drop in network throughput.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content=' Decreas-  ing the value of A2-threshold, on the other hand, may cause  a poor experience for edge UEs and lead to repeated connec-  tion loss due to weak signal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content=' In attempt to solve the problem  of optimization of the parameters of a wireless network, dif-  ferent techniques such as fuzzy systems, deep reinforcement  learning (DRL), and contextual bandit have been used in the  literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content=' (see Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content='2 for some details).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content='  The use of graph convolutoinal networks (GCNs) (Hamil-  ton et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content=' 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content=' Kipf et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content=' 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content=' Fan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content=' 2019) has also  yielded significantly well-designed models to predict the  network traffic and optimize the corresponding parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content='  For example, in (Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content=' 2020), the authors introduce a  novel handover strategy based on GCNs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content=' The handover pro-  cess is modeled as a directed graph by which the user tries to  predict its future signal strength.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content=' Other works such as (Zhao  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content=' 2020) introduce novel methods of network traffic pre-  diction combined with a greedy search or action configu-  ration method to optimize handover parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content=' However,  these works fail to consider the heterogeneous aspect of the  cellular networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content='  Despite being effective, none of the above-mentioned  methods uses the capacity of the neighbors’ information to  fully tailor the model to adapt to the spatial characteristics  of a cellular network, where the interaction is complex and  the network is heterogeneous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content=' Also, despite the fact that  these techniques consider some important measures of op-  timization, none of them approaches the problem at hand  by considering two of the most important measures simul-  taneously (especially from the users’ perspective): load bal-  ancing and throughput.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content=' In this article, we propose an effec-  tive and efficient framework that models the network as a  heterogeneous graph where we learn an implicit interaction  type for each neighboring cell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content=' Then, it incorporates the im-  pact of neighboring cells from each interaction group in a  unique way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content=' Moreover, in contrast to the available methods  in the literature, we exploit two important measures in the  network simultaneously, to configure the parameters effec-  tively: throughput and load balancing, which are directly re-  lated to the user experience in the network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content='  3  Problem Formulation  Let us consider a network with N cells, and form N clusters  each composed of one of the network cells as its center cell  along with its neighboring cells.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content=' As an example, we choose  the optimization of the A2-threshold to investigate the per-  formance of our algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content=' According to the 3GPP standard  (3GPP TS36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content='331 2016), an A2 event is triggered when the  received power at user u from cell n, P u,n, satisfies  P u,n + Hys < Thresh,  (1)  where Hys is the hysteresis parameter to avoid frequent han-  dovers and Thresh is the A2-threshold we are optimizing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content='  We consider an online optimization process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content=' In real prac-  tice, network operators are often conservative and only allow  a limited number of experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content=' During the optimization  period of L days, and the A2-threshold can be adjusted once  for each cell at the beginning of each day.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content=' For day t, let Dt  be the total bits transmitted by all the cells, and Tt be the to-  tal transmission time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content=' We would like to maximize the accu-  mulated network throughput of the optimization period, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content=',  max �L  t=1  Dt  Tt .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content=' Maximizing the overall network throughput  by jointly optimizing the A2-threshold of all cells is diffi-  cult.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content=' The problem becomes even more complicated as the  network size increases, which makes a centralized solution  not scalable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content=' The adjustment of the A2-threshold of one cell  only affects its local neighborhood and thus, we convert the  centralized problem into a local decision problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content=' That is,  each cell only examines its local performance metrics and  chooses its own parameter configuration value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content='  The adjustment of the A2-threshold affects the network  throughput via two means: better resource utilization by load  balancing, and improved cell throughput with less connec-  tion loss and measurement reporting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content=' Consequently, in order  to configure it, these two metrics must be considered in the  local decision problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content=' The throughput of cell i on day t is  highly dependent on its A2-threshold, formulated as ai  t, de-  noted as αi  t(ai  t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content=' The load balancing factor in the i-th cluster  with center cell i on day t with ai  t is defined as the ratio  of the center cell throughput to the average throughput of  its neighboring cells, denoted by βi  t(ai  t) and formulated as  βi  t(ai  t) = αi  t(ai  t)/¯αi  t, where ¯αi  t is the average throughput  of the neighbors of cell i with action ai  t and, denoting by  Nt(i) the set of all neighbors of cell i on day t, it can be  formulated as ¯αi  t =  1  |Nt(i)|  �  j∈Nt(i) αj  t(aj  t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content=' The through-  put ratio (rather than traffic/user ratio) is used since different  cells have different capacities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content=' This value approaches 1 when  loads of different cells match their capacities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content='  Our goal is to maximize the overall network throughput  by optimizing the two important network performance met-  rics, namely, throughput ratio βi  t(ai  t) and cell throughput  αi  t(ai  t) for each cell i ∈ [1, Nt], where Nt is the total number  of cells on day t, at the same time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content=' Therefore, we propose the  following optimization problem for tuning the A2-threshold  for cell i:  arg max  ai  t∈A  �  −  ���1 − βi  t(ai  t)  ��, αi  t(ai  t)  �  ,  (2)  where A is the set of all possible values for the A2-threshold  in the cellular network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content='  The challenge of solving the above problem lies in sev-  eral folds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content=' First, since the network performance function is  complex, dynamic and unknown, obtaining accurate βi  t(ai  t)  and αi  t(ai  t) is difficult.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content=' Instead, in this work, we adopt a  data-driven approach to learn reward models and estimate  the performance metrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content=' Second, in real-world cases, only  a limited experimental budget is allowed by network opera-  tors leading to insufficient diverse historical data (state, ac-  tion pairs) to train a data-driven learning model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content=' In our de-  sign, we use a data augmentation technique in the form of  neighbor cell augmentation to enrich the features from each  cell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content=' Third, the handover parameter configuration is affected  by adjacent cells.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content=' Thus, it is essential to model the informa-  tion coming from the adjacent cells to achieve accurate re-  ward modeling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content=' Lastly, optimizing one performance metric  greedily might hinder another, thus, how to jointly optimize  different performance metrics needs careful consideration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content='  4  Temporal Auto-Grouping GCN for  Reward Modeling  In order to better capture the dependency between each  cell and its neighboring cells, we first introduce our novel  method for neighboring cell feature aggregation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content=' Second, we  propose a temporal feature aggregation step with recurrent  neural networks (RNN) to model the temporal correlation  from the historical sequence of the network states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content=' Third, we  elaborate the overall training process, considering the im-  pact from the neighboring cells, the temporal correlation in  the network and the action we aim to optimize.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content='1  Spatial Feature Modeling  The handover parameters heavily impact the learning prob-  lem on the graph of the center cells as well as the neighbor-  ing cells, hence, we aim to capture the neighboring cells in-  formation during our modeling process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content=' Recently, message-  passing neural networks (MPNNs) in the form of graph neu-  ral networks (GNNs) have been introduced and showed to  be effective in modeling real world applications with struc-  tural information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content=' The dependencies in the dataset are mod-  eled using a graph (Hamilton et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content=' 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content=' Ying et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content=' 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content='  Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content=' 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content=' In each layer of a GNN, each node’s rep-  resentation includes the features from itself as well as the  features from its neighboring nodes (messages sent from the  neighborhood).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content=' We believe the GNN framework is suitable  for handling the dependencies between the center cell and  the neighboring cells in cellular networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content=' We present more  details on GNN and recent works on homogeneous and het-  erogeneous graphs in the Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content='  Graph-Based Cellular Network Modeling  We construct  a graph Gt=(Vt, Et, Xt) for day t, where each node v∈Vt  represents one cell and is associated with a feature vector  xv  t ∈ Rd (v-th column of Xt ∈ Rd×|Vt|), including the sta-  tistical properties of node v measured on day t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content=' The sta-  tistical properties could include several features such as the  antenna transmission power, physical resource block (PRB)  usage ratio, the amount of data traffic, and the transmission  bandwidth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content=' These features serve as the node attributes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content=' The  edge set Et encodes the interactions between cells based on  the handover events between pairs of cells.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content=' Based on his-  torical data, if any pair of cells has an average number of  handover events above a threshold τ, we assume an edge  between those two cells.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content=' The neighboring set for node v is  denoted as N g  t (v)={u|u ∈ Vt, (u, v) ∈ Et}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content='  Due to the heterogeneous nature of the cellular network,  the relationships between the neighboring cells can be com-  plex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content=' Concretely, there might be an implicit M latent rela-  tionship types R = {r1, r2, · · · , rM} that can be learned  to better handle the complex interactions in the cellular net-  works.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content=' Assuming each cell is represented by its states such  as PRB usage, traffic, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content=' in the network graph, we aim at  dividing the neighboring cells into different groups, each of  which will provide some information that is shared between  the neighbors in that group and help to better capture the  rich information from neighboring cells in a distinguishable  way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content=' Thus, inspired by the above motivation and a recent  work (Pei et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content=' 2020), we propose a novel GCN approach  called auto-grouping GCN (AG-GCN) to characterize this  special property of cellular networks when handling the in-  teractions between neighboring cells.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content=' In the following, we  elaborate upon the detailed steps to realize our design.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content='  Neighborhood Augmentation  In cellular network mod-  eling, since the experiment budget is limited, the historical  data (state-action pairs) is not diverse enough to train our  data-driven model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content=' Besides, since we construct the graph  based on the handover events, there are cells that have a very  limited number of neighboring cells.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content=' Thus, in our design, we  use a data augmentation technique in the form of neighbor  cell augmentation based on the similarity between cells in a  latent space, to enrich the features of each cell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content='  We define a feature transformation function f(·) : Rd →  Rl which maps the input node feature xv  t ∈ Rd to a la-  tent space yv  t = f(xv  t ) ∈ Rl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content=' In order to capture the long-  range dependencies and similarity in the cellular network,  we design an additional neighborhood in the latent repre-  sentation space based on Euclidean distance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content=' For each node  v ∈ Vt, we form the augmented neighborhood Nt(v) =  N g  t (v) ∪ N s  t (v), where N g  t (v) and N s  t (v) are the neigh-  bors of node v in the original graph and in the latent space,  respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content=' The neighbors in the latent space are selected  based on their Euclidean distance to the center cell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content=' The n  nearest nodes in the latent space are selected to create N s  t (v)  for cell v, where the number of nodes we select based on the  feature similarity is equal to the neighborhood size in the  original graph |N g  t (v)|=|N s  t (v)|=n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content=' The neighbor augmen-  tation module in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content=' 1 illustrates this process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content='  Neighborhood Auto-Grouping  Once we have obtained  the augmented neighborhood set, the neighbors in the  augmented neighborhood Nt(v) are divided into different  groups by a geometric operator γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content=' Consider node v and  its neighbor node u ∈ Nt(v).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content=' The relation between them  on day t is denoted as γ(yv  t , yu  t ) : (Rl, Rl) → R =  {r1, r2, · · · , rM}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content=' This grouping aims at combining neigh-  bors’ information in groups with similar inter-group fea-  tures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content=' For each group ri ∈ R, the neighborhood feature set  on day t is defined as N ri  t (v) = {u|u ∈ Nt(v), γ(yv  t , yu  t ) =  ri}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content=' The auto-grouping module in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content=' 1 demonstrates this  process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content=' Note that yellow neighbors (marked with *) are the  projected counterparts of the neighbors in the graph space,  while the green neighbors (marked with +) correspond to the  augmented neighbors from the latent space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content='  Conditional Message Passing  Since the order within  each neighbor group should not impact the output of the  representation, we apply a permutation invariant function  π(·) on the neighbors within each group (mean pooling  across each feature dimension) and aggregate them sepa-  rately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content=' 1 shows an example of the AG-GCN, where  l = 2 and |R| = 4 and the representation after the per-  mutation invariant function π(·) is shown by black dashed  arrows ended to nodes 1, 2, 3, and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content=' Then for each group  ri ∈ R, a non-linear transform is further applied as:  zv,ri  t  = σ  �  Wv,ri  t  π  �  {xu  t |u ∈ N ri  t (v)}  ��  ,  (3)  where Wv,ri  t  is a learnable weight matrix for the neighbors  in group ri of node v on day t, and σ(·) is a non-linear  function, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content=', tanh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content=' Then for each node we aim to aggre-  gate the transformed neighborhood features from their dif-  ferent groups of neighbors in a distinguishable way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content=' The  vectors zv,ri  t  for ri ∈ R are further aggregated as hv  t =  [zv,r1  t  ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content=' · · · ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content=' zv,rM  t  ], where [ ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content=' ] represents concatenation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content='2  Temporal Feature Modeling  Capturing the trend in the evolution of the states of each cell  within a day properly can benefit the prediction of the per-  formance metric for the following day.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content=' We propose to use  additional temporal features for each center cell to extract  the changing dynamic pattern of its states within each day to  further improve the reward model performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content=' We assume  the samples of the center cell v on day t can be divided into  K groups by their temporal order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content=' For all the samples in each  group k, we take the average network state for each group of  samples and denote it as xv  t,k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content=' We use an RNN layer to cap-  ture this temporal dependency of the features from different  groups by feeding all the network states as an input sequence  Pv  t =  �  xv  t,1  T ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content=' xv  t,2  T ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content=' · · · ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content=' xv  t,K  T �T ∈ RK×d, to obtain  cv  t = RNN(Pv  t , δ) ∈ Rd′,  (4)  where δ and d′ are the set of trainable parameters and the  output dimension of the RNN layer, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content='3  Overall Training Pipeline  The main purpose of the model is to estimate the real  network‘s response, and predict the throughput ratio and  throughput of the center cell for the next day based on the  observed network states in the current day.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content=' These perfor-  mance metrics are not only affected by the current day’s  states, but also highly correlated with the action we choose  to configure for the next day.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content=' Thus, we also consider the ac-  tions of the next day.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content=' Furthermore, the throughput ratio and  throughput of the next day are highly dependent on the pre-  vious performance metrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content=' Hence, we consider the current  throughput ratio, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content=', βv  t , in the prediction process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content='  To make the final prediction,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content=' the learned representation of  the neighborhood by the AG-GCN aggregation,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content=' the tempo-  ral features of the center cell,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content=' and the throughput ratio of the  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content='RNN  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content='Predict  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content='Throughout  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content='or  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content='Throughput  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content='Ratio  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content='Neighbor  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content='Augmentation Module  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content='Auto-Grouping  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content='Module  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content='Daily Average  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content='Graph Space  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content='Euclidean Space  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content='Embedding function  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content='Group 1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content='Group 2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content='Group 3  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content='Group 4  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content='1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content='4  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content='3  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content='2 +  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content='Neighbors from graph space  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content='Neighbors from Euclidean space  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content='TAG-GCN  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content='Neighbor Augmentation Module  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content='Auto-Grouping Module  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content='Capturing Spatial Information  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content='Capturing Temporal Information +  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content='+  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content='+  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content='+ Temporal Feature  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content='Modeling  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content='Neighbor  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content='Information  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content='Modeling  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content='Concatenation  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content='Concatenation  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content='Concatenation  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content='Figure 1: The flow of information from graph structure to final prediction,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content=' used to form the training pipeline of two models for  predicting the throughput ˆα and the throughput ratio ˆβ for Tin = {1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content=' 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content=' · · · ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content=' t − 1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content=' In this demo example, the auto-grouping  module constructs M = 4 groups of neighbors, where l = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content=' Empty groups are filled with the average of other groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content='  current day, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content=' βv  t , are concatenated to form the state vec-  tor of cell v as sv  t = Ψ(Wv  t ·  �  βv  t ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content=' cv  t ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content=' hv  t  �  ), where Wv  t is  a learnable weight matrix for node v on day t, and Ψ(·) is  a non-linear function, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content=', tanh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content=' Since the final representa-  tion should be sensitive to the chosen input action (of which  the decision making process will be elaborated in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content=' 5),  the throughput ratio and throughput of the next day for cell  v are formulated as the output of a non-linear transformation  Λ(·) function of state and action:  ˆβv  t+1 = Λ  �  Wv  β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content=' ([sv  t ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content=' av  t ])  �  ,  (5)  ˆαv  t+1 = Λ  �  Wv  α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content=' ([sv  t ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content=' av  t ])  �  ,  (6)  where Wv  β and Wv  α are trainable matrices of node v for  throughput ratio and throughput models, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content=' The  overall flow of data from the graph structure to the final pre-  diction is represented in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content=' Note that we train two sepa-  rate models for predicting the throughput and the throughput  ratio simultaneously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content='  In order to properly use the A2-threshold for the predic-  tion, we use the change in this parameter compared to the  previous day as the action av  t = A2v  t+1 −A2v  t , where A2v  t+1  and A2v  t are the A2-thresholds for cell v on day t + 1 and t,  respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content=' The reason for this design choice has twofold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content='  First, the original action space of A2 is large, but the range  of the change of action can be smaller by controlling the  adjustment steps, making it easier for the model to learn  and conduct the decision making step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content=' Besides, the delta ac-  tion directly reflects the change in the cell coverage/loads, so  they are more sensitive to the performance metrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content=' To form  the training objective,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content=' we consider data of T + 1 consecu-  tive days and form the pairs (t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content=' t + 1),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content=' t ∈ {1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content=' 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content=' · · · ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content=' T},' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content='  to predict the throughput ratio and throughput of the center  cell in day T + 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content=' trained by minimizing the following loss  functions respectively:  1  T  T  �  t=1  1  Nt  Nt  �  v=1  (ˆβv  t+1 − βv  t+1)2 + λ1||Θ1||2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content='  (7)  1  T  T  �  t=1  1  Nt  Nt  �  v=1  (ˆαv  t+1 − αv  t+1)2 + λ2||Θ2||2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content='  (8)  where λ1 and λ2 are the hyperparameters chosen for reg-  ularization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content=' Θ1 and Θ2 represent all the trainable parame-  ters in the models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content=' The trained reward model is now able to  mimic the real network and predict both throughput ratio and  throughput of each center cell for the coming day and can be  used to check the impact of actions towards the performance  metrics we are considering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content='  5  Action Configuration  As discussed in the earlier sections the main objectives to  consider in the action configuration process are load balanc-  ing, identified by the throughput ratio, and the cell through-  put.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content=' Hence, the best action for cell v on day t, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content=' av  t ∈ A, is  the one that optimizes the problem in (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content=' In general, when  dealing with a multi-objective problem, different objectives  are often conflicting, and we may not be able to optimize  them simultaneously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content=' One common way to tackle this prob-  lem is to give different objectives weights and optimize the  weighted objective value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content=' However, in our scenario, it is dif-  ficult to determine the weights and different clusters may  require cluster-specific weights.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content=' Here we break the problem  in (2) into two sub-problems, and solve them sequentially.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content='  We first optimize the action with respect to the predicted  throughput ratio, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content=', ˆβv  t+1(av  t ) for cell v on day t, where  av  t ∈ A, and then optimize the throughput ˆαv  t+1(av  t ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content='  Specifically, the throughput ratio is optimized and we find  the set of best c values for av  t , denoted Av  c, such that  min  av  t ∈Av  c  −  ���1 − ˆβv  t+1(av  t )  �� ≥ max  av  t ∈A−Avc  −  ���1 − ˆβv  t+1(av  t )  ��.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content=' (9)  Then, our goal is to achieve the maximum possible through-  put for cell v on day t and this is through  ˆav  t = arg max  av  t ∈Av  c  ˆαv  t+1(av  t ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content='  (10)  ˆav  t is then the final recommended action for cell v on day t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content='  This procedure for all the Nt cells of the network on day t is  presented in Algorithm 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content='  6  Experimental Results  The experiments are conducted on a large-scale cellular net-  work simulator constructed from real-world data which pre-  sented in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content=' We use principal component analysis  γ(yv  t , yu  t )  yv  t [0] > yu  t [0]  yv  t [0] ≤ yu  t [0]  yv  t [1] ≤ yu  t [1]  2  1  yv  t [1] > yu  t [1]  3  4  Table 1: The relationship operator γ  Algorithm 1: TAG-GCN for Action Configuration  Input: Pv  t and xu  t where, u ∈ Nt(v) ∀v ∈ [1, Nt]  Output: av  t for ∀v ∈ [1, Nt]  1: Let v = 1 and ∀j ∈ [1, Nt] set Aj  c = ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content='  2: while v ≤ Nt do  3:  Feed the TAG-GCN model with Pv  t and xu  t , for all  neighbor u ∈ Nt(v).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content='  4:  Freeze all inputs of TAG-GCN except actions and  predict the performance metrics ˆαv  t+1(·) and ˆβv  t+1(·)  for the input actions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content='  5:  for |Av  c| ≤ ν do  6:  x = arg maxa∈{A−Avc} −  ���1 − ˆβv  t+1(a)  ��  7:  Av  c = Av  c ∪ {x}  8:  end for  9:  av  t = arg maxa∈Avc ˆαv  t+1(a)  10:  v = v + 1  11: end while  12: return av  t for ∀v ∈ [1, Nt]  (PCA) as the mapping function f(·), as defined in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content=' 4, for  obtaining the latent representation in the AG-GCN step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content=' It  transforms the original feature into a 2-dimensional space to  perform the neighborhood augmentation and the neighbors  group assignment process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content=' After this transformation, the re-  lationship operator γ for the auto-grouping assigns a group  to each subset of points in each quadrant of this two dimen-  sional space presented in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content=' The permutation-invariant  function π applied on each group of neighbors is average in  our experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content='  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content='1  Datasets  To perform our experiments and evaluate the proposed  model, two datasets are used in this study (see Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content='2 for  more details):  Dataset-A:  A real metropolitan cellular network contain-  ing around 1500 cells sampled hourly and collected from  Oct.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content=' 17 to Oct.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content=' 31, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content=' Each data sample contains infor-  mation such as the cell ID, sample time, configuration of cell  parameters, and measurements of the cell states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content='  Dataset-B:  Also a real metropolitan cellular network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content=' The  network contains 1459 cells, and the data is collected from  Sep.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content=' 1 to Sep.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content=' 29, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content=' Each data sample contains similar  information as above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content='  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content='2  Reward Model Accuracy Evaluation  Dataset Generation  In order to evaluate the prediction ac-  curacy of our model, we use a simulator to modify Dataset-A  with a random policy to diversify our network configuration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content='  On each day, the A2-threshold for each cell is randomly se-  lected around the default action -100 dBm within the range  [−105, −95].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content=' This approach provides us a fix data buffer  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content='with diverse action dataset to train all models and have a fair  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content='2019-10-22  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content='2019-10-23  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content='2019-10-24  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content='2019-10-25  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content='2019-10-28  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content='2019-10-29  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content='Time  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content='6  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content='7  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content='8  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content='9  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content='10  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content='11  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content='Throughput MSE  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content='MLP  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content='GCN  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content='AG-GCN  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content='TAG-GCN  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content='(a)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content='2021-09-05 2021-09-06 2021-09-07 2021-09-08 2021-09-09 2021-09-10 2021-09-11 2021-09-12  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content='Time  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content='6  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content='7  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content='8  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content='9  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content='10  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content='11  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content='Throughput MSE  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content='MLP  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content='GCN  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content='AG-GCN  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content='TAG-GCN  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content='(b)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content='Figure 2: Achieved MSE of the throughput for test data of  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content='(a) Dataset-A and (b) Dataset-B for different methods  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content='comparison of their accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content=' For Dataset-B, there exists a  reasonable amount of the diversity in the handover parame-  ter configuration, thus we directly use the raw dataset from  the live network to perform the training and evaluation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content='  Training Process and Metrics  As samples generated  hourly, we aggregate them within each day as described in  Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content=' To evaluate the model accuracy in predicting cell  throughput and throughput ratio, we train the model with  the generated pairs {(1, 2), · · · , (t − 1, t)} for t = 9 and 12  days for Dataset-A and B, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content=' At each day t > 2,  data pairs {(1, 2), · · · , (t−2, t−1)} are used as training and  validation sets, and (t − 1, t) serves as testing set for eval-  uation across different models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content=' We report the mean square  error (MSE) to measure the reward model performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content='  Comparison with Benchmark Models  It is important to  note that, the social impact of this work has not been ad-  dressed by ML approaches the same way as we propose.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content='  Due to the uniqueness of our problem, existing solutions  for optimizing handover parameters are either not appropri-  ate to solve it or there is no apparent way to directly adapt  them to our problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content=' For instance, traditional handover op-  timization methods rely on designing fuzzy rules based on  different measures of QoS in the network (Vasu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content=' 2012),  however, designing proper rules is complex and cannot han-  dle the change in highly dynamic systems well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content=' Instead, we  hope to use a data-driven approach, among which the (deep)  RL method gains the most attention (Cao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content=' 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content=' Wang  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content=' 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content=' However, in this type of problems, the network  provider only allows limited exploration of the parameter  values (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content=', allows changing the A2 value once a day) to en-  sure the stability of the network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content=' Thus, we only have limited  days for exploring the best action.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content=' RL models, nevertheless,  usually need longer episodes to optimize the accumulated  return.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content=' Despite this fact, we made some preliminary attempt,  presented in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content='3, to adapt the RL paradigm from the lit-  erature into our problem which did not show any advantage  over our simpler design.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content='  In order to show the effectiveness of our proposed re-  ward model, we compare it with alternative designs for the  prediction model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content=' It should be mentioned that all of these  models are our contribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content=' In the Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content='4, we summarize  our benchmarks and the properties of each model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content=' The first  model is MLP, where we only use the features of the cen-  ter cells and ignore the neighboring cells’ features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content=' In GCN  model, we follow the typical GCN formulation (Hamilton  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content=' 2017) and process the network as a homogeneous  Model  Action  Configuration  Simulator  𝐱𝑡  𝑣, {𝐱𝑡  𝑢|𝑢 ∈ 𝒩𝑡(𝑣)}  𝐱𝑡+1  𝑣  , {𝐱𝑡+1  𝑢  |𝑢 ∈ 𝒩𝑡+1(𝑣)}  𝑎𝑡  𝑣  𝛽𝑡+1  𝑣  ,  𝛼𝑡+1  𝑣  Figure 3: The process of action recommendation by the  trained model and the simulator  graph where the neighbor information is aggregated jointly  without distinction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content=' The AG-GCN model ignores the tem-  poral dependencies of the data which we consider in TAG-  GCN model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content=' 2, we compare the prediction accuracy  of these models for throughput in Dataset-A and B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content=' We ob-  serve on average the best accuracy for the test set is achieved  by AG-GCN and TAG-GCN, with TAG-GCN performing  marginally better on the average rank metric across the eval-  uation days, indicating that our neighbor aggregation and  temporal features extraction have a considerable impact on  the reward modeling for cellular networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content=' The same results  also achieved for the throughput ratio model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content=' The same test  is performed for throughput ratio and included in the sup-  plementary materials in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content=' 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content='  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content='3  Overall Parameter Optimization  Performance  The Action Recommendation Process  In the following  experiments we use the presented models to recommend the  actions for Dataset-A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content=' The actions in day 1, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content=', Oct.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content=' 17,  has been set to the default action which is -100 dBm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content=' Unless  otherwise stated, the action for the second day, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content=', Oct.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content=' 18,  is initialized by a set of random actions around the default  action in the range of [−105, −95].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content=' The model is trained it-  eratively on each day and used to recommend actions for the  next day.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content=' The process is depicted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content=' 3, where states of  the cells on day t are given to the trained model to predict  performance metrics of the network on day t + 1 and the  action av  t is adjusted for each cell based on the predictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content='  Finally, the network states and performance measurements  for day t + 1 are computed according to the new selected  action by the cellular network simulator and used for model  training and action recommendation in the following day.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content='  Baseline Performance Bounds  In addition to the result  achieved by the actions recommended by the models, we use  three baseline performance bounds achieved by the default  A2-threshold, the expert rule, and the optimal actions of the  simulator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content=' As stated before the default A2-threshold value  is −100 dBm and this is used as the lower bound in the fol-  lowing experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content=' The optimal actions in the simulator are  obtained by brute-force search and it introduces the upper  performance bound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content=' The expert rule-based method provided  by experienced network operator is a simple rule presented  in the Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content=' The performance achieved by the expert rule  is better than the default action.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content=' We hope to use our proposed  learning based framework to further fill the gap with the re-  ward achieved by optimal actions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content='  Results  In the following experiments, we compare the per-  formance of the network under the actions recommended  2019-10-17 2019-10-18 2019-10-21 2019-10-22 2019-10-23 2019-10-24 2019-10-25 2019-10-28 2019-10-29 2019-10-30 2019-10-31  Time  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content='00  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content='75  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content='50  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content='25  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content='00  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content='25  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content='50  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content='75  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content='00  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content='25  Reward  MLP  GCN  AG-GCN  TAG-GCN  Best Action  Experts Actions  Default Action  Figure 4: Performance comparison of different models along  with optimal action curve initialized with random actions  by different models in terms of the network throughput as  defined in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content=' We plot the trajectory of the throughput  difference to the default A2-threshold baseline (dash black  line) in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content=' We repeat all the experiments 20 times for  all models, where each run uses the same set of random ac-  tions on the first action exploration day (Oct.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content=' 18) for all the  models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content=' We also show the performance achieved through the  expert rule action recommendation, default action, and the  optimal actions of the simulator (random actions are also  used on Oct.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content=' 18 for the curve of the optimal action).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content=' The  quantitative results for Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content=' 4 are summarized in Table 6  in the supplementary materials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content=' TAG-GCN can achieve bet-  ter average throughput in the final days which indicates the  importance of our auto-grouping GCN design to tailor the  heterogeneous property of the cellular networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content=' Besides, as  expected, all the learning-based models can beat the expert  rule algorithm which is highly dependent on human experi-  ence and is unable to recover from the performance degra-  dation due to bad random initialization on the first day.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content=' Fur-  thermore, to show the effectiveness of our proposed model  in terms of load balancing and enhancing cluster through-  put ratio, we illustrate the progress of this ratio achieved by  TAG-GCN for some selected severely unbalanced cells in  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content=' As it can be seen, the throughput ratio of the clusters  form a trajectory that converges to 1 which is the ideal target  value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content=' More ablation studies are presented in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content='  Based on the above experiments, our proposed ML-based  solutions can improve the network performance and opti-  mize the handover process compared to the conventional  methods such as using the default action or human experts  rule-based methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content=' Moreover, the automation of the param-  eter optimization process achieved by our ML-based solu-  tions reduces the domain expert’s intervention and, hence,  the management cost of network operators and improves the  maintenance efficiency of cellular networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content=' Consequently,  the proposed solutions open up the possibilities to provide  reliable and high-quality network access even to geograph-  ical areas that are currently underserved by the cellular net-  work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content=' This can bring exciting new opportunities to these  regions such as remote education, remote working, health  monitoring, video streaming, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content='  7  Conclusion  In this paper, we study the handover parameter configura-  tion problem in cellular networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content=' We propose a reward pre-  diction model to accurately imitate the cellular network and  estimate the performance metrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content=' Our proposed model, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content=',  2019-10-17 2019-10-18 2019-10-21 2019-10-22 2019-10-23 2019-10-24 2019-10-25 2019-10-28 2019-10-29 2019-10-30 2019-10-31  Time  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content='8  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content='0  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content='2  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content='4  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content='6  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content='8  Throughput Ratio  CELL_CELLNAME  cell132  cell2575  cell1304  cell3141  cell2955  cell2790  cell2250  cell3872  cell1015  cell3140  Figure 5: Trend of the throughput ratio for sample clusters  TAG-GCN, investigates the impact of the adjacent cells and  differentiate their impact on the center cell of each clus-  ter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content=' We also consider the network changing dynamics in our  model to learn the temporal dependencies in the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content=' Based  on the reward model, a novel multi-objective parameter con-  figuration strategy is proposed to perform the optimization  for each cluster and balance the performance metrics in each  neighborhood.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content=' The conducted simulations shows the superi-  ority of TAG-GCN which has a huge potential social impact  by improving the cellular network parameters and providing  massive connectivity and high coverage with a balanced traf-  fic across the network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content=' Hence, this can help the widespread  adaptation of new technologies to benefit many sectors such  as health and education.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content='  References  3GPP TS36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
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+page_content='Radio Resource Control (RRC);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content=' Protocol  specification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
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+page_content='  IEEE Communications  Surveys & Tutorials, 18(3): 1617–1655.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content='  Attaran, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
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+page_content=' The impact of 5G on the evolution of  intelligent automation and industry digitization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content=' Journal of  Ambient Intelligence and Humanized Computing, 1–17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content='  Bhat, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
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+page_content='  IEEE Communications Magazine, 50(2): 104–114.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content='  Cao, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
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+page_content=' AIF:An Artificial Intelligence Frame-  work for Smart Wireless Network Management.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content=' IEEE Com-  munications Letters, 22(2): 400–403.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content='  Chuai, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
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+page_content=' A Collaborative Learning Based Ap-  proach for Parameter Configuration of Cellular Networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content=' In  IEEE Int.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content=' Conf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content=' on Computer Communications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content='  Dahlman, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
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+page_content=' 4G: LTE/LTE-advanced for mobile  broadband.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content=' Academic press.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content='  Fan, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content=' et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
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+page_content=' Graph Neural Networks for Social Rec-  ommendation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content=' In Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content=' Int.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content=' Conf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content=' World Wide Web.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content='  Gilmer, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content=' Schoenholz, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
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+page_content=' Riley, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
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+page_content=' Vinyals, O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content=' and  Dahl, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content=' 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content='  Neural message passing for quantum  chemistry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content=' In International conference on machine learn-  ing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content='  Gohar, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content=' and Nencioni, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content=' 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content=' The role of 5G technolo-  gies in a smart city: The case for intelligent transportation  system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content=' Sustainability, 13(9): 5188.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content='  Goodfellow, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content=' et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
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+page_content=' Deep learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content=' MIT press.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content='  Hamilton, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content=' et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content=' 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content='  Inductive Representation  Learning on Large Graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content=' In Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content=' Adv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content=' Neural Inf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content=' Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content='  Systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content='  Imran, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content=' et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content=' 2014.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content=' Challenges in 5G: how to empower  SON with big data for enabling 5G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content=' IEEE Network, 28(6):  27–33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content='  Jiang, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content=' et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content=' 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content=' Machine learning paradigms for next-  generation wireless networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content=' IEEE Wireless Communica-  tions, 24(2): 98–105.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content='  Jiang, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content=' et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content=' 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content='  A decoupled learning strategy for  massive access optimization in cellular iot networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content=' IEEE  Journal on Selected Areas in Communications, 39(3): 668–  685.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content='  Kipf, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content=' et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
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+page_content=' In Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
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+page_content=' Learning  Representations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
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+page_content=' Intelligent spectrum management  based on transfer actor-critic learning for rateless transmis-  sions in cognitive radio networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content='  IEEE Transactions on  Mobile Computing, 17(5): 1204–1215.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
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+page_content=' IEEE transactions on wireless communications,  13(4): 2000–2011.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
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+page_content=' Barco, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
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+page_content=' Fortes, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
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+page_content=' Conflict resolution  between load balancing and handover optimization in LTE  networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content='  IEEE Communications Letters, 18(10): 1795–  1798.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content='  Pei, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content=' et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
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+page_content=' Geom-GCN: Geometric Graph Convo-  lutional Networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content=' In International Conference on Learning  Representations (ICLR).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
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+page_content=' Impact of 5G technologies  on industry 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
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+page_content=' Wireless personal communications, 100(1):  145–159.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
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+page_content=' IEEE Computational  Intelligence Magazine, 13(1): 51–60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
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+page_content=' Springer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
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+page_content=' The role of 5G for digital healthcare against COVID-  19 pandemic: Opportunities and challenges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content=' ICT Express,  7(2): 244–252.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
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+page_content='  Number  of  smartphone  users  worldwide  from  2016  to  2027.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
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+page_content=' In 2012  IEEE International Conference on Communications (ICC),  3217–3222.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
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+page_content=' IEEE Internet of  Things Journal, 6(2): 2061–2073.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
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+page_content=' United Nations Sustainable Develop-  ment Goals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content='  World Economic Forum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
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+page_content=' The Impact of 5G: Creating  New Value across Industries and Society.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
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+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content=' et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content=' 2013.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content=' User Association for Load Balancing in  Heterogeneous Cellular Networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content=' IEEE Trans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content=' on Wireless  Communications, 12(6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content='  Ying, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content=' et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content=' 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content='  Graph Convolutional Neural Net-  works for Web-Scale Recommender Systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content=' In Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content=' ACM  SIGKDD Int.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content=' Conf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content=' Knowledge Discovery and Data Mining.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content='  Yu, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content=' et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content=' 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content='  Modeling User Activity Patterns for  Next-Place Prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content=' IEEE Systems Journal, 11(2): 1060–  1071.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content='  Zhang, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content=' et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content=' 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content=' Heterogeneous graph neural network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content='  In Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content=' ACM SIGKDD Int.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content=' Conf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content=' Knowledge Discovery and  Data Mining.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content='  Zhang, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content=' et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content=' 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content='  An AI-Based Optimization of  Handover Strategy in Non-Terrestrial Networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content='  In 2020  ITU Kaleidoscope: Industry-Driven Digital Transformation  (ITU K), 1–6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content='  Zhao, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content=' et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content=' 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content=' Cellular Network Traffic Prediction In-  corporating Handover: A Graph Convolutional Approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content=' In  2020 17th Annual IEEE International Conference on Sens-  ing, Communication, and Networking (SECON), 1–9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content='  Supplementary Materials  A  Extended Background and Related Work  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content='1  Handover in Cellular Networks  A cellular network is formed by many cells distributed over  land areas, each served by at least one fixed-location base  station (BS) providing wireless links with a wide geographic  area coverage to support many users.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content=' A cluster in a cellular  network (consisting of a cell and its neighboring cells) is  shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content=' 6 where, user u is crossing one cell’s cov-  erage area and monitors the received power from cell n to  check the handover criteria.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content=' The cell BS is responsible for  monitoring the strength of the signals received by served  users, which can degrade when a user travels from one cell  to another.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content=' The BS should trigger a handover process at the  proper moment to avoid any service interruption and trans-  fer the user to another cell BS that is receiving the strongest  signals (Bhat et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content=' 2012;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content=' Tekinay et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content=' 1991).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content='  There are multiple parameters that impact handover in a  network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content=' These parameters generally are grouped in three  categories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content=' RSRP and RSRQ parameters are indicators of the  signal strength and quality of the serving station, respec-  tively, Hysteresis parameters act as a tolerance margin to  avoid the Ping Pong (PP) effect, and Time To Trigger (TTT)  parameters are set such that short term violations of han-  dover conditions are ignored to avoid the PP effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content=' Also,  different measures such as handover failure (HOF) rate, han-  dover frequency, Ping-Pong (PP) rate, network throughput,  and load balancing have been used in the literature to assess  the effectiveness of the proposed algorithms for optimizing  handover parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content=' When performing optimization in a  real network, some of these measures may conflict.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content=' There-  fore, it is crucial to consider a combination of these quanti-  ties to find a solution that satisfies different requirements of  the network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content='  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content='2  Related Work  Different methods have been used in the literature for solv-  ing the problem of optimization of handover parameters in  a wireless network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content=' Fuzzy system handover algorithms, for  example, have been used in (Vasu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content=' 2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content=' Such tech-  niques, however, are not scalable despite being accurate and  stable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content=' Deep reinforcement learning (DRL) is another tech-  nique that has been used to solve the handover optimization  problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content=' In (Cao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content=' 2018), the authors propose a frame-  work based on DRL where actions are flexible and can be  chosen by the user, and the objective of the optimization  is the throughput and the handover count.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content=' In (Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content='  2018), the authors propose a framework in which the UEs  first are clustered based on their usage pattern and then the  handover process is optimized in each cluster by a DRL  method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content=' Another newly introduced approach to the problem  of cellular network parameter optimization is the Contex-  tual Bandit model proposed in (Chuai et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content=' 2019)in which  the throughput of the cells is used as a performance metric  for optimizing some of the cell parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content='  The above mentioned methods, despite being effective, do  not exploit the information from the neighboring cells which  <A2-threshold  <A2-threshold  <A2-threshold  <A2-threshold  <A2-threshold  <A2-threshold  <A2-threshold  Center Cell 𝑛  User 𝑢  𝑃𝑢,𝑛  Figure 6: A typical structure of a cluster in the cellular net-  work serving different types of users.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content='  Model each sub-network as a graph where  nodes are cells and each edge represents heavy  handover events between this pair of cells  𝒢𝑡 = (𝒱𝑡, ℰ𝑡)  Graph Construction  Typical GCN Aggregation  All neighbors are using the same  transformation function to perform  the feature mapping  Figure 7: Commonly used GNN design.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content='  can be used to develop a more effective model of the net-  work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content=' Moreover, two of the most important measures from  the user’s standpoint are throughput and load balancing,  none of which are considered in the above models simulta-  neously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content=' Thus, in the current work, we propose a technique  which remedies these shortcomings by utilizing the neigh-  bors’ information, and optimizing the parameters of the net-  work using two significantly important objectives, namely,  load balancing and throughput.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content='  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content='3  Graph Neural Networks for Cellular  Network  Successful applications of GNNs mostly rely on a homoge-  neous graph structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content=' For example, in social networks or  recommender systems, each edge represents a positive cor-  relation between the adjacent nodes (Kipf et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content=' 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content=' Fan  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content=' 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content=' The MPNN framework (Gilmer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content=' 2017) ap-  plies the same transformation function to every neighbor and  aggregates across them to obtain the neighbor representa-  tion, shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content=' There is a rich literature that studies  how to process a heterogeneous graph with multiple relation  types (Schlichtkrull et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content=' 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content=' Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content=' 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content=' They  are established on the prior knowledge of the given relation  type between each pair of nodes and then use a meta-path  design to distinguish the information coming from different  types of neighbors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content=' Such an approach is not applicable to our  problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content='  B  Experimental Details, Results, and  Ablations  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content='1  Simulator Construction  The experiments are run through a proprietary cellular net-  work simulator which uses statistical models to simulate the  Figure 8: The internal workflow of the simulator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content='  network response to any change in the parameter values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content='  This simulator is designed based on the network correspond-  ing to Dataset-A and uses the states of the network cells  along with their action to simulate the throughput and the  traffic of the cells.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content='  Dataset-A is collected under the default network configu-  ration and the value of A2-threshold is not changed during  the collection period.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content=' To evaluate our model and the action  configuration strategy, the simulator needs to simulate the  network performance given arbitrary A2-threshold values of  the cells.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content=' More precisely, let xi  t represents cell i’s states at  day t under the default A2-threshold configuration, and A2i  t  be the configured A2-threshold value for cell i at day t, the  simulator will output  ˜x0  t, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content=', ˜xNt  t , r0  t , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content=', rNt  t  = g(x0  t, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content=', xNt  t , A20  t, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content=', A2Nt  t ),  (11)  where ri  t is cell i’s throughput, and ˜xi  t is the observed states  of cell i under the new configurations at t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content='  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content=' 8 gives a simple sketch on the internal workflow of  the simulator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content=' During simulation, when A20  t, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content=', A2Nt  t  are  configured at day t, the cell states x0  t, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content=', xNt  t  are read from  the collected data, and inputted to the simulator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content=' The cell  coverage Ci for each i is first computed by a built-in func-  tion that considers cell i’s frequency, bandwidth and A2i  t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content='  Then cell i’s traffic load (including the number of users and  the amount of data bits for transmission) at day t is redis-  tributed among itself and its neighbors based on the change  of Ci and Cj with j ∈ Nt(i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content=' After each cell’s traffic load is  updated, the throughput ri  t is predicted based on cell i’s traf-  fic load.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content=' The load-throughput prediction model for each cell  is pre-trained from the collected historical data and stored  when the simulator is initialized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content=' The value ri  t is further ad-  justed by a factor that considers the throughput loss due to  measurement reporting or connection loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content=' Finally, the sim-  ulator outputs r0  t , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content=', rNt  t .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content=' As each cell’s traffic load (part of  the cell states) is modified, the updated states ˜x0  t, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content=', ˜xNt  t  are  also outputted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content='  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content='2  Dataset Samples  Dataset-A and B contain wide range of cell states (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content=', the  number of total users within the cell, the number of active  users, the cell average CQI, the cell traffic load, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content=') and  performance indicators (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content=', the average cell throughput,  the edge user throughput, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content=').' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content=' The neighbor relation infor-  mation between cells in both datasets are collected and the  hourly average handover counts between neighbor cells are  recorded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content=' The weekend dates are excluded from Dataset-A  since these days are not considered as rush hour usage for  this region of cellular networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content="  We present the values of four different features of three  0  Nt  N  Nt0  N  Estimate each cell  Update each cell  Predict each cell  i's coverage ci  i's traffic load  i's throughput口  A2-threshold  Center Cell n  A2." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content='threshold  thres  A2-threshold  <A2-threshold  A2-threshold  UseruTime  Cell Throughput  AVG Traffic  Cell Bandwidth  AVG CQI  00 - 01  17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
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+page_content='32  Table 2: Samples from Dataset-A and B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content='  random cells in Dataset-A in Tables 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content=' Note that the repre-  sented values are collected in the first day.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content=' As seen the range  of variation across different cells and different hours for each  single cell is too high and that is the reason we consider the  modules to handle spatial and temporal dependencies in our  model to better imitate the real-world environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content=' We have  almost the same pattern in Dataset-B as well for these pre-  sented features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content='  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content='3  Reinforcement Learning Modeling  As we mentioned in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content=' 6, for problems similar to ours  which require following a policy to choose an action to inter-  act with an environment, reinforcement learning (RL) mod-  eling is a good candidate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content=' Among all the RL methods, the  actor-critic (AC) method is a potential match to our prob-  lem due to its continuous action space and modeling flexibil-  ity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content=' AC methods have been widely used in cellular network  problems and for different tasks such as power allocation  and traffic forecasting (Li et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content=' 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content=' Koushik et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content=' 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content='  Jiang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content=' 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content=' Wei et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content=' 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content=' Hence, as an additional  benchmark we compare our proposed algorithm with a ba-  sic AC algorithm we implemented to adapt to our problem  where the actor model recommends actions in order to max-  imize the reward predicted by the critic model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content='  We use TAG-GCN as the critic model to predict the re-  ward which can be either throughput ratio or throughput to  train the actor model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content=' For the actor model, we use the same  modules presented for TAG-GCN to do the neighbor auto-  grouping and capture the temporal dependencies in the data  2019-10-17 2019-10-18 2019-10-21 2019-10-22 2019-10-23 2019-10-24 2019-10-25 2019-10-28 2019-10-29 2019-10-30 2019-10-31  Time  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content='00  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content='75  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content='50  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content='25  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content='00  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content='25  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content='50  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content='75  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content='00  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content='25  Reward  TAG-GCN  Actor-Critic  Figure 9: Performance comparison between our proposed  design and actor-critic design for action exploration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content='  Model List  Model Name  Graph  Heterogeneous  Graph  Daily Temporal  Features  MLP  \x17  \x17  \x17  GCN  \x13  \x17  \x17  AG-GCN  \x13  \x13  \x17  TAG-GCN  \x13  \x13  \x13  Table 3: Properties of Different Models  and train it with the following loss function  Loss(actor) = 1  T  T  �  t=1  1  Nt  Nt  �  v=1  ���1 − ˆβv  t (ˆav  t )  �� + λ3||Θ3||2,  (12)  where ˆav  t is the recommended action by the actor model and  ˆβv  t (ˆav  t ) is the predicted reward by the critic model using the  recommended action.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content=' λ3 is the hyperparameter chosen for  regularization and Θ3 represents all the trainable parameters  in the actor model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content='  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content=' 9 shows that TAG-GCN outperforms this basic RL  model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content=' The proposed AC model is a very preliminary one  and obviously it can be improved even more which is out of  the scope of this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content=' However, this potential improvement  comes with the cost of increasing complexity and requires  a highly diverse data and more interactions with the envi-  ronment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content=' The RL approach generally requires frequent and  plenty of interactions with the environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content=' However, in our  case, the action is configured once a day, and the total explo-  ration duration only consists of small number of days, and  is far less than what is generally required by the RL mod-  els.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content=' Furthermore, RL models usually consider accumulated  returns, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content=', impact of the current action on the future per-  formance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content=' However, in our scenario, the current action only  affects the loads distribution of the current hour, and does  not affect the future performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content=' Therefore the standard RL  approach does not apply here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content='  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content='4  Alternative reward model comparison  We summarize the alternative designs for the reward model  we consider in our paper in Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content='  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content='5  Additional Model Evaluation Metrics (avg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content='  Rank)  We present the average rank of different models across dif-  ferent evaluation days in terms of the achieved test MSE for  both throughput ratio and cell throughput models in Dataset-  A and B, in Table 4 and 5, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content=' As seen, considering  Model Name  Ratio Model  Throughput Model  Average  MLP  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content='83  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content='83  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content='83  GCN  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content='17  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content='00  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content='08  AG-GCN  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content='67  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content='67  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content='67  TAG-GCN  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content='33  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content='50  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content='42  Table 4: Average rank of the learning models in terms of the  achieved test MSE across the evaluation days for Dataset-A  Model Name  Ratio Model  Throughput Model  Average  MLP  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content='50  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content='83  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content='67  GCN  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content='00  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content='00  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content='00  AG-GCN  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content='00  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content='50  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content='75  TAG-GCN  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content='50  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content='67  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content='58  Table 5: Average rank of the learning models in terms of the  achieved test MSE across the evaluation days for Dataset-B  2019-10-22  2019-10-23  2019-10-24  2019-10-25  2019-10-28  2019-10-29  Time  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content='008  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content='010  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content='012  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content='014  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content='016  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content='018  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content='020  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content='022  Ratio MSE  MLP  GCN  AG-GCN  TAG-GCN  (a)  2021-09-05 2021-09-06 2021-09-07 2021-09-08 2021-09-09 2021-09-10 2021-09-11 2021-09-12  Time  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content='008  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content='010  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content='012  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content='014  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content='016  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content='018  Ratio MSE  MLP  GCN  AG-GCN  TAG-GCN  (b)  Figure 10: Mean squared error of the throughput ratio for  test data of (a) Dataset-A and (b) Dataset-B for TAG-GCN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content='  the results of both datasets, TAG-GCN can achieve better  performance than other proposed model which is due to its  capacity to learn the spatial and temporal dependencies in  the network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content='  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content='6  Expert Rule-Based Parameter Optimization  Strategy  The experts rule-based method can set the actions for each  cell v ∈ Nt for day t + 1 based on the load balancing as:  av  t =  �rv  1(βv  t − 1)  βv  t ≥ 1  rv  2(βv  t − 1)  βv  t < 1 ,  (13)  where rv  1, rv  2 < 0 are the weights set by human experts based  on domain knowledge and assumptions about the network  dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content=' The intuition is to increase the A2-threshold to  release traffic when a cell is overloaded and decrease the  value in the opposite case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content='  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content='7  Additional Experimental Results  Reward Modeling Performance for Throughput Ratio  We repeat the same experiment as Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content=' 2 in the paper but  for the throughput ratio reward and compare the perfor-  mance measured in the achieved MSE by different models  for Dataset-A and B in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content=' 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content='  Ablation Study on the Multi-Objective Action Recom-  mendation Strategy  In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content=' 11, we show an ablation study  on the effectiveness of our proposed multi-objective action  recommendation strategy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content=' For each setting, we repeat our ex-  periment five times with TAG-GCN as the reward model but  follow different action configuration strategies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content=' Other than  the multi-object formulation, we consider two alternatives  2019-10-17 2019-10-18 2019-10-21 2019-10-22 2019-10-23 2019-10-24 2019-10-25 2019-10-28 2019-10-29 2019-10-30 2019-10-31  Time  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content='00  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content='75  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content='50  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content='25  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content='00  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content='25  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content='50  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content='75  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content='00  Reward  TAG-GCN (multi-objective)  TAG-GCN (single objective by throughput ratio)  TAG-GCN (single objective by cell throughput)  Figure 11: The impact of different optimization objectives  for the parameter configuration in our proposed TAG-GCN  model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content='  for action configuration, one by considering only the cell  throughput in the optimization problem, and one by consid-  ering only the load-balancing objective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content=' As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content=' 11,  our proposed TAG-GCN model with both load-balancing  and throughput as optimization objectives can outperform  others, achieving the best average throughput in the final  days across random trials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content='  The Quantitative Results  The quantitative results for  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content=' 4 and Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content=' 11 are summarized in Table 6 and Table 7, re-  spectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content=' We present the achieved network throughput by  different models divided by the performance achieved by the  best actions of the simulator for the last 4 days of the exper-  iments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content=' It should be noted that in all these experiments the  model is trained up to Oct.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content=' 29 and after that the model is  not updated and only performs the inference step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content=' As seen  in Table 6, the proposed method is outperforming the rest of  the methods and yields better average throughput.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content=' Further-  more, our method also yields lower variance with respect to  change in the initial values of the actions, which is a signif-  icant advantage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content=' In general, our method proves to be very  robust in this regard.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content=' The second table demonstrates the ef-  fectiveness of the multi-objective optimization strategy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content=' In  this table, we have denoted three different variants of TAG-  GCN which represent three optimization strategies, namely,  multi-objective strategy using both throughput and through-  put ratio models (TAG-GCN1), single objective strategies  using throughput ratio (TAG-GCN2), and throughput mod-  els (TAG-GCN3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content=' As the results in the table show, the multi-  objective strategy outperforms the single objective ones and  therefore is superior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content='  The Average Rank of Different Models (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content=' 4)  As  stated earlier the models for the experiment depicted in  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content=' 4 are trained up to Oct.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content=' 29 and for the rest of days they  are not updated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content=' In Table 8, we compare the performance of  the proposed learning-based models in terms of the achieved  average rank.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content=' We separate the results for before and after  Oct.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content=' 29 to show the generalization of different methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content=' As  seen TAG-GCN has the best performance both before and  after freezing the learning model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content='  Different action initialization strategies  In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content=' 12, we  present an ablation study on the different initialization  schemes on the first action exploration day.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content=' In addition to the  random initialization, we use two more action initialization  schemes including the expert rule and the negative slope.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content='  Model Name  10-28  10-29  10-30  10-31  Experts Rule-Based  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content='47 ± .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content='0026  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content='37 ± .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content='0173  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content='62 ± .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content='0007  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content='41 ± .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content='0090  MLP  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content='68 ± .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content='0073  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content='76 ± .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content='0080  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content='87 ± .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content='0102  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content='66 ± .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content='0110  GCN  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content='66 ± .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content='0148  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content='76 ± .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content='0077  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content='86 ± .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content='0002  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content='65 ± .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content='0118  AG-GCN  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content='68 ± .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content='0345  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content='77 ± .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content='0055  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content='86 ± .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content='0019  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content='67 ± .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content='0068  TAG-GCN  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content='71 ± .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content='0136  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content='78 ± .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content='0005  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content='93 ± .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content='0074  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content='73 ± .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content='0069  Table 6: The Average throughput divided by the throughput  achieved from the best action designed by the simulator (in  last four days) for different reward models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content='  Model Name  10-28  10-29  10-30  10-31  TAG-GCN1  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content='71 ± .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content='0136  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content='78 ± .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content='0005  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content='93 ± .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content='0074  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content='73 ± .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content='0069  TAG-GCN2  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content='66 ± .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content='0259  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content='75 ± .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content='0212  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content='85 ± .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content='0131  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content='65 ± .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content='0060  TAG-GCN3  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content='63 ± .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content='0199  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content='71 ± .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content='0094  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content='74 ± .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content='0089  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content='61 ± .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content='0170  Table 7: The 95% confidence interval for the average  throughput divided by the throughput achieved from the best  action designed by the simulator (in last four days) for dif-  ferent optimization methods using TAG-GCN model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content='  Model Name  Before Oct.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content=' 29  After Oct.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content=' 29  All Days  MLP  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content='80  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content='67  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content='13  GCN  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content='80  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content='33  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content='00  AG-GCN  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content='20  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content='00  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content='75  TAG-GCN  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content='20  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content='00  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content='13  Table 8: Average rank of different models in terms of the  achieved reward before and after freezing the learning mod-  els  2019-10-17 2019-10-18 2019-10-21 2019-10-22 2019-10-23 2019-10-24 2019-10-25 2019-10-28 2019-10-29 2019-10-30 2019-10-31  Time  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content='00  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content='75  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content='50  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content='25  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content='00  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content='25  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content='50  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content='75  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content='00  Reward  TAG-GCN (random initialization)  TAG-GCN (negative slope initialization)  TAG-GCN (experts rule initialization)  Figure 12: Performance comparison of the TAG-GCN model  under different types of action initialization for Oct.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content=' 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content='  The expert rule initialization is given in (13).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content=' The negative  slope procedure for the action initialization is defined based  on the throughput ratio of the first day of each cell v, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content=', βv  t1  as  av  t1 =  �2(βv  t1 − ρ)φ  ρ − τ  + φ  �  ,  (14)  where ρ = minv∈Nt1 (βv  t1), and τ = maxv∈Nt1 (βv  t1) and  [·] is the rounding operation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content=' For controlling the variation  of action, the maximum range of change is defined as φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
+page_content='  We can observe that regardless of the type of initialization,  the model can converge to almost the same performance in  the final days, which indicates that even without strong prior  information from the network, our design can still quickly  converge to the desired model performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfwgWm/content/2301.03412v1.pdf'}
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+1
+Active-IRS-Aided Wireless Communication:
+Fundamentals, Designs and Open Issues
+Zhenyu Kang, Changsheng You, and Rui Zhang
+Abstract—Intelligent reflecting surface (IRS) has emerged as a
+promising technology to realize smart radio environment for future
+wireless communication systems. Existing works in this line of
+research have mainly considered the conventional passive IRS that
+reflects wireless signals without power amplification, while in this
+article, we give an overview of a new type of IRS, called active IRS,
+which enables simultaneous signal reflection and amplification,
+thus significantly extending the signal coverage of passive IRS.
+We first present the fundamentals of active IRS, including its
+hardware architecture, signal and channel models, as well as
+practical constraints, in comparison with those of passive IRS.
+Then, we discuss new considerations and open issues in designing
+active-IRS-aided wireless communications, such as the reflection
+optimization, channel estimation, and deployment for active IRS,
+as well as its integrated design with passive IRS. Finally, numerical
+results are provided to show the potential performance gains of
+active IRS as compared to passive IRS and traditional active relay.
+I. INTRODUCTION
+To accommodate assorted emerging new applications (e.g.,
+holographic communication, extended reality, and tactile Inter-
+net), the next-generation wireless networks need to meet more
+stringent performance requirements than today’s fifth-generation
+(5G) networks, such as unprecedentedly high throughput, super-
+high reliability, ultra-low latency, extremely low power con-
+sumption, etc. These requirements, however, may not be fully
+achieved by existing 5G technologies such as massive multi-
+input multi-output (MIMO) and millimeter wave (mmWave)
+communications cost-effectively [1], since they enhance com-
+munication performance with significantly increased energy
+consumption and hardware cost. Moreover, the performance of
+5G networks is still fundamentally constrained by the random
+and uncontrollable wireless channels due to various propagation
+impairments and user mobility. To address the above issues, a
+promising new technology, called intelligent reflecting surface
+(IRS), has been proposed to dynamically control the radio
+propagation environment and improve the wireless network
+performance in a cost-effective manner [2]. Specifically, IRS
+is a planar meta-surface consisting of massive low-cost passive
+reflecting elements, which are able to reflect the incident signals
+with individually tuned phase shifts and/or amplitudes. As
+such, the reflected signals by all reflecting elements can be
+added constructively for enhancing the signal power in desired
+direction (so-called passive beamforming) or destructively for
+mitigating undesired interference. These useful functions have
+Z. Kang is with National University of Singapore; C. You is with Southern
+University of Science and Technology; R. Zhang is with The Chinese University
+of Hong Kong, Shenzhen, Shenzhen Research Institute of Big Data, and National
+University of Singapore.
+spurred growing research interests to apply IRS in various wire-
+less systems for improving their performance, such as passive
+relaying, wireless sensing, simultaneous wireless information
+and power transfer (SWIPT), etc [2].
+However, conventional IRS is mounted with fully passive
+reflecting elements (referred to as passive IRS), thus facing
+several issues in practice. In particular, the reflected signal by
+IRS suffers severe product-distance path loss, i.e., the cascaded
+path loss in the transmitter-IRS-receiver link is the product of
+those in the transmitter-IRS and IRS-receiver links, respectively.
+This fundamentally constrains the signal power at the receiver
+and hence the effective coverage of each IRS. Two practical
+methods to resolve this issue have been proposed, which are,
+respectively, deploying more passive elements at the IRS to
+enhance its passive beamforming gain and placing the passive
+IRS closer to either the transmitter or the receiver to reduce
+the cascaded channel path loss. However, these methods may
+not be applicable in practice for the following reasons. First,
+equipping IRS with massive reflecting elements or placing it
+too close to the transceivers will impose great difficulty in
+practical IRS deployment due to limited space and unavailable
+site. Second, more reflecting elements generally incur higher
+training overhead for their corresponding channel estimation and
+higher complexity for optimizing their reflection coefficients in
+real time [2].
+To tackle this problem, a new type of IRS, called active
+IRS, has been recently proposed to enable simultaneous signal
+reflection and amplification, thus more effectively compensating
+the severe cascaded path loss as compared to passive IRS
+without signal amplification [3]–[5]. As shown in Fig. 1, active
+IRS comprises a number of meta-atoms with reflection-type
+amplifiers that can simultaneously alter the signal’s phase and
+amplify its amplitude to enhance the signal power at the receiver,
+albeit at modestly higher power consumption and hardware cost
+compared to conventional passive IRS. Moreover, the amplitude
+control with power amplification by active IRS provides more
+degrees-of-freedom (DoFs) to achieve finer-grained reflective
+beamforming than passive IRS. In addition, the reflection-type
+amplifiers of active IRS directly amplify the incident signals
+in a full-duplex manner yet without the need of high-cost and
+power-hungry radio frequency (RF) chains, which is thus more
+cost and energy efficient than traditional half-duplex or more
+advanced full-duplex active relay that requires sophisticated RF
+chains for signal processing and amplification. Despite the above
+advantages, active IRS also brings new design issues, which
+need to be studied to evaluate its practical performance gain
+over passive IRS and traditional active relay. For example, due
+to signal amplification, the reflecting components of active IRS
+arXiv:2301.04311v1  [cs.IT]  11 Jan 2023
+
+2
+IRS controller
+Active IRS, Type I
+Radiating patch
+Amplifier
+Phase 
+shifter
++
+-
+Matching/coupling 
+circuit
+Biasing 
+source
+Tunnel 
+diode
+Load modulation circuit
+Active IRS, Type II
+PIN diode
+Active relay (full duplex) 
+Antenna array
+Amplify
+Noise
+Self-interference
+Passive IRS
+Phase shifter
+Fig. 1. Hardware architectures of active IRS, passive IRS, and active relay.
+generate non-negligible amplification noise, which results in
+degraded signal-to-noise ratio (SNR) at the receiver and thus
+calls for efficient techniques to balance the trade-off between
+signal and noise amplification by active IRS. Moreover, as
+active IRS usually has limited power supply in practice, it is of
+paramount importance to design its size (number of reflecting
+elements) and placement for optimizing the communication
+performance.
+Motivated by the above, this article aims to provide a com-
+prehensive overview of active-IRS-aided wireless communica-
+tions. To this end, we first present the fundamentals of active
+IRS, including its hardware architecture, signal and channel
+models, as well as practical constraints. Next, we discuss new
+considerations and open issues in designing active-IRS-aided
+communication systems, such as the reflection optimization,
+channel estimation, and deployment for active IRS, as well as
+how to efficiently integrate it with conventional passive IRS
+to achieve superior performance than stand-alone active/passive
+IRS only. Finally, numerical results are presented to show
+the potential performance gains of active IRS as compared to
+conventional passive IRS and active relay.
+II. FUNDAMENTALS OF ACTIVE IRS
+In this section, we present the fundamentals of active IRS,
+and compare it with conventional passive IRS and active relay
+for wireless relaying.
+A. Hardware Architecture
+As shown in Fig. 1, active IRS mainly consists of a
+smart controller and a number of properly designed reflecting
+elements/meta-atoms. Specifically, the IRS controller provides
+appropriate stimulus to adjust the reflection coefficients (i.e.,
+amplitude and phase shift) of each reflecting element via e.g.,
+a field programmable gate array (FPGA)-based control board.
+In addition, it is also responsible for communicating with the
+base station (BS)/access point (AP) through dedicated wireless
+control links. On the other hand, each active-IRS reflecting
+element can reflect the incident signal with desired phase shift
+and amplitude adjustment based on the electromagnetic (EM)
+scattering principle, which can be achieved by e.g., the cascaded
+amplifying and phase shifting (APS) circuits [6], [7], and load
+modulation circuits [8], [9]. In Fig. 1, we illustrate two typical
+hardware architectures for implementing active-IRS reflecting
+elements. The first one (type I) is implemented by the APS
+circuit, which is the integration of an operational amplifier and
+a phase shifter [6], [7]. Specifically, triggered by a biasing power
+source, the operational amplifier amplifies the weak electric
+signal with desired amplification factor. Meanwhile, the phase
+shifting circuit (e.g., positive-intrinsic-negative (PIN) diode,
+varactor diode [6]) connected to the amplifier enables flexible
+phase modulation to the reflected signal. Moreover, to achieve
+maximum power transfer to the radiating structure, a match-
+ing/coupling circuit is added to maintain a good impedance
+match between the APS circuit and the radiating structure.
+In contrast, the second implementation of active-IRS reflecting
+element (type II) can be achieved by load modulation circuits
+based on e.g., tunnel diodes [8] and complementary metal-oxide
+semiconductor (CMOS) [9]. Taking the tunnel diode [8] as
+an example, by imposing proper bias voltage, it can operate
+in the region with negative differential resistance such that
+the reflecting element has an active load impedance (negative
+resistance). As such, the reflection amplitude of incident signals
+can be increased by converting the direct current (DC) power
+into the RF power, while the phase shift is adjusted by tuning
+the circuit capacitance.
+
+3
+B. Signal and Channel Models
+The signal and channel models for active IRS are more
+complicated than those for passive IRS. First, different from
+the noise-free passive IRS, the noise introduced by active
+IRS is composed of both dynamic noise and static noise.
+Specifically, the dynamic noise is related to the input noise
+which is determined by the source resistance of the antenna
+and inherent device noise; while the static noise is independent
+of the transmit/amplification power, which is usually negli-
+gible compared to the dynamic noise. Second, the signal at
+the receiver via active IRS is superposed by the desired sig-
+nals over both the cascaded transmitter-IRS-receiver and direct
+transmitter-receiver links, the amplification noise generated at
+each reflecting element over the IRS-receiver link, and the
+additive noise at the receiver. Let M denote the number of
+active-IRS elements and x denote the transmitted signal. Then
+the received signal y at the receiver can be mathematically
+expressed as y
+=
+�
+hHΨg + t
+�
+x + hHΨzI + z0, where
+hH, g, and t denote the equivalent baseband channels of
+the IRS→receiver, transmitter→IRS, and transmitter→receiver
+links; Ψ ≜ diag (α1, · · · , αM) diag
+�
+eȷφ1, · · · , eȷφM �
+is the
+active-IRS reflection matrix with αm and φm representing
+the amplification factor and phase shift of the m-th reflecting
+element, respectively; zI and z0 denote the amplification noise
+induced by active IRS and the additive noise at the receiver.
+C. Practical Constraints
+Active IRS usually has smaller amplification power than
+active relay, which thus limits its maximum amplification factors
+at the reflecting elements in practice. Generally speaking, there
+are two practical amplification power models applicable to
+active IRS. In the first model, the amplification factors of all
+reflecting elements are subject to a total power constraint as
+they share a common power source [3]. In this case, the total
+amplification power can be allocated to reflecting elements based
+on the corresponding IRS-related channels to achieve optimal
+performance. While for the second model, in contrast, each
+active-IRS reflecting element is attached with a separate power
+source with independent power control. As such, the per-element
+maximum amplification factor is constrained by the individual
+per-element amplification power.
+Moreover, in practice, other hardware constraints should be
+considered for modeling active IRS. First, it is worth noting that
+the reflection amplitude based on the negative resistance circuit
+is generally phase-dependent, because both the amplification
+factor and phase shift are functions of the effective capacitance
+and resistance of each reflecting element. Second, for the active-
+IRS element based on tunnel diode, its load impedance is
+determined by the effective capacitance and resistance; thus, its
+maximum amplification factor is constrained by the inherent
+load impedance, which cannot be tuned arbitrarily large regard-
+less of the power supply. Last but not least, the resolutions of
+active-IRS phase shift and amplification factor depend on the
+specific hardware implementation, which need to take discrete
+values in practice.
+D. Comparison with Passive IRS and Active Relay
+In Table I, we compare the main characteristics of active
+IRS with its two counterpart technologies for wireless relaying,
+namely, passive IRS and active relay. First, in terms of hardware
+cost and power consumption, active IRS surpasses passive IRS
+due to the additional reflection-type amplifiers, while it falls
+below active relay since the latter requires power-hungry and
+costly RF chains for signal processing and amplification.
+Next, compared to passive IRS, active IRS endows additional
+amplitude amplification for the reflected signal, thus attaining
+more flexible beam steering and larger coverage (see Fig. 2(a)).
+However, its beamforming performance is generally inferior to
+half-/full-duplex active relay, since active IRS only reflects and
+amplifies signals in the RF band with an equivalent diagonal
+baseband reflection matrix (i.e., Ψ), while active relay with
+multiple antennas have more sophisticated signal processing
+capability with typically a full-rank MIMO signal processing
+matrix at the baseband.
+Last, wireless communication systems aided by active IRS
+generally attain higher spectral efficiency than those aided by
+passive IRS and half-duplex active relay. On one hand, active
+IRS enables power amplification when reflecting signals, thus
+leading to a higher received SNR at the receiver than passive
+IRS. On the other hand, practical active relay usually operates
+in the half-duplex mode1, thus suffering considerable rate per-
+formance loss compared to full-duplex active IRS.
+III. NEW DESIGN CONSIDERATIONS AND CHALLENGES
+In this section, we present new considerations and open
+issues in designing active-IRS-aided wireless communications,
+including the reflection optimization, channel estimation, and
+deployment for active IRS, as well as its efficient integration
+with passive IRS. Moreover, promising solutions for tackling
+these issues are also proposed to inspire future work.
+A. Active-IRS Reflection Optimization
+One key challenge in active-IRS-aided communication design
+is how to optimize the reflection coefficients of active IRS for
+achieving the optimal beamforming and SNR performance. To
+address it, several practical considerations need to be taken
+into account such as the phase-amplitude dependent control,
+maximum amplification factor, discrete phase shifts/amplitude
+levels (cf. Section II-C). These considerations usually result
+in non-convex constraints over the reflection coefficients and
+hence non-convex optimization problems [5]. In addition, as the
+amplification factor of active-IRS element can be controlled, the
+resulting design variables are significantly more than those of
+passive IRS for which the reflection amplitude is usually set
+as one or its maximum value. Several algorithms have been
+proposed in the literature to solve the above problem (see, e.g.,
+[3]–[6]). For example, the alternating optimization (AO) method
+can be applied to iteratively optimize the amplification factor
+and phase shift, until the convergence is achieved. However, the
+performance of AO critically depends on its initialization, which,
+1Full-duplex relay requires higher hardware and implementation cost than
+half-duplex relay due to the need to eliminate the loop-back self-interference.
+
+4
+TABLE I
+COMPARISONS OF ACTIVE IRS, PASSIVE IRS, AND ACTIVE RELAY.
+Active IRS
+Passive IRS
+Active relay
+Hardware cost
+Low
+Very low
+High
+Power consumption
+Moderate
+Very low
+High
+Operation mechanism
+Reflect and amplify
+Reflect
+Receive and amplify
+Duplex mode
+Full duplex
+Full duplex
+Full/half duplex
+Amplification noise
+Yes
+No
+Yes
+if not properly designed, can result in considerable performance
+loss. Moreover, existing works (e.g., [3]–[6]) usually assumed
+ideal reflection coefficients with continuous and independent
+phase-shift and amplitude control, which cannot be realized in
+practice due to hardware limitations. Thus, efficient algorithms
+that can handle more practical constraints are worthy of future
+investigation.
+Several results on the active-IRS reflection design under
+different channel conditions and/or system setups have been
+reported in the literature (e.g., [4], [10]). Taking the single-user
+case as an example, it was shown that the reflected signals over
+different IRS elements should be phase-aligned such that they
+can be constructively added at the receiver, for both the cases
+with active or passive IRS [10]. Moreover, the amplification
+factor should be set to the maximum available value for com-
+pensating the cascaded channel path loss. Next, for the general
+multi-user case, there exists a new trade-off between signal
+power enhancement and interference mitigation in designing
+the active-IRS reflection coefficients given a total power budget
+constraint. In this case, the amplification factors of reflecting
+elements should be properly set, which may not necessarily
+equal to their maximum values. Generally speaking, active IRS
+is expected to achieve better interference mitigation performance
+than passive IRS, since its amplitude control endows more DoFs
+to eliminate the undesired interference more effectively.
+An important performance metric in the active-IRS reflection
+design is how the received SNR scales with the number of
+reflecting elements, M, since such scaling orders characterize
+the SNR performance when the number of IRS elements is
+asymptotically large. First, consider the case with a total power
+constraint over all active-IRS elements. It was reported in [3]
+that the received SNR with an active IRS linearly increases with
+M when M is sufficiently large. This is expected since the signal
+power at the receiver quadratically increases with M thanks to
+the IRS beamforming gain, while the amplification noise power
+also increases linearly with M, thus leading to the SNR scaling
+order of M. However, it is worth noting that although active
+IRS has a smaller SNR scaling order than passive IRS without
+amplification noise (i.e., O (M) versus O
+�
+M 2�
+[2]), it provides
+superior rate performance than passive IRS when the number of
+reflecting elements is moderate thanks to the additional signal
+power amplification gain. Next, consider the case with fixed
+per-element power for each active-IRS element. In this case,
+the total power budget linearly increases with the number of
+reflecting elements. As such, the active IRS is expected to
+achieve additional amplification power gain in the order of M,
+hence yielding the same received SNR scaling order with the
+conventional passive IRS, i.e., O
+�
+M 2�
+(such SNR comparisons
+will be numerically shown in Fig. 5).
+B. Active-IRS Channel Estimation
+Active-IRS reflection design requires accurate channel state
+information (CSI) of IRS-related channels for fully exploiting its
+beamforming and amplification gains. Compared with passive
+IRS, the CSI acquisition for active IRS is generally more
+challenging due to the following reasons. First, the active-IRS
+reflection design generally requires explicit CSI of the separate
+transmitter-IRS and IRS-receiver links due to the new consider-
+ation of amplification noise [10], which greatly differs from the
+passive-IRS case that requires cascaded CSI only. Specifically,
+although the phase shift can be designed based on cascaded
+CSI, the amplification factor control is generally dependent on
+the incident signal power at active IRS and its power budget,
+while the incident signal power depends on the transmitter-IRS
+CSI. Second, in active-IRS channel estimation, it is difficult to
+set the amplification factor to meet the given total/per-element
+power constraint exactly, since this requires the knowledge of
+incident signal power that depends on the transmitter-IRS CSI,
+which is yet to be estimated. Last, the amplification noise also
+affects the channel estimation performance, while this issue does
+not exist in the channel estimation for passive IRS.
+In the following, we introduce two practical active-IRS chan-
+nel estimation approaches, with and without IRS-mounted sen-
+sors, respectively. The first approach requires low-cost sensors
+installed on the active IRS, which are interleaved with the
+reflecting elements. Equipped with these sensors that can receive
+signals sent by the BS/users, active IRS can estimate the BS-
+IRS and IRS-user channels separately by exploiting the strong
+channel correlations between the IRS elements and sensors. It
+is worth noting that active IRS in general requires much fewer
+reflecting elements than passive IRS for achieving comparable
+rate performance, thus it is easier to place reflecting elements
+with sensors close to each other to achieve high channel cor-
+relation. Second, for active IRS without sensors installed, one
+promising approach is to combine the separate and cascaded
+channel estimation methods. Specifically, the BS first estimates
+the BS-IRS CSI based on the reflected signal over the BS-
+IRS-BS link (i.e., by wireless sensing of the BS). Next, the
+BS estimates the cascaded BS-IRS-user CSI based on the pilot
+signals from the user, and then resolves the IRS-user CSI based
+on the estimated cascaded CSI and BS-IRS CSI. Moreover,
+for the amplification factor control in channel estimation, one
+practical method is by setting an equal IRS reflection amplitude
+over all reflecting elements based on the received signal powers
+
+5
+BS
+User
+(d) Multi-IRS reflection with active and passive IRSs
+BS
+User
+Downlink communication
+Uplink communication
+(c) Hybrid active-passive IRS
+(b) Active-IRS deployment
+(a) Active-IRS reflection design
+Optimal location: closer to user
+Optimal location: closer to BS
+Passive IRS
+Active IRS
+Narrower beam and larger coverage 
+than passive IRS
+Fig. 2. New design considerations for active-IRS-aided wireless communications.
+(with sensors) or simply setting the amplification factor equal
+to one (without sensors) similar to the case of passive IRS, to
+facilitate the estimation of IRS-user CSI.
+C. Active-IRS Deployment
+How to deploy active IRSs in wireless networks to achieve the
+largest network coverage and maximum throughput is another
+crucial problem to solve. Generally speaking, several factors
+need to be considered in the active-IRS deployment design, such
+as the locations of users, uplink or downlink communication,
+number of reflecting elements, amplification power budget,
+amplification noise power, etc.
+First, consider the single-user case where the active-IRS
+location needs to be optimized for achieving the maximum
+received SNR at the user. In this case, it has been shown in
+[10] that active IRS should be properly deployed between the
+transmitter and receiver to balance the large-scale path loss
+of the transmitter-IRS and IRS-receiver channels. In particular,
+with a small amplification power or a large number of reflecting
+elements, active IRS should be deployed closer to the receiver;
+otherwise, the amplification factors are too small to compensate
+the amplification noise. This is different from the passive-IRS
+case where the IRS should be deployed near either the trans-
+mitter or receiver to minimize the cascaded path loss. Besides,
+the optimal active-IRS deployment locations are different in the
+uplink and downlink for two-way communications in general,
+and thus their performance should be balanced by choosing an
+appropriate location for active IRS (see Fig. 2(b)). Second, for
+the multi-user case, the active-IRS deployment design is more
+complicated. On one hand, it needs to strike the performance
+balance among different users. On the other hand, the active
+IRS may not be deployed at the BS/AP side as in the passive-
+IRS case, since it also needs to balance the powers of desired
+signal and amplification noise. To tackle this problem, instead of
+employing a computationally prohibitive brute-force search, one
+efficient and low-complexity approach is by leveraging machine
+learning (ML) and geometry-based optimization methods to
+deploy the active IRS at a location with good statistical channel
+conditions. Besides active-IRS location, the reflection gain of
+active IRS is also affected by its rotation angle. As shown
+in [11], considerable rate performance improvement can be
+attained by properly rotating the IRS without changing its
+locations; thus, the IRS rotation should be carefully tuned to
+maximize the number of served users and IRS-user channel
+gains.
+Compared with single-IRS deployment, the multi-active-IRS
+deployment is generally more challenging. In particular, there
+exists a fundamental trade-off between the deployment cost
+(e.g., hardware cost, power consumption) and communication
+performance (e.g., network coverage, received SNR). As such,
+besides deployment locations, the number of active IRSs needs
+to be carefully chosen to minimize the total deployment cost,
+while achieving the communication performance requirements.
+Furthermore, for indoor environment, how to deploy multiple
+IRSs to maximize the network connectivity subject to the indoor
+topology constraints is a challenging problem, which deserves
+future investigation.
+D. Integrated Design with Passive IRS
+While the existing works mainly considered the wireless
+communication systems aided by active or passive IRS only,
+it has been recently shown that the integrated design with
+both active and passive IRSs has the potential to achieve their
+combined advantages at the same time [12]–[14], as discussed
+below.
+First, it was shown in [12] that given the total deployment
+budget and amplification power, a hybrid IRS with both passive
+and active elements (e.g., see Fig. 2(c)) can achieve better
+communication performance than the conventional IRS with
+active or passive elements only. Specifically, there generally
+exists a trade-off between deploying more active elements to
+achieve the appealing power amplification gain and deploying
+more passive elements to attain a higher beamforming gain,
+thus leading to the new design issue of active versus passive
+elements allocation [12]. Generally speaking, when the number
+of reflecting elements is small and/or the amplification power
+is large, more active elements should be deployed to harness
+the power amplification gain; otherwise, deploying more passive
+elements is expected to achieve better performance thanks to
+their higher-order beamforming gain [12].
+Another type of the integrated design is deploying both
+stand-alone active and passive IRSs in the wireless network,
+
+()6
+BS
+IRS/Relay
+User
+x
+z
+H
+D
+Fig. 3. Simulation setup
+as illustrated in Fig. 2(d), to improve the communication
+performance [13], [14]. The advantages are two-folds. First,
+deploying multiple cooperative IRSs provides the opportunity
+to construct a multi-hop reflection path between the transmitter
+and receiver to bypass environmental obstacles [15], which is
+particularly appealing in complex environment (e.g., indoor).
+Second, compared with the conventional multi-passive-IRS net-
+work, adding a number of active IRSs in the network can
+substantially enhance the communication performance. This is
+because active IRSs can opportunistically amplify the reflected
+signal along the multi-reflection path, thus effectively compen-
+sating the severe end-to-end product-distance path loss. Despite
+the advantages, the design of multi-active and multi-passive
+(MAMP) IRS aided communications is also more challenging
+[13]. For example, the multi-IRS beam routing path with both
+active and passive IRSs should be carefully devised to maximize
+the power amplification gain, and at the same time, minimize
+the accumulated amplification noise power arising from active
+IRSs. Moreover, the numbers of active and passive IRSs should
+also be carefully chosen given limited deployment budget and
+their locations should be optimally designed [14], which deserve
+further studies.
+IV. NUMERICAL RESULTS
+Numerical results are presented in this section to demonstrate
+the potential performance gains of active IRS as compared
+to passive IRS and active relay. Specifically, we consider a
+typical point-to-point communication system as shown in Fig.
+3, where an IRS or a relay is deployed at an altitude of 2
+meters (m) to assist the communication between a BS and a user
+(both equipped with single antenna), while the direct BS-user
+channel is assumed being blocked. For each reflecting element,
+we consider the continuous and independently controlled phase
+shift and reflection amplitude (the amplitude is set to be one for
+passive IRS). Moreover, the line-of-sight (LoS) channel model
+is assumed for all involved links, where the path loss exponent
+is 2 and the reference channel power gain at a distance of 1 m
+is −30 dB. Both powers of the receiver noise and per-element
+active-IRS amplification noise are set as −80 dBm.
+Fig. 4 shows the user’s maximum achievable rates versus the
+BS-user distance in four wireless communication systems aided
+by: 1) an active IRS with 128 reflecting elements and a total
+amplification power of 10 dBm; 2) a passive IRS with 128
+reflecting elements; 3) an ideal full-duplex amplify-and-forward
+(AF) relay with perfect self-interference cancellation and a total
+amplification power of 15 dBm; and 4) a half-duplex AF relay
+with optimal time division and a total amplification power of
+15 dBm. It is observed that given the same BS-user distance,
+150
+200
+250
+300
+350
+400
+450
+Distance between BS and user (m)
+0
+5
+10
+15
+Achievable Rate (bps/Hz)
+Active IRS
+Passive IRS
+Full-duplex AF Relay
+Half-duplex AF Relay
+Fig. 4. Achievable rate versus user location.
+50
+100
+150
+200
+250
+300
+350
+400
+450
+Number of reflecting elements
+0
+5
+10
+15
+20
+25
+30
+35
+SNR (dB)
+Active IRS with FTP
+Active IRS with FPP
+Passive IRS
+Fig. 5. Received SNR versus number of reflecting elements.
+active IRS always achieves a higher rate than passive IRS and
+half-duplex relay. This is expected since active IRS provides an
+additional power amplification gain compared to passive IRS,
+and it achieves the full-duplex gain compared to half-duplex
+relay. Besides, active IRS is inferior to full-duplex active relay
+due to its limited signal processing capability and less power
+supply than active relay.
+Next, we show in Fig. 5 how the received SNR scales with the
+number of reflecting elements in three wireless communication
+systems aided by: 1) an active IRS with a fixed total power (FTP)
+of 0.2 W; 2) an active IRS with a fixed per-element power (FPP)
+of 1 mW; and 3) a passive IRS. For the above three systems, the
+corresponding IRS reflections are optimized for maximizing the
+received SNR. First, it is observed that both active-IRS-aided
+systems achieve much higher received SNRs than the passive-
+IRS counterpart. This is expected because active IRS provides
+both the beamforming gain and power amplification gain, while
+passive IRS only offers the beamforming gain. Second, one can
+observe that with an increasing number of reflecting elements,
+the received SNR of active IRS with FPP is firstly lower than and
+then exceeds that with FTP. This is because in the latter case,
+the average per-element power decreases with the number of
+reflecting elements, thus yielding a lower received SNR scaling
+order.
+
+7
+V. CONCLUSIONS
+In this article, we provide a comprehensive overview of the
+newly proposed active IRS that enables simultaneous signal
+reflection and amplification. Specifically, we first present the
+fundamentals of active IRS, including its hardware architecture,
+signal and channel models, as well as practical constraints. Then,
+new considerations and open issues in designing active-IRS-
+aided wireless communications are discussed to shed new light
+to promising solutions, including the reflection optimization,
+channel estimation, and deployment for active IRS, as well as its
+integrated design with passive IRS. Last, we numerically show
+the potential performance gains of active IRS over passive IRS
+and traditional half-duplex active relay, and the SNR scaling
+orders of active IRS under different power constraints versus
+passive IRS.
+REFERENCES
+[1] M. Giordani, M. Polese, M. Mezzavilla, S. Rangan, and M. Zorzi, “Toward
+6G networks: Use cases and technologies,” IEEE Commun. Mag., vol. 58,
+no. 3, pp. 55–61, Mar. 2020.
+[2] Q. Wu, S. Zhang, B. Zheng, C. You, and R. Zhang, “Intelligent reflecting
+surface-aided wireless communications: A tutorial,” IEEE Trans. Com-
+mun., vol. 69, no. 5, pp. 3313–3351, May 2021.
+[3] R. Long, Y.-C. Liang, Y. Pei, and E. G. Larsson, “Active reconfigurable
+intelligent surface-aided wireless communications,” IEEE Trans. Wireless
+Commun., vol. 20, no. 8, pp. 4962–4975, Aug. 2021.
+[4] K. Zhi, C. Pan, H. Ren, K. K. Chai, and M. Elkashlan, “Active RIS
+versus passive RIS: Which is superior with the same power budget?” IEEE
+Commun. Lett., vol. 26, no. 5, pp. 1150–1154, May 2022.
+[5] Z. Zhang, L. Dai, X. Chen, C. Liu, F. Yang, R. Schober, and H. Vin-
+cent Poor, “Active RIS vs. passive RIS: which will prevail in 6G?” in
+IEEE Trans. Commun., Early Access.
+[6] K. K. Kishor and S. V. Hum, “An amplifying reconfigurable reflectarray
+antenna,” IEEE Trans. Antennas Propag., vol. 60, no. 1, pp. 197–205, Sep.
+2012.
+[7] J. Lonˇcar, A. Grbic, and S. Hrabar, “Ultrathin active polarization-selective
+metasurface at X-band frequencies,” Physical Review B, vol. 100, no. 7,
+p. 075131, 2019.
+[8] F. Amato, C. W. Peterson, B. P. Degnan, and G. D. Durgin, “Tunneling
+RFID tags for long-range and low-power microwave applications,” IEEE
+J. Radio Freq. Identif., vol. 2, no. 2, pp. 93–103, Jun. 2018.
+[9] N. Landsberg and E. Socher, “A low-power 28-nm CMOS FD-SOI
+reflection amplifier for an active F-band reflectarray,” IEEE Trans. Microw.
+Theory Tech., vol. 65, no. 10, pp. 3910–3921, May 2017.
+[10] C. You and R. Zhang, “Wireless communication aided by intelligent
+reflecting surface: Active or passive?” IEEE Wireless Commun. Lett.,
+vol. 10, no. 12, pp. 2659–2663, Dec. 2021.
+[11] Y. Cheng, W. Peng, C. Huang, G. C. Alexandropoulos, C. Yuen, and
+M. Debbah, “RIS-aided wireless communications: Extra degrees of free-
+dom via rotation and location optimization,” IEEE Trans. Wireless Com-
+mun., vol. 21, no. 8, pp. 6656–6671, Feb. 2022.
+[12] Z. Kang, C. You, and R. Zhang, “Active-passive IRS aided wireless
+communication: New hybrid architecture and elements allocation optimiza-
+tion,” [Online]. Available: https://arxiv.org/pdf/2207.01244.pdf, accessed
+on Jul. 2022.
+[13] Y. Zhang, C. You, and B. Zheng, “Multi-active multi-passive (MAMP)-
+IRS aided wireless communication: A multi-hop beam routing design,”
+[Online]. Available: https://arxiv.org/abs/2209.06390, accessed on Sep.
+2022.
+[14] M. Fu and R. Zhang, “Active and passive IRS jointly aided communication:
+Deployment design and achievable rate,” in IEEE Wireless Commun. Lett.,
+Early Access.
+[15] W. Mei and R. Zhang, “Multi-beam multi-hop routing for intelligent
+reflecting surfaces aided massive MIMO,” IEEE Trans. Wireless Commun.,
+vol. 21, no. 3, pp. 1897–1912, Sep. 2022.
+BIOGRAPHIES
+Zhenyu Kang (zhenyu kang@u.nus.edu) is a Ph.D. candidate
+with National University of Singapore, Singapore.
+Changsheng You (youcs@sustech.edu.cn) received his Ph.D.
+degree from The University of Hong Kong. He is currently
+an Assistant Professor with Southern University of Science and
+Technology, China.
+Rui Zhang [F’17] (rzhang@cuhk.edu.cn) received the Ph.D. de-
+gree from Stanford University. He is now a Principal’s Diligence
+Chair Professor with The Chinese University of Hong Kong,
+Shenzhen, and also a Provost’s Chair Professor with National
+University of Singapore. He is a Fellow of the IEEE and the
+Academy of Engineering, Singapore.
+
diff --git a/b9E3T4oBgHgl3EQfGAm_/content/tmp_files/load_file.txt b/b9E3T4oBgHgl3EQfGAm_/content/tmp_files/load_file.txt
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+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E3T4oBgHgl3EQfGAm_/content/2301.04311v1.pdf,len=429
+page_content='1  Active-IRS-Aided Wireless Communication:  Fundamentals, Designs and Open Issues  Zhenyu Kang, Changsheng You, and Rui Zhang  Abstract—Intelligent reflecting surface (IRS) has emerged as a  promising technology to realize smart radio environment for future  wireless communication systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E3T4oBgHgl3EQfGAm_/content/2301.04311v1.pdf'}
+page_content=' Existing works in this line of  research have mainly considered the conventional passive IRS that  reflects wireless signals without power amplification, while in this  article, we give an overview of a new type of IRS, called active IRS,  which enables simultaneous signal reflection and amplification,  thus significantly extending the signal coverage of passive IRS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E3T4oBgHgl3EQfGAm_/content/2301.04311v1.pdf'}
+page_content='  We first present the fundamentals of active IRS, including its  hardware architecture, signal and channel models, as well as  practical constraints, in comparison with those of passive IRS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E3T4oBgHgl3EQfGAm_/content/2301.04311v1.pdf'}
+page_content='  Then, we discuss new considerations and open issues in designing  active-IRS-aided wireless communications, such as the reflection  optimization, channel estimation, and deployment for active IRS,  as well as its integrated design with passive IRS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E3T4oBgHgl3EQfGAm_/content/2301.04311v1.pdf'}
+page_content=' Finally, numerical  results are provided to show the potential performance gains of  active IRS as compared to passive IRS and traditional active relay.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E3T4oBgHgl3EQfGAm_/content/2301.04311v1.pdf'}
+page_content='  I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E3T4oBgHgl3EQfGAm_/content/2301.04311v1.pdf'}
+page_content=' INTRODUCTION  To accommodate assorted emerging new applications (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E3T4oBgHgl3EQfGAm_/content/2301.04311v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E3T4oBgHgl3EQfGAm_/content/2301.04311v1.pdf'}
+page_content=',  holographic communication, extended reality, and tactile Inter-  net), the next-generation wireless networks need to meet more  stringent performance requirements than today’s fifth-generation  (5G) networks, such as unprecedentedly high throughput, super-  high reliability, ultra-low latency, extremely low power con-  sumption, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E3T4oBgHgl3EQfGAm_/content/2301.04311v1.pdf'}
+page_content=' These requirements, however, may not be fully  achieved by existing 5G technologies such as massive multi-  input multi-output (MIMO) and millimeter wave (mmWave)  communications cost-effectively [1], since they enhance com-  munication performance with significantly increased energy  consumption and hardware cost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E3T4oBgHgl3EQfGAm_/content/2301.04311v1.pdf'}
+page_content=' Moreover, the performance of  5G networks is still fundamentally constrained by the random  and uncontrollable wireless channels due to various propagation  impairments and user mobility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E3T4oBgHgl3EQfGAm_/content/2301.04311v1.pdf'}
+page_content=' To address the above issues, a  promising new technology, called intelligent reflecting surface  (IRS), has been proposed to dynamically control the radio  propagation environment and improve the wireless network  performance in a cost-effective manner [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E3T4oBgHgl3EQfGAm_/content/2301.04311v1.pdf'}
+page_content=' Specifically, IRS  is a planar meta-surface consisting of massive low-cost passive  reflecting elements, which are able to reflect the incident signals  with individually tuned phase shifts and/or amplitudes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E3T4oBgHgl3EQfGAm_/content/2301.04311v1.pdf'}
+page_content=' As  such, the reflected signals by all reflecting elements can be  added constructively for enhancing the signal power in desired  direction (so-called passive beamforming) or destructively for  mitigating undesired interference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E3T4oBgHgl3EQfGAm_/content/2301.04311v1.pdf'}
+page_content=' These useful functions have  Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E3T4oBgHgl3EQfGAm_/content/2301.04311v1.pdf'}
+page_content=' Kang is with National University of Singapore;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E3T4oBgHgl3EQfGAm_/content/2301.04311v1.pdf'}
+page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E3T4oBgHgl3EQfGAm_/content/2301.04311v1.pdf'}
+page_content=' You is with Southern  University of Science and Technology;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E3T4oBgHgl3EQfGAm_/content/2301.04311v1.pdf'}
+page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E3T4oBgHgl3EQfGAm_/content/2301.04311v1.pdf'}
+page_content=' Zhang is with The Chinese University  of Hong Kong, Shenzhen, Shenzhen Research Institute of Big Data, and National  University of Singapore.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E3T4oBgHgl3EQfGAm_/content/2301.04311v1.pdf'}
+page_content='  spurred growing research interests to apply IRS in various wire-  less systems for improving their performance, such as passive  relaying, wireless sensing, simultaneous wireless information  and power transfer (SWIPT), etc [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E3T4oBgHgl3EQfGAm_/content/2301.04311v1.pdf'}
+page_content='  However, conventional IRS is mounted with fully passive  reflecting elements (referred to as passive IRS), thus facing  several issues in practice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E3T4oBgHgl3EQfGAm_/content/2301.04311v1.pdf'}
+page_content=' In particular, the reflected signal by  IRS suffers severe product-distance path loss, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E3T4oBgHgl3EQfGAm_/content/2301.04311v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E3T4oBgHgl3EQfGAm_/content/2301.04311v1.pdf'}
+page_content=', the cascaded  path loss in the transmitter-IRS-receiver link is the product of  those in the transmitter-IRS and IRS-receiver links, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E3T4oBgHgl3EQfGAm_/content/2301.04311v1.pdf'}
+page_content='  This fundamentally constrains the signal power at the receiver  and hence the effective coverage of each IRS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E3T4oBgHgl3EQfGAm_/content/2301.04311v1.pdf'}
+page_content=' Two practical  methods to resolve this issue have been proposed, which are,  respectively, deploying more passive elements at the IRS to  enhance its passive beamforming gain and placing the passive  IRS closer to either the transmitter or the receiver to reduce  the cascaded channel path loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E3T4oBgHgl3EQfGAm_/content/2301.04311v1.pdf'}
+page_content=' However, these methods may  not be applicable in practice for the following reasons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E3T4oBgHgl3EQfGAm_/content/2301.04311v1.pdf'}
+page_content=' First,  equipping IRS with massive reflecting elements or placing it  too close to the transceivers will impose great difficulty in  practical IRS deployment due to limited space and unavailable  site.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E3T4oBgHgl3EQfGAm_/content/2301.04311v1.pdf'}
+page_content=' Second, more reflecting elements generally incur higher  training overhead for their corresponding channel estimation and  higher complexity for optimizing their reflection coefficients in  real time [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E3T4oBgHgl3EQfGAm_/content/2301.04311v1.pdf'}
+page_content='  To tackle this problem, a new type of IRS, called active  IRS, has been recently proposed to enable simultaneous signal  reflection and amplification, thus more effectively compensating  the severe cascaded path loss as compared to passive IRS  without signal amplification [3]–[5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E3T4oBgHgl3EQfGAm_/content/2301.04311v1.pdf'}
+page_content=' As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E3T4oBgHgl3EQfGAm_/content/2301.04311v1.pdf'}
+page_content=' 1, active  IRS comprises a number of meta-atoms with reflection-type  amplifiers that can simultaneously alter the signal’s phase and  amplify its amplitude to enhance the signal power at the receiver,  albeit at modestly higher power consumption and hardware cost  compared to conventional passive IRS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E3T4oBgHgl3EQfGAm_/content/2301.04311v1.pdf'}
+page_content=' Moreover, the amplitude  control with power amplification by active IRS provides more  degrees-of-freedom (DoFs) to achieve finer-grained reflective  beamforming than passive IRS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E3T4oBgHgl3EQfGAm_/content/2301.04311v1.pdf'}
+page_content=' In addition, the reflection-type  amplifiers of active IRS directly amplify the incident signals  in a full-duplex manner yet without the need of high-cost and  power-hungry radio frequency (RF) chains, which is thus more  cost and energy efficient than traditional half-duplex or more  advanced full-duplex active relay that requires sophisticated RF  chains for signal processing and amplification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E3T4oBgHgl3EQfGAm_/content/2301.04311v1.pdf'}
+page_content=' Despite the above  advantages, active IRS also brings new design issues, which  need to be studied to evaluate its practical performance gain  over passive IRS and traditional active relay.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E3T4oBgHgl3EQfGAm_/content/2301.04311v1.pdf'}
+page_content=' For example, due  to signal amplification, the reflecting components of active IRS  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E3T4oBgHgl3EQfGAm_/content/2301.04311v1.pdf'}
+page_content='04311v1  [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E3T4oBgHgl3EQfGAm_/content/2301.04311v1.pdf'}
+page_content='IT]  11 Jan 2023  2  IRS controller  Active IRS, Type I  Radiating patch  Amplifier  Phase  shifter  + Matching/coupling  circuit  Biasing  source  Tunnel  diode  Load modulation circuit  Active IRS, Type II  PIN diode  Active relay (full duplex)  Antenna array  Amplify  Noise  Self-interference  Passive IRS  Phase shifter  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E3T4oBgHgl3EQfGAm_/content/2301.04311v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E3T4oBgHgl3EQfGAm_/content/2301.04311v1.pdf'}
+page_content=' Hardware architectures of active IRS, passive IRS, and active relay.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E3T4oBgHgl3EQfGAm_/content/2301.04311v1.pdf'}
+page_content='  generate non-negligible amplification noise, which results in  degraded signal-to-noise ratio (SNR) at the receiver and thus  calls for efficient techniques to balance the trade-off between  signal and noise amplification by active IRS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E3T4oBgHgl3EQfGAm_/content/2301.04311v1.pdf'}
+page_content=' Moreover, as  active IRS usually has limited power supply in practice, it is of  paramount importance to design its size (number of reflecting  elements) and placement for optimizing the communication  performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E3T4oBgHgl3EQfGAm_/content/2301.04311v1.pdf'}
+page_content='  Motivated by the above, this article aims to provide a com-  prehensive overview of active-IRS-aided wireless communica-  tions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E3T4oBgHgl3EQfGAm_/content/2301.04311v1.pdf'}
+page_content=' To this end, we first present the fundamentals of active  IRS, including its hardware architecture, signal and channel  models, as well as practical constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E3T4oBgHgl3EQfGAm_/content/2301.04311v1.pdf'}
+page_content=' Next, we discuss new  considerations and open issues in designing active-IRS-aided  communication systems, such as the reflection optimization,  channel estimation, and deployment for active IRS, as well as  how to efficiently integrate it with conventional passive IRS  to achieve superior performance than stand-alone active/passive  IRS only.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E3T4oBgHgl3EQfGAm_/content/2301.04311v1.pdf'}
+page_content=' Finally, numerical results are presented to show  the potential performance gains of active IRS as compared to  conventional passive IRS and active relay.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E3T4oBgHgl3EQfGAm_/content/2301.04311v1.pdf'}
+page_content='  II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E3T4oBgHgl3EQfGAm_/content/2301.04311v1.pdf'}
+page_content=' FUNDAMENTALS OF ACTIVE IRS  In this section, we present the fundamentals of active IRS,  and compare it with conventional passive IRS and active relay  for wireless relaying.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E3T4oBgHgl3EQfGAm_/content/2301.04311v1.pdf'}
+page_content='  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E3T4oBgHgl3EQfGAm_/content/2301.04311v1.pdf'}
+page_content=' Hardware Architecture  As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E3T4oBgHgl3EQfGAm_/content/2301.04311v1.pdf'}
+page_content=' 1, active IRS mainly consists of a  smart controller and a number of properly designed reflecting  elements/meta-atoms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E3T4oBgHgl3EQfGAm_/content/2301.04311v1.pdf'}
+page_content=' Specifically, the IRS controller provides  appropriate stimulus to adjust the reflection coefficients (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E3T4oBgHgl3EQfGAm_/content/2301.04311v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E3T4oBgHgl3EQfGAm_/content/2301.04311v1.pdf'}
+page_content=',  amplitude and phase shift) of each reflecting element via e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E3T4oBgHgl3EQfGAm_/content/2301.04311v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E3T4oBgHgl3EQfGAm_/content/2301.04311v1.pdf'}
+page_content=',  a field programmable gate array (FPGA)-based control board.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E3T4oBgHgl3EQfGAm_/content/2301.04311v1.pdf'}
+page_content='  In addition, it is also responsible for communicating with the  base station (BS)/access point (AP) through dedicated wireless  control links.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E3T4oBgHgl3EQfGAm_/content/2301.04311v1.pdf'}
+page_content=' On the other hand, each active-IRS reflecting  element can reflect the incident signal with desired phase shift  and amplitude adjustment based on the electromagnetic (EM)  scattering principle, which can be achieved by e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E3T4oBgHgl3EQfGAm_/content/2301.04311v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E3T4oBgHgl3EQfGAm_/content/2301.04311v1.pdf'}
+page_content=', the cascaded  amplifying and phase shifting (APS) circuits [6], [7], and load  modulation circuits [8], [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E3T4oBgHgl3EQfGAm_/content/2301.04311v1.pdf'}
+page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E3T4oBgHgl3EQfGAm_/content/2301.04311v1.pdf'}
+page_content=' 1, we illustrate two typical  hardware architectures for implementing active-IRS reflecting  elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E3T4oBgHgl3EQfGAm_/content/2301.04311v1.pdf'}
+page_content=' The first one (type I) is implemented by the APS  circuit, which is the integration of an operational amplifier and  a phase shifter [6], [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E3T4oBgHgl3EQfGAm_/content/2301.04311v1.pdf'}
+page_content=' Specifically, triggered by a biasing power  source, the operational amplifier amplifies the weak electric  signal with desired amplification factor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E3T4oBgHgl3EQfGAm_/content/2301.04311v1.pdf'}
+page_content=' Meanwhile, the phase  shifting circuit (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E3T4oBgHgl3EQfGAm_/content/2301.04311v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E3T4oBgHgl3EQfGAm_/content/2301.04311v1.pdf'}
+page_content=', positive-intrinsic-negative (PIN) diode,  varactor diode [6]) connected to the amplifier enables flexible  phase modulation to the reflected signal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E3T4oBgHgl3EQfGAm_/content/2301.04311v1.pdf'}
+page_content=' Moreover, to achieve  maximum power transfer to the radiating structure, a match-  ing/coupling circuit is added to maintain a good impedance  match between the APS circuit and the radiating structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E3T4oBgHgl3EQfGAm_/content/2301.04311v1.pdf'}
+page_content='  In contrast, the second implementation of active-IRS reflecting  element (type II) can be achieved by load modulation circuits  based on e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E3T4oBgHgl3EQfGAm_/content/2301.04311v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E3T4oBgHgl3EQfGAm_/content/2301.04311v1.pdf'}
+page_content=', tunnel diodes [8] and complementary metal-oxide  semiconductor (CMOS) [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E3T4oBgHgl3EQfGAm_/content/2301.04311v1.pdf'}
+page_content=' Taking the tunnel diode [8] as  an example, by imposing proper bias voltage, it can operate  in the region with negative differential resistance such that  the reflecting element has an active load impedance (negative  resistance).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E3T4oBgHgl3EQfGAm_/content/2301.04311v1.pdf'}
+page_content=' As such, the reflection amplitude of incident signals  can be increased by converting the direct current (DC) power  into the RF power, while the phase shift is adjusted by tuning  the circuit capacitance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E3T4oBgHgl3EQfGAm_/content/2301.04311v1.pdf'}
+page_content='  3  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E3T4oBgHgl3EQfGAm_/content/2301.04311v1.pdf'}
+page_content=' Signal and Channel Models  The signal and channel models for active IRS are more  complicated than those for passive IRS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E3T4oBgHgl3EQfGAm_/content/2301.04311v1.pdf'}
+page_content=' First, different from  the noise-free passive IRS, the noise introduced by active  IRS is composed of both dynamic noise and static noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E3T4oBgHgl3EQfGAm_/content/2301.04311v1.pdf'}
+page_content='  Specifically, the dynamic noise is related to the input noise  which is determined by the source resistance of the antenna  and inherent device noise;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E3T4oBgHgl3EQfGAm_/content/2301.04311v1.pdf'}
+page_content=' while the static noise is independent  of the transmit/amplification power, which is usually negli-  gible compared to the dynamic noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E3T4oBgHgl3EQfGAm_/content/2301.04311v1.pdf'}
+page_content=' Second, the signal at  the receiver via active IRS is superposed by the desired sig-  nals over both the cascaded transmitter-IRS-receiver and direct  transmitter-receiver links, the amplification noise generated at  each reflecting element over the IRS-receiver link, and the  additive noise at the receiver.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E3T4oBgHgl3EQfGAm_/content/2301.04311v1.pdf'}
+page_content=' Let M denote the number of  active-IRS elements and x denote the transmitted signal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E3T4oBgHgl3EQfGAm_/content/2301.04311v1.pdf'}
+page_content=' Then  the received signal y at the receiver can be mathematically  expressed as y  =  �  hHΨg + t  �  x + hHΨzI + z0, where  hH, g, and t denote the equivalent baseband channels of  the IRS→receiver, transmitter→IRS, and transmitter→receiver  links;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E3T4oBgHgl3EQfGAm_/content/2301.04311v1.pdf'}
+page_content=' Ψ ≜ diag (α1, · · · , αM) diag  �  eȷφ1, · · · , eȷφM �  is the  active-IRS reflection matrix with αm and φm representing  the amplification factor and phase shift of the m-th reflecting  element, respectively;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E3T4oBgHgl3EQfGAm_/content/2301.04311v1.pdf'}
+page_content=' zI and z0 denote the amplification noise  induced by active IRS and the additive noise at the receiver.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E3T4oBgHgl3EQfGAm_/content/2301.04311v1.pdf'}
+page_content='  C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E3T4oBgHgl3EQfGAm_/content/2301.04311v1.pdf'}
+page_content=' Practical Constraints  Active IRS usually has smaller amplification power than  active relay, which thus limits its maximum amplification factors  at the reflecting elements in practice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E3T4oBgHgl3EQfGAm_/content/2301.04311v1.pdf'}
+page_content=' Generally speaking, there  are two practical amplification power models applicable to  active IRS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E3T4oBgHgl3EQfGAm_/content/2301.04311v1.pdf'}
+page_content=' In the first model, the amplification factors of all  reflecting elements are subject to a total power constraint as  they share a common power source [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E3T4oBgHgl3EQfGAm_/content/2301.04311v1.pdf'}
+page_content=' In this case, the total  amplification power can be allocated to reflecting elements based  on the corresponding IRS-related channels to achieve optimal  performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E3T4oBgHgl3EQfGAm_/content/2301.04311v1.pdf'}
+page_content=' While for the second model, in contrast, each  active-IRS reflecting element is attached with a separate power  source with independent power control.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E3T4oBgHgl3EQfGAm_/content/2301.04311v1.pdf'}
+page_content=' As such, the per-element  maximum amplification factor is constrained by the individual  per-element amplification power.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E3T4oBgHgl3EQfGAm_/content/2301.04311v1.pdf'}
+page_content='  Moreover, in practice, other hardware constraints should be  considered for modeling active IRS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E3T4oBgHgl3EQfGAm_/content/2301.04311v1.pdf'}
+page_content=' First, it is worth noting that  the reflection amplitude based on the negative resistance circuit  is generally phase-dependent, because both the amplification  factor and phase shift are functions of the effective capacitance  and resistance of each reflecting element.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E3T4oBgHgl3EQfGAm_/content/2301.04311v1.pdf'}
+page_content=' Second, for the active-  IRS element based on tunnel diode, its load impedance is  determined by the effective capacitance and resistance;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E3T4oBgHgl3EQfGAm_/content/2301.04311v1.pdf'}
+page_content=' thus, its  maximum amplification factor is constrained by the inherent  load impedance, which cannot be tuned arbitrarily large regard-  less of the power supply.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E3T4oBgHgl3EQfGAm_/content/2301.04311v1.pdf'}
+page_content=' Last but not least, the resolutions of  active-IRS phase shift and amplification factor depend on the  specific hardware implementation, which need to take discrete  values in practice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E3T4oBgHgl3EQfGAm_/content/2301.04311v1.pdf'}
+page_content='  D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E3T4oBgHgl3EQfGAm_/content/2301.04311v1.pdf'}
+page_content=' Comparison with Passive IRS and Active Relay  In Table I, we compare the main characteristics of active  IRS with its two counterpart technologies for wireless relaying,  namely, passive IRS and active relay.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E3T4oBgHgl3EQfGAm_/content/2301.04311v1.pdf'}
+page_content=' First, in terms of hardware  cost and power consumption, active IRS surpasses passive IRS  due to the additional reflection-type amplifiers, while it falls  below active relay since the latter requires power-hungry and  costly RF chains for signal processing and amplification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E3T4oBgHgl3EQfGAm_/content/2301.04311v1.pdf'}
+page_content='  Next, compared to passive IRS, active IRS endows additional  amplitude amplification for the reflected signal, thus attaining  more flexible beam steering and larger coverage (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E3T4oBgHgl3EQfGAm_/content/2301.04311v1.pdf'}
+page_content=' 2(a)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E3T4oBgHgl3EQfGAm_/content/2301.04311v1.pdf'}
+page_content='  However, its beamforming performance is generally inferior to  half-/full-duplex active relay, since active IRS only reflects and  amplifies signals in the RF band with an equivalent diagonal  baseband reflection matrix (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E3T4oBgHgl3EQfGAm_/content/2301.04311v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E3T4oBgHgl3EQfGAm_/content/2301.04311v1.pdf'}
+page_content=', Ψ), while active relay with  multiple antennas have more sophisticated signal processing  capability with typically a full-rank MIMO signal processing  matrix at the baseband.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E3T4oBgHgl3EQfGAm_/content/2301.04311v1.pdf'}
+page_content='  Last, wireless communication systems aided by active IRS  generally attain higher spectral efficiency than those aided by  passive IRS and half-duplex active relay.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E3T4oBgHgl3EQfGAm_/content/2301.04311v1.pdf'}
+page_content=' On one hand, active  IRS enables power amplification when reflecting signals, thus  leading to a higher received SNR at the receiver than passive  IRS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E3T4oBgHgl3EQfGAm_/content/2301.04311v1.pdf'}
+page_content=' On the other hand, practical active relay usually operates  in the half-duplex mode1, thus suffering considerable rate per-  formance loss compared to full-duplex active IRS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E3T4oBgHgl3EQfGAm_/content/2301.04311v1.pdf'}
+page_content='  III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E3T4oBgHgl3EQfGAm_/content/2301.04311v1.pdf'}
+page_content=' NEW DESIGN CONSIDERATIONS AND CHALLENGES  In this section, we present new considerations and open  issues in designing active-IRS-aided wireless communications,  including the reflection optimization, channel estimation, and  deployment for active IRS, as well as its efficient integration  with passive IRS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E3T4oBgHgl3EQfGAm_/content/2301.04311v1.pdf'}
+page_content=' Moreover, promising solutions for tackling  these issues are also proposed to inspire future work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E3T4oBgHgl3EQfGAm_/content/2301.04311v1.pdf'}
+page_content='  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E3T4oBgHgl3EQfGAm_/content/2301.04311v1.pdf'}
+page_content=' Active-IRS Reflection Optimization  One key challenge in active-IRS-aided communication design  is how to optimize the reflection coefficients of active IRS for  achieving the optimal beamforming and SNR performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E3T4oBgHgl3EQfGAm_/content/2301.04311v1.pdf'}
+page_content=' To  address it, several practical considerations need to be taken  into account such as the phase-amplitude dependent control,  maximum amplification factor, discrete phase shifts/amplitude  levels (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E3T4oBgHgl3EQfGAm_/content/2301.04311v1.pdf'}
+page_content=' Section II-C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E3T4oBgHgl3EQfGAm_/content/2301.04311v1.pdf'}
+page_content=' These considerations usually result  in non-convex constraints over the reflection coefficients and  hence non-convex optimization problems [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E3T4oBgHgl3EQfGAm_/content/2301.04311v1.pdf'}
+page_content=' In addition, as the  amplification factor of active-IRS element can be controlled, the  resulting design variables are significantly more than those of  passive IRS for which the reflection amplitude is usually set  as one or its maximum value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E3T4oBgHgl3EQfGAm_/content/2301.04311v1.pdf'}
+page_content=' Several algorithms have been  proposed in the literature to solve the above problem (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E3T4oBgHgl3EQfGAm_/content/2301.04311v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E3T4oBgHgl3EQfGAm_/content/2301.04311v1.pdf'}
+page_content=',  [3]–[6]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E3T4oBgHgl3EQfGAm_/content/2301.04311v1.pdf'}
+page_content=' For example, the alternating optimization (AO) method  can be applied to iteratively optimize the amplification factor  and phase shift, until the convergence is achieved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E3T4oBgHgl3EQfGAm_/content/2301.04311v1.pdf'}
+page_content=' However, the  performance of AO critically depends on its initialization, which,  1Full-duplex relay requires higher hardware and implementation cost than  half-duplex relay due to the need to eliminate the loop-back self-interference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E3T4oBgHgl3EQfGAm_/content/2301.04311v1.pdf'}
+page_content='  4  TABLE I  COMPARISONS OF ACTIVE IRS, PASSIVE IRS, AND ACTIVE RELAY.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E3T4oBgHgl3EQfGAm_/content/2301.04311v1.pdf'}
+page_content='  Active IRS  Passive IRS  Active relay  Hardware cost  Low  Very low  High  Power consumption  Moderate  Very low  High  Operation mechanism  Reflect and amplify  Reflect  Receive and amplify  Duplex mode  Full duplex  Full duplex  Full/half duplex  Amplification noise  Yes  No  Yes  if not properly designed, can result in considerable performance  loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E3T4oBgHgl3EQfGAm_/content/2301.04311v1.pdf'}
+page_content=' Moreover, existing works (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E3T4oBgHgl3EQfGAm_/content/2301.04311v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E3T4oBgHgl3EQfGAm_/content/2301.04311v1.pdf'}
+page_content=', [3]–[6]) usually assumed  ideal reflection coefficients with continuous and independent  phase-shift and amplitude control, which cannot be realized in  practice due to hardware limitations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E3T4oBgHgl3EQfGAm_/content/2301.04311v1.pdf'}
+page_content=' Thus, efficient algorithms  that can handle more practical constraints are worthy of future  investigation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E3T4oBgHgl3EQfGAm_/content/2301.04311v1.pdf'}
+page_content='  Several results on the active-IRS reflection design under  different channel conditions and/or system setups have been  reported in the literature (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E3T4oBgHgl3EQfGAm_/content/2301.04311v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E3T4oBgHgl3EQfGAm_/content/2301.04311v1.pdf'}
+page_content=', [4], [10]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E3T4oBgHgl3EQfGAm_/content/2301.04311v1.pdf'}
+page_content=' Taking the single-user  case as an example, it was shown that the reflected signals over  different IRS elements should be phase-aligned such that they  can be constructively added at the receiver, for both the cases  with active or passive IRS [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E3T4oBgHgl3EQfGAm_/content/2301.04311v1.pdf'}
+page_content=' Moreover, the amplification  factor should be set to the maximum available value for com-  pensating the cascaded channel path loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E3T4oBgHgl3EQfGAm_/content/2301.04311v1.pdf'}
+page_content=' Next, for the general  multi-user case, there exists a new trade-off between signal  power enhancement and interference mitigation in designing  the active-IRS reflection coefficients given a total power budget  constraint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E3T4oBgHgl3EQfGAm_/content/2301.04311v1.pdf'}
+page_content=' In this case, the amplification factors of reflecting  elements should be properly set, which may not necessarily  equal to their maximum values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E3T4oBgHgl3EQfGAm_/content/2301.04311v1.pdf'}
+page_content=' Generally speaking, active IRS  is expected to achieve better interference mitigation performance  than passive IRS, since its amplitude control endows more DoFs  to eliminate the undesired interference more effectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E3T4oBgHgl3EQfGAm_/content/2301.04311v1.pdf'}
+page_content='  An important performance metric in the active-IRS reflection  design is how the received SNR scales with the number of  reflecting elements, M, since such scaling orders characterize  the SNR performance when the number of IRS elements is  asymptotically large.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E3T4oBgHgl3EQfGAm_/content/2301.04311v1.pdf'}
+page_content=' First, consider the case with a total power  constraint over all active-IRS elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E3T4oBgHgl3EQfGAm_/content/2301.04311v1.pdf'}
+page_content=' It was reported in [3]  that the received SNR with an active IRS linearly increases with  M when M is sufficiently large.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E3T4oBgHgl3EQfGAm_/content/2301.04311v1.pdf'}
+page_content=' This is expected since the signal  power at the receiver quadratically increases with M thanks to  the IRS beamforming gain, while the amplification noise power  also increases linearly with M, thus leading to the SNR scaling  order of M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E3T4oBgHgl3EQfGAm_/content/2301.04311v1.pdf'}
+page_content=' However, it is worth noting that although active  IRS has a smaller SNR scaling order than passive IRS without  amplification noise (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E3T4oBgHgl3EQfGAm_/content/2301.04311v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E3T4oBgHgl3EQfGAm_/content/2301.04311v1.pdf'}
+page_content=', O (M) versus O  �  M 2�  [2]), it provides  superior rate performance than passive IRS when the number of  reflecting elements is moderate thanks to the additional signal  power amplification gain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E3T4oBgHgl3EQfGAm_/content/2301.04311v1.pdf'}
+page_content=' Next, consider the case with fixed  per-element power for each active-IRS element.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E3T4oBgHgl3EQfGAm_/content/2301.04311v1.pdf'}
+page_content=' In this case,  the total power budget linearly increases with the number of  reflecting elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E3T4oBgHgl3EQfGAm_/content/2301.04311v1.pdf'}
+page_content=' As such, the active IRS is expected to  achieve additional amplification power gain in the order of M,  hence yielding the same received SNR scaling order with the  conventional passive IRS, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E3T4oBgHgl3EQfGAm_/content/2301.04311v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E3T4oBgHgl3EQfGAm_/content/2301.04311v1.pdf'}
+page_content=', O  �  M 2�  (such SNR comparisons  will be numerically shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E3T4oBgHgl3EQfGAm_/content/2301.04311v1.pdf'}
+page_content=' 5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E3T4oBgHgl3EQfGAm_/content/2301.04311v1.pdf'}
+page_content='  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E3T4oBgHgl3EQfGAm_/content/2301.04311v1.pdf'}
+page_content=' Active-IRS Channel Estimation  Active-IRS reflection design requires accurate channel state  information (CSI) of IRS-related channels for fully exploiting its  beamforming and amplification gains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E3T4oBgHgl3EQfGAm_/content/2301.04311v1.pdf'}
+page_content=' Compared with passive  IRS, the CSI acquisition for active IRS is generally more  challenging due to the following reasons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E3T4oBgHgl3EQfGAm_/content/2301.04311v1.pdf'}
+page_content=' First, the active-IRS  reflection design generally requires explicit CSI of the separate  transmitter-IRS and IRS-receiver links due to the new consider-  ation of amplification noise [10], which greatly differs from the  passive-IRS case that requires cascaded CSI only.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E3T4oBgHgl3EQfGAm_/content/2301.04311v1.pdf'}
+page_content=' Specifically,  although the phase shift can be designed based on cascaded  CSI, the amplification factor control is generally dependent on  the incident signal power at active IRS and its power budget,  while the incident signal power depends on the transmitter-IRS  CSI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E3T4oBgHgl3EQfGAm_/content/2301.04311v1.pdf'}
+page_content=' Second, in active-IRS channel estimation, it is difficult to  set the amplification factor to meet the given total/per-element  power constraint exactly, since this requires the knowledge of  incident signal power that depends on the transmitter-IRS CSI,  which is yet to be estimated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E3T4oBgHgl3EQfGAm_/content/2301.04311v1.pdf'}
+page_content=' Last, the amplification noise also  affects the channel estimation performance, while this issue does  not exist in the channel estimation for passive IRS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E3T4oBgHgl3EQfGAm_/content/2301.04311v1.pdf'}
+page_content='  In the following, we introduce two practical active-IRS chan-  nel estimation approaches, with and without IRS-mounted sen-  sors, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E3T4oBgHgl3EQfGAm_/content/2301.04311v1.pdf'}
+page_content=' The first approach requires low-cost sensors  installed on the active IRS, which are interleaved with the  reflecting elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E3T4oBgHgl3EQfGAm_/content/2301.04311v1.pdf'}
+page_content=' Equipped with these sensors that can receive  signals sent by the BS/users, active IRS can estimate the BS-  IRS and IRS-user channels separately by exploiting the strong  channel correlations between the IRS elements and sensors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E3T4oBgHgl3EQfGAm_/content/2301.04311v1.pdf'}
+page_content=' It  is worth noting that active IRS in general requires much fewer  reflecting elements than passive IRS for achieving comparable  rate performance, thus it is easier to place reflecting elements  with sensors close to each other to achieve high channel cor-  relation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E3T4oBgHgl3EQfGAm_/content/2301.04311v1.pdf'}
+page_content=' Second, for active IRS without sensors installed, one  promising approach is to combine the separate and cascaded  channel estimation methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E3T4oBgHgl3EQfGAm_/content/2301.04311v1.pdf'}
+page_content=' Specifically, the BS first estimates  the BS-IRS CSI based on the reflected signal over the BS-  IRS-BS link (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E3T4oBgHgl3EQfGAm_/content/2301.04311v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E3T4oBgHgl3EQfGAm_/content/2301.04311v1.pdf'}
+page_content=', by wireless sensing of the BS).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E3T4oBgHgl3EQfGAm_/content/2301.04311v1.pdf'}
+page_content=' Next, the  BS estimates the cascaded BS-IRS-user CSI based on the pilot  signals from the user, and then resolves the IRS-user CSI based  on the estimated cascaded CSI and BS-IRS CSI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E3T4oBgHgl3EQfGAm_/content/2301.04311v1.pdf'}
+page_content=' Moreover,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E3T4oBgHgl3EQfGAm_/content/2301.04311v1.pdf'}
+page_content='  for the amplification factor control in channel estimation,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E3T4oBgHgl3EQfGAm_/content/2301.04311v1.pdf'}
+page_content=' one  practical method is by setting an equal IRS reflection amplitude  over all reflecting elements based on the received signal powers  5  BS  User  (d) Multi-IRS reflection with active and passive IRSs  BS  User  Downlink communication  Uplink communication  (c) Hybrid active-passive IRS  (b) Active-IRS deployment  (a) Active-IRS reflection design  Optimal location: closer to user  Optimal location: closer to BS  Passive IRS  Active IRS  Narrower beam and larger coverage  than passive IRS  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E3T4oBgHgl3EQfGAm_/content/2301.04311v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E3T4oBgHgl3EQfGAm_/content/2301.04311v1.pdf'}
+page_content=' New design considerations for active-IRS-aided wireless communications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E3T4oBgHgl3EQfGAm_/content/2301.04311v1.pdf'}
+page_content='  (with sensors) or simply setting the amplification factor equal  to one (without sensors) similar to the case of passive IRS, to  facilitate the estimation of IRS-user CSI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E3T4oBgHgl3EQfGAm_/content/2301.04311v1.pdf'}
+page_content='  C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E3T4oBgHgl3EQfGAm_/content/2301.04311v1.pdf'}
+page_content=' Active-IRS Deployment  How to deploy active IRSs in wireless networks to achieve the  largest network coverage and maximum throughput is another  crucial problem to solve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E3T4oBgHgl3EQfGAm_/content/2301.04311v1.pdf'}
+page_content=' Generally speaking, several factors  need to be considered in the active-IRS deployment design, such  as the locations of users, uplink or downlink communication,  number of reflecting elements, amplification power budget,  amplification noise power, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E3T4oBgHgl3EQfGAm_/content/2301.04311v1.pdf'}
+page_content='  First, consider the single-user case where the active-IRS  location needs to be optimized for achieving the maximum  received SNR at the user.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E3T4oBgHgl3EQfGAm_/content/2301.04311v1.pdf'}
+page_content=' In this case, it has been shown in  [10] that active IRS should be properly deployed between the  transmitter and receiver to balance the large-scale path loss  of the transmitter-IRS and IRS-receiver channels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E3T4oBgHgl3EQfGAm_/content/2301.04311v1.pdf'}
+page_content=' In particular,  with a small amplification power or a large number of reflecting  elements, active IRS should be deployed closer to the receiver;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E3T4oBgHgl3EQfGAm_/content/2301.04311v1.pdf'}
+page_content='  otherwise, the amplification factors are too small to compensate  the amplification noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E3T4oBgHgl3EQfGAm_/content/2301.04311v1.pdf'}
+page_content=' This is different from the passive-IRS  case where the IRS should be deployed near either the trans-  mitter or receiver to minimize the cascaded path loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E3T4oBgHgl3EQfGAm_/content/2301.04311v1.pdf'}
+page_content=' Besides,  the optimal active-IRS deployment locations are different in the  uplink and downlink for two-way communications in general,  and thus their performance should be balanced by choosing an  appropriate location for active IRS (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E3T4oBgHgl3EQfGAm_/content/2301.04311v1.pdf'}
+page_content=' 2(b)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E3T4oBgHgl3EQfGAm_/content/2301.04311v1.pdf'}
+page_content=' Second, for  the multi-user case, the active-IRS deployment design is more  complicated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E3T4oBgHgl3EQfGAm_/content/2301.04311v1.pdf'}
+page_content=' On one hand, it needs to strike the performance  balance among different users.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E3T4oBgHgl3EQfGAm_/content/2301.04311v1.pdf'}
+page_content=' On the other hand, the active  IRS may not be deployed at the BS/AP side as in the passive-  IRS case, since it also needs to balance the powers of desired  signal and amplification noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E3T4oBgHgl3EQfGAm_/content/2301.04311v1.pdf'}
+page_content=' To tackle this problem, instead of  employing a computationally prohibitive brute-force search, one  efficient and low-complexity approach is by leveraging machine  learning (ML) and geometry-based optimization methods to  deploy the active IRS at a location with good statistical channel  conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E3T4oBgHgl3EQfGAm_/content/2301.04311v1.pdf'}
+page_content=' Besides active-IRS location, the reflection gain of  active IRS is also affected by its rotation angle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E3T4oBgHgl3EQfGAm_/content/2301.04311v1.pdf'}
+page_content=' As shown  in [11], considerable rate performance improvement can be  attained by properly rotating the IRS without changing its  locations;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E3T4oBgHgl3EQfGAm_/content/2301.04311v1.pdf'}
+page_content=' thus, the IRS rotation should be carefully tuned to  maximize the number of served users and IRS-user channel  gains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E3T4oBgHgl3EQfGAm_/content/2301.04311v1.pdf'}
+page_content='  Compared with single-IRS deployment, the multi-active-IRS  deployment is generally more challenging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E3T4oBgHgl3EQfGAm_/content/2301.04311v1.pdf'}
+page_content=' In particular, there  exists a fundamental trade-off between the deployment cost  (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E3T4oBgHgl3EQfGAm_/content/2301.04311v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E3T4oBgHgl3EQfGAm_/content/2301.04311v1.pdf'}
+page_content=', hardware cost, power consumption) and communication  performance (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E3T4oBgHgl3EQfGAm_/content/2301.04311v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E3T4oBgHgl3EQfGAm_/content/2301.04311v1.pdf'}
+page_content=', network coverage, received SNR).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E3T4oBgHgl3EQfGAm_/content/2301.04311v1.pdf'}
+page_content=' As such,  besides deployment locations, the number of active IRSs needs  to be carefully chosen to minimize the total deployment cost,  while achieving the communication performance requirements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E3T4oBgHgl3EQfGAm_/content/2301.04311v1.pdf'}
+page_content='  Furthermore, for indoor environment, how to deploy multiple  IRSs to maximize the network connectivity subject to the indoor  topology constraints is a challenging problem, which deserves  future investigation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E3T4oBgHgl3EQfGAm_/content/2301.04311v1.pdf'}
+page_content='  D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E3T4oBgHgl3EQfGAm_/content/2301.04311v1.pdf'}
+page_content=' Integrated Design with Passive IRS  While the existing works mainly considered the wireless  communication systems aided by active or passive IRS only,  it has been recently shown that the integrated design with  both active and passive IRSs has the potential to achieve their  combined advantages at the same time [12]–[14], as discussed  below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E3T4oBgHgl3EQfGAm_/content/2301.04311v1.pdf'}
+page_content='  First, it was shown in [12] that given the total deployment  budget and amplification power, a hybrid IRS with both passive  and active elements (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E3T4oBgHgl3EQfGAm_/content/2301.04311v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E3T4oBgHgl3EQfGAm_/content/2301.04311v1.pdf'}
+page_content=', see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E3T4oBgHgl3EQfGAm_/content/2301.04311v1.pdf'}
+page_content=' 2(c)) can achieve better  communication performance than the conventional IRS with  active or passive elements only.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E3T4oBgHgl3EQfGAm_/content/2301.04311v1.pdf'}
+page_content=' Specifically, there generally  exists a trade-off between deploying more active elements to  achieve the appealing power amplification gain and deploying  more passive elements to attain a higher beamforming gain,  thus leading to the new design issue of active versus passive  elements allocation [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E3T4oBgHgl3EQfGAm_/content/2301.04311v1.pdf'}
+page_content=' Generally speaking, when the number  of reflecting elements is small and/or the amplification power  is large, more active elements should be deployed to harness  the power amplification gain;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E3T4oBgHgl3EQfGAm_/content/2301.04311v1.pdf'}
+page_content=' otherwise, deploying more passive  elements is expected to achieve better performance thanks to  their higher-order beamforming gain [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E3T4oBgHgl3EQfGAm_/content/2301.04311v1.pdf'}
+page_content='  Another type of the integrated design is deploying both  stand-alone active and passive IRSs in the wireless network,  ()6  BS  IRS/Relay  User  x  z  H  D  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E3T4oBgHgl3EQfGAm_/content/2301.04311v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E3T4oBgHgl3EQfGAm_/content/2301.04311v1.pdf'}
+page_content=' Simulation setup  as illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E3T4oBgHgl3EQfGAm_/content/2301.04311v1.pdf'}
+page_content=' 2(d), to improve the communication  performance [13], [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E3T4oBgHgl3EQfGAm_/content/2301.04311v1.pdf'}
+page_content=' The advantages are two-folds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E3T4oBgHgl3EQfGAm_/content/2301.04311v1.pdf'}
+page_content=' First,  deploying multiple cooperative IRSs provides the opportunity  to construct a multi-hop reflection path between the transmitter  and receiver to bypass environmental obstacles [15], which is  particularly appealing in complex environment (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E3T4oBgHgl3EQfGAm_/content/2301.04311v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E3T4oBgHgl3EQfGAm_/content/2301.04311v1.pdf'}
+page_content=', indoor).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E3T4oBgHgl3EQfGAm_/content/2301.04311v1.pdf'}
+page_content='  Second, compared with the conventional multi-passive-IRS net-  work, adding a number of active IRSs in the network can  substantially enhance the communication performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E3T4oBgHgl3EQfGAm_/content/2301.04311v1.pdf'}
+page_content=' This is  because active IRSs can opportunistically amplify the reflected  signal along the multi-reflection path, thus effectively compen-  sating the severe end-to-end product-distance path loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E3T4oBgHgl3EQfGAm_/content/2301.04311v1.pdf'}
+page_content=' Despite  the advantages, the design of multi-active and multi-passive  (MAMP) IRS aided communications is also more challenging  [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E3T4oBgHgl3EQfGAm_/content/2301.04311v1.pdf'}
+page_content=' For example, the multi-IRS beam routing path with both  active and passive IRSs should be carefully devised to maximize  the power amplification gain, and at the same time, minimize  the accumulated amplification noise power arising from active  IRSs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E3T4oBgHgl3EQfGAm_/content/2301.04311v1.pdf'}
+page_content=' Moreover, the numbers of active and passive IRSs should  also be carefully chosen given limited deployment budget and  their locations should be optimally designed [14], which deserve  further studies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E3T4oBgHgl3EQfGAm_/content/2301.04311v1.pdf'}
+page_content='  IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E3T4oBgHgl3EQfGAm_/content/2301.04311v1.pdf'}
+page_content=' NUMERICAL RESULTS  Numerical results are presented in this section to demonstrate  the potential performance gains of active IRS as compared  to passive IRS and active relay.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E3T4oBgHgl3EQfGAm_/content/2301.04311v1.pdf'}
+page_content=' Specifically, we consider a  typical point-to-point communication system as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E3T4oBgHgl3EQfGAm_/content/2301.04311v1.pdf'}
+page_content='  3, where an IRS or a relay is deployed at an altitude of 2  meters (m) to assist the communication between a BS and a user  (both equipped with single antenna), while the direct BS-user  channel is assumed being blocked.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E3T4oBgHgl3EQfGAm_/content/2301.04311v1.pdf'}
+page_content=' For each reflecting element,  we consider the continuous and independently controlled phase  shift and reflection amplitude (the amplitude is set to be one for  passive IRS).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E3T4oBgHgl3EQfGAm_/content/2301.04311v1.pdf'}
+page_content=' Moreover, the line-of-sight (LoS) channel model  is assumed for all involved links, where the path loss exponent  is 2 and the reference channel power gain at a distance of 1 m  is −30 dB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E3T4oBgHgl3EQfGAm_/content/2301.04311v1.pdf'}
+page_content=' Both powers of the receiver noise and per-element  active-IRS amplification noise are set as −80 dBm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E3T4oBgHgl3EQfGAm_/content/2301.04311v1.pdf'}
+page_content='  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E3T4oBgHgl3EQfGAm_/content/2301.04311v1.pdf'}
+page_content=' 4 shows the user’s maximum achievable rates versus the  BS-user distance in four wireless communication systems aided  by: 1) an active IRS with 128 reflecting elements and a total  amplification power of 10 dBm;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E3T4oBgHgl3EQfGAm_/content/2301.04311v1.pdf'}
+page_content=' 2) a passive IRS with 128  reflecting elements;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E3T4oBgHgl3EQfGAm_/content/2301.04311v1.pdf'}
+page_content=' 3) an ideal full-duplex amplify-and-forward  (AF) relay with perfect self-interference cancellation and a total  amplification power of 15 dBm;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E3T4oBgHgl3EQfGAm_/content/2301.04311v1.pdf'}
+page_content=' and 4) a half-duplex AF relay  with optimal time division and a total amplification power of  15 dBm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E3T4oBgHgl3EQfGAm_/content/2301.04311v1.pdf'}
+page_content=' It is observed that given the same BS-user distance,  150  200  250  300  350  400  450  Distance between BS and user (m)  0  5  10  15  Achievable Rate (bps/Hz)  Active IRS  Passive IRS  Full-duplex AF Relay  Half-duplex AF Relay  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E3T4oBgHgl3EQfGAm_/content/2301.04311v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E3T4oBgHgl3EQfGAm_/content/2301.04311v1.pdf'}
+page_content=' Achievable rate versus user location.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E3T4oBgHgl3EQfGAm_/content/2301.04311v1.pdf'}
+page_content='  50  100  150  200  250  300  350  400  450  Number of reflecting elements  0  5  10  15  20  25  30  35  SNR (dB)  Active IRS with FTP  Active IRS with FPP  Passive IRS  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E3T4oBgHgl3EQfGAm_/content/2301.04311v1.pdf'}
+page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E3T4oBgHgl3EQfGAm_/content/2301.04311v1.pdf'}
+page_content=' Received SNR versus number of reflecting elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E3T4oBgHgl3EQfGAm_/content/2301.04311v1.pdf'}
+page_content='  active IRS always achieves a higher rate than passive IRS and  half-duplex relay.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E3T4oBgHgl3EQfGAm_/content/2301.04311v1.pdf'}
+page_content=' This is expected since active IRS provides an  additional power amplification gain compared to passive IRS,  and it achieves the full-duplex gain compared to half-duplex  relay.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E3T4oBgHgl3EQfGAm_/content/2301.04311v1.pdf'}
+page_content=' Besides, active IRS is inferior to full-duplex active relay  due to its limited signal processing capability and less power  supply than active relay.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E3T4oBgHgl3EQfGAm_/content/2301.04311v1.pdf'}
+page_content='  Next, we show in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E3T4oBgHgl3EQfGAm_/content/2301.04311v1.pdf'}
+page_content=' 5 how the received SNR scales with the  number of reflecting elements in three wireless communication  systems aided by: 1) an active IRS with a fixed total power (FTP)  of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E3T4oBgHgl3EQfGAm_/content/2301.04311v1.pdf'}
+page_content='2 W;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E3T4oBgHgl3EQfGAm_/content/2301.04311v1.pdf'}
+page_content=' 2) an active IRS with a fixed per-element power (FPP)  of 1 mW;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E3T4oBgHgl3EQfGAm_/content/2301.04311v1.pdf'}
+page_content=' and 3) a passive IRS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E3T4oBgHgl3EQfGAm_/content/2301.04311v1.pdf'}
+page_content=' For the above three systems, the  corresponding IRS reflections are optimized for maximizing the  received SNR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E3T4oBgHgl3EQfGAm_/content/2301.04311v1.pdf'}
+page_content=' First, it is observed that both active-IRS-aided  systems achieve much higher received SNRs than the passive-  IRS counterpart.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E3T4oBgHgl3EQfGAm_/content/2301.04311v1.pdf'}
+page_content=' This is expected because active IRS provides  both the beamforming gain and power amplification gain, while  passive IRS only offers the beamforming gain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E3T4oBgHgl3EQfGAm_/content/2301.04311v1.pdf'}
+page_content=' Second, one can  observe that with an increasing number of reflecting elements,  the received SNR of active IRS with FPP is firstly lower than and  then exceeds that with FTP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E3T4oBgHgl3EQfGAm_/content/2301.04311v1.pdf'}
+page_content=' This is because in the latter case,  the average per-element power decreases with the number of  reflecting elements, thus yielding a lower received SNR scaling  order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E3T4oBgHgl3EQfGAm_/content/2301.04311v1.pdf'}
+page_content='  7  V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E3T4oBgHgl3EQfGAm_/content/2301.04311v1.pdf'}
+page_content=' CONCLUSIONS  In this article, we provide a comprehensive overview of the  newly proposed active IRS that enables simultaneous signal  reflection and amplification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E3T4oBgHgl3EQfGAm_/content/2301.04311v1.pdf'}
+page_content=' Specifically, we first present the  fundamentals of active IRS, including its hardware architecture,  signal and channel models, as well as practical constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E3T4oBgHgl3EQfGAm_/content/2301.04311v1.pdf'}
+page_content=' Then,  new considerations and open issues in designing active-IRS-  aided wireless communications are discussed to shed new light  to promising solutions, including the reflection optimization,  channel estimation, and deployment for active IRS, as well as its  integrated design with passive IRS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E3T4oBgHgl3EQfGAm_/content/2301.04311v1.pdf'}
+page_content=' Last, we numerically show  the potential performance gains of active IRS over passive IRS  and traditional half-duplex active relay, and the SNR scaling  orders of active IRS under different power constraints versus  passive IRS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E3T4oBgHgl3EQfGAm_/content/2301.04311v1.pdf'}
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+page_content=' 8, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E3T4oBgHgl3EQfGAm_/content/2301.04311v1.pdf'}
+page_content=' 6656–6671, Feb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E3T4oBgHgl3EQfGAm_/content/2301.04311v1.pdf'}
+page_content=' 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E3T4oBgHgl3EQfGAm_/content/2301.04311v1.pdf'}
+page_content='  [12] Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E3T4oBgHgl3EQfGAm_/content/2301.04311v1.pdf'}
+page_content=' Kang, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E3T4oBgHgl3EQfGAm_/content/2301.04311v1.pdf'}
+page_content=' You, and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E3T4oBgHgl3EQfGAm_/content/2301.04311v1.pdf'}
+page_content=' Zhang, “Active-passive IRS aided wireless  communication: New hybrid architecture and elements allocation optimiza-  tion,” [Online].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E3T4oBgHgl3EQfGAm_/content/2301.04311v1.pdf'}
+page_content=' Available: https://arxiv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E3T4oBgHgl3EQfGAm_/content/2301.04311v1.pdf'}
+page_content='org/pdf/2207.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E3T4oBgHgl3EQfGAm_/content/2301.04311v1.pdf'}
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+page_content=' 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E3T4oBgHgl3EQfGAm_/content/2301.04311v1.pdf'}
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+page_content=' You, and B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E3T4oBgHgl3EQfGAm_/content/2301.04311v1.pdf'}
+page_content=' Zheng, “Multi-active multi-passive (MAMP)-  IRS aided wireless communication: A multi-hop beam routing design,”  [Online].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E3T4oBgHgl3EQfGAm_/content/2301.04311v1.pdf'}
+page_content=' Available: https://arxiv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E3T4oBgHgl3EQfGAm_/content/2301.04311v1.pdf'}
+page_content='org/abs/2209.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E3T4oBgHgl3EQfGAm_/content/2301.04311v1.pdf'}
+page_content='06390, accessed on Sep.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E3T4oBgHgl3EQfGAm_/content/2301.04311v1.pdf'}
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+page_content='  [14] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E3T4oBgHgl3EQfGAm_/content/2301.04311v1.pdf'}
+page_content=' Fu and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E3T4oBgHgl3EQfGAm_/content/2301.04311v1.pdf'}
+page_content=' Zhang, “Active and passive IRS jointly aided communication:  Deployment design and achievable rate,” in IEEE Wireless Commun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E3T4oBgHgl3EQfGAm_/content/2301.04311v1.pdf'}
+page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E3T4oBgHgl3EQfGAm_/content/2301.04311v1.pdf'}
+page_content=',  Early Access.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E3T4oBgHgl3EQfGAm_/content/2301.04311v1.pdf'}
+page_content='  [15] W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E3T4oBgHgl3EQfGAm_/content/2301.04311v1.pdf'}
+page_content=' Mei and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E3T4oBgHgl3EQfGAm_/content/2301.04311v1.pdf'}
+page_content=' Zhang, “Multi-beam multi-hop routing for intelligent  reflecting surfaces aided massive MIMO,” IEEE Trans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E3T4oBgHgl3EQfGAm_/content/2301.04311v1.pdf'}
+page_content=' Wireless Commun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E3T4oBgHgl3EQfGAm_/content/2301.04311v1.pdf'}
+page_content=',  vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E3T4oBgHgl3EQfGAm_/content/2301.04311v1.pdf'}
+page_content=' 21, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E3T4oBgHgl3EQfGAm_/content/2301.04311v1.pdf'}
+page_content=' 3, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E3T4oBgHgl3EQfGAm_/content/2301.04311v1.pdf'}
+page_content=' 1897–1912, Sep.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E3T4oBgHgl3EQfGAm_/content/2301.04311v1.pdf'}
+page_content=' 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E3T4oBgHgl3EQfGAm_/content/2301.04311v1.pdf'}
+page_content='  BIOGRAPHIES  Zhenyu Kang (zhenyu kang@u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E3T4oBgHgl3EQfGAm_/content/2301.04311v1.pdf'}
+page_content='nus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E3T4oBgHgl3EQfGAm_/content/2301.04311v1.pdf'}
+page_content='edu) is a Ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E3T4oBgHgl3EQfGAm_/content/2301.04311v1.pdf'}
+page_content='D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E3T4oBgHgl3EQfGAm_/content/2301.04311v1.pdf'}
+page_content=' candidate  with National University of Singapore, Singapore.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E3T4oBgHgl3EQfGAm_/content/2301.04311v1.pdf'}
+page_content='  Changsheng You (youcs@sustech.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E3T4oBgHgl3EQfGAm_/content/2301.04311v1.pdf'}
+page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E3T4oBgHgl3EQfGAm_/content/2301.04311v1.pdf'}
+page_content='cn) received his Ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E3T4oBgHgl3EQfGAm_/content/2301.04311v1.pdf'}
+page_content='D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E3T4oBgHgl3EQfGAm_/content/2301.04311v1.pdf'}
+page_content='  degree from The University of Hong Kong.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E3T4oBgHgl3EQfGAm_/content/2301.04311v1.pdf'}
+page_content=' He is currently  an Assistant Professor with Southern University of Science and  Technology, China.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E3T4oBgHgl3EQfGAm_/content/2301.04311v1.pdf'}
+page_content='  Rui Zhang [F’17] (rzhang@cuhk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E3T4oBgHgl3EQfGAm_/content/2301.04311v1.pdf'}
+page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E3T4oBgHgl3EQfGAm_/content/2301.04311v1.pdf'}
+page_content='cn) received the Ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E3T4oBgHgl3EQfGAm_/content/2301.04311v1.pdf'}
+page_content='D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E3T4oBgHgl3EQfGAm_/content/2301.04311v1.pdf'}
+page_content=' de-  gree from Stanford University.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E3T4oBgHgl3EQfGAm_/content/2301.04311v1.pdf'}
+page_content=' He is now a Principal’s Diligence  Chair Professor with The Chinese University of Hong Kong,  Shenzhen, and also a Provost’s Chair Professor with National  University of Singapore.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E3T4oBgHgl3EQfGAm_/content/2301.04311v1.pdf'}
+page_content=' He is a Fellow of the IEEE and the  Academy of Engineering, Singapore.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E3T4oBgHgl3EQfGAm_/content/2301.04311v1.pdf'}
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+Force-Field-Enhanced Neural Network Interactions: from Local Equivariant
+Embedding to Atom-in-Molecule properties and long-range effects
+Thomas Plé,1, ∗ Louis Lagardère,1, † and Jean-Philip Piquemal1, ‡
+1Sorbonne Université, LCT, UMR 7616 CNRS, F-75005, Paris, France
+We introduce FENNIX (Force-Field-Enhanced Neural Network InteraXions), an hybrid approach
+between machine-learning and force-fields. We leverage state-of-the-art equivariant neural networks
+to predict local energy contributions and multiple atom-in-molecule properties that are then used as
+geometry-dependent parameters for physically-motivated energy terms which account for long-range
+electrostatics and dispersion. Using high-accuracy ab initio data (small organic molecules/dimers),
+we trained a first version of the model. Exhibiting accurate gas-phase energy predictions, FENNIX
+is transferable to the condensed phase. It is able to produce stable Molecular Dynamics simulations,
+including nuclear quantum effects, for water predicting accurate liquid properties. The extrapo-
+lating power of the hybrid physically-driven machine learning FENNIX approach is exemplified by
+computing: i) the solvated alanine dipeptide free energy landscape; ii) the reactive dissociation of
+small molecules.
+I.
+INTRODUCTION
+Over the last few decades, the accuracy and perfor-
+mance of molecular dynamics (MD) simulations have
+greatly increased, thanks to both hardware improve-
+ments (parallel architectures, graphical accelerators...)
+and intense methodological research.
+Calculations in-
+volving tens of thousands of atoms are currently rou-
+tinely performed and multi-million atoms simulations are
+now reachable. Molecular dynamics can be used to com-
+pute properties of materials and molecular systems and is
+now widespread in academic studies as a tool to interpret
+or even predict experimental results. Furthermore, this
+method is progressively being adopted in the industry,
+for example to guide and accelerate drug design[1].
+In large-scale simulations, interactions between atoms
+cannot generally be computed from first principles be-
+cause of the high numerical cost of quantum methods. In-
+stead, they are generally modeled using force fields (FFs)
+that postulate a physically-motivated functional form of
+the potential energy and are parameterized in order to
+match ab initio energies and/or reproduce experimental
+data. The most widespread FFs are the so-called classical
+force fields (such as AMBER [2] or CHARMM [3]) which
+use a combination of fixed-charge Coulomb potential and
+Lennard-Jones interactions to model the inter-molecular
+potential. These models are extremely efficient numeri-
+cally, allowing the simulation of very large systems over
+long time scales. Their simple functional form, however,
+lacks polarization and many-body effects which can be
+critical to correctly describe some systems (for example
+solvation in a polar solvent, pi-stacking or complex pro-
+tein structures [4]). More advanced force fields – such
+as AMOEBA [5], TTM [6], CHARMM Drude [7], AR-
+ROW [8] or SIBFA [9, 10] – have thus been developed
+∗ Corresponding author: thomas.ple@sorbonne-université.fr
+† Corresponding author: louis.lagardere@sorbonne-université.fr
+‡ Corresponding
+author:
+jean-philip.piquemal@sorbonne-
+université.fr
+in order to explicitly include these effects. These polar-
+izable force fields (PFFs) [11, 12] are much more flexible
+and accurate but are significantly costlier. Nonetheless,
+advances in high-performance computing (HPC), the in-
+crease in GPU (Graphical Processing Units) availability
+and recent methodological developments (advanced iter-
+ative solvers) now allow large-scale PFF simulations[13].
+Both classical and polarizable FFs however assume a
+fixed connectivity between atoms (i.e.
+covalent bonds
+cannot be broken), making them unsuitable to study
+chemical reactions. Some reactive force fields – such as
+ReaxFF [14] or Empirical Valence Bond [15] – are actively
+being developed but are generally specialized towards a
+relatively narrow class of systems.
+From this brief overview of the domain of force fields,
+it is clear that a general, many-body reactive model is
+highly desirable but its design remains an outstanding
+challenge for the current frameworks.
+In recent years,
+considerable attention and resource have been devoted
+to the development of machine-learning potentials that
+promise to bridge the accuracy and generality gap be-
+tween force fields and ab initio methods. These models
+use flexible functional forms from the domain of machine-
+learning (such as deep neural networks, graph networks
+or kernel models) in order to accurately fit ab initio en-
+ergies, with a numerical cost comparable to standard
+FFs.
+A large variety of such models have been de-
+veloped over the last few years (for example the HD-
+NNP [16], ANI [17], AIMNet [18, 19], DeePMD [20],
+ACE [21], sGDML [22], Nequip [23], etc...)
+and have
+been applied to small molecular systems [17], periodic
+crystals [24, 25] and more general condensed-phase sys-
+tems [26, 27]. Among these, the ANI models occupy a
+particular place as they aim to provide a generic pre-
+trained potential for a whole class of organic molecules.
+In this work, we follow a similar strategy.
+In order to respect the inherent symmetries of molec-
+ular systems, most architectures are designed to be in-
+variant with respect to rotations, translations and ex-
+change of identical atoms.
+These models have shown
+good accuracy on many systems but usually require large
+arXiv:2301.08734v1  [physics.chem-ph]  20 Jan 2023
+
+2
+amounts of data to be trained on (of the order of a
+million molecular configurations). More recently, equiv-
+ariant models (for example Nequip [23], Allegro [28],
+SpookyNet [29], UNiTE [30] or GemNet [31]) have at-
+tracted much attention because of their impressive data
+efficiency and their ability to generalize more accurately
+to out-of-distribution configurations [23, 32].
+Most ML models, however, assume a purely local func-
+tional form and tend to neglect or implicitly account for
+long range effects from the training data. The accurate
+description of long-range interactions is however critical
+to correctly simulate condensed-phase systems and to de-
+scribe the structure of large molecular formations (e.g.
+protein or DNA structure [33, 34]). The framework of
+message-passing neural networks [35] in principle allows
+to describe long-range effects by iteratively exchanging
+information with neighbouring atoms.
+This approach
+however have been shown to pose difficulties when ap-
+plied to large systems as this iterative process is not well
+suited for parallel architectures.
+Some other pure ML
+multi-scale models are being developed (for example the
+LODE descriptor [36, 37]) but this area is still in its in-
+fancy. On the other hand, quantum perturbation theory
+(for example SAPT) gives solid grounds to the descrip-
+tion of long-range effects in terms of classical electrostat-
+ics [38] which can be well captured in the FF framework
+(via multipolar Coulomb interactions and dispersion ef-
+fects for example). It thus seems advantageous to com-
+bine an ML model – which excel at predicting short-range
+properties – with long-range FF interactions, in order to
+obtain the best of both approaches. A few models apply-
+ing this idea have recently been developed (for example
+HDNNP-Gen4 [39], PhysNet [40] and others [26, 41]) and
+have shown good results across multiple systems. Hybrid
+FF/ML models thus provide a promising route to more
+physics-aware ML potentials.
+In this paper, we propose a general framework for
+building force-field-enhanced ML models.
+We leverage
+the latest advances in local equivariant neural networks in
+order to accurately predict short-range energy contribu-
+tions as well as multiple atom-in-molecule properties that
+are then used to dynamically parameterize QM-inspired
+FF energy terms that account for long-range interactions.
+We show that this architecture allows for highly transfer-
+able models that are able to accurately generalize on large
+molecular systems, as well as in the condensed phase af-
+ter being trained on small monomers and dimers only.
+This paper is organized as follows. Section II describes
+the model architecture, from the Allegro [28] equivariant
+embedding to the output and physics modules. It also
+describes the particular FF terms that we used for the
+pretrained model, named FENNIX-OP1, that we provide
+with this work. Section III focuses on the FENNIX-OP1
+model and provides details on its construction, its target
+properties, the datasets used for training and the differ-
+ent training stages that were required.
+In section IV,
+we validate the model via several applications.
+First,
+we show that the model predicts accurate dissociation
+energy curves of some simple molecules. We then com-
+pute structural properties of liquid water in molecular
+dynamics simulations including nuclear quantum effects
+(NQEs) via the recently developed adaptive quantum
+thermal bath (adQTB) [42, 43]. Indeed, since the model
+is trained purely on ab initio data, the explicit inclusion
+of NQEs is critical for the correct calculation of thermo-
+dynamical properties, as was shown in numerous previ-
+ous studies [44, 45]. For this purpose, the adQTB was
+shown to provide robust approximations of NQEs while
+being numerically affordable (similar to a classical MD)
+and thus constitutes a very efficient tool for quickly test-
+ing ML models. We then show that the model produces
+stable dynamics of alanine dipetide in solution and pro-
+vides a qualitatively correct description of the torsional
+free energy profile (computed using an enhanced sam-
+pling method[46]). Finally, section V provides some con-
+clusions and outlooks for future extensions of the model.
+II.
+MODEL ARCHITECTURE
+The FENNIX (Force-field-Enhanced Neural Network
+Interactions) model is based on a local multi-output
+equivariant model that processes atomic neighborhoods
+and predicts multiple atomic or pairwise properties. As
+an example for this work, we will present a model that
+outputs local pairwise energy contributions, charges and
+atomic volumes. The output is subsequently enriched by
+a "physical" module that computes force field terms such
+as electrostatic and dispersion energy terms. The core of
+our model is a slightly modified version of the Allegro
+local equivariant model presented in ref. [28]. We use the
+Allegro model as a general embedding of atomic pairs
+which is then fed into independent neural networks that
+predict the target properties.
+In this section, we will briefly describe the Allegro ar-
+chitecture and our modifications to the model. The the-
+oretical analysis of this architecture was thoroughly done
+in the original paper [28] so that we will only review here
+the main points necessary to the understanding of the
+model and our improvements to the architecture.
+We
+will then present the output module that processes the
+Allegro embedding. Finally, we will describe the phys-
+ical module and the particular functional form for the
+FENNIX-OP1 potential energy surface that we used in
+this work.
+A.
+Equivariant embedding using the Allegro
+architecture
+1.
+Review of the Allegro architecture
+The Allegro model provides a local many-body descrip-
+tor
+�
+xij, V nlp
+ij
+�
+– interchangeably referred to as embed-
+ding in the following – for each directed pair of atoms
+
+3
+with source atom i and destination atom j in the neigh-
+borhood N(i) defined by all the atoms located at a dis-
+tance shorter than a cutoff radius rc from atom i:
+N(i) =
+�
+k
+s.t.
+���⃗Rik
+��� < rc
+�
+(1)
+with ⃗Rik = ⃗Rk− ⃗Ri the vector going from the position of
+atom i to atom k. The first part of the descriptor xij is
+built so that it is invariant under the action of certain geo-
+metric symmetries (i.e. global rotations, translations and
+inversions of the system). On the other hand, the second
+descriptor V nlp
+ij
+is composed of features that are equivari-
+ant with respect to these symmetries. These features take
+the form of tensors that are labeled with a channel index
+n ∈ 1, . . . , Nchannels, a rotational index l ∈ 0, 1, . . . , lmax
+and a parity index p ∈ −1, 1. The rotational index in-
+dicates how the tensor transforms under rotation oper-
+ations: l = 0 corresponds to scalar/invariant quantities,
+l = 1 corresponds to vector-like objects and we refer to
+l >= 2 objects as higher-order tensors. The parity in-
+dex, on the other hand, indicates how the tensor’s sign
+changes under inversion of the coordinate system. In our
+implementation of the Allegro model, these tensorial ob-
+jects are handled by the e3nn python package [47] which
+provides high-level classes that represent them and easy-
+to-use functions to manipulate and combine them while
+preserving symmetries and global equivariance.
+In the following paragraphs, we describe how initial
+two-body features are computed, how features from
+neighbors are combined through Nlayers layers of interac-
+tions to enrich them with many-body information and
+how they are filtered at each layer to control the size of
+the embedding.
+Initial two-body features
+The Allegro model starts by decomposing each inter-
+atomic vector {⃗Rij}j∈N (i) into fingerprints that are more
+suitably processed by the network. The interatomic dis-
+tance Rij is projected onto a radial basis B(Rij) =
+[B1(Rij), . . . , BNbasis(Rij)] (we use the Bessel basis func-
+tion with a polynomial envelope [48] that we normalize
+as in the original paper) and we compute the two-body
+scalar embedding as:
+x
+2B
+ij = MLP2B
+�1(Zi) || 1(Zj) || B(Rij)
+�
+fc(Rij)
+(2)
+where || denotes concatenation, MLP2B is a multilayer
+perceptron (i.e. a fully connected scalar neural network),
+fc(Rij) is a cutoff function going smoothly to zero as Rij
+approaches rc (we use the same polynomial envelope as
+for the radial basis) and 1(Zi) (resp. 1(Zj)) is a vector
+representing the chemical species of the source atom i
+(resp. destination atom j). In the original paper 1(Zi)
+is a direct one-hot encoding of the atomic number Zi,
+meaning that one has to fix in advance the number of
+species the model will be able to process (the one-hot en-
+coding then defines a simple orthonormal basis which has
+the same dimensions as the chosen number of species). In
+section II A 2, we propose to modify this one-hot encod-
+ing by a positional encoding of coordinates in the periodic
+table which allows for more flexibility in the treatment
+of atomic species.
+We obtain the two-body equivariant features by pro-
+jecting the unit vector ˆRij = ⃗Rij/Rij onto a basis of
+real spherical harmonics Y lp
+ij . We then mix them with
+radial information with a linear embedding on Nchannels
+channels:
+V nlp,2B
+ij
+=
+�
+MLP
+2B
+embed(x
+2B
+ij )
+�nlp Y lp
+ij
+(3)
+Interaction with the local environment
+The two-body embedding
+�
+x2B
+ij , V nlp,2B
+ij
+�
+is then pro-
+cessed through multiple "interaction" layers that allow
+to combine information with other atoms in the vicinity
+of atom i.
+Each interaction layer starts by building a
+global equivariant neighborhood embedding for atom i
+from the current scalar embeddings xik and the spherical
+harmonics projections Y lp
+ik :
+Γnlp,(L)
+i
+=
+�
+k∈N (i)
+�
+MLP (L)
+embed(x(L−1)
+ik
+)
+�nlp
+Y lp
+ik
+(4)
+with L = 1, . . . , Nlayers the layer index and x(0)
+ik = x2B
+ik and
+V nlp,(0)
+ij
+= V nlp,2B
+ij
+. The interaction is then performed via
+a tensor product of Γnlp,(L)
+i
+with each equivariant em-
+bedding V nlp,(L−1)
+ij
+(the tensor product is done indepen-
+dently for each channel n). The resulting "latent space"
+Lnmlp,(L)
+ij
+=
+�
+Γnl1p1,(L)
+i
+⊗ V nl2p2,(L)
+ij
+�nmlp
+(5)
+contains all possible combinations of rotational and
+parity indices that are allowed by symmetry (i.e. such
+that |l1 − l2| ≤ l ≤ |l1 + l2| and p = p1p2).
+Note
+that since multiple combinations of (l1, p1), (l2, p2) may
+produce outputs of indices (l, p), we need to add a
+multiplicity index m that distinguishes these paths.
+Feature filtering and channel mixing
+Finally, the latent space is filtered to obtain the new pair-
+wise embedding. The scalar embedding is combined with
+the scalar part of the latent space (with every channels
+and all multiplicities concatenated) to obtain:
+x(L)
+ij
+= α x(L−1)
+ij
++
+�
+1 − α2 fc(Rij)
+× MLP (L)
+latent
+�
+x(L−1)
+ij
+||
+n,m
+Lnm01,(L)
+ij
+�
+(6)
+with 0 ≤ α < 1 a mixing coefficient that allows to easily
+propagate scalar information from a layer to the next. In
+our implementation, the value of α can be set as a hyper-
+parameter (for example to the value α = 2/
+√
+5 proposed
+
+4
+in the original Allegro paper) or can be optimized inde-
+pendently for each layer during the training procedure.
+The new equivariant features are obtained by linearly
+combining the elements of the latent space with same
+indices (l, p) from all channels and multiplicities:
+V nlp,(L)
+ij
+=
+�
+n′,m
+wnlp,(L)
+n′,m
+Ln′mlp,(L)
+ij
+(7)
+which results in features with the same number of ele-
+ments as the previous layer.
+The weights wnlp,(L)
+n′,m
+are
+optimized in the training procedure.
+The
+output
+features
+of
+the
+last
+layer
+�
+x(Nlayers)
+ij
+, V nlp,(Nlayers)
+ij
+�
+compose
+the
+many-body
+embedding of our model which is passed to the output
+module to predict the different atomic or pairwise
+properties that the model is trained on.
+2.
+Positional encoding of chemical species
+While a one-hot encoding allows to represent chemi-
+cal species in a simple manner, it fixes from the start
+the number of different species that the model can treat.
+Thus, if more data becomes available for new species, one
+would have to retrain the model from scratch in order to
+accommodate for the new data.
+Furthermore, in such
+encoding, all species are treated equally and no similar-
+ities between species (for example closeness in the peri-
+odic table) are provided: the network must learn these
+correlations purely from data. This encoding is thus suit-
+able when targeting a specific system but might not be
+the best choice when building a more general chemical
+model.
+In our implementation, the default encoding is a po-
+sitional encoding (as defined in ref. [49]) that encodes
+coordinates in the periodic table using sine and cosine
+functions of different frequencies. For example, the col-
+umn index c is encoded as a vector ec of dimension dcol
+as:
+∀k ∈ 0, . . . , dcol, (ec)k =
+�
+�
+�
+�
+�
+sin
+�
+c/γ2i/dcol
+col
+�
+if k=2i
+cos
+�
+c/γ2i/dcol
+col
+�
+if k=2i+1
+(8)
+and similarly for the row index with dimension drow and
+frequency parameter γrow. In our implementation, the di-
+mensions and frequency parameters are fixed at dcol = 10,
+drow = 5, γcol = 1000 and γrow = 100.
+These could
+be also be treated as hyperparameters or even learned
+during training.
+The row and column encodings are
+then concatenated to obtain the full encoding vector
+1(Zi) =
+�
+erow(Zi) || ecol(Zi)
+�
+Figure 1 shows a heatmap
+of the positional encoding of the species H,C,N,O and S
+(from top to bottom). We see that the first five columns
+are the same for all the heavy atoms as they represent
+the row encoding (the second row of the periodic table
+in this case) while they are different from the first line
+FIG. 1. Heatmap of the positional encodings of the chemi-
+cal species H,C,N,O,F (from top to bottom). The first five
+columns represent the encoding of the row index in the peri-
+odic table, while the last ten columns represent the encoding
+of the column index.
+corresponding to the Hydrogen. We also see that the last
+ten columns are different for all the species shown here
+as they are all on different columns of the periodic table.
+The motivation behind using this positional encoding is
+that we hypothesized that having similar encodings for
+species sharing a row or a column might help with gen-
+eralization and allow to transfer learned knowledge from
+a species to another, thus requiring less training data.
+Furthermore, as stated in ref. [49] the encoding for index
+c+k can be represented as a linear function of the encod-
+ing for index c, which might further help with inferring
+similarities.
+Additional features such as the ionization
+state could be encoded in the same manner (though we
+restrict ourselves to neutral atoms in this work, thus only
+requiring the knowledge of chemical species).
+B.
+Output module
+After computing the embedding from the Allegro
+model, we use it as input for independent MLPs for each
+target property. In the following, we will simply denote
+�
+xij, V nlp
+ij
+�
+the output from the last Allegro layer (thus
+dropping the (L) layer index). The current implementa-
+tion also allows some modularity in the composition of
+inputs and on the operations done on the outputs. For
+example, the input can exploit either the scalar embed-
+ding to obtain invariant properties via a standard MLP
+as
+oij = MLPout[xij]
+(9)
+, or both the scalar and tensorial embeddings via a linear
+projection of V nlp
+ij
+Omlp
+ij
+=
+�
+n
+[MLPout(xij]mlp
+n
+V nlp
+ij
+(10)
+.
+For atom-wise properties, the pairwise outputs are
+simply summed up on the central atom. For properties
+that should sum up to zero (for example partial atomic
+charges), the outputs oij and oji can be antisymmetrized
+(which for partial charges is equivalent to charge ex-
+change between neighbouring atom pairs). Futhermore,
+
+0
+1
+2
+3
+45
+in order to impose constraints on invariant outputs, a fi-
+nal activation function can optionally be applied. This
+is for example useful when the output targets a positive
+quantity (for instance an atomic volume) for which we
+can apply a softplus function, a probability for which
+we may apply a sigmoid function or a discrete proba-
+bility distribution (in the case of multidimensional oij)
+for which a softmax function can be used. Finally, the
+output is optionally shifted and scaled.
+Further modifications can optionally be applied. For
+instance, the input can be filtered according to an addi-
+tional shorter-range cutoff for example one distinguish-
+ing between bonded and non-bonded pairs. Finally, the
+two-body embedding x2B
+ij can be used in place of or con-
+catenated to (as it is done in the FENNIX-OP1 model)
+the final embedding to use as input.
+This allows the
+output MLP to easily access simple pairwise informa-
+tion and should let the Allegro embedding specialize in
+finer many-body correlations.
+This compositional ap-
+proach allows for a great flexibility in the model’s output,
+which is especially useful when experimenting with a ML
+parametrization of physical models.
+The output module for the FENNIX-OP1 model is
+composed of three targets: a local energy contribution
+V NN
+i
+= �
+j∈N (i) V NN
+ij
+(which is a simple scalar out-
+put), an atomic partial charge qNN
+i
+= �
+j∈N (i) ∆qij −
+∆qji (through antisymmetrized charge exchange) and an
+atomic volume vNN
+i
+(constrained to be positive).
+C.
+Physics module and energy functional form
+Finally, the physics module uses the results from the
+output module and feed them into physically-motivated
+models to enrich the output.
+In the case of FENNIX-OP1, the force field module is
+composed of an electrostatic energy term V
+elec,CP
+ij
+, and a
+pairwise dispersion term V
+disp
+ij
+. The functional form of
+FENNIX-OP1 is then given by:
+VOP1 =
+�
+i
+V
+NN
+i
++
+�
+i,j<i
+V
+elec,CP
+ij
++
+�
+i,j<i
+V
+disp
+ij
+(11)
+The first term V NN
+i
+is the neural network contribution
+that accounts for short-range interactions that we intro-
+duced in section II B. We model the electrostatic inter-
+action V
+elec,CP
+ij
+via a Coulomb potential with fluctuating
+charges and the Piquemal charge penetration model [50]:
+V
+elec,CP
+ij
+=
+1
+Rij
+�
+NiNj + Nj(qi − Ni)f i
+damp(Rij)
++ Ni(qj − Nj)f j
+damp(Rij)
++ (qi − Ni)(qj − Nj)f i
+ov(Rij)f j
+ov(Rij)
+�
+(12)
+where Ni is the number of valence electrons of atom i ,
+qi = ϵqNN
+i
+is the environment-dependent charge of atom i
+predicted by the neural network (with an adjustable uni-
+versal scaling parameter ϵ) and f i
+damp(r) = 1 − e−αr/rvdw
+i
+and f i
+ov(r) = 1 − e−βr/rvdw
+i
+are damping functions with
+α and β adjustable parameters that are assumed to be
+universal and where
+r
+vdw
+i
+=
+� vNN
+i
+vfree
+i
+� 1
+3
+r
+vdw,free
+i
+(13)
+is the environment-dependent Van der Waals radius of
+atom i with vNN
+i /vfree
+i
+the atomic volume ratio predicted
+by the neural network.
+Finally, the dispersion interaction V
+disp
+ij
+is computed
+using the pairwise Tkatchenko-Scheffler model [51]:
+V
+disp
+ij
+= −C6,ij
+R6
+ij
+σij(Rij)
+(14)
+with the combination rule:
+C6,ij =
+2C6,iC6,j
+C6,i
+αj
+αi + C6,j αi
+αj
+(15)
+and the environment-dependent homonuclear parame-
+ters:
+C6,i =
+� vNN
+i
+vfree
+i
+�2
+C
+free
+6,i
+;
+αi =
+� vNN
+i
+vfree
+i
+�
+α
+free
+i
+(16)
+with αfree
+i
+the isolated atom polarizabilities, Cfree
+6,i the iso-
+lated atom sixth order dispersion coefficients and the sig-
+moid damping function:
+σij(Rij) =
+�
+1 + e
+−γ
+�
+1
+s
+Rij
+rvdw
+i
++rvdw
+j
+−1
+��−1
+(17)
+with γ and s adjustable parameters that we assume to
+be universal.
+We furthermore mention that the physics module is
+not limited in principle to compute energy terms. The
+aim of this module is to be a general physical interface
+between the ML embedding and the final target proper-
+ties. For example, we use it in the FENNIX-OP1 model
+to correct for ML-predicted charge exchanges that would
+deplete the valence shell of an atom. The implementation
+also provides charge exchange via a simple bond-capacity
+model [52] that leverages ML-predicted atom-in-molecule
+electronegativities and which capabilities will be explored
+in future iterations of the FENNIX model. Each phys-
+ical model is implemented as a simple Pytorch Module
+that takes as input a dictionary which contains previously
+computed properties. It then adds its contribution and
+outputs the enriched dictionnary.
+These physical sub-
+modules can then easily be chained, and it facilitates the
+implementation of additional models.
+
+6
+III.
+DATASETS AND TRAINING PROCEDURE
+In this section, we start by providing some details
+on the construction of FENNIX-OP1 model. We then
+review the datasets that we used for training the model
+and its target properties. Finally, we will focus on the
+non-trivial task of training a multi-output FENN model.
+A.
+FENNIX-OP1 model architecture
+In the FENNIX-OP1, chemical species are encoded
+using the positional encoding defined in section II A 2,
+with 5 dimensions for the row encoding and 10 dimen-
+sions for the column encoding.
+The use Nbasis = 10
+Bessel basis functions for the radial embedding, with a
+cutoff distance rc = 5.2 Åand a smoothing parameter
+p = 3 for the polynomial envelope.
+We used 256
+features for the scalar embedding and 10 channels with
+a maximum rotational index lmax = 2 for all equivariant
+features. The two-body scalar MLP is composed of two
+hidden layers with 64 and 128 neurons respectively with
+SiLU activation function.
+The two-body embedding
+MLP for the initial equivariant features is a simple
+linear projection of the two-body embedding with no
+activation function. The embedding is constructed using
+3 Allegro layers. In each layer, the embedding MLP is a
+simple linear projection with no activation function and
+the latent MLP contains two hidden layers both with
+256 neurons and SiLU activation function. The mixing
+coefficient α is initially set to the original 2/
+√
+5 and is
+optimized independently for each layer during training.
+In total, the Allegro model has NNN parameters.
+The output module is composed of three independent
+identical MLPs for the short-range energy contribution,
+the charge exchange and the atomic volume. The input of
+these MLPs is the concatenation of the two-body scalar
+embedding and the final scalar embedding. They have 5
+hidden layers with 256, 128, 64, 32 and 16 neurons with
+a total of NNN parameters each.
+The output module also has a "constant" submodule
+that simply provides the atomic reference energies.
+Importantly,
+we used the CCSD(T) isolated atom
+energies as a reference instead of the average atomic
+energy over the dataset that is typically advised when
+training a ML model. Indeed, this reference energy has a
+physical meaning – which is the energy of an atom when
+it has no neighbors to interact with – and is not solely
+a convenience parameter for facilitating the training.
+In particular, this choice has huge consequences for the
+description of molecular dissociation, as will become
+clear in section IV A.
+Finally, the physics module is composed of four sub-
+modules: a charge correction module, a charge scaling
+module, the Coulomb interaction module and the dis-
+persion module. These last two modules implement the
+physical interactions using the corresponding equations
+described in section II C. The charge scaling module sim-
+ply scales all charges by a constant factor in order to
+compensate for the lack of higher order multipoles in the
+permanent electrostatics. The charge correction module
+antisymmetrizes the charge exchanges and ensures that
+atoms do not unphysically deplete their valence shell. In-
+deed, if we assume that only valence electrons can be
+transferred, an atom cannot have a partial charge larger
+than +Ni (which is particularly important for the charge
+penetration model that we use). This constraint is not
+ensured by the neural network model so we need to en-
+force it a posteriori. The constraint is achieved by trans-
+ferring back some electrons from neighbouring atoms that
+drained too much charge. First the unphysical "hole" in
+the valence shell is computed using a smooth function (to
+ensure smooth gradients of the final coulomb energy) on
+the basis that an atom cannot lose more than 95 percent
+of its valence electrons. Charges that sum up to the va-
+lence hole are then transferred from neighbouring atoms
+that took charges, proportional to the quantity drained.
+This procedure should enforce the constraint in a single
+iteration for most non-pathological cases. Reassuringly
+however, the charge correction is almost never needed af-
+ter training (i.e. the valence hole is almost always zero),
+even in condensed-phase MD simulations for which the
+model was not explicitly trained.
+B.
+Datasets and target properties
+For the FENNIX-OP1 model, we chose to reproduce
+high-level coupled-cluster energies and we thus selected
+three of the few freely-available and generalist datasets
+that provide such data:
+the ANI-1ccx [17] dataset,
+the DES370K [53] dataset and the q-AQUA dimers
+dataset [54].
+The
+ANI-1ccx
+dataset
+provides
+approximately
+500,000 CCSD(T)/CBS total energies of various neu-
+tral monomers and dimers (composed of the elements
+H,C,N,O) in equilibrium and out-of-equilibrium configu-
+rations. It also provides atomic forces at DFT level that
+were crucial to speed-up the beginning of the training
+procedure. Importantly, it provides partial charges and
+atomic volumes obtained from a Minimal Basis Iterative
+Stockholder [55] (MBIS) partitioning of the DFT elec-
+tronic density that were used to parametrize the force
+field terms.
+The DES370K dataset is composed of approxi-
+mately 370,000 CCSD(T)/CBS interaction energies of
+diverse dimers in various configurations.
+It comprises
+both neutral and charged molecules. The latter were dis-
+carded for the training of the FENNIX-OP1 model as
+out-of-scope for this study. It also provides SAPT de-
+compositions of the interaction energies that we used to
+regularize the Coulomb interactions.
+The q-AQUA
+dimers
+dataset is composed of
+more than 70000 water dimer interaction energies at
+
+7
+CCSD(T)/CBS level. We used this dataset in the end of
+the training in order to fine-tune the model for handling
+water.
+The FENNIX-OP1 model has then three target prop-
+erties that use the available data: the CCSD(T) total
+energy of the system as modelled by VOP1 described in
+section II C, the MBIS partial charges qNN
+i
+and the ratio
+of MBIS volumes to free atom volumes vNN
+i /vfree
+i
+.
+C.
+Training procedure
+The training of a multi-output force-field-enhanced
+neural network revealed to be a non-trivial task.
+In-
+deed, the inter-dependencies between target properties
+(for example partial charges and the electrostatic contri-
+bution to the total energy) seemed to pose difficulties for
+the standard optimization methods, which implied that
+a brute-force optimization of the whole model would not
+give satisfactory results. To overcome this difficulty, we
+resorted to a training procedure in multiple stages that
+is described in the following of this section.
+Furthermore, in order to obtain a final model that was
+stable when performing molecular dynamics, we found
+that strong regularization was needed, both in the form
+of standard weight decay and also physically-motivated
+loss contributions, as detailed later.
+Throughout, we
+used the AdamW optimizer [56] implemented in Pytorch,
+as its algorithm for weight decay provides one of the best
+compromise between training speed and accuracy. We
+used a fairly strong weight decay parameter of 0.5 for
+all the parameters in the output module and no weight
+decay for the Allegro parameters. In all stages, we used
+the same random 10% of the dataset as a validation
+set and optimized the model using mini-batches of 256
+configurations from the ANI-1ccx and 64 configurations
+from DES370K and q-AQUA when they are used.
+The training procedure for the FENNIX-OP1 model
+required four stages. In the first stage, the model was
+trained to reproduce DFT total energies and forces using
+the short-range contribution V NN only. We also trained
+at the same time the charges and atomic volumes to re-
+produce the MBIS targets. At this stage, only the ANI-
+1ccx dataset was used. The loss function for this stage is
+given by:
+L(1) = λE(E
+DFT − V
+NN)2 + λF
+3
+�
+j=1
+Nat
+�
+i=1
+�
+F
+DFT
+ij
+− F
+NN
+ij
+�2
++ λq
+Nat
+�
+i=1
+�
+q
+MBIS
+i
+− qNN
+i
+�2 + λv
+Nat
+�
+i=1
+�vMBIS
+i
+vfree
+i
+− vNN
+i
+vfree
+i
+�2
+(18)
+with λE = 0.001, λF = 1 and λq = λv = 1000 and F NN is
+the model’s predicted force obtained by automatic differ-
+entiation (Pytorch’s autograd) of the total energy V NN.
+As suggested in previous studies [28], we strongly favour
+learning forces over energies in the beginning to acceler-
+ate the training procedure. We note that the provided
+loss is for a single configuration and that, in practice,
+it is averaged over the configurations in the mini-batch.
+We further added a regularization in order to minimize
+the off-diagonal elements of the covariance matrix of the
+scalar embedding’s features over a batch. This promotes
+learning statistically independent features in the embed-
+ding which should be favourable for a multi-output net-
+work and we found that it led to models with better
+generalization capabilities. In this stage, we train all the
+parameters in the the embedding and output modules
+with a starting learning rate of 10−3. Furthermore, we
+used a learning rate scheduler that reduces the learning
+rate when the error on the training set stops diminishing
+for a few steps (we set the patience of the scheduler to 10
+epochs and the learning rate scaling factor to 0.8). After
+about 100 epochs, progress of both training and valida-
+tion steps slowed down and we modified the energy and
+force parameters to λE = 0.01 and λF = 0.1 in order to
+obtain a more balanced training. We stopped the first
+stage when the learning rate reached 10−4.
+In the second stage, we freeze the embedding parame-
+ters and output MLPs for charge and volumes and acti-
+vate the Coulomb and dispersion energy terms. We then
+retrain the short-range energy MLP so that the full V OP1
+of eq. (11) reproduces DFT energies and forces. The loss
+function for this stage is:
+L(2) = λE(E
+DFT − V
+OP1)2 + λF
+3
+�
+j=1
+Nat
+�
+i=1
+�
+F
+DFT
+ij
+− F
+OP1
+ij
+�2
+(19)
+with the same weights as in the end of the previous stage.
+Freezing the embedding ensures that the predicted vol-
+umes and charges are not modified in this training stage.
+Since the energy target is modified, the errors starts much
+higher than at the end of the previous stage and quickly
+decreased. When the error on the train and validation
+sets drop to the same order as in the previous stage, the
+full model is unfrozen and training is resumed until the
+error stops decreasing.
+In the third stage, the embedding and charge and vol-
+umes MLPs are frozen again and the energy MLP is
+finally retrained to reproduce CCSD(T) energies from
+both ANI-1ccx total energies and DES370K interaction
+energies.
+We used the same type of mean-square loss
+functions with λE = 0.1 and λDES = 5. Again, we train
+the energy MLP until the error is close to the previous
+stage, unfreeze the whole model and optimize again all
+the parameters. For this stage, we also optimize the pa-
+rameters from the physical module in order to reach the
+lowest error possible.
+We stop the training when the
+learning rate reaches 10−5.
+In the last training stage, we refine the model for
+water by including the q-AQUA water dimers interaction
+energies in the loss function. We also generated batches
+of randomly deformed water monomers (with very large
+
+8
+deformations) and trained the model to reproduce forces
+from the highly accurate Partridge-Schwenke potential
+energy surface (that was itself fitted on high-accuracy
+coupled-cluster data). This was particularly helpful to
+reproduce the molecule’s bending energy surface away
+from equilibrium where fewer data are available in the
+ANI-1ccx dataset.
+At the end of the training procedure, the model
+reached a root mean square error (RMSE) of less that
+4 kcal/mol for CCSD(T) total energies on both valida-
+tion and training sets of the ANI-1ccx dataset. While it
+was possible to reach much lower errors with less regular-
+ized models (less than 1 kcal/mol), we found that they
+were unstable when performing MD and concluded that
+they were overfitting the data. We thus favoured a more
+strongly regularized, perhaps slightly underfitted model
+that allowed for better generalization in the condensed
+phase. The model also reached a RMSE of about 0.35
+kcal/mol on both DES370K and q-AQUA datasets. Fi-
+nally, it reached a RMSE of about 0.017 e for charges and
+about 0.017 for volume ratios. As for total energies, less
+regularized models allowed for much lower errors (less
+than 0.008 e for charges for example) but with visible
+overfitting, leading to irregular coulomb interactions and
+unstable dynamics.
+As a validation for the water interactions, we used the
+standard energy benchmarks when training force fields
+for water. The model gives a RMSE of 0.13 kcal/mol on
+the Smith dimer set [57] which are representative of the
+most stable water dimer configurations. This low error
+is not surprising as the Smith dimers are very close to
+configurations present in the q-AQUA dataset on which
+the model was trained.
+For a more challenging test,
+we used a set of typical water clusters up to hexam-
+ers. For these, the model achieved a RMSE lower than 2
+kcal/mol, which is comparable to our recent Q-AMOEBA
+model [45] which was specifically trained to reproduce
+these energies. In the next section, we investigate more
+thoroughly the validity, transferability and robustness of
+the model up to the unforgiving test of condensed-phase
+molecular dynamics including nuclear quantum effects.
+IV.
+MODEL VALIDATION
+In this section, we validate the FENNIX-OP1 model
+using a few examples of applications.
+First, we com-
+pute bond dissociation energy profiles of typical small
+molecules and show that the model is able to consistently
+break covalent bonds.
+Then, we show that the model
+is able to produce stable and accurate MD simulations
+of condensed-phase water including nuclear quantum
+effects. For this study, we included nuclear quantum ef-
+fects using the adaptive quantum thermal bath (adQTB)
+method introduced in ref. [42]. We showed in previous
+studies [43] that the adQTB provides an efficient and
+accurate alternative to path integrals – the gold standard
+method for including NQEs in MD simulations – at a
+cost similar to classical MD. For these two examples, we
+compare our results to the ANI models [17] that have a
+comparable scope as FENNIX-OP1 in terms of chemical
+diversity and were trained on similar datasets. Finally,
+we show that FENNIX-OP1 is able to produce stable
+dynamics of organic molecules solvated in water and
+provides a good qualitative description of the torsional
+free energy landscape of alanine dipeptide in solution.
+All calculations for this work were performed on a
+Nvidia A100 GPU using our custom TorchNFF pack-
+age that is built on top of Pytorch [58] and provides the
+implementation for FENNIX as well as a simple MD al-
+gorithm for NVT simulations in periodic boundary con-
+ditions (with Ewald summation for electrostatic inter-
+actions) and an efficient Pytorch implementation of the
+adQTB. TorchNFF is in an early development version
+and is thus not yet optimized. For example, a FENNIX-
+OP1 simulations of a box of 216 molecules of water in
+periodic boundary conditions can currently reach about
+0.8 ns/day of adQTB simulation on a single A100 GPU.
+A.
+Bond dissociation energy profiles
+We first validate the training and the choice of ref-
+erence energy on potential energy curves for the dis-
+sociation of covalent bonds in small molecules.
+This
+kind of bond breaking is a fundamental step in the pro-
+cess of many chemical reactions, for example in enzyme-
+catalyzed reactions [59], and is key to the description of
+mass spectroscopy experiments [60, 61]. They are how-
+ever difficult to accurately model: force fields usually
+forbid them by design (by assigning a fixed connectiv-
+ity) and ab initio approaches require the use of expensive
+multi-reference calculations to correctly describe the dis-
+sociation process. Their practical description thus usu-
+ally requires the use of specifically designed force fields
+(such as ReaxFF) or low-dimensional potential energy
+surfaces.
+Figure 2 shows the potential energy curves for a) the
+dissociation of a hydrogen atom in methane; b) the sym-
+metric dissociation of the water molecule; c) the dissoci-
+ation of the H-F molecule computed with FENNIX-OP1
+and compared with reference quantum calculations from
+the literature (multi-reference CI/6-31G** from ref [62]
+for CH4, Partridge-Scwhenke PES [63] that was fitted on
+CCSD(T) data and multi-reference CI/au-cc-pV6Z from
+ref [64] for the H-F molecule) and the ANI models. We
+see that FENNIX-OP1 consistently captures a smooth
+dissociation curve thanks to the physically-motivated
+choice of reference energy. On the other hand, the ANI
+models, for which the reference energy was set accord-
+ing to average values over the dataset, fail to reproduce
+the dissociation curves for large distances. FENNIX-OP1
+agrees particularly well with the reference for water as
+
+9
+1
+2
+3
+4
+5
+H-F distance (Angstrom)
+0
+25
+50
+75
+100
+125
+150
+175
+200
+Energy (kcal/mol)
+MRCI/aug-cc-pV6Z
+ANI2x
+FENNIX-OP1
+1
+2
+3
+4
+5
+C-H distance (Angstrom)
+0
+25
+50
+75
+100
+125
+150
+175
+200
+Energy (kcal/mol)
+CI
+ANI2x
+ANI1ccx
+FENNIX-OP1
+a) CH4 ➞ CH3 + H
+b) H2O ➞ OH + H
+c) HF ➞ H + F
+1
+2
+3
+4
+5
+O-H distance (Angstrom)
+0
+25
+50
+75
+100
+125
+150
+175
+200
+Energy (kcal/mol)
+Partridge&Scwhenke
+ANI2x
+ANI1ccx
+QAMOEBA
+FENNIX-OP1
+FIG. 2. Potential energy curves for the dissociations of: a) methane CH4→CH3+H with the CH3 group fixed at the CH4
+equilibrium geometry; b) water H2O→OH+H with OH and angle fixed at the H2O equilibrium geometry; c) HF→H+F
+the Partridge-Schwenke PES was included in the train-
+ing set. For the other two molecules, it tends to under-
+estimate the dissociation barrier. Interestingly, while the
+training data did not contain covalent interaction ener-
+gies for F, the model is still able to reasonably generalize
+its prediction. Indeed, thanks to the positional encoding
+of chemical species that we introduced in section II A 2,
+we expect the model to be able to generalize, at least
+qualitatively, across the periodic table.
+B.
+Structural properties of liquid water
+In order to test the robustness of the model, we per-
+formed molecular dynamics simulations of liquid water.
+Since the model was fitted purely on ab initio data, it
+was critical to explicitly include nuclear quantum effects
+in order to accurately compute thermodynamical prop-
+erties [44]. We used the adaptive quantum thermal bath
+method to include NQES [42], as it was previously shown
+to be a robust alternative to the more costly path inte-
+grals MD [43]. All simulations were performed for a box
+of 216 molecules in the NVT ensemble at 300K and ex-
+perimental density, with a time step of 0.5 fs. Coulomb
+interactions in periodic boundary conditions were han-
+dled using the Ewald summation method [65], that we di-
+rectly implemented using PyTorch operations so that we
+can leverage its automatic differentiation capabilities to
+obtain atomic forces. We equilibrated the system for 100
+ps, which was sufficient to thermalize the system and con-
+verge the adQTB parameters. We then computed the ra-
+dial distribution functions (RDFs) from 1 ns of MD sim-
+ulation. Figure 3 shows the partial RDFs of water simu-
+lated using adQTB and classical MD with the FENNIX-
+OP1 compared to experimental results from [66]. As a
+baseline, we also compare to ANI-2x results with adQTB
+MD. We see that FENNIX-OP1 is able to accurately cap-
+ture the subtle structure of liquid water and agrees well
+with experimental results (from ref. [66]) when nuclear
+quantum effects are included.
+This result is quite re-
+markable as the model was not explicitly trained on con-
+densed phase reference data. On the other hand, ANI-2x
+predicts a largely over-structured liquid. We suspect that
+this is due to both the insufficient ab initio reference for
+ANI-2x (DFT with the wB97x functional) and to the lack
+of long-range effects in the model itself. The ANI-1ccx
+model, that was trained on the same CCSD(T) data as
+FENNIX-OP1, produced very accurate radial distribu-
+tion functions at the beginning of the simulations, thus
+validating the necessity of accurate CCSD(T) reference
+data to be able to describe liquid water. It was however
+not stable for simulations times longer that a few tens
+of picoseconds (even in classical MD with a smaller 0.1
+fs timestep).
+We thus suspect that long-range effects,
+which contribute to an overall energetical stabilization of
+the structure, are important to maintain the structure of
+the liquid.
+Additionally, FENNIX-OP1 predicts an enthalpy of va-
+porization for water around 13 kcal/mol, which is slightly
+overestimated compared to the experimental result closer
+to 11 kcal/mol [67].
+This indicates a tendency of the
+model to form too strong hydrogen bonds in the con-
+densed phase. We expect that training the model using
+data from larger molecular clusters would improve the
+results. Such data at CCSD(T) level is however not yet
+available to our knowledge.
+Another route to improv-
+ing results may be to include a finer description of elec-
+trostatics, including for example multipolar interactions
+and many-body polarization, as is done for example in
+(Q-)AMOEBA [45] or ARROW [8]. Future iterations of
+FENNIX will then focus on refining the physical model
+for including such interactions.
+Finally, we explored the impact of deuteration on the
+structure of the liquid. Such thermodynamical isotope
+effects arise purely from NQEs (since the classical Boltz-
+mann distribution does not depend on the mass of the
+atoms) and can be used to experimentally probe the
+impact of quantum effects on chemical processes.
+We
+
+10
+0
+1
+2
+3
+gOO(r)
+Experimental
+FENNIX classical
+FENNIX adQTB
+ANI-2x adQTB
+0
+1
+2
+3
+gOH(r)
+0
+2
+4
+6
+8
+r (Å)
+0
+1
+2
+3
+gHH(r)
+FIG. 3. Partial radial distribution functions of liquid water
+at 300K simulated using classical MD and adQTB MD with
+the FENNIX-OP1 model, compared to experimental results
+from [66] and ANI-2x (adQTB) results.
+showed in earlier works that the adQTB is able to ac-
+curately capture these isotope effects [43, 45]. Figure 4
+shows the O-O radial distribution function of light and
+heavy water at 300K computed using FENNIX-OP1 and
+compared to experimental results [66] for light water. We
+see that deuteration tends to structure the liquid, in good
+agreement with tendencies observed experimentally (see
+ref. [66]).
+C.
+Enhanced sampling of the torsional free energy
+profile of solvated alanine dipetide
+Lastly, we performed molecular dynamics simulations
+(using enhanced sampling) of alanine dipetide solvated
+in a cubic box of water of length 30 Å(with periodic
+boundary conditions). The alanine dipeptide is a fun-
+damental building block for larger biomolecules and its
+accurate description is thus critical. This system is also
+a typical benchmark for both interaction models and en-
+hanced sampling methods and was recently shown to be
+extremely challenging for ML potentials [32]. It thus con-
+stitutes an interesting test case for our potential.
+Figure 5 shows the joint free energy profile, obtained
+after 3 ns of classical well-tempered metadynamics sim-
+ulation with the FENNIX-OP1 model, for the two dihe-
+dral angles defining the torsional degrees of freedom of
+2
+3
+4
+5
+6
+7
+8
+r (Å)
+0.0
+0.5
+1.0
+1.5
+2.0
+2.5
+gOO(r)
+H2O Experimental
+H2O FENNIX
+D2O FENNIX
+FIG. 4. O-O radial distribution computed using adQTB and
+the FENNIX-OP1 model for light and heavy water, compared
+to experimental results [66] on light water.
+the molecule. The model correctly assigns most of the
+probability to the two expected energy basins denoted 1
+and 2 on figure 5. It was also able to explore the less
+probable states denoted 3 and 4 in the figure. The model
+however seems to underestimate the barrier at Φ = 0,
+thus allowing too many transitions between states 1 and
+2 to 3 and 4. We note that this simulation was performed
+using classical MD, as the adQTB method is not directly
+compatible with enhanced sampling methods. It would
+be interesting to explore the impact of nuclear quantum
+effects on this energy profile. Indeed, as we showed above,
+nuclear quantum effects drastically modify the structure
+of water. Since the torsional free energy profile is strongly
+affected by the solvent (see the difference between sol-
+vated and gas phase alanine dipeptide for example in
+ref. [68]), we can expect important changes when going
+from classical to quantum dynamics.
+This calculation
+would require long and costly path-integrals simulations
+or the development of adequate enhanced sampling tech-
+niques for the adQTB, that we leave for future works.
+V.
+CONCLUSION AND OUTLOOKS
+We introduced FENNIX, a general class of models rep-
+resenting molecular interactions using an hybrid ML/FF
+approach. It builds upon latest equivariant architectures
+to construct an embedding of the local chemical environ-
+ment which is fed into a multi-output module predict-
+ing atom-in-molecule properties. The latter allows for a
+flexible design of potential energy terms. We apply this
+strategy to design a specific model: FENNIX-OP1, cap-
+turing both short range interactions, with a dedicated
+ML energy term, and long range ones using predicted
+atomic volumes and charges that parametrize both a
+
+11
+150
+100
+50
+0
+50
+100
+150
+(°)
+150
+100
+50
+0
+50
+100
+150
+(°)
+2.5
+0.0
+2.5
+5.0
+7.5
+10.0
+12.5
+15.0
+17.5
+20.0
+1
+3
+4
+2
+FIG. 5.
+Contour plot of the free energy profile of the two
+torsional angles of solvated alanine dipeptide computed using
+FENNIX-OP1 model.
+Sampling was enhanced using well-
+tempered metadynamics for the two torsional angles. Ener-
+gies in kcal/mol.
+charge penetration corrected electrostatic energy and a
+Tkatchenko-Scheffler like dispersion one. FENNIX-OP1
+is first trained on both the ANI-1ccx and the DES370K
+datasets which contain monomers, dimers and a few
+multimeric structures focusing on small neutral organic
+molecules. It is then refined for water using the q-AQUA
+dataset and the highly accurate Partridge-Schwenke po-
+tential energy surface for a final RMSE of 0.35 kcal/mol
+on interaction energies. We then showed that the model
+is stable during molecular dynamics (including NQEs via
+adQTB) and able to generalize to condensed phase (de-
+spite the lack of reference data), capturing structural
+properties of bulk water and solvated organic molecules.
+More precisely, it qualitatively reproduces the torsional
+free energy profile of the solvated alanine dipeptide using
+enhanced sampling techniques and yields stable trajecto-
+ries of a protein in gas phase. Interestingly, the model
+is able to capture similarities between chemical species
+thanks to the continuous encoding of their positions in
+the periodic table. Furthermore, it allows to overcome
+an intrinsic limitation of traditional force fields as it sat-
+isfactorily reproduces covalent bond dissociation. All in
+all, this work shows the extrapolating power of hybrid
+physically driven machine learning approaches and paves
+the way to even more refined models using enriched data
+sets (in particular for charged systems) and additional
+energy terms in the spirit of advanced polarizable force-
+fields such as SIBFA that include multipolar electrostat-
+ics and many-body polarization and dispersion. These
+will enable the study of reactive complex systems such as
+proton transfers between DNA pairs or enzyme catalyzed
+reactions where such an accurate description of molecu-
+lar interactions is mandatory. The FENNIX framework
+is available through a dedicated python package which
+includes the FENNIX-OP1 model. Further work will fo-
+cus on the extension of the approach and its inclusion in
+the optimized Deep-HP platform[69, 70].
+ACKNOWLEDGEMENTS
+This work was made possible thanks to funding
+from the European Research Council (ERC) under the
+European Union’s Horizon 2020 research and innovation
+program (grant agreement No 810367), project EMC2.
+Computations have been performed at GENCI (IDRIS,
+Orsay, France and TGCC, Bruyères le Chatel) on grant
+no A0130712052.
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+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQf5h65/content/2301.08734v1.pdf,len=1051
+page_content='Force-Field-Enhanced Neural Network Interactions: from Local Equivariant  Embedding to Atom-in-Molecule properties and long-range effects  Thomas Plé,1, ∗ Louis Lagardère,1, † and Jean-Philip Piquemal1, ‡  1Sorbonne Université, LCT, UMR 7616 CNRS, F-75005, Paris, France  We introduce FENNIX (Force-Field-Enhanced Neural Network InteraXions), an hybrid approach  between machine-learning and force-fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQf5h65/content/2301.08734v1.pdf'}
+page_content=' We leverage state-of-the-art equivariant neural networks  to predict local energy contributions and multiple atom-in-molecule properties that are then used as  geometry-dependent parameters for physically-motivated energy terms which account for long-range  electrostatics and dispersion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQf5h65/content/2301.08734v1.pdf'}
+page_content=' Using high-accuracy ab initio data (small organic molecules/dimers),  we trained a first version of the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQf5h65/content/2301.08734v1.pdf'}
+page_content=' Exhibiting accurate gas-phase energy predictions, FENNIX  is transferable to the condensed phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQf5h65/content/2301.08734v1.pdf'}
+page_content=' It is able to produce stable Molecular Dynamics simulations,  including nuclear quantum effects, for water predicting accurate liquid properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQf5h65/content/2301.08734v1.pdf'}
+page_content=' The extrapo-  lating power of the hybrid physically-driven machine learning FENNIX approach is exemplified by  computing: i) the solvated alanine dipeptide free energy landscape;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQf5h65/content/2301.08734v1.pdf'}
+page_content=' ii) the reactive dissociation of  small molecules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQf5h65/content/2301.08734v1.pdf'}
+page_content='  I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQf5h65/content/2301.08734v1.pdf'}
+page_content='  INTRODUCTION  Over the last few decades, the accuracy and perfor-  mance of molecular dynamics (MD) simulations have  greatly increased, thanks to both hardware improve-  ments (parallel architectures, graphical accelerators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQf5h65/content/2301.08734v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQf5h65/content/2301.08734v1.pdf'}
+page_content=')  and intense methodological research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQf5h65/content/2301.08734v1.pdf'}
+page_content='  Calculations in-  volving tens of thousands of atoms are currently rou-  tinely performed and multi-million atoms simulations are  now reachable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQf5h65/content/2301.08734v1.pdf'}
+page_content=' Molecular dynamics can be used to com-  pute properties of materials and molecular systems and is  now widespread in academic studies as a tool to interpret  or even predict experimental results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQf5h65/content/2301.08734v1.pdf'}
+page_content=' Furthermore, this  method is progressively being adopted in the industry,  for example to guide and accelerate drug design[1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQf5h65/content/2301.08734v1.pdf'}
+page_content='  In large-scale simulations, interactions between atoms  cannot generally be computed from first principles be-  cause of the high numerical cost of quantum methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQf5h65/content/2301.08734v1.pdf'}
+page_content=' In-  stead, they are generally modeled using force fields (FFs)  that postulate a physically-motivated functional form of  the potential energy and are parameterized in order to  match ab initio energies and/or reproduce experimental  data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQf5h65/content/2301.08734v1.pdf'}
+page_content=' The most widespread FFs are the so-called classical  force fields (such as AMBER [2] or CHARMM [3]) which  use a combination of fixed-charge Coulomb potential and  Lennard-Jones interactions to model the inter-molecular  potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQf5h65/content/2301.08734v1.pdf'}
+page_content=' These models are extremely efficient numeri-  cally, allowing the simulation of very large systems over  long time scales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQf5h65/content/2301.08734v1.pdf'}
+page_content=' Their simple functional form, however,  lacks polarization and many-body effects which can be  critical to correctly describe some systems (for example  solvation in a polar solvent, pi-stacking or complex pro-  tein structures [4]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQf5h65/content/2301.08734v1.pdf'}
+page_content=' More advanced force fields – such  as AMOEBA [5], TTM [6], CHARMM Drude [7], AR-  ROW [8] or SIBFA [9, 10] – have thus been developed  ∗ Corresponding author: thomas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQf5h65/content/2301.08734v1.pdf'}
+page_content='ple@sorbonne-université.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQf5h65/content/2301.08734v1.pdf'}
+page_content='fr  † Corresponding author: louis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQf5h65/content/2301.08734v1.pdf'}
+page_content='lagardere@sorbonne-université.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQf5h65/content/2301.08734v1.pdf'}
+page_content='fr  ‡ Corresponding  author:  jean-philip.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQf5h65/content/2301.08734v1.pdf'}
+page_content='piquemal@sorbonne-  université.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQf5h65/content/2301.08734v1.pdf'}
+page_content='fr  in order to explicitly include these effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQf5h65/content/2301.08734v1.pdf'}
+page_content=' These polar-  izable force fields (PFFs) [11, 12] are much more flexible  and accurate but are significantly costlier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQf5h65/content/2301.08734v1.pdf'}
+page_content=' Nonetheless,  advances in high-performance computing (HPC), the in-  crease in GPU (Graphical Processing Units) availability  and recent methodological developments (advanced iter-  ative solvers) now allow large-scale PFF simulations[13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQf5h65/content/2301.08734v1.pdf'}
+page_content='  Both classical and polarizable FFs however assume a  fixed connectivity between atoms (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQf5h65/content/2301.08734v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQf5h65/content/2301.08734v1.pdf'}
+page_content='  covalent bonds  cannot be broken), making them unsuitable to study  chemical reactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQf5h65/content/2301.08734v1.pdf'}
+page_content=' Some reactive force fields – such as  ReaxFF [14] or Empirical Valence Bond [15] – are actively  being developed but are generally specialized towards a  relatively narrow class of systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQf5h65/content/2301.08734v1.pdf'}
+page_content='  From this brief overview of the domain of force fields,  it is clear that a general, many-body reactive model is  highly desirable but its design remains an outstanding  challenge for the current frameworks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQf5h65/content/2301.08734v1.pdf'}
+page_content='  In recent years,  considerable attention and resource have been devoted  to the development of machine-learning potentials that  promise to bridge the accuracy and generality gap be-  tween force fields and ab initio methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQf5h65/content/2301.08734v1.pdf'}
+page_content=' These models  use flexible functional forms from the domain of machine-  learning (such as deep neural networks, graph networks  or kernel models) in order to accurately fit ab initio en-  ergies, with a numerical cost comparable to standard  FFs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQf5h65/content/2301.08734v1.pdf'}
+page_content='  A large variety of such models have been de-  veloped over the last few years (for example the HD-  NNP [16], ANI [17], AIMNet [18, 19], DeePMD [20],  ACE [21], sGDML [22], Nequip [23], etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQf5h65/content/2301.08734v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQf5h65/content/2301.08734v1.pdf'}
+page_content=')  and have  been applied to small molecular systems [17], periodic  crystals [24, 25] and more general condensed-phase sys-  tems [26, 27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQf5h65/content/2301.08734v1.pdf'}
+page_content=' Among these, the ANI models occupy a  particular place as they aim to provide a generic pre-  trained potential for a whole class of organic molecules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQf5h65/content/2301.08734v1.pdf'}
+page_content='  In this work, we follow a similar strategy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQf5h65/content/2301.08734v1.pdf'}
+page_content='  In order to respect the inherent symmetries of molec-  ular systems, most architectures are designed to be in-  variant with respect to rotations, translations and ex-  change of identical atoms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQf5h65/content/2301.08734v1.pdf'}
+page_content='  These models have shown  good accuracy on many systems but usually require large  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQf5h65/content/2301.08734v1.pdf'}
+page_content='08734v1  [physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQf5h65/content/2301.08734v1.pdf'}
+page_content='chem-ph]  20 Jan 2023  2  amounts of data to be trained on (of the order of a  million molecular configurations).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQf5h65/content/2301.08734v1.pdf'}
+page_content=' More recently, equiv-  ariant models (for example Nequip [23], Allegro [28],  SpookyNet [29], UNiTE [30] or GemNet [31]) have at-  tracted much attention because of their impressive data  efficiency and their ability to generalize more accurately  to out-of-distribution configurations [23, 32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQf5h65/content/2301.08734v1.pdf'}
+page_content='  Most ML models, however, assume a purely local func-  tional form and tend to neglect or implicitly account for  long range effects from the training data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQf5h65/content/2301.08734v1.pdf'}
+page_content=' The accurate  description of long-range interactions is however critical  to correctly simulate condensed-phase systems and to de-  scribe the structure of large molecular formations (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQf5h65/content/2301.08734v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQf5h65/content/2301.08734v1.pdf'}
+page_content='  protein or DNA structure [33, 34]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQf5h65/content/2301.08734v1.pdf'}
+page_content=' The framework of  message-passing neural networks [35] in principle allows  to describe long-range effects by iteratively exchanging  information with neighbouring atoms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQf5h65/content/2301.08734v1.pdf'}
+page_content='  This approach  however have been shown to pose difficulties when ap-  plied to large systems as this iterative process is not well  suited for parallel architectures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQf5h65/content/2301.08734v1.pdf'}
+page_content='  Some other pure ML  multi-scale models are being developed (for example the  LODE descriptor [36, 37]) but this area is still in its in-  fancy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQf5h65/content/2301.08734v1.pdf'}
+page_content=' On the other hand, quantum perturbation theory  (for example SAPT) gives solid grounds to the descrip-  tion of long-range effects in terms of classical electrostat-  ics [38] which can be well captured in the FF framework  (via multipolar Coulomb interactions and dispersion ef-  fects for example).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQf5h65/content/2301.08734v1.pdf'}
+page_content=' It thus seems advantageous to com-  bine an ML model – which excel at predicting short-range  properties – with long-range FF interactions, in order to  obtain the best of both approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQf5h65/content/2301.08734v1.pdf'}
+page_content=' A few models apply-  ing this idea have recently been developed (for example  HDNNP-Gen4 [39], PhysNet [40] and others [26, 41]) and  have shown good results across multiple systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQf5h65/content/2301.08734v1.pdf'}
+page_content=' Hybrid  FF/ML models thus provide a promising route to more  physics-aware ML potentials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQf5h65/content/2301.08734v1.pdf'}
+page_content='  In this paper, we propose a general framework for  building force-field-enhanced ML models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQf5h65/content/2301.08734v1.pdf'}
+page_content='  We leverage  the latest advances in local equivariant neural networks in  order to accurately predict short-range energy contribu-  tions as well as multiple atom-in-molecule properties that  are then used to dynamically parameterize QM-inspired  FF energy terms that account for long-range interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQf5h65/content/2301.08734v1.pdf'}
+page_content='  We show that this architecture allows for highly transfer-  able models that are able to accurately generalize on large  molecular systems, as well as in the condensed phase af-  ter being trained on small monomers and dimers only.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQf5h65/content/2301.08734v1.pdf'}
+page_content='  This paper is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQf5h65/content/2301.08734v1.pdf'}
+page_content=' Section II describes  the model architecture, from the Allegro [28] equivariant  embedding to the output and physics modules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQf5h65/content/2301.08734v1.pdf'}
+page_content=' It also  describes the particular FF terms that we used for the  pretrained model, named FENNIX-OP1, that we provide  with this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQf5h65/content/2301.08734v1.pdf'}
+page_content=' Section III focuses on the FENNIX-OP1  model and provides details on its construction, its target  properties, the datasets used for training and the differ-  ent training stages that were required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQf5h65/content/2301.08734v1.pdf'}
+page_content='  In section IV,  we validate the model via several applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQf5h65/content/2301.08734v1.pdf'}
+page_content='  First,  we show that the model predicts accurate dissociation  energy curves of some simple molecules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQf5h65/content/2301.08734v1.pdf'}
+page_content=' We then com-  pute structural properties of liquid water in molecular  dynamics simulations including nuclear quantum effects  (NQEs) via the recently developed adaptive quantum  thermal bath (adQTB) [42, 43].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQf5h65/content/2301.08734v1.pdf'}
+page_content=' Indeed, since the model  is trained purely on ab initio data, the explicit inclusion  of NQEs is critical for the correct calculation of thermo-  dynamical properties, as was shown in numerous previ-  ous studies [44, 45].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQf5h65/content/2301.08734v1.pdf'}
+page_content=' For this purpose, the adQTB was  shown to provide robust approximations of NQEs while  being numerically affordable (similar to a classical MD)  and thus constitutes a very efficient tool for quickly test-  ing ML models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQf5h65/content/2301.08734v1.pdf'}
+page_content=' We then show that the model produces  stable dynamics of alanine dipetide in solution and pro-  vides a qualitatively correct description of the torsional  free energy profile (computed using an enhanced sam-  pling method[46]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQf5h65/content/2301.08734v1.pdf'}
+page_content=' Finally, section V provides some con-  clusions and outlooks for future extensions of the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQf5h65/content/2301.08734v1.pdf'}
+page_content='  II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQf5h65/content/2301.08734v1.pdf'}
+page_content='  MODEL ARCHITECTURE  The FENNIX (Force-field-Enhanced Neural Network  Interactions) model is based on a local multi-output  equivariant model that processes atomic neighborhoods  and predicts multiple atomic or pairwise properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQf5h65/content/2301.08734v1.pdf'}
+page_content=' As  an example for this work, we will present a model that  outputs local pairwise energy contributions, charges and  atomic volumes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQf5h65/content/2301.08734v1.pdf'}
+page_content=' The output is subsequently enriched by  a "physical" module that computes force field terms such  as electrostatic and dispersion energy terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQf5h65/content/2301.08734v1.pdf'}
+page_content=' The core of  our model is a slightly modified version of the Allegro  local equivariant model presented in ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQf5h65/content/2301.08734v1.pdf'}
+page_content=' [28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQf5h65/content/2301.08734v1.pdf'}
+page_content=' We use the  Allegro model as a general embedding of atomic pairs  which is then fed into independent neural networks that  predict the target properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQf5h65/content/2301.08734v1.pdf'}
+page_content='  In this section, we will briefly describe the Allegro ar-  chitecture and our modifications to the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQf5h65/content/2301.08734v1.pdf'}
+page_content=' The the-  oretical analysis of this architecture was thoroughly done  in the original paper [28] so that we will only review here  the main points necessary to the understanding of the  model and our improvements to the architecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQf5h65/content/2301.08734v1.pdf'}
+page_content='  We  will then present the output module that processes the  Allegro embedding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQf5h65/content/2301.08734v1.pdf'}
+page_content=' Finally, we will describe the phys-  ical module and the particular functional form for the  FENNIX-OP1 potential energy surface that we used in  this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQf5h65/content/2301.08734v1.pdf'}
+page_content='  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQf5h65/content/2301.08734v1.pdf'}
+page_content='  Equivariant embedding using the Allegro  architecture  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQf5h65/content/2301.08734v1.pdf'}
+page_content='  Review of the Allegro architecture  The Allegro model provides a local many-body descrip-  tor  �  xij, V nlp  ij  �  – interchangeably referred to as embed-  ding in the following – for each directed pair of atoms  3  with source atom i and destination atom j in the neigh-  borhood N(i) defined by all the atoms located at a dis-  tance shorter than a cutoff radius rc from atom i:  N(i) =  �  k  s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQf5h65/content/2301.08734v1.pdf'}
+page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQf5h65/content/2301.08734v1.pdf'}
+page_content='  ���⃗Rik  ��� < rc  �  (1)  with ⃗Rik = ⃗Rk− ⃗Ri the vector going from the position of  atom i to atom k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQf5h65/content/2301.08734v1.pdf'}
+page_content=' The first part of the descriptor xij is  built so that it is invariant under the action of certain geo-  metric symmetries (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQf5h65/content/2301.08734v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQf5h65/content/2301.08734v1.pdf'}
+page_content=' global rotations, translations and  inversions of the system).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQf5h65/content/2301.08734v1.pdf'}
+page_content=' On the other hand, the second  descriptor V nlp  ij  is composed of features that are equivari-  ant with respect to these symmetries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQf5h65/content/2301.08734v1.pdf'}
+page_content=' These features take  the form of tensors that are labeled with a channel index  n ∈ 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQf5h65/content/2301.08734v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQf5h65/content/2301.08734v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQf5h65/content/2301.08734v1.pdf'}
+page_content=' , Nchannels, a rotational index l ∈ 0, 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQf5h65/content/2301.08734v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQf5h65/content/2301.08734v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQf5h65/content/2301.08734v1.pdf'}
+page_content=' , lmax  and a parity index p ∈ −1, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQf5h65/content/2301.08734v1.pdf'}
+page_content=' The rotational index in-  dicates how the tensor transforms under rotation oper-  ations: l = 0 corresponds to scalar/invariant quantities,  l = 1 corresponds to vector-like objects and we refer to  l >= 2 objects as higher-order tensors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQf5h65/content/2301.08734v1.pdf'}
+page_content=' The parity in-  dex, on the other hand, indicates how the tensor’s sign  changes under inversion of the coordinate system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQf5h65/content/2301.08734v1.pdf'}
+page_content=' In our  implementation of the Allegro model, these tensorial ob-  jects are handled by the e3nn python package [47] which  provides high-level classes that represent them and easy-  to-use functions to manipulate and combine them while  preserving symmetries and global equivariance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQf5h65/content/2301.08734v1.pdf'}
+page_content='  In the following paragraphs, we describe how initial  two-body features are computed, how features from  neighbors are combined through Nlayers layers of interac-  tions to enrich them with many-body information and  how they are filtered at each layer to control the size of  the embedding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQf5h65/content/2301.08734v1.pdf'}
+page_content='  Initial two-body features  The Allegro model starts by decomposing each inter-  atomic vector {⃗Rij}j∈N (i) into fingerprints that are more  suitably processed by the network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQf5h65/content/2301.08734v1.pdf'}
+page_content=' The interatomic dis-  tance Rij is projected onto a radial basis B(Rij) =  [B1(Rij), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQf5h65/content/2301.08734v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQf5h65/content/2301.08734v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQf5h65/content/2301.08734v1.pdf'}
+page_content=' , BNbasis(Rij)] (we use the Bessel basis func-  tion with a polynomial envelope [48] that we normalize  as in the original paper) and we compute the two-body  scalar embedding as:  x  2B  ij = MLP2B  �1(Zi) || 1(Zj) || B(Rij)  �  fc(Rij)  (2)  where || denotes concatenation, MLP2B is a multilayer  perceptron (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQf5h65/content/2301.08734v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQf5h65/content/2301.08734v1.pdf'}
+page_content=' a fully connected scalar neural network),  fc(Rij) is a cutoff function going smoothly to zero as Rij  approaches rc (we use the same polynomial envelope as  for the radial basis) and 1(Zi) (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQf5h65/content/2301.08734v1.pdf'}
+page_content=' 1(Zj)) is a vector  representing the chemical species of the source atom i  (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQf5h65/content/2301.08734v1.pdf'}
+page_content=' destination atom j).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQf5h65/content/2301.08734v1.pdf'}
+page_content=' In the original paper 1(Zi)  is a direct one-hot encoding of the atomic number Zi,  meaning that one has to fix in advance the number of  species the model will be able to process (the one-hot en-  coding then defines a simple orthonormal basis which has  the same dimensions as the chosen number of species).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQf5h65/content/2301.08734v1.pdf'}
+page_content=' In  section II A 2, we propose to modify this one-hot encod-  ing by a positional encoding of coordinates in the periodic  table which allows for more flexibility in the treatment  of atomic species.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQf5h65/content/2301.08734v1.pdf'}
+page_content='  We obtain the two-body equivariant features by pro-  jecting the unit vector ˆRij = ⃗Rij/Rij onto a basis of  real spherical harmonics Y lp  ij .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQf5h65/content/2301.08734v1.pdf'}
+page_content=' We then mix them with  radial information with a linear embedding on Nchannels  channels:  V nlp,2B  ij  =  �  MLP  2B  embed(x  2B  ij )  �nlp Y lp  ij  (3)  Interaction with the local environment  The two-body embedding  �  x2B  ij , V nlp,2B  ij  �  is then pro-  cessed through multiple "interaction" layers that allow  to combine information with other atoms in the vicinity  of atom i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQf5h65/content/2301.08734v1.pdf'}
+page_content='  Each interaction layer starts by building a  global equivariant neighborhood embedding for atom i  from the current scalar embeddings xik and the spherical  harmonics projections Y lp  ik :  Γnlp,(L)  i  =  �  k∈N (i)  �  MLP (L)  embed(x(L−1)  ik  )  �nlp  Y lp  ik  (4)  with L = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQf5h65/content/2301.08734v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQf5h65/content/2301.08734v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQf5h65/content/2301.08734v1.pdf'}
+page_content=' , Nlayers the layer index and x(0)  ik = x2B  ik and  V nlp,(0)  ij  = V nlp,2B  ij  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQf5h65/content/2301.08734v1.pdf'}
+page_content=' The interaction is then performed via  a tensor product of Γnlp,(L)  i  with each equivariant em-  bedding V nlp,(L−1)  ij  (the tensor product is done indepen-  dently for each channel n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQf5h65/content/2301.08734v1.pdf'}
+page_content=' The resulting "latent space"  Lnmlp,(L)  ij  =  �  Γnl1p1,(L)  i  ⊗ V nl2p2,(L)  ij  �nmlp  (5)  contains all possible combinations of rotational and  parity indices that are allowed by symmetry (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQf5h65/content/2301.08734v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQf5h65/content/2301.08734v1.pdf'}
+page_content=' such  that |l1 − l2| ≤ l ≤ |l1 + l2| and p = p1p2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQf5h65/content/2301.08734v1.pdf'}
+page_content='  Note  that since multiple combinations of (l1, p1), (l2, p2) may  produce outputs of indices (l, p), we need to add a  multiplicity index m that distinguishes these paths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQf5h65/content/2301.08734v1.pdf'}
+page_content='  Feature filtering and channel mixing  Finally, the latent space is filtered to obtain the new pair-  wise embedding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQf5h65/content/2301.08734v1.pdf'}
+page_content=' The scalar embedding is combined with  the scalar part of the latent space (with every channels  and all multiplicities concatenated) to obtain:  x(L)  ij  = α x(L−1)  ij  +  �  1 − α2 fc(Rij)  × MLP (L)  latent  �  x(L−1)  ij  ||  n,m  Lnm01,(L)  ij  �  (6)  with 0 ≤ α < 1 a mixing coefficient that allows to easily  propagate scalar information from a layer to the next.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQf5h65/content/2301.08734v1.pdf'}
+page_content=' In  our implementation, the value of α can be set as a hyper-  parameter (for example to the value α = 2/  √  5 proposed  4  in the original Allegro paper) or can be optimized inde-  pendently for each layer during the training procedure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQf5h65/content/2301.08734v1.pdf'}
+page_content='  The new equivariant features are obtained by linearly  combining the elements of the latent space with same  indices (l, p) from all channels and multiplicities:  V nlp,(L)  ij  =  �  n′,m  wnlp,(L)  n′,m  Ln′mlp,(L)  ij  (7)  which results in features with the same number of ele-  ments as the previous layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQf5h65/content/2301.08734v1.pdf'}
+page_content='  The weights wnlp,(L)  n′,m  are  optimized in the training procedure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQf5h65/content/2301.08734v1.pdf'}
+page_content='  The  output  features  of  the  last  layer  �  x(Nlayers)  ij  , V nlp,(Nlayers)  ij  �  compose  the  many-body  embedding of our model which is passed to the output  module to predict the different atomic or pairwise  properties that the model is trained on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQf5h65/content/2301.08734v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQf5h65/content/2301.08734v1.pdf'}
+page_content='  Positional encoding of chemical species  While a one-hot encoding allows to represent chemi-  cal species in a simple manner, it fixes from the start  the number of different species that the model can treat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQf5h65/content/2301.08734v1.pdf'}
+page_content='  Thus, if more data becomes available for new species, one  would have to retrain the model from scratch in order to  accommodate for the new data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQf5h65/content/2301.08734v1.pdf'}
+page_content='  Furthermore, in such  encoding, all species are treated equally and no similar-  ities between species (for example closeness in the peri-  odic table) are provided: the network must learn these  correlations purely from data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQf5h65/content/2301.08734v1.pdf'}
+page_content=' This encoding is thus suit-  able when targeting a specific system but might not be  the best choice when building a more general chemical  model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQf5h65/content/2301.08734v1.pdf'}
+page_content='  In our implementation, the default encoding is a po-  sitional encoding (as defined in ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQf5h65/content/2301.08734v1.pdf'}
+page_content=' [49]) that encodes  coordinates in the periodic table using sine and cosine  functions of different frequencies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQf5h65/content/2301.08734v1.pdf'}
+page_content=' For example, the col-  umn index c is encoded as a vector ec of dimension dcol  as:  ∀k ∈ 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQf5h65/content/2301.08734v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQf5h65/content/2301.08734v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQf5h65/content/2301.08734v1.pdf'}
+page_content=' , dcol, (ec)k =  �  �  �  �  �  sin  �  c/γ2i/dcol  col  �  if k=2i  cos  �  c/γ2i/dcol  col  �  if k=2i+1  (8)  and similarly for the row index with dimension drow and  frequency parameter γrow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQf5h65/content/2301.08734v1.pdf'}
+page_content=' In our implementation, the di-  mensions and frequency parameters are fixed at dcol = 10,  drow = 5, γcol = 1000 and γrow = 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQf5h65/content/2301.08734v1.pdf'}
+page_content='  These could  be also be treated as hyperparameters or even learned  during training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQf5h65/content/2301.08734v1.pdf'}
+page_content='  The row and column encodings are  then concatenated to obtain the full encoding vector  1(Zi) =  �  erow(Zi) || ecol(Zi)  �  Figure 1 shows a heatmap  of the positional encoding of the species H,C,N,O and S  (from top to bottom).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQf5h65/content/2301.08734v1.pdf'}
+page_content=' We see that the first five columns  are the same for all the heavy atoms as they represent  the row encoding (the second row of the periodic table  in this case) while they are different from the first line  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQf5h65/content/2301.08734v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQf5h65/content/2301.08734v1.pdf'}
+page_content=' Heatmap of the positional encodings of the chemi-  cal species H,C,N,O,F (from top to bottom).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQf5h65/content/2301.08734v1.pdf'}
+page_content=' The first five  columns represent the encoding of the row index in the peri-  odic table, while the last ten columns represent the encoding  of the column index.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQf5h65/content/2301.08734v1.pdf'}
+page_content='  corresponding to the Hydrogen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQf5h65/content/2301.08734v1.pdf'}
+page_content=' We also see that the last  ten columns are different for all the species shown here  as they are all on different columns of the periodic table.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQf5h65/content/2301.08734v1.pdf'}
+page_content='  The motivation behind using this positional encoding is  that we hypothesized that having similar encodings for  species sharing a row or a column might help with gen-  eralization and allow to transfer learned knowledge from  a species to another, thus requiring less training data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQf5h65/content/2301.08734v1.pdf'}
+page_content='  Furthermore, as stated in ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQf5h65/content/2301.08734v1.pdf'}
+page_content=' [49] the encoding for index  c+k can be represented as a linear function of the encod-  ing for index c, which might further help with inferring  similarities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQf5h65/content/2301.08734v1.pdf'}
+page_content='  Additional features such as the ionization  state could be encoded in the same manner (though we  restrict ourselves to neutral atoms in this work, thus only  requiring the knowledge of chemical species).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQf5h65/content/2301.08734v1.pdf'}
+page_content='  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQf5h65/content/2301.08734v1.pdf'}
+page_content='  Output module  After computing the embedding from the Allegro  model, we use it as input for independent MLPs for each  target property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQf5h65/content/2301.08734v1.pdf'}
+page_content=' In the following, we will simply denote  �  xij, V nlp  ij  �  the output from the last Allegro layer (thus  dropping the (L) layer index).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQf5h65/content/2301.08734v1.pdf'}
+page_content=' The current implementa-  tion also allows some modularity in the composition of  inputs and on the operations done on the outputs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQf5h65/content/2301.08734v1.pdf'}
+page_content=' For  example, the input can exploit either the scalar embed-  ding to obtain invariant properties via a standard MLP  as  oij = MLPout[xij]  (9)  , or both the scalar and tensorial embeddings via a linear  projection of V nlp  ij  Omlp  ij  =  �  n  [MLPout(xij]mlp  n  V nlp  ij  (10)  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQf5h65/content/2301.08734v1.pdf'}
+page_content='  For atom-wise properties, the pairwise outputs are  simply summed up on the central atom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQf5h65/content/2301.08734v1.pdf'}
+page_content=' For properties  that should sum up to zero (for example partial atomic  charges), the outputs oij and oji can be antisymmetrized  (which for partial charges is equivalent to charge ex-  change between neighbouring atom pairs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQf5h65/content/2301.08734v1.pdf'}
+page_content=' Futhermore,  0  1  2  3  45  in order to impose constraints on invariant outputs, a fi-  nal activation function can optionally be applied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQf5h65/content/2301.08734v1.pdf'}
+page_content=' This  is for example useful when the output targets a positive  quantity (for instance an atomic volume) for which we  can apply a softplus function, a probability for which  we may apply a sigmoid function or a discrete proba-  bility distribution (in the case of multidimensional oij)  for which a softmax function can be used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQf5h65/content/2301.08734v1.pdf'}
+page_content=' Finally, the  output is optionally shifted and scaled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQf5h65/content/2301.08734v1.pdf'}
+page_content='  Further modifications can optionally be applied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQf5h65/content/2301.08734v1.pdf'}
+page_content=' For  instance, the input can be filtered according to an addi-  tional shorter-range cutoff for example one distinguish-  ing between bonded and non-bonded pairs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQf5h65/content/2301.08734v1.pdf'}
+page_content=' Finally, the  two-body embedding x2B  ij can be used in place of or con-  catenated to (as it is done in the FENNIX-OP1 model)  the final embedding to use as input.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQf5h65/content/2301.08734v1.pdf'}
+page_content='  This allows the  output MLP to easily access simple pairwise informa-  tion and should let the Allegro embedding specialize in  finer many-body correlations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQf5h65/content/2301.08734v1.pdf'}
+page_content='  This compositional ap-  proach allows for a great flexibility in the model’s output,  which is especially useful when experimenting with a ML  parametrization of physical models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQf5h65/content/2301.08734v1.pdf'}
+page_content='  The output module for the FENNIX-OP1 model is  composed of three targets: a local energy contribution  V NN  i  = �  j∈N (i) V NN  ij  (which is a simple scalar out-  put), an atomic partial charge qNN  i  = �  j∈N (i) ∆qij −  ∆qji (through antisymmetrized charge exchange) and an  atomic volume vNN  i  (constrained to be positive).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQf5h65/content/2301.08734v1.pdf'}
+page_content='  C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQf5h65/content/2301.08734v1.pdf'}
+page_content='  Physics module and energy functional form  Finally, the physics module uses the results from the  output module and feed them into physically-motivated  models to enrich the output.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQf5h65/content/2301.08734v1.pdf'}
+page_content='  In the case of FENNIX-OP1, the force field module is  composed of an electrostatic energy term V  elec,CP  ij  , and a  pairwise dispersion term V  disp  ij  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQf5h65/content/2301.08734v1.pdf'}
+page_content=' The functional form of  FENNIX-OP1 is then given by:  VOP1 =  �  i  V  NN  i  +  �  i,j<i  V  elec,CP  ij  +  �  i,j<i  V  disp  ij  (11)  The first term V NN  i  is the neural network contribution  that accounts for short-range interactions that we intro-  duced in section II B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQf5h65/content/2301.08734v1.pdf'}
+page_content=' We model the electrostatic inter-  action V  elec,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQf5h65/content/2301.08734v1.pdf'}
+page_content='CP  ij  via a Coulomb potential with fluctuating  charges and the Piquemal charge penetration model [50]:  V  elec,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQf5h65/content/2301.08734v1.pdf'}
+page_content='CP  ij  =  1  Rij  �  NiNj + Nj(qi − Ni)f i  damp(Rij)  + Ni(qj − Nj)f j  damp(Rij)  + (qi − Ni)(qj − Nj)f i  ov(Rij)f j  ov(Rij)  �  (12)  where Ni is the number of valence electrons of atom i ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQf5h65/content/2301.08734v1.pdf'}
+page_content='  qi = ϵqNN  i  is the environment-dependent charge of atom i  predicted by the neural network (with an adjustable uni-  versal scaling parameter ϵ) and f i  damp(r) = 1 − e−αr/rvdw  i  and f i  ov(r) = 1 − e−βr/rvdw  i  are damping functions with  α and β adjustable parameters that are assumed to be  universal and where  r  vdw  i  =  � vNN  i  vfree  i  � 1  3  r  vdw,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQf5h65/content/2301.08734v1.pdf'}
+page_content='free  i  (13)  is the environment-dependent Van der Waals radius of  atom i with vNN  i /vfree  i  the atomic volume ratio predicted  by the neural network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQf5h65/content/2301.08734v1.pdf'}
+page_content='  Finally, the dispersion interaction V  disp  ij  is computed  using the pairwise Tkatchenko-Scheffler model [51]:  V  disp  ij  = −C6,ij  R6  ij  σij(Rij)  (14)  with the combination rule:  C6,ij =  2C6,iC6,j  C6,i  αj  αi + C6,j αi  αj  (15)  and the environment-dependent homonuclear parame-  ters:  C6,i =  � vNN  i  vfree  i  �2  C  free  6,i  ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQf5h65/content/2301.08734v1.pdf'}
+page_content='  αi =  � vNN  i  vfree  i  �  α  free  i  (16)  with αfree  i  the isolated atom polarizabilities, Cfree  6,i the iso-  lated atom sixth order dispersion coefficients and the sig-  moid damping function:  σij(Rij) =  �  1 + e  −γ  �  1  s  Rij  rvdw  i  +rvdw  j  −1  ��−1  (17)  with γ and s adjustable parameters that we assume to  be universal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQf5h65/content/2301.08734v1.pdf'}
+page_content='  We furthermore mention that the physics module is  not limited in principle to compute energy terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQf5h65/content/2301.08734v1.pdf'}
+page_content=' The  aim of this module is to be a general physical interface  between the ML embedding and the final target proper-  ties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQf5h65/content/2301.08734v1.pdf'}
+page_content=' For example, we use it in the FENNIX-OP1 model  to correct for ML-predicted charge exchanges that would  deplete the valence shell of an atom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQf5h65/content/2301.08734v1.pdf'}
+page_content=' The implementation  also provides charge exchange via a simple bond-capacity  model [52] that leverages ML-predicted atom-in-molecule  electronegativities and which capabilities will be explored  in future iterations of the FENNIX model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQf5h65/content/2301.08734v1.pdf'}
+page_content=' Each phys-  ical model is implemented as a simple Pytorch Module  that takes as input a dictionary which contains previously  computed properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQf5h65/content/2301.08734v1.pdf'}
+page_content=' It then adds its contribution and  outputs the enriched dictionnary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQf5h65/content/2301.08734v1.pdf'}
+page_content='  These physical sub-  modules can then easily be chained, and it facilitates the  implementation of additional models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQf5h65/content/2301.08734v1.pdf'}
+page_content='  6  III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQf5h65/content/2301.08734v1.pdf'}
+page_content='  DATASETS AND TRAINING PROCEDURE  In this section, we start by providing some details  on the construction of FENNIX-OP1 model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQf5h65/content/2301.08734v1.pdf'}
+page_content=' We then  review the datasets that we used for training the model  and its target properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQf5h65/content/2301.08734v1.pdf'}
+page_content=' Finally, we will focus on the  non-trivial task of training a multi-output FENN model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQf5h65/content/2301.08734v1.pdf'}
+page_content='  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQf5h65/content/2301.08734v1.pdf'}
+page_content='  FENNIX-OP1 model architecture  In the FENNIX-OP1, chemical species are encoded  using the positional encoding defined in section II A 2,  with 5 dimensions for the row encoding and 10 dimen-  sions for the column encoding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQf5h65/content/2301.08734v1.pdf'}
+page_content='  The use Nbasis = 10  Bessel basis functions for the radial embedding, with a  cutoff distance rc = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQf5h65/content/2301.08734v1.pdf'}
+page_content='2 Åand a smoothing parameter  p = 3 for the polynomial envelope.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQf5h65/content/2301.08734v1.pdf'}
+page_content='  We used 256  features for the scalar embedding and 10 channels with  a maximum rotational index lmax = 2 for all equivariant  features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQf5h65/content/2301.08734v1.pdf'}
+page_content=' The two-body scalar MLP is composed of two  hidden layers with 64 and 128 neurons respectively with  SiLU activation function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQf5h65/content/2301.08734v1.pdf'}
+page_content='  The two-body embedding  MLP for the initial equivariant features is a simple  linear projection of the two-body embedding with no  activation function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQf5h65/content/2301.08734v1.pdf'}
+page_content=' The embedding is constructed using  3 Allegro layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQf5h65/content/2301.08734v1.pdf'}
+page_content=' In each layer, the embedding MLP is a  simple linear projection with no activation function and  the latent MLP contains two hidden layers both with  256 neurons and SiLU activation function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQf5h65/content/2301.08734v1.pdf'}
+page_content=' The mixing  coefficient α is initially set to the original 2/  √  5 and is  optimized independently for each layer during training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQf5h65/content/2301.08734v1.pdf'}
+page_content='  In total, the Allegro model has NNN parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQf5h65/content/2301.08734v1.pdf'}
+page_content='  The output module is composed of three independent  identical MLPs for the short-range energy contribution,  the charge exchange and the atomic volume.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQf5h65/content/2301.08734v1.pdf'}
+page_content=' The input of  these MLPs is the concatenation of the two-body scalar  embedding and the final scalar embedding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQf5h65/content/2301.08734v1.pdf'}
+page_content=' They have 5  hidden layers with 256, 128, 64, 32 and 16 neurons with  a total of NNN parameters each.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQf5h65/content/2301.08734v1.pdf'}
+page_content='  The output module also has a "constant" submodule  that simply provides the atomic reference energies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQf5h65/content/2301.08734v1.pdf'}
+page_content='  Importantly,  we used the CCSD(T) isolated atom  energies as a reference instead of the average atomic  energy over the dataset that is typically advised when  training a ML model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQf5h65/content/2301.08734v1.pdf'}
+page_content=' Indeed, this reference energy has a  physical meaning – which is the energy of an atom when  it has no neighbors to interact with – and is not solely  a convenience parameter for facilitating the training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQf5h65/content/2301.08734v1.pdf'}
+page_content='  In particular, this choice has huge consequences for the  description of molecular dissociation, as will become  clear in section IV A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQf5h65/content/2301.08734v1.pdf'}
+page_content='  Finally, the physics module is composed of four sub-  modules: a charge correction module, a charge scaling  module, the Coulomb interaction module and the dis-  persion module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQf5h65/content/2301.08734v1.pdf'}
+page_content=' These last two modules implement the  physical interactions using the corresponding equations  described in section II C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQf5h65/content/2301.08734v1.pdf'}
+page_content=' The charge scaling module sim-  ply scales all charges by a constant factor in order to  compensate for the lack of higher order multipoles in the  permanent electrostatics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQf5h65/content/2301.08734v1.pdf'}
+page_content=' The charge correction module  antisymmetrizes the charge exchanges and ensures that  atoms do not unphysically deplete their valence shell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQf5h65/content/2301.08734v1.pdf'}
+page_content=' In-  deed, if we assume that only valence electrons can be  transferred, an atom cannot have a partial charge larger  than +Ni (which is particularly important for the charge  penetration model that we use).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQf5h65/content/2301.08734v1.pdf'}
+page_content=' This constraint is not  ensured by the neural network model so we need to en-  force it a posteriori.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQf5h65/content/2301.08734v1.pdf'}
+page_content=' The constraint is achieved by trans-  ferring back some electrons from neighbouring atoms that  drained too much charge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQf5h65/content/2301.08734v1.pdf'}
+page_content=' First the unphysical "hole" in  the valence shell is computed using a smooth function (to  ensure smooth gradients of the final coulomb energy) on  the basis that an atom cannot lose more than 95 percent  of its valence electrons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQf5h65/content/2301.08734v1.pdf'}
+page_content=' Charges that sum up to the va-  lence hole are then transferred from neighbouring atoms  that took charges, proportional to the quantity drained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQf5h65/content/2301.08734v1.pdf'}
+page_content='  This procedure should enforce the constraint in a single  iteration for most non-pathological cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQf5h65/content/2301.08734v1.pdf'}
+page_content=' Reassuringly  however, the charge correction is almost never needed af-  ter training (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQf5h65/content/2301.08734v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQf5h65/content/2301.08734v1.pdf'}
+page_content=' the valence hole is almost always zero),  even in condensed-phase MD simulations for which the  model was not explicitly trained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQf5h65/content/2301.08734v1.pdf'}
+page_content='  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQf5h65/content/2301.08734v1.pdf'}
+page_content='  Datasets and target properties  For the FENNIX-OP1 model, we chose to reproduce  high-level coupled-cluster energies and we thus selected  three of the few freely-available and generalist datasets  that provide such data:  the ANI-1ccx [17] dataset,  the DES370K [53] dataset and the q-AQUA dimers  dataset [54].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQf5h65/content/2301.08734v1.pdf'}
+page_content='  The  ANI-1ccx  dataset  provides  approximately  500,000 CCSD(T)/CBS total energies of various neu-  tral monomers and dimers (composed of the elements  H,C,N,O) in equilibrium and out-of-equilibrium configu-  rations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQf5h65/content/2301.08734v1.pdf'}
+page_content=' It also provides atomic forces at DFT level that  were crucial to speed-up the beginning of the training  procedure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQf5h65/content/2301.08734v1.pdf'}
+page_content=' Importantly, it provides partial charges and  atomic volumes obtained from a Minimal Basis Iterative  Stockholder [55] (MBIS) partitioning of the DFT elec-  tronic density that were used to parametrize the force  field terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQf5h65/content/2301.08734v1.pdf'}
+page_content='  The DES370K dataset is composed of approxi-  mately 370,000 CCSD(T)/CBS interaction energies of  diverse dimers in various configurations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQf5h65/content/2301.08734v1.pdf'}
+page_content='  It comprises  both neutral and charged molecules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQf5h65/content/2301.08734v1.pdf'}
+page_content=' The latter were dis-  carded for the training of the FENNIX-OP1 model as  out-of-scope for this study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQf5h65/content/2301.08734v1.pdf'}
+page_content=' It also provides SAPT de-  compositions of the interaction energies that we used to  regularize the Coulomb interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQf5h65/content/2301.08734v1.pdf'}
+page_content='  The q-AQUA  dimers  dataset is composed of  more than 70000 water dimer interaction energies at  7  CCSD(T)/CBS level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQf5h65/content/2301.08734v1.pdf'}
+page_content=' We used this dataset in the end of  the training in order to fine-tune the model for handling  water.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQf5h65/content/2301.08734v1.pdf'}
+page_content='  The FENNIX-OP1 model has then three target prop-  erties that use the available data: the CCSD(T) total  energy of the system as modelled by VOP1 described in  section II C, the MBIS partial charges qNN  i  and the ratio  of MBIS volumes to free atom volumes vNN  i /vfree  i  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQf5h65/content/2301.08734v1.pdf'}
+page_content='  C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQf5h65/content/2301.08734v1.pdf'}
+page_content='  Training procedure  The training of a multi-output force-field-enhanced  neural network revealed to be a non-trivial task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQf5h65/content/2301.08734v1.pdf'}
+page_content='  In-  deed, the inter-dependencies between target properties  (for example partial charges and the electrostatic contri-  bution to the total energy) seemed to pose difficulties for  the standard optimization methods, which implied that  a brute-force optimization of the whole model would not  give satisfactory results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQf5h65/content/2301.08734v1.pdf'}
+page_content=' To overcome this difficulty, we  resorted to a training procedure in multiple stages that  is described in the following of this section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQf5h65/content/2301.08734v1.pdf'}
+page_content='  Furthermore, in order to obtain a final model that was  stable when performing molecular dynamics, we found  that strong regularization was needed, both in the form  of standard weight decay and also physically-motivated  loss contributions, as detailed later.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQf5h65/content/2301.08734v1.pdf'}
+page_content='  Throughout, we  used the AdamW optimizer [56] implemented in Pytorch,  as its algorithm for weight decay provides one of the best  compromise between training speed and accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQf5h65/content/2301.08734v1.pdf'}
+page_content=' We  used a fairly strong weight decay parameter of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQf5h65/content/2301.08734v1.pdf'}
+page_content='5 for  all the parameters in the output module and no weight  decay for the Allegro parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQf5h65/content/2301.08734v1.pdf'}
+page_content=' In all stages, we used  the same random 10% of the dataset as a validation  set and optimized the model using mini-batches of 256  configurations from the ANI-1ccx and 64 configurations  from DES370K and q-AQUA when they are used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQf5h65/content/2301.08734v1.pdf'}
+page_content='  The training procedure for the FENNIX-OP1 model  required four stages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQf5h65/content/2301.08734v1.pdf'}
+page_content=' In the first stage, the model was  trained to reproduce DFT total energies and forces using  the short-range contribution V NN only.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQf5h65/content/2301.08734v1.pdf'}
+page_content=' We also trained  at the same time the charges and atomic volumes to re-  produce the MBIS targets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQf5h65/content/2301.08734v1.pdf'}
+page_content=' At this stage, only the ANI-  1ccx dataset was used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQf5h65/content/2301.08734v1.pdf'}
+page_content=' The loss function for this stage is  given by:  L(1) = λE(E  DFT − V  NN)2 + λF  3  �  j=1  Nat  �  i=1  �  F  DFT  ij  − F  NN  ij  �2  + λq  Nat  �  i=1  �  q  MBIS  i  − qNN  i  �2 + λv  Nat  �  i=1  �vMBIS  i  vfree  i  − vNN  i  vfree  i  �2  (18)  with λE = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQf5h65/content/2301.08734v1.pdf'}
+page_content='001, λF = 1 and λq = λv = 1000 and F NN is  the model’s predicted force obtained by automatic differ-  entiation (Pytorch’s autograd) of the total energy V NN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQf5h65/content/2301.08734v1.pdf'}
+page_content='  As suggested in previous studies [28], we strongly favour  learning forces over energies in the beginning to acceler-  ate the training procedure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQf5h65/content/2301.08734v1.pdf'}
+page_content=' We note that the provided  loss is for a single configuration and that, in practice,  it is averaged over the configurations in the mini-batch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQf5h65/content/2301.08734v1.pdf'}
+page_content='  We further added a regularization in order to minimize  the off-diagonal elements of the covariance matrix of the  scalar embedding’s features over a batch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQf5h65/content/2301.08734v1.pdf'}
+page_content=' This promotes  learning statistically independent features in the embed-  ding which should be favourable for a multi-output net-  work and we found that it led to models with better  generalization capabilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQf5h65/content/2301.08734v1.pdf'}
+page_content=' In this stage, we train all the  parameters in the the embedding and output modules  with a starting learning rate of 10−3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQf5h65/content/2301.08734v1.pdf'}
+page_content=' Furthermore, we  used a learning rate scheduler that reduces the learning  rate when the error on the training set stops diminishing  for a few steps (we set the patience of the scheduler to 10  epochs and the learning rate scaling factor to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQf5h65/content/2301.08734v1.pdf'}
+page_content='8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQf5h65/content/2301.08734v1.pdf'}
+page_content=' After  about 100 epochs, progress of both training and valida-  tion steps slowed down and we modified the energy and  force parameters to λE = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQf5h65/content/2301.08734v1.pdf'}
+page_content='01 and λF = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQf5h65/content/2301.08734v1.pdf'}
+page_content='1 in order to  obtain a more balanced training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQf5h65/content/2301.08734v1.pdf'}
+page_content=' We stopped the first  stage when the learning rate reached 10−4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQf5h65/content/2301.08734v1.pdf'}
+page_content='  In the second stage, we freeze the embedding parame-  ters and output MLPs for charge and volumes and acti-  vate the Coulomb and dispersion energy terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQf5h65/content/2301.08734v1.pdf'}
+page_content=' We then  retrain the short-range energy MLP so that the full V OP1  of eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQf5h65/content/2301.08734v1.pdf'}
+page_content=' (11) reproduces DFT energies and forces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQf5h65/content/2301.08734v1.pdf'}
+page_content=' The loss  function for this stage is:  L(2) = λE(E  DFT − V  OP1)2 + λF  3  �  j=1  Nat  �  i=1  �  F  DFT  ij  − F  OP1  ij  �2  (19)  with the same weights as in the end of the previous stage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQf5h65/content/2301.08734v1.pdf'}
+page_content='  Freezing the embedding ensures that the predicted vol-  umes and charges are not modified in this training stage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQf5h65/content/2301.08734v1.pdf'}
+page_content='  Since the energy target is modified, the errors starts much  higher than at the end of the previous stage and quickly  decreased.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQf5h65/content/2301.08734v1.pdf'}
+page_content=' When the error on the train and validation  sets drop to the same order as in the previous stage, the  full model is unfrozen and training is resumed until the  error stops decreasing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQf5h65/content/2301.08734v1.pdf'}
+page_content='  In the third stage, the embedding and charge and vol-  umes MLPs are frozen again and the energy MLP is  finally retrained to reproduce CCSD(T) energies from  both ANI-1ccx total energies and DES370K interaction  energies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQf5h65/content/2301.08734v1.pdf'}
+page_content='  We used the same type of mean-square loss  functions with λE = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQf5h65/content/2301.08734v1.pdf'}
+page_content='1 and λDES = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQf5h65/content/2301.08734v1.pdf'}
+page_content=' Again, we train  the energy MLP until the error is close to the previous  stage, unfreeze the whole model and optimize again all  the parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQf5h65/content/2301.08734v1.pdf'}
+page_content=' For this stage, we also optimize the pa-  rameters from the physical module in order to reach the  lowest error possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQf5h65/content/2301.08734v1.pdf'}
+page_content='  We stop the training when the  learning rate reaches 10−5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQf5h65/content/2301.08734v1.pdf'}
+page_content='  In the last training stage, we refine the model for  water by including the q-AQUA water dimers interaction  energies in the loss function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQf5h65/content/2301.08734v1.pdf'}
+page_content=' We also generated batches  of randomly deformed water monomers (with very large  8  deformations) and trained the model to reproduce forces  from the highly accurate Partridge-Schwenke potential  energy surface (that was itself fitted on high-accuracy  coupled-cluster data).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQf5h65/content/2301.08734v1.pdf'}
+page_content=' This was particularly helpful to  reproduce the molecule’s bending energy surface away  from equilibrium where fewer data are available in the  ANI-1ccx dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQf5h65/content/2301.08734v1.pdf'}
+page_content='  At the end of the training procedure, the model  reached a root mean square error (RMSE) of less that  4 kcal/mol for CCSD(T) total energies on both valida-  tion and training sets of the ANI-1ccx dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQf5h65/content/2301.08734v1.pdf'}
+page_content=' While it  was possible to reach much lower errors with less regular-  ized models (less than 1 kcal/mol), we found that they  were unstable when performing MD and concluded that  they were overfitting the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQf5h65/content/2301.08734v1.pdf'}
+page_content=' We thus favoured a more  strongly regularized, perhaps slightly underfitted model  that allowed for better generalization in the condensed  phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQf5h65/content/2301.08734v1.pdf'}
+page_content=' The model also reached a RMSE of about 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQf5h65/content/2301.08734v1.pdf'}
+page_content='35  kcal/mol on both DES370K and q-AQUA datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQf5h65/content/2301.08734v1.pdf'}
+page_content=' Fi-  nally, it reached a RMSE of about 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQf5h65/content/2301.08734v1.pdf'}
+page_content='017 e for charges and  about 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQf5h65/content/2301.08734v1.pdf'}
+page_content='017 for volume ratios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQf5h65/content/2301.08734v1.pdf'}
+page_content=' As for total energies, less  regularized models allowed for much lower errors (less  than 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQf5h65/content/2301.08734v1.pdf'}
+page_content='008 e for charges for example) but with visible  overfitting, leading to irregular coulomb interactions and  unstable dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQf5h65/content/2301.08734v1.pdf'}
+page_content='  As a validation for the water interactions, we used the  standard energy benchmarks when training force fields  for water.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQf5h65/content/2301.08734v1.pdf'}
+page_content=' The model gives a RMSE of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQf5h65/content/2301.08734v1.pdf'}
+page_content='13 kcal/mol on  the Smith dimer set [57] which are representative of the  most stable water dimer configurations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQf5h65/content/2301.08734v1.pdf'}
+page_content=' This low error  is not surprising as the Smith dimers are very close to  configurations present in the q-AQUA dataset on which  the model was trained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQf5h65/content/2301.08734v1.pdf'}
+page_content='  For a more challenging test,  we used a set of typical water clusters up to hexam-  ers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQf5h65/content/2301.08734v1.pdf'}
+page_content=' For these, the model achieved a RMSE lower than 2  kcal/mol, which is comparable to our recent Q-AMOEBA  model [45] which was specifically trained to reproduce  these energies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQf5h65/content/2301.08734v1.pdf'}
+page_content=' In the next section, we investigate more  thoroughly the validity, transferability and robustness of  the model up to the unforgiving test of condensed-phase  molecular dynamics including nuclear quantum effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQf5h65/content/2301.08734v1.pdf'}
+page_content='  IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQf5h65/content/2301.08734v1.pdf'}
+page_content='  MODEL VALIDATION  In this section, we validate the FENNIX-OP1 model  using a few examples of applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQf5h65/content/2301.08734v1.pdf'}
+page_content='  First, we com-  pute bond dissociation energy profiles of typical small  molecules and show that the model is able to consistently  break covalent bonds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQf5h65/content/2301.08734v1.pdf'}
+page_content='  Then, we show that the model  is able to produce stable and accurate MD simulations  of condensed-phase water including nuclear quantum  effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQf5h65/content/2301.08734v1.pdf'}
+page_content=' For this study, we included nuclear quantum ef-  fects using the adaptive quantum thermal bath (adQTB)  method introduced in ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQf5h65/content/2301.08734v1.pdf'}
+page_content=' [42].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQf5h65/content/2301.08734v1.pdf'}
+page_content=' We showed in previous  studies [43] that the adQTB provides an efficient and  accurate alternative to path integrals – the gold standard  method for including NQEs in MD simulations – at a  cost similar to classical MD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQf5h65/content/2301.08734v1.pdf'}
+page_content=' For these two examples, we  compare our results to the ANI models [17] that have a  comparable scope as FENNIX-OP1 in terms of chemical  diversity and were trained on similar datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQf5h65/content/2301.08734v1.pdf'}
+page_content=' Finally,  we show that FENNIX-OP1 is able to produce stable  dynamics of organic molecules solvated in water and  provides a good qualitative description of the torsional  free energy landscape of alanine dipeptide in solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQf5h65/content/2301.08734v1.pdf'}
+page_content='  All calculations for this work were performed on a  Nvidia A100 GPU using our custom TorchNFF pack-  age that is built on top of Pytorch [58] and provides the  implementation for FENNIX as well as a simple MD al-  gorithm for NVT simulations in periodic boundary con-  ditions (with Ewald summation for electrostatic inter-  actions) and an efficient Pytorch implementation of the  adQTB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQf5h65/content/2301.08734v1.pdf'}
+page_content=' TorchNFF is in an early development version  and is thus not yet optimized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQf5h65/content/2301.08734v1.pdf'}
+page_content=' For example, a FENNIX-  OP1 simulations of a box of 216 molecules of water in  periodic boundary conditions can currently reach about  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQf5h65/content/2301.08734v1.pdf'}
+page_content='8 ns/day of adQTB simulation on a single A100 GPU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQf5h65/content/2301.08734v1.pdf'}
+page_content='  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQf5h65/content/2301.08734v1.pdf'}
+page_content='  Bond dissociation energy profiles  We first validate the training and the choice of ref-  erence energy on potential energy curves for the dis-  sociation of covalent bonds in small molecules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQf5h65/content/2301.08734v1.pdf'}
+page_content='  This  kind of bond breaking is a fundamental step in the pro-  cess of many chemical reactions, for example in enzyme-  catalyzed reactions [59], and is key to the description of  mass spectroscopy experiments [60, 61].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQf5h65/content/2301.08734v1.pdf'}
+page_content=' They are how-  ever difficult to accurately model: force fields usually  forbid them by design (by assigning a fixed connectiv-  ity) and ab initio approaches require the use of expensive  multi-reference calculations to correctly describe the dis-  sociation process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQf5h65/content/2301.08734v1.pdf'}
+page_content=' Their practical description thus usu-  ally requires the use of specifically designed force fields  (such as ReaxFF) or low-dimensional potential energy  surfaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQf5h65/content/2301.08734v1.pdf'}
+page_content='  Figure 2 shows the potential energy curves for a) the  dissociation of a hydrogen atom in methane;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQf5h65/content/2301.08734v1.pdf'}
+page_content=' b) the sym-  metric dissociation of the water molecule;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQf5h65/content/2301.08734v1.pdf'}
+page_content=' c) the dissoci-  ation of the H-F molecule computed with FENNIX-OP1  and compared with reference quantum calculations from  the literature (multi-reference CI/6-31G** from ref [62]  for CH4, Partridge-Scwhenke PES [63] that was fitted on  CCSD(T) data and multi-reference CI/au-cc-pV6Z from  ref [64] for the H-F molecule) and the ANI models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQf5h65/content/2301.08734v1.pdf'}
+page_content=' We  see that FENNIX-OP1 consistently captures a smooth  dissociation curve thanks to the physically-motivated  choice of reference energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQf5h65/content/2301.08734v1.pdf'}
+page_content=' On the other hand, the ANI  models, for which the reference energy was set accord-  ing to average values over the dataset, fail to reproduce  the dissociation curves for large distances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQf5h65/content/2301.08734v1.pdf'}
+page_content=' FENNIX-OP1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQf5h65/content/2301.08734v1.pdf'}
+page_content='agrees particularly well with the reference for water as  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQf5h65/content/2301.08734v1.pdf'}
+page_content='9  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQf5h65/content/2301.08734v1.pdf'}
+page_content='1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQf5h65/content/2301.08734v1.pdf'}
+page_content='2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQf5h65/content/2301.08734v1.pdf'}
+page_content='3  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQf5h65/content/2301.08734v1.pdf'}
+page_content='4  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQf5h65/content/2301.08734v1.pdf'}
+page_content='5  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQf5h65/content/2301.08734v1.pdf'}
+page_content='H-F distance (Angstrom)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQf5h65/content/2301.08734v1.pdf'}
+page_content='0  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQf5h65/content/2301.08734v1.pdf'}
+page_content='25  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQf5h65/content/2301.08734v1.pdf'}
+page_content='50  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQf5h65/content/2301.08734v1.pdf'}
+page_content='75  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQf5h65/content/2301.08734v1.pdf'}
+page_content='100  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQf5h65/content/2301.08734v1.pdf'}
+page_content='125  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQf5h65/content/2301.08734v1.pdf'}
+page_content='150  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQf5h65/content/2301.08734v1.pdf'}
+page_content='175  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQf5h65/content/2301.08734v1.pdf'}
+page_content='200  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQf5h65/content/2301.08734v1.pdf'}
+page_content='Energy (kcal/mol)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQf5h65/content/2301.08734v1.pdf'}
+page_content='MRCI/aug-cc-pV6Z  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQf5h65/content/2301.08734v1.pdf'}
+page_content='ANI2x  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQf5h65/content/2301.08734v1.pdf'}
+page_content='FENNIX-OP1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQf5h65/content/2301.08734v1.pdf'}
+page_content='1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQf5h65/content/2301.08734v1.pdf'}
+page_content='2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQf5h65/content/2301.08734v1.pdf'}
+page_content='3  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQf5h65/content/2301.08734v1.pdf'}
+page_content='4  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQf5h65/content/2301.08734v1.pdf'}
+page_content='5  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQf5h65/content/2301.08734v1.pdf'}
+page_content='C-H distance (Angstrom)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQf5h65/content/2301.08734v1.pdf'}
+page_content='0  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQf5h65/content/2301.08734v1.pdf'}
+page_content='25  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQf5h65/content/2301.08734v1.pdf'}
+page_content='50  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQf5h65/content/2301.08734v1.pdf'}
+page_content='75  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQf5h65/content/2301.08734v1.pdf'}
+page_content='100  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQf5h65/content/2301.08734v1.pdf'}
+page_content='125  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQf5h65/content/2301.08734v1.pdf'}
+page_content='150  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQf5h65/content/2301.08734v1.pdf'}
+page_content='175  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQf5h65/content/2301.08734v1.pdf'}
+page_content='200  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQf5h65/content/2301.08734v1.pdf'}
+page_content='Energy (kcal/mol)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQf5h65/content/2301.08734v1.pdf'}
+page_content='CI  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQf5h65/content/2301.08734v1.pdf'}
+page_content='ANI2x  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQf5h65/content/2301.08734v1.pdf'}
+page_content='ANI1ccx  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQf5h65/content/2301.08734v1.pdf'}
+page_content='FENNIX-OP1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQf5h65/content/2301.08734v1.pdf'}
+page_content='a) CH4 ➞ CH3 + H  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQf5h65/content/2301.08734v1.pdf'}
+page_content='b) H2O ➞ OH + H  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQf5h65/content/2301.08734v1.pdf'}
+page_content='c) HF ➞ H + F  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQf5h65/content/2301.08734v1.pdf'}
+page_content='1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQf5h65/content/2301.08734v1.pdf'}
+page_content='2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQf5h65/content/2301.08734v1.pdf'}
+page_content='3  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQf5h65/content/2301.08734v1.pdf'}
+page_content='4  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQf5h65/content/2301.08734v1.pdf'}
+page_content='5  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQf5h65/content/2301.08734v1.pdf'}
+page_content='O-H distance (Angstrom)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQf5h65/content/2301.08734v1.pdf'}
+page_content='0  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQf5h65/content/2301.08734v1.pdf'}
+page_content='25  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQf5h65/content/2301.08734v1.pdf'}
+page_content='50  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQf5h65/content/2301.08734v1.pdf'}
+page_content='75  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQf5h65/content/2301.08734v1.pdf'}
+page_content='100  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQf5h65/content/2301.08734v1.pdf'}
+page_content='125  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQf5h65/content/2301.08734v1.pdf'}
+page_content='150  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQf5h65/content/2301.08734v1.pdf'}
+page_content='175  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQf5h65/content/2301.08734v1.pdf'}
+page_content='200  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQf5h65/content/2301.08734v1.pdf'}
+page_content='Energy (kcal/mol)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQf5h65/content/2301.08734v1.pdf'}
+page_content='Partridge&Scwhenke  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQf5h65/content/2301.08734v1.pdf'}
+page_content='ANI2x  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQf5h65/content/2301.08734v1.pdf'}
+page_content='ANI1ccx  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQf5h65/content/2301.08734v1.pdf'}
+page_content='QAMOEBA  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQf5h65/content/2301.08734v1.pdf'}
+page_content='FENNIX-OP1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQf5h65/content/2301.08734v1.pdf'}
+page_content='FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQf5h65/content/2301.08734v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQf5h65/content/2301.08734v1.pdf'}
+page_content=' Potential energy curves for the dissociations of: a) methane CH4→CH3+H with the CH3 group fixed at the CH4  equilibrium geometry;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQf5h65/content/2301.08734v1.pdf'}
+page_content=' b) water H2O→OH+H with OH and angle fixed at the H2O equilibrium geometry;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQf5h65/content/2301.08734v1.pdf'}
+page_content=' c) HF→H+F  the Partridge-Schwenke PES was included in the train-  ing set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQf5h65/content/2301.08734v1.pdf'}
+page_content=' For the other two molecules, it tends to under-  estimate the dissociation barrier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQf5h65/content/2301.08734v1.pdf'}
+page_content=' Interestingly, while the  training data did not contain covalent interaction ener-  gies for F, the model is still able to reasonably generalize  its prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQf5h65/content/2301.08734v1.pdf'}
+page_content=' Indeed, thanks to the positional encoding  of chemical species that we introduced in section II A 2,  we expect the model to be able to generalize, at least  qualitatively, across the periodic table.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQf5h65/content/2301.08734v1.pdf'}
+page_content='  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQf5h65/content/2301.08734v1.pdf'}
+page_content='  Structural properties of liquid water  In order to test the robustness of the model, we per-  formed molecular dynamics simulations of liquid water.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQf5h65/content/2301.08734v1.pdf'}
+page_content='  Since the model was fitted purely on ab initio data, it  was critical to explicitly include nuclear quantum effects  in order to accurately compute thermodynamical prop-  erties [44].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQf5h65/content/2301.08734v1.pdf'}
+page_content=' We used the adaptive quantum thermal bath  method to include NQES [42], as it was previously shown  to be a robust alternative to the more costly path inte-  grals MD [43].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQf5h65/content/2301.08734v1.pdf'}
+page_content=' All simulations were performed for a box  of 216 molecules in the NVT ensemble at 300K and ex-  perimental density, with a time step of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQf5h65/content/2301.08734v1.pdf'}
+page_content='5 fs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQf5h65/content/2301.08734v1.pdf'}
+page_content=' Coulomb  interactions in periodic boundary conditions were han-  dled using the Ewald summation method [65], that we di-  rectly implemented using PyTorch operations so that we  can leverage its automatic differentiation capabilities to  obtain atomic forces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQf5h65/content/2301.08734v1.pdf'}
+page_content=' We equilibrated the system for 100  ps, which was sufficient to thermalize the system and con-  verge the adQTB parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQf5h65/content/2301.08734v1.pdf'}
+page_content=' We then computed the ra-  dial distribution functions (RDFs) from 1 ns of MD sim-  ulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQf5h65/content/2301.08734v1.pdf'}
+page_content=' Figure 3 shows the partial RDFs of water simu-  lated using adQTB and classical MD with the FENNIX-  OP1 compared to experimental results from [66].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQf5h65/content/2301.08734v1.pdf'}
+page_content=' As a  baseline, we also compare to ANI-2x results with adQTB  MD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQf5h65/content/2301.08734v1.pdf'}
+page_content=' We see that FENNIX-OP1 is able to accurately cap-  ture the subtle structure of liquid water and agrees well  with experimental results (from ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQf5h65/content/2301.08734v1.pdf'}
+page_content=' [66]) when nuclear  quantum effects are included.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQf5h65/content/2301.08734v1.pdf'}
+page_content='  This result is quite re-  markable as the model was not explicitly trained on con-  densed phase reference data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQf5h65/content/2301.08734v1.pdf'}
+page_content=' On the other hand, ANI-2x  predicts a largely over-structured liquid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQf5h65/content/2301.08734v1.pdf'}
+page_content=' We suspect that  this is due to both the insufficient ab initio reference for  ANI-2x (DFT with the wB97x functional) and to the lack  of long-range effects in the model itself.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQf5h65/content/2301.08734v1.pdf'}
+page_content=' The ANI-1ccx  model, that was trained on the same CCSD(T) data as  FENNIX-OP1, produced very accurate radial distribu-  tion functions at the beginning of the simulations, thus  validating the necessity of accurate CCSD(T) reference  data to be able to describe liquid water.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQf5h65/content/2301.08734v1.pdf'}
+page_content=' It was however  not stable for simulations times longer that a few tens  of picoseconds (even in classical MD with a smaller 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQf5h65/content/2301.08734v1.pdf'}
+page_content='1  fs timestep).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQf5h65/content/2301.08734v1.pdf'}
+page_content='  We thus suspect that long-range effects,  which contribute to an overall energetical stabilization of  the structure, are important to maintain the structure of  the liquid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQf5h65/content/2301.08734v1.pdf'}
+page_content='  Additionally, FENNIX-OP1 predicts an enthalpy of va-  porization for water around 13 kcal/mol, which is slightly  overestimated compared to the experimental result closer  to 11 kcal/mol [67].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQf5h65/content/2301.08734v1.pdf'}
+page_content='  This indicates a tendency of the  model to form too strong hydrogen bonds in the con-  densed phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQf5h65/content/2301.08734v1.pdf'}
+page_content=' We expect that training the model using  data from larger molecular clusters would improve the  results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQf5h65/content/2301.08734v1.pdf'}
+page_content=' Such data at CCSD(T) level is however not yet  available to our knowledge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQf5h65/content/2301.08734v1.pdf'}
+page_content='  Another route to improv-  ing results may be to include a finer description of elec-  trostatics, including for example multipolar interactions  and many-body polarization, as is done for example in  (Q-)AMOEBA [45] or ARROW [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQf5h65/content/2301.08734v1.pdf'}
+page_content=' Future iterations of  FENNIX will then focus on refining the physical model  for including such interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQf5h65/content/2301.08734v1.pdf'}
+page_content='  Finally, we explored the impact of deuteration on the  structure of the liquid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQf5h65/content/2301.08734v1.pdf'}
+page_content=' Such thermodynamical isotope  effects arise purely from NQEs (since the classical Boltz-  mann distribution does not depend on the mass of the  atoms) and can be used to experimentally probe the  impact of quantum effects on chemical processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQf5h65/content/2301.08734v1.pdf'}
+page_content='  We  10  0  1  2  3  gOO(r)  Experimental  FENNIX classical  FENNIX adQTB  ANI-2x adQTB  0  1  2  3  gOH(r)  0  2  4  6  8  r (Å)  0  1  2  3  gHH(r)  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQf5h65/content/2301.08734v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQf5h65/content/2301.08734v1.pdf'}
+page_content=' Partial radial distribution functions of liquid water  at 300K simulated using classical MD and adQTB MD with  the FENNIX-OP1 model, compared to experimental results  from [66] and ANI-2x (adQTB) results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQf5h65/content/2301.08734v1.pdf'}
+page_content='  showed in earlier works that the adQTB is able to ac-  curately capture these isotope effects [43, 45].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQf5h65/content/2301.08734v1.pdf'}
+page_content=' Figure 4  shows the O-O radial distribution function of light and  heavy water at 300K computed using FENNIX-OP1 and  compared to experimental results [66] for light water.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQf5h65/content/2301.08734v1.pdf'}
+page_content=' We  see that deuteration tends to structure the liquid, in good  agreement with tendencies observed experimentally (see  ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQf5h65/content/2301.08734v1.pdf'}
+page_content=' [66]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQf5h65/content/2301.08734v1.pdf'}
+page_content='  C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQf5h65/content/2301.08734v1.pdf'}
+page_content='  Enhanced sampling of the torsional free energy  profile of solvated alanine dipetide  Lastly, we performed molecular dynamics simulations  (using enhanced sampling) of alanine dipetide solvated  in a cubic box of water of length 30 Å(with periodic  boundary conditions).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQf5h65/content/2301.08734v1.pdf'}
+page_content=' The alanine dipeptide is a fun-  damental building block for larger biomolecules and its  accurate description is thus critical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQf5h65/content/2301.08734v1.pdf'}
+page_content=' This system is also  a typical benchmark for both interaction models and en-  hanced sampling methods and was recently shown to be  extremely challenging for ML potentials [32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQf5h65/content/2301.08734v1.pdf'}
+page_content=' It thus con-  stitutes an interesting test case for our potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQf5h65/content/2301.08734v1.pdf'}
+page_content='  Figure 5 shows the joint free energy profile, obtained  after 3 ns of classical well-tempered metadynamics sim-  ulation with the FENNIX-OP1 model, for the two dihe-  dral angles defining the torsional degrees of freedom of  2  3  4  5  6  7  8  r (Å)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQf5h65/content/2301.08734v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQf5h65/content/2301.08734v1.pdf'}
+page_content='5  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQf5h65/content/2301.08734v1.pdf'}
+page_content='0  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQf5h65/content/2301.08734v1.pdf'}
+page_content='5  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQf5h65/content/2301.08734v1.pdf'}
+page_content='0  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQf5h65/content/2301.08734v1.pdf'}
+page_content='5  gOO(r)  H2O Experimental  H2O FENNIX  D2O FENNIX  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQf5h65/content/2301.08734v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQf5h65/content/2301.08734v1.pdf'}
+page_content=' O-O radial distribution computed using adQTB and  the FENNIX-OP1 model for light and heavy water, compared  to experimental results [66] on light water.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQf5h65/content/2301.08734v1.pdf'}
+page_content='  the molecule.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQf5h65/content/2301.08734v1.pdf'}
+page_content=' The model correctly assigns most of the  probability to the two expected energy basins denoted 1  and 2 on figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQf5h65/content/2301.08734v1.pdf'}
+page_content=' It was also able to explore the less  probable states denoted 3 and 4 in the figure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQf5h65/content/2301.08734v1.pdf'}
+page_content=' The model  however seems to underestimate the barrier at Φ = 0,  thus allowing too many transitions between states 1 and  2 to 3 and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQf5h65/content/2301.08734v1.pdf'}
+page_content=' We note that this simulation was performed  using classical MD, as the adQTB method is not directly  compatible with enhanced sampling methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQf5h65/content/2301.08734v1.pdf'}
+page_content=' It would  be interesting to explore the impact of nuclear quantum  effects on this energy profile.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQf5h65/content/2301.08734v1.pdf'}
+page_content=' Indeed, as we showed above,  nuclear quantum effects drastically modify the structure  of water.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQf5h65/content/2301.08734v1.pdf'}
+page_content=' Since the torsional free energy profile is strongly  affected by the solvent (see the difference between sol-  vated and gas phase alanine dipeptide for example in  ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQf5h65/content/2301.08734v1.pdf'}
+page_content=' [68]), we can expect important changes when going  from classical to quantum dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQf5h65/content/2301.08734v1.pdf'}
+page_content='  This calculation  would require long and costly path-integrals simulations  or the development of adequate enhanced sampling tech-  niques for the adQTB, that we leave for future works.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQf5h65/content/2301.08734v1.pdf'}
+page_content='  V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQf5h65/content/2301.08734v1.pdf'}
+page_content='  CONCLUSION AND OUTLOOKS  We introduced FENNIX, a general class of models rep-  resenting molecular interactions using an hybrid ML/FF  approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQf5h65/content/2301.08734v1.pdf'}
+page_content=' It builds upon latest equivariant architectures  to construct an embedding of the local chemical environ-  ment which is fed into a multi-output module predict-  ing atom-in-molecule properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQf5h65/content/2301.08734v1.pdf'}
+page_content=' The latter allows for a  flexible design of potential energy terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQf5h65/content/2301.08734v1.pdf'}
+page_content=' We apply this  strategy to design a specific model: FENNIX-OP1, cap-  turing both short range interactions, with a dedicated  ML energy term, and long range ones using predicted  atomic volumes and charges that parametrize both a  11  150  100  50  0  50  100  150  (°)  150  100  50  0  50  100  150  (°)  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQf5h65/content/2301.08734v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQf5h65/content/2301.08734v1.pdf'}
+page_content='0  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQf5h65/content/2301.08734v1.pdf'}
+page_content='5  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQf5h65/content/2301.08734v1.pdf'}
+page_content='0  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQf5h65/content/2301.08734v1.pdf'}
+page_content='5  10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQf5h65/content/2301.08734v1.pdf'}
+page_content='0  12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQf5h65/content/2301.08734v1.pdf'}
+page_content='5  15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQf5h65/content/2301.08734v1.pdf'}
+page_content='0  17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQf5h65/content/2301.08734v1.pdf'}
+page_content='5  20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQf5h65/content/2301.08734v1.pdf'}
+page_content='0  1  3  4  2  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQf5h65/content/2301.08734v1.pdf'}
+page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQf5h65/content/2301.08734v1.pdf'}
+page_content='  Contour plot of the free energy profile of the two  torsional angles of solvated alanine dipeptide computed using  FENNIX-OP1 model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQf5h65/content/2301.08734v1.pdf'}
+page_content='  Sampling was enhanced using well-  tempered metadynamics for the two torsional angles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQf5h65/content/2301.08734v1.pdf'}
+page_content=' Ener-  gies in kcal/mol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQf5h65/content/2301.08734v1.pdf'}
+page_content='  charge penetration corrected electrostatic energy and a  Tkatchenko-Scheffler like dispersion one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQf5h65/content/2301.08734v1.pdf'}
+page_content=' FENNIX-OP1  is first trained on both the ANI-1ccx and the DES370K  datasets which contain monomers, dimers and a few  multimeric structures focusing on small neutral organic  molecules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQf5h65/content/2301.08734v1.pdf'}
+page_content=' It is then refined for water using the q-AQUA  dataset and the highly accurate Partridge-Schwenke po-  tential energy surface for a final RMSE of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQf5h65/content/2301.08734v1.pdf'}
+page_content='35 kcal/mol  on interaction energies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQf5h65/content/2301.08734v1.pdf'}
+page_content=' We then showed that the model  is stable during molecular dynamics (including NQEs via  adQTB) and able to generalize to condensed phase (de-  spite the lack of reference data), capturing structural  properties of bulk water and solvated organic molecules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQf5h65/content/2301.08734v1.pdf'}
+page_content='  More precisely, it qualitatively reproduces the torsional  free energy profile of the solvated alanine dipeptide using  enhanced sampling techniques and yields stable trajecto-  ries of a protein in gas phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQf5h65/content/2301.08734v1.pdf'}
+page_content=' Interestingly, the model  is able to capture similarities between chemical species  thanks to the continuous encoding of their positions in  the periodic table.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQf5h65/content/2301.08734v1.pdf'}
+page_content=' Furthermore, it allows to overcome  an intrinsic limitation of traditional force fields as it sat-  isfactorily reproduces covalent bond dissociation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQf5h65/content/2301.08734v1.pdf'}
+page_content=' All in  all, this work shows the extrapolating power of hybrid  physically driven machine learning approaches and paves  the way to even more refined models using enriched data  sets (in particular for charged systems) and additional  energy terms in the spirit of advanced polarizable force-  fields such as SIBFA that include multipolar electrostat-  ics and many-body polarization and dispersion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQf5h65/content/2301.08734v1.pdf'}
+page_content=' These  will enable the study of reactive complex systems such as  proton transfers between DNA pairs or enzyme catalyzed  reactions where such an accurate description of molecu-  lar interactions is mandatory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQf5h65/content/2301.08734v1.pdf'}
+page_content=' The FENNIX framework  is available through a dedicated python package which  includes the FENNIX-OP1 model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQf5h65/content/2301.08734v1.pdf'}
+page_content=' Further work will fo-  cus on the extension of the approach and its inclusion in  the optimized Deep-HP platform[69, 70].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQf5h65/content/2301.08734v1.pdf'}
+page_content='  ACKNOWLEDGEMENTS  This work was made possible thanks to funding  from the European Research Council (ERC) under the  European Union’s Horizon 2020 research and innovation  program (grant agreement No 810367), project EMC2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQf5h65/content/2301.08734v1.pdf'}
+page_content='  Computations have been performed at GENCI (IDRIS,  Orsay, France and TGCC, Bruyères le Chatel) on grant  no A0130712052.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQf5h65/content/2301.08734v1.pdf'}
+page_content='  [1] L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQf5h65/content/2301.08734v1.pdf'}
+page_content=' El Khoury,  Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQf5h65/content/2301.08734v1.pdf'}
+page_content=' Jing,  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQf5h65/content/2301.08734v1.pdf'}
+page_content=' Cuzzolin,  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFAT4oBgHgl3EQf5h65/content/2301.08734v1.pdf'}
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+JOURNAL OF LATEX CLASS FILES, 2023 - PREPRINT
+1
+Optical Flow Estimation in 360◦ Videos:
+Dataset, Model and Application
+Bin Duan, Keshav Bhandari, Gaowen Liu and Yan Yan
+Abstract— Optical flow estimation has been a long-lasting and fundamental problem in the computer vision community. However,
+despite the advances of optical flow estimation in perspective videos, the 360◦ videos counterpart remains in its infancy, primarily due
+to the shortage of benchmark datasets and the failure to accommodate the omnidirectional nature of 360◦ videos. We propose the first
+perceptually realistic 360◦ filed-of-view video benchmark dataset, namely FLOW360, with 40 different videos and 4,000 video frames.
+We then conduct comprehensive characteristic analysis and extensive comparisons with existing datasets, manifesting FLOW360’s
+perceptual realism, uniqueness, and diversity. Moreover, we present a novel Siamese representation Learning framework for
+Omnidirectional Flow (SLOF) estimation, which is trained in a contrastive manner via a hybrid loss that combines siamese contrastive
+and optical flow losses. By training the model on random rotations of the input omnidirectional frames, our proposed contrastive
+scheme accommodates the omnidirectional nature of optical flow estimation in 360◦ videos, resulting in significantly reduced prediction
+errors. The learning scheme is further proven to be efficient by expanding our siamese learning scheme and omnidirectional optical
+flow estimation to the egocentric activity recognition task, where the classification accuracy is boosted up to ∼26%. To summarize, we
+study the optical flow estimation in 360◦ videos problem from perspectives of the benchmark dataset, learning model, and also
+practical application. The FLOW360 dataset and code are available at https://siamlof.github.io.
+Index Terms—360◦ Optical Flow Dataset, Siamese Representation Learning, Optical Flow Estimation, Egocentric Activity Recognition
+!
+1
+INTRODUCTION
+O
+Ptical flow estimation, as a long-lasting and fundamen-
+tal problem in the computer vision community, has
+been studied over decades where early works [1], [2] can
+date back to 80s. Nowadays, with the advances of cutting-
+edge 360◦1 sensors and cameras, 360◦ videos pose a great
+challenge for optical flow estimation due to most studies
+being limited to perspective videos, not only benchmark
+datasets but also methodologies. In fact, the insufficiency
+of such resources holds back the evolution of optical flow
+estimation in 360◦ videos, resulting in surprisingly little
+work among the computer vision community.
+Before the era of modern deep learning, traditional
+optical flow estimation methods relied on hand-crafted
+feature designs [3], [4], [5], energy-based optimizations [6],
+[7], [8] and variational approaches [9], [10], [11]. Although
+deep learning-based approaches [12], [13], [14], [15], [16],
+[17] have shown impressive advantages over classical ap-
+proaches, most works are specially tailored for perspective
+videos. The easy access to such perspective datasets [18],
+[19], [20], [21], [22] further stimulates advances of these
+modern deep learning-based approaches for perspective
+optical flow estimation, which is not the case for omnidirec-
+tional flow estimation research. The disproportional num-
+bers of omnidirectional research is an implicit indicator that
+•
+B. Duan and Y. Yan are with Department of Computer Science, Illinois
+Tech, Chicago, IL, USA 60616.
+Contact: bduan2@hawk.iit.edu; yyan34@iit.edu
+•
+K. Bhandari was with Texas State University, present with Tesla Inc,
+USA.
+•
+G. Liu is with Cisco Research, USA.
+1. 360◦ and omnidirectional are using interchangeably in this paper.
+omnidirectional optical flow estimation research remains in
+its infancy.
+To stimulate omnidirectional optical flow estimation re-
+search, a high-quality benchmark dataset is in high demand.
+This need for a benchmark dataset brings up the first
+challenge: there is no such reliable, i.e., perceptually natural
+and complex 360◦ video dataset in the literature for om-
+nidirectional optical flow estimation. This paper addresses
+the first challenge of reliable benchmark dataset shortage by
+proposing a new dataset named FLOW360. To the best of
+our knowledge, this is the very first perceptually natural-
+synthetic 360◦ video dataset for omnidirectional flow es-
+timation. Unlike existing datasets, our proposed dataset
+advances mainly in two aspects, i.e., full 360◦ field of view
+(FOV) and high perceptual realism. Comparatively, Om-
+niFlow [23] dataset only has 180◦ FOV failing to address
+the omnidirectional nature, while the dataset proposed in
+OmniFlowNet [24] lacks perceptual realism in scene and
+motion. We conduct comprehensive characteristic analysis
+on the proposed dataset and extensive comparisons with
+existing datasets, which further manifest FLOW360’s per-
+ceptual realism, uniqueness, and diversity. Given the suc-
+cess of perspective optical flow datasets such as [18], [19],
+[22] facilitating researchers’ investigation on perspective
+optical flow estimation methods [13], [14], [16], [17], [25],
+the availability of such omnidirectional video dataset is
+essential to advance this particular field. Moreover, it is
+worth noting that FLOW360 can also be used in various
+other areas, such as continuous flow estimation in 3-frame
+settings with forward/backward consistency [26], [27], [28],
+depth estimation [29], [30] and normal map estimation [31].
+Another challenge of omnidirectional optical flow es-
+timation is that current perspective video-based deep
+networks fail to accommodate the nature of 360◦ videos.
+arXiv:2301.11880v1  [cs.CV]  27 Jan 2023
+
+JOURNAL OF LATEX CLASS FILES, 2023 - PREPRINT
+2
+These perspective methods inevitably require fine-tuning
+due to the presence of radial distortion [32] in 360◦ videos,
+which is caused by projecting 360◦ videos (spherical) to
+an equirectangular plane. This fine-tuning is onerous and
+requires tricky transformative techniques to adapt the radial
+distortion [33], [34], such as spherical convolution [33], [35],
+[36], [37], spectral convolution [38], [39] and tangent convo-
+lution [34]. Generally speaking, these methods require sig-
+nificant modifications of convolution layers, i.e., from per-
+spective kernel to omnidirectional-aware kernel. Not only
+they require immense effort to redo layer-wise architecture
+design, but the dramatic changes of underlying kernels also
+restrict the transition from fruitful perspective studies to the
+less-explored omnidirectional field. Other than modifying
+the underlying convolutional kernel, an intuitive solution is
+to fine-tune perspective-based deep networks under omni-
+directional supervision. However, this brute-force migration
+of perspective-based networks often requires enormous data
+and still leads to significant performance degradation [40].
+Instead of switching to new convolution layers and
+also to ease the amount of supervision, we design a novel
+Siamese representation Learning for Omnidirectional Flow
+(SLOF) framework, which leverages the rotation-invariant
+property of omnidirectional videos to address the radial
+distortion problem. The term rotation-invariant here implies
+that we can recover the original projection of 360◦ videos
+given the reverse rotation of the applied projection. The
+good thing about this rotation-invariant property is that
+we can project omnidirectional videos on any plane by
+rotating the spherical videos on three different axes XY Z,
+namely pitch, roll and yaw, while these rotation operations
+still preserve the overall information. That is, any projected
+planar representation can be converted back to the spheri-
+cal representation since we define the rotation beforehand.
+This rotation-invariant property motivates our design of
+the proposed SLOF, where we estimate the omnidirectional
+optical flow from a pair of consecutive frames and their
+rotated counterparts, assuming that the representations of
+these two cases are similar enough so that it can generate
+nearly identical optical flow in the spherical domain. We
+also design and compare different combinations of rota-
+tional strategies and derive guidelines for selecting the most
+effective augmentation scheme.
+When it comes to practical applications such as egocen-
+tric activity recognition, 360◦ videos impose a challenge of
+significant performance drop due to random FOV projec-
+tions. For example, since the trained model typically only
+saw one projected planar videos, it fails to capture an overall
+omnidirectional representation for prediction. This paper in-
+troduces our siamese representation learning framework in
+the egocentric activity recognition task. We propose Vision
+Transformer 360 (VIT360) to address the aforementioned is-
+sue. Compared to accuracy-centric convolution-based archi-
+tecture trained on fixed FOV, VIT360 is rotational-invariant
+and therefore does not suffer from performance degradation
+when the field-of-view projection is randomized. Besides,
+VIT360 provides the foundation to leverage motion infer-
+ence techniques from off-the-shelf optical flow architecture,
+where researchers are free to choose their favorite networks.
+We hope this egocentric activity recognition task showcases
+that omnidirectional learning can significantly benefit from
+incorporating the rotation-invariant property in the network
+architecture design for practical application.
+We summarize our main contribution as follows: (1)
+we introduce FLOW360, a new optical flow dataset for
+omnidirectional videos, to fill the need for a benchmark
+dataset to advance the omnidirectional flow estimation
+field. Comprehensive characteristic analysis of FLOW360
+and extensive comparisons with existing datasets manifests
+FLOW360’s perceptual realism, uniqueness, and diversity;
+(2) We propose SLOF, a novel framework for optical flow
+estimation in omnidirectional videos, to mitigate the cum-
+bersome framework adjustments for omnidirectional flow
+estimation, and well-accommodated to the nature of 360◦
+videos; (3) We propose VIT360 and introduce siamese learn-
+ing into the egocentric activity recognition task to mitigate
+the performance drop problem. This design of VIT360 tack-
+les the challenges of processing 360◦ videos with a fully
+convolution-based framework while learning the rotational
+invariant representation to keep up the high performance.
+Overall, the FLOW360 dataset, the SLOF framework, and
+the VIT364 network, as well as our experimental results,
+provide a solid foundation for future exploration in this
+interesting and important omnidirectional research field.
+2
+RELATED WORK
+Optical Flow Datasets. Perspective datasets such as [22],
+[41], [42], [43], [44], [45] comprise synthetic image sequences
+along with the synthetic and hand-crafted optical flow.
+However, these datasets fall short in terms of perceptual
+realism and complexities. Even though several optical flow
+datasets have been published recently in [19], [20], [21], [46],
+they are primarily used in automotive driving scenarios.
+The other relevant dataset in the literature was Sintel [18],
+which provided a bridge to contemporary optical flow esti-
+mation and synthetic datasets that can be used in real-world
+situations.
+All datasets, as mentioned earlier, are introduced for
+perspective videos and thus cannot be used for omnidi-
+rectional flow estimation. To address this problem, Lite-
+FlowNet360 [40] on omnidirectional flow estimation was
+released to augment the Sintel dataset by introducing distor-
+tion artifacts for the domain adaptation task. Nevertheless,
+these augmented datasets are discontinuous around the
+edges and violate the 360◦ nature of omnidirectional videos.
+The closest datasets to ours are OmniFlow [23] and Omni-
+FlowNet [24]. OmniFlow introduced a synthetic 180◦ field-
+of-view (FOV) dataset, which is limited to indoor scenes and
+lacks full 360◦ FOV. Similarly, OmniFlowNet introduced a
+full 360◦ FOV dataset. However, both datasets lack complex-
+ities and evidence for perceptual realism. We show detailed
+comparisons to OmniFlow, and OmniFlowNet in the dataset
+characteristic study. Compared to existing datasets in the
+literature, FLOW360 is the first perceptually natural bench-
+mark 360◦ dataset and fills the void in current research.
+Optical Flow Estimation. Advancements in optical flow
+estimation techniques largely rely on the success of data-
+driven deep learning frameworks. Flownet [25] marked one
+of the initial adoption of CNN- based deep learning frame-
+works for optical flow estimation. Several other works [16],
+[17], [47], [48], [49], [50], [51], [52] followed the footsteps
+
+JOURNAL OF LATEX CLASS FILES, 2023 - PREPRINT
+3
+with improved results. Generally, these networks adopt
+an encoder-decoder framework to learn optical flow in a
+coarse-to-fine manner. The current framework RAFT [13]
+has shown improvements with correlation learning.
+The methods mentioned above are insufficient on om-
+nidirectional flow field estimation as they are designed
+and trained for perspective datasets. One initial work [53]
+on omnidirectional flow estimation was presented as flow
+estimation by back-projecting image points to the virtually
+curved retina, thus called back-projection flow. It showed
+an improvement over classical algorithms. Similarly, an-
+other classical approach [54] relied on spherical wavelet
+to compute optical flow on omnidirectional videos. How-
+ever, these methods are limited to classical approaches
+as they are not relevant in existing deep learning-based
+approaches. One of the recent works, LiteFlowNet360 [40]
+tried to compute optical flow on omnidirectional videos
+using domain adaptation. This method utilized the kernel
+transformer technique (KTN [33]) to adapt convolution
+layers on LiteFlowNet [17] and learn correct convolution
+mapping on spherical data. Similarly, OmniFlowNet [24]
+proposed a deep learning-based optical flow estimation
+technique for omnidirectional videos. The major draw-
+back of these methods is the requirement to adapt con-
+volution layers, which takes a substantial amount of time
+and makes portability a significant issue. For example,
+in LiteFlowNet360, each convolution layer in LiteFlowNet
+was transformed using KTN with additional training and
+adjustments. Similar to OmniFlowNet, every convolution
+layer in LiteFlowNet2 [55] was transformed using kernel
+mapping [56] based on different locations of the spherical
+image. These techniques incur computational overheads
+and limit the use of existing architectures. Such approaches
+demand explicit adaptation of convolution layers, which is
+hard to maintain when more up-to-date methods are pub-
+lished constantly. Contrary to these methods, we propose a
+Siamese Representation Learning for Omnidirectional Flow
+(SLOF) method to learn omnidirectional flow by exploiting
+existing architectures with designed representation learning
+objectives, significantly reducing the unnecessary effort of
+transforming or redesigning the convolution layer.
+Action/Activity Recognition. The convolution-based ac-
+tion/activity recognition framework has been the standard
+practice in the activity recognition field. Though 2D CNN
+shows massive success in still image-based applications,
+incorporating the temporal aspects for the downstream
+task is challenging because the CNN alone does not ac-
+count for time. The initial work [57] shows a way to use
+CNN to incorporate temporal aspects for the video clas-
+sification task. Similarly, the two-stream architecture [58]
+learns spatio-temporal features from input RGB and optical
+flow information for video classification. Many other ap-
+proaches [59], [60], [61], [62] leverage the recurrent neural
+network to model long-term dependencies across input
+video frames. Following the first introduction of 3D CNN
+based video approach [63], several other variants of 3D
+CNN based approaches [64], [65], [66], [67] were proposed.
+Our transformer-based approach, a hybrid method, takes
+inspiration from these methods to extract video features.
+Though initially introduced in NLP [68], the attention
+mechanism in deep learning has changed several aspects of
+computer vision research. Ranging from still image-based
+vision tasks [69] to video action classification task [70], the
+attention-based mechanism is drawing increasing interest in
+application areas like the activity recognition domain. The
+introduction of the paper, “Attention is all you need” [71]
+marks a significant turning point in transformer-based ap-
+proaches. This attention framework led to the notable out-
+come of the transformer-based vision applications as Vision
+Transformer [72] and activity recognition frameworks [73],
+[74], [75], [76]. However, research in egocentric activity
+recognition in 360◦ videos is still in its infancy. Inspired
+by recent advancements in transformer-based vision tasks,
+we combine the potential of vision transformers with tradi-
+tional convolution-based techniques for egocentric activity
+recognition.
+Although there are affluent datasets in the egocen-
+tric activity recognition literature, most of the published
+datasets [77], [78], [79], [80], [81] are limited to the perspec-
+tive field-of-view. The 360◦ videos based egocentric activity
+recognition datasets are relatively scarce. EGOK360 [82] is
+a recently published egocentric dataset with 360◦ field-of-
+view, which includes two confusing terminologies called
+activity and actions. The activity is defined as a collection
+of minor actions, where the actions are more fine-grained
+motions related to egocentric activities. We perform our
+experiments on these datasets by focusing only on activity
+recognition (twelve different classes).
+Siamese Representation Learning. Siamese representation
+learning is a powerful approach in the unsupervised learn-
+ing category. Siamese networks have shown great success in
+different vision-related tasks such as verification [83], [84],
+[85] and tracking [86]. A recent approach [87] in siamese rep-
+resentation learning showed impressive results in unsuper-
+vised visual representation learning via exploiting different
+augmentation views of the same data. They presented their
+work in pre-training and fine-tuning stages, where the for-
+mer being the unsupervised representation learning. We use
+the representation learning scheme on omnidirectional data
+via rotational augmentations, maximizing the similarity for
+latent representations and minimizing the flow loss.
+3
+FLOW360 DATASET
+FLOW360 is an optical flow dataset tailored for 360◦
+videos using Blender [88]. This dataset contains naturalis-
+tic 360◦ videos, forward and backward optical flow, and
+dynamic depth information. The dataset comprises 40 dif-
+ferent videos extracted from huge 3D-World ‘The Room’,
+‘Modern’, ‘Alien Planet’, and ‘City Rush’. Due to their size,
+this 3D-World cannot be rendered at once in a single video.
+We render several parts of this 3D-World, which provides
+enough qualitative variation in motion and visual percep-
+tion like 3D assets, textures, and illuminations. The nature
+of this large and diverse animated world provides rela-
+tively enough diversity to qualify for a standard benchmark
+dataset. Fig. 2 shows some of the examples of motion and
+scene diversity of FLOW360. Similarly, samples from the
+dataset of different 3D-World are shown in Fig. 1. We build
+this 3D-World using publicly available 3D models [89], [90],
+[91] and 3D animated characters [92], [93], [94]. Meanwhile,
+
+JOURNAL OF LATEX CLASS FILES, 2023 - PREPRINT
+4
+Fig. 1. The FLOW360 Dataset. Sample frames (first and second column, respectively) from some of the videos with corresponding forward optical
+flow and dynamic depth information. Motion in the 3D Sphere (fourth column) is computed by transforming the motion vectors from Equirectangular
+plane (θ, φ) to unit sphere f(x, y, z). Motion in the sphere is represented in RGBA color notation. RGB color representation (as suggested in
+Middlebury [22]) is encoded using (x, y) components, and the alpha color is encoded from z of a unit sphere. RGB encoding (fifth column) is an
+RGB color map of flow in 3D space. Note: flow fields are clipped for better visualization.
+we adopt Blender [88] for additional rigging and animation
+for the dataset.
+FLOW360 contains 40 video clips extracted from differ-
+ent parts of huge 3D-World, ‘The Room’, ‘Alien Planet’,
+‘City Rush’, and Modern’. The datasets also contain other
+information like depth maps and normal fields extracted
+from the 3D-World. The FLOW360 dataset has 4,000 video
+frames, 4,000 depth maps, and 3,960 flow fields. We divide
+the video frames into 2700/1300 train/test split. We render
+the video frames with the dimension of (512, 1024) to save
+the rendering time. However, FLOW360 can be rendered
+with higher resolution, as 3D models and Blender add-ons
+will also be public.
+Diversity. We design FLOW360 datasets to include a diverse
+situation that resembles the real-world scenario as much
+as possible. The statistical validity of the datasets in terms
+of perceptual realism of scene and motion is presented in
+Fig. 4. The datasets contain a wide range of motion complex-
+ity from smaller to larger displacement, occlusion, motion
+blur, and similar complexities on the scene using camera
+focus-defocus, shadow, reflections, and several distortion
+combinations. As these complexities are quite common in
+natural videos, the FLOW360 provides similar complexities.
+Similarly, the datasets cover diverse scenarios like environ-
+mental effects, textures, 3D assets, and diverse illumina-
+tions. The qualitative presentation of these diversities and
+complexities are presented in Fig. 2 and Fig. 3 respectively.
+Fairness. The FLOW360 dataset contains custom-tailored
+animated 360 videos. We plan to release the dataset with
+the 3D models and our custom Blender add-ons to provide
+researchers a platform to create their custom optical flow
+datasets for all kinds of environments (perspective, 180◦ and
+360◦ FOV). However, the release of 3D world scenes can
+raise questions regarding fairness. To mitigate this issue, we
+will perturb certain parts of 3D world scenes and not release
+any camera information related to the test set.
+Flow-generator with Blender Add-ons. Flow-generator is
+a custom Blender add-on written for Blender-v2.92. The
+flow-generator serves two basic purposes. First, it creates
+a Blender compositor pipeline to collect frames, depth maps
+and optical flow information. This add-on can also collect
+additional information, such as normal maps. Second, it sets
+up a camera configuration for 360◦ FOV. We will describe
+the details of the add-ons in the supplementary material.
+Render Passes. We exploit several modern features from
+Blender-v2.92 like advanced ray-tracing as a render engine
+along with render passes like vector, normal, depth, mist,
+and so on to produce realistic 3D scenes. Additionally, we
+incorporate features like ambient occlusion, motion blur,
+camera focus/defocus, smooth shading, specular reflection,
+shadow, and camera distortion to introduce naturalistic
+complexity (shown in Fig. 3) in our dataset. Besides optical
+flow information, the FLOW360 3D-world may be used to
+collect several other helpful information like depth, normal
+maps, and semantic segmentation.
+Dataset Statistics. We conduct a comprehensive analysis
+and compare our dataset with Sintel [18], Lookalikes (pre-
+sented in the original Sintel paper to compare the image
+
+cleanframesamples
+frames
+forwardopticalflow
+motion
+3D-flow
+_depth
+3D-flow
+depth
+3D-flow
+depth
+3D-flow
+depthJOURNAL OF LATEX CLASS FILES, 2023 - PREPRINT
+5
+frame
+flow
+frame
+flow
+frame
+flow
+Fig. 2. Motion and Scene Diversity. Samples from our FLOW360 Dataset with random projections, showing scene and motion diversity. The
+FLOW360 dataset has a vast scene consisting of several lighting scenarios, textures, diverse 3D assets, and motion complexity in different regions.
+motion blur
+lens distortion
+camera focus & defocus
+camera defocus
+camera focus
+shadow and reflection
+reflection on surface
+reflection on water
+Fig. 3. Complexity of FLOW360 Dataset. Frames in FLOW360 Dataset
+include complex characteristics like camera focus/defocus, motion blur,
+lens distortion, shadow, and reflections. Our dataset provides ambiance
+occlusion and environmental effects for a realistic visual appearance.
+statistics with the simulated dataset), Middlebury [22], Om-
+niFlow [23] and OmniFlowNet [24]. The analysis shown in
+Fig. 4 shows the image and motion statistics in the top and
+bottom rows, respectively.
+Based on analysis from Sintel, we present frame statistics
+with three different analysis: luminance histogram, power
+spectrum, and spatial derivative. For luminance statistics,
+we convert the frames to gray-scale, I(x, y) ∈ [0, 255]
+then we compute histograms of gray-scale images across
+all pixels in the entire dataset. The luminance statistics
+show the FLOW360 has a similar distribution with the peak
+in the range between [0−100] and decreasing luminosity
+beyond that range. Similarly, we estimate power spectra
+from the 2D FFT of the 512×512 in the center of each
+frame. We compute the average of these power spectra
+across all the datasets. We present power spectra analysis
+separately for the training and test set in this analysis.
+The power spectra analysis closely resembles the Sintel,
+Lookalikes, and Middlebury datasets. Based on [45], [95],
+the real-world movies exhibit a characteristic of a power
+spectrum slope around -2, which is equivalent to a 1/f 2
+falloff. FLOW360 with the slope (−2.30, −2.36) on test and
+training split shows such characteristics. We do not claim
+
+JOURNAL OF LATEX CLASS FILES, 2023 - PREPRINT
+6
+luminance histogram
+spatial derivative
+power spectrum
+OmniFlow dI/dx
+OmniFlow dI/dy
+F360-test dI/dx
+F360-test dI/dy
+F360-train dI/dx
+F360-train dI/dy
+0
+-20
+-200
+200
+Sintel
+Lookalikes
+Middlebury
+l
+l
+l
+l
+l
+l
+l
+-1024
+-512
+-200 0 200
+512
+1024
+0-
+-5-
+-10-
+-15-
+log (fraction of pixels)
+Sintel
+Lookalikes
+Middlebury
+l
+l
+l
+10-2
+10-1
+100
+108-
+108-
+108-
+108-
+power (x)
+-2.27
+-2.36
+-2.17
+-2.30
+-2.36
+Slopes
+-1.79
+OmniFlow dI/dx
+F360-test dI/dx
+F360-train dI/dx
+*OmniFlowNet 
+scaled by 10 
+0
+250
+0.020
+OmniFlow dI/dx
+F360-test dI/dx
+F360-train dI/dx
+Sintel
+Lookalikes
+Middlebury
+l
+l
+l
+l
+l
+l
+0.920-
+0.015-
+0.010-
+0.005-
+0.000-
+0
+50
+100
+150
+200
+250
+fraction of pixels
+sintel
+-14-
+-12-
+-10-
+-8-
+-6-
+-4-
+-2-
+0-
+l
+l
+l
+l
+l
+-300
+-150
+0
+150
+300
+log (fraction of pixels)
+sintel
+-12-
+-10-
+-8-
+-6-
+-4-
+-2-
+0-
+log (fraction of pixels)
+l
+l
+l
+l
+0
+100
+200
+300
+*OmniFlow 
+clipped at y=-14, 
+minimum value 
+range upto -23
+-14-
+-12-
+-10-
+-8-
+-6-
+-4-
+-2-
+0-
+l
+l
+l
+l
+l
+-50
+-100
+0
+50
+100
+log (fraction of pixels)
+sintel
+*OmniFlow 
+clipped at y=-16, 
+minimum value 
+range upto -23
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+F360-train (
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+speed (pixels/frame)
+u (pixels/frame)
+flow/dx (1/frame)
+spatial flow derivative
+-2.0-
+-3.0-
+-4.0-
+-5.0-
+-6.0-
+-7.0-
+l
+l
+l
+l
+l
+log (fraction of pixels)
+-100
+-50
+0
+50
+100
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+frequency (cycle/pixel)
+luminance
+motion (u)
+flow direction
+speed (s)
+simple low 
+poly
+complex
+nature
+simple
+indoor
+scene
+outdoor
+indoor/
+outdoor
+Yes
+360o
+FLOW360
+Yes
+moving
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+motion due 
+to camera
+No
+Yes
+camera
+360o
+entities in 
+motion
+fixed
+moving
+180o
+Yes
+No
+FOV
+FLOW360 vs. 
+OmniFlow & OmniFlowNet
+OmniFlowNet dI/dx
+0.211-
+OmniFlowNet dI/dx
+-1.42
+OmniFlowNet dI/dx
+Kurtosis
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+44.97
+41.16
+32.74
+  58.61
+-49.94
+  23.93
+62.53
+55.94
+391.09
+373.55
+*OmniFlow 
+clipped at y=-14, 
+minimum value 
+range upto -23
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+<latexit sha1_base64="GQr36rEgJrKkloNsbobrLqWD0=">AB+3icbVA9SwNBEJ3zM8avM5Y2i0GwCncWamfARrsI5gOSI+xt9pIle7vH7l40HPkrNhaK2Po/xM5/4+aSQhMfDzem2FmXphwpo3nfTsrq2vrG5uFreL2zu7evntQamiZKkLrRHKpWiHWlDNB64YZTluJojgOW
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+OmniFlowNet (
+<latexit sha1_base64="Q+J/QevfRzv8ShX7RW6JSCb6s+s=">AB+3icbVA9SwNBEJ2LXzF+nbG0WQyCVbizUDsDNtpFMB+QHGFvs0mW7O0eu3tqO6v2FgoYuv/EDv/jZtLCk18MPB4b4aZeWHMmTae9+0UVlbX1jeKm6Wt7Z3dPXe/3NQyUYQ2iORStUOsKWeCNgwznLZjRXEUct
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+<latexit sha1_base64="Q+J/QevfRzv8ShX7RW6JSCb6s+s=">AB+3icbVA9SwNBEJ2LXzF+nbG0WQyCVbizUDsDNtpFMB+QHGFvs0mW7O0eu3tqO6v2FgoYuv/EDv/jZtLCk18MPB4b4aZeWHMmTae9+0UVlbX1jeKm6Wt7Z3dPXe/3NQyUYQ2iORStUOsKWeCNgwznLZjRXEUct
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+*OmniFlow and 
+OmniFlowNet 
+clipped at y=-7, 
+minimum value 
+range up to -11 
+and -8 
+respectively
+Fig. 4. Comparision of frames and flow statistics. The top row represents the statistics and comparison of the frame with Sintel, Lookalikes,
+Middlebury, OmniFlow [23] and OmniFlowNet [24]. Bottom row represents flow statistics and comparison with Sintel (red), OmniFlow (magenta)
+and OmniFlowNet (turquoise). The table on the top-right shows a brief comparison of OmniFlow & OmniFlowNet with the FLOW360 dataset. Note:
+(→, ←) represents forward and backward flow fields, respectively.
+that FLOW360 is realistic, but it certainly exhibits perceptual
+similarity with natural movies. The spatial and temporal
+derivative analysis additionally supports this characteristic.
+The Kurtosis of frames’ spatial derivatives range from 32.74
+to 57.27, peaking at zero. This characteristic shows that
+FLOW360 has a resemblance to natural scenes [45].
+Regarding the flow field analysis, we directly com-
+pare the distribution of motion u(x, y), speed defined as
+s(x, y) =
+�
+u(x, y)2 + v(x, y)2, flow direction Θ(x, y) =
+tan−1 (v(x, y)/u(x, y)) and spatial flow derivative of u and
+v. The close resemblance of the flow field statistics between
+Sintel and FLOW360 suggests motion field resemblance
+with natural movies. Based on these comparisons, FLOW360
+exhibits sufficient properties evident enough for its percep-
+tual realism and complexities.
+Comparison with OmniFlow and OmniFlowNet. Omni-
+Flow [23] presents an omnidirectional flow dataset that is
+roughly similar to FLOW360. However, the major distinc-
+tion between these datasets is the FOV. FLOW360 provides
+immersive 360◦ FOV, whereas OmniFlow provides only
+180◦ FOV showing FLOW360 compared to OmniFlow is the
+true omnidirectional dataset. Similarly, OmniFlowNet [24]
+presents synthetic omnidirectional flow dataset with 360◦
+FOV. However, this dataset contains low poly unnatural
+scenes, which can be explained by relatively larger kurtosis
+(373.55, 391.09), characteristic of a power spectrum and lu-
+minance distribution (peaked at 255). The overall statistical
+analysis reveals FLOW360’s better perceptual realism and
+diversity.
+Applications. As we mentioned, the FLOW360 dataset con-
+tains frames and forward flow fields and includes backward
+flow fields, depth maps, and 3D-FLOW360 worlds, provid-
+ing potential for applications like continuous flow-field esti-
+mation in 3 frames setting. Besides optical flow estimation,
+the FLOW360 dataset can be used in other applications such
+as depth and normal field estimation. Moreover, given 3D-
+FLOW360 animation data, the researcher can create as many
+optical flow datasets as needed.
+4
+360◦ OPTICAL FLOW ESTIMATION
+4.1
+SLOF for 360◦ Optical Flow Estimation
+SLOF, as shown in Fig. 5, is inspired by the recent work
+on Siamese representation learning [87]. Since the method
+we rely on acts as a hub between several methods like
+contrastive learning, clustering, and siamese networks, it
+exhibits two special properties required for our case. First,
+this method has non-collapsing behavior. Here, the term
+collapsing refers to a situation where an optimizer finds
+possible minimum -1 similarity loss resulting degenerate so-
+lution, characterized by zero standard deviation (std) of l2-
+normalized output z/||z||2 for each channel, while training
+without stop-gradient operations. Stop-gradient yields std
+value near
+1
+√
+d across each channel for all samples prevent-
+ing such behaviour [87]. Second, it is useful when we have
+only positive discriminative cases. SLOF does not consider
+radial distortion mitigation via changing/transforming the
+convolution layers but rather learns the equivariant proper-
+ties of 360 videos via siamese representation. We claim that
+such transformation is trivial, based on the following fact.
+First, the omnidirectional videos are projected in angular
+domain, w.r.t. polar(θ), azimuthal(φ); θ ∈ (− π
+2 , π
+2 ), φ ∈
+(−π, π), so we can learn flow fields in these domains and
+convert these flow fields to the spherical domain using
+
+0
+-5
+log (fraction of pixels)
+-10
+-15
+-1024
+-512　-2000　200
+512
+1024
+image derivative (d)-5
+-10
+-15
+Kurtosis = 49.94
+Kurtosis = 58.61
+Kurtosis = 23.93
+-200
+-100
+0
+100
+200Power spectrum
+8
+10
+10°
+4
+10
+Slope = -2.36
+10
+Slope = -2.27
+Slope = -2.17
+102
+10-2
+10-1108
+Power (x)
+106
+105
+10-2
+10-1
+100
+Frequency (cycles/pixel)0.020
+0.015
+S
+Fraction of pixels
+0.010
+0.005
+0.000
+0
+50
+100
+150
+200
+250
+LuminanceKL-D Sintel to LA: 0.058
+KL-D MB to LA: 0.1760
+-2
+F pixels)
+log (fraction of
+-6
+-8
+-10
+-12
+-14
+-300
+-150
+0
+150
+300
+u (pixels/frame)0
+-2
+log (fraction of pixels)
+-4
+-6
+-8
+-10
+-12
+0
+100
+200
+300
+speed (pixels/frame)0
+2
+log (fraction of pixels)
+-6
+-10
+-12
+-14
+-100
+-50
+0
+50
+100
+flow/dx (1/frame)JOURNAL OF LATEX CLASS FILES, 2023 - PREPRINT
+7
+flow 
+head
+projector
+encoder f
+rotation 
+head
+flow 
+head
+projector
+encoder f
+predictor h
+similarity
+flow 
+upsampler
+spherical projection
+gt-flow
+flow-loss
+stop
+gradient
+frames
+R
+single
+rotation (v1)
+R
+double
+rotation (v3)
+R - switch
+single
+rotation (v2)
+R
+single
+rotation (v1)
+R
+double
+rotation (v3)
+R - switch
+single
+rotation (v2)
+(b) Training Strategy
+(a) SLOF Architecture
+Fig. 5. Siamese Representation Learning for Omnidirectional Flow (SLOF). Pairs of frame sequence (w/ and w/o random rotation) are passed
+as inputs to encoder f (RAFT as a flow head backbone and a standard convolutional projector layer). A predictor layer h is an MLP layer. The entire
+framework is trained by fusing the pretraining and fine-tuning stages to combine the similarity and flow loss in a single stage. The model maximizes
+the similarity between latent representations of flow information from two streams and minimizes the flow loss. Training Strategy (right): Here,
+two different arrows (left, right) represent siamese streams or input pathways to our model. v1 and v2 (either stream is subjected to rotational
+augmentation) are similar strategies achieving overall better performance.
+planar to spherical transformations as shown in Eq. (1)
+and Eq. (2). Second, the intent of a convolution opera-
+tor in optical flow architecture is relatively different from
+other applications like classification, detection, or segmen-
+tation network, where other tasks require convolution to
+learn relevant features (spatially consistent), the relevance
+of these features should stay consistent (strictly for better
+performance) throughout any spatial location of the im-
+ages/videos. However, the convolution operation is dedi-
+cated to computing the pixel-wise displacement regardless
+of spatial inconsistency in the distorted region via equivari-
+ant representation learning [87]. Another important consid-
+eration of such a design is to make this method portable
+to any existing optical flow architecture. This design will
+eliminate the cumbersome architecture re-adjustments tasks
+and make it powerful and portable.
+Mapping Flow Field to Unit Sphere. Input to our model
+are equirectangular images projected in angular domain
+polar(θ), azimuthal(φ), where these angles are defined in
+radian as θ ∈ (− π
+2 , π
+2 ), φ ∈ (−π, π), thus the predicted
+optical flow is in (θ, φ). These flow fields can be converted
+to a unit sphere using planar to spherical coordinate trans-
+formation as shown below:
+(xs, ys, zs) = (sin θ cos φ, sin θ sin φ, cos θ).
+(1)
+We can compute sphere to catadioptric plane [96] projec-
+tions to express the flow field in Cartesian coordinates as:
+(x, y) = (
+xs
+1 − zs
+,
+ys
+1 − zs
+) = (cot θ
+2 cos φ, cot θ
+2 sin φ).
+(2)
+Design of SLOF architecture. Given a pair of input image
+sequence X1 = (x1, x2), the rotation head (R) computes
+augmented view of this sequence as X2 = (x′
+1, x′
+2) with
+rotation r using a random combination of “pitch”, “yaw”
+and “roll” operations. These two augmented views are
+passed as an input to an encoder network f, defined as
+f = P(R′(Θ(E(R(X, r))))) where E is a flow prediction
+module, RAFT [13] in our case, Θ is a mapping of 2D flow
+to the unit sphere, R′ is a reverse rotation operation and P
+is a convolution-based down-sampling head. A prediction
+head presented as h (an MLP head), transforms the output
+from the encoder f from one stream to match the other
+stream. The illustration of this process shown in Eq. (3)
+as maximization of cosine similarity two views from the
+siamese stream:
+D(pleft, zright) = −
+pleft
+||pleft||2
+·
+zright
+||zright||2
+.
+(3)
+Here, pleft
+≜
+h(f left(X1)) and zright
+≜
+f right(X2) de-
+notes the output vectors to match from two different
+streams(f left, f right). This maximization problem can be
+viewed from another direction, with (pright, zleft) as the sec-
+ond matching pair from siamese stream (f right, f left) respec-
+tively. Given two matching pairs, we can use the following
+Eq. 4 symmetrized similarity loss function (note that zleft
+and zright are treated as a constant term using stop-grad
+operations to prevent a degenerate solution due to model
+collapse [87]):
+Lsim = 1
+2D(pleft, zright) + 1
+2D(pright, zleft).
+(4)
+Similarly, the optical flow loss is computed as a sequence
+loss [13] over the predicted flow field and ground truth.
+This loss (l1 distance over predicted and ground truth
+flow fgt) is computed and averaged over a sequence of
+predictions iteratively generated for the same pair of input
+frames {f1, f2, ..., fn} = E(R(X, r)) as shown in Eq. (5),
+where γ = 0.8n−i−1 served as weights over sequence loss.
+Note that (n, i) denotes the number of prediction(n) in
+sequence and prediction id(i) in predicted flow sequences.
+The design of the weighted schemes ensures different levels
+of confidence on predicted flows over time.
+Lflow =
+n
+�
+i=1
+γ||R(fgt, r) − fi||.
+(5)
+Given similarity loss(Lsim) and flow loss(Lflow) we imple-
+ment a hybrid loss function L = Lsim + Lflow. The overall
+objective of this loss function is to maximize the similarity
+
+JOURNAL OF LATEX CLASS FILES, 2023 - PREPRINT
+8
+between the latent representation of flow information while
+minimizing the loss between ground truth and predicted
+optical flow.
+5
+360◦ EGOCENTRIC ACTIVITY RECOGNITION
+5.1
+VIT360 for Egocentric Activity Recognition
+The overall design of egocentric activity recognition in
+360◦ videos comprises four different components: (1) design
+of VIT360, (2) two-stream approach for activity classifi-
+cation, (3) learning projection invariant representation of
+360◦ videos, and (4) computing 360◦ or omnidirectional
+aware motion features from off-the-shelf architecture. The
+following subsections will discuss each of these components
+in chronological order.
+Design of VIT360. VIT360 requires input pre-processing
+which converts an equirectangular image into multiple tan-
+gential patches covering the entire FOV as shown in Fig. 6.
+Instead of processing raw image patches in the time dimen-
+sion, these patches are processed first via the global pro-
+jection layer (L1) to compute feature sequences. Compared
+with the original VIT [72] implementation, input features
+to VIT360 are relatively larger as the patches are fed in the
+time dimension. The projection layers transform the larger
+input feature space into a relatively smaller feature space by
+downsizing the transformed feature dimension. The feature
+extraction layer (L2) acts as a siamese stream for learning
+additional features from independent tangent patches in
+the time dimension to mitigate the low parametrization of
+features. Using a projection layer (L1) and feature extrac-
+tion layer (L2) with stacked convolution architecture makes
+VIT360 a hybrid architecture.
+Two-stream Architecture. Our egocentric activity recogni-
+tion framework (shown in Fig. 7) is based on recent success
+on action/activity recognition task [58] loosely based on two
+stream hypothesis [97]. One stream is designed as a spatial
+stream responsible for object recognition, and another is
+designed as a motion stream responsible for motion recogni-
+tion. Both streams of this framework implement VIT360, as
+discussed above, for different input modalities (e.g. frames
+for spatial and optical flow for motion). These streams are
+trained independently and later fused (late fusion) to make
+a joint prediction based on spatial and motion information.
+Following the best practices in previous work [82], we
+implement two different fusion techniques - average and
+concatenation. The average fusion technique computes the
+average of the scores from the last layer and computes the
+model confidence, and does not require additional training.
+The concatenation-based techniques require concatenation
+of the last MLP head and the introduction of additional
+fully connected layers matching the output class dimension.
+This technique requires additional fine-tuning and achieves
+marginally better results.
+Tangential Projections: VIT360 takes input of 360◦ video
+frames
+or
+optical
+flows
+in
+time
+dimension
+(X
+∈
+RC×T ×H×W ), resulting in spatial modality (where C = 3)
+and motion modality (where C = 2) respectively. Note that
+T represents the number of consecutive frames denoting
+the time dimension. Each input at time t is in fact projected
+in the equirectangular plane(Xt ∈ RC×H×W , 0 ≤ t ≤ T),
+which is obtained by a sphere mesh unwrapped on a flat
+rectangular plane surface. This process maps sphere lati-
+tude and longitude to horizontal and vertical coordinate
+systems expressing the length and height of the plane in
+the range (−π, π) and (−π/2, π/2) respectively. In order to
+create input patches, tangential planes are sampled using
+a spherical-to-cartesian coordinate transformation system.
+This is shown in Eq. (6), where (λ, ψ) represents latitude and
+longitude such that (λ0, ψ0) = (0, 0) represents the centre
+of the plane, and (c) represents the angular distance of the
+point(x, y) from the centre of the projection where
+x = cos ψ sin(λ − λ0)
+cos(c)
+,
+y = cos ψ0 sin ψ − sin ψ0 cos ψ cos (λ − λ0)
+cos c
+,
+cos(c) = sin ψ0 sin ψ + cos ψ0 cos(ψ) cos(λ − λ0).
+(6)
+Similarly, the inverse map from plane to the sphere can be
+computed using the following equation as
+ψ = sin−1
+�
+cos (c) sin ψ0 + y sin(c) cos ψ0
+ρ
+�
+,
+λ = λ0 + tan−1
+�
+x sin (c)
+ρ cos ψ0 cos (c) − y sin ψ0 sin (c)
+�
+,
+s.t.
+ρ =
+�
+x2 + y2, c = tan−1 ρ.
+(7)
+The
+projection
+layer
+samples
+n
+tangential
+planes
+(RC×T ×h×w) covering the entire 360◦ FOV, where (h, w)
+are the height and width of each tangential plane. These
+tangential planes are linearly stacked alongside the width
+to obtain a final output x ∈ RC×T ×h×nw. In summary, the
+tangential projection layers take a series of optical flows or
+frames in a video (X ∈ RC×T ×H×W ) and transform it into
+a series of tangential plane projections (x ∈ RC×T ×h×nw)
+linearly stacked along the width.
+Projection Layer (L1): An overview of the Projection
+Layer (L1) is shown in Fig. 8. This layer takes tangential
+patches (x ∈ RC×T ×h×nw) in time dimension as input. In
+contrast with the original VIT implementation, the input
+to VIT360 is a 3D image (considering the time dimension)
+which requires a modification of the original projection layer
+to 3D convolution-based architecture. The Projection Layer
+(L1) includes a 3D convolution with 3D batch normaliza-
+tion and LeakyReLU layer as an activation function. The
+3D convolution layer is parametrized with the 3D kernel
+size and stride as (TIME, TANGENT H, TANGENT W),
+where TANGENT H = h, TANGENT W = w refers to the
+height and width of tangent patches and TIME = T refers
+to the number of frames in the time dimension. This for-
+mulation of L1 transformed input tangential patches (x ∈
+RC×T ×h×nw) to (x ∈ Rn×EMBED DIM), where EMBED DIM
+refers to the output feature dimension in Fig. 8.
+Feature Extraction Layer (L2):
+The Projection Layer (L1)
+inspired by original VIT implementation maps a huge fea-
+ture space (x ∈ RC×T ×h×nw) into relatively smaller feature
+(x ∈ Rn×EMBED DIM) embedding. This low parametrization
+of the feature space creates VIT360 to suffer from signif-
+icant performance degradation. To mitigate this issue, an
+additional Feature Extraction Layer (L2), as shown in Fig. 8
+is designed to learn additional features. These additional
+features are later used for attention enforcement, resulting
+
+JOURNAL OF LATEX CLASS FILES, 2023 - PREPRINT
+9
+λ
+ψ
+0
+1
+2
+3
+4 *
+n-1
+••••
+Feature 
+Extraction 
+Layer
+(L2)
+Projection Layer (L1)
+Attention Enforcement (AE1)
+Feature Encoder (E2)
+MLP Head
+Class
+Ping Pong
+…..
+Input: Tangential
+Projections
+Corresponding patches of tangential planes stacked 
+on time dimension
+Feature Extraction in 
+time dimension
+Input: 
+Eq. Frames
+VIT360
+XL1
+XL2
+XE2
+XE1
+X0
+X1
+X2
+X3
+X4
+Xn-1
+X
+X
+AE2
+Transformer Encoder (E1)
+VMAP
+••••
+••••
+t0
+t1
+tT-1
+time
+time
+time
+time
+time
+time
+Fig. 6. VIT360 Framework. The sequence of ten consecutive frames (or optical flow) is sampled into six different tangential projections. These
+tangential projections are passed through Projection Layer (L1) and Feature Extraction Layer (L2) to compute features XL1 and XL2. The First
+Attention Enforcement Layer (AE1) computed the attention map between these features and passed it to Transformer Encoder (E1) with positional
+embedding as an input. The final Attention Enforcement Layer (AE2) computes the attention maps between the output XE1 and XE2 and passes it
+as an input to MLP Head for activity classification.
+TIME
+TIME
+TIME
+TIME
+TIME
+TIME
+MOTION
+FEATURES
+TIME
+TIME
+TIME
+TIME
+TIME
+TIME
+SPATIAL 
+FEATURES
+VIT360
+VIT360
+SPATIAL STREAM
+MOTION STREAM
+MLP Head
+MLP Head
+Fusion
+Class
+Ping Pong
+…..
+Time
+Time
+Time
+Time
+Time
+Time
+Time
+Time
+Time
+Time
+Time
+Time
+Time
+Time
+Time
+Time
+Time
+Time
+Spatial Stream
+Motion Stream
+Spatial 
+Input
+Motion 
+Input
+Fig. 7. Two-stream Architecture.
+The motion and spatial stream
+constitute the component of two-stream architecture. The spatial stream
+receives RGB frames, whereas the motion streams receive the pre-
+computed optical flow. These two streams are trained independently
+and later fused to incorporate two different modalities via average and
+concatenation-based late fusion techniques.
+in improved performance. The Feature Extraction Layer (L2)
+is implemented as a siamese network to process individual
+input tangential patches. To achieve the siamese network,
+we use vmap (available in PyTorch framework) operation
+per tangential patches. The Feature Extraction Layer (L2)
+takes an input (x ∈ RC×T ×h×nw), reshapes it to (x ∈
+Rn×C×T ×h×w) and computes features (x ∈ Rn×SEQ DIM)
+where the SEQ DIM is an output embeddings of the Feature
+Extraction Layer (L2).
+Attention Enforcement (AE1 and AE2):
+The Encoder in
+the original implementation of VIT takes input from the
+linear projection layer from 16 × 16 image patches. Such
+design consideration in VIT360 is computationally challeng-
+ing since the input for VIT360 is stacked patches in the
+time dimension. It is possible to generate a relatively larger
+number of tangential planes and subsequently larger em-
+bedding space from the Projection Layer (L1). Such a design
+is computationally infeasible because the input will be rela-
+tively larger than the original VIT. This tradeoff between the
+choice of the number of input patches and computational
+cost directly affects model performance. In addition, the
+Projection Layer (L1) reduces the feature space significantly,
+as we discussed above, leading to lower parametrization
+and reducing the model’s performance. In order to mitigate
+such issues and maintain the optimal input size, we in-
+troduce Attention Enforcement (AE1 and AE2) techniques.
+The first Attention Enforcement (AE1) computes an atten-
+tion map between the output from L1(XL1) and L2(XL2)
+making the input (X ∈ RSEQ DIM×EMB DIM) size of the
+encoder sufficiently optimal retaining the model perfor-
+mance. Similarly, the second Attention Enforcement(AE2)
+computes the attention maps (X
+∈
+Rn×SEQ DIM) be-
+tween the output from E1(XE1 ∈ RSEQ DIM×EMB DIM) and
+E2(XE2 ∈ RSEQ DIM×n), keeping the number of param-
+eter (n × EMB DIM × NUM CLASSES) in MLP Head
+fixed. The overview of Attention Enforcement is presented
+in Fig. 8.
+Encoder Layer (E1 and E2):
+The Transformer Encoder
+layer E1 and E2 contains similar implementation as pre-
+sented in the original VIT implementation. The E1-encoder
+contains two layers of encoder architecture, whereas E2
+contains only one layer of encoder architecture. E1 takes
+an input (XAE1 ∈ RSEQ DIM×EMB DIM) from the Atten-
+tion Enforcement Layer (E1) where EMBED DIM is the
+input dimension of the E1-encoder. Similarly, E2 takes an
+input (XL2 ∈ Rn×SEQ DIM) from the Feature Extraction
+Layer (L2) where SEQ DIM is the input dimension of the
+E2-encoder. Both E1 and E2 computes output feature (XE1 ∈
+RSEQ DIM×EMB DIM, XE2 ∈ Rn×SEQ DIM) vectors similar to
+the input dimension.
+MLP Head: In contrast to the original VIT implementation,
+we introduce a single-layer MLP head as a classification
+head to predict the activity classes. The Attention Enforce-
+ment Layer (AE2) output is passed as an input to the final
+MLP Head.
+Projection Invariant Representation: 360◦ videos in com-
+parison with perspective videos exhibit a unique property of
+unlimited FOV, resulting in infinite projections. Such infinite
+
+JOURNAL OF LATEX CLASS FILES, 2023 - PREPRINT
+10
+Projection Layer (L1)
+Conv3D
+in_channels
+out_channels
+kernel_size
+stride
+{‘spatial’:3, ‘motion’:2}
+EMB_DIM
+(TIME, TANGENT_H, TANGENT_W)
+(TIME, TANGENT_H, TANGENT_W)
+bias True
+BatchNorm3D
+num_features EMB_DIM
+LeakyReLU
+negative_slope 0.01
+LeakyReLU
+LeakyReLU
+LeakyReLU
+LeakyReLU
+BN(SEQ_DIM)
+BN(64)
+BN(32)
+BN(16)
+Conv3D(in=8, out = 16, k = (TIME, 3,3))
+MaxPool3D(k = (1,3,3), s = (1,2,2))
+MaxPool3D(k = (1,3,3), s = (1,2,2))
+MaxPool3D(k = (1,3,3), s = (1,2,2))
+Feature Extraction Layer(L2)
+Conv3D(in={‘spatial’:3, ‘motion:2’}, out = 8, k = (1,1,1))
+Conv3D(in=16, out = 32, k = (1, 3,3))
+Conv3D(in=32, out = 64, k = (1, 3,3))
+Conv3D(in=64, out = SEQ_DIM, k = (1, 3,3))
+MaxPool3D(k = (1,5,5), s = (1,1,1))
+Flatten
+LeakyReLU
+BN(8)
++
++
+MLP
+Norm
+Multi-Head
+Attention
+Norm
+Embedded
+Patches
+Lx
+Encoder
+XE2: [B,SEQ_DIM,n]
+XE1: [B, SEQ_DIM,EMB_DIM]
+AE2 = 
+einsum(‘ijk,ijm->ikm’,XE2,XE1)
+Attention Enforcement
+AE1
+XL2: [B,SEQ_DIM,n]
+XL1: [B,n,EMB_DIM]
+AE1 = 
+einsum(‘ijk,ikm->ijm’,XL2,XL1)
+AE2
+Fig. 8. Layers in VIT360. VIT360 is composed of four major components: Projection Layer (L1) computes the initial flattened features of input
+sequences, and Feature Extraction Layer (L2) computes additional features using the siamese stream. Attention Enforcement Layers (AE1 and
+AE2) compute the attention maps computed across two different feature extraction layers and encoder streams, respectively. Finally, a pair of
+Transformer Encoders (E1 and E2) to learn temporal features for activity recognition.
+λ
+ψ
+Tangential 
+Projection
+••
+encoder f
+VIT360
+encoder f
+VIT360
+predictor (h)
+projector (p)
+similarity
+predictor (h)
+projector (p)
+x
+x
+p1
+p2
+z1
+z2
+stream-1
+stream-2
+Random 
+Rotation
+r1
+r2
+stop
+grad
+stop
+grad
+similarity
+Fig. 9. Siamese Representation Learning for egocentric activity recognition. The random Rotation head (R) computes two different rotational
+views of the same input video sequences. These augmented views are passed as input to VIT360 using tangential projections. The siamese
+network of VIT360 computes latent representation, pi and zi, using projector(p) and predictor (h) respectively. The training policy maximizes the
+similarity between latent representations (p1, z2) and (p2, z1) in an unsupervised manner. The pre-trained network with this policy is then subjected
+to training for egocentric activity recognition.
+projections can be achieved by rotating the 360◦ videos
+on three different axes (X, Y, Z), namely “pitch”, “roll”,
+and “yaw” operations. Regardless of these projections, the
+overall information of 360◦ videos remains intact. This
+rotation-invariant property of 360◦ videos is crucial while
+designing deep learning architecture for 360◦ videos based
+applications. In practice, the ideal architectures should be
+rotationally invariant or projection invariant. To achieve this
+goal, we perform pretraining of VIT360 to learn projection
+invariant representation.
+Pre-training of VIT360 for projection invariant represen-
+tation is loosely based on a contrastive learning approach
+called SimSiam [87], [99]. The overview of pre-training stage
+is illustrated in Fig. 9. This representation learning based on
+SimSiam is completed using siamese representation learn-
+ing, formulated as a pair of weight-sharing VIT360 streams
+(stream-1, stream-2) to maximize the similarity between two
+different representations of the same 360◦ videos informa-
+tion (both frames and optical flow). Given a sequence of
+frames X = (x0, x1, ..., xT −1), where T is the number of
+frames in sequence in time, the rotation head (R) computes
+a pair of rotational augmentation Xi = R(X, ri) using
+r ∈ (r1, r2) defined as a random configuration of “pitch”,
+“yaw” and “roll” operations. These rotationally augmented
+representations are now passed as an input to an encoder
+network f = h(p(VIT360(R(X, r)))), where p and h are
+MLP layers, defined as the projector and predictor’s head
+respectively. A projector head (p) appends the MLP Head
+(shown in Fig. 5) with additional MLP layers to create a
+bottleneck MLP module, which constitutes the final VIT360
+architecture. Similarly, a predictor head (h) transforms the
+output (pi
+∈
+Rn×EMBED DIM) from the encoder (f) to
+(zi ∈ Rn×EMBED DIM) where i = 1, 2 represents the first and
+second stream.
+Given output (pi, zi) from respective streams, we for-
+mulate a pre-training stage to maximize cosine similarity
+between these representations across siamese streams. The
+illustration of the maximization process is shown in Eq. (8).
+D(p1, z2) = −
+p1
+||p1||2
+·
+z2
+||z2||2
+,
+Lsim = 1
+2D(p1, z2) + 1
+2D(p2, z1).
+(8)
+Here, p1 = f(X1) and z2 = h(f(X2)) denote latent repre-
+sentations computed at different levels for the same input
+X using rotationally augmented different view (X1, X2).
+Siamese VIT360 is trained so that both projector and pre-
+dictor layers are subjected to learn a similar latent represen-
+tation of the given input across two different sets of inputs.
+This representation learning aims to maximize the similarity
+between these two outputs at different levels across the
+siamese streams. The maximization problem can be con-
+sidered in both directions, another being the maximization
+between (p2, z1). Given output pairs (p1, z2) and (p2, z1),
+we use the symmetric similarity loss function (Lsim) as
+shown in Eq. (8). As discussed in SimSiam [87], we treat
+(p1, p2) as a constant via stop-grad operations to prevent a
+degenerate solution due to model collapse.
+
+JOURNAL OF LATEX CLASS FILES, 2023 - PREPRINT
+11
+Fig. 10. 360◦ flow inference using PercieverIO. The pairs of consecutive frames are projected with six different FOV covering the entire 360◦
+field. The PercieverIO [98] is implemented as Flow Head, which computes the optical flow in each pair of projected FOV. The computed flow fields
+are then projected using tangential projections to extract the central FOV. This central FOV is then warped in a spherical representation and the
+final omnidirectional flow is computed using sphere-to-plane projections of the warped spherical flow.
+5.2
+Omnidirectional-aware Optical Flow Estimation
+The overview of the optical flow inference for 360◦ videos
+using PerceiverIO is shown in Fig. 10. Similar to VIT360,
+this approach also considers different projections of pairs of
+input 360◦ frames for calculating the final flow between the
+frames. These different projections are considered an input
+patch to PercieverIO, which it calculates motion information
+in each pair. Later these motions in patches are warped
+into one 360◦ FOV to compute the final 360◦ optical flow.
+In order to maintain the computational size, we limit the
+number of patches to six and perform six different projec-
+tions to cover the entire 360◦ FOV. Compared to the original
+PercieverIO implementation, we do not crop these different
+projections but rather resize these projections into input size
+as defined in PercieverIO. We do not limit the flow compu-
+tation across the neighboring patches by avoiding cropping
+beyond the padding zone. This intermediate flow represents
+optical flow per projections over the entire equirectangular
+plane. Given optical flow computed in an equirectangular
+plane, flow information in the highly distorted area (like
+polar regions) suffers the impact of distortions. However,
+the central part of equirectangular plane (λ0, ψ0) = (0, 0) is
+the least distorted region from where the tangential patches
+are sampled with
+π
+2 FOV. These sampled patches cover
+the entire 360◦ FOV. Finally, these patches are warped into
+global 360◦ flow for input frames.
+6
+EXPERIMENTS
+6.1
+360◦ Optical FLow Estimation
+We evaluate SLOF on the FLOW360 test set. We use pre-
+trained RAFT on Sintel [18] and fine-tune on FLOW360
+as a comparison baseline. The fine-tuning process is done
+using training protocols suggested in [13]. Moreover, to
+make a fair comparison with traditional methods, we trans-
+form RAFT (pre-trained) to adapt spherical convolution
+using KTN [33]. KTN transforms the convolution kernel to
+mitigate the radial distortions via estimating the spherical
+convolution function. Additionally, we run ablation studies
+on different training strategies and propose a distortion-
+aware evaluation. We will present details of the training
+procedure in the supplemental material.
+Scope. The scope of our experiments is two folds: First,
+create a baseline for future researchers to explore novel
+methodologies. Second, address the validity of our method
+based on fair comparisons with a flow network designed for
+a spherical dataset. We formulate our baseline experiment
+1.0
+0.8
+0.6
+0.4
+0.2
+0.0
+Fig. 11. Distortion density map. Illustrating different distortion inten-
+sities due to equirectangular projections. Left: upper (red) and lower
+(green) part of projections shows higher distortion in the central part
+whereas the equatorial region (cyan, pink, blue, gray) exhibit higher
+distortion rate away from the center of the tangential plane. Right: shows
+the distortion density from (0, 1). This distortion density map is used to
+evaluate the distortion-aware EPE (EPEd). Note: Each circle patch in
+the left spherical projection has the same area.
+on perspective optical flow network RAFT and a modified
+version of RAFT with KTN [33] to compare the performance.
+The RAFT+KTN architecture simulates a domain adapta-
+tion similar to approaches like [24], [40]. We choose KTN
+because of its success over alternative approaches like [36],
+[37], [38], [39], [100]. It is worth noting that the design of
+omnidirectional flow estimation can be extended to several
+techniques involving the mitigation of radial distortions,
+making it practically impossible to cover all.
+Augmentation Strategy. Given the nature of SLOF, we can
+train it using two different training strategies (v1, v2) as
+shown in Fig. 5 (right). These strategies can be achieved by
+performing different rotational augmentation on the input
+sequences. The first strategy (v1) can be achieved by using
+set of inputs (R(X1, r1), R(X2, r2)) where r1 = (0, 0, 0),
+i.e., X1 does not have any rotational augmentation, whereas
+r2 ̸= (0, 0, 0) has rotation defined with random combina-
+tions of “pitch”, “roll”, and “yaw” operations. This setting
+is kept consistent throughout the training process. Alterna-
+tively, identical augmentation can be achieved by flipping
+this augmentation protocols. The second rotational scheme
+(v2) can be achieved by randomly switching rotation such
+that when r1 is none, the r2 is some random rotational
+augmentation and vice versa. This approach performs on
+par with v1.
+
+Flow Warping
+Flow Head
+Final
+omni-directional
+(0,0,0)
+(0,0,T//2)
+(0,0,-T/2)
+(0,0,r)
+(0,T/2,0)
+(0,-T//2,0)
+flowJOURNAL OF LATEX CLASS FILES, 2023 - PREPRINT
+12
+frame-1
+RAFT+KTN
+RAFT
+SLOF (v2)
+gt-flow
+finetuned - RAFT
+SLOF (v1)
+Fig. 12. Qualitative results on FLOW360 test set. Qualitative results show our best model SLOF(v1) shows better results compared to fine-tuned
+RAFT trained with policy explained in [13]. The dotted (black) rectangle indicates the comparative improvements of our model over fine-tuned RAFT.
+RAFT+KTN method fails to predict flow-field correctly; instead, it only predicts shallow flow fields from camera motion. The weakness of our model
+can be seen on the dotted (red) rectangle where smaller motion segments are missing. Note: Flows information is clipped for better visualization.
+0.0
+2.0
+4.0
+6.0
+8.0
+1.0
+0.8
+0.6
+0.4
+0.2
+0.0
+0.0-0.1
+0.1-0.2
+0.2-0.3
+0.3-0.4
+0.4-0.5
+0.5-0.6
+0.6-0.7
+0.7-0.8
+0.8-0.9
+0.9-1.0
+Avg AE
+Avg EPE
+Distortion Density Range
+0.0
+2.0
+4.0
+6.0
+8.0
+1.0
+0.8
+0.6
+0.4
+0.2
+0.0
+0.0-0.1
+0.1-0.2
+0.2-0.3
+0.3-0.4
+0.4-0.5
+0.5-0.6
+0.6-0.7
+0.7-0.8
+0.8-0.9
+0.9-1.0
+Avg AE
+Avg EPE
+Distortion Density Range
+Avg AE
+0.0
+2.0
+4.0
+6.0
+8.0
+1.0
+0.8
+0.6
+0.4
+0.2
+0.0
+0.0-0.1
+0.1-0.2
+0.2-0.3
+0.3-0.4
+0.4-0.5
+0.5-0.6
+0.6-0.7
+0.7-0.8
+0.8-0.9
+0.9-1.0
+Avg AE
+Avg EPE
+Distortion Density Range
+0.1
+0.8
+0.6
+0.4
+0.2
+0.0
+0.0
+2.0
+4.0
+6.0
+8.0
+1.0
+0.8
+0.6
+0.4
+0.2
+0.0
+0.0-0.1
+0.1-0.2
+0.2-0.3
+0.3-0.4
+0.4-0.5
+0.5-0.6
+0.6-0.7
+0.7-0.8
+0.8-0.9
+0.9-1.0
+Avg AE
+Avg EPE
+Distortion Density Range
+Fig. 13. Error distribution plot of EPE and AE in different distortion density ranges. SLOF performs better in all distortion density ranges.
+TABLE 1
+Quantitave results on FLOW360 test set. ∗ denotes that we use EPEd/AEd as the metrics; otherwise, the normal EPE and AE. Compared to the
+baseline, SLOF achieves lower end-point error and angular error on both distortion aware (EPEd and AEd) and normal scheme. In terms of
+end-point error (lower the better) our model (v1, v2) outperforms all the baselines. Similarly, in terms of angular error (lower the better) our models
+(v1, v2) perform comparatively similarly and outperform all the baselines. Though RAFT+KTN achieves comparable normal EPE, the
+distortion-aware (Weighted) metrics (EPEd and AEd) are significantly larger. Note: metrics in range (all, less than (5, 10, 20) and greater than 20)
+is computed as an average, based on the speed (s(x,y)=
+�
+u(x, y)2 + v(x, y)2) only in the respective pixel regions.
+Method
+Version
+Metric
+Weighted s≥0∗
+s≥0
+s<5
+s<10
+s<20
+s≥20
+Baselines
+RAFT [13]
+EPE
+3.344
+2.058
+0.558
+0.682
+0.838
+71.736
+AE
+1.120
+0.820
+0.825
+0.821
+0.819
+0.868
+Finetuned RAFT [13]
+EPE
+2.635
+1.624
+0.314
+0.393
+0.509
+65.340
+AE
+0.745
+0.522
+0.527
+0.522
+0.520
+0.647
+RAFT + KTN [33]
+EPE
+3.899
+2.222
+0.598
+0.742
+0.924
+76.426
+AE
+2.020
+0.912
+0.912
+0.910
+0.911
+1.0114
+SLOF
+Switch rotation (v2)
+EPE
+2.626
+1.615
+0.326
+0.401
+0.512
+64.678
+AE
+0.691
+0.485
+0.489
+0.484
+0.482
+0.659
+Single rotation (v1)
+EPE
+2.548
+1.568
+0.309
+0.387
+0.502
+62.476
+AE
+0.708
+0.497
+0.501
+0.497
+0.495
+0.607
+AE = arccos(
+ueur + vevr + 1
+�
+u2r + v2r + 1
+�
+u2e + v2e + 1).
+(9)
+EPE = 1
+N
+N
+�
+i
+||fpred − fgt||2.
+(10)
+EPEd = 1
+N
+N
+�
+i
+||fpred − fgt||2
+1 − d
+.
+(11)
+Evaluation Strategy. We evaluate our method based on 2D-
+raw flow. Besides, using EPE (End Point Error in Eq. (10)),
+i.e., Euclidean distance between the predicted flow and
+
+JOURNAL OF LATEX CLASS FILES, 2023 - PREPRINT
+13
+ground truth flow, as a single evaluation metric, we incor-
+porate AE (Angular Error) as shown in Eq. (9) as the second
+measure. To explain the error in the omnidirectional setting,
+we introduce a distortion-aware measure called EPEd as in
+Eq. (11). This metric penalizes the error in the distorted area
+based on the distortion density map.
+As EPEd, AEd is calculated as
+1
+N
+�N
+i
+AE
+1−d where, d
+represents the distortion density map illustrated in Fig. 11,
+fpred = (ue, ve) represents predicted flow, and fgt = (ur, vr)
+represents ground truth flow. Note that, to maintain lower
+metrics scale the distortion density is mapped between
+[0.500, 1.000) from (0.0, 1.0]. Please refer to supplemental
+for additional details on distortion density map.
+Results. Fig. 12, Fig. 13 and Table 1 summarize our exper-
+imental results. The overall summary of qualitative results
+is presented in Fig. 12. SLOF performs better than baseline
+RAFT and kernel transformed RAFT+KTN methods. This
+result is evident enough to show that siamese representation
+learning can exploit the rotational properties of 360◦ videos
+to learn omnidirectional optical flow regardless of explicit
+architecture adjustments.
+Our methods, SLOF (v1, v2) perform better than pre-
+sented baselines. Among these methods v1 has the best EPE
+score whereas, v2 has better AE score. However, AE on both
+v1 and v2 are relatively similar, suggesting v1 as our best
+method, shown in Fig. 12.
+By investigating distortion-aware EPE, we can see that
+RAFT with KTN achieves significantly higher EPE regard-
+less of comparable normal EPE with the other methods.
+This clearly explains why RAFT+KTN methods could not
+predict the motion around the distorted area; instead, it
+predicts shallow flow fields due to camera motion only.
+Moreover, comparing qualitative results in Fig. 12 and EPE
+measure in different distortion ranges in Fig. 13, we can
+see that our best method can predict smoother flow fields
+compared to baseline methods. These fields in the polar
+region are comparatively better and have better motion
+consistency in the edge region. However, our model might
+fail to predict relatively smaller motion regions in some
+cases, which leaves room for future improvements based
+on the proposed method. This concludes that RAFT+KTN
+requires additional re-engineering and domain adaptation,
+which is out of the scope of current work.
+6.2
+360◦ Egocentric Activity Recognition
+We focus our experiments mainly on representation learning
+based egocentric activity recognition on 360◦ videos. We run
+our experiments on EGOK360 [82] dataset, an egocentric
+activity recognition dataset for 360◦ videos. We present a
+series of experiments that demonstrate the efficacy of pre-
+training VIT360 with a siamese representation approach
+maximizing the representation across the different views of
+the same input. In addition, we study the impact of using
+360◦ or omnidirectional aware optical flow compared to
+traditional perspective videos based on optical flow on 360◦
+videos for this motion-based task.
+The first experiment includes egocentric activity recogni-
+tion (using a two-stream approach) without the representa-
+tion learning scheme. Similarly, the follow-up experiments
+include the representation of learning-based egocentric ac-
+tivity recognition, which follows two different consecutive
+experiments of pre-training and two streams approach for
+classification. Similarly, we conducted two additional ex-
+periments based on optical flow and omnidirectional optical
+flow. In addition to these, we also make a comparative study
+of qualitative results on proposed PercieverIO-based 360◦
+optical flow inference techniques with pre-trained perspec-
+tive video-based optical flow technique, RAFT [13]. Note
+that our comparisons based on the two-stream architecture
+are reported in EGOK360 [82] dataset only. Since the ex-
+perimental results presented on EGOK360 are based on a
+random train/test split of entire datasets, we also follow
+a similar approach. However, to make a fair test case, we
+sample train and test frames with a 9:1 split from each clip
+from the entire training data. We achieve marginally similar
+baseline results on EGOK360 compared to experimental
+results reported in the original paper. In addition, we only
+consider activity recognition on the EGOK360 dataset as
+experimental results are sufficient to establish core concepts
+of rotation-invariant egocentric activity recognition.
+Experiment Configuration: VIT360 training follows a sim-
+ilar approach we have seen in other related works [40],
+[58], [67]. We choose Adam [101] as our optimizer with the
+initial learning rate of 10−3, and StepLR as our learning rate
+scheduler for smooth training. We consider ten frames as
+one training input and six tangential patches as input dis-
+cussed as (T, n) in the method section. The experiments are
+conducted on three 2080Ti NVIDIA GPUs with a batch size
+of 16, both in the training and testing phase. The training
+and testing speed approaches 2 seconds/iterations and 1.5
+seconds/iterations, respectively, achieving a throughput of
+nearly 80 and 106 frames per second.
+Discussion on Activity Recognition Results: Table
+2
+summarizes our experimental results on egocentric activity
+recognition on EGOK360 datasets. From these results, we
+can observe that the baseline method achieves marginally
+similar accuracy as reported in [82] on a fairly sampled
+new test set. Though these results are promising for con-
+trolled projections, the weaknesses of the baseline algorithm
+quickly appear when we perform random rotation during
+testing. Its accuracy drops significantly in both motion
+and spatial streams. However, considering the VIT360 for
+a similar effect, we obtain consistent results during fixed
+and random rotations. These achievements of VIT360 are
+useful considering the nature of 360◦ videos in the wild. In
+addition, we see a significant boost in performance using
+optical flow features computed using our inference tech-
+niques on 360◦ videos. The experimental results of VIT360
+compared to the baseline on the same test set have shown
+an impressive performance boost, which provides strong
+evidence for the efficacy of VIT360 in egocentric activity
+recognition on 360◦ videos.
+Discussion on Optical Flow Results:
+Based on results
+from Table 2, the proposed optical flow inference technique
+is comparatively better than the traditional techniques. To
+address this, we also present qualitative results of our
+techniques and compare them with perspective flow-based
+techniques in Fig. 14. The qualitative results show more
+accurate optical flow for 360◦ videos using VIT360, which
+can be further explained by smooth displacement field,
+motion boundaries, and flow continuity around edges.
+
+JOURNAL OF LATEX CLASS FILES, 2023 - PREPRINT
+14
+TABLE 2
+Quantitave Results. The Top-1 accuracy (shown as %) of VIT360 compared with baseline methods in EGOK360 [82] achieves consistent
+performance regardless of projection mode, achieving less than 0.40% accuracy difference. The baseline method shows performance gaps
+between 5-14%. 360◦ optical flow computed using our inference techniques boost the performance of VIT360 by almost 5%. (Note: We use
+Perspective Flow (RAFT [13]) for baseline experiments.)
+Version
+Random Projection
+Flow
+Motion
+Spatial
+Fused Avg.
+Fused Concat.
+ResNet
+✓
+Perspective
+42.53
+59.67
+56.18
+62.79
+Perspective
+56.43
+68.95
+66.79
+69.32
+I3D
+✓
+Perspective
+48.43
+63.61
+60.17
+65.39
+Perspective
+62.19
+69.78
+67.37
+72.68
+VIT360 (Ours)
+✓
+Perspective
+56.52
+67.14
+65.18
+70.89
+Perspective
+56.57
+67.23
+65.23
+70.79
+✓
+360◦(Ours)
+60.34
+71.29
+70.66
+75.87
+360◦(Ours)
+60.43
+71.13
+70.29
+75.59
+Fig. 14. 360◦ optical flow results. We compare 360◦ flow inferred using our techniques and perspective video-based technique using RAFT [13].
+The superiority of our method can be seen in optical flow with smooth motion, distinct motion region, and flow continuity across the edges.
+7
+CONCLUSION
+Omnidirectional flow estimation remains in its infancy be-
+cause of the shortage of reliable benchmark datasets and
+tedious tasks dealing with inescapable radial distortions.
+This paper proposes the first perceptually natural-synthetic
+benchmark dataset, FLOW360, to close the gap, where com-
+prehensive analysis shows excellent advantages over other
+datasets. Our dataset can be extended for other non-motion
+applications like segmentation and normal estimation task
+as well. Moreover, we introduce a siamese representation
+learning approach for omnidirectional flow (SLOF) instead
+of redesigning the convolution layer to adapt omnidirec-
+tional nature. Our method leverages the invariant rotation
+property of 360◦ videos to learn similar flow representation
+on various video augmentations. Meanwhile, we study the
+effect of different rotations on the final flow estimation,
+which provides a guideline for future work. Overall, the
+elimination of network redesigns aids researchers in exploit-
+ing existing architectures without significant modification
+leading to faster deployment in practical applications, such
+as egocentric activity recognition.
+Moreover, as a proof of concept, we propose VIT360 – a
+vision transformer-based network pretrained with siamese
+representation - to achieve rotational invariance in 360◦
+videos for egocentric activity recognition. VIT360 performs
+consistently regardless of the projection mode, achieving
+less than a 0.4% accuracy gap whereas baseline methods
+drop in performance by 5% to 14% between fixed and
+random field-of-view projections. Our 360◦ flow inference
+technique integrates seamlessly with VIT360 to boost the
+performance by almost 26% (from 56.18 to 70.66), compared
+to the fixed projections trained networks. That is, VIT360 is
+more suitable for activity recognition in real-world scenarios
+thanks to its rotational invariant properties.
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+1
+APPENDIX A
+FLOW GENERATOR
+FLOW360 is created using Blender2, a free and open-
+source 3D creation suite. Blender provides an interface to
+write add-ons for automating workflows. We create Flow-
+generator, an add-on written for Blender-v2.92 to collect
+optical flow and several other data like depth information
+and normal maps. Flow-generator can be installed using the
+following steps:
+1) Download Flow-generator3
+2) In Blender-v2.92, follow:
+• Edit → Preferences → Add-ons → Install
+• Select flow generator.py from the Flow-generator
+project folder
+• Install Add-on
+• Search FlowGenerator
+• Checkmark for installation
+3) Setup compositor pipeline in Blender-v2.92
+• Go to Compositing
+• Select “SetFlow Generator” from Custom Node
+Group
+• If Custom Node Group is disabled press “n” to make
+it visible
+• Select the desired configuration and click “SetEnv”
+4) Camera Setup
+• Go to Layout
+• Open tab on the right side of the Layout view by
+pressing “n” if not visible
+• Go to “CameraSetup” on the right tab
+• Select the desired configuration and click “Set Cam-
+era System”
+Note that, we use cloud rendering to speed up the ren-
+dering process which requires crowd-render4 add-on. Flow-
+generator provides two basic functionality: (i) camera setup
+and (ii) compositor pipeline.
+A.1
+Compositor Pipeline
+Compositor in Blender provides a pipeline to process ren-
+der output. Flow-generator creates a basic pipeline (shown
+in Fig. 15) to collect information like optical flow, images,
+depth maps, and so on. It also provides an easy way to cache
+the desired outputs in a structured format. Similarly, it also
+setups basic configurations like selecting render-engine and
+render passes.
+A.2
+Camera Setup
+Camera Setup (shown in Fig. 15) can be accessed via the
+3D-Layout tab in Blender-v2.92 as suggested above. Camera
+Setup creates an omnidirectional camera with a full 360◦
+field of view. The camera setup consists of twelve differ-
+ent cameras (six perspectives and six 360◦) out of which
+any omnidirectional camera (default Camera EQ F) can be
+selected as the main camera for rendering omnidirectional
+videos. It also provides an interface to adjust configurations
+like resolution, dimension, and depth of the scene.
+2. https://www.blender.org/
+3. https://www.dropbox.com/s/v2mvjvs7ze8rzj1/flowgenerator.
+tar.gz?dl=0
+4. https://www.crowd-render.com/
+APPENDIX B
+EXPERIMENTAL SETTING AND MORE EVALUATION
+FOR OPTICAL FLOW ESTIMATION
+We conduct our experiment in PyTorch (1.9.0+cuda10.2) us-
+ing the version of Python (3.8.10). Additional environment
+detail is provided in project5.
+Following is the list of configurations we used for our
+project:
+• Train/Val Batch Size: Ours(16/12 - 8Gpus), Fine-
+tune(8/12 - 4 Gpus)
+• Number of iterations on RAFT: 12
+• Loss: CosineSimilarity, Optical Flow L1-loss
+• Optimizer: AdamW
+• Scheduler: OneCycleLR
+• EarlyStopping: Patience (5) and Min-delta (1e-4)
+• Others: Gradient Clipping, GradScaler
+• Dataset: FLOW3606
+We suggest our readers to refer sample videos7,8 for
+demo purpose.
+B.1
+Distortion Density Map
+We compute distortion mask (Ud, Dd, Fd, Bd, Rd, Ld)
+in a cube-map with six faces: Up(U), Down(D), Front(F),
+Back(B), Right(R) and Left(L) respectively. This distortion
+density cube-map projection is then projected to equirectan-
+gular projection using spherical coordinate transformation
+as
+(xs, ys, zs) = (sin θ cos φ, sin θ sin φ, cos θ).
+(12)
+To compute density mask (Cd, where C∈(U, D, F, B, R,
+L)) in each face, we define a meshgrid for co-oridnates x and
+y ranging from ([−1, 1]) with dimension of (256, 256). The
+coordinates (x, y) are used to compute a radius map (r) of
+size (256, 256) as shown in Eq. (13).
+r =
+�
+x2 + y2
+(13)
+In our paper, we have shown that the radial distortion
+is higher towards the center in the polar region which cor-
+responds to (U, D) faces of cube-map projections. Similarly,
+in the equatorial regions i.e., the rest of the faces (F, R, B,
+L) show a higher distortion rate away from the center. We
+compute two distinct distortion maps (Cd), one for polar
+regions (U, D) and another for equatorial regions (F, B, R, L)
+as shown in Eq. (14)
+Cd =
+�
+1 − r/ max(r)
+if, C ∈ {U,D} ,
+r/ max(r)
+otherwise
+(14)
+Please refer to code9 for additional details.
+5. https://www.dropbox.com/s/a6qioejg6yrxo7s/SLOF.tar.gz?dl=0
+6. https://www.dropbox.com/s/nvzhazq99bg46f2/FLOW360
+train test.zip?dl=0. Note that for better visualization please clip optical
+flow in the range of (-40, 40) or lower.
+7. https://www.dropbox.com/s/54mmjvoz6844mci/trailer.mp4?
+dl=0
+8. https://www.dropbox.com/s/gvihrzj528d92uj/videos.zip?dl=0
+We recommend to use VLC-Media player to play these videos.
+9. https://www.dropbox.com/s/q1d4eoqvj2a30ij/distortion
+weight.ipynb?dl=0
+
+JOURNAL OF LATEX CLASS FILES, 2023 - PREPRINT
+2
+Compositor Pipeline
+Camera Setup
+Fig. 15. FlowGenerator. Flow-generator serves two major purposes: (Left) Setting up a compositor pipeline which involves setting up file paths,
+pipelining desired output, and setting initial configurations like render-engine and render passes. (Right) Setting up camera configurations and
+additional information like resolution, dimensions, and depth of the scene.
+B.2
+Impact of Rotational Invariance
+We perform additional experiments to quantify the impact
+of rotational invariance (see table below, Table.3). The im-
+provement is noticeable as seen in LiteFLowNet360 [40]
+and Tangent Images [34]-based optical flow estimation with
+rotational invariance.
+TABLE 3
+Impact of rotational invariance
+Method
+w/o Rot.Inv (EPE)
+w Rot.Inv (EPE)
+LiteFLowNet360 [40]
+3.95
+2.52
+Tangent Images [34]
+3.57
+1.78
+
+Options
+Camera Setup Node Group
+Iten
+Resolution
+100
+Cube width
+512
+Equi Wicth
+1024
+Min Depth
+0.05
+Max Depth
+100.00
+Camera:
+Camera EQ_F
+Set Camera System File Output
+Base Path:
+/depth/Camera_EQ_F/
+Image
+OpenEXR
+File Output
+ Render Layers
+Base Path:
+Image
+/normal/Camera_EQ_F/
+Alpha
+Image
+OpenEXR
+Depth
+Mist
+ Composite
+Normal
+Y Use Alpha
+Vector
+Image
+IndexOB
+ Alpha
+1.000
+Noisy Image
+1.000
+Separate RGBA
+ Combine RGBA
+ File Output
+Image
+Base Path:
+R
+/flow/Camera_EQ_F
++ Image
+OpenEXR
+.Image
+ File Output
+tovScene
+Base Path:
+ View Layer
+/raw_flow/Camera_EQ_F/
+ Image
+OpenEXRCustomNodeGroup
+Item
+ROOT:
+Resolution
+100
+Cube width
+512
+Equi Width
+1024
+Min Depth
+0.05
+Max Depth
+100.00
+Camera:
+Camera EQ F
+Set Flow Generator
+SetEnvJOURNAL OF LATEX CLASS FILES, 2023 - PREPRINT
+3
+frame-1
+flow
+frame-1
+flow
+frame-1
+flow
+frame-1
+flow
+Fig. 16. More Motion and Scene Diversity - Train Set. Illustrating motion and scene diversity via randomly sampled tangential plane from
+FLOW360 dataset. Flow360 contains a range of scene complexity governing varied properties like texture, illumination, human, building, cars, and
+other 3D assets. Similarly, the various level of motion complexity can be seen for similar scenes ranging from smaller to larger displacement.
+As explained in the main paper the dataset also contains other complexities like motion blur, camera focus/defocus, camera distortion, and
+environmental effects.
+
+JOURNAL OF LATEX CLASS FILES, 2023 - PREPRINT
+4
+frame-1
+flow
+frame-1
+flow
+frame-1
+flow
+frame-1
+flow
+Fig. 17. More Motion and Scene Diversity - Test Set. Illustrating motion and scene diversity via randomly sampled tangential plane from FLOW360
+dataset. Flow360 contains a range of scene complexity governing varied properties like texture, illumination, human, building, cars, and other 3D
+assets. Similarly, the various level of motion complexity can be seen for similar scenes ranging from smaller to larger displacement. As explained in
+the main paper the dataset also contains other complexities like motion blur, camera focus/defocus, camera distortion, and environmental effects.
+
diff --git a/etFKT4oBgHgl3EQfry7P/content/tmp_files/load_file.txt b/etFKT4oBgHgl3EQfry7P/content/tmp_files/load_file.txt
new file mode 100644
index 0000000000000000000000000000000000000000..5348e604f5a951e935d46240faf16488ae9ea60c
--- /dev/null
+++ b/etFKT4oBgHgl3EQfry7P/content/tmp_files/load_file.txt
@@ -0,0 +1,1948 @@
+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf,len=1947
+page_content='JOURNAL OF LATEX CLASS FILES, 2023 - PREPRINT  1  Optical Flow Estimation in 360◦ Videos:  Dataset, Model and Application  Bin Duan, Keshav Bhandari, Gaowen Liu and Yan Yan  Abstract— Optical flow estimation has been a long-lasting and fundamental problem in the computer vision community.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content=' However,  despite the advances of optical flow estimation in perspective videos, the 360◦ videos counterpart remains in its infancy, primarily due  to the shortage of benchmark datasets and the failure to accommodate the omnidirectional nature of 360◦ videos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content=' We propose the first  perceptually realistic 360◦ filed-of-view video benchmark dataset, namely FLOW360, with 40 different videos and 4,000 video frames.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content='  We then conduct comprehensive characteristic analysis and extensive comparisons with existing datasets, manifesting FLOW360’s  perceptual realism, uniqueness, and diversity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content=' Moreover, we present a novel Siamese representation Learning framework for  Omnidirectional Flow (SLOF) estimation, which is trained in a contrastive manner via a hybrid loss that combines siamese contrastive  and optical flow losses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content=' By training the model on random rotations of the input omnidirectional frames, our proposed contrastive  scheme accommodates the omnidirectional nature of optical flow estimation in 360◦ videos, resulting in significantly reduced prediction  errors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content=' The learning scheme is further proven to be efficient by expanding our siamese learning scheme and omnidirectional optical  flow estimation to the egocentric activity recognition task, where the classification accuracy is boosted up to ∼26%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content=' To summarize, we  study the optical flow estimation in 360◦ videos problem from perspectives of the benchmark dataset, learning model, and also  practical application.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content=' The FLOW360 dataset and code are available at https://siamlof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content='github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content='io.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content='  Index Terms—360◦ Optical Flow Dataset, Siamese Representation Learning, Optical Flow Estimation, Egocentric Activity Recognition  !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content='  1  INTRODUCTION  O  Ptical flow estimation, as a long-lasting and fundamen-  tal problem in the computer vision community, has  been studied over decades where early works [1], [2] can  date back to 80s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content=' Nowadays, with the advances of cutting-  edge 360◦1 sensors and cameras, 360◦ videos pose a great  challenge for optical flow estimation due to most studies  being limited to perspective videos, not only benchmark  datasets but also methodologies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content=' In fact, the insufficiency  of such resources holds back the evolution of optical flow  estimation in 360◦ videos, resulting in surprisingly little  work among the computer vision community.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content='  Before the era of modern deep learning, traditional  optical flow estimation methods relied on hand-crafted  feature designs [3], [4], [5], energy-based optimizations [6],  [7], [8] and variational approaches [9], [10], [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content=' Although  deep learning-based approaches [12], [13], [14], [15], [16],  [17] have shown impressive advantages over classical ap-  proaches, most works are specially tailored for perspective  videos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content=' The easy access to such perspective datasets [18],  [19], [20], [21], [22] further stimulates advances of these  modern deep learning-based approaches for perspective  optical flow estimation, which is not the case for omnidirec-  tional flow estimation research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content=' The disproportional num-  bers of omnidirectional research is an implicit indicator that B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content=' Duan and Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content=' Yan are with Department of Computer Science, Illinois  Tech, Chicago, IL, USA 60616.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content='  Contact: bduan2@hawk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content='iit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content='edu;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content=' yyan34@iit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content='edu K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content=' Bhandari was with Texas State University, present with Tesla Inc,  USA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content=' Liu is with Cisco Research, USA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content=' 360◦ and omnidirectional are using interchangeably in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content='  omnidirectional optical flow estimation research remains in  its infancy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content='  To stimulate omnidirectional optical flow estimation re-  search, a high-quality benchmark dataset is in high demand.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content='  This need for a benchmark dataset brings up the first  challenge: there is no such reliable, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content=', perceptually natural  and complex 360◦ video dataset in the literature for om-  nidirectional optical flow estimation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content=' This paper addresses  the first challenge of reliable benchmark dataset shortage by  proposing a new dataset named FLOW360.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content=' To the best of  our knowledge, this is the very first perceptually natural-  synthetic 360◦ video dataset for omnidirectional flow es-  timation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content=' Unlike existing datasets, our proposed dataset  advances mainly in two aspects, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content=', full 360◦ field of view  (FOV) and high perceptual realism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content=' Comparatively, Om-  niFlow [23] dataset only has 180◦ FOV failing to address  the omnidirectional nature, while the dataset proposed in  OmniFlowNet [24] lacks perceptual realism in scene and  motion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content=' We conduct comprehensive characteristic analysis  on the proposed dataset and extensive comparisons with  existing datasets, which further manifest FLOW360’s per-  ceptual realism, uniqueness, and diversity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content=' Given the suc-  cess of perspective optical flow datasets such as [18], [19],  [22] facilitating researchers’ investigation on perspective  optical flow estimation methods [13], [14], [16], [17], [25],  the availability of such omnidirectional video dataset is  essential to advance this particular field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content=' Moreover, it is  worth noting that FLOW360 can also be used in various  other areas, such as continuous flow estimation in 3-frame  settings with forward/backward consistency [26], [27], [28],  depth estimation [29], [30] and normal map estimation [31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content='  Another challenge of omnidirectional optical flow es-  timation is that current perspective video-based deep  networks fail to accommodate the nature of 360◦ videos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content='  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content='11880v1  [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content='CV]  27 Jan 2023  JOURNAL OF LATEX CLASS FILES, 2023 - PREPRINT  2  These perspective methods inevitably require fine-tuning  due to the presence of radial distortion [32] in 360◦ videos,  which is caused by projecting 360◦ videos (spherical) to  an equirectangular plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content=' This fine-tuning is onerous and  requires tricky transformative techniques to adapt the radial  distortion [33], [34], such as spherical convolution [33], [35],  [36], [37], spectral convolution [38], [39] and tangent convo-  lution [34].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content=' Generally speaking, these methods require sig-  nificant modifications of convolution layers, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content=', from per-  spective kernel to omnidirectional-aware kernel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content=' Not only  they require immense effort to redo layer-wise architecture  design, but the dramatic changes of underlying kernels also  restrict the transition from fruitful perspective studies to the  less-explored omnidirectional field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content=' Other than modifying  the underlying convolutional kernel, an intuitive solution is  to fine-tune perspective-based deep networks under omni-  directional supervision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content=' However, this brute-force migration  of perspective-based networks often requires enormous data  and still leads to significant performance degradation [40].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content='  Instead of switching to new convolution layers and  also to ease the amount of supervision, we design a novel  Siamese representation Learning for Omnidirectional Flow  (SLOF) framework, which leverages the rotation-invariant  property of omnidirectional videos to address the radial  distortion problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content=' The term rotation-invariant here implies  that we can recover the original projection of 360◦ videos  given the reverse rotation of the applied projection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content=' The  good thing about this rotation-invariant property is that  we can project omnidirectional videos on any plane by  rotating the spherical videos on three different axes XY Z,  namely pitch, roll and yaw, while these rotation operations  still preserve the overall information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content=' That is, any projected  planar representation can be converted back to the spheri-  cal representation since we define the rotation beforehand.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content='  This rotation-invariant property motivates our design of  the proposed SLOF, where we estimate the omnidirectional  optical flow from a pair of consecutive frames and their  rotated counterparts, assuming that the representations of  these two cases are similar enough so that it can generate  nearly identical optical flow in the spherical domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content=' We  also design and compare different combinations of rota-  tional strategies and derive guidelines for selecting the most  effective augmentation scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content='  When it comes to practical applications such as egocen-  tric activity recognition, 360◦ videos impose a challenge of  significant performance drop due to random FOV projec-  tions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content=' For example, since the trained model typically only  saw one projected planar videos, it fails to capture an overall  omnidirectional representation for prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content=' This paper in-  troduces our siamese representation learning framework in  the egocentric activity recognition task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content=' We propose Vision  Transformer 360 (VIT360) to address the aforementioned is-  sue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content=' Compared to accuracy-centric convolution-based archi-  tecture trained on fixed FOV, VIT360 is rotational-invariant  and therefore does not suffer from performance degradation  when the field-of-view projection is randomized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content=' Besides,  VIT360 provides the foundation to leverage motion infer-  ence techniques from off-the-shelf optical flow architecture,  where researchers are free to choose their favorite networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content='  We hope this egocentric activity recognition task showcases  that omnidirectional learning can significantly benefit from  incorporating the rotation-invariant property in the network  architecture design for practical application.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content='  We summarize our main contribution as follows: (1)  we introduce FLOW360, a new optical flow dataset for  omnidirectional videos, to fill the need for a benchmark  dataset to advance the omnidirectional flow estimation  field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content=' Comprehensive characteristic analysis of FLOW360  and extensive comparisons with existing datasets manifests  FLOW360’s perceptual realism, uniqueness, and diversity;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content='  (2) We propose SLOF, a novel framework for optical flow  estimation in omnidirectional videos, to mitigate the cum-  bersome framework adjustments for omnidirectional flow  estimation, and well-accommodated to the nature of 360◦  videos;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content=' (3) We propose VIT360 and introduce siamese learn-  ing into the egocentric activity recognition task to mitigate  the performance drop problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content=' This design of VIT360 tack-  les the challenges of processing 360◦ videos with a fully  convolution-based framework while learning the rotational  invariant representation to keep up the high performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content='  Overall, the FLOW360 dataset, the SLOF framework, and  the VIT364 network, as well as our experimental results,  provide a solid foundation for future exploration in this  interesting and important omnidirectional research field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content='  2  RELATED WORK  Optical Flow Datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content=' Perspective datasets such as [22],  [41], [42], [43], [44], [45] comprise synthetic image sequences  along with the synthetic and hand-crafted optical flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content='  However, these datasets fall short in terms of perceptual  realism and complexities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content=' Even though several optical flow  datasets have been published recently in [19], [20], [21], [46],  they are primarily used in automotive driving scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content='  The other relevant dataset in the literature was Sintel [18],  which provided a bridge to contemporary optical flow esti-  mation and synthetic datasets that can be used in real-world  situations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content='  All datasets, as mentioned earlier, are introduced for  perspective videos and thus cannot be used for omnidi-  rectional flow estimation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content=' To address this problem, Lite-  FlowNet360 [40] on omnidirectional flow estimation was  released to augment the Sintel dataset by introducing distor-  tion artifacts for the domain adaptation task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content=' Nevertheless,  these augmented datasets are discontinuous around the  edges and violate the 360◦ nature of omnidirectional videos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content='  The closest datasets to ours are OmniFlow [23] and Omni-  FlowNet [24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content=' OmniFlow introduced a synthetic 180◦ field-  of-view (FOV) dataset, which is limited to indoor scenes and  lacks full 360◦ FOV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content=' Similarly, OmniFlowNet introduced a  full 360◦ FOV dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content=' However, both datasets lack complex-  ities and evidence for perceptual realism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content=' We show detailed  comparisons to OmniFlow, and OmniFlowNet in the dataset  characteristic study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content=' Compared to existing datasets in the  literature, FLOW360 is the first perceptually natural bench-  mark 360◦ dataset and fills the void in current research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content='  Optical Flow Estimation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content=' Advancements in optical flow  estimation techniques largely rely on the success of data-  driven deep learning frameworks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content=' Flownet [25] marked one  of the initial adoption of CNN- based deep learning frame-  works for optical flow estimation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content=' Several other works [16],  [17], [47], [48], [49], [50], [51], [52] followed the footsteps  JOURNAL OF LATEX CLASS FILES, 2023 - PREPRINT  3  with improved results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content=' Generally, these networks adopt  an encoder-decoder framework to learn optical flow in a  coarse-to-fine manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content=' The current framework RAFT [13]  has shown improvements with correlation learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content='  The methods mentioned above are insufficient on om-  nidirectional flow field estimation as they are designed  and trained for perspective datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content=' One initial work [53]  on omnidirectional flow estimation was presented as flow  estimation by back-projecting image points to the virtually  curved retina, thus called back-projection flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content=' It showed  an improvement over classical algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content=' Similarly, an-  other classical approach [54] relied on spherical wavelet  to compute optical flow on omnidirectional videos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content=' How-  ever, these methods are limited to classical approaches  as they are not relevant in existing deep learning-based  approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content=' One of the recent works, LiteFlowNet360 [40]  tried to compute optical flow on omnidirectional videos  using domain adaptation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content=' This method utilized the kernel  transformer technique (KTN [33]) to adapt convolution  layers on LiteFlowNet [17] and learn correct convolution  mapping on spherical data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content=' Similarly, OmniFlowNet [24]  proposed a deep learning-based optical flow estimation  technique for omnidirectional videos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content=' The major draw-  back of these methods is the requirement to adapt con-  volution layers, which takes a substantial amount of time  and makes portability a significant issue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content=' For example,  in LiteFlowNet360, each convolution layer in LiteFlowNet  was transformed using KTN with additional training and  adjustments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content=' Similar to OmniFlowNet, every convolution  layer in LiteFlowNet2 [55] was transformed using kernel  mapping [56] based on different locations of the spherical  image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content=' These techniques incur computational overheads  and limit the use of existing architectures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content=' Such approaches  demand explicit adaptation of convolution layers, which is  hard to maintain when more up-to-date methods are pub-  lished constantly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content=' Contrary to these methods, we propose a  Siamese Representation Learning for Omnidirectional Flow  (SLOF) method to learn omnidirectional flow by exploiting  existing architectures with designed representation learning  objectives, significantly reducing the unnecessary effort of  transforming or redesigning the convolution layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content='  Action/Activity Recognition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content=' The convolution-based ac-  tion/activity recognition framework has been the standard  practice in the activity recognition field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content=' Though 2D CNN  shows massive success in still image-based applications,  incorporating the temporal aspects for the downstream  task is challenging because the CNN alone does not ac-  count for time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content=' The initial work [57] shows a way to use  CNN to incorporate temporal aspects for the video clas-  sification task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content=' Similarly, the two-stream architecture [58]  learns spatio-temporal features from input RGB and optical  flow information for video classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content=' Many other ap-  proaches [59], [60], [61], [62] leverage the recurrent neural  network to model long-term dependencies across input  video frames.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content=' Following the first introduction of 3D CNN  based video approach [63], several other variants of 3D  CNN based approaches [64], [65], [66], [67] were proposed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content='  Our transformer-based approach, a hybrid method, takes  inspiration from these methods to extract video features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content='  Though initially introduced in NLP [68], the attention  mechanism in deep learning has changed several aspects of  computer vision research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content=' Ranging from still image-based  vision tasks [69] to video action classification task [70], the  attention-based mechanism is drawing increasing interest in  application areas like the activity recognition domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content=' The  introduction of the paper, “Attention is all you need” [71]  marks a significant turning point in transformer-based ap-  proaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content=' This attention framework led to the notable out-  come of the transformer-based vision applications as Vision  Transformer [72] and activity recognition frameworks [73],  [74], [75], [76].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content=' However, research in egocentric activity  recognition in 360◦ videos is still in its infancy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content=' Inspired  by recent advancements in transformer-based vision tasks,  we combine the potential of vision transformers with tradi-  tional convolution-based techniques for egocentric activity  recognition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content='  Although there are affluent datasets in the egocen-  tric activity recognition literature, most of the published  datasets [77], [78], [79], [80], [81] are limited to the perspec-  tive field-of-view.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content=' The 360◦ videos based egocentric activity  recognition datasets are relatively scarce.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content=' EGOK360 [82] is  a recently published egocentric dataset with 360◦ field-of-  view, which includes two confusing terminologies called  activity and actions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content=' The activity is defined as a collection  of minor actions, where the actions are more fine-grained  motions related to egocentric activities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content=' We perform our  experiments on these datasets by focusing only on activity  recognition (twelve different classes).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content='  Siamese Representation Learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content=' Siamese representation  learning is a powerful approach in the unsupervised learn-  ing category.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content=' Siamese networks have shown great success in  different vision-related tasks such as verification [83], [84],  [85] and tracking [86].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content=' A recent approach [87] in siamese rep-  resentation learning showed impressive results in unsuper-  vised visual representation learning via exploiting different  augmentation views of the same data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content=' They presented their  work in pre-training and fine-tuning stages, where the for-  mer being the unsupervised representation learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content=' We use  the representation learning scheme on omnidirectional data  via rotational augmentations, maximizing the similarity for  latent representations and minimizing the flow loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content='  3  FLOW360 DATASET  FLOW360 is an optical flow dataset tailored for 360◦  videos using Blender [88].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content=' This dataset contains naturalis-  tic 360◦ videos, forward and backward optical flow, and  dynamic depth information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content=' The dataset comprises 40 dif-  ferent videos extracted from huge 3D-World ‘The Room’,  ‘Modern’, ‘Alien Planet’, and ‘City Rush’.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content=' Due to their size,  this 3D-World cannot be rendered at once in a single video.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content='  We render several parts of this 3D-World, which provides  enough qualitative variation in motion and visual percep-  tion like 3D assets, textures, and illuminations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content=' The nature  of this large and diverse animated world provides rela-  tively enough diversity to qualify for a standard benchmark  dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content=' 2 shows some of the examples of motion and  scene diversity of FLOW360.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content=' Similarly, samples from the  dataset of different 3D-World are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content=' We build  this 3D-World using publicly available 3D models [89], [90],  [91] and 3D animated characters [92], [93], [94].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content=' Meanwhile,  JOURNAL OF LATEX CLASS FILES, 2023 - PREPRINT  4  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content=' The FLOW360 Dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content=' Sample frames (first and second column, respectively) from some of the videos with corresponding forward optical  flow and dynamic depth information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content=' Motion in the 3D Sphere (fourth column) is computed by transforming the motion vectors from Equirectangular  plane (θ, φ) to unit sphere f(x, y, z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content=' Motion in the sphere is represented in RGBA color notation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content=' RGB color representation (as suggested in  Middlebury [22]) is encoded using (x, y) components, and the alpha color is encoded from z of a unit sphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content=' RGB encoding (fifth column) is an  RGB color map of flow in 3D space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content=' Note: flow fields are clipped for better visualization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content='  we adopt Blender [88] for additional rigging and animation  for the dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content='  FLOW360 contains 40 video clips extracted from differ-  ent parts of huge 3D-World, ‘The Room’, ‘Alien Planet’,  ‘City Rush’, and Modern’.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content=' The datasets also contain other  information like depth maps and normal fields extracted  from the 3D-World.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content=' The FLOW360 dataset has 4,000 video  frames, 4,000 depth maps, and 3,960 flow fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content=' We divide  the video frames into 2700/1300 train/test split.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content=' We render  the video frames with the dimension of (512, 1024) to save  the rendering time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content=' However, FLOW360 can be rendered  with higher resolution, as 3D models and Blender add-ons  will also be public.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content='  Diversity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content=' We design FLOW360 datasets to include a diverse  situation that resembles the real-world scenario as much  as possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content=' The statistical validity of the datasets in terms  of perceptual realism of scene and motion is presented in  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content=' The datasets contain a wide range of motion complex-  ity from smaller to larger displacement, occlusion, motion  blur, and similar complexities on the scene using camera  focus-defocus, shadow, reflections, and several distortion  combinations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content=' As these complexities are quite common in  natural videos, the FLOW360 provides similar complexities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content='  Similarly, the datasets cover diverse scenarios like environ-  mental effects, textures, 3D assets, and diverse illumina-  tions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content=' The qualitative presentation of these diversities and  complexities are presented in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content=' 2 and Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content=' 3 respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content='  Fairness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content=' The FLOW360 dataset contains custom-tailored  animated 360 videos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content=' We plan to release the dataset with  the 3D models and our custom Blender add-ons to provide  researchers a platform to create their custom optical flow  datasets for all kinds of environments (perspective, 180◦ and  360◦ FOV).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content=' However, the release of 3D world scenes can  raise questions regarding fairness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content=' To mitigate this issue, we  will perturb certain parts of 3D world scenes and not release  any camera information related to the test set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content='  Flow-generator with Blender Add-ons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content=' Flow-generator is  a custom Blender add-on written for Blender-v2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content='92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content=' The  flow-generator serves two basic purposes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content=' First, it creates  a Blender compositor pipeline to collect frames, depth maps  and optical flow information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content=' This add-on can also collect  additional information, such as normal maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content=' Second, it sets  up a camera configuration for 360◦ FOV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content=' We will describe  the details of the add-ons in the supplementary material.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content='  Render Passes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content=' We exploit several modern features from  Blender-v2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content='92 like advanced ray-tracing as a render engine  along with render passes like vector, normal, depth, mist,  and so on to produce realistic 3D scenes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content=' Additionally, we  incorporate features like ambient occlusion, motion blur,  camera focus/defocus, smooth shading, specular reflection,  shadow, and camera distortion to introduce naturalistic  complexity (shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content=' 3) in our dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content=' Besides optical  flow information, the FLOW360 3D-world may be used to  collect several other helpful information like depth, normal  maps, and semantic segmentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content='  Dataset Statistics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content=' We conduct a comprehensive analysis  and compare our dataset with Sintel [18], Lookalikes (pre-  sented in the original Sintel paper to compare the image  cleanframesamples  frames  forwardopticalflow  motion  3D-flow  _depth  3D-flow  depth  3D-flow  depth  3D-flow  depthJOURNAL OF LATEX CLASS FILES, 2023 - PREPRINT  5  frame  flow  frame  flow  frame  flow  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content=' Motion and Scene Diversity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content=' Samples from our FLOW360 Dataset with random projections, showing scene and motion diversity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content=' The  FLOW360 dataset has a vast scene consisting of several lighting scenarios, textures, diverse 3D assets, and motion complexity in different regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content='  motion blur  lens distortion  camera focus & defocus  camera defocus  camera focus  shadow and reflection  reflection on surface  reflection on water  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content=' Complexity of FLOW360 Dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content=' Frames in FLOW360 Dataset  include complex characteristics like camera focus/defocus, motion blur,  lens distortion, shadow, and reflections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content=' Our dataset provides ambiance  occlusion and environmental effects for a realistic visual appearance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content='  statistics with the simulated dataset), Middlebury [22], Om-  niFlow [23] and OmniFlowNet [24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content=' The analysis shown in  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content=' 4 shows the image and motion statistics in the top and  bottom rows, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content='  Based on analysis from Sintel, we present frame statistics  with three different analysis: luminance histogram, power  spectrum, and spatial derivative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content=' For luminance statistics,  we convert the frames to gray-scale, I(x, y) ∈ [0, 255]  then we compute histograms of gray-scale images across  all pixels in the entire dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content=' The luminance statistics  show the FLOW360 has a similar distribution with the peak  in the range between [0−100] and decreasing luminosity  beyond that range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content=' Similarly, we estimate power spectra  from the 2D FFT of the 512×512 in the center of each  frame.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content=' We compute the average of these power spectra  across all the datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content=' We present power spectra analysis  separately for the training and test set in this analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content='  The power spectra analysis closely resembles the Sintel,  Lookalikes, and Middlebury datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content=' Based on [45], [95],  the real-world movies exhibit a characteristic of a power  spectrum slope around -2, which is equivalent to a 1/f 2  falloff.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content=' FLOW360 with the slope (−2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content='30, −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content='36) on test and  training split shows such characteristics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content=' We do not claim  JOURNAL OF LATEX CLASS FILES, 2023 - PREPRINT  6  luminance histogram  spatial derivative  power spectrum  OmniFlow dI/dx  OmniFlow dI/dy  F360-test dI/dx  F360-test dI/dy  F360-train dI/dx  F360-train dI/dy  0  20  200  200  Sintel  Lookalikes  Middlebury  l  l  l  l  l  l  l  1024  512  200 0 200  512  1024  0-  5-  10-  15-  log (fraction of pixels)  Sintel  Lookalikes  Middlebury  l  l  l  10-2  10-1  100  108-  108-  108-  108-  power (x)  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content='27  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content='36  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content='17  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content='30  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content='36  Slopes  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content='79  OmniFlow dI/dx  F360-test dI/dx  F360-train dI/dx  OmniFlowNet  scaled by 10  0  250  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content='020  OmniFlow dI/dx  F360-test dI/dx  F360-train dI/dx  Sintel  Lookalikes  Middlebury  l  l  l  l  l  l  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content='920-  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content='015-  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content='010-  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content='005-  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content='000-  0  50  100  150  200  250  fraction of pixels  sintel  14-  12-  10-  8-  6-  4-  2-  0-  l  l  l  l  l  300  150  0  150  300  log (fraction of pixels)  sintel  12-  10-  8-  6-  4-  2-  0-  log (fraction of pixels)  l  l  l  l  0  100  200  300  OmniFlow  clipped at y=-14,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content='  minimum value  range upto -23  14-  12-  10-  8-  6-  4-  2-  0-  l  l  l  l  l  50  100  0  50  100  log (fraction of pixels)  sintel  OmniFlow  clipped at y=-16,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content='  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content='minimum value  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content='range upto -23  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content='F360-test  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content='(  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
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+page_content='OmniFlowNet (  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
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+page_content='u )  OmniFlow and  OmniFlowNet  clipped at y=-7,  minimum value  range up to -11  and -8  respectively  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content=' Comparision of frames and flow statistics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content=' The top row represents the statistics and comparison of the frame with Sintel, Lookalikes,  Middlebury, OmniFlow [23] and OmniFlowNet [24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content=' Bottom row represents flow statistics and comparison with Sintel (red), OmniFlow (magenta)  and OmniFlowNet (turquoise).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content=' The table on the top-right shows a brief comparison of OmniFlow & OmniFlowNet with the FLOW360 dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content=' Note:  (→, ←) represents forward and backward flow fields, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content='  that FLOW360 is realistic, but it certainly exhibits perceptual  similarity with natural movies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content=' The spatial and temporal  derivative analysis additionally supports this characteristic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content='  The Kurtosis of frames’ spatial derivatives range from 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content='74  to 57.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content='27, peaking at zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content=' This characteristic shows that  FLOW360 has a resemblance to natural scenes [45].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content='  Regarding the flow field analysis, we directly com-  pare the distribution of motion u(x, y), speed defined as  s(x, y) =  �  u(x, y)2 + v(x, y)2, flow direction Θ(x, y) =  tan−1 (v(x, y)/u(x, y)) and spatial flow derivative of u and  v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content=' The close resemblance of the flow field statistics between  Sintel and FLOW360 suggests motion field resemblance  with natural movies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content=' Based on these comparisons, FLOW360  exhibits sufficient properties evident enough for its percep-  tual realism and complexities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content='  Comparison with OmniFlow and OmniFlowNet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content=' Omni-  Flow [23] presents an omnidirectional flow dataset that is  roughly similar to FLOW360.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content=' However, the major distinc-  tion between these datasets is the FOV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content=' FLOW360 provides  immersive 360◦ FOV, whereas OmniFlow provides only  180◦ FOV showing FLOW360 compared to OmniFlow is the  true omnidirectional dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content=' Similarly, OmniFlowNet [24]  presents synthetic omnidirectional flow dataset with 360◦  FOV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content=' However, this dataset contains low poly unnatural  scenes, which can be explained by relatively larger kurtosis  (373.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content='55, 391.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content='09), characteristic of a power spectrum and lu-  minance distribution (peaked at 255).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content=' The overall statistical  analysis reveals FLOW360’s better perceptual realism and  diversity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content='  Applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content=' As we mentioned, the FLOW360 dataset con-  tains frames and forward flow fields and includes backward  flow fields, depth maps, and 3D-FLOW360 worlds, provid-  ing potential for applications like continuous flow-field esti-  mation in 3 frames setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content=' Besides optical flow estimation,  the FLOW360 dataset can be used in other applications such  as depth and normal field estimation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content=' Moreover, given 3D-  FLOW360 animation data, the researcher can create as many  optical flow datasets as needed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content='  4  360◦ OPTICAL FLOW ESTIMATION  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content='1  SLOF for 360◦ Optical Flow Estimation  SLOF, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content=' 5, is inspired by the recent work  on Siamese representation learning [87].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content=' Since the method  we rely on acts as a hub between several methods like  contrastive learning, clustering, and siamese networks, it  exhibits two special properties required for our case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content=' First,  this method has non-collapsing behavior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content=' Here, the term  collapsing refers to a situation where an optimizer finds  possible minimum -1 similarity loss resulting degenerate so-  lution, characterized by zero standard deviation (std) of l2-  normalized output z/||z||2 for each channel, while training  without stop-gradient operations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content=' Stop-gradient yields std  value near  1  √  d across each channel for all samples prevent-  ing such behaviour [87].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content=' Second, it is useful when we have  only positive discriminative cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content=' SLOF does not consider  radial distortion mitigation via changing/transforming the  convolution layers but rather learns the equivariant proper-  ties of 360 videos via siamese representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content=' We claim that  such transformation is trivial, based on the following fact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content='  First, the omnidirectional videos are projected in angular  domain, w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content=' polar(θ), azimuthal(φ);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content=' θ ∈ (− π  2 , π  2 ), φ ∈  (−π, π), so we can learn flow fields in these domains and  convert these flow fields to the spherical domain using  0  5  log (fraction of pixels)  10  15  1024  512 -2000 200  512  1024  image derivative (d)-5  10  15  Kurtosis = 49.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content='94  Kurtosis = 58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content='61  Kurtosis = 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content='93  200  100  0  100  200Power spectrum  8  10  10°  4  10  Slope = -2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content='36  10  Slope = -2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content='27  Slope = -2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content='17  102  10-2  10-1108  Power (x)  106  105  10-2  10-1  100  Frequency (cycles/pixel)0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content='020  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content='015  S  Fraction of pixels  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content='010  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content='005  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content='000  0  50  100  150  200  250  LuminanceKL-D Sintel to LA: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content='058  KL-D MB to LA: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content='1760  2  F pixels)  log (fraction of  6  8  10  12  14  300  150  0  150  300  u (pixels/frame)0  2  log (fraction of pixels)  4  6  8  10  12  0  100  200  300  speed (pixels/frame)0  2  log (fraction of pixels)  6  10  12  14  100  50  0  50  100  flow/dx (1/frame)JOURNAL OF LATEX CLASS FILES,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content=' 2023 - PREPRINT  7  flow  head  projector  encoder f  rotation  head  flow  head  projector  encoder f  predictor h  similarity  flow  upsampler  spherical projection  gt-flow  flow-loss  stop  gradient  frames  R  single  rotation (v1)  R  double  rotation (v3)  R - switch  single  rotation (v2)  R  single  rotation (v1)  R  double  rotation (v3)  R - switch  single  rotation (v2)  (b) Training Strategy  (a) SLOF Architecture  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content=' Siamese Representation Learning for Omnidirectional Flow (SLOF).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content=' Pairs of frame sequence (w/ and w/o random rotation) are passed  as inputs to encoder f (RAFT as a flow head backbone and a standard convolutional projector layer).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content=' A predictor layer h is an MLP layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content=' The entire  framework is trained by fusing the pretraining and fine-tuning stages to combine the similarity and flow loss in a single stage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content=' The model maximizes  the similarity between latent representations of flow information from two streams and minimizes the flow loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content=' Training Strategy (right): Here,  two different arrows (left, right) represent siamese streams or input pathways to our model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content=' v1 and v2 (either stream is subjected to rotational  augmentation) are similar strategies achieving overall better performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content='  planar to spherical transformations as shown in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content=' (1)  and Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content=' (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content=' Second, the intent of a convolution opera-  tor in optical flow architecture is relatively different from  other applications like classification, detection, or segmen-  tation network, where other tasks require convolution to  learn relevant features (spatially consistent), the relevance  of these features should stay consistent (strictly for better  performance) throughout any spatial location of the im-  ages/videos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content=' However, the convolution operation is dedi-  cated to computing the pixel-wise displacement regardless  of spatial inconsistency in the distorted region via equivari-  ant representation learning [87].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content=' Another important consid-  eration of such a design is to make this method portable  to any existing optical flow architecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content=' This design will  eliminate the cumbersome architecture re-adjustments tasks  and make it powerful and portable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content='  Mapping Flow Field to Unit Sphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content=' Input to our model  are equirectangular images projected in angular domain  polar(θ), azimuthal(φ), where these angles are defined in  radian as θ ∈ (− π  2 , π  2 ), φ ∈ (−π, π), thus the predicted  optical flow is in (θ, φ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content=' These flow fields can be converted  to a unit sphere using planar to spherical coordinate trans-  formation as shown below:  (xs, ys, zs) = (sin θ cos φ, sin θ sin φ, cos θ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content='  (1)  We can compute sphere to catadioptric plane [96] projec-  tions to express the flow field in Cartesian coordinates as:  (x, y) = (  xs  1 − zs  ,  ys  1 − zs  ) = (cot θ  2 cos φ, cot θ  2 sin φ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content='  (2)  Design of SLOF architecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content=' Given a pair of input image  sequence X1 = (x1, x2), the rotation head (R) computes  augmented view of this sequence as X2 = (x′  1, x′  2) with  rotation r using a random combination of “pitch”, “yaw”  and “roll” operations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content=' These two augmented views are  passed as an input to an encoder network f, defined as  f = P(R′(Θ(E(R(X, r))))) where E is a flow prediction  module, RAFT [13] in our case, Θ is a mapping of 2D flow  to the unit sphere, R′ is a reverse rotation operation and P  is a convolution-based down-sampling head.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content=' A prediction  head presented as h (an MLP head), transforms the output  from the encoder f from one stream to match the other  stream.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content=' The illustration of this process shown in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content=' (3)  as maximization of cosine similarity two views from the  siamese stream:  D(pleft, zright) = −  pleft  ||pleft||2 zright  ||zright||2  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content='  (3)  Here, pleft  ≜  h(f left(X1)) and zright  ≜  f right(X2) de-  notes the output vectors to match from two different  streams(f left, f right).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content=' This maximization problem can be  viewed from another direction, with (pright, zleft) as the sec-  ond matching pair from siamese stream (f right, f left) respec-  tively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content=' Given two matching pairs, we can use the following  Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content=' 4 symmetrized similarity loss function (note that zleft  and zright are treated as a constant term using stop-grad  operations to prevent a degenerate solution due to model  collapse [87]):  Lsim = 1  2D(pleft, zright) + 1  2D(pright, zleft).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content='  (4)  Similarly, the optical flow loss is computed as a sequence  loss [13] over the predicted flow field and ground truth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content='  This loss (l1 distance over predicted and ground truth  flow fgt) is computed and averaged over a sequence of  predictions iteratively generated for the same pair of input  frames {f1, f2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content=', fn} = E(R(X, r)) as shown in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content=' (5),  where γ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content='8n−i−1 served as weights over sequence loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content='  Note that (n, i) denotes the number of prediction(n) in  sequence and prediction id(i) in predicted flow sequences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content='  The design of the weighted schemes ensures different levels  of confidence on predicted flows over time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content='  Lflow =  n  �  i=1  γ||R(fgt, r) − fi||.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content='  (5)  Given similarity loss(Lsim) and flow loss(Lflow) we imple-  ment a hybrid loss function L = Lsim + Lflow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content=' The overall  objective of this loss function is to maximize the similarity  JOURNAL OF LATEX CLASS FILES, 2023 - PREPRINT  8  between the latent representation of flow information while  minimizing the loss between ground truth and predicted  optical flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content='  5  360◦ EGOCENTRIC ACTIVITY RECOGNITION  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content='1  VIT360 for Egocentric Activity Recognition  The overall design of egocentric activity recognition in  360◦ videos comprises four different components: (1) design  of VIT360, (2) two-stream approach for activity classifi-  cation, (3) learning projection invariant representation of  360◦ videos, and (4) computing 360◦ or omnidirectional  aware motion features from off-the-shelf architecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content=' The  following subsections will discuss each of these components  in chronological order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content='  Design of VIT360.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content=' VIT360 requires input pre-processing  which converts an equirectangular image into multiple tan-  gential patches covering the entire FOV as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content='  Instead of processing raw image patches in the time dimen-  sion, these patches are processed first via the global pro-  jection layer (L1) to compute feature sequences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content=' Compared  with the original VIT [72] implementation, input features  to VIT360 are relatively larger as the patches are fed in the  time dimension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content=' The projection layers transform the larger  input feature space into a relatively smaller feature space by  downsizing the transformed feature dimension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content=' The feature  extraction layer (L2) acts as a siamese stream for learning  additional features from independent tangent patches in  the time dimension to mitigate the low parametrization of  features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content=' Using a projection layer (L1) and feature extrac-  tion layer (L2) with stacked convolution architecture makes  VIT360 a hybrid architecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content='  Two-stream Architecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content=' Our egocentric activity recogni-  tion framework (shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content=' 7) is based on recent success  on action/activity recognition task [58] loosely based on two  stream hypothesis [97].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content=' One stream is designed as a spatial  stream responsible for object recognition, and another is  designed as a motion stream responsible for motion recogni-  tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content=' Both streams of this framework implement VIT360, as  discussed above, for different input modalities (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content=' frames  for spatial and optical flow for motion).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content=' These streams are  trained independently and later fused (late fusion) to make  a joint prediction based on spatial and motion information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content='  Following the best practices in previous work [82], we  implement two different fusion techniques - average and  concatenation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content=' The average fusion technique computes the  average of the scores from the last layer and computes the  model confidence, and does not require additional training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content='  The concatenation-based techniques require concatenation  of the last MLP head and the introduction of additional  fully connected layers matching the output class dimension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content='  This technique requires additional fine-tuning and achieves  marginally better results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content='  Tangential Projections: VIT360 takes input of 360◦ video  frames  or  optical  flows  in  time  dimension  (X  ∈  RC×T ×H×W ), resulting in spatial modality (where C = 3)  and motion modality (where C = 2) respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content=' Note that  T represents the number of consecutive frames denoting  the time dimension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content=' Each input at time t is in fact projected  in the equirectangular plane(Xt ∈ RC×H×W , 0 ≤ t ≤ T),  which is obtained by a sphere mesh unwrapped on a flat  rectangular plane surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content=' This process maps sphere lati-  tude and longitude to horizontal and vertical coordinate  systems expressing the length and height of the plane in  the range (−π, π) and (−π/2, π/2) respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content=' In order to  create input patches, tangential planes are sampled using  a spherical-to-cartesian coordinate transformation system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content='  This is shown in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content=' (6), where (λ, ψ) represents latitude and  longitude such that (λ0, ψ0) = (0, 0) represents the centre  of the plane, and (c) represents the angular distance of the  point(x, y) from the centre of the projection where  x = cos ψ sin(λ − λ0)  cos(c)  ,  y = cos ψ0 sin ψ − sin ψ0 cos ψ cos (λ − λ0)  cos c  ,  cos(c) = sin ψ0 sin ψ + cos ψ0 cos(ψ) cos(λ − λ0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content='  (6)  Similarly, the inverse map from plane to the sphere can be  computed using the following equation as  ψ = sin−1  �  cos (c) sin ψ0 + y sin(c) cos ψ0  ρ  �  ,  λ = λ0 + tan−1  �  x sin (c)  ρ cos ψ0 cos (c) − y sin ψ0 sin (c)  �  ,  s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content='  ρ =  �  x2 + y2, c = tan−1 ρ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content='  (7)  The  projection  layer  samples  n  tangential  planes  (RC×T ×h×w) covering the entire 360◦ FOV, where (h, w)  are the height and width of each tangential plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content=' These  tangential planes are linearly stacked alongside the width  to obtain a final output x ∈ RC×T ×h×nw.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content=' In summary, the  tangential projection layers take a series of optical flows or  frames in a video (X ∈ RC×T ×H×W ) and transform it into  a series of tangential plane projections (x ∈ RC×T ×h×nw)  linearly stacked along the width.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content='  Projection Layer (L1): An overview of the Projection  Layer (L1) is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content=' This layer takes tangential  patches (x ∈ RC×T ×h×nw) in time dimension as input.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content=' In  contrast with the original VIT implementation, the input  to VIT360 is a 3D image (considering the time dimension)  which requires a modification of the original projection layer  to 3D convolution-based architecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content=' The Projection Layer  (L1) includes a 3D convolution with 3D batch normaliza-  tion and LeakyReLU layer as an activation function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content=' The  3D convolution layer is parametrized with the 3D kernel  size and stride as (TIME, TANGENT H, TANGENT W),  where TANGENT H = h, TANGENT W = w refers to the  height and width of tangent patches and TIME = T refers  to the number of frames in the time dimension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content=' This for-  mulation of L1 transformed input tangential patches (x ∈  RC×T ×h×nw) to (x ∈ Rn×EMBED DIM), where EMBED DIM  refers to the output feature dimension in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content='  Feature Extraction Layer (L2):  The Projection Layer (L1)  inspired by original VIT implementation maps a huge fea-  ture space (x ∈ RC×T ×h×nw) into relatively smaller feature  (x ∈ Rn×EMBED DIM) embedding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content=' This low parametrization  of the feature space creates VIT360 to suffer from signif-  icant performance degradation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content=' To mitigate this issue, an  additional Feature Extraction Layer (L2), as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content=' 8  is designed to learn additional features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content=' These additional  features are later used for attention enforcement, resulting  JOURNAL OF LATEX CLASS FILES, 2023 - PREPRINT  9  λ  ψ  0  1  2  3  4 *  n-1  ••••  Feature  Extraction  Layer  (L2)  Projection Layer (L1)  Attention Enforcement (AE1)  Feature Encoder (E2)  MLP Head  Class  Ping Pong  ….' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content='.  Input: Tangential  Projections  Corresponding patches of tangential planes stacked  on time dimension  Feature Extraction in  time dimension  Input:  Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content=' Frames  VIT360  XL1  XL2  XE2  XE1  X0  X1  X2  X3  X4  Xn-1  X  X  AE2  Transformer Encoder (E1)  VMAP  ••••  ••••  t0  t1  tT-1  time  time  time  time  time  time  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content=' VIT360 Framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content=' The sequence of ten consecutive frames (or optical flow) is sampled into six different tangential projections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content=' These  tangential projections are passed through Projection Layer (L1) and Feature Extraction Layer (L2) to compute features XL1 and XL2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content=' The First  Attention Enforcement Layer (AE1) computed the attention map between these features and passed it to Transformer Encoder (E1) with positional  embedding as an input.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content=' The final Attention Enforcement Layer (AE2) computes the attention maps between the output XE1 and XE2 and passes it  as an input to MLP Head for activity classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content='  TIME  TIME  TIME  TIME  TIME  TIME  MOTION  FEATURES  TIME  TIME  TIME  TIME  TIME  TIME  SPATIAL  FEATURES  VIT360  VIT360  SPATIAL STREAM  MOTION STREAM  MLP Head  MLP Head  Fusion  Class  Ping Pong  ….' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content='.  Time  Time  Time  Time  Time  Time  Time  Time  Time  Time  Time  Time  Time  Time  Time  Time  Time  Time  Spatial Stream  Motion Stream  Spatial  Input  Motion  Input  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content=' Two-stream Architecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content='  The motion and spatial stream  constitute the component of two-stream architecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content=' The spatial stream  receives RGB frames, whereas the motion streams receive the pre-  computed optical flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content=' These two streams are trained independently  and later fused to incorporate two different modalities via average and  concatenation-based late fusion techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content='  in improved performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content=' The Feature Extraction Layer (L2)  is implemented as a siamese network to process individual  input tangential patches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content=' To achieve the siamese network,  we use vmap (available in PyTorch framework) operation  per tangential patches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content=' The Feature Extraction Layer (L2)  takes an input (x ∈ RC×T ×h×nw), reshapes it to (x ∈  Rn×C×T ×h×w) and computes features (x ∈ Rn×SEQ DIM)  where the SEQ DIM is an output embeddings of the Feature  Extraction Layer (L2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content='  Attention Enforcement (AE1 and AE2):  The Encoder in  the original implementation of VIT takes input from the  linear projection layer from 16 × 16 image patches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content=' Such  design consideration in VIT360 is computationally challeng-  ing since the input for VIT360 is stacked patches in the  time dimension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content=' It is possible to generate a relatively larger  number of tangential planes and subsequently larger em-  bedding space from the Projection Layer (L1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content=' Such a design  is computationally infeasible because the input will be rela-  tively larger than the original VIT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content=' This tradeoff between the  choice of the number of input patches and computational  cost directly affects model performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content=' In addition, the  Projection Layer (L1) reduces the feature space significantly,  as we discussed above, leading to lower parametrization  and reducing the model’s performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content=' In order to mitigate  such issues and maintain the optimal input size, we in-  troduce Attention Enforcement (AE1 and AE2) techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content='  The first Attention Enforcement (AE1) computes an atten-  tion map between the output from L1(XL1) and L2(XL2)  making the input (X ∈ RSEQ DIM×EMB DIM) size of the  encoder sufficiently optimal retaining the model perfor-  mance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content=' Similarly, the second Attention Enforcement(AE2)  computes the attention maps (X  ∈  Rn×SEQ DIM) be-  tween the output from E1(XE1 ∈ RSEQ DIM×EMB DIM) and  E2(XE2 ∈ RSEQ DIM×n), keeping the number of param-  eter (n × EMB DIM × NUM CLASSES) in MLP Head  fixed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content=' The overview of Attention Enforcement is presented  in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content='  Encoder Layer (E1 and E2):  The Transformer Encoder  layer E1 and E2 contains similar implementation as pre-  sented in the original VIT implementation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content=' The E1-encoder  contains two layers of encoder architecture, whereas E2  contains only one layer of encoder architecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content=' E1 takes  an input (XAE1 ∈ RSEQ DIM×EMB DIM) from the Atten-  tion Enforcement Layer (E1) where EMBED DIM is the  input dimension of the E1-encoder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content=' Similarly, E2 takes an  input (XL2 ∈ Rn×SEQ DIM) from the Feature Extraction  Layer (L2) where SEQ DIM is the input dimension of the  E2-encoder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content=' Both E1 and E2 computes output feature (XE1 ∈  RSEQ DIM×EMB DIM, XE2 ∈ Rn×SEQ DIM) vectors similar to  the input dimension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content='  MLP Head: In contrast to the original VIT implementation,  we introduce a single-layer MLP head as a classification  head to predict the activity classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content=' The Attention Enforce-  ment Layer (AE2) output is passed as an input to the final  MLP Head.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content='  Projection Invariant Representation: 360◦ videos in com-  parison with perspective videos exhibit a unique property of  unlimited FOV, resulting in infinite projections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content=' Such infinite  JOURNAL OF LATEX CLASS FILES, 2023 - PREPRINT  10  Projection Layer (L1)  Conv3D  in_channels  out_channels  kernel_size  stride  {‘spatial’:3, ‘motion’:2}  EMB_DIM  (TIME, TANGENT_H, TANGENT_W)  (TIME, TANGENT_H, TANGENT_W)  bias True  BatchNorm3D  num_features EMB_DIM  LeakyReLU  negative_slope 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content='01  LeakyReLU  LeakyReLU  LeakyReLU  LeakyReLU  BN(SEQ_DIM)  BN(64)  BN(32)  BN(16)  Conv3D(in=8,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content=' out = 16,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content=' k = (TIME,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content=' 3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content='3))  MaxPool3D(k = (1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content='3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content='3),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content=' s = (1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content='2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content='2))  MaxPool3D(k = (1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content='3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content='3),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content=' s = (1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content='2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content='2))  MaxPool3D(k = (1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content='3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content='3),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content=' s = (1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content='2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content='2))  Feature Extraction Layer(L2)  Conv3D(in={‘spatial’:3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content=' ‘motion:2’},' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content=' out = 8,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content=' k = (1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content='1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content='1))  Conv3D(in=16,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content=' out = 32,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content=' k = (1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content=' 3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content='3))  Conv3D(in=32,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content=' out = 64,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content=' k = (1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content=' 3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content='3))  Conv3D(in=64,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content=' out = SEQ_DIM,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content=' k = (1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content=' 3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content='3))  MaxPool3D(k = (1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content='5,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content='5),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content=' s = (1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content='1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content='1))  Flatten  LeakyReLU  BN(8)  +  +  MLP  Norm  Multi-Head  Attention  Norm  Embedded  Patches  Lx  Encoder  XE2: [B,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content='SEQ_DIM,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content='n]  XE1: [B,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content=' SEQ_DIM,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content='EMB_DIM]  AE2 =  einsum(‘ijk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content='ijm->ikm’,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content='XE2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content='XE1)  Attention Enforcement  AE1  XL2: [B,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content='SEQ_DIM,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content='n]  XL1: [B,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content='n,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content='EMB_DIM]  AE1 =  einsum(‘ijk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content='ikm->ijm’,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content='XL2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content='XL1)  AE2  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content=' Layers in VIT360.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content=' VIT360 is composed of four major components: Projection Layer (L1) computes the initial flattened features of input  sequences, and Feature Extraction Layer (L2) computes additional features using the siamese stream.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content=' Attention Enforcement Layers (AE1 and  AE2) compute the attention maps computed across two different feature extraction layers and encoder streams, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content=' Finally, a pair of  Transformer Encoders (E1 and E2) to learn temporal features for activity recognition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content='  λ  ψ  Tangential  Projection  ••  encoder f  VIT360  encoder f  VIT360  predictor (h)  projector (p)  similarity  predictor (h)  projector (p)  x  x  p1  p2  z1  z2  stream-1  stream-2  Random  Rotation  r1  r2  stop  grad  stop  grad  similarity  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content=' Siamese Representation Learning for egocentric activity recognition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content=' The random Rotation head (R) computes two different rotational  views of the same input video sequences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content=' These augmented views are passed as input to VIT360 using tangential projections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content=' The siamese  network of VIT360 computes latent representation, pi and zi, using projector(p) and predictor (h) respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content=' The training policy maximizes the  similarity between latent representations (p1, z2) and (p2, z1) in an unsupervised manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content=' The pre-trained network with this policy is then subjected  to training for egocentric activity recognition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content='  projections can be achieved by rotating the 360◦ videos  on three different axes (X, Y, Z), namely “pitch”, “roll”,  and “yaw” operations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content=' Regardless of these projections, the  overall information of 360◦ videos remains intact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content=' This  rotation-invariant property of 360◦ videos is crucial while  designing deep learning architecture for 360◦ videos based  applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content=' In practice, the ideal architectures should be  rotationally invariant or projection invariant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content=' To achieve this  goal, we perform pretraining of VIT360 to learn projection  invariant representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content='  Pre-training of VIT360 for projection invariant represen-  tation is loosely based on a contrastive learning approach  called SimSiam [87], [99].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content=' The overview of pre-training stage  is illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content=' This representation learning based on  SimSiam is completed using siamese representation learn-  ing, formulated as a pair of weight-sharing VIT360 streams  (stream-1, stream-2) to maximize the similarity between two  different representations of the same 360◦ videos informa-  tion (both frames and optical flow).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content=' Given a sequence of  frames X = (x0, x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content=', xT −1), where T is the number of  frames in sequence in time, the rotation head (R) computes  a pair of rotational augmentation Xi = R(X, ri) using  r ∈ (r1, r2) defined as a random configuration of “pitch”,  “yaw” and “roll” operations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content=' These rotationally augmented  representations are now passed as an input to an encoder  network f = h(p(VIT360(R(X, r)))), where p and h are  MLP layers, defined as the projector and predictor’s head  respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content=' A projector head (p) appends the MLP Head  (shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content=' 5) with additional MLP layers to create a  bottleneck MLP module, which constitutes the final VIT360  architecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content=' Similarly, a predictor head (h) transforms the  output (pi  ∈  Rn×EMBED DIM) from the encoder (f) to  (zi ∈ Rn×EMBED DIM) where i = 1, 2 represents the first and  second stream.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content='  Given output (pi, zi) from respective streams, we for-  mulate a pre-training stage to maximize cosine similarity  between these representations across siamese streams.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content=' The  illustration of the maximization process is shown in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content=' (8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content='  D(p1, z2) = −  p1  ||p1||2 z2  ||z2||2  ,  Lsim = 1  2D(p1, z2) + 1  2D(p2, z1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content='  (8)  Here, p1 = f(X1) and z2 = h(f(X2)) denote latent repre-  sentations computed at different levels for the same input  X using rotationally augmented different view (X1, X2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content='  Siamese VIT360 is trained so that both projector and pre-  dictor layers are subjected to learn a similar latent represen-  tation of the given input across two different sets of inputs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content='  This representation learning aims to maximize the similarity  between these two outputs at different levels across the  siamese streams.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content=' The maximization problem can be con-  sidered in both directions, another being the maximization  between (p2, z1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content=' Given output pairs (p1, z2) and (p2, z1),  we use the symmetric similarity loss function (Lsim) as  shown in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content=' (8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content=' As discussed in SimSiam [87], we treat  (p1, p2) as a constant via stop-grad operations to prevent a  degenerate solution due to model collapse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content='  JOURNAL OF LATEX CLASS FILES, 2023 - PREPRINT  11  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content=' 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content=' 360◦ flow inference using PercieverIO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content=' The pairs of consecutive frames are projected with six different FOV covering the entire 360◦  field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content=' The PercieverIO [98] is implemented as Flow Head, which computes the optical flow in each pair of projected FOV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content=' The computed flow fields  are then projected using tangential projections to extract the central FOV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content=' This central FOV is then warped in a spherical representation and the  final omnidirectional flow is computed using sphere-to-plane projections of the warped spherical flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content='2  Omnidirectional-aware Optical Flow Estimation  The overview of the optical flow inference for 360◦ videos  using PerceiverIO is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content=' 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content=' Similar to VIT360,  this approach also considers different projections of pairs of  input 360◦ frames for calculating the final flow between the  frames.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content=' These different projections are considered an input  patch to PercieverIO, which it calculates motion information  in each pair.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content=' Later these motions in patches are warped  into one 360◦ FOV to compute the final 360◦ optical flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content='  In order to maintain the computational size, we limit the  number of patches to six and perform six different projec-  tions to cover the entire 360◦ FOV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content=' Compared to the original  PercieverIO implementation, we do not crop these different  projections but rather resize these projections into input size  as defined in PercieverIO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content=' We do not limit the flow compu-  tation across the neighboring patches by avoiding cropping  beyond the padding zone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content=' This intermediate flow represents  optical flow per projections over the entire equirectangular  plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content=' Given optical flow computed in an equirectangular  plane, flow information in the highly distorted area (like  polar regions) suffers the impact of distortions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content=' However,  the central part of equirectangular plane (λ0, ψ0) = (0, 0) is  the least distorted region from where the tangential patches  are sampled with  π  2 FOV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content=' These sampled patches cover  the entire 360◦ FOV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content=' Finally, these patches are warped into  global 360◦ flow for input frames.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content='  6  EXPERIMENTS  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content='1  360◦ Optical FLow Estimation  We evaluate SLOF on the FLOW360 test set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content=' We use pre-  trained RAFT on Sintel [18] and fine-tune on FLOW360  as a comparison baseline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content=' The fine-tuning process is done  using training protocols suggested in [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content=' Moreover, to  make a fair comparison with traditional methods, we trans-  form RAFT (pre-trained) to adapt spherical convolution  using KTN [33].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content=' KTN transforms the convolution kernel to  mitigate the radial distortions via estimating the spherical  convolution function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content=' Additionally, we run ablation studies  on different training strategies and propose a distortion-  aware evaluation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content=' We will present details of the training  procedure in the supplemental material.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content='  Scope.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content=' The scope of our experiments is two folds: First,  create a baseline for future researchers to explore novel  methodologies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content=' Second, address the validity of our method  based on fair comparisons with a flow network designed for  a spherical dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content=' We formulate our baseline experiment  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content='8  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content='0  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content=' 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content=' Distortion density map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content=' Illustrating different distortion inten-  sities due to equirectangular projections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content=' Left: upper (red) and lower  (green) part of projections shows higher distortion in the central part  whereas the equatorial region (cyan, pink, blue, gray) exhibit higher  distortion rate away from the center of the tangential plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content=' Right: shows  the distortion density from (0, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content=' This distortion density map is used to  evaluate the distortion-aware EPE (EPEd).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content=' Note: Each circle patch in  the left spherical projection has the same area.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content='  on perspective optical flow network RAFT and a modified  version of RAFT with KTN [33] to compare the performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content='  The RAFT+KTN architecture simulates a domain adapta-  tion similar to approaches like [24], [40].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content=' We choose KTN  because of its success over alternative approaches like [36],  [37], [38], [39], [100].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content=' It is worth noting that the design of  omnidirectional flow estimation can be extended to several  techniques involving the mitigation of radial distortions,  making it practically impossible to cover all.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content='  Augmentation Strategy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content=' Given the nature of SLOF, we can  train it using two different training strategies (v1, v2) as  shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content=' 5 (right).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content=' These strategies can be achieved by  performing different rotational augmentation on the input  sequences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content=' The first strategy (v1) can be achieved by using  set of inputs (R(X1, r1), R(X2, r2)) where r1 = (0, 0, 0),  i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content=', X1 does not have any rotational augmentation, whereas  r2 ̸= (0, 0, 0) has rotation defined with random combina-  tions of “pitch”, “roll”, and “yaw” operations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content=' This setting  is kept consistent throughout the training process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content=' Alterna-  tively, identical augmentation can be achieved by flipping  this augmentation protocols.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content=' The second rotational scheme  (v2) can be achieved by randomly switching rotation such  that when r1 is none, the r2 is some random rotational  augmentation and vice versa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content=' This approach performs on  par with v1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content='  Flow Warping  Flow Head  Final  omni-directional  (0,0,0)  (0,0,T//2)  (0,0,-T/2)  (0,0,r)  (0,T/2,0)  (0,-T//2,0)  flowJOURNAL OF LATEX CLASS FILES, 2023 - PREPRINT  12  frame-1  RAFT+KTN  RAFT  SLOF (v2)  gt-flow  finetuned - RAFT  SLOF (v1)  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content=' 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content=' Qualitative results on FLOW360 test set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content=' Qualitative results show our best model SLOF(v1) shows better results compared to fine-tuned  RAFT trained with policy explained in [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content=' The dotted (black) rectangle indicates the comparative improvements of our model over fine-tuned RAFT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
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+page_content=' instead, it only predicts shallow flow fields from camera motion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content=' The weakness of our model  can be seen on the dotted (red) rectangle where smaller motion segments are missing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
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+page_content='0  Avg AE  Avg EPE  Distortion Density Range  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content=' 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content=' Error distribution plot of EPE and AE in different distortion density ranges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content=' SLOF performs better in all distortion density ranges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content='  TABLE 1  Quantitave results on FLOW360 test set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content=' ∗ denotes that we use EPEd/AEd as the metrics;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content=' otherwise, the normal EPE and AE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content=' Compared to the  baseline, SLOF achieves lower end-point error and angular error on both distortion aware (EPEd and AEd) and normal scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content=' In terms of  end-point error (lower the better) our model (v1, v2) outperforms all the baselines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content=' Similarly, in terms of angular error (lower the better) our models  (v1, v2) perform comparatively similarly and outperform all the baselines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content=' Though RAFT+KTN achieves comparable normal EPE, the  distortion-aware (Weighted) metrics (EPEd and AEd) are significantly larger.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content=' Note: metrics in range (all, less than (5, 10, 20) and greater than 20)  is computed as an average, based on the speed (s(x,y)=  �  u(x, y)2 + v(x, y)2) only in the respective pixel regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content='  Method  Version  Metric  Weighted s≥0∗  s≥0  s<5  s<10  s<20  s≥20  Baselines  RAFT [13]  EPE  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
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+page_content='736  AE  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
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+page_content='340  AE  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
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+page_content='647  RAFT + KTN [33]  EPE  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content='899  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
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+page_content='426  AE  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
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+page_content='0114  SLOF  Switch rotation (v2)  EPE  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
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+page_content='512  64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content='678  AE  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content='691  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
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+page_content='482  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content='659  Single rotation (v1)  EPE  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content='548  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content='568  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
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+page_content='502  62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content='476  AE  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content='708  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content='497  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content='501  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content='497  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content='495  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content='607  AE = arccos(  ueur + vevr + 1  �  u2r + v2r + 1  �  u2e + v2e + 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content='  (9)  EPE = 1  N  N  �  i  ||fpred − fgt||2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content='  (10)  EPEd = 1  N  N  �  i  ||fpred − fgt||2  1 − d  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content='  (11)  Evaluation Strategy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content=' We evaluate our method based on 2D-  raw flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content=' Besides, using EPE (End Point Error in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content=' (10)),  i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content=', Euclidean distance between the predicted flow and  JOURNAL OF LATEX CLASS FILES, 2023 - PREPRINT  13  ground truth flow, as a single evaluation metric, we incor-  porate AE (Angular Error) as shown in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content=' (9) as the second  measure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content=' To explain the error in the omnidirectional setting,  we introduce a distortion-aware measure called EPEd as in  Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content=' (11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content=' This metric penalizes the error in the distorted area  based on the distortion density map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content='  As EPEd, AEd is calculated as  1  N  �N  i  AE  1−d where, d  represents the distortion density map illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content=' 11,  fpred = (ue, ve) represents predicted flow, and fgt = (ur, vr)  represents ground truth flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content=' Note that, to maintain lower  metrics scale the distortion density is mapped between  [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content='500, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content='000) from (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content='0, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content='0].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content=' Please refer to supplemental  for additional details on distortion density map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content='  Results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content=' 12, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content=' 13 and Table 1 summarize our exper-  imental results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content=' The overall summary of qualitative results  is presented in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content=' 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content=' SLOF performs better than baseline  RAFT and kernel transformed RAFT+KTN methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content=' This  result is evident enough to show that siamese representation  learning can exploit the rotational properties of 360◦ videos  to learn omnidirectional optical flow regardless of explicit  architecture adjustments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content='  Our methods, SLOF (v1, v2) perform better than pre-  sented baselines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content=' Among these methods v1 has the best EPE  score whereas, v2 has better AE score.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content=' However, AE on both  v1 and v2 are relatively similar, suggesting v1 as our best  method, shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content=' 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content='  By investigating distortion-aware EPE, we can see that  RAFT with KTN achieves significantly higher EPE regard-  less of comparable normal EPE with the other methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content='  This clearly explains why RAFT+KTN methods could not  predict the motion around the distorted area;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content=' instead, it  predicts shallow flow fields due to camera motion only.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content='  Moreover, comparing qualitative results in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content=' 12 and EPE  measure in different distortion ranges in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content=' 13, we can  see that our best method can predict smoother flow fields  compared to baseline methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content=' These fields in the polar  region are comparatively better and have better motion  consistency in the edge region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content=' However, our model might  fail to predict relatively smaller motion regions in some  cases, which leaves room for future improvements based  on the proposed method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content=' This concludes that RAFT+KTN  requires additional re-engineering and domain adaptation,  which is out of the scope of current work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content='  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content='2  360◦ Egocentric Activity Recognition  We focus our experiments mainly on representation learning  based egocentric activity recognition on 360◦ videos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content=' We run  our experiments on EGOK360 [82] dataset, an egocentric  activity recognition dataset for 360◦ videos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content=' We present a  series of experiments that demonstrate the efficacy of pre-  training VIT360 with a siamese representation approach  maximizing the representation across the different views of  the same input.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content=' In addition, we study the impact of using  360◦ or omnidirectional aware optical flow compared to  traditional perspective videos based on optical flow on 360◦  videos for this motion-based task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content='  The first experiment includes egocentric activity recogni-  tion (using a two-stream approach) without the representa-  tion learning scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content=' Similarly, the follow-up experiments  include the representation of learning-based egocentric ac-  tivity recognition, which follows two different consecutive  experiments of pre-training and two streams approach for  classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content=' Similarly, we conducted two additional ex-  periments based on optical flow and omnidirectional optical  flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content=' In addition to these, we also make a comparative study  of qualitative results on proposed PercieverIO-based 360◦  optical flow inference techniques with pre-trained perspec-  tive video-based optical flow technique, RAFT [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content=' Note  that our comparisons based on the two-stream architecture  are reported in EGOK360 [82] dataset only.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content=' Since the ex-  perimental results presented on EGOK360 are based on a  random train/test split of entire datasets, we also follow  a similar approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content=' However, to make a fair test case, we  sample train and test frames with a 9:1 split from each clip  from the entire training data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content=' We achieve marginally similar  baseline results on EGOK360 compared to experimental  results reported in the original paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content=' In addition, we only  consider activity recognition on the EGOK360 dataset as  experimental results are sufficient to establish core concepts  of rotation-invariant egocentric activity recognition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content='  Experiment Configuration: VIT360 training follows a sim-  ilar approach we have seen in other related works [40],  [58], [67].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content=' We choose Adam [101] as our optimizer with the  initial learning rate of 10−3, and StepLR as our learning rate  scheduler for smooth training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content=' We consider ten frames as  one training input and six tangential patches as input dis-  cussed as (T, n) in the method section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content=' The experiments are  conducted on three 2080Ti NVIDIA GPUs with a batch size  of 16, both in the training and testing phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content=' The training  and testing speed approaches 2 seconds/iterations and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content='5  seconds/iterations, respectively, achieving a throughput of  nearly 80 and 106 frames per second.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content='  Discussion on Activity Recognition Results: Table  2  summarizes our experimental results on egocentric activity  recognition on EGOK360 datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content=' From these results, we  can observe that the baseline method achieves marginally  similar accuracy as reported in [82] on a fairly sampled  new test set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content=' Though these results are promising for con-  trolled projections, the weaknesses of the baseline algorithm  quickly appear when we perform random rotation during  testing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content=' Its accuracy drops significantly in both motion  and spatial streams.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content=' However, considering the VIT360 for  a similar effect, we obtain consistent results during fixed  and random rotations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content=' These achievements of VIT360 are  useful considering the nature of 360◦ videos in the wild.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content=' In  addition, we see a significant boost in performance using  optical flow features computed using our inference tech-  niques on 360◦ videos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content=' The experimental results of VIT360  compared to the baseline on the same test set have shown  an impressive performance boost, which provides strong  evidence for the efficacy of VIT360 in egocentric activity  recognition on 360◦ videos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content='  Discussion on Optical Flow Results:  Based on results  from Table 2, the proposed optical flow inference technique  is comparatively better than the traditional techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content=' To  address this, we also present qualitative results of our  techniques and compare them with perspective flow-based  techniques in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content=' 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content=' The qualitative results show more  accurate optical flow for 360◦ videos using VIT360, which  can be further explained by smooth displacement field,  motion boundaries, and flow continuity around edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content='  JOURNAL OF LATEX CLASS FILES, 2023 - PREPRINT  14  TABLE 2  Quantitave Results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content=' The Top-1 accuracy (shown as %) of VIT360 compared with baseline methods in EGOK360 [82] achieves consistent  performance regardless of projection mode, achieving less than 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content='40% accuracy difference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content=' The baseline method shows performance gaps  between 5-14%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content=' 360◦ optical flow computed using our inference techniques boost the performance of VIT360 by almost 5%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content=' (Note: We use  Perspective Flow (RAFT [13]) for baseline experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content=')  Version  Random Projection  Flow  Motion  Spatial  Fused Avg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content='  Fused Concat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content='  ResNet  ✓  Perspective  42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content='53  59.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content='67  56.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content='18  62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content='79  Perspective  56.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content='43  68.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content='95  66.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content='79  69.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content='32  I3D  ✓  Perspective  48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content='43  63.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content='61  60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content='17  65.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content='39  Perspective  62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content='19  69.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content='78  67.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content='37  72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content='68  VIT360 (Ours)  ✓  Perspective  56.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content='52  67.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content='14  65.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content='18  70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content='89  Perspective  56.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content='57  67.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content='23  65.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content='23  70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content='79  ✓  360◦(Ours)  60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content='34  71.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content='29  70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content='66  75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content='87  360◦(Ours)  60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content='43  71.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content='13  70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content='29  75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content='59  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content=' 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content=' 360◦ optical flow results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content=' We compare 360◦ flow inferred using our techniques and perspective video-based technique using RAFT [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content='  The superiority of our method can be seen in optical flow with smooth motion, distinct motion region, and flow continuity across the edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content='  7  CONCLUSION  Omnidirectional flow estimation remains in its infancy be-  cause of the shortage of reliable benchmark datasets and  tedious tasks dealing with inescapable radial distortions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content='  This paper proposes the first perceptually natural-synthetic  benchmark dataset, FLOW360, to close the gap, where com-  prehensive analysis shows excellent advantages over other  datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content=' Our dataset can be extended for other non-motion  applications like segmentation and normal estimation task  as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content=' Moreover, we introduce a siamese representation  learning approach for omnidirectional flow (SLOF) instead  of redesigning the convolution layer to adapt omnidirec-  tional nature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content=' Our method leverages the invariant rotation  property of 360◦ videos to learn similar flow representation  on various video augmentations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content=' Meanwhile, we study the  effect of different rotations on the final flow estimation,  which provides a guideline for future work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content=' Overall, the  elimination of network redesigns aids researchers in exploit-  ing existing architectures without significant modification  leading to faster deployment in practical applications, such  as egocentric activity recognition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content='  Moreover, as a proof of concept, we propose VIT360 – a  vision transformer-based network pretrained with siamese  representation - to achieve rotational invariance in 360◦  videos for egocentric activity recognition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content=' VIT360 performs  consistently regardless of the projection mode, achieving  less than a 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content='4% accuracy gap whereas baseline methods  drop in performance by 5% to 14% between fixed and  random field-of-view projections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content=' Our 360◦ flow inference  technique integrates seamlessly with VIT360 to boost the  performance by almost 26% (from 56.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
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+page_content=' That is, VIT360 is  more suitable for activity recognition in real-world scenarios  thanks to its rotational invariant properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
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+page_content=' Ionescu,  D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content=' Ding, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content=' Koppula, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content=' Zoran, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content=' Brock, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content=' Shelhamer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content=',  “Perceiver io: A general architecture for structured inputs &  outputs,” arXiv:2107.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content='14795, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content=' 11  [99]  K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content=' Bhandari, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content=' Duan, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content=' Liu, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content=' Latapie, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content=' Zong, and Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content=' Yan,  “Learning omnidirectional flow in 360circ video via siamese  representation,” in ECCV, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content=' 10  [100] Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content=' Zhang, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content=' Xu, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content=' Yu, and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content=' Gao, “Saliency detection in 360  videos,” in ECCV, 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content=' 11  [101] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content=' Kingma and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content=' Ba, “Adam: A method for stochastic opti-  mization,” arXiv:1412.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content='6980, 2014.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content=' 13  JOURNAL OF LATEX CLASS FILES, 2023 - PREPRINT  1  APPENDIX A  FLOW GENERATOR  FLOW360 is created using Blender2, a free and open-  source 3D creation suite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content=' Blender provides an interface to  write add-ons for automating workflows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content=' We create Flow-  generator, an add-on written for Blender-v2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content='92 to collect  optical flow and several other data like depth information  and normal maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content=' Flow-generator can be installed using the  following steps:  1) Download Flow-generator3  2) In Blender-v2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content='92, follow:  Edit → Preferences → Add-ons → Install  Select flow generator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content='py from the Flow-generator  project folder  Install Add-on  Search FlowGenerator  Checkmark for installation  3) Setup compositor pipeline in Blender-v2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content='92  Go to Compositing  Select “SetFlow Generator” from Custom Node  Group  If Custom Node Group is disabled press “n” to make  it visible  Select the desired configuration and click “SetEnv”  4) Camera Setup  Go to Layout  Open tab on the right side of the Layout view by  pressing “n” if not visible  Go to “CameraSetup” on the right tab  Select the desired configuration and click “Set Cam-  era System”  Note that,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content=' we use cloud rendering to speed up the ren-  dering process which requires crowd-render4 add-on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content=' Flow-  generator provides two basic functionality: (i) camera setup  and (ii) compositor pipeline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content='  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content='1  Compositor Pipeline  Compositor in Blender provides a pipeline to process ren-  der output.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content=' Flow-generator creates a basic pipeline (shown  in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content=' 15) to collect information like optical flow, images,  depth maps, and so on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content=' It also provides an easy way to cache  the desired outputs in a structured format.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content=' Similarly, it also  setups basic configurations like selecting render-engine and  render passes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content='  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content='2  Camera Setup  Camera Setup (shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content=' 15) can be accessed via the  3D-Layout tab in Blender-v2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content='92 as suggested above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content=' Camera  Setup creates an omnidirectional camera with a full 360◦  field of view.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content=' The camera setup consists of twelve differ-  ent cameras (six perspectives and six 360◦) out of which  any omnidirectional camera (default Camera EQ F) can be  selected as the main camera for rendering omnidirectional  videos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content=' It also provides an interface to adjust configurations  like resolution, dimension, and depth of the scene.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content=' https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content='blender.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content='org/  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content=' https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content='dropbox.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content='com/s/v2mvjvs7ze8rzj1/flowgenerator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content='  tar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content='gz?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content='dl=0  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content=' https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content='crowd-render.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content='com/  APPENDIX B  EXPERIMENTAL SETTING AND MORE EVALUATION  FOR OPTICAL FLOW ESTIMATION  We conduct our experiment in PyTorch (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content='0+cuda10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content='2) us-  ing the version of Python (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content='10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content=' Additional environment  detail is provided in project5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content='  Following is the list of configurations we used for our  project:  Train/Val Batch Size: Ours(16/12 - 8Gpus), Fine-  tune(8/12 - 4 Gpus)  Number of iterations on RAFT: 12  Loss: CosineSimilarity, Optical Flow L1-loss  Optimizer: AdamW  Scheduler: OneCycleLR  EarlyStopping: Patience (5) and Min-delta (1e-4)  Others: Gradient Clipping, GradScaler  Dataset: FLOW3606  We suggest our readers to refer sample videos7,8 for  demo purpose.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content='  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content='1  Distortion Density Map  We compute distortion mask (Ud, Dd, Fd, Bd, Rd, Ld)  in a cube-map with six faces: Up(U), Down(D), Front(F),  Back(B), Right(R) and Left(L) respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content=' This distortion  density cube-map projection is then projected to equirectan-  gular projection using spherical coordinate transformation  as  (xs, ys, zs) = (sin θ cos φ, sin θ sin φ, cos θ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content='  (12)  To compute density mask (Cd, where C∈(U, D, F, B, R,  L)) in each face, we define a meshgrid for co-oridnates x and  y ranging from ([−1, 1]) with dimension of (256, 256).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content=' The  coordinates (x, y) are used to compute a radius map (r) of  size (256, 256) as shown in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content=' (13).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content='  r =  �  x2 + y2  (13)  In our paper, we have shown that the radial distortion  is higher towards the center in the polar region which cor-  responds to (U, D) faces of cube-map projections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content=' Similarly,  in the equatorial regions i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content=', the rest of the faces (F, R, B,  L) show a higher distortion rate away from the center.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content=' We  compute two distinct distortion maps (Cd), one for polar  regions (U, D) and another for equatorial regions (F, B, R, L)  as shown in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content=' (14)  Cd =  �  1 − r/ max(r)  if, C ∈ {U,D} ,  r/ max(r)  otherwise  (14)  Please refer to code9 for additional details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content=' https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content='dropbox.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content='com/s/a6qioejg6yrxo7s/SLOF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content='tar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content='gz?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content='dl=0  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content=' https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content='dropbox.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content='com/s/nvzhazq99bg46f2/FLOW360  train test.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content='zip?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content='dl=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content=' Note that for better visualization please clip optical  flow in the range of (-40, 40) or lower.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content='  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content=' https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content='dropbox.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content='com/s/54mmjvoz6844mci/trailer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content='mp4?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content='  dl=0  8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content=' https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content='dropbox.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content='com/s/gvihrzj528d92uj/videos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content='zip?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content='dl=0  We recommend to use VLC-Media player to play these videos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content='  9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content=' https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content='dropbox.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content='com/s/q1d4eoqvj2a30ij/distortion  weight.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content='ipynb?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content='dl=0  JOURNAL OF LATEX CLASS FILES, 2023 - PREPRINT  2  Compositor Pipeline  Camera Setup  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content=' 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content=' FlowGenerator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content=' Flow-generator serves two major purposes: (Left) Setting up a compositor pipeline which involves setting up file paths,  pipelining desired output, and setting initial configurations like render-engine and render passes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content=' (Right) Setting up camera configurations and  additional information like resolution, dimensions, and depth of the scene.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content='  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content='2  Impact of Rotational Invariance  We perform additional experiments to quantify the impact  of rotational invariance (see table below, Table.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content=' The im-  provement is noticeable as seen in LiteFLowNet360 [40]  and Tangent Images [34]-based optical flow estimation with  rotational invariance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content='  TABLE 3  Impact of rotational invariance  Method  w/o Rot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content='Inv (EPE)  w Rot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content='Inv (EPE)  LiteFLowNet360 [40]  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content='95  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content='52  Tangent Images [34]  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content='57  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content='78  Options  Camera Setup Node Group  Iten  Resolution  100  Cube width  512  Equi Wicth  1024  Min Depth  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content='05  Max Depth  100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content='00  Camera:  Camera EQ_F  Set Camera System File Output  Base Path:  /depth/Camera_EQ_F/  Image  OpenEXR  File Output  Render Layers  Base Path:  Image  /normal/Camera_EQ_F/  Alpha  Image  OpenEXR  Depth  Mist  Composite  Normal  Y Use Alpha  Vector  Image  IndexOB  Alpha  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content='000  Noisy Image  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content='000  Separate RGBA  Combine RGBA  File Output  Image  Base Path:  R  /flow/Camera_EQ_F  + Image  OpenEXR  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content='Image  File Output  tovScene  Base Path:  View Layer  /raw_flow/Camera_EQ_F/  Image  OpenEXRCustomNodeGroup  Item  ROOT:  Resolution  100  Cube width  512  Equi Width  1024  Min Depth  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content='05  Max Depth  100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content='00  Camera:  Camera EQ F  Set Flow Generator  SetEnvJOURNAL OF LATEX CLASS FILES, 2023 - PREPRINT  3  frame-1  flow  frame-1  flow  frame-1  flow  frame-1  flow  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content=' 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content=' More Motion and Scene Diversity - Train Set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content=' Illustrating motion and scene diversity via randomly sampled tangential plane from  FLOW360 dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content=' Flow360 contains a range of scene complexity governing varied properties like texture, illumination, human, building, cars, and  other 3D assets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content=' Similarly, the various level of motion complexity can be seen for similar scenes ranging from smaller to larger displacement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content='  As explained in the main paper the dataset also contains other complexities like motion blur, camera focus/defocus, camera distortion, and  environmental effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content='  JOURNAL OF LATEX CLASS FILES, 2023 - PREPRINT  4  frame-1  flow  frame-1  flow  frame-1  flow  frame-1  flow  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content=' 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content=' More Motion and Scene Diversity - Test Set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content=' Illustrating motion and scene diversity via randomly sampled tangential plane from FLOW360  dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content=' Flow360 contains a range of scene complexity governing varied properties like texture, illumination, human, building, cars, and other 3D  assets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content=' Similarly, the various level of motion complexity can be seen for similar scenes ranging from smaller to larger displacement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
+page_content=' As explained in  the main paper the dataset also contains other complexities like motion blur, camera focus/defocus, camera distortion, and environmental effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFKT4oBgHgl3EQfry7P/content/2301.11880v1.pdf'}
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+Generative Quantile Regression with
+Variability Penalty
+Shijie Wang1, Minsuk Shin2, and Ray Bai1
+1Department of Statistics, University of South Carolina, Columbia, SC 29208
+E-mails: shijiew@email.sc.edu, rbai@mailbox.sc.edu
+2Gauss Labs, Palo Alto, CA 94301
+Abstract
+Quantile regression and conditional density estimation can reveal structure that is missed
+by mean regression, such as multimodality and skewness. In this paper, we introduce a deep
+learning generative model for joint quantile estimation called Penalized Generative Quantile
+Regression (PGQR). Our approach simultaneously generates samples from many random
+quantile levels, allowing us to infer the conditional distribution of a response variable given
+a set of covariates. Our method employs a novel variability penalty to avoid the problem of
+vanishing variability, or memorization, in deep generative models. Further, we introduce a new
+family of partial monotonic neural networks (PMNN) to circumvent the problem of crossing
+quantile curves. A major benefit of PGQR is that it can be fit using a single optimization,
+thus bypassing the need to repeatedly train the model at multiple quantile levels or use
+computationally expensive cross-validation to tune the penalty parameter. We illustrate the
+efficacy of PGQR through extensive simulation studies and analysis of real datasets. Code to
+implement our method is available at https://github.com/shijiew97/PGQR.
+Keywords: conditional quantile, deep generative model, generative learning, joint quantile model,
+neural networks, nonparametric quantile regression
+1
+Introduction
+Quantile regression is a popular alternative to classical mean regression (Koenker and Bassett Jr,
+1982). For a response variable Y ∈ R and covariates X = (X1, X2, . . . , Xp)⊤ ∈ Rp, we define the
+τ-th conditional quantile, τ ∈ (0, 1), as the τ-th quantile of the conditional distribution of Y given
+X, i.e.
+QY |X(τ) = inf{y : FY |X(y) ≥ τ},
+(1)
+where FY |X is the conditional cumulative distribution function of Y given X. For a fixed quantile
+level τ ∈ (0, 1), linear quantile regression aims to model (1) as a linear combination of X, i.e.
+1
+arXiv:2301.03661v1  [stat.ME]  9 Jan 2023
+
+QY |X(τ) = X⊤βτ. Linear quantile regression is less sensitive to outliers than least squares regres-
+sion and is more robust when the assumption of iid Gaussian residual errors is violated (Koenker
+and Hallock, 2001; Koenker et al., 2017). Thus, it is widely used for modeling heterogeneous data
+and heterogeneous covariate-response associations.
+Numerous extensions and variants of quantile regression have been proposed.
+In order to
+reveal complex nonlinear structures, nonparametric quantile regression is frequently considered
+(Chaudhuri and Loh, 2002; Li et al., 2021; Koenker et al., 1994).
+For some function class F,
+nonparametric quantile regression aims to estimate f(X, τ) ∈ F for QY |X(τ) = f(X, τ) at a
+given quantile level τ. Many existing nonparametric quantile methods are based on either kernel
+smoothing or basis expansions. While quantile regression has traditionally focused on estimating
+the conditional quantile function at a single τ-th quantile level, composite (or joint) quantile
+regression aims to estimate multiple quantile levels simultaneously for a set of K ≥ 2 quantiles
+{τ1, τ2, . . . , τK}. Some nonparametric methods for composite quantile regression are proposed in
+Zou and Yuan (2008), Jiang et al. (2012b), Jiang et al. (2012a), and Xu et al. (2017).
+Despite its flexibility, nonparametric quantile regression carries the risk that the estimated
+curves at different quantile levels might cross each other. For instance, the estimate of the 95th
+conditional quantile of Yi given some covariates Xi might be smaller than the estimate of the 90th
+conditional quantile. This is known as the crossing quantile phenomenon. The left two panels
+of Figure 4 illustrate this crossing problem. When quantile curves cross each other, the quantile
+estimates violate the laws of probability and are no longer reasonable. To tackle this issue, many
+approaches have been proposed, such as constraining the model space (Takeuchi et al., 2006;
+Sangnier et al., 2016; Moon et al., 2021) or the model parameters (Meinshausen, 2006; Cannon,
+2018).
+Recently in the area of deep learning, neural networks have also been applied to nonparametric
+quantile regression. Taylor (2000) used feedforward neural networks (FNNs) to estimate condi-
+tional quantiles. Takeuchi et al. (2006) considered support vector machines in combination with
+quantile regression. Neural networks have also been used for composite quantile regression by Xu
+et al. (2017) and Jin and Zhao (2021). Existing neural network-based approaches for joint quantile
+estimation are restricted at a prespecified quantile set. If only several quantiles (e.g. the interquar-
+tiles τ ∈ {0.25, 0.5, 0.75}) are of interest, then these methods are adequate to produce desirable
+2
+
+inference. Otherwise, one typically has to refit the model at new quantiles or specify a large enough
+quantile candidate set in order to infer the full conditional density p(Y | X). Dabney et al. (2018)
+introduced the implicit quantile network (IQN), which takes a grid of quantile levels as inputs and
+approximates the conditional density at any quantile level (not just the given inputs). However,
+IQN cannot guarantee that quantile functions at different levels do not cross each other.
+Composite quantile regression is closely related to the problem of conditional density estimation
+(CDE) of p(Y | X). Apart from traditional CDE methods that estimate the unknown probability
+density curve (Izbicki and Lee, 2017; Pospisil and Lee, 2019), several deep generative approaches
+besides IQN have also been proposed. Zhou et al. (2022) introduced the generative conditional
+distribution sampler (GCDS), while Liu et al. (2021) introduced the Wasserstein generative con-
+ditional sampler (WGCS) for CDE. GCDS and WGCS employ the idea of generative adversarial
+networks (GANs) (Goodfellow et al., 2014) to generate samples from the conditional distribution
+p(Y | X). In these deep generative models, random noise inputs are used to learn and generate
+samples from the target distribution. However, a well-known problem with deep generative net-
+works is that the generator may simply memorize the training samples instead of generalizing to
+new test data (Arpit et al., 2017; van den Burg and Williams, 2021). When memorization occurs,
+the random noise variable no longer reflects any variability, and the predicted density p(Y | X)
+approaches a point mass. We refer to this phenomenon as vanishing variability.
+To achieve the goal of joint quantile estimation as well as conditional density estimation, we
+propose a new deep generative approach called penalized generative quantile regression (PGQR).
+PGQR employs a novel variability penalty to avoid vanishing variability. Although introduced in
+the context of joint quantile regression, we believe that our penalty formulation is broadly applicable
+to other deep generative models that may suffer from memorization. We further guarantee the
+monotonicity (i.e.
+non-crossing) of the estimated quantile functions from PGQR by designing
+a new family of partial monotonic neural networks (PMNNs). The PMNN architecture ensures
+partial monotonicity with respect to quantile inputs, while retaining the expressiveness of neural
+networks.
+The performance of PGQR depends crucially on carefully choosing a hyperparameter λ, which
+controls the amount of regularization by our variability penalty. However, common tuning proce-
+dures such as cross-validation are impractical for deep learning, since this would involve repetitive
+3
+
+evaluations of the network for different choices of λ and different training sets. Inspired by the
+generative idea from Shin et al. (2022), we construct our deep generative network in such a way
+that λ is included as an additional random input. This way, only a single optimization is needed
+to learn the neural network parameters, and then it is effortless to generate samples from a set of
+candidate values for λ. Finally, we introduce a criterion for selecting the optimal choice of λ which
+we use to generate the final desired samples from multiple quantiles of p(Y | X) simultaneously.
+Our main contributions can be summarized as follows:
+1. We propose a deep generative approach for composite quantile regression and conditional
+density estimation called penalized generative quantile regression.
+PGQR simultaneously
+generates samples from multiple random quantile levels, thus precluding the need to refit the
+model at different quantiles.
+2. We introduce a novel variability penalty to avoid the vanishing variability phenomenon in
+deep generative models and apply this regularization technique to PGQR.
+3. We construct a new family of partial monotonic neural networks (PMNNs) to circumvent the
+problem of crossing quantile curves.
+4. To facilitate scalable computation for PGQR and bypass computationally expensive cross-
+validation for tuning the penalty parameter, we devise a strategy that allows PGQR to be
+implemented using only a single optimization.
+The rest of the article is structured as follows. Section 2 motivates our PGQR framework with a
+real data application on body composition and strength in older adults. Section 3 introduces the
+generative quantile regression framework and our variability penalty. In Section 4, we introduce
+the PMNN family for preventing quantile curves from crossing each other. Section 5 discusses
+scalable computation for PGQR, namely how to tune the variability penalty parameter with only
+single-model training. In Section 6, we demonstrate the utility of PGQR through simulation studies
+and analyses of additional real datasets. Section 7 concludes the paper with some discussion and
+directions for future research.
+4
+
+Figure 1: Using PGQR to model the conditional density of AP-L/F ratio given weight in older adults.
+Left panel: Weight = 45.4 kg, Right panel: Weight = 77.5 kg.
+2
+Motivating Application: Discovering Subpopulations
+As discussed in Section 1, we aim to generate samples from the conditional quantiles QY |X(τ) of
+p(Y | X) at different quantile levels τ ∈ (0, 1). An automatic byproduct of joint nonparametric
+quantile regression (as opposed to linear quantile regression) is that if the conditional quantiles
+QY |X(τ) are estimated well for a large number of quantiles, then we can also infer the entire
+conditional distribution for Y given X.
+To motivate our methodology, we apply the proposed PGQR method (introduced in Section 3)
+to a dataset on body composition and strength in older adults (RoyChoudhury and Xu, 2020). The
+data was collected over a period of 12 years for 1466 subjects as part of the Rancho Bernardo Study
+(RBS), a longitudinal observational cohort study. We are interested in modeling the appendicular
+lean/fat (AP-L/F) ratio, i.e.
+AP-L/F Ratio = Weight on legs and arms
+Fat weight
+,
+as a function of weight (kg). Accurately predicting the AP-L/F ratio is of practical clinical interest,
+since the AP-L/F ratio provides information about limb tissue quality and is used to diagnose
+sarcopenia (age-related, involuntary loss of skeletal muscle mass and strength) in adults over the
+age of 30 (Evans, 2010; Scafoglieri et al., 2017).
+5
+
+Rati
+Rati
+[]
+[
+AP-L/
+40
+60
+80
+100
+120
+140
+40
+60
+80
+100
+120
+140
+Weight
+WeightFigure 1 plots the approximated conditional density of AP-L/F ratio given weight of 45.4 kg (left
+panel) and 77.5 kg (right panel) under the PGQR model. We see evidence of data heterogeneity
+(actually, depending on an unobserved factor of gender), as the estimated conditional density is
+unimodal when the weight of older adults is 45.4 kg but bimodal when the weight of older adults
+is 77.5 kg. In short, our method discovers the presence of two heterogeneous subpopulations of
+adults that weigh around 78 kg. In contrast, mean regression (e.g. simple linear regression or
+nonparametric mean regression) of AP-L/F ratio given weight might obscure the presence of two
+modes and miss the fact that weight affects AP-L/F ratio differently for these two clusters of adults.
+3
+Penalized Generative Quantile Regression
+3.1
+Generative Quantile Regression
+Before introducing PGQR, we first introduce our framework for generative quantile regression
+(without variability penalty). Given n training samples (X1, Y1), (X2, Y2), . . . , (Xn, Yn) and quan-
+tile level τ ∈ (0, 1), nonparametric quantile regression minimizes the empirical quantile loss,
+argmin
+f∈F
+1
+n
+n
+�
+i=1
+ρτ (Yi − f(Xi, τ)) ,
+(2)
+where ρτ(u) = u(τ −I(u < 0)) is the check function, and F is a function class such as a reproducing
+kernel Hilbert space or a family of neural networks. Neural networks are a particularly attractive
+way to model the quantile function f(X, τ) in (2), because they are universal approximators for
+any Lebesgue integrable function (Lu et al., 2017).
+In this paper, we choose to model the quantile function in (2) using deep neural networks
+(DNNs). We define a DNN as a neural network with at least two hidden layers and a fairly large
+number of nodes. We refer to Emmert-Streib et al. (2020) for a comprehensive review of DNNs.
+Despite the universal approximation properties of DNNs, we must also take care to ensure that
+our estimated quantile functions do not cross each other. Therefore, we have to consider a wide
+enough monotonic function class Gm to cover the true QY |X(·). We formally introduce this class
+Gm in Section 4.
+Let G denote the constructed DNN, which is defined as a feature map function {G ∈ Gm :
+6
+
+Rp+1 �→ R1} that takes (Xi, τ) as input and generates the conditional quantile for Yi given Xi at
+level τ as output. We refer to this generative framework as Generative Quantile Regression (GQR).
+The optimization problem for GQR is
+�G = argmin
+G∈Gm
+1
+n
+n
+�
+i=1
+Eτ
+�
+ρτ
+�
+Yi − G(Xi, τ)
+��
+,
+(3)
+where �G denotes the estimated quantile function with optimized parameters (i.e. the weights and
+biases) of the DNN. Note that optimizing the integrative loss Eτ{·} over τ in (3) is justified by
+Proposition 3.1 introduced in the next section (i.e. we set λ = 0 in Proposition 3.1).
+In order to solve (3), we can use stochastic gradient descent (SGD) with mini-batching (Emmert-
+Streib et al., 2020). For each mini-batch evaluation, Xi is paired with a quantile level τ sampled
+from Uniform(0, 1). Consequently, if there are M mini-batches, the expectation in (3) can be ap-
+proximated by a Monte Carlo average of the random τ’s, i.e. we approximate Eτ
+�
+ρτ
+�
+Yi−G(Xi, τ)
+��
+with M −1 �M
+k=1
+�
+ρτk
+�
+Yi − G(Xi, τk)
+��
+. Once we have solved (3), it is straightforward to use �G
+to generate new samples �G(X, ξk), k = 1, 2, . . . , b, at various quantile levels ξ = {ξ1, ξ2, . . . , ξb},
+where the ξ’s are random Uniform(0, 1) noise inputs. Provided that b is large enough, the generated
+samples at ξ ∈ (0, 1)b can be used to reconstruct the full conditional density p(Y | X).
+As discussed in Section 1, there are several other deep generative models (Zhou et al., 2022; Liu
+et al., 2021) for generating samples from the underlying conditional distribution. These generative
+approaches also take random noise z as an input (typically z ∼ N(0, 1)) to reflect variability, but
+there is no specific statistical meaning for z. In contrast, the random noise τ in GQR (3) has a
+clear interpretation as a quantile level τ ∈ (0, 1). Although GQR and these other deep generative
+approaches are promising approaches for conditional sampling, these methods are all unfortunately
+prone to memorization of the training data. When this occurs, the random noise does not generate
+any variability. To remedy this, we now introduce our variability penalty in conjunction with our
+GQR loss function (3).
+3.2
+Variability Penalty for GQR
+When a DNN is too overparameterized to capture the underlying structure in the training data,
+the random noise in DNN is likely to reflect no variability, no matter what value it inputs. We refer
+7
+
+this phenomenon as vanishing variability, and it is a common problem in deep generative models
+(Arpit et al., 2017; Arora et al., 2017; van den Burg and Williams, 2021). To be more specific,
+let G(·, z) be the generator function constructed by a DNN, where z is a random noise variable
+following some reference distribution such as a standard Gaussian or a standard uniform. The
+vanishing variability phenomenon occurs when
+�G(Xi, z) = Yi, i = 1, ..., n.
+(4)
+In other words, there is no variability when inputting different random noise z, generating almost
+surely a discrete point mass at the training data. Additionally, given a new feature vector Xnew,
+G(Xnew, z) can only generate one novel sample from the true data distribution because of vanishing
+variability.
+Since GQR takes the training data X ∈ Rp as input and the target quantile estimate lies in
+R1, the weights in the DNN associated with X ∈ Rp are very likely to overwhelm those associated
+with the noise input τ. As a result, GQR is very prone to encountering vanishing variability, as
+are other generative approaches with multidimensional features. From another point of view, we
+can see that due to the nonnegativity of the check function, the GQR loss (3) achieves a minimum
+value of zero when we have vanishing variability (4).
+To remedy this problem, we propose a new regularization term that encourages the network
+to have more variability when vanishing variability occurs. Given a set of features X ∈ Rp, the
+proposed variability penalty is formulated as follows:
+penλ,α(G(X, τ), G(X, τ ′)) = −λ log {∥G(X, τ) − G(X, τ ′)∥1 + 1/α} ,
+(5)
+where λ ≥ 0 is the hyperparameter controlling the degree of penalization, α > 0 is a fixed hyper-
+parameter, and τ, τ ′ ∼ Uniform(0, 1). The addition of 1/α inside the logarithmic term of (5) is
+mainly to ensure that the penalty function is always well-defined (i.e. the quantity inside of log{·}
+of (5) cannot equal zero). Our sensitivity analysis in Section C of the Appendix shows that our
+method is not usually sensitive to the choice of α. We find that fixing α = 1 or α = 5 works well
+in practice.
+With the addition of the variability penalty (5) to our GQR loss function (3), our penalized
+8
+
+GQR (PGQR) method solves the optimization,
+�G = argmin
+G∈Gm
+1
+n
+n
+�
+i=1
+�
+Eτ
+�
+ρτ
+�
+Yi − G(Xi, τ)
+��
++ Eτ,τ ′ �
+penλ,α (G(Xi, τ), G(Xi, τ ′))
+��
+,
+(6)
+where Gm denotes the PMNN family introduced in Section 4, and τ and τ ′ independently follow
+a uniform distribution on (0, 1). The expectation Eτ,τ ′ is again approximated by a Monte Carlo
+average. Note that when λ = 0, the PGQR loss (6) reduces to the (non-penalized) GQR loss (3).
+The next proposition states that an estimated generator function �G(X, τ) under the PGQR
+objective (6) is equivalent to a neural network estimator �gτ(X) based on the individual (non-
+integrative) quantile loss function, provided that the family Gm in (6) is large enough.
+Proposition 3.1 (equivalence of �gτ(X) and �G(X, τ)). For fixed λ ≥ 0 and α > 0, let �gτ(X) =
+argming∈H
+�n
+i=1
+�
+ρτ(Yi − g(Xi)) + Eτ,τ ′{penλ,α(g(Xi, τ), g(Xi, τ ′))}
+�
+, for a class of neural networks
+H, and τ, τ ′
+iid∼ Uniform(0, 1).
+Consider a class of generator functions G, where {G ∈ G :
+Rp × R × R �→ R}.
+Assume that for all X ∈ X ⊂ Rp and quantile levels τ ∈ (0, 1), there
+exists G ∈ G such that the target neural network �gτ(X) can be represented by a hyper-network
+G(X, τ). Then, for i = 1, . . . , n,
+�gτ(Xi) = �G(Xi, τ) a.s.,
+with respect to the probability law related to Eτ, where �G is a solution to (6).
+Proof. See Appendix A.
+Proposition 3.1 justifies optimizing the integrative quantile loss over τ in the PGQR objective
+(6). The next proposition justifies adding the variability penalty (5) to the GQR loss (3) by showing
+that there exists λ > 0 so that memorization of the training data does not occur under PGQR.
+Proposition 3.2 (PGQR does not memorize the training data). Suppose that α > 0 is fixed in
+the variability penalty (5). Denote any minimizer of the PGQR objective function (6) by �G. Then,
+there exists λ > 0 such that
+�G(Xi, τ) ̸= Yi for at least one i = 1, . . . , n.
+9
+
+Figure 2: A plot of the penalty function penλ,α(V ) = −λ log(|V | + 1/α).
+Proof. See Appendix A.
+We now provide some intuition into the variability penalty (5). We can quantify the amount of
+variability in the network by V := G(X, τ) − G(X, τ ′). Clearly, V = 0 when vanishing variability
+(4) occurs. To avoid data memorization, we should thus penalize V more heavily when V ≈ 0, with
+the maximum amount of penalization being applied when V = 0. Figure 2 plots the variability
+penalty, penλ,α(V ) = −λ log(|V |+1/α), for several pairs of (α, λ). Figure 2 shows that penλ,α(V ) is
+sharply peaked at zero and that penλ,α(V ) is strictly convex decreasing in |Vi|. Therefore, maximum
+penalization occurs when each V = 0, and there is less penalization for larger |V |. As λ increases,
+penλ,α(V ) also increases for all values of V ∈ (−∞, ∞), indicating that larger values of λ lead to
+more penalization.
+Our penalty function takes a specific form (5).
+However, any other penalty on G(X, τ) −
+G(X, τ ′) for X ∈ X that has a similar shape as the PGQR penalty (i.e. sharply peaked around
+zero, as in Figure 2) would also conceivably encourage the network to have greater variability.
+Another way to control the variability is to directly penalize the empirical variance s2 = (n −
+1)−1 �n
+i=1[G(Xi, τ) − ¯G]2, where ¯G = n−1 �n
+i=1 G(Xi, τ), or the empirical standard deviation
+s =
+√
+s2. However, penalties on s2 or s do not have an additive form and individual penalty terms
+depend on each other, and therefore, these are not conducive to SGD-based optimization.
+We now pause to highlight the novelty of our constructed variability penalty. In many other
+nonparametric models, e.g. those based on nonparametric smoothing, a roughness penalty is often
+10
+
+Variability Penality
+α= 1.0, 2= 0.5
+α= 1.0, 1= 2.5
+α= 5.0, 1= 0.5
+2
+α= 5.0, 1= 2.5
+(A)
+aa
+2
+4
+-4
+-2
+0
+2Figure 3: Plots of the estimated conditional densities of p(Y | Xtest) for three different test observations
+under non-penalized GQR and PGQR with two different choices of λ ∈ {0.01, 2.5}. λ⋆ = 0.01 is the
+optimal λ chosen by our tuning parameter selection method in Section 5.2.
+added to an empirical loss function in order to control the smoothness of the function or density
+estimate (Green and Silverman, 1994). These roughness penalties encourage greater smoothness
+of the target function in the spirit of reducing variance in the bias-variance trade-off. In contrast,
+our variability penalty (5) directly penalizes generators with too small variance. By adding the
+penalty to the GQR loss (3), we are encouraging the estimated network to have more variability.
+To illustrate how PGQR prevents vanishing variability, we carry out a small simulation study
+under the model, Yi = X⊤
+i β + ϵi, i = 1, . . . , n, where Xi
+iid
+∼ N(0, I20), ϵi
+iid
+∼ N(0, 1), and the
+coefficients in β are equispaced over [−2, 2]. We used 2000 samples to train GQR (3) and PGQR (6),
+and an additional 200 validation samples were used for tuning parameter selection of λ (described
+in Section 5.2).
+To train the generative network, we fixed α = 1 and tuned λ from a set of
+100 equispaced values between 0 and 2.5. We then generated 1000 samples { �G(Xtest, ξk, λ)}1000
+k=1 ,
+ξk
+iid
+∼ Uniform(0, 1), from the estimated conditional density of p(Y | Xtest) for three different choices
+of out-of-sample test data for Xtest. In our simulation, the tuning parameter selection procedure
+introduced in Section 5.2 chose an optimal λ of λ⋆ = 0.01.
+As shown in Figure 3, the non-penalized GQR model introduced in Section 3.1 (dashed orange
+line) suffers from vanishing variability. Namely, GQR generates values near the true test sample
+Ytest almost surely, despite the fact that we used 1000 different inputs for ξ to generate the �G
+samples. In contrast, the PGQR model with optimal λ⋆ = 0.01 (solid blue line with filled circles)
+approximates the true conditional density (solid black line) very closely, capturing the Gaussian
+shape and matching the true underlying variance.
+On the other hand, Figure 3 also illustrates that if λ is chosen to be too large in PGQR (6),
+11
+
+PGQR(.01)
+True
+to
+PGQR(2.5)
+GQR
+:
+:
+0
+-10
+5
+10
+-10
+-5
+5
+10
+-20
+15
+-10
+-5
+10then the subsequent approximated conditional density will exhibit larger variance than it should.
+In particular, when λ = 2.5, Figure 3 shows that PGQR (dashed green line with hollow triangles)
+overestimates the true conditional variance for Y given X. This demonstrates that the choice of
+λ in the variability penalty (2) plays a very crucial role in the practical performance of PGQR. In
+Section 5.2, we describe how to select an “optimal” λ so that PGQR neither underestimates nor
+overestimates the true conditional variance. Our method for tuning λ also requires only a single
+optimization, making it an attractive and scalable alternative to cross-validation.
+4
+Partial Monotonic Neural Network
+Without any constraints on the network, PGQR is just as prone as other unconstrained nonpara-
+metric quantile regression models to suffer from crossing quantiles. For a fixed data point Xi and
+an estimated network �G, the crossing problem occurs when
+�G(Xi, τ1) > �G(Xi, τ2) when 0 < τ1 < τ2 < 1.
+(7)
+In the neural network literature, one popular way to address the crossing problem is to add a
+penalty term to the loss such as max(0, −∂G(Xi; τ)/∂τ) (Tagasovska and Lopez-Paz, 2018; Liu
+et al., 2020). Through regularization, the network is encouraged to have larger partial derivatives
+with respect to τ. However, adding this penalty merely alleviates the crossing problem. There is
+no guarantee that the final model is free of crossing quantiles (Tagasovska and Lopez-Paz, 2018),
+or we may have to keep training the model until the final network is certified to be monotonic
+with respect to τ at the expense of heavy computational cost (Liu et al., 2020). Another natural
+way to construct a monotonic neural network (MNN) is by restricting the weights of network to be
+nonnegative through a transformation or through weight clipping (Zhang and Zhang, 1999; Daniels
+and Velikova, 2010; Mikulincer and Reichman, 2022). Because all the weights are constrained to
+be nonnegative, this class of MNNs might require longer optimization (i.e. more epochs in SGD)
+and/or a large amount of hidden neurons to ensure the network’s final expressiveness.
+Instead of constraining all the weights in the neural network, we make a simple modification to
+the MNN architecture which we call the partial monotonic neural network (PMNN). The PMNN
+12
+
+family consists of two feedforward neural networks (FNN) as sub-networks which divide the input
+into two segments.
+The first sub-network is a weight-constrained quantile network R1 �→ Rh
+where the only input is the quantile level τ and h denotes the number of hidden neurons. The
+FNN structure for one hidden layer in this sub-network is g(τ) = σ(Uposτ + b), where Upos is
+the weights matrix of nonnegative weights, b is the bias matrix, and σ is the activation function,
+such as the hyperbolic tangent (tanh) function σ(x) = tanh(x) or the rectified linear unit (ReLU)
+function σ(x) = max{0, x}. The K1-layer constrained sub-network gc can then be constructed as
+gc(τ) = gK ◦ · · · ◦ g1.
+The second sub-network is a K2-layer unconstrained network guc : Rp �→ Rh, taking the data X
+as inputs. The structure of one hidden layer in this sub-network is g′(X) = σ(UX +b), where U is
+an unconstrained weights matrix. We construct guc as guc(X) = g′
+K2 ◦· · ·◦g′
+1. Finally, we construct
+a single weight-constrained connection layer f : Rh �→ R1 that connects the two sub-networks to
+quantile estimators,
+G(X, τ) = f ◦ (gc + guc).
+(8)
+With our PMNN architecture, the monotonicity of G(·, τ) with respect to τ is guaranteed by the
+positive weights in the quantile sub-network gc and the final layer f. At the same time, since the
+data sub-network guc is unconstrained in its weights, guc can learn the features of the training data
+well. In particular, the PMNN family is more flexible in its ability to learn features of the data and
+is easier to optimize than MNN because of this unconstrained sub-network. We use our proposed
+PMNN architecture as the family Gm over which to optimize the PGQR objective for all of the
+simulation studies and real data analyses in this manuscript.
+To investigate the performance of our proposed PMNN family of neural networks, we fit the
+PGQR model with PMNN as the function class Gm on two benchmark datasets. The first dataset
+is the motorcycles data (Silverman, 1985) where the response variable is head acceleration (in g)
+and the predictor is time from crash impact (in ms). The second application is a bone mineral
+density (BMD) dataset (Takeuchi et al., 2006) where the response is the standardized relative
+change in spinal BMD in adolescents and the predictor is the standardized age of the adoles-
+cents.
+For these two datasets, we estimated the quantile functions at different quantile levels
+τ ∈ {0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7, 0.8, 0.9} over the domain of the predictor. We compared PGQR
+13
+
+Figure 4: Estimated quantile curves at levels τ ∈ {0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7, 0.8, 0.9} for the motorcycle
+and BMD datasets. The left two panels plot the estimated curves for unconstrained nonparametric quantile
+regression, and the right two panels plot the estimated curves for PGQR with the PMNN family.
+under PMNN to the unconstrained nonparametric quantile regression approach implemented in
+the R package quantreg. The left two panels of Figure 4 demonstrate that the estimated quantile
+curves under unconstrained nonparametric quantile regression are quite problematic for both the
+motorcycle and BMD datasets; the quantile curves cross each other at multiple points in the pre-
+dictor domain. In comparison, the right two panels of Figure 4 show that PGQR with the PMNN
+family ensures no crossing quantiles for either dataset.
+5
+Scalable Computation for PGQR
+5.1
+Single-Model Training for PGQR
+As illustrated in Section 3.2, PGQR’s performance depends greatly on the regularization parameter
+λ ≥ 0 in the variability penalty (5). If λ is too small or if λ = 0, then we may underestimate the
+true conditional variance of p(Y | X) and/or encounter the vanishing variability phenomenon. On
+the other hand, if λ is too large, then we may overestimate the conditional variance.
+14
+
+(1)
+Quantile Regression
+(2) PGQR
+Motorcycles Dataset
+Motorcycles
+Dataset
+8
+8
+0.2
+0.1
+Acceleration
+Acceleration
+0.3
+0
+0
+09-
+09-
+0.8
+0.9
+-100
+-100
+10
+20
+30
+40
+50
+10
+20
+30
+40
+50
+Time
+Time
+BMD Dataset
+BMD Dataset
+(standardized)
+(standardized)
+0.1
+0.2
+2
+0
+0.8
+0.9
+BMD
+BMD
+2
+2
+-1
+0
+1
+2
+-1
+0
+1
+2
+(standardized)
+(standardized)Due to the large number of parameters in a DNN, tuning λ would be quite burdensome if
+we had to repeatedly evaluate the deep generative network G(·, τ) for multiple choices of λ and
+training sets. In particular, using cross-validation to tune λ is computationally infeasible for DNNs,
+especially if the size of the training set is large.
+Inspired by the idea of Generative Multiple-purpose Sampler (GMS) introduced by Shin et al.
+(2022), we instead propose to include λ as an additional input in the generator, along with data
+X and quantile τ. In other words, we impose a discrete uniform distribution p(λ) for λ whose
+support Λ is a grid of candidate values for λ. We then estimate the network G(·, τ, λ) with the
+modified PGQR loss function,
+�G = argmin
+G∈Gm
+1
+n
+n
+�
+i=1
+Eτ,λ
+�
+ρτ
+�
+Yi − G(Xi, τ, λ)
+�
++ Eτ,τ ′ �
+penλ,α (G(Xi, τ, λ), G(Xi, τ ′, λ))
+� �
+.
+(9)
+Note that (9) differs from the earlier formulation (6) in that λ ∈ Λ is not fixed a priori. In the
+PMNN family Gm, λ is included in the unconstrained sub-network guc : Rp+1 �→ Rh.
+The next corollary to Proposition 3.1 justifies including λ in the generator and optimizing the
+integrative loss over {τ, λ} in the modified PGQR objective (9). The proof of Corollary 5.1 is a
+straightforward extension of the proof of Proposition 3.1 and is therefore omitted.
+Corollary 5.1. Let �gτ,λ(X) = argming∈H
+�n
+i=1
+�
+ρτ(Yi − g(Xi)) + Eτ,τ ′{penλ,α (g(Xi, τ), g(Xi, τ ′))}
+�
+,
+for a class of neural networks H, where α > 0 is fixed and τ, τ ′ iid
+∼ Uniform(0, 1). Consider a class
+of generator functions G where {G ∈ G : Rp × R × R �→ R}. Suppose that for all X ∈ X ⊂ Rp,
+quantile levels τ ∈ (0, 1), and tuning parameters λ ∈ Λ, there exists G ∈ G such that the target
+neural networks �gτ,λ(X) can be represented by some G(X, τ, λ). Then, for i = 1, . . . , n,
+�gτ,λ(Xi) = �G(Xi, τ, λ) a.s.,
+with respect to the probability law related to Eτ,λ, where �G is a solution to (9).
+We note that Shin et al. (2022) proposed to use GMS for inference in linear quantile regression.
+In contrast, PGQR is a method for nonparametric joint quantile estimation and conditional density
+estimation (CDE). Linear quantile regression cannot be used for CDE; as a linear model, GMS
+linear quantile regression also does not suffer from vanishing variability. This is not the case for
+15
+
+GQR, which motivates us to introduce the variability penalty in Section 3.2. Finally, we employ
+the GMS idea for tuning parameter selection in PGQR, rather than for constructing pointwise
+confidence bands (as in Shin et al. (2022)).
+By including λ ∼ p(λ) in the generator, PGQR only needs to perform one single optimization
+in order to estimate the network G. We can then use the estimated network �G to tune λ, as we
+detail in the next section. This single-model training stands in contrast to traditional smoothing
+methods for nonparametric quantile regression, which typically require repeated model evaluations
+via (generalized) cross-validation to tune hyperparameters such as bandwidth or roughness penalty.
+5.2
+Selecting An Optimal Regularization Parameter
+We now introduce our method for selecting the regularization parameter λ in our variability penalty
+(5). Considering the candidate set Λ, the basic idea is to select the best λ⋆ ∈ Λ that minimizes the
+distance between the generated conditional distribution and the true conditional distribution.
+Denote the trained network under the modified PGQR objective (9) as �G(·, τ, λ), and let
+(Xval, Yval) denote a validation sample. After optimizing (9), it is effortless to generate the con-
+ditional quantile functions for each Xval in the validation set, each λ ∈ Λ, and any quantile level
+ξ ∈ (0, 1). In order to select the optimal λ⋆ ∈ Λ, we first prove the following proposition.
+Proposition 5.2. Suppose that two univariate random variables Q and W are independent. Then,
+Q and W have the same distribution if and only if P(Q < W | W) follows a standard uniform
+distribution.
+Proof. See Appendix A.
+Based on Proposition 5.2, we know that, given a validation sample (Xval, Yval), the best
+λ⋆ ∈ Λ should satisfy Pτ,λ( �G(Xval, τ, λ⋆) < Yval
+�� Yval) ∼ Uniform(0, 1). In practice, if we have
+M random quantile levels τ = (τ1, . . . , τM) ∈ (0, 1)M, we can estimate the probability Pτ,λ :=
+Pτ,λ( �G(Xval, τ, λ⋆) < Yval
+�� Yval) with a Monte Carlo approximation,
+�P (i)
+τ,λ = M −1
+M
+�
+k=1
+I{ �G(X(i)
+val, τk, λ) < Y (i)
+val } ≈ Pτ,λ( �G(Xval, τ, λ⋆) < Yval
+�� Yval),
+(10)
+16
+
+Figure 5: Histogram of the estimated �P (i)
+τ,λ⋆’s in the validation set for the optimal λ⋆.
+for each ith validation sample (X(i)
+val, Y (i)
+val ). With nval validation samples, we proceed to estimate
+�P (i)
+τ,λ for all nval validation samples (X(i)
+val, Y (i)
+val )’s and each λ ∈ Λ. We then compare the empirical
+distribution of the �P (i)
+τ,λ’s to a standard uniform distribution for each λ ∈ Λ and select the λ⋆ that
+minimizes the distance between �Pτ,λ and Uniform(0, 1). That is, given a valid distance measure d,
+we select the λ⋆ which satisfies
+λ⋆ = argmin
+λ∈Λ
+d( �Pτ,λ, Uniform(0, 1)).
+(11)
+There are many possible choices for d in (11). In this paper, we use the Cramer–von Mises (CvM)
+criterion due to the simplicity of its computation and its good empirical performance. The CvM
+criterion is straightforward to compute as
+d( �Pτ,λ, Uniform(0, 1)) =
+nval
+�
+i=1
+(i/nval − �P (i)
+τ,λ − 2/nval)2/nval.
+(12)
+After selecting λ∗ according to (11), we can easily generate samples from the conditional quantiles
+of p(Y | X) for any feature data vector X. We simply generate �G(X, ξk, λ⋆), where k = 1, . . . , b,
+for random quantiles ξ1, . . . , ξb
+iid
+∼ Uniform(0, 1). For sufficiently large b, the conditional density
+p(Y | X) can be inferred from { �G(X, ξk, λ⋆)}b
+k=1. The complete algorithm for scalable computation
+of PGQR is given in Algorithm 1.
+We now illustrate our proposed selection method for λ ∈ Λ in the simulated linear regression
+example from Section 3.2. Figure 5 plots the histogram of the estimated �P (i)
+τ,λ⋆’s in the validation
+set for the λ⋆ which minimizes the CvM criterion. We see that the empirical distribution of the
+17
+
+ (i)
+Histogram of estiamted
+t, 1.
+2
+0
+4.
+0
+9.
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0Figure 6: The conditional standard deviation (left panel), CvM statistic (middle panel), and coverage
+rate (right panel) in the validation set vs. log(λ) for each λ ∈ Λ. The red dashed line corresponds to the
+λ⋆ which minimizes the CvM criterion.
+�P (i)
+τ,λ⋆’s closely follows a standard uniform distribution.
+Figure 6 illustrates how the PGQR solution changes as log(λ) increases for every λ ∈ Λ. Figure
+6 plots the conditional standard deviation (left panel), the CvM criterion (middle panel), and the
+coverage rate of the 95% confidence intervals (right panel) for the PGQR samples in the validation
+set. The vertical dashed red line denotes the optimal λ⋆ = 0.01. We see that the λ⋆ which minimizes
+the CvM (middle panel) captures the true conditional standard deviation of σ = 1 (left panel) and
+attains coverage probability close to the nominal rate (right panel). This example demonstrates
+that our proposed selection method provides a practical alternative to computationally burdensome
+cross-validation for tuning λ in the variability penalty (5). In Section B.1 of the Appendix, we
+further demonstrate that our tuning parameter selection method is suitable to use even when the
+true conditional variance is very small (true σ2 = 0.01). In this scenario, our procedure selects a
+very small λ⋆ ≈ 0, so that PGQR does not overestimate the conditional variance.
+6
+Numerical Experiments and Real Data Analysis
+We evaluated the performance of PGQR on several simulated and real datasets. We fixed α = 1
+or α = 5 in the variability penalty (5). We optimized the modified PGQR loss (9) over the PMNN
+family (Section 4), where each sub-network had three hidden layers with 1000 neurons per layer.
+The support for Λ was chosen to be 100 equispaced values between 0 and 1. We compared PGQR
+to several other state-of-the-art methods:
+18
+
+Conditional Standard Deviation
+CvM Statistics
+Coverage Rate
+×
+2.5
+0.010
+X
+2.0
+5.
+900'0
+8.
+x
+*
+: ×
+0
+-80-52
+-38
+-28
+-19.3-9.5 -1.5
+-60
+09-
+-40
+-30
+-20
+-10
+0
+-60
+09-
+-40
+-30
+-20
+-10
+0
+log(2.)
+log(.)
+log(2.)Algorithm 1 Scalable Implementation of PGQR
+1: Split non-test data into non-overlapping training set (Xtrain, Ytrain) and validation set (Xval, Yval)
+2: Initialize G parameters φ, learning rate γ, width h of DNN hidden layers, and total epoches T
+3: procedure: Optimizing G
+4:
+for epoch t in 1, . . . , T do
+5:
+Sample τ, τ ′ ∼ Uniform(0, 1) and λ ∈ Λ
+6:
+Evaluate loss (9) with (Xtrain, Ytrain)
+7:
+Update G parameters φ via SGD
+8:
+end for
+9:
+return �G
+10: end procedure
+11: procedure: Tuning λ
+12:
+Set M = 1000 and sample τ1, . . . , τM
+iid
+∼ Uniform(0, 1)
+13:
+Generate { �G(X(i)
+val, τ1, λl), . . . , �G(X(i)
+val, τM, λl)} for each λl ∈ Λ and each X(i)
+val, i = 1, . . . , nval
+14:
+Compute �P (i)
+τ,λl as in (10) on (X(i)
+val, Y (i)
+val ) for each i = 1, . . . , nval and each λl ∈ Λ
+15:
+Select λ∗ according to (11) with CvM criterion (12) as d
+16:
+return λ∗
+17: end procedure
+18: procedure: Estimating p(Y | Xtest) for test data Xtest
+19:
+Set b = 1000 and sample ξ1, . . . , ξb
+iid
+∼ Uniform(0, 1)
+20:
+Estimate ξk-th quantile of Y | Xtest as �G(Xtest, ξk, λ∗) for k = 1, . . . , b
+21:
+return { �G(Xtest, ξ1, λ∗), . . . , �G(Xtest, ξb, λ∗)}
+22: end procedure
+• GCDS (Zhou et al., 2022). Following Zhou et al. (2022), we trained both a generator and a
+discriminator using FNNs with one hidden layer of 25 neurons. Despite this simple architecture,
+we found that GCDS might still encounter vanishing variability. For fair comparison to PGQR,
+we also increased the number of hidden layers to three and the number of nodes per layer to
+1000. We refer to this modification as deep-GCDS.
+• WGCS (Liu et al., 2021). We adopted the gradient penalty recommended by Liu et al. (2021)
+and set the hyperparameter associated with the gradient penalty as 0.1.
+• FlexCoDE (Izbicki and Lee, 2017). We considered three ways of estimating the basis func-
+tions βj(X): nearest-neighbor regression (NNR), sparse additive model (SAM), and XGBoost
+(FlexZboost).
+• Random Forest CDE, or RFCDE (Pospisil and Lee, 2019).
+For FlexCoDE and RFCDE, we adopted the default hyperparameter settings of Izbicki and Lee
+(2017) and Pospisil and Lee (2019) respectively.
+19
+
+Figure 7: Plots of the estimated conditional densities of p(Y | Xtest) for three different test observations
+from one replication of Simulation 3. Plotted are the estimated conditional densities for PGQR (α = 1),
+GCDS, WGCS, and deep-GCDS.
+6.1
+Simulation Studies
+For our simulation studies, we generated data from Yi = g(Xi) + ϵi, i = 1, . . . , 2000, for some
+function g, where Xi
+iid
+∼ N(0, Ip) and the residual errors ϵi’s were independent.
+The specific
+simulation settings are described below.
+1. Simulation 1: Multimodal and heteroscedastic. Yi = βiXi + ϵi, where βi = {−1, 0, 1}
+with equal probability and ϵi = (0.25 · |Xi|)1/2.
+2. Simulation 2: Mixture of left-skewed and right-skewed. Yi = X⊤
+i β + ϵi, where β ∈ R5
+is equispaced between [−2, 2], and ϵi = χ2(1, 1)I(X1 >= 0.5) + log[χ2(1, 1)]I(X1 < 0.5). Here,
+the skewness is controlled by the covariate X1.
+3. Simulation 3: Mixture of unimodal and bimodal. Yi = X⊤
+i βi + ϵi, where βi ∈ R5 with
+βi1 ∈ {−2, 2} with equal probability, (βi2, βi3, βi4, βi5)⊤ are equispaced between [−2, 2], and
+ϵi
+iid
+∼ N(0, 1). Here, p(Y | X) is unimodal when X1 ≈ 0, and otherwise, it is bimodal.
+In our simulations, 80% of the data was used to train the model, 10% was used as the validation
+set for tuning parameter selection, and the remaining 10% was used as test data to evaluate model
+performance.
+In Section B.1 of the Appendix, we present additional results where g(Xi) is a
+nonlinear function of Xi (Simulation 4) and where the error variance is very small (Simulation 5).
+Figure 7 compares the estimated conditional density of p(Y | Xtest) for three test observations
+from one replication of Simulation 3. We see that PGQR (solid blue line with filled circles) is
+able to capture both the unimodality and the bimodality of the ground truth condtional densities
+(solid black line).
+Meanwhile, GCDS (dashed red line with hollow triangles), WGCS (dashed
+20
+
+4
+4.
+0
+T
+0
+True
+PGQR
+GCDS
+0
++
+WGCS
+deep-GCDS
+zo
+2
+0
+0
+0
+:
+9
+0
+5
+10
+-15
+-10
+15Simulation 1
+Simulation 2
+Simulation 3
+Method
+E(Y | X)
+sd(Y | X)
+Cov (Width)
+E(Y | X)
+sd(Y | X)
+Cov (Width)
+E(Y | X)
+sd(Y | X)
+Cov (Width)
+PGQR (α=1)
+0.41
+0.34
+0.95 (23.48)
+0.38
+0.11
+0.93 (8.14)
+0.30
+0.08
+0.92 (6.61)
+PGQR (α=5)
+0.36
+0.31
+0.95 (23.41)
+0.31
+0.07
+0.95 (8.83)
+0.25
+0.06
+0.96 (6.60)
+GCDS
+10.49
+25.82
+0.68 (15.48)
+0.53
+0.27
+0.92 (8.55)
+0.33
+0.12
+0.84 (5.78)
+WGCS
+229.91
+73.68
+0.15 (4.88)
+6.57
+1.45
+0.80 (9.17)
+6.25
+1.98
+0.71 (8.08)
+deep-GCDS
+7.99
+56.87
+0.38 (7.42)
+5.41
+2.89
+0.42 (2.12)
+6.11
+2.78
+0.28 (2.04)
+FlexCoDE-NNR
+0.97
+0.54
+0.96 (23.94)
+0.83
+0.36
+0.91 (9.03)
+1.12
+0.75
+0.92 (9.11)
+FlexCoDE-SAM
+0.37
+0.62
+0.97 (25.07)
+0.73
+1.03
+0.93 (11.15)
+1.01
+1.99
+0.93 (10.91)
+FlexZBoost
+0.77
+63.81
+1.00 (46.11)
+1.29
+0.36
+0.91 (8.28)
+1.77
+0.74
+0.85 (7.88)
+RFCDE
+1.98
+0.72
+0.44 (25.46)
+0.61
+0.34
+0.56 (6.26)
+0.83
+0.65
+0.96 (23.06)
+Table 1: Table reporting the PMSE for the conditional expectation and standard deviation, as well as
+the coverage rate (Cov) and average width of the 95% prediction intervals, for Simulations 1 through 3.
+Results were averaged across 20 replicates.
+green line with crosses), and deep-GCDS (dashed purple line with plusses) struggled to capture
+the true conditional densities for at least some test points. In particular, Figure 7 shows some
+evidence of variance underestimation for WGCS and deep-GCDS, whereas this is counteracted by
+the variability penalty in PGQR. Additional figures from our simulation studies are provided in
+Section B.2 of the Appendix.
+We repeated our simulations for 20 replications. For each experiment, we computed the pre-
+dicted mean squared error (PMSE) for different summaries of the conditional densities for the test
+data. We define the PMSE as
+PMSE =
+1
+ntest
+ntest
+�
+i=1
+( �m(Xtest,i) − m(Xtest,i))2 ,
+(13)
+where m(X) generically refers to the conditional mean of Y given X, i.e.
+E(Y | X), or the
+conditional standard deviation, i.e. sd(Y | X). For the generative models (PGQR, GCDS, WGCS,
+and deep-GCDS), we approximated �m in (13) by Monte Carlo simulation using 1000 generated
+samples, while for FlexCoDE and RFCDE, we approximated �m using numerical integration. In
+addition to PMSE, we also used the 2.5th and 97.5th percentiles of the predicted conditional
+densities to construct 95% prediction intervals for the test data. We then calculated the coverage
+probability (Cov) and the average width for these prediction intervals.
+Table 1 summarizes the results for PGQR with α ∈ {1, 5} in (5) and all competing methods,
+averaged across 20 experiments. There was not much substantial difference between α = 1 and
+α = 5 for PGQR. Table 1 shows that PGQR had the lowest PMSE in all three simulations
+21
+
+Figure 8:
+Plot of the average PMSE across 20 replicates for the τ-th quantiles QY |X(τ), τ
+∈
+{0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7, 0.8, 0.9} for Simulations 1 through 3.
+and attained coverage close to the nominal rate. In Simulations 2 and 3, the prediction intervals
+produced by PGQR had the highest coverage rate. In Simulation 1, FlexZBoost had 100% coverage,
+but the average width of the prediction intervals for FlexZBoost was considerably larger than that of
+the other methods, suggesting that the intervals produced by FlexZBoost may be too conservative
+to be informative.
+We also computed the PMSE for the τ-th quantile. Figure 8 plots the PMSE for these nine
+quantiles for PGQR (α = 1), GCDS, deep-GCDS, and WGCS, averaged across 20 experiments.
+We see that PGQR (blue line with crosses) had the lowest PMSE for most of the quantile levels.
+PGQR also had uniformly lower PMSE in our quantile set than WGCS (green line with plusses)
+and deep-GCDS (red line with triangles), suggesting that WGCS and deep-GCDS may have been
+more prone to vanishing variability.
+6.2
+Real Data Analysis
+We also examined the performance of PGQR on three real datasets from the UCI Machine Learning
+Repository, which we denote as: machine, fish, and noise.1 The machine dataset comes from a
+real experiment that collected the excitation current (Y ) and four machine attributes (X) for a set
+of synchronous motors (Kahraman, 2014). The fish dataset contains the concentration of aquatic
+toxicity (Y ) that can cause death in fathead minnows and six molecular descriptors (X) described
+in (Cassotti et al., 2015). The noise dataset measures the scaled sound pressure in decibels (Y )
+1Accessed from https://archive.ics.uci.edu/ml/index.php.
+22
+
+Simulation 1
+Simulation 2
+simulation 3
+5
+PGQR
+GCDS
+WGCS
+ deep-GCDS
+T
+5
+5 -
+0 :
+0 :
+0.1
+0.2
+0.3
+0.5
+0.6
+0.7
+8.0
+0.9
+0.1
+0.2
+0.3
+0.4
+0.5
+0.6
+0.8
+0.9
+0.10.20.3
+0.6
+0.70.80.9
+Quantile
+Quantile
+QuantilePGQR
+GCDS
+deep-GCDS
+WGCS
+Dataset
+n
+p
+Cov
+Width
+Cov
+Width
+Cov
+Width
+Cov
+Width
+machine
+557
+4
+0.96
+2.82
+0.91
+1.92
+0.87
+1.52
+0.69
+1.66
+fish
+908
+6
+0.96
+3.29
+0.79
+2.38
+0.54
+1.08
+0.80
+2.36
+noise
+1503
+5
+0.92
+7.58
+0.00
+1.68
+0.00
+0.24
+0.00
+22.41
+Table 2: Results from our real data analysis. Cov and width denote the coverage rate and average width
+respectively of the 95% prediction intervals for the test observations.
+at different frequencies, angles of attacks, wind speed, chord length, and suction side displacement
+thickness (X) for a set of airfoils (Lopez et al., 2008). Table 2 reports the sample size n and
+covariate dimension p for these three benchmark datasets.
+We examined the out-of-sample performance for PGQR (with fixed α = 1), GCDS, deep-GCDS,
+and WGCS. In particular, 80% of each dataset was randomly selected as training data, 10% was
+used as validation data for tuning parameter selection, and the remaining 10% was used as test data
+for model evaluation. To compare these deep generative methods, we considered the out-of-sample
+coverage rate (Cov) and the average width of the 95% prediction intervals in the test data.
+The results from our real data analysis are summarized in Table 2. We see that on all three
+datasets, PGQR achieved higher coverage that was closer to the nominal rate than the competing
+methods. It seems as though the other generative approaches may have been impacted by the
+vanishing variability phenomenon, resulting in too narrow prediction intervals that did not cover as
+many test samples. In particular, GCDS, deep-GCDS, and WGCS all performed extremely poorly
+on the noise dataset, with an out-of-sample coverage rate of zero. On the other hand, with the
+help of the variability penalty (5), PGQR did not underestimate the variance and demonstrated
+an overwhelming advantage over these other methods in terms of predictive power. Moreover,
+the average widths of the PGQR prediction intervals were not overwhelmingly large so as to be
+uninformative.
+Figure 9 plots the PGQR 95% prediction intervals for the test observations ordered in ascending
+order. The red crosses depict test samples that were not captured by their corresponding PGQR
+prediction intervals. We do not detect any specific pattern for the uncaptured test points, indicating
+reasonable generalizability for the PQGR model.
+23
+
+Figure 9: The PGQR 95% prediction intervals for the test samples in the three benchmark datasets. The
+red crosses indicate the test points that were not captured by their corresponding prediction intervals.
+7
+Discussion
+In this paper, we have made contributions to both the quantile regression and deep learning lit-
+erature. Specifically, we proposed PGQR as a new deep generative approach to joint quantile
+estimation and conditional density estimation. Different from existing conditional sampling meth-
+ods (Zhou et al., 2022; Liu et al., 2021), PGQR employs a novel variability penalty to counteract
+the vanishing variability phenomenon in deep generative networks. We introduced the PMNN
+family of neural networks to enforce the monotonicity of quantile functions. Finally, we provided
+a scalable implementation of PGQR which requires solving only a single optimization to select the
+regularization term in the variability penalty. Through analyses of real and simulated datasets,
+we demonstrated PGQR’s ability to capture critical aspects of the conditional distribution such as
+multimodality, heteroscedasticity, and skewness.
+We anticipate that our penalty on vanishing variability in generative networks is broadly ap-
+plicable for a wide number of loss functions besides the check function. In the future, we will
+extend the variability penalty to other deep generative models and other statistical problems be-
+sides quantile regression. In addition, we plan to pursue variable selection, so that our method
+can also identify the most relevant covariates when the number of features p is large. Owing to its
+single-model training, PGQR is scalable for large n. However, further improvements are needed in
+order for PGQR to avoid the curse of dimensionality for large p.
+24
+
+machine
+fish
+noise
+10
+20
+30
+40
+50
+20
+40
+60
+80
+50
+100
+150Acknowledgments
+The last author was generously supported by NSF grant DMS-2015528. The authors are grateful
+to Dr. Jun Liu for helpful comments.
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+A
+Proofs of Propositions
+Proof of Proposition 3.1. Suppose that for some ϵ > 0, there exists a set C ⊂ (0, 1) with
+Pτ(C) > ϵ such that for some i∗ ∈ {1, . . . , n}, �gτ(Xi∗) ̸= �G(Xi∗, τ) for all τ ∈ C. Then we can
+construct another optimal generator �G such that
+1
+n
+n
+�
+i=1
+Eτ
+�
+ρτ
+�
+yi − �G(Xi, τ)
+��
++ E�τ,�τ ′
+�
+penλ,α
+�
+�G(Xi, �τ), �G(Xi, �τ ′)
+��
+≥
+1
+n
+n
+�
+i=1
+Eτ
+�
+ρτ
+�
+yi − �G(Xi, τ)
+��
++ E�τ,�τ ′
+�
+penλ,α
+�
+�G(Xi, �τ), �G(Xi, �τ ′)
+��
+,
+where
+�G(Xi∗, τ) =
+�
+�
+�
+�
+�
+�G(Xi∗, τ) for τ ̸∈ C,
+�gτ(Xi∗) for τ ∈ C.
+This is a contradiction due to the fact that �gτ is the minimizer of
+1
+n
+�n
+i=1 ρτ(yi − �G(Xi, τ)) +
+E�τ,�τ ′{penλ,α( �G(Xi, �τ), �G(Xi, �τ ′))}.
+Proof of Proposition 3.2. Suppose that for all λ ≥ 0, all τ ∈ (0, 1), and all i ∈ {1, . . . , n},
+�G(Xi, τ) = Yi.
+27
+
+Then the penalty part in the PGQR loss (6) attains Eτ,τ ′{penλ,α( �G(Xi, τ), �G(Xi, τ ′))} = λ log(α),
+while the first term in (6) satisfies Eτ[ρτ
+�
+yi − �G(Xi, τ)
+�
+] = 0. As a result, the total loss is λ log(α).
+Since the case of �G(Xi, τ) = Yi is included in cases of Varτ{ �G(Xi, τ)} = 0, we focus on the
+variance. When there exists some i ∈ {1, . . . , n} and some τ ∈ (0, 1) such that
+Varτ
+�
+�G(Xi, τ)
+�
+> 0,
+the resulting total loss can be made less than λ log(α) by choosing an appropriate λ > 0. This
+contradicts the fact that �G is the minimizer as in (6).
+Proof of Proposition 5.2. It is trivial that FQ(W) := P(Q ≤ W | W) ∼ Uniform(0, 1) when
+Q
+d= W. WLOG, we assume that FQ is invertible. We shall show that FQ(W) ∼ Uniform(0, 1)
+implies that the two distributions for Q and W are identical.
+Suppose that FQ(W) follows a
+standard uniform distribution. Then,
+x = P(FQ(W) < x) = P(W < F −1
+Q (x)) = FQ(F −1
+Q (x)),
+which implies that FW(F −1
+W (x)) = x.
+Thus, it follows that the distributions of Q and W are
+identical.
+B
+More Simulation Results
+B.1
+Additional Simulation Studies
+In addition to the three simulations described in Section 6 of the main manuscript, we also con-
+ducted simulation studies under the following scenarios:
+• Simulation 4: Nonlinear function with an interaction term and one irrelevant
+covariate. Yi = 0.5 log(10−X2
+i1)+0.75 exp(Xi2X3/5)−0.25|Xi4/2|+ϵi, where ϵi ∼ N(0, 1).
+Note that there is a (nonlinear) interaction between X2 and X3, while X5 is irrelevant.
+• Simulation 5: Very small conditional variance. Yi = βXi+ϵi, i = 1, . . . , n, where β = 1
+and ϵi
+iid
+∼ N(0, 0.01).
+28
+
+Simulation 4
+Simulation 5
+Method
+E(Y | X)
+sd(Y | X)
+Cov (Width)
+E(Y | X)
+sd(Y | X)
+Cov (Width)
+PGQR (α = 1)
+0.15
+0.07
+0.95 (4.33)
+0.004
+0.0001
+0.93 (0.42)
+PGQR (α = 5)
+0.14
+0.08
+0.96 (4.50)
+0.005
+0.03
+0.99 (1.01)
+GCDS
+0.23
+0.05
+0.87 (3.39)
+0.002
+0.0011
+0.73 (0.27)
+deep-GCDS
+0.43
+0.20
+0.76 (2.88)
+0.004
+0.0022
+0.99 (0.56)
+WGCS
+0.92
+0.15
+0.79 (4.41)
+0.640
+0.2372
+0.70 (1.15)
+FlexCoDE-NNR
+0.23
+0.003
+0.92 (3.82)
+0.069
+0.0168
+0.89 (0.27)
+FlexCoDE-SAM
+0.23
+0.002
+0.93 (3.83)
+0.043
+0.0130
+0.93 (4.04)
+FlexZBoost
+0.45
+0.06
+0.82 (3.50)
+1.244
+0.2824
+0.81 (4.13)
+RFCDE
+0.24
+0.003
+0.94 (3.97)
+0.067
+0.0292
+0.92 (3.97)
+Table 3: Table reporting the PMSE for the conditional expectation and standard deviation, as well as the
+coverage rate (Cov) and average width of the 95% prediction intervals, for Simulations 4 and 5. Results
+were averaged across 20 replicates.
+The results from these two simulations averaged across 20 replicates are shown below in Table 3.
+The average PMSE for the τ-th quantile levels QY |X(τ), τ ∈ {0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7, 0.8, 0.9}
+are plotted in Figure 10.
+Figure 10:
+Plot of the average PMSE across 20 replicates for the τ-th quantiles QY |X(τ), τ
+∈
+{0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7, 0.8, 0.9} for Simulations 4 and 5. In Simulation 5, the PMSE for WGCS
+is not displayed, as it lies outside of the plot area (i.e. the average PMSE for WGCS is much larger than
+that of the other methods).
+One may be concerned whether the regularized PGQR overestimates the conditional variance
+when the true conditional density has a very small variance. To illustrate the flexibility of PGQR,
+Figure 11 plots the estimated conditional densities for three test observations from one replication
+of Simulation 5. Recall that in Simulation 5, the true conditional variance is very small (σ2 = 0.01).
+With the optimal λ⋆ selected using the method introduced in Section 5.2, Figure 11 shows that the
+estimated PGQR conditional density still manages to capture the Gaussian shape while matching
+29
+
+Simulation 4
+Simulation 5
+0.010
+5.
+PGQR
+GCDS
+PMSE
+0.008
+WGCS
+deep-GCDS
+900'0
+ntile
+0.004
+Quar
+0.5
+0.002
+0.000
+0'0
+0.1
+0.2
+0.3
+0.4
+0.5
+0.6
+0.7
+0.8
+0.9
+0.1
+0.2
+0.3
+0.4
+0.5
+0.6
+0.7
+0.8
+0.9
+Quantile
+QuantileFigure 11: Plots of the estimated PGQR (α = 1) conditional densities for three different test observations
+in Simulation 5. The optimal λ⋆ is chosen by our tuning parameter selection method in Section 5.2.
+the true variance of 0.01. If the true conditional variance is very small (as in Simulation 5), then
+PGQR selects a tiny λ⋆ ≈ 0. In this scenario, PGQR only applies a small amount of variability
+penalization and thus does not overestimate the variance.
+B.2
+Additional Figures
+Here, we provide additional figures from one replication each of Simulations 1, 2, and 4 (with α = 1
+in PGQR). Figures 12-14 illustrate that PGQR (blue solid line with filled circles) is better able
+to estimate the true conditional densities (solid black line) than GCDS, WGCS, and deep-GCDS
+(dashed lines). In particular, PGQR does a better job of capturing critical aspects of the true
+conditional distributions such as multimodality, heteroscedasticity, and skewness.
+• Simulation 1: Multimodal and heteroscedastic.
+Figure 12: Plots of the estimated conditional densities of p(Y | Xtest) for three different test observations
+from one replication of Simulation 1.
+30
+
+15 
+1
+True
+:PGQR
+4
+2:
+2-
+2-
+0
+0
+-3.0
+-2.5
+-2.0
+-1.5
+-1.0
+-0.5
+0.0
+-3.0
+-2.5
+-2.0
+-1.5
+-1.0
+-0.5
+0.0
+-3.0
+-2.5
+-2.0
+-1.5
+-1.0
+-0.5
+0.02
+2
+True
+PGQR
+ -
+GCDS
+.
+WGCS.
+0
+deep-GCDS
+N
+4
+0
+8
+8
+0
+-30
+-20 
+10
+10
+20
+30
+09-
+-40
+-20
+20
+09• Simulation 2: Mixture of left-skewed and right-skewed.
+Figure 13: Plots of the estimated conditional densities of p(Y | Xtest) for three different test observations
+from one replication of Simulation 2.
+• Simulation 4: Nonlinear function with an interaction term and one irrelevant co-
+variate.
+Figure 14: Plots of the estimated conditional densities of p(Y | Xtest) for three different test observations
+from one replication of Simulation 4.
+C
+Sensitivity Analysis of PGQR to the Choice of α
+As mentioned in Section 3.2 of the main manuscript, we choose to fix α > 0 in the variability
+penalty (5). The main purpose of α is to ensure that the logarithmic term in the penalty is always
+well-defined.
+In this section, we conduct a sensitivity analysis to the choice of α. To do this, we generated
+data using the same settings from Simulations 1 through 5. We then fit the PGQR model with
+five different choices for α ∈ {0.5, 1, 5, 10, 50} and evaluated the performance of these five PGQR
+models on out-of-sample test data. Table 4 shows the results from our sensitivity analysis averaged
+across 20 replicates.
+31
+
+4.
+1:
+4.
+4
+0
+0
+0
+True
+PGQR
+GCDS
+0
+0
++
+deep-GCDS
+zo
+2
+0
+:
+6
+5
+0
+0
+0.
+9
+-10
+-5
+5
+10
+15
+20
+-20
+-15
+10
+-5
+10
+-20
+-10
+0
+10
+206.
+0
+11
+0
+True
+90
+90
+90
+PGQR
+GCDS
+to
+0.4
++
+deep-GCDS
+2
+.
+0
+0
+0.1
+6
+0.
+-5
+10
+5
+10
+-5
+-
+10We found that in Simulations 1 through 4, PGQR was not particularly sensitive to the choice
+of α.
+PGQR was somewhat more sensitive to the choice of α in Simulation 5 (i.e.
+when the
+true conditional variance is very small), with larger values of α leading to higher PMSE for the
+conditional expectation and conditional standard deviation.
+In practice, we recommend fixing
+α = 1 as the default α for PGQR to perform well. This choice of α = 1 leads to good empirical
+performance across many different scenarios.
+32
+
+Simulation 1
+Simulation 2
+Simulation 3
+Simulation 4
+Simulation 5
+α
+E(Y | X)
+sd(Y | X)
+Cov (Width)
+E(Y | X)
+sd(Y | X)
+Cov (Width)
+E(Y | X)
+sd(Y | X)
+Cov (Width)
+E(Y | X)
+sd(Y | X)
+Cov (Width)
+E(Y | X)
+sd(Y | X)
+Cov (Width)
+0.5
+0.39
+0.41
+0.95 (23.60)
+0.43
+0.18
+0.91 (7.69)
+0.36
+0.09
+0.89 (5.93)
+0.14
+0.01
+0.96 (3.95)
+0.004
+0.0001
+0.92 (0.38)
+1
+0.42
+0.34
+0.95 (23.49)
+0.38
+0.11
+0.93 (8.14)
+0.30
+0.09
+0.92 (6.61)
+0.15
+0.07
+0.95 (4.33)
+0.004
+0.0001
+0.93 (0.42)
+5
+0.36
+0.31
+0.95 (23.41)
+0.31
+0.07
+0.94 (8.83)
+0.25
+0.06
+0.96 (6.60)
+0.14
+0.08
+0.96 (4.50)
+0.005
+0.03
+0.99 (1.01)
+10
+0.32
+0.30
+0.95 (23.40)
+0.29
+0.06
+0.95 (9.02)
+0.25
+0.07
+0.95 (6.55)
+0.13
+0.08
+0.95 (4.39)
+0.007
+0.06
+0.99 (1.33)
+50
+0.33
+0.34
+0.97 (24.15)
+0.29
+0.07
+0.95 (9.15)
+0.25
+0.06
+0.95 (6.63)
+0.15
+0.14
+0.94 (4.42)
+0.009
+0.101
+0.99 (1.81)
+Table 4: PGQR results with different choices of α ∈ {0.5, 1.0, 5.0, 10.0, 50.0} in the variability penalty for Simulations 1 through 5.
+This table reports the average PMSE for the conditional expectation and standard deviation, as well as the coverage rate (Cov) and the
+average width of the 95% prediction intervals. Results were averaged across 20 replicates.
+33
+
diff --git a/fdE2T4oBgHgl3EQfGwaV/content/tmp_files/load_file.txt b/fdE2T4oBgHgl3EQfGwaV/content/tmp_files/load_file.txt
new file mode 100644
index 0000000000000000000000000000000000000000..cf8f7e0cd74f8f59357c9b3582f0c67ffecfdd87
--- /dev/null
+++ b/fdE2T4oBgHgl3EQfGwaV/content/tmp_files/load_file.txt
@@ -0,0 +1,1478 @@
+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf,len=1477
+page_content='Generative Quantile Regression with  Variability Penalty  Shijie Wang1, Minsuk Shin2, and Ray Bai1  1Department of Statistics, University of South Carolina, Columbia, SC 29208  E-mails: shijiew@email.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content='sc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content='edu, rbai@mailbox.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content='sc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content='edu  2Gauss Labs, Palo Alto, CA 94301  Abstract  Quantile regression and conditional density estimation can reveal structure that is missed  by mean regression, such as multimodality and skewness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content=' In this paper, we introduce a deep  learning generative model for joint quantile estimation called Penalized Generative Quantile  Regression (PGQR).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content=' Our approach simultaneously generates samples from many random  quantile levels, allowing us to infer the conditional distribution of a response variable given  a set of covariates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content=' Our method employs a novel variability penalty to avoid the problem of  vanishing variability, or memorization, in deep generative models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content=' Further, we introduce a new  family of partial monotonic neural networks (PMNN) to circumvent the problem of crossing  quantile curves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content=' A major benefit of PGQR is that it can be fit using a single optimization,  thus bypassing the need to repeatedly train the model at multiple quantile levels or use  computationally expensive cross-validation to tune the penalty parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content=' We illustrate the  efficacy of PGQR through extensive simulation studies and analysis of real datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content=' Code to  implement our method is available at https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content='com/shijiew97/PGQR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content='  Keywords: conditional quantile, deep generative model, generative learning, joint quantile model,  neural networks, nonparametric quantile regression  1  Introduction  Quantile regression is a popular alternative to classical mean regression (Koenker and Bassett Jr,  1982).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content=' For a response variable Y ∈ R and covariates X = (X1, X2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content=' , Xp)⊤ ∈ Rp, we define the  τ-th conditional quantile, τ ∈ (0, 1), as the τ-th quantile of the conditional distribution of Y given  X, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content='  QY |X(τ) = inf{y : FY |X(y) ≥ τ},  (1)  where FY |X is the conditional cumulative distribution function of Y given X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content=' For a fixed quantile  level τ ∈ (0, 1), linear quantile regression aims to model (1) as a linear combination of X, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content='  1  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content='03661v1  [stat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content='ME]  9 Jan 2023  QY |X(τ) = X⊤βτ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content=' Linear quantile regression is less sensitive to outliers than least squares regres-  sion and is more robust when the assumption of iid Gaussian residual errors is violated (Koenker  and Hallock, 2001;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content=' Koenker et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content=', 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content=' Thus, it is widely used for modeling heterogeneous data  and heterogeneous covariate-response associations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content='  Numerous extensions and variants of quantile regression have been proposed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content='  In order to  reveal complex nonlinear structures, nonparametric quantile regression is frequently considered  (Chaudhuri and Loh, 2002;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content=' Li et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content=', 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content=' Koenker et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content=', 1994).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content='  For some function class F,  nonparametric quantile regression aims to estimate f(X, τ) ∈ F for QY |X(τ) = f(X, τ) at a  given quantile level τ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content=' Many existing nonparametric quantile methods are based on either kernel  smoothing or basis expansions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content=' While quantile regression has traditionally focused on estimating  the conditional quantile function at a single τ-th quantile level, composite (or joint) quantile  regression aims to estimate multiple quantile levels simultaneously for a set of K ≥ 2 quantiles  {τ1, τ2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content=' , τK}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content=' Some nonparametric methods for composite quantile regression are proposed in  Zou and Yuan (2008), Jiang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content=' (2012b), Jiang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content=' (2012a), and Xu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content=' (2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content='  Despite its flexibility, nonparametric quantile regression carries the risk that the estimated  curves at different quantile levels might cross each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content=' For instance, the estimate of the 95th  conditional quantile of Yi given some covariates Xi might be smaller than the estimate of the 90th  conditional quantile.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content=' This is known as the crossing quantile phenomenon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content=' The left two panels  of Figure 4 illustrate this crossing problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content=' When quantile curves cross each other, the quantile  estimates violate the laws of probability and are no longer reasonable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content=' To tackle this issue, many  approaches have been proposed, such as constraining the model space (Takeuchi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content=', 2006;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content='  Sangnier et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content=', 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content=' Moon et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content=', 2021) or the model parameters (Meinshausen, 2006;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content=' Cannon,  2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content='  Recently in the area of deep learning, neural networks have also been applied to nonparametric  quantile regression.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content=' Taylor (2000) used feedforward neural networks (FNNs) to estimate condi-  tional quantiles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content=' Takeuchi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content=' (2006) considered support vector machines in combination with  quantile regression.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content=' Neural networks have also been used for composite quantile regression by Xu  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content=' (2017) and Jin and Zhao (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content=' Existing neural network-based approaches for joint quantile  estimation are restricted at a prespecified quantile set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content=' If only several quantiles (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content=' the interquar-  tiles τ ∈ {0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content='25, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content='5, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content='75}) are of interest, then these methods are adequate to produce desirable  2  inference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content=' Otherwise, one typically has to refit the model at new quantiles or specify a large enough  quantile candidate set in order to infer the full conditional density p(Y | X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content=' Dabney et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content=' (2018)  introduced the implicit quantile network (IQN), which takes a grid of quantile levels as inputs and  approximates the conditional density at any quantile level (not just the given inputs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content=' However,  IQN cannot guarantee that quantile functions at different levels do not cross each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content='  Composite quantile regression is closely related to the problem of conditional density estimation  (CDE) of p(Y | X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content=' Apart from traditional CDE methods that estimate the unknown probability  density curve (Izbicki and Lee, 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content=' Pospisil and Lee, 2019), several deep generative approaches  besides IQN have also been proposed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content=' Zhou et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content=' (2022) introduced the generative conditional  distribution sampler (GCDS), while Liu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content=' (2021) introduced the Wasserstein generative con-  ditional sampler (WGCS) for CDE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content=' GCDS and WGCS employ the idea of generative adversarial  networks (GANs) (Goodfellow et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content=', 2014) to generate samples from the conditional distribution  p(Y | X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content=' In these deep generative models, random noise inputs are used to learn and generate  samples from the target distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content=' However, a well-known problem with deep generative net-  works is that the generator may simply memorize the training samples instead of generalizing to  new test data (Arpit et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content=', 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content=' van den Burg and Williams, 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content=' When memorization occurs,  the random noise variable no longer reflects any variability, and the predicted density p(Y | X)  approaches a point mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content=' We refer to this phenomenon as vanishing variability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content='  To achieve the goal of joint quantile estimation as well as conditional density estimation, we  propose a new deep generative approach called penalized generative quantile regression (PGQR).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content='  PGQR employs a novel variability penalty to avoid vanishing variability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content=' Although introduced in  the context of joint quantile regression, we believe that our penalty formulation is broadly applicable  to other deep generative models that may suffer from memorization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content=' We further guarantee the  monotonicity (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content='  non-crossing) of the estimated quantile functions from PGQR by designing  a new family of partial monotonic neural networks (PMNNs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content=' The PMNN architecture ensures  partial monotonicity with respect to quantile inputs, while retaining the expressiveness of neural  networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content='  The performance of PGQR depends crucially on carefully choosing a hyperparameter λ, which  controls the amount of regularization by our variability penalty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content=' However, common tuning proce-  dures such as cross-validation are impractical for deep learning, since this would involve repetitive  3  evaluations of the network for different choices of λ and different training sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content=' Inspired by the  generative idea from Shin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content=' (2022), we construct our deep generative network in such a way  that λ is included as an additional random input.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content=' This way, only a single optimization is needed  to learn the neural network parameters, and then it is effortless to generate samples from a set of  candidate values for λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content=' Finally, we introduce a criterion for selecting the optimal choice of λ which  we use to generate the final desired samples from multiple quantiles of p(Y | X) simultaneously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content='  Our main contributions can be summarized as follows:  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content=' We propose a deep generative approach for composite quantile regression and conditional  density estimation called penalized generative quantile regression.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content='  PGQR simultaneously  generates samples from multiple random quantile levels, thus precluding the need to refit the  model at different quantiles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content=' We introduce a novel variability penalty to avoid the vanishing variability phenomenon in  deep generative models and apply this regularization technique to PGQR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content=' We construct a new family of partial monotonic neural networks (PMNNs) to circumvent the  problem of crossing quantile curves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content=' To facilitate scalable computation for PGQR and bypass computationally expensive cross-  validation for tuning the penalty parameter, we devise a strategy that allows PGQR to be  implemented using only a single optimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content='  The rest of the article is structured as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content=' Section 2 motivates our PGQR framework with a  real data application on body composition and strength in older adults.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content=' Section 3 introduces the  generative quantile regression framework and our variability penalty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content=' In Section 4, we introduce  the PMNN family for preventing quantile curves from crossing each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content=' Section 5 discusses  scalable computation for PGQR, namely how to tune the variability penalty parameter with only  single-model training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content=' In Section 6, we demonstrate the utility of PGQR through simulation studies  and analyses of additional real datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content=' Section 7 concludes the paper with some discussion and  directions for future research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content='  4  Figure 1: Using PGQR to model the conditional density of AP-L/F ratio given weight in older adults.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content='  Left panel: Weight = 45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content='4 kg, Right panel: Weight = 77.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content='5 kg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content='  2  Motivating Application: Discovering Subpopulations  As discussed in Section 1, we aim to generate samples from the conditional quantiles QY |X(τ) of  p(Y | X) at different quantile levels τ ∈ (0, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content=' An automatic byproduct of joint nonparametric  quantile regression (as opposed to linear quantile regression) is that if the conditional quantiles  QY |X(τ) are estimated well for a large number of quantiles, then we can also infer the entire  conditional distribution for Y given X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content='  To motivate our methodology, we apply the proposed PGQR method (introduced in Section 3)  to a dataset on body composition and strength in older adults (RoyChoudhury and Xu, 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content=' The  data was collected over a period of 12 years for 1466 subjects as part of the Rancho Bernardo Study  (RBS), a longitudinal observational cohort study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content=' We are interested in modeling the appendicular  lean/fat (AP-L/F) ratio, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content='  AP-L/F Ratio = Weight on legs and arms  Fat weight  ,  as a function of weight (kg).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content=' Accurately predicting the AP-L/F ratio is of practical clinical interest,  since the AP-L/F ratio provides information about limb tissue quality and is used to diagnose  sarcopenia (age-related, involuntary loss of skeletal muscle mass and strength) in adults over the  age of 30 (Evans, 2010;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content=' Scafoglieri et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content=', 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content='  5  Rati  Rati  []  [  AP-L/  40  60  80  100  120  140  40  60  80  100  120  140  Weight  WeightFigure 1 plots the approximated conditional density of AP-L/F ratio given weight of 45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content='4 kg (left  panel) and 77.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content='5 kg (right panel) under the PGQR model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content=' We see evidence of data heterogeneity  (actually, depending on an unobserved factor of gender), as the estimated conditional density is  unimodal when the weight of older adults is 45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content='4 kg but bimodal when the weight of older adults  is 77.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content='5 kg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content=' In short, our method discovers the presence of two heterogeneous subpopulations of  adults that weigh around 78 kg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content=' In contrast, mean regression (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content=' simple linear regression or  nonparametric mean regression) of AP-L/F ratio given weight might obscure the presence of two  modes and miss the fact that weight affects AP-L/F ratio differently for these two clusters of adults.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content='  3  Penalized Generative Quantile Regression  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content='1  Generative Quantile Regression  Before introducing PGQR, we first introduce our framework for generative quantile regression  (without variability penalty).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content=' Given n training samples (X1, Y1), (X2, Y2), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content=' , (Xn, Yn) and quan-  tile level τ ∈ (0, 1), nonparametric quantile regression minimizes the empirical quantile loss,  argmin  f∈F  1  n  n  �  i=1  ρτ (Yi − f(Xi, τ)) ,  (2)  where ρτ(u) = u(τ −I(u < 0)) is the check function, and F is a function class such as a reproducing  kernel Hilbert space or a family of neural networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content=' Neural networks are a particularly attractive  way to model the quantile function f(X, τ) in (2), because they are universal approximators for  any Lebesgue integrable function (Lu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content=', 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content='  In this paper, we choose to model the quantile function in (2) using deep neural networks  (DNNs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content=' We define a DNN as a neural network with at least two hidden layers and a fairly large  number of nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content=' We refer to Emmert-Streib et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content=' (2020) for a comprehensive review of DNNs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content='  Despite the universal approximation properties of DNNs, we must also take care to ensure that  our estimated quantile functions do not cross each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content=' Therefore, we have to consider a wide  enough monotonic function class Gm to cover the true QY |X(·).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content=' We formally introduce this class  Gm in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content='  Let G denote the constructed DNN, which is defined as a feature map function {G ∈ Gm :  6  Rp+1 �→ R1} that takes (Xi, τ) as input and generates the conditional quantile for Yi given Xi at  level τ as output.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content=' We refer to this generative framework as Generative Quantile Regression (GQR).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content='  The optimization problem for GQR is  �G = argmin  G∈Gm  1  n  n  �  i=1  Eτ  �  ρτ  �  Yi − G(Xi, τ)  ��  ,  (3)  where �G denotes the estimated quantile function with optimized parameters (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content=' the weights and  biases) of the DNN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content=' Note that optimizing the integrative loss Eτ{·} over τ in (3) is justified by  Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content='1 introduced in the next section (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content=' we set λ = 0 in Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content='  In order to solve (3), we can use stochastic gradient descent (SGD) with mini-batching (Emmert-  Streib et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content=', 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content=' For each mini-batch evaluation, Xi is paired with a quantile level τ sampled  from Uniform(0, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content=' Consequently, if there are M mini-batches, the expectation in (3) can be ap-  proximated by a Monte Carlo average of the random τ’s, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content=' we approximate Eτ  �  ρτ  �  Yi−G(Xi, τ)  ��  with M −1 �M  k=1  �  ρτk  �  Yi − G(Xi, τk)  ��  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content=' Once we have solved (3), it is straightforward to use �G  to generate new samples �G(X, ξk), k = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content=' , b, at various quantile levels ξ = {ξ1, ξ2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content=' , ξb},  where the ξ’s are random Uniform(0, 1) noise inputs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content=' Provided that b is large enough, the generated  samples at ξ ∈ (0, 1)b can be used to reconstruct the full conditional density p(Y | X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content='  As discussed in Section 1, there are several other deep generative models (Zhou et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content=', 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content=' Liu  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content=', 2021) for generating samples from the underlying conditional distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content=' These generative  approaches also take random noise z as an input (typically z ∼ N(0, 1)) to reflect variability, but  there is no specific statistical meaning for z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content=' In contrast, the random noise τ in GQR (3) has a  clear interpretation as a quantile level τ ∈ (0, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content=' Although GQR and these other deep generative  approaches are promising approaches for conditional sampling, these methods are all unfortunately  prone to memorization of the training data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content=' When this occurs, the random noise does not generate  any variability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content=' To remedy this, we now introduce our variability penalty in conjunction with our  GQR loss function (3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content='2  Variability Penalty for GQR  When a DNN is too overparameterized to capture the underlying structure in the training data,  the random noise in DNN is likely to reflect no variability, no matter what value it inputs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content=' We refer  7  this phenomenon as vanishing variability, and it is a common problem in deep generative models  (Arpit et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content=', 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content=' Arora et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content=', 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content=' van den Burg and Williams, 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content=' To be more specific,  let G(·, z) be the generator function constructed by a DNN, where z is a random noise variable  following some reference distribution such as a standard Gaussian or a standard uniform.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content=' The  vanishing variability phenomenon occurs when  �G(Xi, z) = Yi, i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content=', n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content='  (4)  In other words, there is no variability when inputting different random noise z, generating almost  surely a discrete point mass at the training data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content=' Additionally, given a new feature vector Xnew,  G(Xnew, z) can only generate one novel sample from the true data distribution because of vanishing  variability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content='  Since GQR takes the training data X ∈ Rp as input and the target quantile estimate lies in  R1, the weights in the DNN associated with X ∈ Rp are very likely to overwhelm those associated  with the noise input τ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content=' As a result, GQR is very prone to encountering vanishing variability, as  are other generative approaches with multidimensional features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content=' From another point of view, we  can see that due to the nonnegativity of the check function, the GQR loss (3) achieves a minimum  value of zero when we have vanishing variability (4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content='  To remedy this problem, we propose a new regularization term that encourages the network  to have more variability when vanishing variability occurs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content=' Given a set of features X ∈ Rp, the  proposed variability penalty is formulated as follows:  penλ,α(G(X, τ), G(X, τ ′)) = −λ log {∥G(X, τ) − G(X, τ ′)∥1 + 1/α} ,  (5)  where λ ≥ 0 is the hyperparameter controlling the degree of penalization, α > 0 is a fixed hyper-  parameter, and τ, τ ′ ∼ Uniform(0, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content=' The addition of 1/α inside the logarithmic term of (5) is  mainly to ensure that the penalty function is always well-defined (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content=' the quantity inside of log{·}  of (5) cannot equal zero).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content=' Our sensitivity analysis in Section C of the Appendix shows that our  method is not usually sensitive to the choice of α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content=' We find that fixing α = 1 or α = 5 works well  in practice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content='  With the addition of the variability penalty (5) to our GQR loss function (3), our penalized  8  GQR (PGQR) method solves the optimization,  �G = argmin  G∈Gm  1  n  n  �  i=1  �  Eτ  �  ρτ  �  Yi − G(Xi, τ)  ��  + Eτ,τ ′ �  penλ,α (G(Xi, τ), G(Xi, τ ′))  ��  ,  (6)  where Gm denotes the PMNN family introduced in Section 4, and τ and τ ′ independently follow  a uniform distribution on (0, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content=' The expectation Eτ,τ ′ is again approximated by a Monte Carlo  average.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content=' Note that when λ = 0, the PGQR loss (6) reduces to the (non-penalized) GQR loss (3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content='  The next proposition states that an estimated generator function �G(X, τ) under the PGQR  objective (6) is equivalent to a neural network estimator �gτ(X) based on the individual (non-  integrative) quantile loss function, provided that the family Gm in (6) is large enough.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content='  Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content='1 (equivalence of �gτ(X) and �G(X, τ)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content=' For fixed λ ≥ 0 and α > 0, let �gτ(X) =  argming∈H  �n  i=1  �  ρτ(Yi − g(Xi)) + Eτ,τ ′{penλ,α(g(Xi, τ), g(Xi, τ ′))}  �  , for a class of neural networks  H, and τ, τ ′  iid∼ Uniform(0, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content='  Consider a class of generator functions G, where {G ∈ G :  Rp × R × R �→ R}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content='  Assume that for all X ∈ X ⊂ Rp and quantile levels τ ∈ (0, 1), there  exists G ∈ G such that the target neural network �gτ(X) can be represented by a hyper-network  G(X, τ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content=' Then, for i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content=' , n,  �gτ(Xi) = �G(Xi, τ) a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content=',  with respect to the probability law related to Eτ, where �G is a solution to (6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content=' See Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content='  Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content='1 justifies optimizing the integrative quantile loss over τ in the PGQR objective  (6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content=' The next proposition justifies adding the variability penalty (5) to the GQR loss (3) by showing  that there exists λ > 0 so that memorization of the training data does not occur under PGQR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content='  Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content='2 (PGQR does not memorize the training data).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content=' Suppose that α > 0 is fixed in  the variability penalty (5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content=' Denote any minimizer of the PGQR objective function (6) by �G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content=' Then,  there exists λ > 0 such that  �G(Xi, τ) ̸= Yi for at least one i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content=' , n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content='  9  Figure 2: A plot of the penalty function penλ,α(V ) = −λ log(|V | + 1/α).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content=' See Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content='  We now provide some intuition into the variability penalty (5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content=' We can quantify the amount of  variability in the network by V := G(X, τ) − G(X, τ ′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content=' Clearly, V = 0 when vanishing variability  (4) occurs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content=' To avoid data memorization, we should thus penalize V more heavily when V ≈ 0, with  the maximum amount of penalization being applied when V = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content=' Figure 2 plots the variability  penalty, penλ,α(V ) = −λ log(|V |+1/α), for several pairs of (α, λ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content=' Figure 2 shows that penλ,α(V ) is  sharply peaked at zero and that penλ,α(V ) is strictly convex decreasing in |Vi|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content=' Therefore, maximum  penalization occurs when each V = 0, and there is less penalization for larger |V |.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content=' As λ increases,  penλ,α(V ) also increases for all values of V ∈ (−∞, ∞), indicating that larger values of λ lead to  more penalization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content='  Our penalty function takes a specific form (5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content='  However, any other penalty on G(X, τ) −  G(X, τ ′) for X ∈ X that has a similar shape as the PGQR penalty (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content=' sharply peaked around  zero, as in Figure 2) would also conceivably encourage the network to have greater variability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content='  Another way to control the variability is to directly penalize the empirical variance s2 = (n −  1)−1 �n  i=1[G(Xi, τ) − ¯G]2, where ¯G = n−1 �n  i=1 G(Xi, τ), or the empirical standard deviation  s =  √  s2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content=' However, penalties on s2 or s do not have an additive form and individual penalty terms  depend on each other, and therefore, these are not conducive to SGD-based optimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content='  We now pause to highlight the novelty of our constructed variability penalty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content=' In many other  nonparametric models, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content=' those based on nonparametric smoothing, a roughness penalty is often  10  Variability Penality  α= 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content='0, 2= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content='5  α= 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content='0, 1= 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content='5  α= 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content='0, 1= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content='5  2  α= 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content='0, 1= 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content='5  (A)  aa  2  4  4  2  0  2Figure 3: Plots of the estimated conditional densities of p(Y | Xtest) for three different test observations  under non-penalized GQR and PGQR with two different choices of λ ∈ {0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content='01, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content='5}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content=' λ⋆ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content='01 is the  optimal λ chosen by our tuning parameter selection method in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content='  added to an empirical loss function in order to control the smoothness of the function or density  estimate (Green and Silverman, 1994).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content=' These roughness penalties encourage greater smoothness  of the target function in the spirit of reducing variance in the bias-variance trade-off.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content=' In contrast,  our variability penalty (5) directly penalizes generators with too small variance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content=' By adding the  penalty to the GQR loss (3), we are encouraging the estimated network to have more variability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content='  To illustrate how PGQR prevents vanishing variability, we carry out a small simulation study  under the model, Yi = X⊤  i β + ϵi, i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content=' , n, where Xi  iid  ∼ N(0, I20), ϵi  iid  ∼ N(0, 1), and the  coefficients in β are equispaced over [−2, 2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content=' We used 2000 samples to train GQR (3) and PGQR (6),  and an additional 200 validation samples were used for tuning parameter selection of λ (described  in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content='  To train the generative network, we fixed α = 1 and tuned λ from a set of  100 equispaced values between 0 and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content=' We then generated 1000 samples { �G(Xtest, ξk, λ)}1000  k=1 ,  ξk  iid  ∼ Uniform(0, 1), from the estimated conditional density of p(Y | Xtest) for three different choices  of out-of-sample test data for Xtest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content=' In our simulation, the tuning parameter selection procedure  introduced in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content='2 chose an optimal λ of λ⋆ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content='01.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content='  As shown in Figure 3, the non-penalized GQR model introduced in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content='1 (dashed orange  line) suffers from vanishing variability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content=' Namely, GQR generates values near the true test sample  Ytest almost surely, despite the fact that we used 1000 different inputs for ξ to generate the �G  samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content=' In contrast, the PGQR model with optimal λ⋆ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content='01 (solid blue line with filled circles)  approximates the true conditional density (solid black line) very closely, capturing the Gaussian  shape and matching the true underlying variance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content='  On the other hand, Figure 3 also illustrates that if λ is chosen to be too large in PGQR (6),  11  PGQR(.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content='01)  True  to  PGQR(2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content='5)  GQR  :  :  0  10  5  10  10  5  5  10  20  15  10  5  10then the subsequent approximated conditional density will exhibit larger variance than it should.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content='  In particular, when λ = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content='5, Figure 3 shows that PGQR (dashed green line with hollow triangles)  overestimates the true conditional variance for Y given X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content=' This demonstrates that the choice of  λ in the variability penalty (2) plays a very crucial role in the practical performance of PGQR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content=' In  Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content='2, we describe how to select an “optimal” λ so that PGQR neither underestimates nor  overestimates the true conditional variance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content=' Our method for tuning λ also requires only a single  optimization, making it an attractive and scalable alternative to cross-validation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content='  4  Partial Monotonic Neural Network  Without any constraints on the network, PGQR is just as prone as other unconstrained nonpara-  metric quantile regression models to suffer from crossing quantiles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content=' For a fixed data point Xi and  an estimated network �G, the crossing problem occurs when  �G(Xi, τ1) > �G(Xi, τ2) when 0 < τ1 < τ2 < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content='  (7)  In the neural network literature, one popular way to address the crossing problem is to add a  penalty term to the loss such as max(0, −∂G(Xi;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content=' τ)/∂τ) (Tagasovska and Lopez-Paz, 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content=' Liu  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content=', 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content=' Through regularization, the network is encouraged to have larger partial derivatives  with respect to τ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content=' However, adding this penalty merely alleviates the crossing problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content=' There is  no guarantee that the final model is free of crossing quantiles (Tagasovska and Lopez-Paz, 2018),  or we may have to keep training the model until the final network is certified to be monotonic  with respect to τ at the expense of heavy computational cost (Liu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content=', 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content=' Another natural  way to construct a monotonic neural network (MNN) is by restricting the weights of network to be  nonnegative through a transformation or through weight clipping (Zhang and Zhang, 1999;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content=' Daniels  and Velikova, 2010;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content=' Mikulincer and Reichman, 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content=' Because all the weights are constrained to  be nonnegative, this class of MNNs might require longer optimization (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content=' more epochs in SGD)  and/or a large amount of hidden neurons to ensure the network’s final expressiveness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content='  Instead of constraining all the weights in the neural network, we make a simple modification to  the MNN architecture which we call the partial monotonic neural network (PMNN).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content=' The PMNN  12  family consists of two feedforward neural networks (FNN) as sub-networks which divide the input  into two segments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content='  The first sub-network is a weight-constrained quantile network R1 �→ Rh  where the only input is the quantile level τ and h denotes the number of hidden neurons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content=' The  FNN structure for one hidden layer in this sub-network is g(τ) = σ(Uposτ + b), where Upos is  the weights matrix of nonnegative weights, b is the bias matrix, and σ is the activation function,  such as the hyperbolic tangent (tanh) function σ(x) = tanh(x) or the rectified linear unit (ReLU)  function σ(x) = max{0, x}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content=' The K1-layer constrained sub-network gc can then be constructed as  gc(τ) = gK ◦ · · · ◦ g1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content='  The second sub-network is a K2-layer unconstrained network guc : Rp �→ Rh, taking the data X  as inputs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content=' The structure of one hidden layer in this sub-network is g′(X) = σ(UX +b), where U is  an unconstrained weights matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content=' We construct guc as guc(X) = g′  K2 ◦· · ·◦g′  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content=' Finally, we construct  a single weight-constrained connection layer f : Rh �→ R1 that connects the two sub-networks to  quantile estimators,  G(X, τ) = f ◦ (gc + guc).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content='  (8)  With our PMNN architecture, the monotonicity of G(·, τ) with respect to τ is guaranteed by the  positive weights in the quantile sub-network gc and the final layer f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content=' At the same time, since the  data sub-network guc is unconstrained in its weights, guc can learn the features of the training data  well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content=' In particular, the PMNN family is more flexible in its ability to learn features of the data and  is easier to optimize than MNN because of this unconstrained sub-network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content=' We use our proposed  PMNN architecture as the family Gm over which to optimize the PGQR objective for all of the  simulation studies and real data analyses in this manuscript.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content='  To investigate the performance of our proposed PMNN family of neural networks, we fit the  PGQR model with PMNN as the function class Gm on two benchmark datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content=' The first dataset  is the motorcycles data (Silverman, 1985) where the response variable is head acceleration (in g)  and the predictor is time from crash impact (in ms).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content=' The second application is a bone mineral  density (BMD) dataset (Takeuchi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content=', 2006) where the response is the standardized relative  change in spinal BMD in adolescents and the predictor is the standardized age of the adoles-  cents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content='  For these two datasets, we estimated the quantile functions at different quantile levels  τ ∈ {0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content='1, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content='2, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content='3, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content='4, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content='5, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content='6, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content='7, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content='8, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content='9} over the domain of the predictor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content=' We compared PGQR  13  Figure 4: Estimated quantile curves at levels τ ∈ {0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content='1, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content='2, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content='3, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content='4, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content='5, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content='6, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content='7, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content='8, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content='9} for the motorcycle  and BMD datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content=' The left two panels plot the estimated curves for unconstrained nonparametric quantile  regression, and the right two panels plot the estimated curves for PGQR with the PMNN family.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content='  under PMNN to the unconstrained nonparametric quantile regression approach implemented in  the R package quantreg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content=' The left two panels of Figure 4 demonstrate that the estimated quantile  curves under unconstrained nonparametric quantile regression are quite problematic for both the  motorcycle and BMD datasets;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content=' the quantile curves cross each other at multiple points in the pre-  dictor domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content=' In comparison, the right two panels of Figure 4 show that PGQR with the PMNN  family ensures no crossing quantiles for either dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content='  5  Scalable Computation for PGQR  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content='1  Single-Model Training for PGQR  As illustrated in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content='2, PGQR’s performance depends greatly on the regularization parameter  λ ≥ 0 in the variability penalty (5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content=' If λ is too small or if λ = 0, then we may underestimate the  true conditional variance of p(Y | X) and/or encounter the vanishing variability phenomenon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content=' On  the other hand, if λ is too large, then we may overestimate the conditional variance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content='  14  (1)  Quantile Regression  (2) PGQR  Motorcycles Dataset  Motorcycles  Dataset  8  8  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content='1  Acceleration  Acceleration  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content='3  0  0  09-  09-  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content='8  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content='9  100  100  10  20  30  40  50  10  20  30  40  50  Time  Time  BMD Dataset  BMD Dataset  (standardized)  (standardized)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content='1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content='2  2  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content='8  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content='9  BMD  BMD  2  2  1  0  1  2  1  0  1  2  (standardized)  (standardized)Due to the large number of parameters in a DNN, tuning λ would be quite burdensome if  we had to repeatedly evaluate the deep generative network G(·, τ) for multiple choices of λ and  training sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content=' In particular, using cross-validation to tune λ is computationally infeasible for DNNs,  especially if the size of the training set is large.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content='  Inspired by the idea of Generative Multiple-purpose Sampler (GMS) introduced by Shin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content='  (2022), we instead propose to include λ as an additional input in the generator, along with data  X and quantile τ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content=' In other words, we impose a discrete uniform distribution p(λ) for λ whose  support Λ is a grid of candidate values for λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content=' We then estimate the network G(·, τ, λ) with the  modified PGQR loss function,  �G = argmin  G∈Gm  1  n  n  �  i=1  Eτ,λ  �  ρτ  �  Yi − G(Xi, τ, λ)  �  + Eτ,τ ′ �  penλ,α (G(Xi, τ, λ), G(Xi, τ ′, λ))  � �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content='  (9)  Note that (9) differs from the earlier formulation (6) in that λ ∈ Λ is not fixed a priori.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content=' In the  PMNN family Gm, λ is included in the unconstrained sub-network guc : Rp+1 �→ Rh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content='  The next corollary to Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content='1 justifies including λ in the generator and optimizing the  integrative loss over {τ, λ} in the modified PGQR objective (9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content=' The proof of Corollary 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content='1 is a  straightforward extension of the proof of Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content='1 and is therefore omitted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content='  Corollary 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content=' Let �gτ,λ(X) = argming∈H  �n  i=1  �  ρτ(Yi − g(Xi)) + Eτ,τ ′{penλ,α (g(Xi, τ), g(Xi, τ ′))}  �  ,  for a class of neural networks H, where α > 0 is fixed and τ, τ ′ iid  ∼ Uniform(0, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content=' Consider a class  of generator functions G where {G ∈ G : Rp × R × R �→ R}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content=' Suppose that for all X ∈ X ⊂ Rp,  quantile levels τ ∈ (0, 1), and tuning parameters λ ∈ Λ, there exists G ∈ G such that the target  neural networks �gτ,λ(X) can be represented by some G(X, τ, λ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content=' Then, for i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content=' , n,  �gτ,λ(Xi) = �G(Xi, τ, λ) a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content=',  with respect to the probability law related to Eτ,λ, where �G is a solution to (9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content='  We note that Shin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content=' (2022) proposed to use GMS for inference in linear quantile regression.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content='  In contrast, PGQR is a method for nonparametric joint quantile estimation and conditional density  estimation (CDE).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content=' Linear quantile regression cannot be used for CDE;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content=' as a linear model, GMS  linear quantile regression also does not suffer from vanishing variability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content=' This is not the case for  15  GQR, which motivates us to introduce the variability penalty in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content=' Finally, we employ  the GMS idea for tuning parameter selection in PGQR, rather than for constructing pointwise  confidence bands (as in Shin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content=' (2022)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content='  By including λ ∼ p(λ) in the generator, PGQR only needs to perform one single optimization  in order to estimate the network G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content=' We can then use the estimated network �G to tune λ, as we  detail in the next section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content=' This single-model training stands in contrast to traditional smoothing  methods for nonparametric quantile regression, which typically require repeated model evaluations  via (generalized) cross-validation to tune hyperparameters such as bandwidth or roughness penalty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content='2  Selecting An Optimal Regularization Parameter  We now introduce our method for selecting the regularization parameter λ in our variability penalty  (5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content=' Considering the candidate set Λ, the basic idea is to select the best λ⋆ ∈ Λ that minimizes the  distance between the generated conditional distribution and the true conditional distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content='  Denote the trained network under the modified PGQR objective (9) as �G(·, τ, λ), and let  (Xval, Yval) denote a validation sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content=' After optimizing (9), it is effortless to generate the con-  ditional quantile functions for each Xval in the validation set, each λ ∈ Λ, and any quantile level  ξ ∈ (0, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content=' In order to select the optimal λ⋆ ∈ Λ, we first prove the following proposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content='  Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content=' Suppose that two univariate random variables Q and W are independent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content=' Then,  Q and W have the same distribution if and only if P(Q < W | W) follows a standard uniform  distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content=' See Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content='  Based on Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content='2, we know that, given a validation sample (Xval, Yval), the best  λ⋆ ∈ Λ should satisfy Pτ,λ( �G(Xval, τ, λ⋆) < Yval  �� Yval) ∼ Uniform(0, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content=' In practice, if we have  M random quantile levels τ = (τ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content=' , τM) ∈ (0, 1)M, we can estimate the probability Pτ,λ :=  Pτ,λ( �G(Xval, τ, λ⋆) < Yval  �� Yval) with a Monte Carlo approximation,  �P (i)  τ,λ = M −1  M  �  k=1  I{ �G(X(i)  val, τk, λ) < Y (i)  val } ≈ Pτ,λ( �G(Xval, τ, λ⋆) < Yval  �� Yval),  (10)  16  Figure 5: Histogram of the estimated �P (i)  τ,λ⋆’s in the validation set for the optimal λ⋆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content='  for each ith validation sample (X(i)  val, Y (i)  val ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content=' With nval validation samples, we proceed to estimate  �P (i)  τ,λ for all nval validation samples (X(i)  val, Y (i)  val )’s and each λ ∈ Λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content=' We then compare the empirical  distribution of the �P (i)  τ,λ’s to a standard uniform distribution for each λ ∈ Λ and select the λ⋆ that  minimizes the distance between �Pτ,λ and Uniform(0, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content=' That is, given a valid distance measure d,  we select the λ⋆ which satisfies  λ⋆ = argmin  λ∈Λ  d( �Pτ,λ, Uniform(0, 1)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content='  (11)  There are many possible choices for d in (11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content=' In this paper, we use the Cramer–von Mises (CvM)  criterion due to the simplicity of its computation and its good empirical performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content=' The CvM  criterion is straightforward to compute as  d( �Pτ,λ, Uniform(0, 1)) =  nval  �  i=1  (i/nval − �P (i)  τ,λ − 2/nval)2/nval.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content='  (12)  After selecting λ∗ according to (11), we can easily generate samples from the conditional quantiles  of p(Y | X) for any feature data vector X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content=' We simply generate �G(X, ξk, λ⋆), where k = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content=' , b,  for random quantiles ξ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content=' , ξb  iid  ∼ Uniform(0, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content=' For sufficiently large b, the conditional density  p(Y | X) can be inferred from { �G(X, ξk, λ⋆)}b  k=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content=' The complete algorithm for scalable computation  of PGQR is given in Algorithm 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content='  We now illustrate our proposed selection method for λ ∈ Λ in the simulated linear regression  example from Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content=' Figure 5 plots the histogram of the estimated �P (i)  τ,λ⋆’s in the validation  set for the λ⋆ which minimizes the CvM criterion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content=' We see that the empirical distribution of the  17  (i)  Histogram of estiamted  t, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content='  2  0  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content='  0  9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content='  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content='8  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content='0Figure 6: The conditional standard deviation (left panel), CvM statistic (middle panel), and coverage  rate (right panel) in the validation set vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content=' log(λ) for each λ ∈ Λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content=' The red dashed line corresponds to the  λ⋆ which minimizes the CvM criterion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content='  �P (i)  τ,λ⋆’s closely follows a standard uniform distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content='  Figure 6 illustrates how the PGQR solution changes as log(λ) increases for every λ ∈ Λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content=' Figure  6 plots the conditional standard deviation (left panel), the CvM criterion (middle panel), and the  coverage rate of the 95% confidence intervals (right panel) for the PGQR samples in the validation  set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content=' The vertical dashed red line denotes the optimal λ⋆ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content='01.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content=' We see that the λ⋆ which minimizes  the CvM (middle panel) captures the true conditional standard deviation of σ = 1 (left panel) and  attains coverage probability close to the nominal rate (right panel).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content=' This example demonstrates  that our proposed selection method provides a practical alternative to computationally burdensome  cross-validation for tuning λ in the variability penalty (5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content=' In Section B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content='1 of the Appendix, we  further demonstrate that our tuning parameter selection method is suitable to use even when the  true conditional variance is very small (true σ2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content='01).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content=' In this scenario, our procedure selects a  very small λ⋆ ≈ 0, so that PGQR does not overestimate the conditional variance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content='  6  Numerical Experiments and Real Data Analysis  We evaluated the performance of PGQR on several simulated and real datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content=' We fixed α = 1  or α = 5 in the variability penalty (5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content=' We optimized the modified PGQR loss (9) over the PMNN  family (Section 4), where each sub-network had three hidden layers with 1000 neurons per layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content='  The support for Λ was chosen to be 100 equispaced values between 0 and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content=' We compared PGQR  to several other state-of-the-art methods:  18  Conditional Standard Deviation  CvM Statistics  Coverage Rate  ×  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content='010  X  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content='0  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content="  900'0  8." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content='  x : ×  0  80-52  38  28  19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content='3-9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content='5 -1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content='5  60  09-  40  30  20  10  0  60  09-  40  30  20  10  0  log(2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content=')  log(.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content=')  log(2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content=' )Algorithm 1 Scalable Implementation of PGQR  1: Split non-test data into non-overlapping training set (Xtrain, Ytrain) and validation set (Xval, Yval)  2: Initialize G parameters φ, learning rate γ, width h of DNN hidden layers, and total epoches T  3: procedure: Optimizing G  4:  for epoch t in 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content=' , T do  5:  Sample τ, τ ′ ∼ Uniform(0, 1) and λ ∈ Λ  6:  Evaluate loss (9) with (Xtrain, Ytrain)  7:  Update G parameters φ via SGD  8:  end for  9:  return �G  10: end procedure  11: procedure: Tuning λ  12:  Set M = 1000 and sample τ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content=' , τM  iid  ∼ Uniform(0, 1)  13:  Generate { �G(X(i)  val, τ1, λl), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content=' , �G(X(i)  val, τM, λl)} for each λl ∈ Λ and each X(i)  val, i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content=' , nval  14:  Compute �P (i)  τ,λl as in (10) on (X(i)  val, Y (i)  val ) for each i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content=' , nval and each λl ∈ Λ  15:  Select λ∗ according to (11) with CvM criterion (12) as d  16:  return λ∗  17: end procedure  18: procedure: Estimating p(Y | Xtest) for test data Xtest  19:  Set b = 1000 and sample ξ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content=' , ξb  iid  ∼ Uniform(0, 1)  20:  Estimate ξk-th quantile of Y | Xtest as �G(Xtest, ξk, λ∗) for k = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content=' , b  21:  return { �G(Xtest, ξ1, λ∗), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content=' , �G(Xtest, ξb, λ∗)}  22: end procedure  GCDS (Zhou et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content=', 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content=' Following Zhou et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content=' (2022), we trained both a generator and a  discriminator using FNNs with one hidden layer of 25 neurons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content=' Despite this simple architecture,  we found that GCDS might still encounter vanishing variability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content=' For fair comparison to PGQR,  we also increased the number of hidden layers to three and the number of nodes per layer to  1000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content=' We refer to this modification as deep-GCDS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content='  WGCS (Liu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content=', 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content=' We adopted the gradient penalty recommended by Liu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content=' (2021)  and set the hyperparameter associated with the gradient penalty as 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content='  FlexCoDE (Izbicki and Lee, 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content=' We considered three ways of estimating the basis func-  tions βj(X): nearest-neighbor regression (NNR), sparse additive model (SAM), and XGBoost  (FlexZboost).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content='  Random Forest CDE, or RFCDE (Pospisil and Lee, 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content='  For FlexCoDE and RFCDE, we adopted the default hyperparameter settings of Izbicki and Lee  (2017) and Pospisil and Lee (2019) respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content='  19  Figure 7: Plots of the estimated conditional densities of p(Y | Xtest) for three different test observations  from one replication of Simulation 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content=' Plotted are the estimated conditional densities for PGQR (α = 1),  GCDS, WGCS, and deep-GCDS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content='  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content='1  Simulation Studies  For our simulation studies, we generated data from Yi = g(Xi) + ϵi, i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content=' , 2000, for some  function g, where Xi  iid  ∼ N(0, Ip) and the residual errors ϵi’s were independent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content='  The specific  simulation settings are described below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content=' Simulation 1: Multimodal and heteroscedastic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content=' Yi = βiXi + ϵi, where βi = {−1, 0, 1}  with equal probability and ϵi = (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content='25 · |Xi|)1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content=' Simulation 2: Mixture of left-skewed and right-skewed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content=' Yi = X⊤  i β + ϵi, where β ∈ R5  is equispaced between [−2, 2], and ϵi = χ2(1, 1)I(X1 >= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content='5) + log[χ2(1, 1)]I(X1 < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content=' Here,  the skewness is controlled by the covariate X1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content=' Simulation 3: Mixture of unimodal and bimodal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content=' Yi = X⊤  i βi + ϵi, where βi ∈ R5 with  βi1 ∈ {−2, 2} with equal probability, (βi2, βi3, βi4, βi5)⊤ are equispaced between [−2, 2], and  ϵi  iid  ∼ N(0, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content=' Here, p(Y | X) is unimodal when X1 ≈ 0, and otherwise, it is bimodal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content='  In our simulations, 80% of the data was used to train the model, 10% was used as the validation  set for tuning parameter selection, and the remaining 10% was used as test data to evaluate model  performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content='  In Section B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content='1 of the Appendix, we present additional results where g(Xi) is a  nonlinear function of Xi (Simulation 4) and where the error variance is very small (Simulation 5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content='  Figure 7 compares the estimated conditional density of p(Y | Xtest) for three test observations  from one replication of Simulation 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content=' We see that PGQR (solid blue line with filled circles) is  able to capture both the unimodality and the bimodality of the ground truth condtional densities  (solid black line).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content='  Meanwhile, GCDS (dashed red line with hollow triangles), WGCS (dashed  20  4  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content='  0  T  0  True  PGQR  GCDS  0  +  WGCS  deep-GCDS  zo  2  0  0  0  :  9  0  5  10  15  10  15Simulation 1  Simulation 2  Simulation 3  Method  E(Y | X)  sd(Y | X)  Cov (Width)  E(Y | X)  sd(Y | X)  Cov (Width)  E(Y | X)  sd(Y | X)  Cov (Width)  PGQR (α=1)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
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+page_content='26)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content='83  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content='65  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content='96 (23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content='06)  Table 1: Table reporting the PMSE for the conditional expectation and standard deviation, as well as  the coverage rate (Cov) and average width of the 95% prediction intervals, for Simulations 1 through 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content='  Results were averaged across 20 replicates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content='  green line with crosses), and deep-GCDS (dashed purple line with plusses) struggled to capture  the true conditional densities for at least some test points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content=' In particular, Figure 7 shows some  evidence of variance underestimation for WGCS and deep-GCDS, whereas this is counteracted by  the variability penalty in PGQR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content=' Additional figures from our simulation studies are provided in  Section B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content='2 of the Appendix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content='  We repeated our simulations for 20 replications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content=' For each experiment, we computed the pre-  dicted mean squared error (PMSE) for different summaries of the conditional densities for the test  data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content=' We define the PMSE as  PMSE =  1  ntest  ntest  �  i=1  ( �m(Xtest,i) − m(Xtest,i))2 ,  (13)  where m(X) generically refers to the conditional mean of Y given X, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content='  E(Y | X), or the  conditional standard deviation, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content=' sd(Y | X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content=' For the generative models (PGQR, GCDS, WGCS,  and deep-GCDS), we approximated �m in (13) by Monte Carlo simulation using 1000 generated  samples, while for FlexCoDE and RFCDE, we approximated �m using numerical integration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content=' In  addition to PMSE, we also used the 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content='5th and 97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content='5th percentiles of the predicted conditional  densities to construct 95% prediction intervals for the test data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content=' We then calculated the coverage  probability (Cov) and the average width for these prediction intervals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content='  Table 1 summarizes the results for PGQR with α ∈ {1, 5} in (5) and all competing methods,  averaged across 20 experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content=' There was not much substantial difference between α = 1 and  α = 5 for PGQR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content=' Table 1 shows that PGQR had the lowest PMSE in all three simulations  21  Figure 8:  Plot of the average PMSE across 20 replicates for the τ-th quantiles QY |X(τ), τ  ∈  {0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content='1, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
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+page_content='4, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content='5, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content='6, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content='7, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content='8, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content='9} for Simulations 1 through 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content='  and attained coverage close to the nominal rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content=' In Simulations 2 and 3, the prediction intervals  produced by PGQR had the highest coverage rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content=' In Simulation 1, FlexZBoost had 100% coverage,  but the average width of the prediction intervals for FlexZBoost was considerably larger than that of  the other methods, suggesting that the intervals produced by FlexZBoost may be too conservative  to be informative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content='  We also computed the PMSE for the τ-th quantile.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content=' Figure 8 plots the PMSE for these nine  quantiles for PGQR (α = 1), GCDS, deep-GCDS, and WGCS, averaged across 20 experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content='  We see that PGQR (blue line with crosses) had the lowest PMSE for most of the quantile levels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content='  PGQR also had uniformly lower PMSE in our quantile set than WGCS (green line with plusses)  and deep-GCDS (red line with triangles), suggesting that WGCS and deep-GCDS may have been  more prone to vanishing variability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content='  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content='2  Real Data Analysis  We also examined the performance of PGQR on three real datasets from the UCI Machine Learning  Repository, which we denote as: machine, fish, and noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content='1 The machine dataset comes from a  real experiment that collected the excitation current (Y ) and four machine attributes (X) for a set  of synchronous motors (Kahraman, 2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content=' The fish dataset contains the concentration of aquatic  toxicity (Y ) that can cause death in fathead minnows and six molecular descriptors (X) described  in (Cassotti et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content=', 2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content=' The noise dataset measures the scaled sound pressure in decibels (Y )  1Accessed from https://archive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content='ics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content='uci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content='edu/ml/index.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content='php.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content='  22  Simulation 1  Simulation 2  simulation 3  5  PGQR  GCDS  WGCS  deep-GCDS  T  5  5 -  0 :  0 :  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
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+page_content='9  Quantile  Quantile  QuantilePGQR  GCDS  deep-GCDS  WGCS  Dataset  n  p  Cov  Width  Cov  Width  Cov  Width  Cov  Width  machine  557  4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
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+page_content='66  fish  908  6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
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+page_content='36  noise  1503  5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content='92  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
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+page_content='00  22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content='41  Table 2: Results from our real data analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content=' Cov and width denote the coverage rate and average width  respectively of the 95% prediction intervals for the test observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content='  at different frequencies, angles of attacks, wind speed, chord length, and suction side displacement  thickness (X) for a set of airfoils (Lopez et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content=', 2008).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content=' Table 2 reports the sample size n and  covariate dimension p for these three benchmark datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content='  We examined the out-of-sample performance for PGQR (with fixed α = 1), GCDS, deep-GCDS,  and WGCS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content=' In particular, 80% of each dataset was randomly selected as training data, 10% was  used as validation data for tuning parameter selection, and the remaining 10% was used as test data  for model evaluation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content=' To compare these deep generative methods, we considered the out-of-sample  coverage rate (Cov) and the average width of the 95% prediction intervals in the test data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content='  The results from our real data analysis are summarized in Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content=' We see that on all three  datasets, PGQR achieved higher coverage that was closer to the nominal rate than the competing  methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content=' It seems as though the other generative approaches may have been impacted by the  vanishing variability phenomenon, resulting in too narrow prediction intervals that did not cover as  many test samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content=' In particular, GCDS, deep-GCDS, and WGCS all performed extremely poorly  on the noise dataset, with an out-of-sample coverage rate of zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content=' On the other hand, with the  help of the variability penalty (5), PGQR did not underestimate the variance and demonstrated  an overwhelming advantage over these other methods in terms of predictive power.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content=' Moreover,  the average widths of the PGQR prediction intervals were not overwhelmingly large so as to be  uninformative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content='  Figure 9 plots the PGQR 95% prediction intervals for the test observations ordered in ascending  order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content=' The red crosses depict test samples that were not captured by their corresponding PGQR  prediction intervals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content=' We do not detect any specific pattern for the uncaptured test points, indicating  reasonable generalizability for the PQGR model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content='  23  Figure 9: The PGQR 95% prediction intervals for the test samples in the three benchmark datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content=' The  red crosses indicate the test points that were not captured by their corresponding prediction intervals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content='  7  Discussion  In this paper, we have made contributions to both the quantile regression and deep learning lit-  erature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content=' Specifically, we proposed PGQR as a new deep generative approach to joint quantile  estimation and conditional density estimation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content=' Different from existing conditional sampling meth-  ods (Zhou et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content=', 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content=' Liu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content=', 2021), PGQR employs a novel variability penalty to counteract  the vanishing variability phenomenon in deep generative networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content=' We introduced the PMNN  family of neural networks to enforce the monotonicity of quantile functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content=' Finally, we provided  a scalable implementation of PGQR which requires solving only a single optimization to select the  regularization term in the variability penalty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content=' Through analyses of real and simulated datasets,  we demonstrated PGQR’s ability to capture critical aspects of the conditional distribution such as  multimodality, heteroscedasticity, and skewness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content='  We anticipate that our penalty on vanishing variability in generative networks is broadly ap-  plicable for a wide number of loss functions besides the check function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content=' In the future, we will  extend the variability penalty to other deep generative models and other statistical problems be-  sides quantile regression.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content=' In addition, we plan to pursue variable selection, so that our method  can also identify the most relevant covariates when the number of features p is large.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content=' Owing to its  single-model training, PGQR is scalable for large n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content=' However, further improvements are needed in  order for PGQR to avoid the curse of dimensionality for large p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content='  24  machine  fish  noise  10  20  30  40  50  20  40  60  80  50  100  150Acknowledgments  The last author was generously supported by NSF grant DMS-2015528.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content=' The authors are grateful  to Dr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content=' Jun Liu for helpful comments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
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+page_content=' Composite quantile regression neural network with  applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content=' Expert Systems with Applications, 76:129–139.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content='  Zhang, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content=' and Zhang, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content=' (1999).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content=' Feedforward networks with monotone constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content=' In IJCNN’99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content=' International  Joint Conference on Neural Networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content=' Proceedings (Cat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content=' No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content='99CH36339), volume 3, pages 1820–1823 vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content='  Zhou, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content=', Jiao, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content=', Liu, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content=', and Huang, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content=' (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content=' A deep generative approach to conditional sampling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content=' Journal of  the American Statistical Association (to appear).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content='  Zou, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content=' and Yuan, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content=' (2008).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content=' Composite quantile regression and the oracle model selection theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content=' The Annals of  Statistics, 36(3):1108–1126.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content='  A  Proofs of Propositions  Proof of Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content=' Suppose that for some ϵ > 0, there exists a set C ⊂ (0, 1) with  Pτ(C) > ϵ such that for some i∗ ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content=' , n}, �gτ(Xi∗) ̸= �G(Xi∗, τ) for all τ ∈ C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content=' Then we can  construct another optimal generator �G such that  1  n  n  �  i=1  Eτ  �  ρτ  �  yi − �G(Xi, τ)  ��  + E�τ,�τ ′  �  penλ,α  �  �G(Xi, �τ), �G(Xi, �τ ′)  ��  ≥  1  n  n  �  i=1  Eτ  �  ρτ  �  yi − �G(Xi, τ)  ��  + E�τ,�τ ′  �  penλ,α  �  �G(Xi, �τ), �G(Xi, �τ ′)  ��  ,  where  �G(Xi∗, τ) =  �  �  �  �  �  �G(Xi∗, τ) for τ ̸∈ C,  �gτ(Xi∗) for τ ∈ C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content='  This is a contradiction due to the fact that �gτ is the minimizer of  1  n  �n  i=1 ρτ(yi − �G(Xi, τ)) +  E�τ,�τ ′{penλ,α( �G(Xi, �τ), �G(Xi, �τ ′))}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content='  Proof of Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content=' Suppose that for all λ ≥ 0, all τ ∈ (0, 1), and all i ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content=' , n},  �G(Xi, τ) = Yi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content='  27  Then the penalty part in the PGQR loss (6) attains Eτ,τ ′{penλ,α( �G(Xi, τ), �G(Xi, τ ′))} = λ log(α),  while the first term in (6) satisfies Eτ[ρτ  �  yi − �G(Xi, τ)  �  ] = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content=' As a result, the total loss is λ log(α).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content='  Since the case of �G(Xi, τ) = Yi is included in cases of Varτ{ �G(Xi, τ)} = 0, we focus on the  variance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content=' When there exists some i ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content=' , n} and some τ ∈ (0, 1) such that  Varτ  �  �G(Xi, τ)  �  > 0,  the resulting total loss can be made less than λ log(α) by choosing an appropriate λ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content=' This  contradicts the fact that �G is the minimizer as in (6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content='  Proof of Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content=' It is trivial that FQ(W) := P(Q ≤ W | W) ∼ Uniform(0, 1) when  Q  d= W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content=' WLOG, we assume that FQ is invertible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content=' We shall show that FQ(W) ∼ Uniform(0, 1)  implies that the two distributions for Q and W are identical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content='  Suppose that FQ(W) follows a  standard uniform distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content=' Then,  x = P(FQ(W) < x) = P(W < F −1  Q (x)) = FQ(F −1  Q (x)),  which implies that FW(F −1  W (x)) = x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content='  Thus, it follows that the distributions of Q and W are  identical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content='  B  More Simulation Results  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content='1  Additional Simulation Studies  In addition to the three simulations described in Section 6 of the main manuscript, we also con-  ducted simulation studies under the following scenarios:  Simulation 4: Nonlinear function with an interaction term and one irrelevant  covariate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content=' Yi = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content='5 log(10−X2  i1)+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content='75 exp(Xi2X3/5)−0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content='25|Xi4/2|+ϵi, where ϵi ∼ N(0, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content='  Note that there is a (nonlinear) interaction between X2 and X3, while X5 is irrelevant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content='  Simulation 5: Very small conditional variance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content=' Yi = βXi+ϵi, i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content=' , n, where β = 1  and ϵi  iid  ∼ N(0, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content='01).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content='  28  Simulation 4  Simulation 5  Method  E(Y | X)  sd(Y | X)  Cov (Width)  E(Y | X)  sd(Y | X)  Cov (Width)  PGQR (α = 1)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content='15  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
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+page_content='42)  PGQR (α = 5)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
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+page_content='97)  Table 3: Table reporting the PMSE for the conditional expectation and standard deviation, as well as the  coverage rate (Cov) and average width of the 95% prediction intervals, for Simulations 4 and 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content=' Results  were averaged across 20 replicates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content='  The results from these two simulations averaged across 20 replicates are shown below in Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content='  The average PMSE for the τ-th quantile levels QY |X(τ), τ ∈ {0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
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+page_content='9}  are plotted in Figure 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content='  Figure 10:  Plot of the average PMSE across 20 replicates for the τ-th quantiles QY |X(τ), τ  ∈  {0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
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+page_content='9} for Simulations 4 and 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content=' In Simulation 5, the PMSE for WGCS  is not displayed, as it lies outside of the plot area (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content=' the average PMSE for WGCS is much larger than  that of the other methods).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content='  One may be concerned whether the regularized PGQR overestimates the conditional variance  when the true conditional density has a very small variance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content=' To illustrate the flexibility of PGQR,  Figure 11 plots the estimated conditional densities for three test observations from one replication  of Simulation 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content=' Recall that in Simulation 5, the true conditional variance is very small (σ2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content='01).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content='  With the optimal λ⋆ selected using the method introduced in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content='2, Figure 11 shows that the  estimated PGQR conditional density still manages to capture the Gaussian shape while matching  29  Simulation 4  Simulation 5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content='010  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content='  PGQR  GCDS  PMSE  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content="008  WGCS  deep-GCDS  900'0  ntile  0." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content='004  Quar  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
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+page_content='9  Quantile  QuantileFigure 11: Plots of the estimated PGQR (α = 1) conditional densities for three different test observations  in Simulation 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content=' The optimal λ⋆ is chosen by our tuning parameter selection method in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content='  the true variance of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content='01.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content=' If the true conditional variance is very small (as in Simulation 5), then  PGQR selects a tiny λ⋆ ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content=' In this scenario, PGQR only applies a small amount of variability  penalization and thus does not overestimate the variance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content='  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content='2  Additional Figures  Here, we provide additional figures from one replication each of Simulations 1, 2, and 4 (with α = 1  in PGQR).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content=' Figures 12-14 illustrate that PGQR (blue solid line with filled circles) is better able  to estimate the true conditional densities (solid black line) than GCDS, WGCS, and deep-GCDS  (dashed lines).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content=' In particular, PGQR does a better job of capturing critical aspects of the true  conditional distributions such as multimodality, heteroscedasticity, and skewness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content='  Simulation 1: Multimodal and heteroscedastic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content='  Figure 12: Plots of the estimated conditional densities of p(Y | Xtest) for three different test observations  from one replication of Simulation 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content='  30  15  1  True  :PGQR  4  2:  2-  2-  0  0  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
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+page_content='02  2  True  PGQR GCDS  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content='  WGCS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content='  0  deep-GCDS  N  4  0  8  8  0  30  20  10  10  20  30  09-  40  20  20  09• Simulation 2: Mixture of left-skewed and right-skewed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content='  Figure 13: Plots of the estimated conditional densities of p(Y | Xtest) for three different test observations  from one replication of Simulation 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content='  Simulation 4: Nonlinear function with an interaction term and one irrelevant co-  variate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content='  Figure 14: Plots of the estimated conditional densities of p(Y | Xtest) for three different test observations  from one replication of Simulation 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content='  C  Sensitivity Analysis of PGQR to the Choice of α  As mentioned in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content='2 of the main manuscript, we choose to fix α > 0 in the variability  penalty (5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content=' The main purpose of α is to ensure that the logarithmic term in the penalty is always  well-defined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content='  In this section, we conduct a sensitivity analysis to the choice of α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content=' To do this, we generated  data using the same settings from Simulations 1 through 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content=' We then fit the PGQR model with  five different choices for α ∈ {0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content='5, 1, 5, 10, 50} and evaluated the performance of these five PGQR  models on out-of-sample test data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content=' Table 4 shows the results from our sensitivity analysis averaged  across 20 replicates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content='  31  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content='  1:  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content='  4  0  0  0  True  PGQR  GCDS  0  0  +  deep-GCDS  zo  2  0  :  6  5  0  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content='  9  10  5  5  10  15  20  20  15  10  5  10  20  10  0  10  206.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content='  0  11  0  True  90  90  90  PGQR  GCDS  to  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content='4  +  deep-GCDS  2  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content='  0  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content='1  6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content='  5  10  5  10  5 10We found that in Simulations 1 through 4, PGQR was not particularly sensitive to the choice  of α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content='  PGQR was somewhat more sensitive to the choice of α in Simulation 5 (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content='  when the  true conditional variance is very small), with larger values of α leading to higher PMSE for the  conditional expectation and conditional standard deviation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content='  In practice, we recommend fixing  α = 1 as the default α for PGQR to perform well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content=' This choice of α = 1 leads to good empirical  performance across many different scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content='  32  Simulation 1  Simulation 2  Simulation 3  Simulation 4  Simulation 5  α  E(Y | X)  sd(Y | X)  Cov (Width)  E(Y | X)  sd(Y | X)  Cov (Width)  E(Y | X)  sd(Y | X)  Cov (Width)  E(Y | X)  sd(Y | X)  Cov (Width)  E(Y | X)  sd(Y | X)  Cov (Width)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
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+page_content='81)  Table 4: PGQR results with different choices of α ∈ {0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content='5, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content='0, 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content='0, 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
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+page_content='0} in the variability penalty for Simulations 1 through 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content='  This table reports the average PMSE for the conditional expectation and standard deviation, as well as the coverage rate (Cov) and the  average width of the 95% prediction intervals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content=' Results were averaged across 20 replicates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
+page_content='  33' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE2T4oBgHgl3EQfGwaV/content/2301.03661v1.pdf'}
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+RCsearcher: Reaction Center Identification in Retrosynthesis
+via Deep Q-Learning
+Zixun Lan 1 Zuo Zeng 2 Binjie Hong 3 Zhenfu Liu 2 Fei Ma� 1
+Abstract
+The reaction center consists of atoms in the prod-
+uct whose local properties are not identical to the
+corresponding atoms in the reactants. Prior stud-
+ies on reaction center identification are mainly
+on semi-templated retrosynthesis methods. More-
+over, they are limited to single reaction center
+identification. However, many reaction centers
+are comprised of multiple bonds or atoms in re-
+ality. We refer to it as the multiple reaction cen-
+ter. This paper presents RCsearcher, a unified
+framework for single and multiple reaction cen-
+ter identification that combines the advantages of
+the graph neural network and deep reinforcement
+learning. The critical insight in this framework is
+that the single or multiple reaction center must be
+a node-induced subgraph of the molecular prod-
+uct graph. At each step, it considers choosing one
+node in the molecular product graph and adding
+it to the explored node-induced subgraph as an
+action. Comprehensive experiments demonstrate
+that RCsearcher consistently outperforms other
+baselines and can extrapolate the reaction center
+patterns that have not appeared in the training set.
+Ablation experiments verify the effectiveness of
+individual components, including the beam search
+and one-hop constraint of action space.
+1. Intruduction
+Reaction center identification plays a vital role in single-step
+retrosynthesis. The reaction center (RC) consists of atoms
+in the product whose local properties are not identical to the
+corresponding atoms in the reactants (Coley et al., 2019).
+Experienced chemists first disconnect the target molecule
+1Department of Applied Mathematics, Xi’an Jiaotong-
+Liverpool University, Suzhou, China 2Hours Technology Co., Ltd.,
+Suzhou, China 3Department of Information and Computing Sci-
+ence, Xi’an Jiaotong-Liverpool University, Suzhou, China. Corre-
+spondence to: Fei ma <Fei.Ma@xjtlu.edu.cn>.
+RCsearcher: Reaction Center Identification in Retrosynthesis via
+Deep Q-Learning
+single reaction center:
+a.
+multiple reaction center:
+incorrect reaction center
+identical
+b.
+incorrect
+Figure 1. (a) Examples of Single reaction center and multiple reac-
+tion center. The red highlight is the reaction center. (b) The case
+of the same intermediate molecules are visited many times in the
+multi-step retrosynthesis. The red highlight and arrow denote the
+predicted reaction center and retrosynthetic prediction respectively.
+through the potential reaction center when performing ret-
+rosynthesis analysis (Corey, 1991). In computer-aided ret-
+rosynthesis, template-based single-step retrosynthesis meth-
+ods (Coley et al., 2017; Dai et al., 2019; Chen & Jung, 2021)
+rank the reaction templates, where the reaction center is part
+of the reaction template. The first step of the semi-template-
+based retrosynthesis methods (Shi et al., 2020; Yan et al.,
+2020; Somnath et al., 2021) is to use one graph neural net-
+work to predict the reaction center of the target molecule.
+Moreover, some template-free retrosynthesis methods (Wan
+et al., 2022; Wang et al., 2021) begin explicitly or implicitly
+adding reaction center information to improve performance
+and interpretability.
+Prior studies on reaction center identification are mainly
+on semi-templated retrosynthesis methods. Moreover, they
+are limited to single reaction center identification. Here, a
+single reaction center refers to the reaction center composed
+of a bond or an atom (Fig. 1(a)). G2G (Shi et al., 2020) and
+RetroXpert (Yan et al., 2020) regards the bond that appears
+in the product but not in the reactants as the reaction center.
+arXiv:2301.12071v1  [cs.LG]  28 Jan 2023
+
+RCsearcher: Reaction Center Identification in Retrosynthesis via Deep Q-Learning
+environment
+Action
+Reward, New State
+Agent
+Ours:
+Prior GNN-based method:
+node1
+node2
+node3
+...
+edge1
+edge2
+edge3
+...
+GNN
+representation
+...
+...
+Reaction center
+identification
+Reaction Center
+Ground Truth
+Classification
+update
+Read-out
+Graph level
+embedding
+3
+Infer
+Top 3
+Node level
+embedding
+Reaction center
+probability
+Figure 2. Left: An illustration of Prior GNN-based method. Right: An overview of our proposed RCsearcher.
+They use R-GCN (Schlichtkrull et al., 2018) and EGAT
+(Kami´nski et al., 2022) respectively to predict the probabil-
+ity of each bond as a single reaction center. GRAPHRETRO
+(Somnath et al., 2021) also considers the atom whose num-
+ber of attached hydrogens changes in the product as a single
+reaction center. Then, it uses MPN (Gilmer et al., 2017)
+to model the probability of each bond or atom as a single
+reaction center. Despite the excellent performance of semi-
+templated retrosynthesis methods, they are still not widely
+used. The main reason is that the current reaction center
+identification method cannot detect multiple reaction center
+effectively.
+In reality, many reaction centers in reactions’ products are
+composed of multiple bonds or atoms, which is not fully rep-
+resented in the USPTO-50k dataset. In this paper, we refer
+to these as the multiple reaction center (Fig. 1(a)). (Dai et al.,
+2019) creates a large dataset USPTO-full from the entire set
+of USPTO 1976-2016 (1,808,937 raw reactions). USPTO-
+full has roughly 1M unique reactions, and the proportion
+of samples with multiple reaction center reaches 38.84%.
+For example, the reaction center of the product of most
+ring-opening reactions is the multiple reaction center (Coley
+et al., 2019). In retrosynthesis, the prior models sometimes
+misidentify multiple reaction center as the single reaction
+center leading to predictions deviating from the ground truth.
+It results in the same intermediate molecules being visited
+many times in the process of multi-step retrosynthesis (Xie
+et al., 2022). As shown in Fig. 1(b), due to misidentification
+of multiple reaction center, the ring-opening operation is
+not performed, which leads to an infinite loop in multi-step
+retrosynthesis.
+To the best of our knowledge, the only existing method
+that can simultaneously address single and multiple reaction
+center identification task appear in RetroXpert (Yan et al.,
+2020). It uses one shared EGAT (Kami´nski et al., 2022)
+to output two tensors (Fig. 3). One tensor represents the
+probability that each bond is the reaction center. Another
+tensor indicates the result of multi-classification, where
+each category indicates the number of bonds forming the
+reaction center. During inference, the prediction of the
+multi-classification task is used to select the corresponding
+number of bonds as the reaction center according to the
+predicted probabilities. There is a gap between its two
+joint optimization objectives and the purpose of finding the
+right single or multiple reaction center. Hence, designing a
+unified and reasonable framework for single and multiple
+reaction center identification remains a challenge.
+In this paper, we present RCsearcher, a unified framework
+for single and multiple reaction center identification that
+combines the advantages of graph neural network and deep
+reinforcement learning (Fig. 3). The critical insight in
+this framework is that the single or multiple reaction center
+must be a node-induced subgraph of the molecular product
+graph (Coley et al., 2019). RCsearcher represents states and
+actions in continuous embeddings by graph representation
+learning (Kami´nski et al., 2022) and uses a Deep Q-Network
+(DQN) (Mnih et al., 2013) to predict action distributions. At
+each step, it considers choosing one node in the molecular
+product graph and adding it to the explored node-induced
+subgraph as an action. After satisfying the stop condition,
+RCsearcher regards the explored node-induced subgraph as
+the predicted reaction center. For effective and fair evalua-
+tion, we stratifingly sample 40k reactions from USPTO-full
+to build a general dataset USPTO-40k, where the proportion
+of samples with multiple reaction center is also 38.84%.
+Comprehensive experiments demonstrate that RCsearcher
+consistently outperforms other baselines for the reaction
+centre identification task, and it can extrapolate the reac-
+tion center patterns that have not appeared in the training
+set. Ablation experiments verify the effectiveness of indi-
+vidual components, including the beam search and one-hop
+constraint of action space. In brief, we highlight our main
+contributions as follows:
+• We address the important yet challenging task of reac-
+tion center identification and propose a unified frame-
+
+RCsearcher: Reaction Center Identification in Retrosynthesis via Deep Q-Learning
+work RCsearcher both for single and multiple reaction
+center as the solution.
+• The key novelty is the GNN-based DQN which pro-
+vides a strategy to select the node-induced subgraph
+as the predicted reaction center, and the core insight in
+this framework is that single or multiple reaction cen-
+ter must be a node-induced subgraph of the molecular
+product graph.
+• For fair and effective evaluation, we build a general
+dataset USPTO-40k. We conduct extensive experi-
+ments to demonstrate the effectiveness and general-
+ization of the RCsearcher. Ablation experiments also
+verify the effectiveness of individual components.
+2. Problem Definition
+The reaction center (RC) consists of atoms in the product
+whose local properties are not identical to the corresponding
+atoms in the reactants (Coley et al., 2019). In this paper,
+we use RDChiral (Coley et al., 2019) to extract the super
+general reaction center. Hence, regardless of whether the
+single or multiple reaction center must be a node-induced
+subgraph of the molecular product graph.
+Notation A molecular product graph Gp = (Vp, Ep) is
+represented as a set of |Vp| nodes (atoms) and a set of
+|Ep| edges (bonds).
+The reaction center graph Grc =
+(Vrc, Erc) is a node-induced subgraph of the molecu-
+lar product graph Gp such that Vrc ⊆ Vp and Erc =
+{(u, v) | u, v ∈ Vrc, (u, v) ∈ Ep}.
+Reaction Center Identification Given the molecular prod-
+uct graph Gp, reaction center identification aims to detect
+the corresponding reaction center graph Grc, i.e. Grc’s node
+set Vrc.
+Evaluation There may be several isomorphic node-induced
+subgraphs in a graph. Therefore, the condition for correct
+identification is that the node set of the predicted reaction
+center graph is consistent with the ground-truth Vrc.
+3. Related work
+Efforts on Reaction Center Identification in Retrosyn-
+thesis
+G2G (Shi et al., 2020) and RetroXpert (Yan et al.,
+2020) regard the bond that appears in the product but not
+in the reactants as the reaction center. G2G uses R-GCN
+(Schlichtkrull et al., 2018) to predict the probability of each
+bond as a single reaction center. In contrast, RetroXpert
+uses EGAT (Kami´nski et al., 2022) to model the probability
+of each bond as the reaction center, and it adds a graph-level
+auxiliary task to predict the total number of disconnection
+bonds (Fig. 3). Hence, RetroXpert is not limited to single
+reaction center identification. GRAPHRETRO (Somnath
+et al., 2021) also considers the atom whose number of at-
+tached hydrogens changes in the product as a single reaction
+center. Then, it uses MPN (Gilmer et al., 2017) to model
+the probability of each bond or atom as a single reaction
+center. It is worth noting that GRAPHRETRO roughly pro-
+posed an autoregressive model for multiple reaction center
+identification in its appendix. However, GRAPHRETRO
+does not elaborate on the specific details of the autoregres-
+sive model, and the code of the autoregressive model is
+not available in the official GitHub link of GRAPHRETRO
+(https://github.com/vsomnath/graphretro).
+Efforts
+on
+Single-step
+Retrosynthesis
+Prediction
+Single-step prediction models can be categorized into
+three main classes, i.e., template-based, template-free and
+semi-template-based method. Template-based methods rely
+on templates that encode the core of chemical reactions,
+converting product molecules into reactants. The key is to
+rank the templates and select the appropriate one to apply,
+and recent attempts (Coley et al., 2017; Dai et al., 2019)
+have addressed the template selection problem by similarity
+and neural networks. Despite their superior interpretability,
+template-based approaches are disadvantaged by poor
+generalization to structures beyond templates. On the other
+hand, template-free methods (Liu et al., 2017; Schwaller
+et al., 2019) regard single-step retrosynthesis prediction as a
+translation task and translate a product molecule represented
+in SMILES strings (Weininger, 1988) to reactant SMILES
+strings. To combine the benefits of template-based and
+template-free methods, recent works (Shi et al., 2020; Yan
+et al., 2020; Somnath et al., 2021) seek semi-template-based
+methods where they first predict single reaction center via
+graph neural networks and then intermediate synthons are
+secondly translated into reactants via seq2seq or graph
+translation models.
+Efforts
+on
+Multi-step
+Retrosynthetic
+planning
+HgSearch (Schwaller et al., 2020) and Proof Num-
+ber Search (Kishimoto et al., 2019) are traditional heuristic
+search algorithms in which chemical feasibility and failed
+pathway values are not considered. (Segler et al., 2018)
+adopt Monte Carlo tree search to generate search trees and
+explore multiple synthetic paths dynamically. (Chen et al.,
+2020) devise a neural-based A*-like algorithm that learns
+an additional value network with automatically constructed
+and only successful paths to bias the search prior. (Han
+et al., 2022) and (Xie et al., 2022) use a GNN-based value
+network to capture intermolecular/intra-pathway level
+information to further improve A*-like retrosynthesis
+planning algorithms. Notably, (Xie et al., 2022) observe
+that the same intermediate molecules are visited many times
+in the searching process. Sometimes, the ring-opening
+operation is not performed due to misidentifying multiple
+reaction center as single reaction center, which leads to an
+infinite loop in multi-step retrosynthesis (Fig. 1(b)).
+
+RCsearcher: Reaction Center Identification in Retrosynthesis via Deep Q-Learning
+4. Preliminaries and Proposed Method
+In this section, we present our RL based reaction center iden-
+tification in retrosynthesis method, RCsearcher. The rest of
+Section 4 is organized as follows. Section 4.1 introduces the
+preliminaries. Section 4.2 presents a high-level overview
+of how to leverage Deep Q-Network (DQN) (Mnih et al.,
+2013) to tackle the reaction center identification in retrosyn-
+thesis (Fig. 4.2). Section 4.3 describes the details of the
+state, action, reward (Fig. 4.2 Left). Section 4.4 explains
+how to train the DQN efficiently. Section 4.5 shows how
+RCsearcher does inference with beam search mechanism
+(Fig. 4.2 Right). Section 4.6 analyzes the time complexity.
+4.1. Preliminaries
+In this paper, we adopt the EGAT (Kami´nski et al., 2022) to
+compute both the node embeddings and the edge embedding.
+Given H(t) = [h(t)
+1 ; h(t)
+2 ; · · · ; h(t)
+n ] ∈ Rn×d and f (t)
+ij
+∈
+R1×d are the node embedding matrix and the embedding of
+edge (i, j) at the t-th layer repectively, where h(t)
+i
+∈ R1×d
+is the node-level embedding for node i of the graph and
+is also the i-th row of H(t), d is the dimension of node-
+level embedding and n is the number of nodes. h(0)
+i
+and
+f (0)
+ij are the initial features of node and edge in molecular
+product graph Gp. The details of the initial features can be
+found in the appendix. EGAT injects the graph structure into
+the attention mechanism by performing masked attention,
+namely it only computes αij for nodes j ∈ Ni, where Ni is
+the first-order neighbors of node i in the graph:
+f (t+1)
+ij
+= LeakyReLU
+��
+h(t)
+i W
+���f (t)
+ij
+��� h(t)
+j W
+�
+A
+�
+,
+eij = a · f (t+1)
+ij
+T , αij =
+exp (eij)
+�
+k∈Ni exp (eik),
+(1)
+where eij ∈ R and αij ∈ R are a non-normalized attention
+coefficient and a normalized attention coefficient represent-
+ing the weight of message aggregated from node j to node
+i respectively in the t-th layer of EGAT, and ∥ is the con-
+catenation operation. Besides,W ∈ Rd×d, A ∈ R3d×d and
+a ∈ R1×d are learnable parameters in the t-th layer.
+EGAT employs multi-head attention to stabilize the learning
+process of self-attention, similar to Transformer (Vaswani
+et al., 2017). If there are K heads, K independent attention
+mechanisms execute the Eq. 1, and then their features are
+concatenated:
+h(t+1)
+i
+= MLP
+�
+�∥K
+k=1σ
+�
+� �
+j∈Ni
+αk
+ijh(t)
+j Wk
+�
+�
+�
+�
+(2)
+where ∥ represents concatenation, αk
+ij are normalized at-
+tention coefficients computed by the k-th learnable Wk ∈
+Rd×d, Ak ∈ R3d×d and ak ∈ R1×d following Eq. 1. Be-
+sides, MLP denotes multi-perceptron. For simplicity, we
+denote the encoding process by EGAT(·) in this study.
+4.2. Overview of RCSeacher
+RCSeacher enables graph representation learning techniques
+to tackle the reaction center identification in retrosynthesis
+and uses deep Q-learning to select one node that is added
+to the current explored subgraph in each search state. RC-
+Seacher represents states and actions in continuous embed-
+dings, and maps (st, at) to a score Q(st, at) via a DQN
+which consists of a Graph Neural Network encoder and
+learnable components to project the representations into the
+final score. Here, st and at denote the state and action re-
+spectively in the step t. Once RCSeacher is trained, it can be
+applied to any new graphs that are unseen during training.
+State st consists of the (1) the molecular product graph Gp,
+(2) current explored reaction center graph ˆGt
+rc = Gp[ˆVt
+rc].
+Action at is defined as selecting one node from the first-
+order neighbor of the current explored subgraph ˆVt
+rc, or a
+stop action implying stop of the exploration process. Since
+most reaction center graphs have only one connected branch,
+we impose a one-hop constraint on the action space to nar-
+row the search space and improve the performance. For
+RCSeacher, given our goal, the intermediate reward is de-
+fined as rt = 0 at any intermediate step t and rT = 1 when
+predicted node-set ˆVT
+rc is consistent with ground-truth Vrc,
+otherwise rT = 0 at the final step T. Therefore, our opti-
+mization goal, maximizing the expected remaining future
+reward, means finding a reaction center that is exactly the
+same as the ground-truth.
+Since the action space can be large, we leverage the repre-
+sentation learning capacity of continuous representations
+for DQN design. At state st, for each action at, our DQN
+predicts a Q(st, at) representing the remaining future re-
+ward after selecting action at.
+Based on the above in-
+sights, we can design a simple DQN leveraging the graph
+encoder EGAT (Kami´nski et al., 2022) to obtain one em-
+bedding per node, {hi | ∀i ∈ Vp}. Denote ∥ as concate-
+nation, READOUT as a readout operation that aggregates
+node-level embeddings into subgraph embeddings h ˆGtrc,
+and whole-graph embedding hGp. A state can then be repre-
+sented as hst = hGp∥h ˆGtrc. An action can be represented as
+hat ∈ {hi | ∀i ∈ Vp} ∪ {hstop}, where hstop denotes the
+stop action. The Q function can then be designed as:
+Q (st, at)
+= MLP (hst∥hat)
+= MLP
+�
+hGp∥h ˆGt
+rc∥hi
+�
+or MLP
+�
+hGp∥h ˆGtrc∥hstop
+�
+,
+(3)
+where MLP denotes multi-perceptron.
+
+RCsearcher: Reaction Center Identification in Retrosynthesis via Deep Q-Learning
+2
+3
+score:0.82
+5
+10
+4
+6
+2
+score:0.78
+score:0.74
+15
+4
+5
+3
+6
+11
+stop
+score:0.72
+score:0.75
+score:0.71
+stop
+score:0.61
+ground truth
+15
+score:0.73
+4
+stop
+stop
+4
+score:0.61
+4
+stop
+score:0.69
+Result of Greedy Search
+Top-1 in BeamSearch
+Terminal
+Pruned
+Node 
+embedding
+State
+Action
+Graph  
+embedding
+ 
+ 
+ 
+ 
+0.65
+0.41
+0.11
+0.09
+Subgraph 
+embedding
+1
+2
+3
+stop
+Q value
+Node1
+Node2
+Node3
+Stop
+1-hop
+constraint
+Figure 3. Left: The details of Q-NET, and representation of state and action. The dashed red line indicates the one-hop constraint. Right:
+An illustration of Beam Search during inference.
+4.3. Details of State, Action and Reward
+State State st consists of the (1) the molecular product
+graph Gp, (2) current explored reaction center graph ˆGt
+rc =
+Gp[ˆVt
+rc]. Firstly, we use EGAT (Kami´nski et al., 2022) to
+obtain the graph-level embedding hGp ∈ R1×d representing
+the molecular product graph Gp:
+{hi | ∀i ∈ Vp} ,
+�
+f ij | ∀(i, j) ∈ Ep
+�
+= EGAT (Gp) ,
+(4)
+hnodes = MEAN ({hi | ∀i ∈ Vp}) ,
+hedges = MEAN
+��
+f ij | ∀(i, j) ∈ Ep
+��
+,
+(5)
+h′
+Gp = ABS (hnodes, hedges) ∥ (hnodes + hedges) ,
+hGp = MLP
+�
+h′
+Gp
+�
+,
+(6)
+where hnodes ∈ R1×d and hedges ∈ R1×d are the whole
+graph information from node and edge perspectives respec-
+tively. ABS denotes absolute difference and ∥ refers to
+concatenation. It ensures our representations are permuta-
+tion invariant. Besides, MLP denotes multi-perceptron.
+Secondly, we use the above node-level embeddings
+{hi | ∀i ∈ Vp} to derive the subgraph embedding h ˆGtrc ∈
+R1×d representing the current explored reaction center
+graph ˆGt
+rc:
+h ˆGtrc =
+�
+�
+�
+⃗0
+, t = 0
+MEAN
+��
+hi | ∀i ∈ ˆVt
+rc
+��
+, t ̸= 0
+.
+(7)
+Since the exploration process did not start at step t = 0
+(ˆV0
+rc = ∅), we use ⃗0 ∈ R1×d to represent ˆV0
+rc.
+Action Since most reaction center graphs have only one
+connected branch, we impose a one-hop constraint on the
+action space to narrow the search space and to improve the
+performance. In other words, the action at is to select one
+node from the first-order neighbour of the current explored
+subgraph ˆVt
+rc or a stop action implying stop of the explo-
+ration process. We use the node i’s embedding hi ∈ R1×d
+and one learnable embedding hstop ∈ R1×d to represent
+the action of the selecting one node and the stop action
+respectively:
+hat =
+�
+�
+�
+hi
+, i ∈ Vp, t = 0
+hi or hstop , i ∈ ONE-HOP
+�
+ˆVt
+rc
+�
+, t ̸= 0
+,
+(8)
+where ONE-HOP is the operation of 1-hop constrain that
+can obtain the first-order neighbour of the current explored
+subgraph.
+Reward When the action at is the stop action, we refer to
+this step t as the final step T. We define the reward rt as:
+rt =
+�
+�
+�
+�
+�
+0
+, t = 0, 1, 2, · · · , T − 1
+0
+, ˆVT
+rc ̸= Vrc, t = T
+1
+, ˆVT
+rc = Vrc, t = T
+.
+(9)
+Therefore, our optimization goal, maximizing the expected
+remaining future reward, means finding a reaction center
+that is exactly the same as the ground truth.
+4.4. Training
+We adopt the standard Deep Q-learning framework (Mnih
+et al., 2013). Since imitation learning is known to help with
+training stability and performance (Levine & Koltun, 2013),
+we allow the agent to follow ground-truth trajectories. For
+each experience (st, at, rt, st+1) in the experience pool, our
+loss function is:
+loss = (yt − Q(st, at))2,
+yt = γrt + MAX (Q(st+1, at+1)) ,
+(10)
+
+RCsearcher: Reaction Center Identification in Retrosynthesis via Deep Q-Learning
+where γ ∈ (0, 1] is the decay rate, and yt is a constant and
+NO-GRAD.
+4.5. Inference
+We use the beam search mechanism (Meister et al., 2020) to
+derive the k best predictions for inference. With a hyperpa-
+rameter budget BEAM SIZE k, the agent can transition to at
+most BEAM SIZE k number of best new states at any given
+state. Thus, we have k2 candidate states before the next
+level in the beam search process. From the k2 candidate
+states, we select the k best states according to the predicted
+value of the Q function at the next level. For example, in Fig.
+4.2, BEAM SIZE = 3, each level of the beam search can
+have up to 3 state nodes (red dashed box). The pseudocode
+is shown in Alg. 1.
+Algorithm 1 Inference with Beam Search
+Input: Q function Q(s, a); Initial State s0; Beam Size k
+Output: k best results K-RESULTS
+BEAM = {s0}
+CANDIDATE = ∅
+repeat
+K-RESULTS = ∅
+for s in BEAM do
+TOP-K-ACTIONS = argtopK
+a
+(Q(s, a))
+for a in TOP-K-ACTIONS do
+add (s, a) to CANDIDATE
+end for
+end for
+BEAM′ =
+argtopK
+(s,a)∈CANDIDATE
+(Q(s, a))
+CANDIDATE = ∅
+BEAM = ∅
+for (s, a) in BEAM′ do
+s′ ← execute a on s
+if a is the stop action then
+add (s, a) to CANDIDATE
+add s to K-RESULTS
+else
+add s′ to BEAM
+end if
+end for
+until BEAM = ∅
+Return: K-RESULTS
+4.6. Complexity Analysis
+At each state, the agent calculates the future values which
+have worst-case time complexity O (|Vp|) due to the action
+space consisting of all atoms in the product at step 0. Overall
+the beam search depth is bounded by |Vp|, and each level of
+the beam search has at most BEAM SIZE k states. Thus,
+the overall time complexity is O
+�
+k × |Vp|2�
+.
+38.84%
+61.16%
+single reaction center
+multiple reaction center
+12.42%
+87.58%
+multiple connected branch
+single connected branch
+Figure 4. The distribution of the USPTO-40k.
+5. Evaluation
+In this section, we evaluate the performance of our RC-
+searcher with comparisons to few baseline approaches for
+the reaction center identification in retrosynthesis, and with
+significant goals of addressing the following questions: Q1:
+How effective and robust is RCsearcher compared to the
+baseline approaches? Q2: How well does RCsearcher per-
+form on the data with the multiple reaction center? Q3:
+How does the proposed one-hop constrain and beam search
+in inference improve performance? Q4: Does RCsearcher
+have more robust generalization than baselines?
+Data Since the data with the multiple reaction center in
+USPTO-50k only accounts for about 5%, for effective eval-
+uation, we created a new dataset USPTO-40k. We adopt
+stratified random sampling for USPTO-full and take out 40k
+samples, ensuring that the data distribution is close enough
+to USPTO-full. In the USPTO-40k, the data with multiple
+reaction center accounts for 38.84%. In the USPTO-40k,
+12.42% of the samples’ reaction center consists of more
+than one connected branch. We randomly select 80% of the
+samples as the training set and divide the rest into validation
+and test sets with equal sizes. Fig. 4 shows the distribution
+of the dataset. The details of the USPTO-40k can be found
+in the appendix.
+Baselines We compare RCsearcher to three baselines for
+evaluating overall performance: GNN-based, Seq2Seq, and
+Similarity-based methods. These include:
+(1) GNN-based: We use the method in RetroXpert. It uses
+one graph encoder to model the probability of each bond as
+the reaction center, and adds a graph-level auxiliary task to
+predict the total number of disconnection bonds. Here, we
+use three graph encoder, including R-GCN (Schlichtkrull
+et al., 2018), EGAT (Kami´nski et al., 2022), MPN (Gilmer
+et al., 2017). Thus, we refer to these three baselines as
+RGCN-based, EGAT-based and MPN-based.
+(2) Seq2Seq: Here, the SMILES string of the product is
+the input sequence, and the SMARTS string of the reaction
+
+RCsearcher: Reaction Center Identification in Retrosynthesis via Deep Q-Learning
+Table 1. Top-k accuracy for Reaction Center Identification on USPTO-40K. The notations ’-20’ and ’-30’ denote 20 sampling experiments
+and 30 sampling experiments respectively.
+Methods
+Top-1(%)
+Top-2(%)
+Top-3(%)
+Top-4(%)
+GNN-based
+MPN-based
+9.30±0.56
+10.28±0.38
+11.58±0.31
+14.07±0.14
+EGAT-based
+18.04±0.79
+23.49±0.66
+25.33±0.43
+25.78±0.25
+RGCN-based
+9.27±0.46
+14.25±0.47
+17.53±0.35
+20.78±0.17
+Seq2Seq
+Product2RC-20
+3.25±0.51
+4.95±0.34
+6.45±0.28
+8.68±0.19
+Product2RC-30
+3.15±0.42
+4.61±0.33
+6.42±0.20
+8.48±0.13
+Product2RC-Aug-20
+7.78±0.37
+10.22±0.20
+12.70±0.16
+15.00±0.06
+Product2RC-Aug-30
+8.51±0.40
+10.36±0.46
+13.33±0.29
+15.85±0.22
+Similarity-based
+Sim-based-20
+19.51±0.41
+23.33±0.20
+26.54±0.17
+27.04±0.06
+Sim-based-30
+19.58±0.39
+23.21±0.20
+26.81±0.19
+27.13±0.07
+Ours
+RCsearcher
+30.75±0.41
+31.45±0.35
+31.92±0.28
+32.38±0.18
+center is the input sequence. We use Transformer (Vaswani
+et al., 2017) as the neural sequence-to-sequence model. We
+follow the same data augmentation tricks used by (Wan
+et al., 2022) for the SMILES generative models. Instead
+of expanding the training dataset off-the-shelf, we perform
+the augmentation on-the-fly. At each iteration, there is a
+probability of 50% to permute the SMILES. We refer to this
+baseline as Product2RC and Product2RC-Aug.
+(3) Similarity-based: We calculate the similarity between
+the input product and each product in the training set and
+then sort the products in training set based on the similarity.
+We regard the reaction center of the corresponding product
+in training set as the prediction. Here, we use molecular
+morgan fingerprints to calculate the similarity. We refer to
+this baseline as Sim-based.
+It is worth noting that Product2RC and Sim-based can
+only predict the reaction center pattern and cannot accurately
+obtain the specific position of the reaction center in the
+product. We regard the reaction center pattern as the query
+graph and perform subgraph matching on the product. When
+there is only one solution in the matching result, the only
+solution is regarded as the predicted result, otherwise, we
+randomly select a solution as the predicted result. In this
+way, we can calculate the top-k results of the overall test
+set once. We repeat this experiment N times, and we finally
+take the average top-k accuracy of these N times as the final
+result.
+Evaluation Metric For different methods, we transform the
+predicted results into the predicted reaction center graph’s
+node set ˆVT
+rc. The condition for correct identification is that
+the node set ˆVT
+rc of the predicted reaction center graph is
+consistent with the ground-truth Vrc. We use the top-k exact
+match accuracy as our evaluation metrics. The k predictions
+containing ground-truth are counted as correct.
+Implementation Settings Our proposed RCsearcher is im-
+plemented with Deep Graph Library (DGL) (Wang et al.,
+2019) and Pytorch (Paszke et al., 2019). As for the EGAT,
+we stack four identical four-head attentive layers of which
+the hidden dimension is 256. All embedding sizes in the
+model are set to 256. We use ELU(x) = α(exp(x) − 1)
+for x ≤ 0 and x for x > 0 as our activation function where
+α = 1. We conduct all the experiments on a machine with
+an Intel Xeon 4114 CPU and one Nvidia Titan GPU. For
+training, we set the learning rate to 0.001, the number of
+training iterations to 100000, and use the Adam optimizer
+(Kingma & Ba, 2015). The first 10000 iterations are imita-
+tion learning. Checkpoints are saved for each 1000 iteration
+to select the best checkpoints on the evaluation set. The
+source code can be found in the supplementary materials.
+5.1. Overall Performance
+We operate the training process five times and report the
+mean and standard deviation of accuracy. The overall per-
+formance is illustrated in Table 1. We have two observations.
+First, our method achieves state-of-the-art performance on
+the dataset. It indicates that our method can better handle
+single and multiple reaction center identification. Second,
+the average performance of Product2RC and Sim-based
+with 20 and 30 sampling is close. It implies that the average
+of 30 results is enough to fairly reflect the capabilities of
+Product2RC and Sim-based.
+5.2. Performance on Multiple Reaction Center
+To address the Q2, we train RCsearcher on the train data
+with multiple reaction center to explore its performance on
+the test data with multiple reaction center. The multiple re-
+action center identification results are shown in Fig. 5. The
+overall top-1 accuracy of Rcsearcher is about eight times
+higher than that of prior GNN-based methods. Also, the
+prior GNN-based methods completely fail on data with the
+reaction center consisting of more than two edges. Con-
+versely, RCsearcher still works. It shows that RCsearcher’s
+performance on multiple reaction center identification is
+
+RCsearcher: Reaction Center Identification in Retrosynthesis via Deep Q-Learning
+2
+3
+4
+5
+6+
+total
+# Edges in Multiple Reaction Center
+0
+10
+20
+30
+40
+50
+Top-1 Accuracy(%)
+MPN-based
+EGAT-based
+RGCN-based
+RCsearcher
+Figure 5. The performance of the multiple reaction center identifi-
+cation.
+MPN-based
+EGAT-based
+RGCN-Based RCsearcher
+Methods
+0
+5
+10
+15
+20
+Numbers
+ 3
+ 1
+ 2
+21
+ 3
+ 1
+ 2
+21
+Figure 6. The performance of the extrapolations.
+significantly better than the baselines. We attribute it to
+consistency between our optimization objectives and the
+purpose of finding the right single or multiple reaction cen-
+ter.
+5.3. Performance of Generalization
+In order to evaluate the generalization ability of the RC-
+searcher, we count the number of reaction center patterns
+that are extrapolated successfully and are not in the training
+set. The results are shown in Figure 6. The number of ex-
+trapolations of Rcsearcher is about twenty times higher than
+that of prior GNN-based methods. It demonstrates that our
+method can predict novel reaction center patterns that have
+not appeared in the training set.
+5.4. Ablation Study
+The results of the ablation study are illustrated in Table 2, on
+which we have the two observations: 1) The setting with the
+beam search results is 0.52 percentage points higher than
+the setting without the beam search, which indicates beam
+search can improve performance. 2) The setting with the
+one-hop constrain results are 3.62 percentage points higher
+than the setting without the one-hop constrain. It demon-
+64
+128
+256
+512
+Dimension
+15
+20
+25
+30
+35
+40
+Top-1 Accuracy(%)
+29.01
+9.13
+2
+75
+30.
+30.22
+2
+8
+15
+20
+25
+30
+35
+40
+Top-1 Accuracy(%)
+30.57
+0.75
+3
+3
+3
+28.
+29.02
+4
+ 
+6
+# Heads
+Figure 7. Hyperparameter Sensitivity Analysis.
+strates that although the one-hop constraint fails in some
+samples with reaction center of multi-connected branches,
+it can improve the model’s performance.
+Table 2. The Results of Ablation Study.
+Ablation Setting
+Top-1 Accuracy
+w/ beam search
+30.75%
+w/o beam search
+30.23%
+w/ one-hop constrain
+30.75%
+w/o one-hop constrain
+27.13%
+5.5. Hyperparameter Sensitivity Analysis
+We respectively fix the number of heads to 4 and the di-
+mension of hidden layers d to 256 to explore the impact of
+several vital hyperparameters (Fig. 7). The performance
+under different hyperparameters is consistent, revealing the
+robustness of our method. We observe that the performance
+of the RCsearcher improves as the dimension of hidden
+layers increases. We hypothesise that the higher dimension
+allows the model to contain more parameters, thus improv-
+ing experimental results.
+6. Conclusion and Future Work
+We present RCsearcher, a unified framework for single and
+multiple reaction center identification that combines the
+advantages of the graph neural network and deep reinforce-
+ment learning. The critical insight in this framework is that
+the single or multiple reaction center must be a node-induced
+subgraph of the molecular product graph. Comprehensive
+experiments demonstrate that RCsearcher consistently out-
+performs other baselines for the reaction centre identifica-
+tion task and can extrapolate the reaction center patterns
+that have not appeared in the training set. Ablation exper-
+iments verify the effectiveness of individual components.
+However, RCsearche is still limited to the reaction center of
+the single connected branch. In the future, we will not only
+take advantage of the one-hop constraint but also improve
+the current mechanism so that the model can generalize to
+the reaction center of multi-connected branches.
+
+RCsearcher: Reaction Center Identification in Retrosynthesis via Deep Q-Learning
+A. Dataset information
+We adopt stratified random sampling for USPTO-full and take out 40k samples, ensuring that the data distribution is close
+enough to USPTO-full. More detailed information about the distribution of the dataset can be found in Figure 8.
+Figure 8. The detailed distribution of the USPTO-40k.
+B. Atom and bond features
+In this paper, initial features of atom and bond can be found in Table 3 and Table 4.
+Table 3. Atom Features used in EGAT. All features are one-hot encoding.
+Feature
+Description
+Size
+Atom type
+Type of an atom by atomic number.
+100
+Total degree
+Degree of an atom including Hs.
+6
+Explicit valence
+Explicit valence of an atom.
+6
+Implicit valence
+Explicit valence of an atom.
+6
+Hybridization
+sp, sp2, sp3, sp3d, or sp3d2.
+5
+# Hs
+Number of bonded Hydrogen atom.
+5
+Formal charge
+Integer electronic charge assigned to atom.
+5
+Aromaticity
+Whether an atom is part of an aromatic system.
+1
+In ring
+Whether an atom is in ring
+1
+Table 4. Bond features used in EGAT. All features are one-hot encoding.
+Feature
+Description
+Size
+Bond type
+Single, double, triple, or aromatic.
+4
+Conjugation
+Whether the bond is conjugated.
+1
+In ring
+Whether the bond is part of a ring.
+1
+Stereo
+None, any, E/Z or cis/trans.
+6
+Direction
+The direction of the bond.
+3
+C. Hyperparameters for baselines
+For fair comparison, we set the number of layers, size of hidden dimensions and the number of attention heads in three
+GNN-based baselines to 4, 156, and 4 respectively. In Seq2Seq-based baseline, we set the number of layers, size of hidden
+dimensions and the number of attention heads in Transformer (Lan et al., 2021; 2022; 2023; Ma et al., 2021) to 4, 2048, 8
+respectively.
+
+2
+28
+2
+2
+5
+24
+2
+19
+17
+MultipleReaction
+16
+15
+4
+2#
+10
+6
+8
+7
+9
+3
+2
+10%
+.0
+0
+0.00
+0
+Proportion in USPTO-40KRCsearcher: Reaction Center Identification in Retrosynthesis via Deep Q-Learning
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diff --git a/hNFLT4oBgHgl3EQfZy_u/content/tmp_files/load_file.txt b/hNFLT4oBgHgl3EQfZy_u/content/tmp_files/load_file.txt
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index 0000000000000000000000000000000000000000..7a1ca323e6b7beb363d2efd864bdae77507af2fb
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@@ -0,0 +1,845 @@
+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFLT4oBgHgl3EQfZy_u/content/2301.12071v1.pdf,len=844
+page_content='RCsearcher: Reaction Center Identification in Retrosynthesis  via Deep Q-Learning  Zixun Lan 1 Zuo Zeng 2 Binjie Hong 3 Zhenfu Liu 2 Fei Ma� 1  Abstract  The reaction center consists of atoms in the prod-  uct whose local properties are not identical to the  corresponding atoms in the reactants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFLT4oBgHgl3EQfZy_u/content/2301.12071v1.pdf'}
+page_content=' Prior stud-  ies on reaction center identification are mainly  on semi-templated retrosynthesis methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFLT4oBgHgl3EQfZy_u/content/2301.12071v1.pdf'}
+page_content=' More-  over, they are limited to single reaction center  identification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFLT4oBgHgl3EQfZy_u/content/2301.12071v1.pdf'}
+page_content=' However, many reaction centers  are comprised of multiple bonds or atoms in re-  ality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFLT4oBgHgl3EQfZy_u/content/2301.12071v1.pdf'}
+page_content=' We refer to it as the multiple reaction cen-  ter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFLT4oBgHgl3EQfZy_u/content/2301.12071v1.pdf'}
+page_content=' This paper presents RCsearcher, a unified  framework for single and multiple reaction cen-  ter identification that combines the advantages of  the graph neural network and deep reinforcement  learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFLT4oBgHgl3EQfZy_u/content/2301.12071v1.pdf'}
+page_content=' The critical insight in this framework is  that the single or multiple reaction center must be  a node-induced subgraph of the molecular prod-  uct graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFLT4oBgHgl3EQfZy_u/content/2301.12071v1.pdf'}
+page_content=' At each step, it considers choosing one  node in the molecular product graph and adding  it to the explored node-induced subgraph as an  action.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFLT4oBgHgl3EQfZy_u/content/2301.12071v1.pdf'}
+page_content=' Comprehensive experiments demonstrate  that RCsearcher consistently outperforms other  baselines and can extrapolate the reaction center  patterns that have not appeared in the training set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFLT4oBgHgl3EQfZy_u/content/2301.12071v1.pdf'}
+page_content='  Ablation experiments verify the effectiveness of  individual components, including the beam search  and one-hop constraint of action space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFLT4oBgHgl3EQfZy_u/content/2301.12071v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFLT4oBgHgl3EQfZy_u/content/2301.12071v1.pdf'}
+page_content=' Intruduction  Reaction center identification plays a vital role in single-step  retrosynthesis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFLT4oBgHgl3EQfZy_u/content/2301.12071v1.pdf'}
+page_content=' The reaction center (RC) consists of atoms  in the product whose local properties are not identical to the  corresponding atoms in the reactants (Coley et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFLT4oBgHgl3EQfZy_u/content/2301.12071v1.pdf'}
+page_content=', 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFLT4oBgHgl3EQfZy_u/content/2301.12071v1.pdf'}
+page_content='  Experienced chemists first disconnect the target molecule  1Department of Applied Mathematics, Xi’an Jiaotong-  Liverpool University, Suzhou, China 2Hours Technology Co.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFLT4oBgHgl3EQfZy_u/content/2301.12071v1.pdf'}
+page_content=', Ltd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFLT4oBgHgl3EQfZy_u/content/2301.12071v1.pdf'}
+page_content=',  Suzhou, China 3Department of Information and Computing Sci-  ence, Xi’an Jiaotong-Liverpool University, Suzhou, China.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFLT4oBgHgl3EQfZy_u/content/2301.12071v1.pdf'}
+page_content=' Corre-  spondence to: Fei ma <Fei.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFLT4oBgHgl3EQfZy_u/content/2301.12071v1.pdf'}
+page_content='Ma@xjtlu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFLT4oBgHgl3EQfZy_u/content/2301.12071v1.pdf'}
+page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFLT4oBgHgl3EQfZy_u/content/2301.12071v1.pdf'}
+page_content='cn>.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFLT4oBgHgl3EQfZy_u/content/2301.12071v1.pdf'}
+page_content='  RCsearcher: Reaction Center Identification in Retrosynthesis via  Deep Q-Learning  single reaction center:  a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFLT4oBgHgl3EQfZy_u/content/2301.12071v1.pdf'}
+page_content='  multiple reaction center:  incorrect reaction center  identical  b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFLT4oBgHgl3EQfZy_u/content/2301.12071v1.pdf'}
+page_content='  incorrect  Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFLT4oBgHgl3EQfZy_u/content/2301.12071v1.pdf'}
+page_content=' (a) Examples of Single reaction center and multiple reac-  tion center.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFLT4oBgHgl3EQfZy_u/content/2301.12071v1.pdf'}
+page_content=' The red highlight is the reaction center.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFLT4oBgHgl3EQfZy_u/content/2301.12071v1.pdf'}
+page_content=' (b) The case  of the same intermediate molecules are visited many times in the  multi-step retrosynthesis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFLT4oBgHgl3EQfZy_u/content/2301.12071v1.pdf'}
+page_content=' The red highlight and arrow denote the  predicted reaction center and retrosynthetic prediction respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFLT4oBgHgl3EQfZy_u/content/2301.12071v1.pdf'}
+page_content='  through the potential reaction center when performing ret-  rosynthesis analysis (Corey, 1991).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFLT4oBgHgl3EQfZy_u/content/2301.12071v1.pdf'}
+page_content=' In computer-aided ret-  rosynthesis, template-based single-step retrosynthesis meth-  ods (Coley et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFLT4oBgHgl3EQfZy_u/content/2301.12071v1.pdf'}
+page_content=', 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFLT4oBgHgl3EQfZy_u/content/2301.12071v1.pdf'}
+page_content=' Dai et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFLT4oBgHgl3EQfZy_u/content/2301.12071v1.pdf'}
+page_content=', 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFLT4oBgHgl3EQfZy_u/content/2301.12071v1.pdf'}
+page_content=' Chen & Jung, 2021)  rank the reaction templates, where the reaction center is part  of the reaction template.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFLT4oBgHgl3EQfZy_u/content/2301.12071v1.pdf'}
+page_content=' The first step of the semi-template-  based retrosynthesis methods (Shi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFLT4oBgHgl3EQfZy_u/content/2301.12071v1.pdf'}
+page_content=', 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFLT4oBgHgl3EQfZy_u/content/2301.12071v1.pdf'}
+page_content=' Yan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFLT4oBgHgl3EQfZy_u/content/2301.12071v1.pdf'}
+page_content=',  2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFLT4oBgHgl3EQfZy_u/content/2301.12071v1.pdf'}
+page_content=' Somnath et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFLT4oBgHgl3EQfZy_u/content/2301.12071v1.pdf'}
+page_content=', 2021) is to use one graph neural net-  work to predict the reaction center of the target molecule.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFLT4oBgHgl3EQfZy_u/content/2301.12071v1.pdf'}
+page_content='  Moreover, some template-free retrosynthesis methods (Wan  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFLT4oBgHgl3EQfZy_u/content/2301.12071v1.pdf'}
+page_content=', 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFLT4oBgHgl3EQfZy_u/content/2301.12071v1.pdf'}
+page_content=' Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFLT4oBgHgl3EQfZy_u/content/2301.12071v1.pdf'}
+page_content=', 2021) begin explicitly or implicitly  adding reaction center information to improve performance  and interpretability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFLT4oBgHgl3EQfZy_u/content/2301.12071v1.pdf'}
+page_content='  Prior studies on reaction center identification are mainly  on semi-templated retrosynthesis methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFLT4oBgHgl3EQfZy_u/content/2301.12071v1.pdf'}
+page_content=' Moreover, they  are limited to single reaction center identification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFLT4oBgHgl3EQfZy_u/content/2301.12071v1.pdf'}
+page_content=' Here, a  single reaction center refers to the reaction center composed  of a bond or an atom (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFLT4oBgHgl3EQfZy_u/content/2301.12071v1.pdf'}
+page_content=' 1(a)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFLT4oBgHgl3EQfZy_u/content/2301.12071v1.pdf'}
+page_content=' G2G (Shi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFLT4oBgHgl3EQfZy_u/content/2301.12071v1.pdf'}
+page_content=', 2020) and  RetroXpert (Yan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFLT4oBgHgl3EQfZy_u/content/2301.12071v1.pdf'}
+page_content=', 2020) regards the bond that appears  in the product but not in the reactants as the reaction center.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFLT4oBgHgl3EQfZy_u/content/2301.12071v1.pdf'}
+page_content='  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFLT4oBgHgl3EQfZy_u/content/2301.12071v1.pdf'}
+page_content='12071v1  [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFLT4oBgHgl3EQfZy_u/content/2301.12071v1.pdf'}
+page_content='LG]  28 Jan 2023  RCsearcher: Reaction Center Identification in Retrosynthesis via Deep Q-Learning  environment  Action  Reward, New State  Agent  Ours:  Prior GNN-based method:  node1  node2  node3  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFLT4oBgHgl3EQfZy_u/content/2301.12071v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFLT4oBgHgl3EQfZy_u/content/2301.12071v1.pdf'}
+page_content='  edge1  edge2  edge3  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFLT4oBgHgl3EQfZy_u/content/2301.12071v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFLT4oBgHgl3EQfZy_u/content/2301.12071v1.pdf'}
+page_content='  GNN  representation  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFLT4oBgHgl3EQfZy_u/content/2301.12071v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFLT4oBgHgl3EQfZy_u/content/2301.12071v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFLT4oBgHgl3EQfZy_u/content/2301.12071v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFLT4oBgHgl3EQfZy_u/content/2301.12071v1.pdf'}
+page_content='  Reaction center  identification  Reaction Center  Ground Truth  Classification  update  Read-out  Graph level  embedding  3  Infer  Top 3  Node level  embedding  Reaction center  probability  Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFLT4oBgHgl3EQfZy_u/content/2301.12071v1.pdf'}
+page_content=' Left: An illustration of Prior GNN-based method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFLT4oBgHgl3EQfZy_u/content/2301.12071v1.pdf'}
+page_content=' Right: An overview of our proposed RCsearcher.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFLT4oBgHgl3EQfZy_u/content/2301.12071v1.pdf'}
+page_content='  They use R-GCN (Schlichtkrull et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFLT4oBgHgl3EQfZy_u/content/2301.12071v1.pdf'}
+page_content=', 2018) and EGAT  (Kami´nski et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFLT4oBgHgl3EQfZy_u/content/2301.12071v1.pdf'}
+page_content=', 2022) respectively to predict the probabil-  ity of each bond as a single reaction center.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFLT4oBgHgl3EQfZy_u/content/2301.12071v1.pdf'}
+page_content=' GRAPHRETRO  (Somnath et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFLT4oBgHgl3EQfZy_u/content/2301.12071v1.pdf'}
+page_content=', 2021) also considers the atom whose num-  ber of attached hydrogens changes in the product as a single  reaction center.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFLT4oBgHgl3EQfZy_u/content/2301.12071v1.pdf'}
+page_content=' Then, it uses MPN (Gilmer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFLT4oBgHgl3EQfZy_u/content/2301.12071v1.pdf'}
+page_content=', 2017)  to model the probability of each bond or atom as a single  reaction center.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFLT4oBgHgl3EQfZy_u/content/2301.12071v1.pdf'}
+page_content=' Despite the excellent performance of semi-  templated retrosynthesis methods, they are still not widely  used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFLT4oBgHgl3EQfZy_u/content/2301.12071v1.pdf'}
+page_content=' The main reason is that the current reaction center  identification method cannot detect multiple reaction center  effectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFLT4oBgHgl3EQfZy_u/content/2301.12071v1.pdf'}
+page_content='  In reality, many reaction centers in reactions’ products are  composed of multiple bonds or atoms, which is not fully rep-  resented in the USPTO-50k dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFLT4oBgHgl3EQfZy_u/content/2301.12071v1.pdf'}
+page_content=' In this paper, we refer  to these as the multiple reaction center (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFLT4oBgHgl3EQfZy_u/content/2301.12071v1.pdf'}
+page_content=' 1(a)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFLT4oBgHgl3EQfZy_u/content/2301.12071v1.pdf'}
+page_content=' (Dai et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFLT4oBgHgl3EQfZy_u/content/2301.12071v1.pdf'}
+page_content=',  2019) creates a large dataset USPTO-full from the entire set  of USPTO 1976-2016 (1,808,937 raw reactions).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFLT4oBgHgl3EQfZy_u/content/2301.12071v1.pdf'}
+page_content=' USPTO-  full has roughly 1M unique reactions, and the proportion  of samples with multiple reaction center reaches 38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFLT4oBgHgl3EQfZy_u/content/2301.12071v1.pdf'}
+page_content='84%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFLT4oBgHgl3EQfZy_u/content/2301.12071v1.pdf'}
+page_content='  For example, the reaction center of the product of most  ring-opening reactions is the multiple reaction center (Coley  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFLT4oBgHgl3EQfZy_u/content/2301.12071v1.pdf'}
+page_content=', 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFLT4oBgHgl3EQfZy_u/content/2301.12071v1.pdf'}
+page_content=' In retrosynthesis, the prior models sometimes  misidentify multiple reaction center as the single reaction  center leading to predictions deviating from the ground truth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFLT4oBgHgl3EQfZy_u/content/2301.12071v1.pdf'}
+page_content='  It results in the same intermediate molecules being visited  many times in the process of multi-step retrosynthesis (Xie  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFLT4oBgHgl3EQfZy_u/content/2301.12071v1.pdf'}
+page_content=', 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFLT4oBgHgl3EQfZy_u/content/2301.12071v1.pdf'}
+page_content=' As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFLT4oBgHgl3EQfZy_u/content/2301.12071v1.pdf'}
+page_content=' 1(b), due to misidentification  of multiple reaction center, the ring-opening operation is  not performed, which leads to an infinite loop in multi-step  retrosynthesis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFLT4oBgHgl3EQfZy_u/content/2301.12071v1.pdf'}
+page_content='  To the best of our knowledge, the only existing method  that can simultaneously address single and multiple reaction  center identification task appear in RetroXpert (Yan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFLT4oBgHgl3EQfZy_u/content/2301.12071v1.pdf'}
+page_content=',  2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFLT4oBgHgl3EQfZy_u/content/2301.12071v1.pdf'}
+page_content=' It uses one shared EGAT (Kami´nski et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFLT4oBgHgl3EQfZy_u/content/2301.12071v1.pdf'}
+page_content=', 2022)  to output two tensors (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFLT4oBgHgl3EQfZy_u/content/2301.12071v1.pdf'}
+page_content=' 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFLT4oBgHgl3EQfZy_u/content/2301.12071v1.pdf'}
+page_content=' One tensor represents the  probability that each bond is the reaction center.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFLT4oBgHgl3EQfZy_u/content/2301.12071v1.pdf'}
+page_content=' Another  tensor indicates the result of multi-classification, where  each category indicates the number of bonds forming the  reaction center.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFLT4oBgHgl3EQfZy_u/content/2301.12071v1.pdf'}
+page_content=' During inference, the prediction of the  multi-classification task is used to select the corresponding  number of bonds as the reaction center according to the  predicted probabilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFLT4oBgHgl3EQfZy_u/content/2301.12071v1.pdf'}
+page_content=' There is a gap between its two  joint optimization objectives and the purpose of finding the  right single or multiple reaction center.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFLT4oBgHgl3EQfZy_u/content/2301.12071v1.pdf'}
+page_content=' Hence, designing a  unified and reasonable framework for single and multiple  reaction center identification remains a challenge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFLT4oBgHgl3EQfZy_u/content/2301.12071v1.pdf'}
+page_content='  In this paper, we present RCsearcher, a unified framework  for single and multiple reaction center identification that  combines the advantages of graph neural network and deep  reinforcement learning (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFLT4oBgHgl3EQfZy_u/content/2301.12071v1.pdf'}
+page_content=' 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFLT4oBgHgl3EQfZy_u/content/2301.12071v1.pdf'}
+page_content=' The critical insight in  this framework is that the single or multiple reaction center  must be a node-induced subgraph of the molecular product  graph (Coley et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFLT4oBgHgl3EQfZy_u/content/2301.12071v1.pdf'}
+page_content=', 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFLT4oBgHgl3EQfZy_u/content/2301.12071v1.pdf'}
+page_content=' RCsearcher represents states and  actions in continuous embeddings by graph representation  learning (Kami´nski et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFLT4oBgHgl3EQfZy_u/content/2301.12071v1.pdf'}
+page_content=', 2022) and uses a Deep Q-Network  (DQN) (Mnih et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFLT4oBgHgl3EQfZy_u/content/2301.12071v1.pdf'}
+page_content=', 2013) to predict action distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFLT4oBgHgl3EQfZy_u/content/2301.12071v1.pdf'}
+page_content=' At  each step, it considers choosing one node in the molecular  product graph and adding it to the explored node-induced  subgraph as an action.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFLT4oBgHgl3EQfZy_u/content/2301.12071v1.pdf'}
+page_content=' After satisfying the stop condition,  RCsearcher regards the explored node-induced subgraph as  the predicted reaction center.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFLT4oBgHgl3EQfZy_u/content/2301.12071v1.pdf'}
+page_content=' For effective and fair evalua-  tion, we stratifingly sample 40k reactions from USPTO-full  to build a general dataset USPTO-40k, where the proportion  of samples with multiple reaction center is also 38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFLT4oBgHgl3EQfZy_u/content/2301.12071v1.pdf'}
+page_content='84%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFLT4oBgHgl3EQfZy_u/content/2301.12071v1.pdf'}
+page_content='  Comprehensive experiments demonstrate that RCsearcher  consistently outperforms other baselines for the reaction  centre identification task, and it can extrapolate the reac-  tion center patterns that have not appeared in the training  set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFLT4oBgHgl3EQfZy_u/content/2301.12071v1.pdf'}
+page_content=' Ablation experiments verify the effectiveness of indi-  vidual components, including the beam search and one-hop  constraint of action space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFLT4oBgHgl3EQfZy_u/content/2301.12071v1.pdf'}
+page_content=' In brief, we highlight our main  contributions as follows:  We address the important yet challenging task of reac-  tion center identification and propose a unified frame-  RCsearcher: Reaction Center Identification in Retrosynthesis via Deep Q-Learning  work RCsearcher both for single and multiple reaction  center as the solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFLT4oBgHgl3EQfZy_u/content/2301.12071v1.pdf'}
+page_content='  The key novelty is the GNN-based DQN which pro-  vides a strategy to select the node-induced subgraph  as the predicted reaction center, and the core insight in  this framework is that single or multiple reaction cen-  ter must be a node-induced subgraph of the molecular  product graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFLT4oBgHgl3EQfZy_u/content/2301.12071v1.pdf'}
+page_content='  For fair and effective evaluation, we build a general  dataset USPTO-40k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFLT4oBgHgl3EQfZy_u/content/2301.12071v1.pdf'}
+page_content=' We conduct extensive experi-  ments to demonstrate the effectiveness and general-  ization of the RCsearcher.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFLT4oBgHgl3EQfZy_u/content/2301.12071v1.pdf'}
+page_content=' Ablation experiments also  verify the effectiveness of individual components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFLT4oBgHgl3EQfZy_u/content/2301.12071v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFLT4oBgHgl3EQfZy_u/content/2301.12071v1.pdf'}
+page_content=' Problem Definition  The reaction center (RC) consists of atoms in the product  whose local properties are not identical to the corresponding  atoms in the reactants (Coley et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFLT4oBgHgl3EQfZy_u/content/2301.12071v1.pdf'}
+page_content=', 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFLT4oBgHgl3EQfZy_u/content/2301.12071v1.pdf'}
+page_content=' In this paper,  we use RDChiral (Coley et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFLT4oBgHgl3EQfZy_u/content/2301.12071v1.pdf'}
+page_content=', 2019) to extract the super  general reaction center.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFLT4oBgHgl3EQfZy_u/content/2301.12071v1.pdf'}
+page_content=' Hence, regardless of whether the  single or multiple reaction center must be a node-induced  subgraph of the molecular product graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFLT4oBgHgl3EQfZy_u/content/2301.12071v1.pdf'}
+page_content='  Notation A molecular product graph Gp = (Vp, Ep) is  represented as a set of |Vp| nodes (atoms) and a set of  |Ep| edges (bonds).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFLT4oBgHgl3EQfZy_u/content/2301.12071v1.pdf'}
+page_content='  The reaction center graph Grc =  (Vrc, Erc) is a node-induced subgraph of the molecu-  lar product graph Gp such that Vrc ⊆ Vp and Erc =  {(u, v) | u, v ∈ Vrc, (u, v) ∈ Ep}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFLT4oBgHgl3EQfZy_u/content/2301.12071v1.pdf'}
+page_content='  Reaction Center Identification Given the molecular prod-  uct graph Gp, reaction center identification aims to detect  the corresponding reaction center graph Grc, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFLT4oBgHgl3EQfZy_u/content/2301.12071v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFLT4oBgHgl3EQfZy_u/content/2301.12071v1.pdf'}
+page_content=' Grc’s node  set Vrc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFLT4oBgHgl3EQfZy_u/content/2301.12071v1.pdf'}
+page_content='  Evaluation There may be several isomorphic node-induced  subgraphs in a graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFLT4oBgHgl3EQfZy_u/content/2301.12071v1.pdf'}
+page_content=' Therefore, the condition for correct  identification is that the node set of the predicted reaction  center graph is consistent with the ground-truth Vrc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFLT4oBgHgl3EQfZy_u/content/2301.12071v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFLT4oBgHgl3EQfZy_u/content/2301.12071v1.pdf'}
+page_content=' Related work  Efforts on Reaction Center Identification in Retrosyn-  thesis  G2G (Shi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFLT4oBgHgl3EQfZy_u/content/2301.12071v1.pdf'}
+page_content=', 2020) and RetroXpert (Yan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFLT4oBgHgl3EQfZy_u/content/2301.12071v1.pdf'}
+page_content=',  2020) regard the bond that appears in the product but not  in the reactants as the reaction center.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFLT4oBgHgl3EQfZy_u/content/2301.12071v1.pdf'}
+page_content=' G2G uses R-GCN  (Schlichtkrull et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFLT4oBgHgl3EQfZy_u/content/2301.12071v1.pdf'}
+page_content=', 2018) to predict the probability of each  bond as a single reaction center.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFLT4oBgHgl3EQfZy_u/content/2301.12071v1.pdf'}
+page_content=' In contrast, RetroXpert  uses EGAT (Kami´nski et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFLT4oBgHgl3EQfZy_u/content/2301.12071v1.pdf'}
+page_content=', 2022) to model the probability  of each bond as the reaction center, and it adds a graph-level  auxiliary task to predict the total number of disconnection  bonds (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFLT4oBgHgl3EQfZy_u/content/2301.12071v1.pdf'}
+page_content=' 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFLT4oBgHgl3EQfZy_u/content/2301.12071v1.pdf'}
+page_content=' Hence, RetroXpert is not limited to single  reaction center identification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFLT4oBgHgl3EQfZy_u/content/2301.12071v1.pdf'}
+page_content=' GRAPHRETRO (Somnath  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFLT4oBgHgl3EQfZy_u/content/2301.12071v1.pdf'}
+page_content=', 2021) also considers the atom whose number of at-  tached hydrogens changes in the product as a single reaction  center.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFLT4oBgHgl3EQfZy_u/content/2301.12071v1.pdf'}
+page_content=' Then, it uses MPN (Gilmer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFLT4oBgHgl3EQfZy_u/content/2301.12071v1.pdf'}
+page_content=', 2017) to model  the probability of each bond or atom as a single reaction  center.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFLT4oBgHgl3EQfZy_u/content/2301.12071v1.pdf'}
+page_content=' It is worth noting that GRAPHRETRO roughly pro-  posed an autoregressive model for multiple reaction center  identification in its appendix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFLT4oBgHgl3EQfZy_u/content/2301.12071v1.pdf'}
+page_content=' However, GRAPHRETRO  does not elaborate on the specific details of the autoregres-  sive model, and the code of the autoregressive model is  not available in the official GitHub link of GRAPHRETRO  (https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFLT4oBgHgl3EQfZy_u/content/2301.12071v1.pdf'}
+page_content='com/vsomnath/graphretro).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFLT4oBgHgl3EQfZy_u/content/2301.12071v1.pdf'}
+page_content='  Efforts  on  Single-step  Retrosynthesis  Prediction  Single-step prediction models can be categorized into  three main classes, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFLT4oBgHgl3EQfZy_u/content/2301.12071v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFLT4oBgHgl3EQfZy_u/content/2301.12071v1.pdf'}
+page_content=', template-based, template-free and  semi-template-based method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFLT4oBgHgl3EQfZy_u/content/2301.12071v1.pdf'}
+page_content=' Template-based methods rely  on templates that encode the core of chemical reactions,  converting product molecules into reactants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFLT4oBgHgl3EQfZy_u/content/2301.12071v1.pdf'}
+page_content=' The key is to  rank the templates and select the appropriate one to apply,  and recent attempts (Coley et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFLT4oBgHgl3EQfZy_u/content/2301.12071v1.pdf'}
+page_content=', 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFLT4oBgHgl3EQfZy_u/content/2301.12071v1.pdf'}
+page_content=' Dai et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFLT4oBgHgl3EQfZy_u/content/2301.12071v1.pdf'}
+page_content=', 2019)  have addressed the template selection problem by similarity  and neural networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFLT4oBgHgl3EQfZy_u/content/2301.12071v1.pdf'}
+page_content=' Despite their superior interpretability,  template-based approaches are disadvantaged by poor  generalization to structures beyond templates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFLT4oBgHgl3EQfZy_u/content/2301.12071v1.pdf'}
+page_content=' On the other  hand, template-free methods (Liu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFLT4oBgHgl3EQfZy_u/content/2301.12071v1.pdf'}
+page_content=', 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFLT4oBgHgl3EQfZy_u/content/2301.12071v1.pdf'}
+page_content=' Schwaller  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFLT4oBgHgl3EQfZy_u/content/2301.12071v1.pdf'}
+page_content=', 2019) regard single-step retrosynthesis prediction as a  translation task and translate a product molecule represented  in SMILES strings (Weininger, 1988) to reactant SMILES  strings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFLT4oBgHgl3EQfZy_u/content/2301.12071v1.pdf'}
+page_content=' To combine the benefits of template-based and  template-free methods, recent works (Shi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFLT4oBgHgl3EQfZy_u/content/2301.12071v1.pdf'}
+page_content=', 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFLT4oBgHgl3EQfZy_u/content/2301.12071v1.pdf'}
+page_content=' Yan  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFLT4oBgHgl3EQfZy_u/content/2301.12071v1.pdf'}
+page_content=', 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFLT4oBgHgl3EQfZy_u/content/2301.12071v1.pdf'}
+page_content=' Somnath et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFLT4oBgHgl3EQfZy_u/content/2301.12071v1.pdf'}
+page_content=', 2021) seek semi-template-based  methods where they first predict single reaction center via  graph neural networks and then intermediate synthons are  secondly translated into reactants via seq2seq or graph  translation models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFLT4oBgHgl3EQfZy_u/content/2301.12071v1.pdf'}
+page_content='  Efforts  on  Multi-step  Retrosynthetic  planning  HgSearch (Schwaller et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFLT4oBgHgl3EQfZy_u/content/2301.12071v1.pdf'}
+page_content=', 2020) and Proof Num-  ber Search (Kishimoto et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFLT4oBgHgl3EQfZy_u/content/2301.12071v1.pdf'}
+page_content=', 2019) are traditional heuristic  search algorithms in which chemical feasibility and failed  pathway values are not considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFLT4oBgHgl3EQfZy_u/content/2301.12071v1.pdf'}
+page_content=' (Segler et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFLT4oBgHgl3EQfZy_u/content/2301.12071v1.pdf'}
+page_content=', 2018)  adopt Monte Carlo tree search to generate search trees and  explore multiple synthetic paths dynamically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFLT4oBgHgl3EQfZy_u/content/2301.12071v1.pdf'}
+page_content=' (Chen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFLT4oBgHgl3EQfZy_u/content/2301.12071v1.pdf'}
+page_content=',  2020) devise a neural-based A*-like algorithm that learns  an additional value network with automatically constructed  and only successful paths to bias the search prior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFLT4oBgHgl3EQfZy_u/content/2301.12071v1.pdf'}
+page_content=' (Han  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFLT4oBgHgl3EQfZy_u/content/2301.12071v1.pdf'}
+page_content=', 2022) and (Xie et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFLT4oBgHgl3EQfZy_u/content/2301.12071v1.pdf'}
+page_content=', 2022) use a GNN-based value  network to capture intermolecular/intra-pathway level  information to further improve A*-like retrosynthesis  planning algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFLT4oBgHgl3EQfZy_u/content/2301.12071v1.pdf'}
+page_content=' Notably, (Xie et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFLT4oBgHgl3EQfZy_u/content/2301.12071v1.pdf'}
+page_content=', 2022) observe  that the same intermediate molecules are visited many times  in the searching process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFLT4oBgHgl3EQfZy_u/content/2301.12071v1.pdf'}
+page_content=' Sometimes, the ring-opening  operation is not performed due to misidentifying multiple  reaction center as single reaction center, which leads to an  infinite loop in multi-step retrosynthesis (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFLT4oBgHgl3EQfZy_u/content/2301.12071v1.pdf'}
+page_content=' 1(b)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFLT4oBgHgl3EQfZy_u/content/2301.12071v1.pdf'}
+page_content='  RCsearcher: Reaction Center Identification in Retrosynthesis via Deep Q-Learning  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFLT4oBgHgl3EQfZy_u/content/2301.12071v1.pdf'}
+page_content=' Preliminaries and Proposed Method  In this section, we present our RL based reaction center iden-  tification in retrosynthesis method, RCsearcher.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFLT4oBgHgl3EQfZy_u/content/2301.12071v1.pdf'}
+page_content=' The rest of  Section 4 is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFLT4oBgHgl3EQfZy_u/content/2301.12071v1.pdf'}
+page_content=' Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFLT4oBgHgl3EQfZy_u/content/2301.12071v1.pdf'}
+page_content='1 introduces the  preliminaries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFLT4oBgHgl3EQfZy_u/content/2301.12071v1.pdf'}
+page_content=' Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFLT4oBgHgl3EQfZy_u/content/2301.12071v1.pdf'}
+page_content='2 presents a high-level overview  of how to leverage Deep Q-Network (DQN) (Mnih et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFLT4oBgHgl3EQfZy_u/content/2301.12071v1.pdf'}
+page_content=',  2013) to tackle the reaction center identification in retrosyn-  thesis (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFLT4oBgHgl3EQfZy_u/content/2301.12071v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFLT4oBgHgl3EQfZy_u/content/2301.12071v1.pdf'}
+page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFLT4oBgHgl3EQfZy_u/content/2301.12071v1.pdf'}
+page_content=' Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFLT4oBgHgl3EQfZy_u/content/2301.12071v1.pdf'}
+page_content='3 describes the details of the  state, action, reward (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFLT4oBgHgl3EQfZy_u/content/2301.12071v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFLT4oBgHgl3EQfZy_u/content/2301.12071v1.pdf'}
+page_content='2 Left).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFLT4oBgHgl3EQfZy_u/content/2301.12071v1.pdf'}
+page_content=' Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFLT4oBgHgl3EQfZy_u/content/2301.12071v1.pdf'}
+page_content='4 explains  how to train the DQN efficiently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFLT4oBgHgl3EQfZy_u/content/2301.12071v1.pdf'}
+page_content=' Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFLT4oBgHgl3EQfZy_u/content/2301.12071v1.pdf'}
+page_content='5 shows how  RCsearcher does inference with beam search mechanism  (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFLT4oBgHgl3EQfZy_u/content/2301.12071v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFLT4oBgHgl3EQfZy_u/content/2301.12071v1.pdf'}
+page_content='2 Right).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFLT4oBgHgl3EQfZy_u/content/2301.12071v1.pdf'}
+page_content=' Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFLT4oBgHgl3EQfZy_u/content/2301.12071v1.pdf'}
+page_content='6 analyzes the time complexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFLT4oBgHgl3EQfZy_u/content/2301.12071v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFLT4oBgHgl3EQfZy_u/content/2301.12071v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFLT4oBgHgl3EQfZy_u/content/2301.12071v1.pdf'}
+page_content=' Preliminaries  In this paper, we adopt the EGAT (Kami´nski et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFLT4oBgHgl3EQfZy_u/content/2301.12071v1.pdf'}
+page_content=', 2022) to  compute both the node embeddings and the edge embedding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFLT4oBgHgl3EQfZy_u/content/2301.12071v1.pdf'}
+page_content='  Given H(t) = [h(t)  1 ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFLT4oBgHgl3EQfZy_u/content/2301.12071v1.pdf'}
+page_content=' h(t)  2 ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFLT4oBgHgl3EQfZy_u/content/2301.12071v1.pdf'}
+page_content=' · · · ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFLT4oBgHgl3EQfZy_u/content/2301.12071v1.pdf'}
+page_content=' h(t)  n ] ∈ Rn×d and f (t)  ij  ∈  R1×d are the node embedding matrix and the embedding of  edge (i, j) at the t-th layer repectively, where h(t)  i  ∈ R1×d  is the node-level embedding for node i of the graph and  is also the i-th row of H(t), d is the dimension of node-  level embedding and n is the number of nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFLT4oBgHgl3EQfZy_u/content/2301.12071v1.pdf'}
+page_content=' h(0)  i  and  f (0)  ij are the initial features of node and edge in molecular  product graph Gp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFLT4oBgHgl3EQfZy_u/content/2301.12071v1.pdf'}
+page_content=' The details of the initial features can be  found in the appendix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFLT4oBgHgl3EQfZy_u/content/2301.12071v1.pdf'}
+page_content=' EGAT injects the graph structure into  the attention mechanism by performing masked attention,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFLT4oBgHgl3EQfZy_u/content/2301.12071v1.pdf'}
+page_content='  namely it only computes αij for nodes j ∈ Ni,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFLT4oBgHgl3EQfZy_u/content/2301.12071v1.pdf'}
+page_content=' where Ni is  the first-order neighbors of node i in the graph:  f (t+1)  ij  = LeakyReLU  ��  h(t)  i W  ���f (t)  ij  ��� h(t)  j W  �  A  �  ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFLT4oBgHgl3EQfZy_u/content/2301.12071v1.pdf'}
+page_content='  eij = a · f (t+1)  ij  T ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFLT4oBgHgl3EQfZy_u/content/2301.12071v1.pdf'}
+page_content=' αij =  exp (eij)  �  k∈Ni exp (eik),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFLT4oBgHgl3EQfZy_u/content/2301.12071v1.pdf'}
+page_content='  (1)  where eij ∈ R and αij ∈ R are a non-normalized attention  coefficient and a normalized attention coefficient represent-  ing the weight of message aggregated from node j to node  i respectively in the t-th layer of EGAT,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFLT4oBgHgl3EQfZy_u/content/2301.12071v1.pdf'}
+page_content=' and ∥ is the con-  catenation operation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFLT4oBgHgl3EQfZy_u/content/2301.12071v1.pdf'}
+page_content=' Besides,W ∈ Rd×d, A ∈ R3d×d and  a ∈ R1×d are learnable parameters in the t-th layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFLT4oBgHgl3EQfZy_u/content/2301.12071v1.pdf'}
+page_content='  EGAT employs multi-head attention to stabilize the learning  process of self-attention, similar to Transformer (Vaswani  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFLT4oBgHgl3EQfZy_u/content/2301.12071v1.pdf'}
+page_content=', 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFLT4oBgHgl3EQfZy_u/content/2301.12071v1.pdf'}
+page_content=' If there are K heads, K independent attention  mechanisms execute the Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFLT4oBgHgl3EQfZy_u/content/2301.12071v1.pdf'}
+page_content=' 1, and then their features are  concatenated:  h(t+1)  i  = MLP  �  �∥K  k=1σ  �  � �  j∈Ni  αk  ijh(t)  j Wk  �  �  �  �  (2)  where ∥ represents concatenation, αk  ij are normalized at-  tention coefficients computed by the k-th learnable Wk ∈  Rd×d, Ak ∈ R3d×d and ak ∈ R1×d following Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFLT4oBgHgl3EQfZy_u/content/2301.12071v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFLT4oBgHgl3EQfZy_u/content/2301.12071v1.pdf'}
+page_content=' Be-  sides, MLP denotes multi-perceptron.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFLT4oBgHgl3EQfZy_u/content/2301.12071v1.pdf'}
+page_content=' For simplicity, we  denote the encoding process by EGAT(·) in this study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFLT4oBgHgl3EQfZy_u/content/2301.12071v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFLT4oBgHgl3EQfZy_u/content/2301.12071v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFLT4oBgHgl3EQfZy_u/content/2301.12071v1.pdf'}
+page_content=' Overview of RCSeacher  RCSeacher enables graph representation learning techniques  to tackle the reaction center identification in retrosynthesis  and uses deep Q-learning to select one node that is added  to the current explored subgraph in each search state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFLT4oBgHgl3EQfZy_u/content/2301.12071v1.pdf'}
+page_content=' RC-  Seacher represents states and actions in continuous embed-  dings, and maps (st, at) to a score Q(st, at) via a DQN  which consists of a Graph Neural Network encoder and  learnable components to project the representations into the  final score.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFLT4oBgHgl3EQfZy_u/content/2301.12071v1.pdf'}
+page_content=' Here, st and at denote the state and action re-  spectively in the step t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFLT4oBgHgl3EQfZy_u/content/2301.12071v1.pdf'}
+page_content=' Once RCSeacher is trained, it can be  applied to any new graphs that are unseen during training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFLT4oBgHgl3EQfZy_u/content/2301.12071v1.pdf'}
+page_content='  State st consists of the (1) the molecular product graph Gp,  (2) current explored reaction center graph ˆGt  rc = Gp[ˆVt  rc].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFLT4oBgHgl3EQfZy_u/content/2301.12071v1.pdf'}
+page_content='  Action at is defined as selecting one node from the first-  order neighbor of the current explored subgraph ˆVt  rc, or a  stop action implying stop of the exploration process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFLT4oBgHgl3EQfZy_u/content/2301.12071v1.pdf'}
+page_content=' Since  most reaction center graphs have only one connected branch,  we impose a one-hop constraint on the action space to nar-  row the search space and improve the performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFLT4oBgHgl3EQfZy_u/content/2301.12071v1.pdf'}
+page_content=' For  RCSeacher, given our goal, the intermediate reward is de-  fined as rt = 0 at any intermediate step t and rT = 1 when  predicted node-set ˆVT  rc is consistent with ground-truth Vrc,  otherwise rT = 0 at the final step T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFLT4oBgHgl3EQfZy_u/content/2301.12071v1.pdf'}
+page_content=' Therefore, our opti-  mization goal, maximizing the expected remaining future  reward, means finding a reaction center that is exactly the  same as the ground-truth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFLT4oBgHgl3EQfZy_u/content/2301.12071v1.pdf'}
+page_content='  Since the action space can be large, we leverage the repre-  sentation learning capacity of continuous representations  for DQN design.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFLT4oBgHgl3EQfZy_u/content/2301.12071v1.pdf'}
+page_content=' At state st, for each action at, our DQN  predicts a Q(st, at) representing the remaining future re-  ward after selecting action at.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFLT4oBgHgl3EQfZy_u/content/2301.12071v1.pdf'}
+page_content='  Based on the above in-  sights, we can design a simple DQN leveraging the graph  encoder EGAT (Kami´nski et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFLT4oBgHgl3EQfZy_u/content/2301.12071v1.pdf'}
+page_content=', 2022) to obtain one em-  bedding per node, {hi | ∀i ∈ Vp}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFLT4oBgHgl3EQfZy_u/content/2301.12071v1.pdf'}
+page_content=' Denote ∥ as concate-  nation, READOUT as a readout operation that aggregates  node-level embeddings into subgraph embeddings h ˆGtrc,  and whole-graph embedding hGp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFLT4oBgHgl3EQfZy_u/content/2301.12071v1.pdf'}
+page_content=' A state can then be repre-  sented as hst = hGp∥h ˆGtrc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFLT4oBgHgl3EQfZy_u/content/2301.12071v1.pdf'}
+page_content=' An action can be represented as  hat ∈ {hi | ∀i ∈ Vp} ∪ {hstop}, where hstop denotes the  stop action.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFLT4oBgHgl3EQfZy_u/content/2301.12071v1.pdf'}
+page_content=' The Q function can then be designed as:  Q (st, at)  = MLP (hst∥hat)  = MLP  �  hGp∥h ˆGt  rc∥hi  �  or MLP  �  hGp∥h ˆGtrc∥hstop  �  ,  (3)  where MLP denotes multi-perceptron.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFLT4oBgHgl3EQfZy_u/content/2301.12071v1.pdf'}
+page_content='  RCsearcher: Reaction Center Identification in Retrosynthesis via Deep Q-Learning  2  3  score:0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFLT4oBgHgl3EQfZy_u/content/2301.12071v1.pdf'}
+page_content='82  5  10  4  6  2  score:0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFLT4oBgHgl3EQfZy_u/content/2301.12071v1.pdf'}
+page_content='78  score:0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFLT4oBgHgl3EQfZy_u/content/2301.12071v1.pdf'}
+page_content='74  15  4  5  3  6  11  stop  score:0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFLT4oBgHgl3EQfZy_u/content/2301.12071v1.pdf'}
+page_content='72  score:0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFLT4oBgHgl3EQfZy_u/content/2301.12071v1.pdf'}
+page_content='75  score:0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFLT4oBgHgl3EQfZy_u/content/2301.12071v1.pdf'}
+page_content='71  stop  score:0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFLT4oBgHgl3EQfZy_u/content/2301.12071v1.pdf'}
+page_content='61  ground truth  15  score:0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFLT4oBgHgl3EQfZy_u/content/2301.12071v1.pdf'}
+page_content='73  4  stop  stop  4  score:0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFLT4oBgHgl3EQfZy_u/content/2301.12071v1.pdf'}
+page_content='61  4  stop  score:0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFLT4oBgHgl3EQfZy_u/content/2301.12071v1.pdf'}
+page_content='69  Result of Greedy Search  Top-1 in BeamSearch  Terminal  Pruned  Node  embedding  State  Action  Graph  embedding  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFLT4oBgHgl3EQfZy_u/content/2301.12071v1.pdf'}
+page_content='65  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFLT4oBgHgl3EQfZy_u/content/2301.12071v1.pdf'}
+page_content='41  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFLT4oBgHgl3EQfZy_u/content/2301.12071v1.pdf'}
+page_content='11  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFLT4oBgHgl3EQfZy_u/content/2301.12071v1.pdf'}
+page_content='09  Subgraph  embedding  1  2  3  stop  Q value  Node1  Node2  Node3  Stop  1-hop  constraint  Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFLT4oBgHgl3EQfZy_u/content/2301.12071v1.pdf'}
+page_content=' Left: The details of Q-NET, and representation of state and action.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFLT4oBgHgl3EQfZy_u/content/2301.12071v1.pdf'}
+page_content=' The dashed red line indicates the one-hop constraint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFLT4oBgHgl3EQfZy_u/content/2301.12071v1.pdf'}
+page_content=' Right:  An illustration of Beam Search during inference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFLT4oBgHgl3EQfZy_u/content/2301.12071v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFLT4oBgHgl3EQfZy_u/content/2301.12071v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFLT4oBgHgl3EQfZy_u/content/2301.12071v1.pdf'}
+page_content=' Details of State, Action and Reward  State State st consists of the (1) the molecular product  graph Gp, (2) current explored reaction center graph ˆGt  rc =  Gp[ˆVt  rc].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFLT4oBgHgl3EQfZy_u/content/2301.12071v1.pdf'}
+page_content=' Firstly, we use EGAT (Kami´nski et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFLT4oBgHgl3EQfZy_u/content/2301.12071v1.pdf'}
+page_content=', 2022) to  obtain the graph-level embedding hGp ∈ R1×d representing  the molecular product graph Gp:  {hi | ∀i ∈ Vp} ,  �  f ij | ∀(i, j) ∈ Ep  �  = EGAT (Gp) ,  (4)  hnodes = MEAN ({hi | ∀i ∈ Vp}) ,  hedges = MEAN  ��  f ij | ∀(i, j) ∈ Ep  ��  ,  (5)  h′  Gp = ABS (hnodes, hedges) ∥ (hnodes + hedges) ,  hGp = MLP  �  h′  Gp  �  ,  (6)  where hnodes ∈ R1×d and hedges ∈ R1×d are the whole  graph information from node and edge perspectives respec-  tively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFLT4oBgHgl3EQfZy_u/content/2301.12071v1.pdf'}
+page_content=' ABS denotes absolute difference and ∥ refers to  concatenation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFLT4oBgHgl3EQfZy_u/content/2301.12071v1.pdf'}
+page_content=' It ensures our representations are permuta-  tion invariant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFLT4oBgHgl3EQfZy_u/content/2301.12071v1.pdf'}
+page_content=' Besides, MLP denotes multi-perceptron.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFLT4oBgHgl3EQfZy_u/content/2301.12071v1.pdf'}
+page_content='  Secondly, we use the above node-level embeddings  {hi | ∀i ∈ Vp} to derive the subgraph embedding h ˆGtrc ∈  R1×d representing the current explored reaction center  graph ˆGt  rc:  h ˆGtrc =  �  �  �  ⃗0  , t = 0  MEAN  ��  hi | ∀i ∈ ˆVt  rc  ��  , t ̸= 0  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFLT4oBgHgl3EQfZy_u/content/2301.12071v1.pdf'}
+page_content='  (7)  Since the exploration process did not start at step t = 0  (ˆV0  rc = ∅), we use ⃗0 ∈ R1×d to represent ˆV0  rc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFLT4oBgHgl3EQfZy_u/content/2301.12071v1.pdf'}
+page_content='  Action Since most reaction center graphs have only one  connected branch, we impose a one-hop constraint on the  action space to narrow the search space and to improve the  performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFLT4oBgHgl3EQfZy_u/content/2301.12071v1.pdf'}
+page_content=' In other words, the action at is to select one  node from the first-order neighbour of the current explored  subgraph ˆVt  rc or a stop action implying stop of the explo-  ration process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFLT4oBgHgl3EQfZy_u/content/2301.12071v1.pdf'}
+page_content=' We use the node i’s embedding hi ∈ R1×d  and one learnable embedding hstop ∈ R1×d to represent  the action of the selecting one node and the stop action  respectively:  hat =  �  �  �  hi  , i ∈ Vp, t = 0  hi or hstop , i ∈ ONE-HOP  �  ˆVt  rc  �  , t ̸= 0  ,  (8)  where ONE-HOP is the operation of 1-hop constrain that  can obtain the first-order neighbour of the current explored  subgraph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFLT4oBgHgl3EQfZy_u/content/2301.12071v1.pdf'}
+page_content='  Reward When the action at is the stop action, we refer to  this step t as the final step T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFLT4oBgHgl3EQfZy_u/content/2301.12071v1.pdf'}
+page_content=' We define the reward rt as:  rt =  �  �  �  �  �  0  , t = 0, 1, 2, · · · , T − 1  0  , ˆVT  rc ̸= Vrc, t = T  1  , ˆVT  rc = Vrc, t = T  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFLT4oBgHgl3EQfZy_u/content/2301.12071v1.pdf'}
+page_content='  (9)  Therefore, our optimization goal, maximizing the expected  remaining future reward, means finding a reaction center  that is exactly the same as the ground truth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFLT4oBgHgl3EQfZy_u/content/2301.12071v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFLT4oBgHgl3EQfZy_u/content/2301.12071v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFLT4oBgHgl3EQfZy_u/content/2301.12071v1.pdf'}
+page_content=' Training  We adopt the standard Deep Q-learning framework (Mnih  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFLT4oBgHgl3EQfZy_u/content/2301.12071v1.pdf'}
+page_content=', 2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFLT4oBgHgl3EQfZy_u/content/2301.12071v1.pdf'}
+page_content=' Since imitation learning is known to help with  training stability and performance (Levine & Koltun, 2013),  we allow the agent to follow ground-truth trajectories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFLT4oBgHgl3EQfZy_u/content/2301.12071v1.pdf'}
+page_content=' For  each experience (st, at, rt, st+1) in the experience pool, our  loss function is:  loss = (yt − Q(st, at))2,  yt = γrt + MAX (Q(st+1, at+1)) ,  (10)  RCsearcher: Reaction Center Identification in Retrosynthesis via Deep Q-Learning  where γ ∈ (0, 1] is the decay rate, and yt is a constant and  NO-GRAD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFLT4oBgHgl3EQfZy_u/content/2301.12071v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFLT4oBgHgl3EQfZy_u/content/2301.12071v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFLT4oBgHgl3EQfZy_u/content/2301.12071v1.pdf'}
+page_content=' Inference  We use the beam search mechanism (Meister et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFLT4oBgHgl3EQfZy_u/content/2301.12071v1.pdf'}
+page_content=', 2020) to  derive the k best predictions for inference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFLT4oBgHgl3EQfZy_u/content/2301.12071v1.pdf'}
+page_content=' With a hyperpa-  rameter budget BEAM SIZE k, the agent can transition to at  most BEAM SIZE k number of best new states at any given  state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFLT4oBgHgl3EQfZy_u/content/2301.12071v1.pdf'}
+page_content=' Thus, we have k2 candidate states before the next  level in the beam search process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFLT4oBgHgl3EQfZy_u/content/2301.12071v1.pdf'}
+page_content=' From the k2 candidate  states, we select the k best states according to the predicted  value of the Q function at the next level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFLT4oBgHgl3EQfZy_u/content/2301.12071v1.pdf'}
+page_content=' For example, in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFLT4oBgHgl3EQfZy_u/content/2301.12071v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFLT4oBgHgl3EQfZy_u/content/2301.12071v1.pdf'}
+page_content='2, BEAM SIZE = 3, each level of the beam search can  have up to 3 state nodes (red dashed box).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFLT4oBgHgl3EQfZy_u/content/2301.12071v1.pdf'}
+page_content=' The pseudocode  is shown in Alg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFLT4oBgHgl3EQfZy_u/content/2301.12071v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFLT4oBgHgl3EQfZy_u/content/2301.12071v1.pdf'}
+page_content='  Algorithm 1 Inference with Beam Search  Input: Q function Q(s, a);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFLT4oBgHgl3EQfZy_u/content/2301.12071v1.pdf'}
+page_content=' Initial State s0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFLT4oBgHgl3EQfZy_u/content/2301.12071v1.pdf'}
+page_content=' Beam Size k  Output: k best results K-RESULTS  BEAM = {s0}  CANDIDATE = ∅  repeat  K-RESULTS = ∅  for s in BEAM do  TOP-K-ACTIONS = argtopK  a  (Q(s, a))  for a in TOP-K-ACTIONS do  add (s, a) to CANDIDATE  end for  end for  BEAM′ =  argtopK  (s,a)∈CANDIDATE  (Q(s, a))  CANDIDATE = ∅  BEAM = ∅  for (s, a) in BEAM′ do  s′ ← execute a on s  if a is the stop action then  add (s, a) to CANDIDATE  add s to K-RESULTS  else  add s′ to BEAM  end if  end for  until BEAM = ∅  Return: K-RESULTS  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFLT4oBgHgl3EQfZy_u/content/2301.12071v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFLT4oBgHgl3EQfZy_u/content/2301.12071v1.pdf'}
+page_content=' Complexity Analysis  At each state, the agent calculates the future values which  have worst-case time complexity O (|Vp|) due to the action  space consisting of all atoms in the product at step 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFLT4oBgHgl3EQfZy_u/content/2301.12071v1.pdf'}
+page_content=' Overall  the beam search depth is bounded by |Vp|, and each level of  the beam search has at most BEAM SIZE k states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFLT4oBgHgl3EQfZy_u/content/2301.12071v1.pdf'}
+page_content=' Thus,  the overall time complexity is O  �  k × |Vp|2�  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFLT4oBgHgl3EQfZy_u/content/2301.12071v1.pdf'}
+page_content='  38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFLT4oBgHgl3EQfZy_u/content/2301.12071v1.pdf'}
+page_content='84%  61.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFLT4oBgHgl3EQfZy_u/content/2301.12071v1.pdf'}
+page_content='16%  single reaction center  multiple reaction center  12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFLT4oBgHgl3EQfZy_u/content/2301.12071v1.pdf'}
+page_content='42%  87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFLT4oBgHgl3EQfZy_u/content/2301.12071v1.pdf'}
+page_content='58%  multiple connected branch  single connected branch  Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFLT4oBgHgl3EQfZy_u/content/2301.12071v1.pdf'}
+page_content=' The distribution of the USPTO-40k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFLT4oBgHgl3EQfZy_u/content/2301.12071v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFLT4oBgHgl3EQfZy_u/content/2301.12071v1.pdf'}
+page_content=' Evaluation  In this section, we evaluate the performance of our RC-  searcher with comparisons to few baseline approaches for  the reaction center identification in retrosynthesis, and with  significant goals of addressing the following questions: Q1:  How effective and robust is RCsearcher compared to the  baseline approaches?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFLT4oBgHgl3EQfZy_u/content/2301.12071v1.pdf'}
+page_content=' Q2: How well does RCsearcher per-  form on the data with the multiple reaction center?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFLT4oBgHgl3EQfZy_u/content/2301.12071v1.pdf'}
+page_content=' Q3:  How does the proposed one-hop constrain and beam search  in inference improve performance?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFLT4oBgHgl3EQfZy_u/content/2301.12071v1.pdf'}
+page_content=' Q4: Does RCsearcher  have more robust generalization than baselines?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFLT4oBgHgl3EQfZy_u/content/2301.12071v1.pdf'}
+page_content='  Data Since the data with the multiple reaction center in  USPTO-50k only accounts for about 5%, for effective eval-  uation, we created a new dataset USPTO-40k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFLT4oBgHgl3EQfZy_u/content/2301.12071v1.pdf'}
+page_content=' We adopt  stratified random sampling for USPTO-full and take out 40k  samples, ensuring that the data distribution is close enough  to USPTO-full.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFLT4oBgHgl3EQfZy_u/content/2301.12071v1.pdf'}
+page_content=' In the USPTO-40k, the data with multiple  reaction center accounts for 38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFLT4oBgHgl3EQfZy_u/content/2301.12071v1.pdf'}
+page_content='84%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFLT4oBgHgl3EQfZy_u/content/2301.12071v1.pdf'}
+page_content=' In the USPTO-40k,  12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFLT4oBgHgl3EQfZy_u/content/2301.12071v1.pdf'}
+page_content='42% of the samples’ reaction center consists of more  than one connected branch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFLT4oBgHgl3EQfZy_u/content/2301.12071v1.pdf'}
+page_content=' We randomly select 80% of the  samples as the training set and divide the rest into validation  and test sets with equal sizes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFLT4oBgHgl3EQfZy_u/content/2301.12071v1.pdf'}
+page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFLT4oBgHgl3EQfZy_u/content/2301.12071v1.pdf'}
+page_content=' 4 shows the distribution  of the dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFLT4oBgHgl3EQfZy_u/content/2301.12071v1.pdf'}
+page_content=' The details of the USPTO-40k can be found  in the appendix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFLT4oBgHgl3EQfZy_u/content/2301.12071v1.pdf'}
+page_content='  Baselines We compare RCsearcher to three baselines for  evaluating overall performance: GNN-based, Seq2Seq, and  Similarity-based methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFLT4oBgHgl3EQfZy_u/content/2301.12071v1.pdf'}
+page_content=' These include:  (1) GNN-based: We use the method in RetroXpert.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFLT4oBgHgl3EQfZy_u/content/2301.12071v1.pdf'}
+page_content=' It uses  one graph encoder to model the probability of each bond as  the reaction center, and adds a graph-level auxiliary task to  predict the total number of disconnection bonds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFLT4oBgHgl3EQfZy_u/content/2301.12071v1.pdf'}
+page_content=' Here, we  use three graph encoder, including R-GCN (Schlichtkrull  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFLT4oBgHgl3EQfZy_u/content/2301.12071v1.pdf'}
+page_content=', 2018), EGAT (Kami´nski et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFLT4oBgHgl3EQfZy_u/content/2301.12071v1.pdf'}
+page_content=', 2022), MPN (Gilmer  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFLT4oBgHgl3EQfZy_u/content/2301.12071v1.pdf'}
+page_content=', 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFLT4oBgHgl3EQfZy_u/content/2301.12071v1.pdf'}
+page_content=' Thus, we refer to these three baselines as  RGCN-based, EGAT-based and MPN-based.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFLT4oBgHgl3EQfZy_u/content/2301.12071v1.pdf'}
+page_content='  (2) Seq2Seq: Here, the SMILES string of the product is  the input sequence, and the SMARTS string of the reaction  RCsearcher: Reaction Center Identification in Retrosynthesis via Deep Q-Learning  Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFLT4oBgHgl3EQfZy_u/content/2301.12071v1.pdf'}
+page_content=' Top-k accuracy for Reaction Center Identification on USPTO-40K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFLT4oBgHgl3EQfZy_u/content/2301.12071v1.pdf'}
+page_content=' The notations ’-20’ and ’-30’ denote 20 sampling experiments  and 30 sampling experiments respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFLT4oBgHgl3EQfZy_u/content/2301.12071v1.pdf'}
+page_content='  Methods  Top-1(%)  Top-2(%)  Top-3(%)  Top-4(%)  GNN-based  MPN-based  9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFLT4oBgHgl3EQfZy_u/content/2301.12071v1.pdf'}
+page_content='30±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFLT4oBgHgl3EQfZy_u/content/2301.12071v1.pdf'}
+page_content='56  10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFLT4oBgHgl3EQfZy_u/content/2301.12071v1.pdf'}
+page_content='28±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFLT4oBgHgl3EQfZy_u/content/2301.12071v1.pdf'}
+page_content='38  11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFLT4oBgHgl3EQfZy_u/content/2301.12071v1.pdf'}
+page_content='58±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFLT4oBgHgl3EQfZy_u/content/2301.12071v1.pdf'}
+page_content='31  14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFLT4oBgHgl3EQfZy_u/content/2301.12071v1.pdf'}
+page_content='07±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFLT4oBgHgl3EQfZy_u/content/2301.12071v1.pdf'}
+page_content='14  EGAT-based  18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFLT4oBgHgl3EQfZy_u/content/2301.12071v1.pdf'}
+page_content='04±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFLT4oBgHgl3EQfZy_u/content/2301.12071v1.pdf'}
+page_content='79  23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFLT4oBgHgl3EQfZy_u/content/2301.12071v1.pdf'}
+page_content='49±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFLT4oBgHgl3EQfZy_u/content/2301.12071v1.pdf'}
+page_content='66  25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFLT4oBgHgl3EQfZy_u/content/2301.12071v1.pdf'}
+page_content='33±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFLT4oBgHgl3EQfZy_u/content/2301.12071v1.pdf'}
+page_content='43  25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFLT4oBgHgl3EQfZy_u/content/2301.12071v1.pdf'}
+page_content='78±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFLT4oBgHgl3EQfZy_u/content/2301.12071v1.pdf'}
+page_content='25  RGCN-based  9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFLT4oBgHgl3EQfZy_u/content/2301.12071v1.pdf'}
+page_content='27±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFLT4oBgHgl3EQfZy_u/content/2301.12071v1.pdf'}
+page_content='46  14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFLT4oBgHgl3EQfZy_u/content/2301.12071v1.pdf'}
+page_content='25±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFLT4oBgHgl3EQfZy_u/content/2301.12071v1.pdf'}
+page_content='47  17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFLT4oBgHgl3EQfZy_u/content/2301.12071v1.pdf'}
+page_content='53±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFLT4oBgHgl3EQfZy_u/content/2301.12071v1.pdf'}
+page_content='35  20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFLT4oBgHgl3EQfZy_u/content/2301.12071v1.pdf'}
+page_content='78±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFLT4oBgHgl3EQfZy_u/content/2301.12071v1.pdf'}
+page_content='17  Seq2Seq  Product2RC-20  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFLT4oBgHgl3EQfZy_u/content/2301.12071v1.pdf'}
+page_content='25±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFLT4oBgHgl3EQfZy_u/content/2301.12071v1.pdf'}
+page_content='51  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFLT4oBgHgl3EQfZy_u/content/2301.12071v1.pdf'}
+page_content='95±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFLT4oBgHgl3EQfZy_u/content/2301.12071v1.pdf'}
+page_content='34  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFLT4oBgHgl3EQfZy_u/content/2301.12071v1.pdf'}
+page_content='45±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFLT4oBgHgl3EQfZy_u/content/2301.12071v1.pdf'}
+page_content='28  8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFLT4oBgHgl3EQfZy_u/content/2301.12071v1.pdf'}
+page_content='68±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFLT4oBgHgl3EQfZy_u/content/2301.12071v1.pdf'}
+page_content='19  Product2RC-30  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFLT4oBgHgl3EQfZy_u/content/2301.12071v1.pdf'}
+page_content='15±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFLT4oBgHgl3EQfZy_u/content/2301.12071v1.pdf'}
+page_content='42  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFLT4oBgHgl3EQfZy_u/content/2301.12071v1.pdf'}
+page_content='61±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFLT4oBgHgl3EQfZy_u/content/2301.12071v1.pdf'}
+page_content='33  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFLT4oBgHgl3EQfZy_u/content/2301.12071v1.pdf'}
+page_content='42±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFLT4oBgHgl3EQfZy_u/content/2301.12071v1.pdf'}
+page_content='20  8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFLT4oBgHgl3EQfZy_u/content/2301.12071v1.pdf'}
+page_content='48±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFLT4oBgHgl3EQfZy_u/content/2301.12071v1.pdf'}
+page_content='13  Product2RC-Aug-20  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFLT4oBgHgl3EQfZy_u/content/2301.12071v1.pdf'}
+page_content='78±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFLT4oBgHgl3EQfZy_u/content/2301.12071v1.pdf'}
+page_content='37  10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFLT4oBgHgl3EQfZy_u/content/2301.12071v1.pdf'}
+page_content='22±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFLT4oBgHgl3EQfZy_u/content/2301.12071v1.pdf'}
+page_content='20  12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFLT4oBgHgl3EQfZy_u/content/2301.12071v1.pdf'}
+page_content='70±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFLT4oBgHgl3EQfZy_u/content/2301.12071v1.pdf'}
+page_content='16  15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFLT4oBgHgl3EQfZy_u/content/2301.12071v1.pdf'}
+page_content='00±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFLT4oBgHgl3EQfZy_u/content/2301.12071v1.pdf'}
+page_content='06  Product2RC-Aug-30  8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFLT4oBgHgl3EQfZy_u/content/2301.12071v1.pdf'}
+page_content='51±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFLT4oBgHgl3EQfZy_u/content/2301.12071v1.pdf'}
+page_content='40  10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFLT4oBgHgl3EQfZy_u/content/2301.12071v1.pdf'}
+page_content='36±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFLT4oBgHgl3EQfZy_u/content/2301.12071v1.pdf'}
+page_content='46  13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFLT4oBgHgl3EQfZy_u/content/2301.12071v1.pdf'}
+page_content='33±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFLT4oBgHgl3EQfZy_u/content/2301.12071v1.pdf'}
+page_content='29  15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFLT4oBgHgl3EQfZy_u/content/2301.12071v1.pdf'}
+page_content='85±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFLT4oBgHgl3EQfZy_u/content/2301.12071v1.pdf'}
+page_content='22  Similarity-based  Sim-based-20  19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFLT4oBgHgl3EQfZy_u/content/2301.12071v1.pdf'}
+page_content='51±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFLT4oBgHgl3EQfZy_u/content/2301.12071v1.pdf'}
+page_content='41  23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFLT4oBgHgl3EQfZy_u/content/2301.12071v1.pdf'}
+page_content='33±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFLT4oBgHgl3EQfZy_u/content/2301.12071v1.pdf'}
+page_content='20  26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFLT4oBgHgl3EQfZy_u/content/2301.12071v1.pdf'}
+page_content='54±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFLT4oBgHgl3EQfZy_u/content/2301.12071v1.pdf'}
+page_content='17  27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFLT4oBgHgl3EQfZy_u/content/2301.12071v1.pdf'}
+page_content='04±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFLT4oBgHgl3EQfZy_u/content/2301.12071v1.pdf'}
+page_content='06  Sim-based-30  19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFLT4oBgHgl3EQfZy_u/content/2301.12071v1.pdf'}
+page_content='58±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFLT4oBgHgl3EQfZy_u/content/2301.12071v1.pdf'}
+page_content='39  23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFLT4oBgHgl3EQfZy_u/content/2301.12071v1.pdf'}
+page_content='21±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFLT4oBgHgl3EQfZy_u/content/2301.12071v1.pdf'}
+page_content='20  26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFLT4oBgHgl3EQfZy_u/content/2301.12071v1.pdf'}
+page_content='81±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFLT4oBgHgl3EQfZy_u/content/2301.12071v1.pdf'}
+page_content='19  27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFLT4oBgHgl3EQfZy_u/content/2301.12071v1.pdf'}
+page_content='13±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFLT4oBgHgl3EQfZy_u/content/2301.12071v1.pdf'}
+page_content='07  Ours  RCsearcher  30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFLT4oBgHgl3EQfZy_u/content/2301.12071v1.pdf'}
+page_content='75±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFLT4oBgHgl3EQfZy_u/content/2301.12071v1.pdf'}
+page_content='41  31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFLT4oBgHgl3EQfZy_u/content/2301.12071v1.pdf'}
+page_content='45±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFLT4oBgHgl3EQfZy_u/content/2301.12071v1.pdf'}
+page_content='35  31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFLT4oBgHgl3EQfZy_u/content/2301.12071v1.pdf'}
+page_content='92±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFLT4oBgHgl3EQfZy_u/content/2301.12071v1.pdf'}
+page_content='28  32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFLT4oBgHgl3EQfZy_u/content/2301.12071v1.pdf'}
+page_content='38±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFLT4oBgHgl3EQfZy_u/content/2301.12071v1.pdf'}
+page_content='18  center is the input sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFLT4oBgHgl3EQfZy_u/content/2301.12071v1.pdf'}
+page_content=' We use Transformer (Vaswani  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFLT4oBgHgl3EQfZy_u/content/2301.12071v1.pdf'}
+page_content=', 2017) as the neural sequence-to-sequence model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFLT4oBgHgl3EQfZy_u/content/2301.12071v1.pdf'}
+page_content=' We  follow the same data augmentation tricks used by (Wan  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFLT4oBgHgl3EQfZy_u/content/2301.12071v1.pdf'}
+page_content=', 2022) for the SMILES generative models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFLT4oBgHgl3EQfZy_u/content/2301.12071v1.pdf'}
+page_content=' Instead  of expanding the training dataset off-the-shelf, we perform  the augmentation on-the-fly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFLT4oBgHgl3EQfZy_u/content/2301.12071v1.pdf'}
+page_content=' At each iteration, there is a  probability of 50% to permute the SMILES.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFLT4oBgHgl3EQfZy_u/content/2301.12071v1.pdf'}
+page_content=' We refer to this  baseline as Product2RC and Product2RC-Aug.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFLT4oBgHgl3EQfZy_u/content/2301.12071v1.pdf'}
+page_content='  (3) Similarity-based: We calculate the similarity between  the input product and each product in the training set and  then sort the products in training set based on the similarity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFLT4oBgHgl3EQfZy_u/content/2301.12071v1.pdf'}
+page_content='  We regard the reaction center of the corresponding product  in training set as the prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFLT4oBgHgl3EQfZy_u/content/2301.12071v1.pdf'}
+page_content=' Here, we use molecular  morgan fingerprints to calculate the similarity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFLT4oBgHgl3EQfZy_u/content/2301.12071v1.pdf'}
+page_content=' We refer to  this baseline as Sim-based.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFLT4oBgHgl3EQfZy_u/content/2301.12071v1.pdf'}
+page_content='  It is worth noting that Product2RC and Sim-based can  only predict the reaction center pattern and cannot accurately  obtain the specific position of the reaction center in the  product.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFLT4oBgHgl3EQfZy_u/content/2301.12071v1.pdf'}
+page_content=' We regard the reaction center pattern as the query  graph and perform subgraph matching on the product.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFLT4oBgHgl3EQfZy_u/content/2301.12071v1.pdf'}
+page_content=' When  there is only one solution in the matching result, the only  solution is regarded as the predicted result, otherwise, we  randomly select a solution as the predicted result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFLT4oBgHgl3EQfZy_u/content/2301.12071v1.pdf'}
+page_content=' In this  way, we can calculate the top-k results of the overall test  set once.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFLT4oBgHgl3EQfZy_u/content/2301.12071v1.pdf'}
+page_content=' We repeat this experiment N times, and we finally  take the average top-k accuracy of these N times as the final  result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFLT4oBgHgl3EQfZy_u/content/2301.12071v1.pdf'}
+page_content='  Evaluation Metric For different methods, we transform the  predicted results into the predicted reaction center graph’s  node set ˆVT  rc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFLT4oBgHgl3EQfZy_u/content/2301.12071v1.pdf'}
+page_content=' The condition for correct identification is that  the node set ˆVT  rc of the predicted reaction center graph is  consistent with the ground-truth Vrc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFLT4oBgHgl3EQfZy_u/content/2301.12071v1.pdf'}
+page_content=' We use the top-k exact  match accuracy as our evaluation metrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFLT4oBgHgl3EQfZy_u/content/2301.12071v1.pdf'}
+page_content=' The k predictions  containing ground-truth are counted as correct.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFLT4oBgHgl3EQfZy_u/content/2301.12071v1.pdf'}
+page_content='  Implementation Settings Our proposed RCsearcher is im-  plemented with Deep Graph Library (DGL) (Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFLT4oBgHgl3EQfZy_u/content/2301.12071v1.pdf'}
+page_content=',  2019) and Pytorch (Paszke et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFLT4oBgHgl3EQfZy_u/content/2301.12071v1.pdf'}
+page_content=', 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFLT4oBgHgl3EQfZy_u/content/2301.12071v1.pdf'}
+page_content=' As for the EGAT,  we stack four identical four-head attentive layers of which  the hidden dimension is 256.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFLT4oBgHgl3EQfZy_u/content/2301.12071v1.pdf'}
+page_content=' All embedding sizes in the  model are set to 256.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFLT4oBgHgl3EQfZy_u/content/2301.12071v1.pdf'}
+page_content=' We use ELU(x) = α(exp(x) − 1)  for x ≤ 0 and x for x > 0 as our activation function where  α = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFLT4oBgHgl3EQfZy_u/content/2301.12071v1.pdf'}
+page_content=' We conduct all the experiments on a machine with  an Intel Xeon 4114 CPU and one Nvidia Titan GPU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFLT4oBgHgl3EQfZy_u/content/2301.12071v1.pdf'}
+page_content=' For  training, we set the learning rate to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFLT4oBgHgl3EQfZy_u/content/2301.12071v1.pdf'}
+page_content='001, the number of  training iterations to 100000, and use the Adam optimizer  (Kingma & Ba, 2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFLT4oBgHgl3EQfZy_u/content/2301.12071v1.pdf'}
+page_content=' The first 10000 iterations are imita-  tion learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFLT4oBgHgl3EQfZy_u/content/2301.12071v1.pdf'}
+page_content=' Checkpoints are saved for each 1000 iteration  to select the best checkpoints on the evaluation set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFLT4oBgHgl3EQfZy_u/content/2301.12071v1.pdf'}
+page_content=' The  source code can be found in the supplementary materials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFLT4oBgHgl3EQfZy_u/content/2301.12071v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFLT4oBgHgl3EQfZy_u/content/2301.12071v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFLT4oBgHgl3EQfZy_u/content/2301.12071v1.pdf'}
+page_content=' Overall Performance  We operate the training process five times and report the  mean and standard deviation of accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFLT4oBgHgl3EQfZy_u/content/2301.12071v1.pdf'}
+page_content=' The overall per-  formance is illustrated in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFLT4oBgHgl3EQfZy_u/content/2301.12071v1.pdf'}
+page_content=' We have two observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFLT4oBgHgl3EQfZy_u/content/2301.12071v1.pdf'}
+page_content='  First, our method achieves state-of-the-art performance on  the dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFLT4oBgHgl3EQfZy_u/content/2301.12071v1.pdf'}
+page_content=' It indicates that our method can better handle  single and multiple reaction center identification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFLT4oBgHgl3EQfZy_u/content/2301.12071v1.pdf'}
+page_content=' Second,  the average performance of Product2RC and Sim-based  with 20 and 30 sampling is close.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFLT4oBgHgl3EQfZy_u/content/2301.12071v1.pdf'}
+page_content=' It implies that the average  of 30 results is enough to fairly reflect the capabilities of  Product2RC and Sim-based.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFLT4oBgHgl3EQfZy_u/content/2301.12071v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFLT4oBgHgl3EQfZy_u/content/2301.12071v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFLT4oBgHgl3EQfZy_u/content/2301.12071v1.pdf'}
+page_content=' Performance on Multiple Reaction Center  To address the Q2, we train RCsearcher on the train data  with multiple reaction center to explore its performance on  the test data with multiple reaction center.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFLT4oBgHgl3EQfZy_u/content/2301.12071v1.pdf'}
+page_content=' The multiple re-  action center identification results are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFLT4oBgHgl3EQfZy_u/content/2301.12071v1.pdf'}
+page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFLT4oBgHgl3EQfZy_u/content/2301.12071v1.pdf'}
+page_content=' The  overall top-1 accuracy of Rcsearcher is about eight times  higher than that of prior GNN-based methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFLT4oBgHgl3EQfZy_u/content/2301.12071v1.pdf'}
+page_content=' Also, the  prior GNN-based methods completely fail on data with the  reaction center consisting of more than two edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFLT4oBgHgl3EQfZy_u/content/2301.12071v1.pdf'}
+page_content=' Con-  versely, RCsearcher still works.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFLT4oBgHgl3EQfZy_u/content/2301.12071v1.pdf'}
+page_content=' It shows that RCsearcher’s  performance on multiple reaction center identification is  RCsearcher: Reaction Center Identification in Retrosynthesis via Deep Q-Learning  2  3  4  5  6+  total  # Edges in Multiple Reaction Center  0  10  20  30  40  50  Top-1 Accuracy(%)  MPN-based  EGAT-based  RGCN-based  RCsearcher  Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFLT4oBgHgl3EQfZy_u/content/2301.12071v1.pdf'}
+page_content=' The performance of the multiple reaction center identifi-  cation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFLT4oBgHgl3EQfZy_u/content/2301.12071v1.pdf'}
+page_content='  MPN-based  EGAT-based  RGCN-Based RCsearcher  Methods  0  5  10  15  20  Numbers  3  1  2  21  3  1  2  21  Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFLT4oBgHgl3EQfZy_u/content/2301.12071v1.pdf'}
+page_content=' The performance of the extrapolations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFLT4oBgHgl3EQfZy_u/content/2301.12071v1.pdf'}
+page_content='  significantly better than the baselines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFLT4oBgHgl3EQfZy_u/content/2301.12071v1.pdf'}
+page_content=' We attribute it to  consistency between our optimization objectives and the  purpose of finding the right single or multiple reaction cen-  ter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFLT4oBgHgl3EQfZy_u/content/2301.12071v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFLT4oBgHgl3EQfZy_u/content/2301.12071v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFLT4oBgHgl3EQfZy_u/content/2301.12071v1.pdf'}
+page_content=' Performance of Generalization  In order to evaluate the generalization ability of the RC-  searcher, we count the number of reaction center patterns  that are extrapolated successfully and are not in the training  set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFLT4oBgHgl3EQfZy_u/content/2301.12071v1.pdf'}
+page_content=' The results are shown in Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFLT4oBgHgl3EQfZy_u/content/2301.12071v1.pdf'}
+page_content=' The number of ex-  trapolations of Rcsearcher is about twenty times higher than  that of prior GNN-based methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFLT4oBgHgl3EQfZy_u/content/2301.12071v1.pdf'}
+page_content=' It demonstrates that our  method can predict novel reaction center patterns that have  not appeared in the training set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFLT4oBgHgl3EQfZy_u/content/2301.12071v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFLT4oBgHgl3EQfZy_u/content/2301.12071v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFLT4oBgHgl3EQfZy_u/content/2301.12071v1.pdf'}
+page_content=' Ablation Study  The results of the ablation study are illustrated in Table 2, on  which we have the two observations: 1) The setting with the  beam search results is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFLT4oBgHgl3EQfZy_u/content/2301.12071v1.pdf'}
+page_content='52 percentage points higher than  the setting without the beam search, which indicates beam  search can improve performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFLT4oBgHgl3EQfZy_u/content/2301.12071v1.pdf'}
+page_content=' 2) The setting with the  one-hop constrain results are 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFLT4oBgHgl3EQfZy_u/content/2301.12071v1.pdf'}
+page_content='62 percentage points higher  than the setting without the one-hop constrain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFLT4oBgHgl3EQfZy_u/content/2301.12071v1.pdf'}
+page_content=' It demon-  64  128  256  512  Dimension  15  20  25  30  35  40  Top-1 Accuracy(%)  29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFLT4oBgHgl3EQfZy_u/content/2301.12071v1.pdf'}
+page_content='01  9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFLT4oBgHgl3EQfZy_u/content/2301.12071v1.pdf'}
+page_content='13  2  75  30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFLT4oBgHgl3EQfZy_u/content/2301.12071v1.pdf'}
+page_content='  30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFLT4oBgHgl3EQfZy_u/content/2301.12071v1.pdf'}
+page_content='22  2  8  15  20  25  30  35  40  Top-1 Accuracy(%)  30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFLT4oBgHgl3EQfZy_u/content/2301.12071v1.pdf'}
+page_content='57  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFLT4oBgHgl3EQfZy_u/content/2301.12071v1.pdf'}
+page_content='75  3  3  3  28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFLT4oBgHgl3EQfZy_u/content/2301.12071v1.pdf'}
+page_content='  29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFLT4oBgHgl3EQfZy_u/content/2301.12071v1.pdf'}
+page_content='02  4  6  # Heads  Figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFLT4oBgHgl3EQfZy_u/content/2301.12071v1.pdf'}
+page_content=' Hyperparameter Sensitivity Analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFLT4oBgHgl3EQfZy_u/content/2301.12071v1.pdf'}
+page_content='  strates that although the one-hop constraint fails in some  samples with reaction center of multi-connected branches,  it can improve the model’s performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFLT4oBgHgl3EQfZy_u/content/2301.12071v1.pdf'}
+page_content='  Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFLT4oBgHgl3EQfZy_u/content/2301.12071v1.pdf'}
+page_content=' The Results of Ablation Study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFLT4oBgHgl3EQfZy_u/content/2301.12071v1.pdf'}
+page_content='  Ablation Setting  Top-1 Accuracy  w/ beam search  30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFLT4oBgHgl3EQfZy_u/content/2301.12071v1.pdf'}
+page_content='75%  w/o beam search  30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFLT4oBgHgl3EQfZy_u/content/2301.12071v1.pdf'}
+page_content='23%  w/ one-hop constrain  30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFLT4oBgHgl3EQfZy_u/content/2301.12071v1.pdf'}
+page_content='75%  w/o one-hop constrain  27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFLT4oBgHgl3EQfZy_u/content/2301.12071v1.pdf'}
+page_content='13%  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFLT4oBgHgl3EQfZy_u/content/2301.12071v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFLT4oBgHgl3EQfZy_u/content/2301.12071v1.pdf'}
+page_content=' Hyperparameter Sensitivity Analysis  We respectively fix the number of heads to 4 and the di-  mension of hidden layers d to 256 to explore the impact of  several vital hyperparameters (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFLT4oBgHgl3EQfZy_u/content/2301.12071v1.pdf'}
+page_content=' 7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFLT4oBgHgl3EQfZy_u/content/2301.12071v1.pdf'}
+page_content=' The performance  under different hyperparameters is consistent, revealing the  robustness of our method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFLT4oBgHgl3EQfZy_u/content/2301.12071v1.pdf'}
+page_content=' We observe that the performance  of the RCsearcher improves as the dimension of hidden  layers increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFLT4oBgHgl3EQfZy_u/content/2301.12071v1.pdf'}
+page_content=' We hypothesise that the higher dimension  allows the model to contain more parameters, thus improv-  ing experimental results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFLT4oBgHgl3EQfZy_u/content/2301.12071v1.pdf'}
+page_content='  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFLT4oBgHgl3EQfZy_u/content/2301.12071v1.pdf'}
+page_content=' Conclusion and Future Work  We present RCsearcher, a unified framework for single and  multiple reaction center identification that combines the  advantages of the graph neural network and deep reinforce-  ment learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFLT4oBgHgl3EQfZy_u/content/2301.12071v1.pdf'}
+page_content=' The critical insight in this framework is that  the single or multiple reaction center must be a node-induced  subgraph of the molecular product graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFLT4oBgHgl3EQfZy_u/content/2301.12071v1.pdf'}
+page_content=' Comprehensive  experiments demonstrate that RCsearcher consistently out-  performs other baselines for the reaction centre identifica-  tion task and can extrapolate the reaction center patterns  that have not appeared in the training set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFLT4oBgHgl3EQfZy_u/content/2301.12071v1.pdf'}
+page_content=' Ablation exper-  iments verify the effectiveness of individual components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFLT4oBgHgl3EQfZy_u/content/2301.12071v1.pdf'}
+page_content='  However, RCsearche is still limited to the reaction center of  the single connected branch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFLT4oBgHgl3EQfZy_u/content/2301.12071v1.pdf'}
+page_content=' In the future, we will not only  take advantage of the one-hop constraint but also improve  the current mechanism so that the model can generalize to  the reaction center of multi-connected branches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFLT4oBgHgl3EQfZy_u/content/2301.12071v1.pdf'}
+page_content='  RCsearcher: Reaction Center Identification in Retrosynthesis via Deep Q-Learning  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFLT4oBgHgl3EQfZy_u/content/2301.12071v1.pdf'}
+page_content=' Dataset information  We adopt stratified random sampling for USPTO-full and take out 40k samples, ensuring that the data distribution is close  enough to USPTO-full.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFLT4oBgHgl3EQfZy_u/content/2301.12071v1.pdf'}
+page_content=' More detailed information about the distribution of the dataset can be found in Figure 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFLT4oBgHgl3EQfZy_u/content/2301.12071v1.pdf'}
+page_content='  Figure 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFLT4oBgHgl3EQfZy_u/content/2301.12071v1.pdf'}
+page_content=' The detailed distribution of the USPTO-40k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFLT4oBgHgl3EQfZy_u/content/2301.12071v1.pdf'}
+page_content='  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFLT4oBgHgl3EQfZy_u/content/2301.12071v1.pdf'}
+page_content=' Atom and bond features  In this paper, initial features of atom and bond can be found in Table 3 and Table 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFLT4oBgHgl3EQfZy_u/content/2301.12071v1.pdf'}
+page_content='  Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFLT4oBgHgl3EQfZy_u/content/2301.12071v1.pdf'}
+page_content=' Atom Features used in EGAT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFLT4oBgHgl3EQfZy_u/content/2301.12071v1.pdf'}
+page_content=' All features are one-hot encoding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFLT4oBgHgl3EQfZy_u/content/2301.12071v1.pdf'}
+page_content='  Feature  Description  Size  Atom type  Type of an atom by atomic number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFLT4oBgHgl3EQfZy_u/content/2301.12071v1.pdf'}
+page_content='  100  Total degree  Degree of an atom including Hs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFLT4oBgHgl3EQfZy_u/content/2301.12071v1.pdf'}
+page_content='  6  Explicit valence  Explicit valence of an atom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFLT4oBgHgl3EQfZy_u/content/2301.12071v1.pdf'}
+page_content='  6  Implicit valence  Explicit valence of an atom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFLT4oBgHgl3EQfZy_u/content/2301.12071v1.pdf'}
+page_content='  6  Hybridization  sp, sp2, sp3, sp3d, or sp3d2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFLT4oBgHgl3EQfZy_u/content/2301.12071v1.pdf'}
+page_content='  5  # Hs  Number of bonded Hydrogen atom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFLT4oBgHgl3EQfZy_u/content/2301.12071v1.pdf'}
+page_content='  5  Formal charge  Integer electronic charge assigned to atom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFLT4oBgHgl3EQfZy_u/content/2301.12071v1.pdf'}
+page_content='  5  Aromaticity  Whether an atom is part of an aromatic system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFLT4oBgHgl3EQfZy_u/content/2301.12071v1.pdf'}
+page_content='  1  In ring  Whether an atom is in ring  1  Table 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFLT4oBgHgl3EQfZy_u/content/2301.12071v1.pdf'}
+page_content=' Bond features used in EGAT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFLT4oBgHgl3EQfZy_u/content/2301.12071v1.pdf'}
+page_content=' All features are one-hot encoding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFLT4oBgHgl3EQfZy_u/content/2301.12071v1.pdf'}
+page_content='  Feature  Description  Size  Bond type  Single, double, triple, or aromatic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFLT4oBgHgl3EQfZy_u/content/2301.12071v1.pdf'}
+page_content='  4  Conjugation  Whether the bond is conjugated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFLT4oBgHgl3EQfZy_u/content/2301.12071v1.pdf'}
+page_content='  1  In ring  Whether the bond is part of a ring.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFLT4oBgHgl3EQfZy_u/content/2301.12071v1.pdf'}
+page_content='  1  Stereo  None, any, E/Z or cis/trans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFLT4oBgHgl3EQfZy_u/content/2301.12071v1.pdf'}
+page_content='  6  Direction  The direction of the bond.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFLT4oBgHgl3EQfZy_u/content/2301.12071v1.pdf'}
+page_content='  3  C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFLT4oBgHgl3EQfZy_u/content/2301.12071v1.pdf'}
+page_content=' Hyperparameters for baselines  For fair comparison, we set the number of layers, size of hidden dimensions and the number of attention heads in three  GNN-based baselines to 4, 156, and 4 respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFLT4oBgHgl3EQfZy_u/content/2301.12071v1.pdf'}
+page_content=' In Seq2Seq-based baseline, we set the number of layers, size of hidden  dimensions and the number of attention heads in Transformer (Lan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFLT4oBgHgl3EQfZy_u/content/2301.12071v1.pdf'}
+page_content=', 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFLT4oBgHgl3EQfZy_u/content/2301.12071v1.pdf'}
+page_content=' 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFLT4oBgHgl3EQfZy_u/content/2301.12071v1.pdf'}
+page_content=' 2023;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFLT4oBgHgl3EQfZy_u/content/2301.12071v1.pdf'}
+page_content=' Ma et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFLT4oBgHgl3EQfZy_u/content/2301.12071v1.pdf'}
+page_content=', 2021) to 4, 2048, 8  respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFLT4oBgHgl3EQfZy_u/content/2301.12071v1.pdf'}
+page_content='  2  28  2  2  5  24  2  19  17  MultipleReaction  16  15  4  2#  10  6  8  7  9  3  2  10%  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFLT4oBgHgl3EQfZy_u/content/2301.12071v1.pdf'}
+page_content='0  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFLT4oBgHgl3EQfZy_u/content/2301.12071v1.pdf'}
+page_content='00  0  Proportion in USPTO-40KRCsearcher: Reaction Center Identification in Retrosynthesis via Deep Q-Learning  References  Chen, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFLT4oBgHgl3EQfZy_u/content/2301.12071v1.pdf'}
+page_content=', Li, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFLT4oBgHgl3EQfZy_u/content/2301.12071v1.pdf'}
+page_content=', Dai, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFLT4oBgHgl3EQfZy_u/content/2301.12071v1.pdf'}
+page_content=', and Song, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFLT4oBgHgl3EQfZy_u/content/2301.12071v1.pdf'}
+page_content=' Retro*: learning  retrosynthetic planning with neural guided a* search.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFLT4oBgHgl3EQfZy_u/content/2301.12071v1.pdf'}
+page_content=' In  International Conference on Machine Learning, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFLT4oBgHgl3EQfZy_u/content/2301.12071v1.pdf'}
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+page_content=' Smiles, a chemical language and information  system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFLT4oBgHgl3EQfZy_u/content/2301.12071v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFLT4oBgHgl3EQfZy_u/content/2301.12071v1.pdf'}
+page_content=' introduction to methodology and encoding  rules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFLT4oBgHgl3EQfZy_u/content/2301.12071v1.pdf'}
+page_content=' Journal of chemical information and computer  sciences, 28(1):31–36, 1988.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFLT4oBgHgl3EQfZy_u/content/2301.12071v1.pdf'}
+page_content='  Xie, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFLT4oBgHgl3EQfZy_u/content/2301.12071v1.pdf'}
+page_content=', Yan, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFLT4oBgHgl3EQfZy_u/content/2301.12071v1.pdf'}
+page_content=', Han, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFLT4oBgHgl3EQfZy_u/content/2301.12071v1.pdf'}
+page_content=', Xia, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFLT4oBgHgl3EQfZy_u/content/2301.12071v1.pdf'}
+page_content=', Wu, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFLT4oBgHgl3EQfZy_u/content/2301.12071v1.pdf'}
+page_content=', Guo, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFLT4oBgHgl3EQfZy_u/content/2301.12071v1.pdf'}
+page_content=', Yang,  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFLT4oBgHgl3EQfZy_u/content/2301.12071v1.pdf'}
+page_content=', and Qin, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFLT4oBgHgl3EQfZy_u/content/2301.12071v1.pdf'}
+page_content=' Retrograph: Retrosynthetic planning with  graph search.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFLT4oBgHgl3EQfZy_u/content/2301.12071v1.pdf'}
+page_content=' In Proceedings of the 28th ACM SIGKDD  Conference on Knowledge Discovery and Data Mining,  pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFLT4oBgHgl3EQfZy_u/content/2301.12071v1.pdf'}
+page_content=' 2120–2129, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFLT4oBgHgl3EQfZy_u/content/2301.12071v1.pdf'}
+page_content='  Yan, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFLT4oBgHgl3EQfZy_u/content/2301.12071v1.pdf'}
+page_content=', Ding, Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFLT4oBgHgl3EQfZy_u/content/2301.12071v1.pdf'}
+page_content=', Zhao, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFLT4oBgHgl3EQfZy_u/content/2301.12071v1.pdf'}
+page_content=', Zheng, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFLT4oBgHgl3EQfZy_u/content/2301.12071v1.pdf'}
+page_content=', Yang, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFLT4oBgHgl3EQfZy_u/content/2301.12071v1.pdf'}
+page_content=', Yu, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFLT4oBgHgl3EQfZy_u/content/2301.12071v1.pdf'}
+page_content=',  and Huang, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFLT4oBgHgl3EQfZy_u/content/2301.12071v1.pdf'}
+page_content=' Retroxpert: Decompose retrosynthesis  prediction like a chemist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFLT4oBgHgl3EQfZy_u/content/2301.12071v1.pdf'}
+page_content=' Advances in Neural Information  Processing Systems, 33:11248–11258, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFLT4oBgHgl3EQfZy_u/content/2301.12071v1.pdf'}
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+Prepared for submission to JHEP
+Spectral form factor in the τ-scaling limit
+Kazumi Okuyamaa and Kazuhiro Sakaib
+aDepartment of Physics, Shinshu University,
+3-1-1 Asahi, Matsumoto 390-8621, Japan
+bInstitute of Physics, Meiji Gakuin University,
+1518 Kamikurata-cho, Totsuka-ku, Yokohama 244-8539, Japan
+E-mail: kazumi@azusa.shinshu-u.ac.jp, kzhrsakai@gmail.com
+Abstract: We study the spectral form factor (SFF) of general topological gravity in the
+limit of large time and fixed temperature. It is observed recently that in this limit, called
+the tau-scaling limit, the genus expansion of the SFF can be summed up and the late-
+time behavior of the SFF such as the ramp-plateau transition can be studied analytically.
+In this paper we develop a technique for the systematic computation of the higher order
+corrections to the SFF in the strict tau-scaling limit. We obtain the first five corrections in
+a closed form for the general background of topological gravity. As concrete examples, we
+present the results for the Airy case and Jackiw-Teitelboim gravity. We find that the above
+higher order corrections are the Fourier transforms of the corrections to the sine-kernel
+approximation of the Christoffel-Darboux kernel in the dual double-scaled matrix integral,
+which naturally explains their structure. Along the way we also develop a technique for
+the systematic computation of the corrections to the sine-kernel formula, which have not
+been fully explored in the literature before.
+arXiv:2301.04773v1  [hep-th]  12 Jan 2023
+
+Contents
+1
+Introduction
+1
+2
+τ-scaling limit of SFF
+3
+3
+Summary of results
+5
+3.1
+SFFg
+6
+3.2
+Relation to the corrections of CD kernel
+8
+4
+Examples of Airy case and JT gravity
+10
+4.1
+Airy case
+10
+4.2
+JT gravity
+10
+5
+Computation of SFF
+12
+5.1
+Derivation of the key differential equation
+12
+5.2
+Small ℏ expansion of one-point function
+13
+5.3
+Small ℏ expansion of SFF
+18
+6
+Higher order corrections to CD kernel
+20
+6.1
+Small ℏ expansion of BA function
+20
+6.2
+Small ℏ expansion of CD kernel
+22
+7
+Conclusions and outlook
+24
+A Airy kernel
+24
+B Coefficient a2 of SFF2 in (5.44)
+26
+1
+Introduction
+Spectral form factor (SFF) is a useful measure of the level statistics of quantum chaotic
+systems [1] and it is widely studied in many areas of physics. In the context of quantum
+gravity and holography, the SFF of the Sachdev-Ye-Kitaev (SYK) model [2–4] was studied
+in [5, 6] as a useful diagnostics of the Maldacena’s version of the information problem [7].
+It is found that the SFF of the SYK model exhibits the behavior of the ramp and plateau
+as a function of time, which is consistent with the conjecture that the level statistics of
+quantum chaotic system is universally described by a random matrix model [8]. As shown
+in [9], Jackiw-Teitelboim (JT) gravity [10, 11], which is holographically dual to the low
+energy sector of the SYK model, is indeed described by a certain double-scaled random
+matrix model. In the bulk gravity picture, the ramp of the SFF of JT gravity comes from
+– 1 –
+
+the contribution of a wormhole connecting two boundaries of spacetime [12]. In the matrix
+model picture, it is described by the connected part of the correlator of two macroscopic
+loop operators [13]. On the other hand, the plateau of the SFF is very mysterious from
+the viewpoint of bulk gravity and it was speculated that the appearance of the plateau is
+related to some non-perturbative effects in quantum gravity [6, 9]. In the matrix model
+picture, it is argued in [6, 9] that the plateau can be explained by the Andreev–Altshuler
+instantons [14]. See also [15] for an explanation of the plateau by eigenvalue instantons.
+In this paper, we will consider the connected part of the SFF in 2d quantum gravity
+SFF = ⟨Z(β + it)Z(β − it)⟩c,
+(1.1)
+where Z(β) = Tr e−βH and the expectation value is defined by averaging over a random
+matrix H. As shown in [16–18], 2d quantum gravity coupled to a conformal matter can
+be defined by a certain double-scaling limit of a random matrix model. More generally,
+one can introduce couplings {tk} (k = 0, 1, 2, . . .) to the matrix model potential. Then the
+free energy of the model is interpreted as a generating function of the intersection numbers
+on the moduli space of Riemann surfaces and the model is called topological gravity [19].
+There is an underlying integrable structure in this model and the free energy serves as
+a tau-function of the KdV hierarchy [19, 20]. It turns out that the matrix model of JT
+gravity in [9] is a special case of topological gravity where infinitely many couplings {tk} are
+turned on in a specific way [21–23] and it corresponds to the computation of Weil-Petersson
+volumes [24, 25].
+SFF exhibits the ramp-plateau transition around the time scale t ∼ ℏ−1, called the
+Heisenberg time, where ℏ is the genus-counting parameter of the matrix model. Recently,
+it is observed in [26–28] that one can focus on the ramp-plateau transition regime by taking
+what is called the “τ-scaling limit”1
+t → ∞,
+ℏ → 0,
+τ = tℏ : fixed,
+(1.2)
+with β fixed in (1.1). In this limit, SFF is expanded as
+SFF =
+∞
+�
+g=0
+ℏ2g−1SFFg(τ, β).
+(1.3)
+Remarkably, it is found that the leading term SFF0 can be computed in a closed form by
+just summing over the original genus expansion in the τ-scaling limit and the resulting
+SFF0 approaches a constant as τ → ∞ [26]. This opens up an interesting avenue for a
+“perturbative plateau.”
+In this paper, we will develop a technique for the systematic computation of the higher
+order corrections SFFg (g ≥ 1). By using our method, we obtain SFFg up to g = 5 for
+arbitrary couplings {tk} of topological gravity. It turns out that SFFg has a structure which
+is a natural generalization of SFF0. In [26] it was shown that SFF0 is essentially determined
+1In [23, 29], another scaling limit where both β and t are of order ℏ−1 was considered. This limit was
+called the “’t Hooft limit” in [23].
+– 2 –
+
+by the Fourier transform of the universal part of the two-body eigenvalue correlation, known
+as the sine kernel formula [30–32]. We find that the higher order correction SFFg is closely
+related to the correction of the Christoffel-Darboux (CD) kernel to the naive sine kernel
+formula. Rather surprisingly, such corrections to the sine kernel formula have not been
+fully explored in the literature before, as far as we know.2 We will also develop a technique
+for the systematic computation of the corrections of the CD kernel and confirm that SFFg
+is correctly reproduced form the Fourier transform of the CD kernel by including the
+corrections to the sine kernel formula.
+This paper is organized as follows. In section 2, we review the known results about the
+SFF in the τ-scaling limit. Along the way we consider the τ-scaling limit based on the genus
+expansion of the SFF and obtain the small τ expansion of SFFg for small g. In section 3,
+we summarize our results of SFFg and the higher order corrections of the CD kernel. We
+also explain their relations. In section 4, based on our general results we compute SFF0 and
+SFF1 for the Airy case and JT gravity, as concrete examples. In section 5, we formulate
+a systematic method of computing SFFg. In section 6, we explain how to compute the
+higher order corrections of the CD kernel beyond the sine kernel approximation. Finally
+we conclude in section 7. In Appendix A, we compute the corrections of the Airy kernel to
+the sine kernel formula. In Appendix B, we present an explicit form of the coefficient that
+determines SFF2.
+2
+τ-scaling limit of SFF
+In this section we will briefly review the known results about the SFF in the τ-scaling
+limit. We will consider the τ-scaling limit for the general background of topological gravity
+based on the known genus expansion of the SFF, which enables us to compute the small τ
+expansion of SFFg for small g.
+For the general background {tk} of topological gravity, the connected two-boundary
+correlator is expanded as [29]
+⟨Z(β1)Z(β2)⟩c =
+√β1β2
+2π
+e−(β1+β2)E0
+�
+1
+β1 + β2
++
+�β2
+1 + β1β2 + β2
+2
+24s2
++ 2(β1 + β2)I2 + I3
+24s3
++
+I2
+2
+12s4
+�
+g2
+s + O(g4
+s )
+�
+,
+(2.1)
+where gs is the genus-counting parameter related to ℏ as3
+gs =
+√
+2ℏ.
+(2.2)
+Ik denote the Itzykson-Zuber variables defined by [33]
+Ik =
+∞
+�
+n=0
+tk+n
+n! (−E0)n,
+k ∈ Z≥0
+(2.3)
+2In [32], general form of the large N limit of the CD kernel was studied from the asymptotic behavior
+of the orthogonal polynomials associated to an arbitrary matrix model potential, before taking the double-
+scaling limit.
+3See [23] for more about our definition of ℏ and its relation to the parameters S0, γ in JT gravity.
+– 3 –
+
+and s in (2.1) is given by
+s = 1 − I1.
+(2.4)
+The threshold energy E0 is determined by the (genus-zero) string equation
+I0 + E0 = 0.
+(2.5)
+In the τ-scaling limit (1.2), after setting β1,2 = β ± iτ/ℏ in the two-boundary correlator
+(2.1) and expanding it in ℏ, we find the small τ expansion of SFF0 and SFF1
+SFF0 = e−2βE0
+2π
+� τ
+2β − τ 3
+12s2 + · · ·
+�
+,
+SFF1 = e−2βE0
+2π
+� β
+4τ +
+� 5β2
+24s2 + 4βI2 + I3
+12s3
++ I2
+2
+6s4
+�
+τ + · · ·
+�
+.
+(2.6)
+In general, the two-boundary correlator is written as
+⟨Z(β1)Z(β2)⟩c = ⟨Z(β1 + β2)⟩ −
+�
+dE1dE2e−β1E1−β2E2K(E1, E2)2,
+(2.7)
+where K(E1, E2) is the Christoffel-Darboux (CD) kernel. Thus, the SFF is written as
+SFF =
+� dE
+2π e−2βEρ(E) −
+�
+dE1dE2e−β(E1+E2)−iτ E1−E2
+ℏ
+K(E1, E2)2.
+(2.8)
+In the τ-scaling limit (1.2), the above integral is dominated by the region
+E1 − E2 ∼ O(ℏ),
+(2.9)
+with finite E1 + E2. In this regime, the CD kernel is approximated by the universal two-
+body correlation of eigenvalues, known as the sine kernel [30–32]
+Ksin(E1, E2) = sin
+� 1
+2ρ0(E)ω
+�
+πℏω
+,
+(2.10)
+where we defined
+E1 = E + 1
+2ℏω,
+E2 = E − 1
+2ℏω.
+(2.11)
+ρ0(E) is the genus-zero part of the eigenvalue density
+ρ(E) =
+∞
+�
+g=0
+ℏ2g−1ρg(E)
+(2.12)
+which is related to the one-point function by
+⟨Z(β)⟩ =
+� ∞
+E0
+dE
+2π e−βEρ(E) =
+∞
+�
+g=0
+ℏ2g−1⟨Z(β)⟩g.
+(2.13)
+For the general background {tk}, ρ0(E) is given by
+ρ0(E) =
+∞
+�
+k=1
+(−1)k(Ik − δk,1)Γ(1/2)
+Γ(k + 1/2)
+(E − E0)k− 1
+2 .
+(2.14)
+– 4 –
+
+In [26–28], it is argued that the above small τ expansion of SFF0 (2.6) can be resummed
+and SFF0 is obtained by replacing the CD kernel by the sine kernel in (2.8). Then, by the
+change of integration variables in (2.11), the leading term of the SFF becomes
+SFF0 =
+� ∞
+E0
+dE
+2π e−2βEρ0(E) −
+� ∞
+E0
+dEe−2βE
+� ∞
+−∞
+dωe−iωτ sin2� 1
+2ρ0(E)ω
+�
+π2ω2
+.
+(2.15)
+The ω-integral is evaluated as [34]
+� ∞
+−∞
+dωe−iωτ sin2� 1
+2ρ0(E)ω
+�
+π2ω2
+= 1
+2π
+�
+ρ0(E) − τ
+�
+θ
+�
+ρ0(E) − τ
+�
+,
+(2.16)
+where θ(x) is the step function
+θ(x) =
+�
+1,
+(x > 0),
+0,
+(x < 0).
+(2.17)
+Assuming that ρ0(E) is a monotonically increasing function of E, there is a unique solution
+Eτ to the equation
+ρ0(Eτ) = τ.
+(2.18)
+In terms of Eτ, SFF0 in (2.15) is written as
+SFF0 =
+� ∞
+E0
+dE
+2π e−2βEρ0(E) −
+� ∞
+Eτ
+dE
+2π e−2βE�
+ρ0(E) − τ
+�
+=
+τ
+4πβ e−2βEτ +
+� Eτ
+E0
+dE
+2π e−2βEρ0(E).
+(2.19)
+The first term corresponds to the ramp and the second term approaches a constant ⟨Z(2β)⟩0
+in the τ → ∞ limit, corresponding to the plateau.
+One can check that (2.19) reproduces the small τ expansion of SFF0 in (2.6). To see
+this, we first notice that Eτ has the following small τ expansion4
+Eτ − E0 =
+1
+4s2 τ 2 −
+I2
+12s5 τ 4 +
+� 7I2
+2
+144s8 +
+I3
+120s7
+�
+τ 6 + O(τ 8).
+(2.20)
+This can be obtained by inverting the relation (2.18) using the expression of ρ0(E) in
+(2.14). Plugging this expansion of Eτ into (2.19), one can show that the small τ expansion
+of SFF0 in (2.6) is indeed reproduced from (2.19).
+In the rest of this paper, we will consider the higher order corrections SFFg (g ≥ 1)
+to the spectral form factor. In the next section we will first summarize the result of SFFg.
+The details of the computation will be postponed to section 5.
+3
+Summary of results
+In this section we will summarize our results of SFFg and discuss their relation to the
+corrections of the CD kernel.
+4A fully explicit expression of this expansion is available (Eτ is given by E(λ) in (5.12) with λ = iτ).
+– 5 –
+
+3.1
+SFFg
+It turns out that the expression of SFF0 in (2.19) has a natural generalization to the higher
+order correction SFFg. We find that SFFg has the structure
+SFFg = 1
+2πfg(τ, β)e−2βEτ +
+� Eτ
+E0
+dE
+2π e−2βEρg(E).
+(3.1)
+By the relation (2.13) we can easily find the first few terms of ρg(E) from the result of
+⟨Z(β)⟩g in [23]. For instance,
+ρ1(E) =
+1
+16sz5 −
+I2
+24s2z3 ,
+ρ2(E) = −
+105
+1024s3z11 +
+203I2
+1536s4z9
++ 1
+z7
+�
+− 7I2
+2
+64s5 − 29I3
+768s4
+�
++ 1
+z5
+� 7I3
+2
+96s6 + 29I2I3
+480s5 +
+I4
+128s4
+�
++ 1
+z3
+�
+− 7I4
+2
+144s7 − 5I2
+2I3
+72s6 − 11I2I4
+720s5 −
+29I2
+3
+2880s5 −
+I5
+576s4
+�
+,
+(3.2)
+where
+z =
+�
+E − E0.
+(3.3)
+Note that the second term of (3.1) is a formal expression since the integral has a diver-
+gence coming from E = E0. This integral should be understood by a certain analytic
+continuation. For instance, the integral involving z−a term is defined as
+� Eτ
+E0
+dEe−2βEz−a =
+� ∞
+E0
+dEe−2βEz−a +
+� Eτ
+∞
+dEe−2βEz−a
+= e−2βE0Γ(1 − a/2)(2β)a/2−1 +
+� Eτ
+∞
+dEe−2βEz−a.
+(3.4)
+Using this prescription, one can show that the second term of (3.1) is written as
+� Eτ
+E0
+dE
+2π e−2βEρg(E) = ⟨Z(2β)⟩gErf(zτ
+�
+2β) + 1
+2πdg(zτ, β)e−2βEτ ,
+(3.5)
+where Erf(z) denotes the error function and zτ is defined by
+zτ =
+�
+Eτ − E0.
+(3.6)
+It turns out that dg(zτ, β) in (3.5) is a polynomial in 1/zτ. For instance, d1(zτ, β) is given
+by
+d1(zτ, β) =
+β
+6szτ
+−
+1
+24sz3τ
++
+I2
+12s2zτ
+.
+(3.7)
+Next, let us consider the first term of (3.1).
+We find that fg(τ, β)e−2βEτ can be
+rewritten as a sum of τ-derivatives
+fg(τ, β)e−2βEτ =
+3g−2
+�
+n=0
+∂n
+τ
+�
+hg,n(τ)e(1)e−2βEτ �
+.
+(3.8)
+– 6 –
+
+Here we introduced the notation
+e(j) = ∂j
+τEτ.
+(3.9)
+The important point is that hg,n(τ) is independent of β; the β-dependence of fg(τ, β) arises
+solely from the τ-derivative in (3.8). We have computed hg,n up to g = 5. For g = 1 we
+find
+h1,1 = −s(2),
+h1,0 =
+1
+16z4τ
+,
+(3.10)
+where s(j) denotes
+s(j) = ρ(j)
+0 (Eτ)
+(j + 1)!2j
+(3.11)
+with ρ(j)
+0 (E) = ∂j
+Eρ0(E). For g = 2 we find
+h2,4 = −1
+2s(2)2,
+h2,3 = s(4) +
+1
+16z4τ
+s(2),
+h2,2 = −
+3
+256z8τ
++ ρ1(Eτ)s(2),
+h2,1 =
+7I2
+768s2z7τ
+−
+73
+1536sz9τ
+,
+h2,0 = −
+I3
+96s3z6τ
+−
+7
+128s2z10
+τ
+− 25
+2 ρ1(Eτ)2.
+(3.12)
+From the definition of Eτ in (2.18), one can show that ρ(j)
+0 (Eτ) can be written as a com-
+bination of e(j). For instance,
+ρ(2)
+0 (Eτ) = − e(2)
+e(1)3 ,
+ρ(4)
+0 (Eτ) = − e(4)
+e(1)5 − 15e(2)3
+e(1)7
++ 10e(3)e(2)
+e(1)6
+.
+(3.13)
+For general g, we find that hg,n with n = 3g − 2 and n = 3g − 3 are given by
+hg,3g−2 = −s(2)g
+g!
+,
+hg,3g−3 =
+1
+16z4τ
+s(2)g−1
+(g − 1)! + s(4) s(2)g−2
+(g − 2)!.
+(3.14)
+In subsection 3.2, we will see that this structure can be naturally understood from the
+correction of the CD kernel to the sine kernel formula.
+Let us take a closer look at the behavior of SFFg in (3.1) as a function of τ. At late
+times, the first term of (3.1) vanishes exponentially and the second term approaches a
+constant
+lim
+τ→∞ SFFg = ⟨Z(2β)⟩g.
+(3.15)
+This gives the higher genus correction to the value of plateau. On the other hand, at early
+times SFFg diverges as
+lim
+τ→0 SFFg ∼ e−2βE0
+�β
+τ
+�2g−1
+.
+(3.16)
+– 7 –
+
+This is just an artifact of the τ-scaling limit. This early time divergence can be traced
+back to the expansion of the original genus-zero term
+√β1β2
+2π(β1 + β2) =
+�
+τ 2 + β2ℏ2
+4πβℏ
+= 1
+4π
+∞
+�
+g=0
+(−1)g−1(2g − 3)!!
+g!2g
+�βℏ
+τ
+�2g−1
+.
+(3.17)
+Before expanding in ℏ, the original expression
+�
+τ 2 + β2ℏ2 is regular at τ = 0, but after
+expanding it by ℏ there appears an apparent divergence at τ = 0. We have checked that
+the coefficient of O(τ 1−2g) term of SFFg in the small τ expansion is indeed given by (3.17).
+3.2
+Relation to the corrections of CD kernel
+From the general relation between the SFF and the CD kernel K(E1, E2) in (2.8), the higher
+order correction SFFg is closely related to the correction of the CD kernel to the naive sine
+kernel formula (2.10). The details of the computation of CD kernel will be presented in
+section 6. It turns out that the τ-derivative structure in (3.8) naturally appears from the
+Fourier transform of the CD kernel squared
+SFF =
+� dE
+2π e−2βEρ(E) − ℏ
+�
+dEe−2βE
+�
+dωe−iωτK
+�
+E + 1
+2ℏω, E − 1
+2ℏω
+�2
+.
+(3.18)
+We find that the corrections of the CD kernel to the sine kernel formula is organized as
+K
+�
+E + 1
+2ℏω, E − 1
+2ℏω
+�
+=
+�
+2
+ℏω +
+∞
+�
+g=1
+ℏ2g−1k(g)
+s (E, ω)
+�
+sin φ
+2π
++
+∞
+�
+g=1
+ℏ2g−1�
+ρg(E) + k(g)
+c (E, ω)
+�cos φ
+2π ,
+(3.19)
+where φ is given by
+φ = 1
+2ℏ
+� E+ 1
+2 ℏω
+E− 1
+2 ℏω
+dEρ0(E) = 1
+2
+∞
+�
+j=0
+ℏ2jω2j+1
+ρ(2j)
+0
+(E)
+(2j + 1)!22j .
+(3.20)
+Note that the diagonal part of the CD kernel is equal to the eigenvalue density
+K(E, E) = 1
+2πρ(E) = 1
+2π
+∞
+�
+g=0
+ℏ2g−1ρg(E).
+(3.21)
+We find that k(g)
+s
+and k(g)
+c
+vanishes as ω → 0 and hence our expansion (3.19) is consistent
+with the diagonal part (3.21). The explicit forms of k(g)
+s
+and k(g)
+c
+for g = 1, 2 are given by
+k(1)
+s
+=
+ω
+16z4 ,
+k(1)
+c
+= 0,
+k(2)
+s
+=
+11ω3
+1024z8 +
+�
+−
+7
+128s2z10 −
+I3
+96s3z6 − 49
+4 ρ1(E)2
+�
+ω,
+k(2)
+c
+=
+�
+35
+768sz9 −
+I2
+128s2z7
+�
+ω2.
+(3.22)
+– 8 –
+
+Expanding sin φ and cos φ further in ℏ, we find the ℏ expansion of the CD kernel in (3.19)
+K
+�
+E + 1
+2ℏω, E − 1
+2ℏω
+�
+=
+∞
+�
+g=0
+ℏ2g−1Kg(E, ω).
+(3.23)
+One can see that the leading term is the sine kernel
+K0(E, ω) = sin
+� 1
+2ρ0(E)ω
+�
+πω
+,
+(3.24)
+and the higher order correction Kg(E, ω) can be obtained from (3.19).
+From the general form of the CD kernel (3.19), one can see that the τ-derivative
+structure of (3.8) can be understood from the Fourier transform of (3.19). Let us consider
+a contribution of the term
+K
+�
+E + 1
+2ℏω, E − 1
+2ℏω
+�2
+⊃ ωneiρ0(E)ω,
+(n ≥ 0).
+(3.25)
+From the Fourier transformation of this term, we find
+� ∞
+−∞
+dω
+2π e−iωτωneiρ0(E)ω = (i∂τ)n
+� ∞
+−∞
+dω
+2π e−iωτ+iρ0(E)ω
+= (i∂τ)nδ
+�
+ρ0(E) − τ
+�
+= (i∂τ)n�
+e(1)δ(E − Eτ)
+�
+,
+(3.26)
+where we used the relation
+ρ(1)(Eτ) =
+1
+e(1).
+(3.27)
+After integrating over E in (3.18), we find the τ-derivative structure of fg(τ, β)e−2βEτ in
+(3.8). We have checked that h1,n in (3.10) and h2,n in (3.12) are correctly reproduced
+from the Fourier transform of the square of the CD kernel in (3.19). In particular, the
+appearance of s(2j) can be naturally understood from the expansion of φ in (3.20).
+The second term of (3.1) arises from the cross-term of sin φ and cos φ in the CD kernel
+squared
+K
+�
+E + 1
+2ℏω, E − 1
+2ℏω
+�2
+⊃ sin
+�
+ρ0(E)ω
+�
+2π2ℏω
+∞
+�
+g=1
+ℏ2g−1ρg(E).
+(3.28)
+Using the relation
+� ∞
+−∞
+dωe−iωτ sin
+�
+ρ0(E)ω
+�
+2π2ω
+= 1
+2πθ
+�
+ρ0(E) − τ
+�
+,
+(3.29)
+we find that the second term of (3.1) is reproduced from (3.18)
+SFFg ⊃
+� ∞
+E0
+dE
+2π e−2βEρg(E) −
+� ∞
+E0
+dE
+2π e−2βEρg(E)θ
+�
+ρ0(E) − τ
+�
+=
+� ∞
+E0
+dE
+2π e−2βEρg(E) −
+� ∞
+Eτ
+dE
+2π e−2βEρg(E)
+=
+� Eτ
+E0
+dE
+2π e−2βEρg(E).
+(3.30)
+– 9 –
+
+To summarize, the structure of SFFg in (3.1) and (3.8) can be naturally understood from
+the Fourier transform of the CD kernel in (3.19). To our knowledge, corrections to the sine
+kernel formula are not known in the literature before and our (3.19) with (3.22) is a new
+result.5
+4
+Examples of Airy case and JT gravity
+In this section, as concrete examples we compute SFF0 and SFF1 for the Airy case and JT
+gravity using our general result in the previous section.
+4.1
+Airy case
+Let us first consider the Airy case. For the Airy case, all couplings tk are set to zero. Then
+one can show that E0 = 0 and Ik = 0 (k ≥ 1), and the genus-zero eigenvalue density (2.14)
+becomes
+ρ0(E) = 2
+√
+E.
+(4.1)
+From the definition of Eτ in (2.18), we find
+Eτ = τ 2
+4 ,
+zτ =
+�
+Eτ = τ
+2.
+(4.2)
+From the general formula in the previous sections, we find that the SFF0 and SFF1 for the
+Airy case are given by
+SFF0 =
+1
+2
+�
+π(2β)3 Erf
+�
+τ
+�
+β
+2
+�
+,
+SFF1 =
+β
+8πτ e− 1
+2 βτ 2 + β3/2
+6
+√
+2πErf
+�
+τ
+�
+β
+2
+�
+.
+(4.3)
+We can compare this with the exact result of two-boundary correlator for the Airy case
+[35]
+⟨Z(β1)Z(β2)⟩c =
+e
+ℏ2
+12 (β1+β2)3
+2ℏ
+�
+π(β1 + β2)3 Erf
+�ℏ
+2
+�
+β1β2(β1 + β2)
+�
+.
+(4.4)
+One can check that the SFF0,1 in (4.3) are indeed reproduced from the τ-scaling limit of
+the exact result (4.4). See also Appendix A for the corrections of the CD kernel in the
+Airy case.
+4.2
+JT gravity
+Next consider the SFF of JT gravity. For JT gravity, the couplings tk are given by [21–23]
+t0 = t1 = 0,
+tk = (−1)k
+(k − 1)!,
+(k ≥ 2).
+(4.5)
+5See also a comment in footnote 2.
+– 10 –
+
+In this case, E0 = 0 and the genus-zero eigenvalue density (2.14) becomes
+ρ0(E) = sinh(2
+√
+E).
+(4.6)
+From (2.18), Eτ and zτ are given by
+Eτ = 1
+4arcsinh(τ)2,
+z =
+�
+Eτ = 1
+2arcsinh(τ).
+(4.7)
+From the general formula (2.19), the leading term SFF0 is evaluated as [26]
+SFF0 = 1
+2⟨Z(2β)⟩g=0
+�
+Erf
+�βarcsinh(τ) + 1
+√2β
+�
++ Erf
+�βarcsinh(τ) − 1
+√2β
+��
+,
+(4.8)
+where the genus-zero one-point function is given by
+⟨Z(β)⟩g=0 =
+e
+1
+β
+2
+�
+πβ3 .
+(4.9)
+The next order correction SFF1 for JT gravity can be found from the general result in the
+previous section. After some algebra, we find
+SFF1 =
+1
+24πe− 1
+2 βarcsinh(τ)2
+�
+τ
+1 + τ 2
+�
+β +
+1
+arcsinh(τ)2
+�
++
+4
+arcsinh(τ)3
+�
+1
+√
+1 + τ 2 − 1
+�
++
+1
+arcsinh(τ)
+�
+2 −
+1
+(1 + τ 2)3/2 + β
+�
+4 −
+1
+√
+1 + τ 2
+���
++ ⟨Z(2β)⟩g=1Erf
+��
+β
+2 arcsinh(τ)
+�
+,
+(4.10)
+where the genus-one one-point function is given by
+⟨Z(β)⟩g=1 = (1 + β)√β
+24√π
+.
+(4.11)
+In Figure 1, we show the plot of SFF0 and SFF1 of JT gravity. From Figure 1a we can
+see that SFF0 exhibits the behavior of the ramp and plateau. From Figure 1b we see that
+SFF1 approaches a constant at late times while it blows up as τ → 0. As we argued in the
+previous section, this diverging behavior of SFF1 at τ = 0 is an artifact of the τ-scaling
+limit. In fact, the small τ expansion of SFF1 in (4.10) reads
+SFF1 = 1
+2π
+�
+β
+4τ + 3 + 8β + 5β2
+24
+τ + O(τ 3)
+�
+.
+(4.12)
+One can see that the negative power term of τ agrees with the O(ℏ) term of (3.17), as
+expected. One can also check that this expansion (4.12) is consistent with the general case
+in (2.6).
+– 11 –
+
+-1
+1
+2
+3
+4
+5
+log(τ)
+0.04
+0.06
+0.08
+0.10
+0.12
+0.14
+0.16
+SFF0
+(a) SFF0
+-1
+0
+1
+2
+3
+4
+5
+log(τ)
+0.09
+0.10
+0.11
+0.12
+0.13
+0.14
+0.15
+SFF1
+(b) SFF1
+Figure 1: Plot of (a) SFF0 and (b) SFF1 for JT gravity. The horizontal axis is log τ. We
+set β = 1 in this figure. SFFg approaches ⟨Z(2β)⟩g as τ → ∞ (shown by orange dashed
+lines).
+5
+Computation of SFF
+In this section we formulate a systematic method of computing the small ℏ expansion of
+the τ-scaling limit of the SFF in the general background {tk} of topological gravity. In
+section 5.1 we will first derive a key relation
+∂τ∂0SFF = i
+ℏ
+�
+W1
+�
+β + iτ
+ℏ
+�
+∂0W1
+�
+β − iτ
+ℏ
+�
+− ∂0W1
+�
+β + iτ
+ℏ
+�
+W1
+�
+β − iτ
+ℏ
+��
+,
+(5.1)
+which relates SFF with (the t0-derivative of) the one-point function
+W1
+�
+β ± iτ
+ℏ
+�
+= ℏ∂0
+�
+Z
+�
+β ± iτ
+ℏ
+��
+.
+(5.2)
+Here ∂k := ∂tk. The relation (5.1) enables us to compute the small ℏ expansion of the SFF
+from that of the one-point function W1(β± iτ
+ℏ ). As we will describe in section 5.2, the small
+ℏ expansion of W1(β ± iτ
+ℏ ) can be obtained by the method developed in [36] with a slight
+modification. This is based on the KdV equation. We will then explain in section 5.3 how
+to integrate ∂τ∂0SFF to obtain the small ℏ expansion of the SFF.
+5.1
+Derivation of the key differential equation
+In this subsection we will derive (5.1). As we have already seen in the previous sections,
+the one- and two-point functions can be expressed in terms of the CD kernel as [29]
+⟨Z(β)⟩ =
+�
+dEe−βEK(E, E),
+⟨Z(β1)Z(β2)⟩c = ⟨Z(β1 + β2)⟩ −
+�
+dE1dE2e−β1E1+β2E2K(E1, E2)2.
+(5.3)
+The CD kernel is written as
+K(E1, E2) = ℏ∂0ψ(E1)ψ(E2) − ℏ∂0ψ(E2)ψ(E1)
+−E1 + E2
+,
+(5.4)
+– 12 –
+
+where ψ(E) is the Baker-Akhiezer function. It satisfies the Schr¨odinger equation
+− (ℏ2∂2
+0 + u)ψ(E) = Eψ(E).
+(5.5)
+By using this equation it immediately follows from (5.4) that
+ℏ∂0K(E1, E2) = ψ(E1)ψ(E2).
+(5.6)
+Then we have
+ℏ∂0⟨Z(β)⟩ = W1(β) =
+�
+dEe−βEψ(E)2,
+ℏ∂0⟨Z(β1)Z(β2)⟩c = W1(β1 + β2) −
+�
+dE1dE2e−β1E1−β2E22ψ(E1)ψ(E2)K(E1, E2).
+(5.7)
+When β1,2 = β ± iτℏ−1, we find
+∂τ∂0⟨Z(β1)Z(β2)⟩c
+= − i
+ℏ2
+�
+dE1dE2e−β1E1−β2E22ψ(E1)ψ(E2)(−E1 + E2)K(E1, E2)
+= − i
+ℏ2
+�
+dE1dE2e−β1E1−β2E22ψ(E1)ψ(E2) [ℏ∂0ψ(E1)ψ(E2) − ℏ∂0ψ(E2)ψ(E1)]
+= − i
+ℏ
+�
+dE1dE2e−β1E1−β2E2 �
+∂0ψ(E1)2ψ(E2)2 − ψ(E1)2∂0ψ(E2)2�
+= − i
+ℏ [∂0W1(β1)W1(β2) − W1(β1)∂0W1(β2)] .
+(5.8)
+Thus we have derived (5.1).
+5.2
+Small ℏ expansion of one-point function
+Let us next consider the small ℏ expansion of W1
+�
+β ± iτ
+ℏ
+�
+. We can restrict ourselves to the
+case of W1
+�
+β + iτ
+ℏ
+�
+without loss of generality, as W1
+�
+β − iτ
+ℏ
+�
+is immediately obtained by
+complex conjugation. The expansion can be done in two ways. One way, which is easier
+to understand, is to use the results of the ’t Hooft expansion computed in [36]:
+W1 (β) = exp
+� ∞
+�
+n=0
+ℏn−1 �Gn(λ)
+�
+,
+λ = ℏβ fixed.
+(5.9)
+Given this expansion, the small ℏ expansion of W1
+�
+β + iτ
+ℏ
+�
+is obtained by merely substi-
+tuting λ = iτ + βℏ into the expansion and then re-expanding it in ℏ as
+W1
+�
+β + iτ
+ℏ
+�
+= exp
+�
+ℏ−1 �G0(iτ) +
+�
+�G1(iτ) + �G′
+0(iτ)β
+�
++ ℏ
+�
+�G2(iτ) + �G′
+1(iτ)β + 1
+2
+�G′′
+0(iτ)β2
+�
++ O(ℏ2)
+�
+.
+(5.10)
+– 13 –
+
+More specifically, the first few �Gn are given by [36]6
+�G0 = −E0λ +
+�
+ja≥0
+�
+a ja=k
+�
+a aja=n
+(2n + k + 1)!
+(2n + 3)!
+λ2n+3
+2n+1s2n+k+2
+∞
+�
+a=1
+Ija
+a+1
+ja!(2a + 1)!!ja
+= −E0λ +
+1
+12s2 λ3 +
+I2
+60s5 λ5 +
+�
+I2
+2
+144s8 +
+I3
+840s7
+�
+λ7 + O(λ9),
+�G1 = 1
+2 log
+� ℏ
+8π
+∂λE(λ)
+E(λ) − E0
+�
+,
+�G2 =
+iI2
+12s2z(λ) −
+5i
+24sz(λ)3 − 3E(1)
+8z(λ)4 +
+E(2)
+4z(λ)2E(1) −
+E(3)
+8
+�
+E(1)�2 +
+�
+E(2)�2
+6
+�
+E(1)�3 ,
+(5.11)
+where
+E(λ) = −∂λ �G0,
+z(λ) =
+�
+E(λ) − E0,
+E(j) = ∂j
+λE(λ).
+(5.12)
+Ik and s = 1 − I1 were defined in section 2. From this one obtains
+W1
+�
+β + iτ
+ℏ
+�
+= eG(β+ iτ
+ℏ ) = exp
+� ∞
+�
+n=0
+ℏn−1Gn(β, τ, {Ik})
+�
+(5.13)
+with
+G0 = �G0(iτ),
+G1 = 1
+2 log
+� ℏ
+8πi
+∂τEτ
+Eτ − E0
+�
+− βEτ,
+G2 = i
+�
+I2
+12s2zτ
+−
+5
+24sz3τ
++ 3e(1)
+8z4τ
+−
+e(2)
+4z2τe(1) + e(3)
+8e(1)2 − e(2)2
+6e(1)3
++
+�e(1)
+2z2τ
+− e(2)
+2e(1)
+�
+β + e(1)
+2 β2
+�
+.
+(5.14)
+Here Eτ = E(iτ), zτ = √Eτ − E0 = z(iτ) and e(j) = ∂j
+τEτ = ijE(j)(iτ).
+The other way to compute the small ℏ expansion of W1
+�
+β + iτ
+ℏ
+�
+, which is technically
+more efficient, is to solve the KdV equation directly with the expansion (5.13). This can be
+done by the method of [36] with only a slight modification. In what follows let us describe
+the method in our present notation. First, recall that W1 = W1
+�
+β + iτ
+ℏ
+�
+satisfies the KdV
+equation [23]
+∂1W1 = u∂0W1 + ℏ2
+6 ∂3
+0W1.
+(5.15)
+6The elements here and there are identified as �Gn = 2(n−1)/2Gn, E0 = −u0, shere = tthere = 1 − I1,
+E(λ) = −ξ∗, E(j) = −2−n/2ξ(n)
+∗
+, z(λ) = −2−1/2iz∗ and λ = 21/2sthere. The normalization of W1 here
+(i.e. the constant part of �G1) is also adjusted accordingly.
+– 14 –
+
+In terms of G = log W1, this is written as
+∂1G = u∂0G + ℏ2
+6
+�
+∂3
+0G + 3∂0G∂2
+0G + (∂0G)3�
+.
+(5.16)
+Here, u is the specific heat of the general topological gravity. It obeys the KdV equation
+and its genus expansion
+u =
+∞
+�
+g=0
+2gugℏ2g
+(5.17)
+was computed by means of a recurrence relation.7 Note, in particular, that the leading
+order part is equal to the threshold energy
+u0 = −E0.
+(5.18)
+By plugging the expansions (5.17) and
+G =
+∞
+�
+n=0
+ℏn−1Gn
+(5.19)
+into (5.16), one obtains the recurrence relation
+−∂sG0 = 1
+6(∂0G0)3,
+(5.20)
+DG1 = 1
+2∂0G0∂2
+0G0
+(5.21)
+for n = 0, 1 and
+DGn = 1
+6
+�
+0≤i,j,k<n
+i+j+k=n
+∂0Gi∂0Gj∂0Gk + 1
+2
+n−1
+�
+k=0
+∂0Gn−k−1∂2
+0Gk + 1
+6∂3
+0Gn−2 +
+⌊ n
+2 ⌋
+�
+k=1
+2kuk∂0Gn−2k
+(5.22)
+for n ≥ 2.8 Here we have introduced the differential operator
+D := −∂s − 1
+2(∂0G0)2∂0.
+(5.23)
+Note also that instead of t0, t1 we regard s and E0 as independent variables and treat tk≥2
+as parameters [23]. From this viewpoint ∂0,1 are understood as
+∂0 = −1
+s (∂E0 + I2∂s) ,
+∂1 = −E0∂0 − ∂s.
+(5.24)
+7See, e.g. [23] with the identification tthere = shere, Bn = (−1)n+1In+1 (n ≥ 1).
+8Gn here is related with Gn in [36] as Ghere
+n
+(β, τ = 0, {Ik}) = 2(n−1)/2Gthere
+n
+(s = 2−1/2ℏβ, {Ik}), up to
+the constant part of G1.
+– 15 –
+
+To solve the recurrence relation (5.20)–(5.22), it is helpful to recall some relevant
+formulas (see [36]):
+∂0s = −I2
+s ,
+∂0In = In+1
+s
+(n ≥ 1),
+∂0G0 = 2izτ,
+∂0zτ =
+1
+2szτ
+− ∂τzτ
+zτ
+,
+∂0e(j) = −2∂j+1
+τ
+zτ,
+∂szτ = −e(1).
+(5.25)
+The differential operator (5.23) is then written as
+D = −∂s + 2z2
+τ∂0.
+(5.26)
+It has the properties
+Dzτ = zτ
+s ,
+DEτ = 0,
+(5.27)
+De(n) = 8
+n−1
+�
+k=0
+k
+�
+ℓ=0
+ℓ
+�
+m=0
+k!
+(k − ℓ)!(ℓ − m)!m!∂(n−ℓ)
+τ
+zτ ∂(ℓ−m)
+τ
+zτ ∂(m+1)
+τ
+zτ.
+(5.28)
+As explained in [37], ∂n
+τ zτ can be expressed in terms of e(j) and zτ:
+∂τzτ = e(1)
+2zτ
+,
+∂2
+τzτ = e(2)
+2zτ
+− e(1)2
+4z3τ
+,
+∂3
+τzτ = e(3)
+2zτ
+− 3e(1)e(2)
+4z3τ
++ 3e(1)3
+8z5τ
+.
+(5.29)
+We are now in a position to solve the recurrence relation. In [36] the recurrence relation
+(5.20)–(5.22) was solved with the initial data
+Gbefore
+0
+= �G0(iτ),
+Gbefore
+1
+= 1
+2 log
+� ℏ
+8πi
+∂τEτ
+Eτ − E0
+�
+.
+(5.30)
+Here, we would like to solve the same recurrence relation with the initial data given in
+(5.14). The only difference from [36] is that here G1 has an extra β-dependent term −βEτ.
+It is clear from (5.27) that (5.21) is satisfied by G1 with this term, given that (5.21) is
+satisfied by Gbefore
+1
+.
+To solve the recurrence relation, we can use almost the same algorithm we developed
+in [36]. The only difference is the step (v), which is modified accordingly so that it works
+for the β-dependent parts as well. The modified algorithm is as follows:
+(i) Compute the r.h.s. of (5.22) and express it as a polynomial in the variables s−1, Ik≥2,
+z−1
+τ , e(1)−1 and e(n), n ≥ 1.
+(ii) Let s−mf
+�
+Ik, zτ, e(n)
+�
+denote the highest-order part in s−1 of the obtained expression.
+This part can arise only from
+D
+�
+f
+�
+Ik, zτ, e(n)
+�
+2(m − 2)sm−2z2τI2
+�
+.
+(5.31)
+Therefore subtract this from the obtained expression.
+– 16 –
+
+(iii) Repeat procedure (ii) down to m = 3. Then all the terms of order s−2 automati-
+cally disappear and the remaining terms are of order s−1 or s0. Note also that the
+expression does not contain any Ik.
+(iv) In the result of (iii), collect all the terms of order s−1 and let s−1zτ∂zτ g
+�
+zτ, e(n)
+�
+denote the sum of them. This part arises from
+Dg
+�
+zτ, e(n)
+�
+.
+(5.32)
+Therefore subtract this from the result of (iii). The remainder turns out to be inde-
+pendent of s.
+(v) In the obtained expression, let
+e(1)mh
+�
+e(n ≥ 2)
+�
+zτ
+(5.33)
+denote the part which is of the order z−1
+τ
+as well as of the highest order in e(1). This
+part arises from
+D
+�
+−
+1
+2e(1)
+� ˜a
+da
+�
+a−1e(1)
+�m h
+�
+e(n) → a−n−1e(n)
+��
+˜a=1
+.
+(5.34)
+(Here, a and ˜a are intermediate formal variables.) Therefore subtract this from the
+obtained expression.
+(vi) Repeat procedure (v) until the resulting expression vanishes.
+(vii) By summing up all the above-obtained primitive functions, we obtain Gn.
+Using the above algorithm we have computed Gn for n ≤ 12. For instance, G3 is computed
+as
+G3 = −
+5
+32s2z6τ
++
+I2
+8s3z4τ
+−
+I3
+24s3z2τ
+−
+I2
+2
+12s4z2τ
+− 5I2e(1)
+96s2z5τ
+− 3e(1)2
+4z8τ
++ 35e(1)
+64sz7τ
++ 11e(2)
+16z6τ
+−
+3e(3)
+16z4τe(1) −
+5e(2)
+32sz5τe(1) +
+I2e(2)
+48s2z3τe(1) +
+e(4)
+16z2τe(1)2 +
+e(2)2
+8z4τe(1)2 −
+e(5)
+48e(1)3
+− 7e(2)e(3)
+24z2τe(1)3 + e(3)2
+8e(1)4 +
+e(2)3
+4z2τe(1)4 + e(2)e(4)
+6e(1)4 − 3e(2)2e(3)
+4e(1)5
++ e(2)4
+2e(1)6
++
+� 5e(1)
+16sz5τ
+− 3e(1)2
+4z6τ
++ 5e(2)
+8z4τ
+− I2e(1)
+24s2z3τ
+−
+e(3)
+4z2τe(1) + e(4)
+8e(1)2 +
+e(2)2
+4z2τe(1)2
+−7e(2)e(3)
+12e(1)3 + e(2)3
+2e(1)4
+�
+β +
+�
+−e(1)2
+4z4τ
++ e(2)
+4z2τ
++ e(2)2
+4e(1)2 − e(3)
+4e(1)
+�
+β2 + e(2)
+6 β3.
+(5.35)
+Let us finally comment on the small ℏ expansion of W1
+�
+β − iτ
+ℏ
+�
+. As we mentioned
+already, it is obtained from (5.13) by complex conjugation:
+W1
+�
+β − iτ
+ℏ
+�
+= exp
+� ∞
+�
+n=0
+ℏn−1Gn
+�
+.
+(5.36)
+– 17 –
+
+Furthermore, Gn is related to Gn by
+Gn = (−1)n+1Gn + πi
+2 δn,1.
+(5.37)
+This can be shown directly for n = 0, 1 and inductively for n ≥ 2 by using the recurrence
+relation (5.22).
+5.3
+Small ℏ expansion of SFF
+Using the results obtained above, one can compute the small ℏ expansion of the SFF.
+Substituting (5.13), (5.36) and the explicit form of G1 in (5.14) into (5.1), one obtains
+∂τ∂0SFF = i
+ℏ(∂0G − ∂0G)eG+G
+= −2i
+ℏ
+∞
+�
+k=0
+ℏ2k−1∂0G2k exp
+�
+G1 + G1 + 2
+∞
+�
+k=1
+ℏ2kG2k+1
+�
+= e−2βEτ
+4πi
+∂τEτ
+Eτ − E0
+∞
+�
+k=0
+ℏ2k−1∂0G2k exp
+� ∞
+�
+k=1
+2G2k+1ℏ2k
+�
+=: e−2βEτ
+2πℏ
+∞
+�
+n=0
+cnℏ2n,
+(5.38)
+where the first two of cn are computed as
+c0 = e(1)
+zτ
+,
+c1 = e(1)e(2)
+3zτ
+β3 +
+�
+−3e(1)3
+8z5τ
++ e(1)e(2)
+4z3τ
+− e(3)
+2zτ
++
+e(2)2
+2zτe(1)
+�
+β2
++
+�
+−15e(1)3
+16z7τ
++ 3e(1)e(2)
+4z5τ
+− e(3)
+4z3τ
++ 3e(1)2
+8sz6τ
+− I2e(1)2
+12s2z4τ
++
+e(4)
+4zτe(1) +
+e(2)2
+4z3τe(1) − 7e(2)e(3)
+6zτe(1)2 +
+e(2)3
+zτe(1)3
+�
+β
+− 5e(1)
+32s2z7τ
++ 25e(1)e(2)
+32z7τ
+− 3e(3)
+16z5τ
++ 9e(1)2
+16sz8τ
++ I2e(1)
+8s3z5τ
+− I2
+2e(1)
+12s4z3τ
+− I2e(1)2
+12s2z6τ
+− 3e(2)
+16sz6τ
+− I3e(1)
+24s3z3τ
++ I2e(2)
+24s2z4τ
+− 105e(1)3
+128z9τ
++
+3e(2)2
+32z5τe(1) +
+e(4)
+16z3τe(1) −
+e(5)
+24zτe(1)2 − 7e(2)e(3)
+24z3τe(1)2
++
+e(2)3
+4z3τe(1)3 +
+e(3)2
+4zτe(1)3 + e(2)e(4)
+3zτe(1)3 − 3e(2)2e(3)
+2zτe(1)4 +
+e(2)4
+zτe(1)5 .
+(5.39)
+Let us next consider the expansion
+∂τSFF = e−2βEτ
+2πℏ
+∞
+�
+n=0
+bnℏ2n.
+(5.40)
+– 18 –
+
+Comparing (5.40) with (5.38), we see that bn and cn are related as
+cn = e2βEτ ∂0e−2βEτ bn
+= (∂0 − 2β∂0Eτ) bn
+=
+�
+∂0 + 2β e(1)
+zτ
+�
+bn.
+(5.41)
+For n = 0, we immediately see that b0 = 1/(2β) gives c0 in (5.39). For n ≥ 1, let us assume
+that bn is a polynomial in β. Then, the highest-order term of this polynomial is determined
+as that of cn divided by 2βe(1)/zτ. Subtracting this contribution from both sides of (5.41),
+one can determine the next to the highest-order term in the same manner. In this way,
+one can determine bn as a polynomial of β without integrating in t0. For instance, the first
+two of them are
+b0 = 1
+2β ,
+b1 = e(2)
+6 β2 +
+�
+−e(1)2
+8z4τ
+− e(3)
+6e(1) + e(2)2
+4e(1)2
+�
+β
+− I2e(1)
+24s2z3τ
++ e(2)
+16z4τ
++ e(1)
+16sz5τ
+− e(1)2
+8z6τ
++
+e(4)
+24e(1)2 − e(2)e(3)
+4e(1)3 + e(2)3
+4e(1)4 .
+(5.42)
+Let us finally consider the small ℏ expansion (1.3) of the SFF:
+SFF = SFFg=0 + SFFg≥1
+= ℏ−1SFF0 +
+∞
+�
+g=1
+ℏ2g−1SFFg.
+(5.43)
+The leading term SFF0 is given in (2.19). It is easy to verify that the τ-derivative of (2.19)
+reproduces the leading order part of (5.40) with b0 = 1/(2β). Here, we are interested in
+the higher order part SFFg≥1. As explained in section 3 (see, in particular (3.5)), a key
+feature of SFFg≥1 is that it takes the form
+SFFg≥1 = Erf
+��
+2βzτ
+�
+⟨Z(2β)⟩g≥1 + e−2βEτ
+2πℏ
+∞
+�
+g=1
+agℏ2g
+(5.44)
+with ag being polynomial in β. The small ℏ expansion of the one-point function ⟨Z(2β)⟩
+was computed in [23]:
+⟨Z(2β)⟩g≥1 =
+e−2βE0
+�
+2π(2β)3√
+2ℏ
+∞
+�
+g=1
+ℏ2g2gZg(2β),
+(5.45)
+where
+Z1(β) =
+1
+24sβ3 +
+I2
+24s2 β2,
+Z2(β) =
+1
+1152s3 β6 +
+29I2
+5760s4 β5 +
+� 7I2
+2
+480s5 +
+29I3
+5760s4
+�
+β4 +
+� 7I3
+2
+288s6 + 29I2I3
+1440s5 +
+I4
+384s4
+�
+β3
++
+� 7I4
+2
+288s7 + 5I2
+2I3
+144s6 +
+29I2
+3
+5760s5 + 11I2I4
+1440s5 +
+I5
+1152s4
+�
+β2.
+(5.46)
+– 19 –
+
+Comparing (5.44) with (5.40), we see that ag and bg are related as
+bg = e(1)
+2βzτ
+2gZg(2β) +
+�
+∂τ − 2βe(1)
+�
+ag.
+(5.47)
+Given that ag are polynomials in β, we can calculate them from bg without integrating in
+τ, in essentially the same way as we obtained bg from cg. For instance, a1 is obtained as
+a1 =
+�
+1
+6szτ
+−
+e(2)
+12e(1)
+�
+β + e(1)
+16z4τ
++
+I2
+12s2zτ
+−
+1
+24sz3τ
++
+e(3)
+24e(1)2 −
+e(2)2
+12e(1)3 .
+(5.48)
+One sees that the terms which do not contain e(j) precisely give d1 in (3.7); the other
+terms comprise f1 given by (3.8) with (3.10). The expression of a2 is already rather long
+and we relegate it to Appendix B. We have computed ag for g ≤ 5.
+6
+Higher order corrections to CD kernel
+In this section we consider the higher order correction of the CD kernel beyond the sine
+kernel approximation.
+That is, we consider a small ℏ expansion of the CD kernel.
+In
+principle, this is not a difficult task since the CD kernel is expressed as a bilinear form (5.4)
+in terms of the BA function ψ(E) whose ℏ-expansion was calculated [29]. The purpose of
+this section is to present a concrete, technically efficient method.
+6.1
+Small ℏ expansion of BA function
+To compute the expansion of the CD kernel, one first needs to compute the small ℏ ex-
+pansion of ψ(E + 1
+2ℏω) rather than ψ(E). Of course, this is obtained by re-expanding the
+results of [29] with the replacement E → E + 1
+2ℏω. For the sake of technical efficiency,
+however, here we compute the expansion
+log ψ(E + 1
+2ℏω) = A(E + 1
+2ℏω) =
+∞
+�
+n=0
+ℏn−1An(E, ω)
+(6.1)
+directly by means of a recurrence relation.
+It was derived in [29] that
+v(E) := ℏ∂0A(E)
+(6.2)
+satisfies the equation
+v2 + ℏ∂0v = −E − u.
+(6.3)
+Here, u is the specific heat of the general topological gravity and we again assume that its
+genus expansion (5.17) is given. Since we want to compute the ℏ expansion of ψ with a
+shifted argument, we replace E in (6.3) by E + 1
+2ℏω and plug the expansion of the form
+v = v(E + 1
+2ℏω) =
+∞
+�
+n=0
+ℏnvn(E, ω)
+(6.4)
+– 20 –
+
+into (6.3). Expanding both sides of the equation in ℏ, we obtain the recurrence relation
+v2
+0 = −E + E0,
+2v0v1 + ∂0v0 = −ω
+2 ,
+vn = − 1
+2v0
+�
+∂0vn−1 +
+n−1
+�
+k=1
+vkvn−k +
+�
+2
+n
+2 u n
+2 (n even)
+0
+(n odd)
+�
+,
+n ≥ 2.
+(6.5)
+The first two equations are solved as
+v0 = iz = i
+�
+E − E0,
+v1 = −
+1
+4sz2 + iω
+4z .
+(6.6)
+vn (n ≥ 2) are also immediately obtained by the recurrence relation given the form of ug.
+For instance, using
+u1 =
+I2
+2
+12s4 +
+I3
+24s3
+(6.7)
+one obtains
+v2 = i
+�
+I2
+2
+12zs4 +
+I3
+24zs3 −
+I2
+8z3s3 +
+5
+32z5s2
+�
++
+1
+8z4sω − i
+1
+32z3 ω2,
+v3 = −
+I3
+2
+6s6z2 − 7I2I3
+48s5z2 + 11I2
+2
+48s5z4 −
+I4
+48s4z2 +
+I3
+12s4z4 −
+9I2
+32s4z6 +
+15
+64s3z8
++ i
+�
+−
+I2
+2
+48s4z3 −
+I3
+96s3z3 +
+3I2
+32s3z5 −
+25
+128s2z7
+�
+ω −
+1
+16sz6 ω2 + i
+1
+128z5 ω3.
+(6.8)
+Note that the form of the recurrence relation (6.5) for vn≥2 is not affected by the shift
+E → E + 1
+2ℏω. The ω-dependence enters entirely through the initial data, i.e. the form of
+v1.
+One can easily obtain An from vn through the relation ∂0An = vn. Due to the structure
+(5.24) of ∂0, the form of the s-dependent part of An can be determined without integrating
+in t0.
+(We use a similar logic as we obtained bn from cn in the last section.)
+The s-
+independent part can also be determined with the help of the formulas (the proofs of
+which are straightforward)
+� t0
+dt0
+1
+sz2 = 2 log z,
+� t0
+dt0
+1
+sz2k = −
+1
+(k − 1)z2(k−1)
+(k ≥ 2),
+� t0
+dt0
+1
+z2k−1 = Γ( 3
+2 − k)
+Γ( 1
+2)
+ρ(k−1)
+0
+= (−2)k−1
+(2k − 3)!!ρ(k−1)
+0
+(k ≥ 0).
+(6.9)
+Here
+ρ(n)
+0
+= ∂n
+Eρ0(E)
+(n ≥ 0),
+ρ(−1)
+0
+=
+� E
+E0
+dEρ0(E).
+(6.10)
+– 21 –
+
+We thus obtain
+A0 = i
+2ρ(−1)
+0
+= i
+2
+� E
+E0
+dEρ0(E),
+A1 = −1
+2 log z + iρ0
+4 ω − 1
+2 log(4π),
+A2 = i
+�
+I2
+24s2z −
+5
+48sz3
+�
+−
+1
+8z2 ω + iρ(1)
+0
+16 ω2,
+A3 = −
+I2
+2
+24s4z2 −
+I3
+48s3z2 +
+I2
+16s3z4 −
+5
+64s2z6 + i
+�
+−
+I2
+96s2z3 +
+5
+64sz5
+�
+ω
++
+1
+32z4 ω2 + iρ(2)
+0
+96 ω3.
+(6.11)
+The constant part of A1 is determined in accordance with the ω = 0 case [29].
+6.2
+Small ℏ expansion of CD kernel
+Let us now compute the small ℏ expansion of the CD kernel. We should first note that
+this is a WKB-type asymptotic expansion. As is known [9], there appear some qualitative
+differences between the cases of E > E0 and E < E0. In this paper we consider the case
+of E > E0. In this case the two independent BA functions are given by
+ψ(E)
+and
+ψ(E) = ψ(E)
+���
+z→−z,
+(6.12)
+i.e. they are complex conjugate to each other. Due to this complex structure, for E > E0
+there are two saddles which equally contribute to the CD kernel [9]. Therefore, as long as
+the perturbative expansion in ℏ concerns, the CD kernel is approximated by
+K(E1, E2) = �K(E1, E2) + �K(E2, E1)
+(6.13)
+with
+�K(E1, E2) = ℏ∂0ψ(E1)ψ(E2) − ψ(E1)ℏ∂0ψ(E2)
+−E1 + E2
+= −v(E1) − v(E2)
+ℏω
+eA(E1)+A(E2).
+(6.14)
+By using the recurrence relation (6.5) it is easy to prove that
+vn(E, −ω) = (−1)n+1vn(E, ω),
+An(E, −ω) = (−1)n+1An(E, ω).
+(6.15)
+Therefore (6.14) is written as
+�K(E1, E2) = − 1
+ℏω
+� ∞
+�
+k=0
+2v2k(E, ω)ℏ2k
+�
+exp
+� ∞
+�
+k=0
+2A2k+1(E, ω)ℏ2k
+�
+.
+(6.16)
+Similarly, we have
+�K(E2, E1) = − 1
+ℏω
+∞
+�
+k=0
+2v2k(E, ω)ℏ2k exp
+� ∞
+�
+k=0
+2A2k+1(E, ω)ℏ2k
+�
+.
+(6.17)
+– 22 –
+
+Thus, the CD kernel is given by
+K(E1, E2) = Re
+�
+2 �K(E1, E2)
+�
+.
+(6.18)
+Substituting the explicit form of A1(E, ω) in (6.11) we obtain
+2πK(E1, E2) = Re
+�
+− 2
+ℏωz
+� ∞
+�
+k=0
+v2k(E, ω)ℏ2k
+�
+exp
+�
+2
+∞
+�
+k=1
+A2k+1(E, ω)ℏ2k
+�
+eiωρ0/2
+�
+= Re
+�
+Xeiωρ0/2�
+= Re{X} cos ωρ0
+2
+− Im{X} sin ωρ0
+2 ,
+(6.19)
+where
+X := − 2
+ℏωz
+� ∞
+�
+k=0
+v2k(E, ω)ℏ2k
+�
+exp
+�
+2
+∞
+�
+k=1
+A2k+1(E, ω)ℏ2k
+�
+.
+(6.20)
+With the forms of vn and An obtained in (6.6), (6.8) and (6.11), this gives the small ℏ
+expansion of the CD kernel.
+A few comments are in order. First, substituting v0 = iz one sees that the expression
+(6.19) correctly reproduces the sine kernel (2.10) at the order of ℏ−1. Second, recall that the
+diagonal part of the CD kernel, i.e. (6.19) in the limit of ω → 0, is equal to the eigenvalue
+density (3.21). This is evident for the genus zero part at the order of ℏ−1, but also implies
+that
+∞
+�
+g=1
+ℏ2g−1ρg(E) = Re{X}
+���
+ω=0.
+(6.21)
+We checked that this indeed reproduces ρg in (3.2) which were obtained from the result of
+⟨Z(β)⟩g in [23].
+By taking (6.21) into account, the expansion (6.19) is written explicitly as
+2πK(E1, E2)
+= 2
+ℏω
+�
+1 + ω2
+32z4 ℏ2 +
+�
++
+�
+−
+49I2
+2
+4608s4z6 −
+I3
+192s3z6 +
+49I2
+1536s3z8 −
+105
+2048s2z10
+�
+ω2
++
+�
+I2ρ(2)
+0
+2304s2z3 −
+ρ(2)
+0
+1536sz5 +
+11
+2048z8
+�
+ω4 − (ρ(2)
+0 )2
+4608 ω6
+�
+ℏ4 + O(ℏ6)
+�
+sin ωρ0
+2
++
+��
+ρ1 + ρ(2)
+0
+24 ω2
+�
+ℏ
++
+�
+ρ2 +
+�
+−
+I2
+128s2z7 +
+35
+768sz9
+�
+ω2 +
+�
+ρ(4)
+0
+1920 + ρ(2)
+0
+768z4
+�
+ω4
+�
+ℏ3 + O(ℏ5)
+�
+cos ωρ0
+2 .
+(6.22)
+Observe that there appear several derivatives of ρ0 at higher orders. These derivatives in
+fact have their origin in the integral (3.20) and are neatly absorbed if one recasts the above
+result into the form (3.19).
+– 23 –
+
+7
+Conclusions and outlook
+In this paper we have computed the higher order corrections of the SFF in the τ-scaling
+limit for an arbitrary background {tk} of topological gravity. We have also shown that
+these corrections of the SFF can be obtained by the Fourier transform of the CD kernel by
+including the corrections to the sine kernel formula. As we can see from Figure 1, SFFg≥1
+approaches a constant at late times which gives a correction to the value of the plateau.
+On the other hand, SFFg≥1 apparently diverges at τ = 0, but this is just an artifact of the
+τ-scaling limit, as we argued at the end of subsection 3.1.
+There are many interesting open questions.
+As far as we know, the corrections of
+the CD kernel to the sine kernel formula have not been fully explored in the literature
+before. In this paper we have computed these corrections of the CD kernel in the double-
+scaled matrix model of general topological gravity. It would be possible to compute the
+corrections of the CD kernel in a random matrix model before taking the double scaling
+limit. It would be interesting to carry out this computation along the lines of [32]. As
+emphasized in [26, 27], the τ-scaling limit enables us to reproduce the plateau of the SFF
+by just summing over the perturbative genus expansion. In general, the perturbative genus
+expansion is an asymptotic series, but it becomes a convergent series with a finite radius of
+convergence after we take the τ-scaling limit. What is happening is that, in the τ-scaling
+limit, the (2g)! growing part of the Weil-Petersson volume is suppressed and only a non-
+growing part of the Weil-Petersson volume survives, and hence we get a convergent series
+for the SFF. In this sense, we throw away most of the contributions of the moduli space
+integral by taking the τ-scaling limit. It is fair to say that we still do not understand the
+non-perturbative effects which might contribute to the appearance of the plateau. In fact,
+on general grounds we expect that the sum over SFFg in (1.3) is an asymptotic series. It
+would be interesting to study its resurgence structure and understand the non-perturbative
+effects associated with this asymptotic series (1.3).
+Acknowledgments
+We are grateful to Douglas Stanford for informing us of the result of [26] prior to its submis-
+sion to arXiv. This work was supported in part by JSPS Grant-in-Aid for Transformative
+Research Areas (A) “Extreme Universe” 21H05187 and JSPS KAKENHI Grant 19K03856
+and 22K03594.
+A
+Airy kernel
+In this appendix, we consider the CD kernel for the Airy case, known as the Airy kernel.
+As the name suggests, the BA function for the Airy case is given by the Airy function
+ψ(E) = ℏ− 1
+6 Ai
+�
+−ℏ− 2
+3 (E + t0)
+�
+.
+(A.1)
+Plugging this into (5.4), the Airy kernel is given by
+K(E1, E2) = Ai′(−ℏ− 2
+3 E1)Ai(−ℏ− 2
+3 E2) − Ai′(−ℏ− 2
+3 E2)Ai(−ℏ− 2
+3 E1)
+E1 − E2
+,
+(A.2)
+– 24 –
+
+where we have set t0 = 0 after taking the t0-derivative.
+To see the corrections to the sine kernel formula, we first recall that Ai(−x) and Ai′(−x)
+have the following asymptotic expansion at large x > 09
+Ai(−x) =
+1
+√πx
+1
+4
+�
+cos
+�
+A − π
+4
+� ∞
+�
+k=0
+(−1)k u2k
+A2k + sin
+�
+A − π
+4
+� ∞
+�
+k=0
+(−1)k u2k+1
+A2k+1
+�
+,
+Ai′(−x) = x
+1
+4
+√π
+�
+sin
+�
+A − π
+4
+� ∞
+�
+k=0
+(−1)k v2k
+A2k − cos
+�
+A − π
+4
+� ∞
+�
+k=0
+(−1)k v2k+1
+A2k+1
+�
+,
+(A.3)
+where A = 2
+3x
+3
+2 and
+uk = Γ(k + 5/6)Γ(k + 1/6)
+2k+1k!π
+,
+vk = 1 + 6k
+1 − 6kuk = −Γ(k + 7/6)Γ(k − 1/6)
+2k+1k!π
+.
+(A.4)
+The first few terms of the expansion (A.3) read
+Ai(−x) =
+1
+√πx
+1
+4
+cos
+�
+A − π
+4
+� �
+1 −
+385
+10368A2 + · · ·
+�
++
+1
+√πx
+1
+4
+sin
+�
+A − π
+4
+� � 5
+72A −
+85085
+2239488A3 + · · ·
+�
+,
+Ai′(−x) = x
+1
+4
+√π sin
+�
+A − π
+4
+� �
+1 +
+455
+10368A2 + · · ·
+�
+− x
+1
+4
+√π cos
+�
+A − π
+4
+� �
+− 7
+72A +
+95095
+2239488A3 + · · ·
+�
+.
+(A.5)
+The corrections to the sine kernel formula can be obtained by plugging the above expansion
+(A.5) into (A.2) with E1,2 = E ± 1
+2ℏω. By keeping the terms proportional to sin(A1 − A2)
+and cos(A1 − A2), we find
+K
+�
+E + ℏω
+2 , E − ℏω
+2
+�
+=
+� 2
+ℏω +
+ω
+16z4 ℏ +
+� 11ω3
+1024z8 −
+105ω
+1024z10
+�
+ℏ3 + O(ℏ5)
+� sin(A1 − A2)
+2π
++
+�
+ℏ
+16z5 +
+� 35ω2
+768z9 −
+105
+1024z11
+�
+ℏ3 + O(ℏ5)
+� cos(A1 − A2)
+2π
+,
+(A.6)
+where z =
+√
+E and
+A1,2 = 2
+3ℏE
+3
+2
+1,2 = 2
+3ℏ
+�
+E ± 1
+2ℏω
+� 3
+2
+.
+(A.7)
+One can check that (A.6) is consistent with our general formula of the CD kernel in (3.19).
+Note that if we plug (A.5) into (A.2), there appear terms proportional to sin(A1 +A2) and
+cos(A1 + A2) as well. However, they are highly oscillating in the limit ℏ → 0 with fixed
+E, ω, and hence these terms can be ignored in the computation of the Fourier transform
+of the CD kernel.
+9See e.g. https://dlmf.nist.gov/9.7.
+– 25 –
+
+B
+Coefficient a2 of SFF2 in (5.44)
+The coefficient a2 of SFF2 in (5.44) is given by (for brevity we abbreviate zτ to z)
+a2 =
+�
+− e(2)2
+72e(1) +
+1
+18s3z
+�
+β4
++
+�e(1)e(2)
+48z4
++
+e(4)
+240e(1) −
+5e(2)3
+144e(1)3 +
+29I2
+180s4z + e(2)e(3)
+72e(1)2 −
+1
+72s3z3
+�
+β3
++
+�
+− e(2)
+96sz5 − e(3)
+32z4 + e(1)e(2)
+16z6
+−
+e(5)
+160e(1)2 +
+e(3)2
+48e(1)3 +
+e(2)4
+24e(1)5 +
+e(2)2
+32z4e(1)
+− 3e(1)3
+64z8 +
+7I2
+2
+30s5z +
+29I3
+360s4z −
+29I2
+720s4z3 + I2e(2)
+144s2z3 + 19e(2)e(4)
+480e(1)3 − 31e(2)2e(3)
+288e(1)4
++
+1
+96z5s3
+�
+β2
++
+� 29I2I3
+180s5z +
+e(3)
+96e(1)sz5 − 7I2e(1)2
+384s2z7 + I2e(2)
+96s2z5 −
+e(2)2
+64sz5e(1)2 −
+I2e(3)
+144s2z3e(1)
++
+I2e(2)2
+96s2z3e(1)2 +
+7I3
+2
+36s6z +
+I4
+48s4z −
+7I2
+2
+120s5z3 −
+29I3
+1440s4z3 +
+29I2
+960s4z5 + 73e(1)2
+768sz9
+− 5e(2)
+192sz7 +
+e(2)2
+16z6e(1) +
+e(4)
+64z4e(1) − 3e(1)3
+16z10 + 13e(2)5
+24e(1)7 − e(3)
+16z6 +
+e(6)
+320e(1)3
+−
+5
+384s3z7 − 7e(2)e(3)
+96z4e(1)2 + 21e(1)e(2)
+128z8
++
+e(2)3
+16z4e(1)3 + 47e(2)e(3)2
+144e(1)5
+− 73e(2)3e(3)
+72e(1)6
++ 181e(2)2e(4)
+720e(1)5
+− 61e(3)e(4)
+960e(1)4 − 113e(2)e(5)
+2880e(1)4
+�
+β
++ 25I2e(1)
+384s3z8 − 49I2e(1)2
+1536s2z9 − I3e(1)
+96s3z6 − 29I2I3
+720s5z3 −
+e(4)
+384sz5e(1)2 +
+5e(3)
+384sz7e(1)
+−
+e(2)3
+64sz5e(1)4 −
+5e(2)2
+256sz7e(1)2 − 25I2
+2e(1)
+1152s4z6 + I2e(2)
+64s2z7 + 5I2
+2I3
+36s6z + 11I2I4
+360s5z
+−
+I2e(3)
+192s2z5e(1) +
+e(2)e(3)
+64sz5e(1)3 +
+I2e(2)2
+128s2z5e(1)2 +
+I2e(2)3
+96s2z3e(1)4 +
+I2e(4)
+576s2z3e(1)2
++ 13e(1)e(2)
+64z10
++
+e(2)3
+16z6e(1)3 +
+3e(2)2
+64z8e(1) +
+e(2)4
+16z4e(1)5 +
+e(3)2
+64z4e(1)3 +
+e(4)
+64z6e(1)
+−
+e(5)
+384z4e(1)2 + e(2)e(4)
+48z4e(1)3 − 7e(2)e(3)
+96z6e(1)2 − 3e(2)2e(3)
+32z4e(1)4 − 119e(2)4e(3)
+48e(1)8
++ 7e(2)e(6)
+720e(1)5
++ 29e(3)e(5)
+1440e(1)5 +
+e(4)2
+80e(1)5 −
+e(7)
+1920e(1)4 + 7e(2)6
+6e(1)9 − 7e(3)3
+96e(1)6 − 9e(2)
+128sz9 −
+53e(1)
+512s2z10
++ 219e(1)2
+1024sz11 −
+29I2
+768s4z7 +
+29I3
+1920s4z5 −
+I4
+192s4z3 +
+7I2
+2
+160s5z5 −
+7I3
+2
+144s6z3 +
+I5
+288s4z
++
+29I2
+3
+1440s5z +
+7I4
+2
+72s7z − 23e(2)e(3)e(4)
+72e(1)6
++ 7e(2)3e(4)
+12e(1)7
++ 19e(2)2e(3)2
+16e(1)7
+− 3e(2)2e(5)
+32e(1)6
+− 15e(1)3
+64z12
+− 15e(3)
+256z8 +
+35
+1536s3z9 − I2e(2)e(3)
+96s2z3e(1)3 .
+(B.1)
+– 26 –
+
+One can check that the e(j)-dependent part of (B.1) reproduces h2,n in (3.12); the e(j)-
+independent part of (B.1) becomes d2(zτ, β) defined in (3.5).
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diff --git a/iNE3T4oBgHgl3EQf4guX/content/tmp_files/load_file.txt b/iNE3T4oBgHgl3EQf4guX/content/tmp_files/load_file.txt
new file mode 100644
index 0000000000000000000000000000000000000000..24353d45b8591669c1b0f875efe17487a5e54852
--- /dev/null
+++ b/iNE3T4oBgHgl3EQf4guX/content/tmp_files/load_file.txt
@@ -0,0 +1,1285 @@
+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf,len=1284
+page_content='Prepared for submission to JHEP  Spectral form factor in the τ-scaling limit  Kazumi Okuyamaa and Kazuhiro Sakaib  aDepartment of Physics, Shinshu University,  3-1-1 Asahi, Matsumoto 390-8621, Japan  bInstitute of Physics, Meiji Gakuin University,  1518 Kamikurata-cho, Totsuka-ku, Yokohama 244-8539, Japan  E-mail: kazumi@azusa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content='shinshu-u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content='jp, kzhrsakai@gmail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content='com  Abstract: We study the spectral form factor (SFF) of general topological gravity in the  limit of large time and fixed temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content=' It is observed recently that in this limit, called  the tau-scaling limit, the genus expansion of the SFF can be summed up and the late-  time behavior of the SFF such as the ramp-plateau transition can be studied analytically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content='  In this paper we develop a technique for the systematic computation of the higher order  corrections to the SFF in the strict tau-scaling limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content=' We obtain the first five corrections in  a closed form for the general background of topological gravity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content=' As concrete examples, we  present the results for the Airy case and Jackiw-Teitelboim gravity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content=' We find that the above  higher order corrections are the Fourier transforms of the corrections to the sine-kernel  approximation of the Christoffel-Darboux kernel in the dual double-scaled matrix integral,  which naturally explains their structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content=' Along the way we also develop a technique for  the systematic computation of the corrections to the sine-kernel formula, which have not  been fully explored in the literature before.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content='  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content='04773v1  [hep-th]  12 Jan 2023  Contents  1  Introduction  1  2  τ-scaling limit of SFF  3  3  Summary of results  5  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content='1  SFFg  6  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content='2  Relation to the corrections of CD kernel  8  4  Examples of Airy case and JT gravity  10  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content='1  Airy case  10  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content='2  JT gravity  10  5  Computation of SFF  12  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content='1  Derivation of the key differential equation  12  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content='2  Small ℏ expansion of one-point function  13  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content='3  Small ℏ expansion of SFF  18  6  Higher order corrections to CD kernel  20  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content='1  Small ℏ expansion of BA function  20  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content='2  Small ℏ expansion of CD kernel  22  7  Conclusions and outlook  24  A Airy kernel  24  B Coefficient a2 of SFF2 in (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content='44)  26  1  Introduction  Spectral form factor (SFF) is a useful measure of the level statistics of quantum chaotic  systems [1] and it is widely studied in many areas of physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content=' In the context of quantum  gravity and holography, the SFF of the Sachdev-Ye-Kitaev (SYK) model [2–4] was studied  in [5, 6] as a useful diagnostics of the Maldacena’s version of the information problem [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content='  It is found that the SFF of the SYK model exhibits the behavior of the ramp and plateau  as a function of time, which is consistent with the conjecture that the level statistics of  quantum chaotic system is universally described by a random matrix model [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content=' As shown  in [9], Jackiw-Teitelboim (JT) gravity [10, 11], which is holographically dual to the low  energy sector of the SYK model, is indeed described by a certain double-scaled random  matrix model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content=' In the bulk gravity picture, the ramp of the SFF of JT gravity comes from  – 1 –  the contribution of a wormhole connecting two boundaries of spacetime [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content=' In the matrix  model picture, it is described by the connected part of the correlator of two macroscopic  loop operators [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content=' On the other hand, the plateau of the SFF is very mysterious from  the viewpoint of bulk gravity and it was speculated that the appearance of the plateau is  related to some non-perturbative effects in quantum gravity [6, 9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content=' In the matrix model  picture, it is argued in [6, 9] that the plateau can be explained by the Andreev–Altshuler  instantons [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content=' See also [15] for an explanation of the plateau by eigenvalue instantons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content='  In this paper, we will consider the connected part of the SFF in 2d quantum gravity  SFF = ⟨Z(β + it)Z(β − it)⟩c,  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content='1)  where Z(β) = Tr e−βH and the expectation value is defined by averaging over a random  matrix H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content=' As shown in [16–18], 2d quantum gravity coupled to a conformal matter can  be defined by a certain double-scaling limit of a random matrix model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content=' More generally,  one can introduce couplings {tk} (k = 0, 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content=') to the matrix model potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content=' Then the  free energy of the model is interpreted as a generating function of the intersection numbers  on the moduli space of Riemann surfaces and the model is called topological gravity [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content='  There is an underlying integrable structure in this model and the free energy serves as  a tau-function of the KdV hierarchy [19, 20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content=' It turns out that the matrix model of JT  gravity in [9] is a special case of topological gravity where infinitely many couplings {tk} are  turned on in a specific way [21–23] and it corresponds to the computation of Weil-Petersson  volumes [24, 25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content='  SFF exhibits the ramp-plateau transition around the time scale t ∼ ℏ−1, called the  Heisenberg time, where ℏ is the genus-counting parameter of the matrix model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content=' Recently,  it is observed in [26–28] that one can focus on the ramp-plateau transition regime by taking  what is called the “τ-scaling limit”1  t → ∞,  ℏ → 0,  τ = tℏ : fixed,  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content='2)  with β fixed in (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content=' In this limit, SFF is expanded as  SFF =  ∞  �  g=0  ℏ2g−1SFFg(τ, β).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content='  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content='3)  Remarkably, it is found that the leading term SFF0 can be computed in a closed form by  just summing over the original genus expansion in the τ-scaling limit and the resulting  SFF0 approaches a constant as τ → ∞ [26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content=' This opens up an interesting avenue for a  “perturbative plateau.”  In this paper, we will develop a technique for the systematic computation of the higher  order corrections SFFg (g ≥ 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content=' By using our method, we obtain SFFg up to g = 5 for  arbitrary couplings {tk} of topological gravity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content=' It turns out that SFFg has a structure which  is a natural generalization of SFF0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content=' In [26] it was shown that SFF0 is essentially determined  1In [23, 29], another scaling limit where both β and t are of order ℏ−1 was considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content=' This limit was  called the “’t Hooft limit” in [23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content='  – 2 –  by the Fourier transform of the universal part of the two-body eigenvalue correlation, known  as the sine kernel formula [30–32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content=' We find that the higher order correction SFFg is closely  related to the correction of the Christoffel-Darboux (CD) kernel to the naive sine kernel  formula.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content=' Rather surprisingly, such corrections to the sine kernel formula have not been  fully explored in the literature before, as far as we know.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content='2 We will also develop a technique  for the systematic computation of the corrections of the CD kernel and confirm that SFFg  is correctly reproduced form the Fourier transform of the CD kernel by including the  corrections to the sine kernel formula.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content='  This paper is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content=' In section 2, we review the known results about the  SFF in the τ-scaling limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content=' Along the way we consider the τ-scaling limit based on the genus  expansion of the SFF and obtain the small τ expansion of SFFg for small g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content=' In section 3,  we summarize our results of SFFg and the higher order corrections of the CD kernel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content=' We  also explain their relations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content=' In section 4, based on our general results we compute SFF0 and  SFF1 for the Airy case and JT gravity, as concrete examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content=' In section 5, we formulate  a systematic method of computing SFFg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content=' In section 6, we explain how to compute the  higher order corrections of the CD kernel beyond the sine kernel approximation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content=' Finally  we conclude in section 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content=' In Appendix A, we compute the corrections of the Airy kernel to  the sine kernel formula.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content=' In Appendix B, we present an explicit form of the coefficient that  determines SFF2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content='  2  τ-scaling limit of SFF  In this section we will briefly review the known results about the SFF in the τ-scaling  limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content=' We will consider the τ-scaling limit for the general background of topological gravity  based on the known genus expansion of the SFF, which enables us to compute the small τ  expansion of SFFg for small g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content='  For the general background {tk} of topological gravity, the connected two-boundary  correlator is expanded as [29]  ⟨Z(β1)Z(β2)⟩c =  √β1β2  2π  e−(β1+β2)E0  �  1  β1 + β2  +  �β2  1 + β1β2 + β2  2  24s2  + 2(β1 + β2)I2 + I3  24s3  +  I2  2  12s4  �  g2  s + O(g4  s )  �  ,  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content='1)  where gs is the genus-counting parameter related to ℏ as3  gs =  √  2ℏ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content='  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content='2)  Ik denote the Itzykson-Zuber variables defined by [33]  Ik =  ∞  �  n=0  tk+n  n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content=' (−E0)n,  k ∈ Z≥0  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content='3)  2In [32], general form of the large N limit of the CD kernel was studied from the asymptotic behavior  of the orthogonal polynomials associated to an arbitrary matrix model potential, before taking the double-  scaling limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content='  3See [23] for more about our definition of ℏ and its relation to the parameters S0, γ in JT gravity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content='  – 3 –  and s in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content='1) is given by  s = 1 − I1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content='  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content='4)  The threshold energy E0 is determined by the (genus-zero) string equation  I0 + E0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content='  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content='5)  In the τ-scaling limit (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content='2), after setting β1,2 = β ± iτ/ℏ in the two-boundary correlator  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content='1) and expanding it in ℏ, we find the small τ expansion of SFF0 and SFF1  SFF0 = e−2βE0  2π  � τ  2β − τ 3  12s2 + · · ·  �  ,  SFF1 = e−2βE0  2π  � β  4τ +  � 5β2  24s2 + 4βI2 + I3  12s3  + I2  2  6s4  �  τ + · · ·  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content='  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content='6)  In general, the two-boundary correlator is written as  ⟨Z(β1)Z(β2)⟩c = ⟨Z(β1 + β2)⟩ −  �  dE1dE2e−β1E1−β2E2K(E1, E2)2,  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content='7)  where K(E1, E2) is the Christoffel-Darboux (CD) kernel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content=' Thus, the SFF is written as  SFF =  � dE  2π e−2βEρ(E) −  �  dE1dE2e−β(E1+E2)−iτ E1−E2  ℏ  K(E1, E2)2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content='  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content='8)  In the τ-scaling limit (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content='2), the above integral is dominated by the region  E1 − E2 ∼ O(ℏ),  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content='9)  with finite E1 + E2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content=' In this regime, the CD kernel is approximated by the universal two-  body correlation of eigenvalues, known as the sine kernel [30–32]  Ksin(E1, E2) = sin  � 1  2ρ0(E)ω  �  πℏω  ,  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content='10)  where we defined  E1 = E + 1  2ℏω,  E2 = E − 1  2ℏω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content='  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content='11)  ρ0(E) is the genus-zero part of the eigenvalue density  ρ(E) =  ∞  �  g=0  ℏ2g−1ρg(E)  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content='12)  which is related to the one-point function by  ⟨Z(β)⟩ =  � ∞  E0  dE  2π e−βEρ(E) =  ∞  �  g=0  ℏ2g−1⟨Z(β)⟩g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content='  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content='13)  For the general background {tk}, ρ0(E) is given by  ρ0(E) =  ∞  �  k=1  (−1)k(Ik − δk,1)Γ(1/2)  Γ(k + 1/2)  (E − E0)k− 1  2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content='  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content='14)  – 4 –  In [26–28], it is argued that the above small τ expansion of SFF0 (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content='6) can be resummed  and SFF0 is obtained by replacing the CD kernel by the sine kernel in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content='8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content=' Then, by the  change of integration variables in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content='11), the leading term of the SFF becomes  SFF0 =  � ∞  E0  dE  2π e−2βEρ0(E) −  � ∞  E0  dEe−2βE  � ∞  −∞  dωe−iωτ sin2� 1  2ρ0(E)ω  �  π2ω2  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content='  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content='15)  The ω-integral is evaluated as [34]  � ∞  −∞  dωe−iωτ sin2� 1  2ρ0(E)ω  �  π2ω2  = 1  2π  �  ρ0(E) − τ  �  θ  �  ρ0(E) − τ  �  ,  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content='16)  where θ(x) is the step function  θ(x) =  �  1,  (x > 0),  0,  (x < 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content='  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content='17)  Assuming that ρ0(E) is a monotonically increasing function of E, there is a unique solution  Eτ to the equation  ρ0(Eτ) = τ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content='  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content='18)  In terms of Eτ, SFF0 in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content='15) is written as  SFF0 =  � ∞  E0  dE  2π e−2βEρ0(E) −  � ∞  Eτ  dE  2π e−2βE�  ρ0(E) − τ  �  =  τ  4πβ e−2βEτ +  � Eτ  E0  dE  2π e−2βEρ0(E).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content='  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content='19)  The first term corresponds to the ramp and the second term approaches a constant ⟨Z(2β)⟩0  in the τ → ∞ limit, corresponding to the plateau.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content='  One can check that (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content='19) reproduces the small τ expansion of SFF0 in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content='6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content=' To see  this, we first notice that Eτ has the following small τ expansion4  Eτ − E0 =  1  4s2 τ 2 −  I2  12s5 τ 4 +  � 7I2  2  144s8 +  I3  120s7  �  τ 6 + O(τ 8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content='  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content='20)  This can be obtained by inverting the relation (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content='18) using the expression of ρ0(E) in  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content='14).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content=' Plugging this expansion of Eτ into (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content='19), one can show that the small τ expansion  of SFF0 in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content='6) is indeed reproduced from (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content='19).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content='  In the rest of this paper, we will consider the higher order corrections SFFg (g ≥ 1)  to the spectral form factor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content=' In the next section we will first summarize the result of SFFg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content='  The details of the computation will be postponed to section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content='  3  Summary of results  In this section we will summarize our results of SFFg and discuss their relation to the  corrections of the CD kernel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content='  4A fully explicit expression of this expansion is available (Eτ is given by E(λ) in (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content='12) with λ = iτ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content='  – 5 –  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content='1  SFFg  It turns out that the expression of SFF0 in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content='19) has a natural generalization to the higher  order correction SFFg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content=' We find that SFFg has the structure  SFFg = 1  2πfg(τ, β)e−2βEτ +  � Eτ  E0  dE  2π e−2βEρg(E).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content='1)  By the relation (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content='13) we can easily find the first few terms of ρg(E) from the result of  ⟨Z(β)⟩g in [23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content=' For instance,  ρ1(E) =  1  16sz5 −  I2  24s2z3 ,  ρ2(E) = −  105  1024s3z11 +  203I2  1536s4z9  + 1  z7  �  − 7I2  2  64s5 − 29I3  768s4  �  + 1  z5  � 7I3  2  96s6 + 29I2I3  480s5 +  I4  128s4  �  + 1  z3  �  − 7I4  2  144s7 − 5I2  2I3  72s6 − 11I2I4  720s5 −  29I2  3  2880s5 −  I5  576s4  �  ,  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content='2)  where  z =  �  E − E0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content='3)  Note that the second term of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content='1) is a formal expression since the integral has a diver-  gence coming from E = E0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content=' This integral should be understood by a certain analytic  continuation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content=' For instance, the integral involving z−a term is defined as  � Eτ  E0  dEe−2βEz−a =  � ∞  E0  dEe−2βEz−a +  � Eτ  ∞  dEe−2βEz−a  = e−2βE0Γ(1 − a/2)(2β)a/2−1 +  � Eτ  ∞  dEe−2βEz−a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content='4)  Using this prescription, one can show that the second term of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content='1) is written as  � Eτ  E0  dE  2π e−2βEρg(E) = ⟨Z(2β)⟩gErf(zτ  �  2β) + 1  2πdg(zτ, β)e−2βEτ ,  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content='5)  where Erf(z) denotes the error function and zτ is defined by  zτ =  �  Eτ − E0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content='6)  It turns out that dg(zτ, β) in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content='5) is a polynomial in 1/zτ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content=' For instance, d1(zτ, β) is given  by  d1(zτ, β) =  β  6szτ  −  1  24sz3τ  +  I2  12s2zτ  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content='7)  Next, let us consider the first term of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content='  We find that fg(τ, β)e−2βEτ can be  rewritten as a sum of τ-derivatives  fg(τ, β)e−2βEτ =  3g−2  �  n=0  ∂n  τ  �  hg,n(τ)e(1)e−2βEτ �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content='8)  – 6 –  Here we introduced the notation  e(j) = ∂j  τEτ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content='9)  The important point is that hg,n(τ) is independent of β;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content=' the β-dependence of fg(τ, β) arises  solely from the τ-derivative in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content='8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content=' We have computed hg,n up to g = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content=' For g = 1 we  find  h1,1 = −s(2),  h1,0 =  1  16z4τ  ,  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content='10)  where s(j) denotes  s(j) = ρ(j)  0 (Eτ)  (j + 1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content='2j  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content='11)  with ρ(j)  0 (E) = ∂j  Eρ0(E).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content=' For g = 2 we find  h2,4 = −1  2s(2)2,  h2,3 = s(4) +  1  16z4τ  s(2),  h2,2 = −  3  256z8τ  + ρ1(Eτ)s(2),  h2,1 =  7I2  768s2z7τ  −  73  1536sz9τ  ,  h2,0 = −  I3  96s3z6τ  −  7  128s2z10  τ  − 25  2 ρ1(Eτ)2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content='12)  From the definition of Eτ in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content='18), one can show that ρ(j)  0 (Eτ) can be written as a com-  bination of e(j).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content=' For instance,  ρ(2)  0 (Eτ) = − e(2)  e(1)3 ,  ρ(4)  0 (Eτ) = − e(4)  e(1)5 − 15e(2)3  e(1)7  + 10e(3)e(2)  e(1)6  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content='13)  For general g, we find that hg,n with n = 3g − 2 and n = 3g − 3 are given by  hg,3g−2 = −s(2)g  g!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content='  ,  hg,3g−3 =  1  16z4τ  s(2)g−1  (g − 1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content=' + s(4) s(2)g−2  (g − 2)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content='.  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content='14)  In subsection 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content='2, we will see that this structure can be naturally understood from the  correction of the CD kernel to the sine kernel formula.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content='  Let us take a closer look at the behavior of SFFg in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content='1) as a function of τ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content=' At late  times, the first term of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content='1) vanishes exponentially and the second term approaches a  constant  lim  τ→∞ SFFg = ⟨Z(2β)⟩g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content='15)  This gives the higher genus correction to the value of plateau.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content=' On the other hand, at early  times SFFg diverges as  lim  τ→0 SFFg ∼ e−2βE0  �β  τ  �2g−1  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content='16)  – 7 –  This is just an artifact of the τ-scaling limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content=' This early time divergence can be traced  back to the expansion of the original genus-zero term  √β1β2  2π(β1 + β2) =  �  τ 2 + β2ℏ2  4πβℏ  = 1  4π  ∞  �  g=0  (−1)g−1(2g − 3)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content=' !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content='  g!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content='2g  �βℏ  τ  �2g−1  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content='17)  Before expanding in ℏ, the original expression  �  τ 2 + β2ℏ2 is regular at τ = 0, but after  expanding it by ℏ there appears an apparent divergence at τ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content=' We have checked that  the coefficient of O(τ 1−2g) term of SFFg in the small τ expansion is indeed given by (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content='17).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content='2  Relation to the corrections of CD kernel  From the general relation between the SFF and the CD kernel K(E1, E2) in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content='8), the higher  order correction SFFg is closely related to the correction of the CD kernel to the naive sine  kernel formula (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content='10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content=' The details of the computation of CD kernel will be presented in  section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content=' It turns out that the τ-derivative structure in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content='8) naturally appears from the  Fourier transform of the CD kernel squared  SFF =  � dE  2π e−2βEρ(E) − ℏ  �  dEe−2βE  �  dωe−iωτK  �  E + 1  2ℏω, E − 1  2ℏω  �2  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content='18)  We find that the corrections of the CD kernel to the sine kernel formula is organized as  K  �  E + 1  2ℏω, E − 1  2ℏω  �  =  �  2  ℏω +  ∞  �  g=1  ℏ2g−1k(g)  s (E, ω)  �  sin φ  2π  +  ∞  �  g=1  ℏ2g−1�  ρg(E) + k(g)  c (E, ω)  �cos φ  2π ,  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content='19)  where φ is given by  φ = 1  2ℏ  � E+ 1  2 ℏω  E− 1  2 ℏω  dEρ0(E) = 1  2  ∞  �  j=0  ℏ2jω2j+1  ρ(2j)  0  (E)  (2j + 1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content='22j .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content='20)  Note that the diagonal part of the CD kernel is equal to the eigenvalue density  K(E, E) = 1  2πρ(E) = 1  2π  ∞  �  g=0  ℏ2g−1ρg(E).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content='21)  We find that k(g)  s  and k(g)  c  vanishes as ω → 0 and hence our expansion (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content='19) is consistent  with the diagonal part (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content='21).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content=' The explicit forms of k(g)  s  and k(g)  c  for g = 1, 2 are given by  k(1)  s  =  ω  16z4 ,  k(1)  c  = 0,  k(2)  s  =  11ω3  1024z8 +  �  −  7  128s2z10 −  I3  96s3z6 − 49  4 ρ1(E)2  �  ω,  k(2)  c  =  �  35  768sz9 −  I2  128s2z7  �  ω2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content='22)  – 8 –  Expanding sin φ and cos φ further in ℏ, we find the ℏ expansion of the CD kernel in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content='19)  K  �  E + 1  2ℏω, E − 1  2ℏω  �  =  ∞  �  g=0  ℏ2g−1Kg(E, ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content='23)  One can see that the leading term is the sine kernel  K0(E, ω) = sin  � 1  2ρ0(E)ω  �  πω  ,  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content='24)  and the higher order correction Kg(E, ω) can be obtained from (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content='19).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content='  From the general form of the CD kernel (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content='19), one can see that the τ-derivative  structure of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content='8) can be understood from the Fourier transform of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content='19).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content=' Let us consider  a contribution of the term  K  �  E + 1  2ℏω, E − 1  2ℏω  �2  ⊃ ωneiρ0(E)ω,  (n ≥ 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content='25)  From the Fourier transformation of this term, we find  � ∞  −∞  dω  2π e−iωτωneiρ0(E)ω = (i∂τ)n  � ∞  −∞  dω  2π e−iωτ+iρ0(E)ω  = (i∂τ)nδ  �  ρ0(E) − τ  �  = (i∂τ)n�  e(1)δ(E − Eτ)  �  ,  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content='26)  where we used the relation  ρ(1)(Eτ) =  1  e(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content='27)  After integrating over E in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content='18), we find the τ-derivative structure of fg(τ, β)e−2βEτ in  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content='8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content=' We have checked that h1,n in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content='10) and h2,n in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content='12) are correctly reproduced  from the Fourier transform of the square of the CD kernel in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content='19).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content=' In particular, the  appearance of s(2j) can be naturally understood from the expansion of φ in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content='20).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content='  The second term of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content='1) arises from the cross-term of sin φ and cos φ in the CD kernel  squared  K  �  E + 1  2ℏω, E − 1  2ℏω  �2  ⊃ sin  �  ρ0(E)ω  �  2π2ℏω  ∞  �  g=1  ℏ2g−1ρg(E).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content='28)  Using the relation  � ∞  −∞  dωe−iωτ sin  �  ρ0(E)ω  �  2π2ω  = 1  2πθ  �  ρ0(E) − τ  �  ,  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content='29)  we find that the second term of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content='1) is reproduced from (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content='18)  SFFg ⊃  � ∞  E0  dE  2π e−2βEρg(E) −  � ∞  E0  dE  2π e−2βEρg(E)θ  �  ρ0(E) − τ  �  =  � ∞  E0  dE  2π e−2βEρg(E) −  � ∞  Eτ  dE  2π e−2βEρg(E)  =  � Eτ  E0  dE  2π e−2βEρg(E).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content='30)  – 9 –  To summarize, the structure of SFFg in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content='1) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content='8) can be naturally understood from  the Fourier transform of the CD kernel in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content='19).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content=' To our knowledge, corrections to the sine  kernel formula are not known in the literature before and our (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content='19) with (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content='22) is a new  result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content='5  4  Examples of Airy case and JT gravity  In this section, as concrete examples we compute SFF0 and SFF1 for the Airy case and JT  gravity using our general result in the previous section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content='1  Airy case  Let us first consider the Airy case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content=' For the Airy case, all couplings tk are set to zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content=' Then  one can show that E0 = 0 and Ik = 0 (k ≥ 1), and the genus-zero eigenvalue density (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content='14)  becomes  ρ0(E) = 2  √  E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content='  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content='1)  From the definition of Eτ in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content='18), we find  Eτ = τ 2  4 ,  zτ =  �  Eτ = τ  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content='  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content='2)  From the general formula in the previous sections, we find that the SFF0 and SFF1 for the  Airy case are given by  SFF0 =  1  2  �  π(2β)3 Erf  �  τ  �  β  2  �  ,  SFF1 =  β  8πτ e− 1  2 βτ 2 + β3/2  6  √  2πErf  �  τ  �  β  2  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content='  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content='3)  We can compare this with the exact result of two-boundary correlator for the Airy case  [35]  ⟨Z(β1)Z(β2)⟩c =  e  ℏ2  12 (β1+β2)3  2ℏ  �  π(β1 + β2)3 Erf  �ℏ  2  �  β1β2(β1 + β2)  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content='  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content='4)  One can check that the SFF0,1 in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content='3) are indeed reproduced from the τ-scaling limit of  the exact result (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content=' See also Appendix A for the corrections of the CD kernel in the  Airy case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content='2  JT gravity  Next consider the SFF of JT gravity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content=' For JT gravity, the couplings tk are given by [21–23]  t0 = t1 = 0,  tk = (−1)k  (k − 1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content=',  (k ≥ 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content='  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content='5)  5See also a comment in footnote 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content='  – 10 –  In this case, E0 = 0 and the genus-zero eigenvalue density (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content='14) becomes  ρ0(E) = sinh(2  √  E).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content='  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content='6)  From (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content='18), Eτ and zτ are given by  Eτ = 1  4arcsinh(τ)2,  z =  �  Eτ = 1  2arcsinh(τ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content='  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content='7)  From the general formula (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content='19), the leading term SFF0 is evaluated as [26]  SFF0 = 1  2⟨Z(2β)⟩g=0  �  Erf  �βarcsinh(τ) + 1  √2β  �  + Erf  �βarcsinh(τ) − 1  √2β  ��  ,  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content='8)  where the genus-zero one-point function is given by  ⟨Z(β)⟩g=0 =  e  1  β  2  �  πβ3 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content='  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content='9)  The next order correction SFF1 for JT gravity can be found from the general result in the  previous section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content=' After some algebra, we find  SFF1 =  1  24πe− 1  2 βarcsinh(τ)2  �  τ  1 + τ 2  �  β +  1  arcsinh(τ)2  �  +  4  arcsinh(τ)3  �  1  √  1 + τ 2 − 1  �  +  1  arcsinh(τ)  �  2 −  1  (1 + τ 2)3/2 + β  �  4 −  1  √  1 + τ 2  ���  + ⟨Z(2β)⟩g=1Erf  ��  β  2 arcsinh(τ)  �  ,  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content='10)  where the genus-one one-point function is given by  ⟨Z(β)⟩g=1 = (1 + β)√β  24√π  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content='  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content='11)  In Figure 1, we show the plot of SFF0 and SFF1 of JT gravity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content=' From Figure 1a we can  see that SFF0 exhibits the behavior of the ramp and plateau.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content=' From Figure 1b we see that  SFF1 approaches a constant at late times while it blows up as τ → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content=' As we argued in the  previous section, this diverging behavior of SFF1 at τ = 0 is an artifact of the τ-scaling  limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content=' In fact, the small τ expansion of SFF1 in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content='10) reads  SFF1 = 1  2π  �  β  4τ + 3 + 8β + 5β2  24  τ + O(τ 3)  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content='  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content='12)  One can see that the negative power term of τ agrees with the O(ℏ) term of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content='17), as  expected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content=' One can also check that this expansion (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content='12) is consistent with the general case  in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content='6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content='  – 11 –  1  1  2  3  4  5  log(τ)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content='04  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content='06  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content='08  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content='10  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content='12  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content='14  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content='16  SFF0  (a) SFF0  1  0  1  2  3  4  5  log(τ)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content='09  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content='10  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content='11  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content='12  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content='13  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content='14  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content='15  SFF1  (b) SFF1  Figure 1: Plot of (a) SFF0 and (b) SFF1 for JT gravity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content=' The horizontal axis is log τ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content=' We  set β = 1 in this figure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content=' SFFg approaches ⟨Z(2β)⟩g as τ → ∞ (shown by orange dashed  lines).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content='  5  Computation of SFF  In this section we formulate a systematic method of computing the small ℏ expansion of  the τ-scaling limit of the SFF in the general background {tk} of topological gravity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content=' In  section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content='1 we will first derive a key relation  ∂τ∂0SFF = i  ℏ  �  W1  �  β + iτ  ℏ  �  ∂0W1  �  β − iτ  ℏ  �  − ∂0W1  �  β + iτ  ℏ  �  W1  �  β − iτ  ℏ  ��  ,  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content='1)  which relates SFF with (the t0-derivative of) the one-point function  W1  �  β ± iτ  ℏ  �  = ℏ∂0  �  Z  �  β ± iτ  ℏ  ��  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content='  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content='2)  Here ∂k := ∂tk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content=' The relation (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content='1) enables us to compute the small ℏ expansion of the SFF  from that of the one-point function W1(β± iτ  ℏ ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content=' As we will describe in section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content='2, the small  ℏ expansion of W1(β ± iτ  ℏ ) can be obtained by the method developed in [36] with a slight  modification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content=' This is based on the KdV equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content=' We will then explain in section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content='3 how  to integrate ∂τ∂0SFF to obtain the small ℏ expansion of the SFF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content='1  Derivation of the key differential equation  In this subsection we will derive (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content=' As we have already seen in the previous sections,  the one- and two-point functions can be expressed in terms of the CD kernel as [29]  ⟨Z(β)⟩ =  �  dEe−βEK(E, E),  ⟨Z(β1)Z(β2)⟩c = ⟨Z(β1 + β2)⟩ −  �  dE1dE2e−β1E1+β2E2K(E1, E2)2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content='  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content='3)  The CD kernel is written as  K(E1, E2) = ℏ∂0ψ(E1)ψ(E2) − ℏ∂0ψ(E2)ψ(E1)  −E1 + E2  ,  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content='4)  – 12 –  where ψ(E) is the Baker-Akhiezer function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content=' It satisfies the Schr¨odinger equation  − (ℏ2∂2  0 + u)ψ(E) = Eψ(E).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content='  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content='5)  By using this equation it immediately follows from (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content='4) that  ℏ∂0K(E1, E2) = ψ(E1)ψ(E2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content='  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content='6)  Then we have  ℏ∂0⟨Z(β)⟩ = W1(β) =  �  dEe−βEψ(E)2,  ℏ∂0⟨Z(β1)Z(β2)⟩c = W1(β1 + β2) −  �  dE1dE2e−β1E1−β2E22ψ(E1)ψ(E2)K(E1, E2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content='  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content='7)  When β1,2 = β ± iτℏ−1, we find  ∂τ∂0⟨Z(β1)Z(β2)⟩c  = − i  ℏ2  �  dE1dE2e−β1E1−β2E22ψ(E1)ψ(E2)(−E1 + E2)K(E1, E2)  = − i  ℏ2  �  dE1dE2e−β1E1−β2E22ψ(E1)ψ(E2) [ℏ∂0ψ(E1)ψ(E2) − ℏ∂0ψ(E2)ψ(E1)]  = − i  ℏ  �  dE1dE2e−β1E1−β2E2 �  ∂0ψ(E1)2ψ(E2)2 − ψ(E1)2∂0ψ(E2)2�  = − i  ℏ [∂0W1(β1)W1(β2) − W1(β1)∂0W1(β2)] .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content='  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content='8)  Thus we have derived (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content='2  Small ℏ expansion of one-point function  Let us next consider the small ℏ expansion of W1  �  β ± iτ  ℏ  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content=' We can restrict ourselves to the  case of W1  �  β + iτ  ℏ  �  without loss of generality, as W1  �  β − iτ  ℏ  �  is immediately obtained by  complex conjugation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content=' The expansion can be done in two ways.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content=' One way, which is easier  to understand, is to use the results of the ’t Hooft expansion computed in [36]:  W1 (β) = exp  � ∞  �  n=0  ℏn−1 �Gn(λ)  �  ,  λ = ℏβ fixed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content='  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content='9)  Given this expansion, the small ℏ expansion of W1  �  β + iτ  ℏ  �  is obtained by merely substi-  tuting λ = iτ + βℏ into the expansion and then re-expanding it in ℏ as  W1  �  β + iτ  ℏ  �  = exp  �  ℏ−1 �G0(iτ) +  �  �G1(iτ) + �G′  0(iτ)β  �  + ℏ  �  �G2(iτ) + �G′  1(iτ)β + 1  2  �G′′  0(iτ)β2  �  + O(ℏ2)  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content='  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content='10)  – 13 –  More specifically, the first few �Gn are given by [36]6  �G0 = −E0λ +  �  ja≥0  �  a ja=k  �  a aja=n  (2n + k + 1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content='  (2n + 3)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content='  λ2n+3  2n+1s2n+k+2  ∞  �  a=1  Ija  a+1  ja!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content=' (2a + 1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content=' !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content='ja  = −E0λ +  1  12s2 λ3 +  I2  60s5 λ5 +  �  I2  2  144s8 +  I3  840s7  �  λ7 + O(λ9),  �G1 = 1  2 log  � ℏ  8π  ∂λE(λ)  E(λ) − E0  �  ,  �G2 =  iI2  12s2z(λ) −  5i  24sz(λ)3 − 3E(1)  8z(λ)4 +  E(2)  4z(λ)2E(1) −  E(3)  8  �  E(1)�2 +  �  E(2)�2  6  �  E(1)�3 ,  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content='11)  where  E(λ) = −∂λ �G0,  z(λ) =  �  E(λ) − E0,  E(j) = ∂j  λE(λ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content='  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content='12)  Ik and s = 1 − I1 were defined in section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content=' From this one obtains  W1  �  β + iτ  ℏ  �  = eG(β+ iτ  ℏ ) = exp  � ∞  �  n=0  ℏn−1Gn(β, τ, {Ik})  �  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content='13)  with  G0 = �G0(iτ),  G1 = 1  2 log  � ℏ  8πi  ∂τEτ  Eτ − E0  �  − βEτ,  G2 = i  �  I2  12s2zτ  −  5  24sz3τ  + 3e(1)  8z4τ  −  e(2)  4z2τe(1) + e(3)  8e(1)2 − e(2)2  6e(1)3  +  �e(1)  2z2τ  − e(2)  2e(1)  �  β + e(1)  2 β2  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content='  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content='14)  Here Eτ = E(iτ), zτ = √Eτ − E0 = z(iτ) and e(j) = ∂j  τEτ = ijE(j)(iτ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content='  The other way to compute the small ℏ expansion of W1  �  β + iτ  ℏ  �  , which is technically  more efficient, is to solve the KdV equation directly with the expansion (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content='13).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content=' This can be  done by the method of [36] with only a slight modification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content=' In what follows let us describe  the method in our present notation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content=' First, recall that W1 = W1  �  β + iτ  ℏ  �  satisfies the KdV  equation [23]  ∂1W1 = u∂0W1 + ℏ2  6 ∂3  0W1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content='  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content='15)  6The elements here and there are identified as �Gn = 2(n−1)/2Gn, E0 = −u0, shere = tthere = 1 − I1,  E(λ) = −ξ∗, E(j) = −2−n/2ξ(n)  ∗  , z(λ) = −2−1/2iz∗ and λ = 21/2sthere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content=' The normalization of W1 here  (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content=' the constant part of �G1) is also adjusted accordingly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content='  – 14 –  In terms of G = log W1, this is written as  ∂1G = u∂0G + ℏ2  6  �  ∂3  0G + 3∂0G∂2  0G + (∂0G)3�  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content='  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content='16)  Here, u is the specific heat of the general topological gravity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content=' It obeys the KdV equation  and its genus expansion  u =  ∞  �  g=0  2gugℏ2g  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content='17)  was computed by means of a recurrence relation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content='7 Note, in particular, that the leading  order part is equal to the threshold energy  u0 = −E0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content='  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content='18)  By plugging the expansions (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content='17) and  G =  ∞  �  n=0  ℏn−1Gn  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content='19)  into (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content='16), one obtains the recurrence relation  −∂sG0 = 1  6(∂0G0)3,  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content='20)  DG1 = 1  2∂0G0∂2  0G0  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content='21)  for n = 0, 1 and  DGn = 1  6  �  0≤i,j,k<n  i+j+k=n  ∂0Gi∂0Gj∂0Gk + 1  2  n−1  �  k=0  ∂0Gn−k−1∂2  0Gk + 1  6∂3  0Gn−2 +  ⌊ n  2 ⌋  �  k=1  2kuk∂0Gn−2k  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content='22)  for n ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content='8 Here we have introduced the differential operator  D := −∂s − 1  2(∂0G0)2∂0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content='  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content='23)  Note also that instead of t0, t1 we regard s and E0 as independent variables and treat tk≥2  as parameters [23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content=' From this viewpoint ∂0,1 are understood as  ∂0 = −1  s (∂E0 + I2∂s) ,  ∂1 = −E0∂0 − ∂s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content='  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content='24)  7See, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content=' [23] with the identification tthere = shere, Bn = (−1)n+1In+1 (n ≥ 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content='  8Gn here is related with Gn in [36] as Ghere  n  (β, τ = 0, {Ik}) = 2(n−1)/2Gthere  n  (s = 2−1/2ℏβ, {Ik}), up to  the constant part of G1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content='  – 15 –  To solve the recurrence relation (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content='20)–(5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content='22), it is helpful to recall some relevant  formulas (see [36]):  ∂0s = −I2  s ,  ∂0In = In+1  s  (n ≥ 1),  ∂0G0 = 2izτ,  ∂0zτ =  1  2szτ  − ∂τzτ  zτ  ,  ∂0e(j) = −2∂j+1  τ  zτ,  ∂szτ = −e(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content='  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content='25)  The differential operator (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content='23) is then written as  D = −∂s + 2z2  τ∂0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content='  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content='26)  It has the properties  Dzτ = zτ  s ,  DEτ = 0,  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content='27)  De(n) = 8  n−1  �  k=0  k  �  ℓ=0  ℓ  �  m=0  k!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content='  (k − ℓ)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content=' (ℓ − m)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content='m!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content='∂(n−ℓ)  τ  zτ ∂(ℓ−m)  τ  zτ ∂(m+1)  τ  zτ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content='  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content='28)  As explained in [37], ∂n  τ zτ can be expressed in terms of e(j) and zτ:  ∂τzτ = e(1)  2zτ  ,  ∂2  τzτ = e(2)  2zτ  − e(1)2  4z3τ  ,  ∂3  τzτ = e(3)  2zτ  − 3e(1)e(2)  4z3τ  + 3e(1)3  8z5τ  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content='  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content='29)  We are now in a position to solve the recurrence relation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content=' In [36] the recurrence relation  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content='20)–(5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content='22) was solved with the initial data  Gbefore  0  = �G0(iτ),  Gbefore  1  = 1  2 log  � ℏ  8πi  ∂τEτ  Eτ − E0  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content='  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content='30)  Here, we would like to solve the same recurrence relation with the initial data given in  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content='14).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content=' The only difference from [36] is that here G1 has an extra β-dependent term −βEτ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content='  It is clear from (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content='27) that (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content='21) is satisfied by G1 with this term, given that (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content='21) is  satisfied by Gbefore  1  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content='  To solve the recurrence relation, we can use almost the same algorithm we developed  in [36].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content=' The only difference is the step (v), which is modified accordingly so that it works  for the β-dependent parts as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content=' The modified algorithm is as follows:  (i) Compute the r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content='h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content=' of (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content='22) and express it as a polynomial in the variables s−1, Ik≥2,  z−1  τ , e(1)−1 and e(n), n ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content='  (ii) Let s−mf  �  Ik, zτ, e(n)  �  denote the highest-order part in s−1 of the obtained expression.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content='  This part can arise only from  D  �  f  �  Ik, zτ, e(n)  �  2(m − 2)sm−2z2τI2  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content='  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content='31)  Therefore subtract this from the obtained expression.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content='  – 16 –  (iii) Repeat procedure (ii) down to m = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content=' Then all the terms of order s−2 automati-  cally disappear and the remaining terms are of order s−1 or s0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content=' Note also that the  expression does not contain any Ik.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content='  (iv) In the result of (iii), collect all the terms of order s−1 and let s−1zτ∂zτ g  �  zτ, e(n)  �  denote the sum of them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content=' This part arises from  Dg  �  zτ, e(n)  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content='  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content='32)  Therefore subtract this from the result of (iii).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content=' The remainder turns out to be inde-  pendent of s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content='  (v) In the obtained expression, let  e(1)mh  �  e(n ≥ 2)  �  zτ  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content='33)  denote the part which is of the order z−1  τ  as well as of the highest order in e(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content=' This  part arises from  D  �  −  1  2e(1)  � ˜a  da  �  a−1e(1)  �m h  �  e(n) → a−n−1e(n)  ��  ˜a=1  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content='  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content='34)  (Here, a and ˜a are intermediate formal variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content=') Therefore subtract this from the  obtained expression.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content='  (vi) Repeat procedure (v) until the resulting expression vanishes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content='  (vii) By summing up all the above-obtained primitive functions, we obtain Gn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content='  Using the above algorithm we have computed Gn for n ≤ 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content=' For instance,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content=' G3 is computed  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content='as  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content='G3 = −  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content='5  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content='32s2z6τ  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content='+  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content='I2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content='8s3z4τ  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content='−  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content='I3  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content='24s3z2τ  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content='−  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content='I2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content='2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content='12s4z2τ  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content='− 5I2e(1)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content='96s2z5τ  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content='− 3e(1)2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content='4z8τ  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content='+ 35e(1)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content='64sz7τ  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content='+ 11e(2)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content='16z6τ  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content='−  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content='3e(3)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content='16z4τe(1) −  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content='5e(2)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content='32sz5τe(1) +  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content='I2e(2)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content='48s2z3τe(1) +  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content='e(4)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content='16z2τe(1)2 +  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content='e(2)2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content='8z4τe(1)2 −  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content='e(5)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content='48e(1)3  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content='− 7e(2)e(3)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content='24z2τe(1)3 + e(3)2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content='8e(1)4 +  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content='e(2)3  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content='4z2τe(1)4 + e(2)e(4)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content='6e(1)4 − 3e(2)2e(3)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content='4e(1)5  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content='+ e(2)4  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content='2e(1)6  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content='+  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content='� 5e(1)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content='16sz5τ  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content='− 3e(1)2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content='4z6τ  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content='+ 5e(2)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content='8z4τ  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content='− I2e(1)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content='24s2z3τ  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content='−  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content='e(3)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content='4z2τe(1) + e(4)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content='8e(1)2 +  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content='e(2)2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content='4z2τe(1)2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content='−7e(2)e(3)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content='12e(1)3 + e(2)3  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content='2e(1)4  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content='β +  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content='−e(1)2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content='4z4τ  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content='+ e(2)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content='4z2τ  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content='+ e(2)2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content='4e(1)2 − e(3)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content='4e(1)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content='β2 + e(2)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content='6 β3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content='  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content='35)  Let us finally comment on the small ℏ expansion of W1  �  β − iτ  ℏ  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content=' As we mentioned  already, it is obtained from (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content='13) by complex conjugation:  W1  �  β − iτ  ℏ  �  = exp  � ∞  �  n=0  ℏn−1Gn  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content='  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content='36)  – 17 –  Furthermore, Gn is related to Gn by  Gn = (−1)n+1Gn + πi  2 δn,1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content='  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content='37)  This can be shown directly for n = 0, 1 and inductively for n ≥ 2 by using the recurrence  relation (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content='22).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content='3  Small ℏ expansion of SFF  Using the results obtained above, one can compute the small ℏ expansion of the SFF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content='  Substituting (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content='13), (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content='36) and the explicit form of G1 in (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content='14) into (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content='1), one obtains  ∂τ∂0SFF = i  ℏ(∂0G − ∂0G)eG+G  = −2i  ℏ  ∞  �  k=0  ℏ2k−1∂0G2k exp  �  G1 + G1 + 2  ∞  �  k=1  ℏ2kG2k+1  �  = e−2βEτ  4πi  ∂τEτ  Eτ − E0  ∞  �  k=0  ℏ2k−1∂0G2k exp  � ∞  �  k=1  2G2k+1ℏ2k  �  =: e−2βEτ  2πℏ  ∞  �  n=0  cnℏ2n,  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content='38)  where the first two of cn are computed as  c0 = e(1)  zτ  ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content='  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content='c1 = e(1)e(2)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content='3zτ  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content='β3 +  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content='−3e(1)3  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content='8z5τ  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content='+ e(1)e(2)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
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+page_content='− e(3)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
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+page_content='+  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
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+page_content='2zτe(1)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content='β2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content='+  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content='−15e(1)3  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content='16z7τ  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content='+ 3e(1)e(2)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content='4z5τ  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content='− e(3)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content='4z3τ  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content='+ 3e(1)2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content='8sz6τ  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content='− I2e(1)2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content='12s2z4τ  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content='+  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content='e(4)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content='4zτe(1) +  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content='e(2)2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content='4z3τe(1) − 7e(2)e(3)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content='6zτe(1)2 +  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content='e(2)3  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content='zτe(1)3  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content='β  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content='− 5e(1)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content='32s2z7τ  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content='+ 25e(1)e(2)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content='32z7τ  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content='− 3e(3)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content='16z5τ  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content='+ 9e(1)2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content='16sz8τ  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content='+ I2e(1)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content='8s3z5τ  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content='− I2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content='2e(1)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content='12s4z3τ  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content='− I2e(1)2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content='12s2z6τ  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content='− 3e(2)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content='16sz6τ  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content='− I3e(1)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content='24s3z3τ  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content='+ I2e(2)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content='24s2z4τ  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content='− 105e(1)3  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content='128z9τ  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content='+  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content='3e(2)2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content='32z5τe(1) +  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content='e(4)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content='16z3τe(1) −  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content='e(5)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content='24zτe(1)2 − 7e(2)e(3)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content='24z3τe(1)2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content='+  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content='e(2)3  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content='4z3τe(1)3 +  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content='e(3)2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content='4zτe(1)3 + e(2)e(4)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content='3zτe(1)3 − 3e(2)2e(3)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content='2zτe(1)4 +  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content='e(2)4  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content='zτe(1)5 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content='  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content='39)  Let us next consider the expansion  ∂τSFF = e−2βEτ  2πℏ  ∞  �  n=0  bnℏ2n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content='  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content='40)  – 18 –  Comparing (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content='40) with (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content='38), we see that bn and cn are related as  cn = e2βEτ ∂0e−2βEτ bn  = (∂0 − 2β∂0Eτ) bn  =  �  ∂0 + 2β e(1)  zτ  �  bn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content='  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content='41)  For n = 0, we immediately see that b0 = 1/(2β) gives c0 in (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content='39).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content=' For n ≥ 1, let us assume  that bn is a polynomial in β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content=' Then, the highest-order term of this polynomial is determined  as that of cn divided by 2βe(1)/zτ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content=' Subtracting this contribution from both sides of (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content='41),  one can determine the next to the highest-order term in the same manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content=' In this way,  one can determine bn as a polynomial of β without integrating in t0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content=' For instance, the first  two of them are  b0 = 1  2β ,  b1 = e(2)  6 β2 +  �  −e(1)2  8z4τ  − e(3)  6e(1) + e(2)2  4e(1)2  �  β  − I2e(1)  24s2z3τ  + e(2)  16z4τ  + e(1)  16sz5τ  − e(1)2  8z6τ  +  e(4)  24e(1)2 − e(2)e(3)  4e(1)3 + e(2)3  4e(1)4 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content='  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content='42)  Let us finally consider the small ℏ expansion (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content='3) of the SFF:  SFF = SFFg=0 + SFFg≥1  = ℏ−1SFF0 +  ∞  �  g=1  ℏ2g−1SFFg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content='  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content='43)  The leading term SFF0 is given in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content='19).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content=' It is easy to verify that the τ-derivative of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content='19)  reproduces the leading order part of (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content='40) with b0 = 1/(2β).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content=' Here, we are interested in  the higher order part SFFg≥1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content=' As explained in section 3 (see, in particular (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content='5)), a key  feature of SFFg≥1 is that it takes the form  SFFg≥1 = Erf  ��  2βzτ  �  ⟨Z(2β)⟩g≥1 + e−2βEτ  2πℏ  ∞  �  g=1  agℏ2g  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content='44)  with ag being polynomial in β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content=' The small ℏ expansion of the one-point function ⟨Z(2β)⟩  was computed in [23]:  ⟨Z(2β)⟩g≥1 =  e−2βE0  �  2π(2β)3√  2ℏ  ∞  �  g=1  ℏ2g2gZg(2β),  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content='45)  where  Z1(β) =  1  24sβ3 +  I2  24s2 β2,  Z2(β) =  1  1152s3 β6 +  29I2  5760s4 β5 +  � 7I2  2  480s5 +  29I3  5760s4  �  β4 +  � 7I3  2  288s6 + 29I2I3  1440s5 +  I4  384s4  �  β3  +  � 7I4  2  288s7 + 5I2  2I3  144s6 +  29I2  3  5760s5 + 11I2I4  1440s5 +  I5  1152s4  �  β2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content='  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content='46)  – 19 –  Comparing (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content='44) with (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content='40), we see that ag and bg are related as  bg = e(1)  2βzτ  2gZg(2β) +  �  ∂τ − 2βe(1)  �  ag.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content='  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content='47)  Given that ag are polynomials in β, we can calculate them from bg without integrating in  τ, in essentially the same way as we obtained bg from cg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content=' For instance, a1 is obtained as  a1 =  �  1  6szτ  −  e(2)  12e(1)  �  β + e(1)  16z4τ  +  I2  12s2zτ  −  1  24sz3τ  +  e(3)  24e(1)2 −  e(2)2  12e(1)3 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content='  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content='48)  One sees that the terms which do not contain e(j) precisely give d1 in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content='7);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content=' the other  terms comprise f1 given by (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content='8) with (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content='10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content=' The expression of a2 is already rather long  and we relegate it to Appendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content=' We have computed ag for g ≤ 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content='  6  Higher order corrections to CD kernel  In this section we consider the higher order correction of the CD kernel beyond the sine  kernel approximation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content='  That is, we consider a small ℏ expansion of the CD kernel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content='  In  principle, this is not a difficult task since the CD kernel is expressed as a bilinear form (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content='4)  in terms of the BA function ψ(E) whose ℏ-expansion was calculated [29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content=' The purpose of  this section is to present a concrete, technically efficient method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content='  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content='1  Small ℏ expansion of BA function  To compute the expansion of the CD kernel, one first needs to compute the small ℏ ex-  pansion of ψ(E + 1  2ℏω) rather than ψ(E).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content=' Of course, this is obtained by re-expanding the  results of [29] with the replacement E → E + 1  2ℏω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content=' For the sake of technical efficiency,  however, here we compute the expansion  log ψ(E + 1  2ℏω) = A(E + 1  2ℏω) =  ∞  �  n=0  ℏn−1An(E, ω)  (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content='1)  directly by means of a recurrence relation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content='  It was derived in [29] that  v(E) := ℏ∂0A(E)  (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content='2)  satisfies the equation  v2 + ℏ∂0v = −E − u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content='  (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content='3)  Here, u is the specific heat of the general topological gravity and we again assume that its  genus expansion (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content='17) is given.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content=' Since we want to compute the ℏ expansion of ψ with a  shifted argument, we replace E in (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content='3) by E + 1  2ℏω and plug the expansion of the form  v = v(E + 1  2ℏω) =  ∞  �  n=0  ℏnvn(E, ω)  (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content='4)  – 20 –  into (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content=' Expanding both sides of the equation in ℏ, we obtain the recurrence relation  v2  0 = −E + E0,  2v0v1 + ∂0v0 = −ω  2 ,  vn = − 1  2v0  �  ∂0vn−1 +  n−1  �  k=1  vkvn−k +  �  2  n  2 u n  2 (n even)  0  (n odd)  �  ,  n ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content='  (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content='5)  The first two equations are solved as  v0 = iz = i  �  E − E0,  v1 = −  1  4sz2 + iω  4z .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content='  (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content='6)  vn (n ≥ 2) are also immediately obtained by the recurrence relation given the form of ug.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content='  For instance, using  u1 =  I2  2  12s4 +  I3  24s3  (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content='7)  one obtains  v2 = i  �  I2  2  12zs4 +  I3  24zs3 −  I2  8z3s3 +  5  32z5s2  �  +  1  8z4sω − i  1  32z3 ω2,  v3 = −  I3  2  6s6z2 − 7I2I3  48s5z2 + 11I2  2  48s5z4 −  I4  48s4z2 +  I3  12s4z4 −  9I2  32s4z6 +  15  64s3z8  + i  �  −  I2  2  48s4z3 −  I3  96s3z3 +  3I2  32s3z5 −  25  128s2z7  �  ω −  1  16sz6 ω2 + i  1  128z5 ω3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content='  (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content='8)  Note that the form of the recurrence relation (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content='5) for vn≥2 is not affected by the shift  E → E + 1  2ℏω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content=' The ω-dependence enters entirely through the initial data, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content=' the form of  v1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content='  One can easily obtain An from vn through the relation ∂0An = vn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content=' Due to the structure  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content='24) of ∂0, the form of the s-dependent part of An can be determined without integrating  in t0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content='  (We use a similar logic as we obtained bn from cn in the last section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content=')  The s-  independent part can also be determined with the help of the formulas (the proofs of  which are straightforward)  � t0  dt0  1  sz2 = 2 log z,  � t0  dt0  1  sz2k = −  1  (k − 1)z2(k−1)  (k ≥ 2),  � t0  dt0  1  z2k−1 = Γ( 3  2 − k)  Γ( 1  2)  ρ(k−1)  0  = (−2)k−1  (2k − 3)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content=' !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content='ρ(k−1)  0  (k ≥ 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content='  (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content='9)  Here  ρ(n)  0  = ∂n  Eρ0(E)  (n ≥ 0),  ρ(−1)  0  =  � E  E0  dEρ0(E).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content='  (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content='10)  – 21 –  We thus obtain  A0 = i  2ρ(−1)  0  = i  2  � E  E0  dEρ0(E),  A1 = −1  2 log z + iρ0  4 ω − 1  2 log(4π),  A2 = i  �  I2  24s2z −  5  48sz3  �  −  1  8z2 ω + iρ(1)  0  16 ω2,  A3 = −  I2  2  24s4z2 −  I3  48s3z2 +  I2  16s3z4 −  5  64s2z6 + i  �  −  I2  96s2z3 +  5  64sz5  �  ω  +  1  32z4 ω2 + iρ(2)  0  96 ω3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content='  (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content='11)  The constant part of A1 is determined in accordance with the ω = 0 case [29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content='  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content='2  Small ℏ expansion of CD kernel  Let us now compute the small ℏ expansion of the CD kernel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content=' We should first note that  this is a WKB-type asymptotic expansion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content=' As is known [9], there appear some qualitative  differences between the cases of E > E0 and E < E0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content=' In this paper we consider the case  of E > E0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content=' In this case the two independent BA functions are given by  ψ(E)  and  ψ(E) = ψ(E)  ���  z→−z,  (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content='12)  i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content=' they are complex conjugate to each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content=' Due to this complex structure, for E > E0  there are two saddles which equally contribute to the CD kernel [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content=' Therefore, as long as  the perturbative expansion in ℏ concerns, the CD kernel is approximated by  K(E1, E2) = �K(E1, E2) + �K(E2, E1)  (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content='13)  with  �K(E1, E2) = ℏ∂0ψ(E1)ψ(E2) − ψ(E1)ℏ∂0ψ(E2)  −E1 + E2  = −v(E1) − v(E2)  ℏω  eA(E1)+A(E2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content='  (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content='14)  By using the recurrence relation (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content='5) it is easy to prove that  vn(E, −ω) = (−1)n+1vn(E, ω),  An(E, −ω) = (−1)n+1An(E, ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content='  (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content='15)  Therefore (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content='14) is written as  �K(E1, E2) = − 1  ℏω  � ∞  �  k=0  2v2k(E, ω)ℏ2k  �  exp  � ∞  �  k=0  2A2k+1(E, ω)ℏ2k  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content='  (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content='16)  Similarly, we have  �K(E2, E1) = − 1  ℏω  ∞  �  k=0  2v2k(E, ω)ℏ2k exp  � ∞  �  k=0  2A2k+1(E, ω)ℏ2k  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content='  (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content='17)  – 22 –  Thus, the CD kernel is given by  K(E1, E2) = Re  �  2 �K(E1, E2)  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content='  (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content='18)  Substituting the explicit form of A1(E, ω) in (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content='11) we obtain  2πK(E1, E2) = Re  �  − 2  ℏωz  � ∞  �  k=0  v2k(E, ω)ℏ2k  �  exp  �  2  ∞  �  k=1  A2k+1(E, ω)ℏ2k  �  eiωρ0/2  �  = Re  �  Xeiωρ0/2�  = Re{X} cos ωρ0  2  − Im{X} sin ωρ0  2 ,  (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content='19)  where  X := − 2  ℏωz  � ∞  �  k=0  v2k(E, ω)ℏ2k  �  exp  �  2  ∞  �  k=1  A2k+1(E, ω)ℏ2k  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content='  (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content='20)  With the forms of vn and An obtained in (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content='6), (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content='8) and (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content='11), this gives the small ℏ  expansion of the CD kernel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content='  A few comments are in order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content=' First, substituting v0 = iz one sees that the expression  (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content='19) correctly reproduces the sine kernel (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content='10) at the order of ℏ−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content=' Second, recall that the  diagonal part of the CD kernel, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content='19) in the limit of ω → 0, is equal to the eigenvalue  density (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content='21).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content=' This is evident for the genus zero part at the order of ℏ−1, but also implies  that  ∞  �  g=1  ℏ2g−1ρg(E) = Re{X}  ���  ω=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content='  (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content='21)  We checked that this indeed reproduces ρg in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content='2) which were obtained from the result of  ⟨Z(β)⟩g in [23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content='  By taking (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content='21) into account, the expansion (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content='19) is written explicitly as  2πK(E1, E2)  = 2  ℏω  �  1 + ω2  32z4 ℏ2 +  �  +  �  −  49I2  2  4608s4z6 −  I3  192s3z6 +  49I2  1536s3z8 −  105  2048s2z10  �  ω2  +  �  I2ρ(2)  0  2304s2z3 −  ρ(2)  0  1536sz5 +  11  2048z8  �  ω4 − (ρ(2)  0 )2  4608 ω6  �  ℏ4 + O(ℏ6)  �  sin ωρ0  2  +  ��  ρ1 + ρ(2)  0  24 ω2  �  ℏ  +  �  ρ2 +  �  −  I2  128s2z7 +  35  768sz9  �  ω2 +  �  ρ(4)  0  1920 + ρ(2)  0  768z4  �  ω4  �  ℏ3 + O(ℏ5)  �  cos ωρ0  2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content='  (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content='22)  Observe that there appear several derivatives of ρ0 at higher orders.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content=' These derivatives in  fact have their origin in the integral (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content='20) and are neatly absorbed if one recasts the above  result into the form (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content='19).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content='  – 23 –  7  Conclusions and outlook  In this paper we have computed the higher order corrections of the SFF in the τ-scaling  limit for an arbitrary background {tk} of topological gravity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content=' We have also shown that  these corrections of the SFF can be obtained by the Fourier transform of the CD kernel by  including the corrections to the sine kernel formula.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content=' As we can see from Figure 1, SFFg≥1  approaches a constant at late times which gives a correction to the value of the plateau.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content='  On the other hand, SFFg≥1 apparently diverges at τ = 0, but this is just an artifact of the  τ-scaling limit, as we argued at the end of subsection 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content='  There are many interesting open questions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content='  As far as we know, the corrections of  the CD kernel to the sine kernel formula have not been fully explored in the literature  before.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content=' In this paper we have computed these corrections of the CD kernel in the double-  scaled matrix model of general topological gravity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content=' It would be possible to compute the  corrections of the CD kernel in a random matrix model before taking the double scaling  limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content=' It would be interesting to carry out this computation along the lines of [32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content=' As  emphasized in [26, 27], the τ-scaling limit enables us to reproduce the plateau of the SFF  by just summing over the perturbative genus expansion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content=' In general, the perturbative genus  expansion is an asymptotic series, but it becomes a convergent series with a finite radius of  convergence after we take the τ-scaling limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content=' What is happening is that, in the τ-scaling  limit, the (2g)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content=' growing part of the Weil-Petersson volume is suppressed and only a non-  growing part of the Weil-Petersson volume survives, and hence we get a convergent series  for the SFF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content=' In this sense, we throw away most of the contributions of the moduli space  integral by taking the τ-scaling limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content=' It is fair to say that we still do not understand the  non-perturbative effects which might contribute to the appearance of the plateau.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content=' In fact,  on general grounds we expect that the sum over SFFg in (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content='3) is an asymptotic series.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content=' It  would be interesting to study its resurgence structure and understand the non-perturbative  effects associated with this asymptotic series (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content='  Acknowledgments  We are grateful to Douglas Stanford for informing us of the result of [26] prior to its submis-  sion to arXiv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content=' This work was supported in part by JSPS Grant-in-Aid for Transformative  Research Areas (A) “Extreme Universe” 21H05187 and JSPS KAKENHI Grant 19K03856  and 22K03594.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content='  A  Airy kernel  In this appendix, we consider the CD kernel for the Airy case, known as the Airy kernel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content='  As the name suggests, the BA function for the Airy case is given by the Airy function  ψ(E) = ℏ− 1  6 Ai  �  −ℏ− 2  3 (E + t0)  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content='  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content='1)  Plugging this into (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content='4), the Airy kernel is given by  K(E1, E2) = Ai′(−ℏ− 2  3 E1)Ai(−ℏ− 2  3 E2) − Ai′(−ℏ− 2  3 E2)Ai(−ℏ− 2  3 E1)  E1 − E2  ,  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content='2)  – 24 –  where we have set t0 = 0 after taking the t0-derivative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content='  To see the corrections to the sine kernel formula, we first recall that Ai(−x) and Ai′(−x)  have the following asymptotic expansion at large x > 09  Ai(−x) =  1  √πx  1  4  �  cos  �  A − π  4  � ∞  �  k=0  (−1)k u2k  A2k + sin  �  A − π  4  � ∞  �  k=0  (−1)k u2k+1  A2k+1  �  ,  Ai′(−x) = x  1  4  √π  �  sin  �  A − π  4  � ∞  �  k=0  (−1)k v2k  A2k − cos  �  A − π  4  � ∞  �  k=0  (−1)k v2k+1  A2k+1  �  ,  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content='3)  where A = 2  3x  3  2 and  uk = Γ(k + 5/6)Γ(k + 1/6)  2k+1k!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content='π  ,  vk = 1 + 6k  1 − 6kuk = −Γ(k + 7/6)Γ(k − 1/6)  2k+1k!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content='π  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content='  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content='4)  The first few terms of the expansion (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content='3) read  Ai(−x) =  1  √πx  1  4  cos  �  A − π  4  � �  1 −  385  10368A2 + · · ·  �  +  1  √πx  1  4  sin  �  A − π  4  � � 5  72A −  85085  2239488A3 + · · ·  �  ,  Ai′(−x) = x  1  4  √π sin  �  A − π  4  � �  1 +  455  10368A2 + · · ·  �  − x  1  4  √π cos  �  A − π  4  � �  − 7  72A +  95095  2239488A3 + · · ·  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content='  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content='5)  The corrections to the sine kernel formula can be obtained by plugging the above expansion  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content='5) into (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content='2) with E1,2 = E ± 1  2ℏω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content=' By keeping the terms proportional to sin(A1 − A2)  and cos(A1 − A2), we find  K  �  E + ℏω  2 , E − ℏω  2  �  =  � 2  ℏω +  ω  16z4 ℏ +  � 11ω3  1024z8 −  105ω  1024z10  �  ℏ3 + O(ℏ5)  � sin(A1 − A2)  2π  +  �  ℏ  16z5 +  � 35ω2  768z9 −  105  1024z11  �  ℏ3 + O(ℏ5)  � cos(A1 − A2)  2π  ,  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content='6)  where z =  √  E and  A1,2 = 2  3ℏE  3  2  1,2 = 2  3ℏ  �  E ± 1  2ℏω  � 3  2  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content='  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content='7)  One can check that (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content='6) is consistent with our general formula of the CD kernel in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content='19).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content='  Note that if we plug (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content='5) into (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content='2), there appear terms proportional to sin(A1 +A2) and  cos(A1 + A2) as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content=' However, they are highly oscillating in the limit ℏ → 0 with fixed  E, ω, and hence these terms can be ignored in the computation of the Fourier transform  of the CD kernel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content='  9See e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content=' https://dlmf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content='nist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content='gov/9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content='  – 25 –  B  Coefficient a2 of SFF2 in (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content='44)  The coefficient a2 of SFF2 in (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content='44) is given by (for brevity we abbreviate zτ to z)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content='a2 =  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content='− e(2)2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content='72e(1) +  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content='1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content='18s3z  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content='β4  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content='+  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content='�e(1)e(2)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content='48z4  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content='+  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content='e(4)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content='240e(1) −  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content='5e(2)3  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content='144e(1)3 +  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
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+page_content=' Leviandier, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content=' Lombardi, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content=' Jost, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content=' Pique, “Fourier Transform: A Tool to  Measure Statistical Level Properties in Very Complex Spectra,” Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content=' 56 (1986)  2449.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content='  [2] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content=' Sachdev and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content=' Ye, “Gapless spin-fluid ground state in a random quantum heisenberg  magnet,” Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content=' 70 no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content=' 21, (1993) 3339–3342, arXiv:cond-mat/9212030.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content='  [3] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content=' Kitaev, “A simple model of quantum holography (part 1),”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content='  https://online.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content='kitp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content='ucsb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content='edu/online/entangled15/kitaev/.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
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+page_content=' Kitaev, “A simple model of quantum holography (part 2),”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
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+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content=' Garc´ıa-Garc´ıa and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQf4guX/content/2301.04773v1.pdf'}
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+arXiv:2301.08557v1  [gr-qc]  18 Jan 2023
+THE LINEAR STABILITY OF WEAKLY CHARGED AND SLOWLY ROTATING
+KERR-NEWMAN FAMILY OF CHARGED BLACK HOLES
+LILI HE
+Abstract. In this paper, we prove the linear stability of weakly charged and slowly rotating Kerr-Newman
+black holes under coupled gravitational and electromagnetic perturbations. We show that the solutions to
+the linearized Einstein-Maxwell equations decay at an inverse polynomial rate to a linearized Kerr-Newman
+solution plus a pure gauge term. This work builds on the framework developed in [36] for the study of the
+Einstein vacuum equations. We work in the generalized wave map and Lorenz gauge. The proof involves the
+analysis of the resolvent of the Fourier transformed linearized Einstein-Maxwell operator on asymptotically
+flat spaces, which relies on recent advances in microlocal analysis and non-elliptic Fredholm theory developed
+in [77]. The most delicate part of the proof is the description of the resolvent at low frequencies.
+Contents
+1. Introduction
+. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
+2
+1.1. Statement of the main result
+. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
+3
+1.2. Related works . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
+4
+1.3. Main ideas of the proof . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
+4
+1.4. Outline
+. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
+8
+1.5. List of notations
+. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
+9
+Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
+10
+2. b- and scattering geometry
+. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
+10
+2.1. b- and scattering vector bundles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
+2.2. Radial compactification
+. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
+11
+2.3. Function spaces
+. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
+12
+3. Initial value problems formalism of Einstein-Maxwell equations
+. . . . . . . . . . . . . . . . . . . . . . 14
+3.1. Einstein-Maxwell equations
+. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
+14
+3.2. Initial value problems for the nonlinear system
+. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
+16
+3.3. Initial value problems for the linearized system
+. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
+4. Kerr-Newman black holes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
+4.1. The Reissner-Nordstr¨om family
+. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
+21
+4.2. The slowly rotating Kerr-Newman family . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
+4.3. Stationarity, vector bundles, and geometric operators
+. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
+4.4. Constructing gauged initial data
+. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
+32
+5. Analysis of the linearized gauge-fixed Einstein-Maxwell operator
+. . . . . . . . . . . . . . . . . . . .
+35
+5.1. Microlocal geometry and dynamics of Kerr-Newman spacetimes
+. . . . . . . . . . . . . . . . . . . . . . . . . . .
+36
+5.2. Uniform Fredholm estimates
+. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
+45
+5.3. Description of the kernel of the wave type operators
+. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
+5.4. Semiclassical behavior of Kerr-Newman spacetimes
+. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
+5.5. High energy estimates
+. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
+5.6. Energy estimates
+. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
+6. Spherical harmonic decompositions
+. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
+77
+7. Mode stability of the Reissner-Nordstr¨om spacetime
+. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
+79
+7.1. Scalar type perturbations
+. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
+80
+1
+
+2
+LILI HE
+7.2. Vector type perturbations
+. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92
+8. Mode analysis of the scalar wave operator
+. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
+95
+8.1. Growing zero modes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98
+9. Mode analysis of the 1-form wave-type operator
+. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100
+9.1. Mode stability of the gauge propagation operator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102
+9.2. Mode stability of the gauge potential wave operator
+. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103
+9.3. Growing zero modes of the gauge potential wave operator
+. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110
+9.4. Generalized zero modes of the gauge potential wave operator
+. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112
+10. Modes of the linearized gauge-fixed Einstein-Maxwell system
+. . . . . . . . . . . . . . . . . . . . . . . 115
+10.1. Modes in the closed upper half plane
+. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116
+10.2. Generalized zero modes: linear growth
+. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121
+10.3. Generalized zero modes: quadratic growth . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125
+11. Structure of the resolvent of the linearized gauge-fixed Einstein-Maxwell operator 127
+11.1. Construction of the reference operator ˇLb(σ).
+. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127
+11.2. Structure of the resolvent near zero energy
+. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129
+12. Regularity of the resolvent of the linearized gauge-fixed Einstein-Maxwell operator
+. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138
+12.1. Regularity at low frequencies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138
+12.2. Regularity for frequencies away from 0.
+. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145
+12.3. Conormal regularity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146
+13. Decay estimates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150
+14. Proof of the main theorem
+. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156
+A. The linearized Einstein-Maxwell operator around Reissner-Nordstr¨om spacetime . . 159
+A.1. Detailed calculation of the linearized Einstein-Maxwell system
+. . . . . . . . . . . . . . . . . . . . . . . . . . . . 159
+A.2. Calculation of Rg ˙g
+. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159
+A.3. Calculation of δ∗
+gδgGg ˙g
+. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160
+A.4. Calculation of □g
+. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161
+A.5. Calculation of D(g,dA)T (˙g, d ˙A)
+. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162
+A.6. Calculation of D(g,dA)(δgdA)(˙g, d ˙A)
+. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162
+B. Calculation of the subprincipal operator at trapping
+. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164
+B.1. Invariant formalism for the subprincipal symbols of operators acting on bundles
+. . . . . . . . . . . 164
+B.2. Calculation of subprincipal operators
+. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165
+C. Calculation of the subprincipal operator at radial points at event horizon
+. . . . . . . . . . 171
+References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175
+1. Introduction
+The Einstein-Maxwell system, which describes the interaction of gravity and electromagnetism, is given
+by
+Ric(g)µν = 2Tµν(g, F),
+dF = 0,
+δgF = 0.
+(1.1)
+where g is a Lorentzian metric g with signature (−, +, +, +), Ric(g) is the Ricci curvature tensor and
+Tµν(g, F) := FµαF α
+ν
+− 1
+4gµνFαβF αβ is the energy momentum tensor associated with an electromagnetic
+field represented by a 2-form F. Thus, the first equation in (1.1), Einstein’s field equation, describes the
+dynamics of the spacetime metric g in the presence of an electromagnetic field F, and the last two equations,
+Maxwell’s equations, describes the propagation of electromagnetic waves on the spacetime g.
+Among the most interesting solutions to Einstein-Maxwell equations are the families of black hole so-
+lutions. The Kerr-Newman (KN) family of solutions of (1.1) describes stationary, charged, and rotating
+black holes. It was discovered by [70], following the discovery of the Kerr [51]. The family of KN solutions
+
+LINEAR STABILITY OF KERR-NEWMAN FAMILY
+3
+(gb, Fb) to the Einstein-Maxwell equations are characterized by the parameters b = (m, a, Qe, Qm) ∈ R6
+where m > 0, a ∈ R3, Qe, Qm are interpreted as the mass, angular momentum, electric charge and magnetic
+charge respectively. The KN spacetime reduces to the Reissner-Nordstr¨om (RN) spacetime for a = 0, and
+RN spacetime reduces to Schwarzschild spacetime when Qe = Qm = 0.
+1.1. Statement of the main result. The stability problem for black holes solutions concerns the long time
+behavior of solutions to the Einstein-Maxwell system with initial data close to known black hole solutions.
+The problem of stability of black hole solutions can be roughly divided into three formulations, each of
+increasing difficulty: mode stability of linearized equations, linear stability, and full nonlinear stability.
+The problem of mode stability of the linearized equations involves separating the solutions into modes
+ψ(t, r, θ, φ) = e−iωteimφR(r)S(θ) and aiming at proving the lack of exponentially growing (in t) modes for
+all metric or curvature components. The proof of linear stability of Einstein-Maxwell system means proving
+boundedness and decay for the solutions of the linearized Einstein-Maxwell equations. Finally, the nonlinear
+stability problem for Einstein-Maxwell equations aims at showing that a small perturbation of a KN black
+hole converges to another member of the KN family.
+In order to understand nonlinear stability, one must first understand linear stability. In this paper, we
+consider the linear stability problem of the linearized Einstein-Maxwell equations under coupled gravitational
+and electromagnetic perturbations for weakly charged and slowly rotating Kerr-Newman black holes, that
+is, the case |a| + |Qe| + |Qm| ≪ m.
+To describe our result, we first recall the initial value problem formalism of Einstein-Maxwell equations
+(see §§3.2–3.3 for detail). Given 5-tuple (Σ0, h, k, E, B) where Σ0 is a smooth 3-manifold, h is a Riemannian
+metric on Σ0, k is a symmetric 2-tensor on Σ0, and E, B are 1-forms on Σ0. One seeks (M, g, F) such that
+(g, F) solve the Einstein-Maxwell system and (h, k) are the metric and second fundamental form induced
+by g, and (E, B) are the electric and magnetic field induced by (g, F) on Σ0. A necessary and sufficient
+condition for local solvability of Einstein-Maxwell equations (see [8, 10] and [9, §6.10]) is that (h, k, E, B)
+satisfy the constraint equations, which consist of Gauss-Codazzi equation and the pull-back of the Maxwell
+equations to Σ0. Linearizing the initial value problem gives rise to the corresponding initial value problem
+of linearized Einstein-Maxwell equations.
+We now state a first version of our linear stability result. For a more formal statement, see Theorem 14.2.
+Theorem 1.1. Let b = (m, a, Qe, Qm) be the black hole parameters satisfying |a| + |Qe| + |Qm| ≪ m. Let
+0 < α < 1. Let (˙h, ˙k, ˙E, ˙B) be an initial data set on a 3-manifold Σ0 satisfying the linearized constraint
+equation, and having decay
+|˙h| ≲ r−1−α,
+|˙k|, | ˙E|, | ˙B| ≲ r−2−α
+and similar bounds after applying a few r∂r, ∂ω derivatives. Let (˙g, ˙F) a solution to the linearized Einstein-
+Maxwell equations and attain the initial data (˙h, ˙k, ˙E, ˙B). Then (˙g, ˙F) decays at an inverse polynomial rate
+in time tb,∗ to a linearized Kerr-Newman solution plus a pure gauge solution. That is, there exists linearized
+black hole parameters ˙b = ( ˙m, ˙a, ˙Qe, ˙Qm) ∈ R × R3 × R × R and a vector field V such that
+˙g = ˙gb(˙b) + LV gb + ˜g,
+˙F = ˙Fb(˙b) + LV Fb + ˜F
+where
+(˙gb(˙b), ˙Fb(˙b)) := d
+ds
+���
+s=0(gb+s˙b, Fb+s˙b)
+is the linearized Kerr-Newman solutions and the tail (˜g, ˜F) satisfies the bound
+|˜g| + | ˜F| ≲ t−α−1+ǫ
+b,∗
+,
+ǫ > 0.
+Remark 1.2. Using the rotation invariance of the Maxwell equation, one can always arrange for the initial
+data to have vanishing magnetic charge, i.e., Qm = 0, and thus the electromagnetic 2-form can be written
+as F = dA (see §3.1 for detail). In this paper, we will restrict to the case where the magnetic charge is 0
+and study Einstein-Maxwell equations of the following form
+Ric(g)µν = 2Tµν(g, dA),
+δgdA = 0.
+We also refer the readers to Remark 4.2 for the relation between the non-magnetically charged RN spacetime
+(gb, Fb = dAb) and the magnetically charged RN spacetime (gb,m, Fb,m).
+
+4
+LILI HE
+Remark 1.3. The hypersurface {tb,∗ = c} terminates at null infinity. We also note that for bounded r, the
+tail (˜g, ˜F) satisfies the bound |˜g| + | ˜F| ≲ t−1−α+ǫ for all ǫ > 0 where t = t for large r.
+Remark 1.4. Upon imposing a suitable linearized generalized wave map gauge condition on ˙g, and replacing
+(˙gb(˙b), ˙Fb(˙b)) by its gauge-fixed version, we can choose V such that V = V1 + V2 with V1 lying in a 6-
+dimensional space (only depending on b ) of smooth vector fields (asymptotic to a linear combination of
+spatial translations and boosts) on M, and V2 decaying (in tb,∗) at the rate t−α+ǫ
+b,∗
+, ǫ > 0.
+For charged asymptotically flat black hole spacetimes (i.e. the cosmology constant Λ = 0), the mode
+stability of Reissner-Nordstr¨om black holes was studied in [6,7,57,67–69]. The linear stability of full subex-
+tremal Reissner-Nordstr¨om spacetime was proved by Giorgi [28, 29]. As for Kerr-Newman spacetime, the
+mode stability of wave equation on the Kerr-Newman metric was established in [12]. The Teukolsky and
+Regge-Wheeler equations governing the linear stability of Kerr-Newman spacetime to coupled electromag-
+netic and gravitational perturbations were derived in [30]. The Carter tensor and the physical-space analysis
+in perturbations of Kerr-Newman spacetime were discussed in [27]. For positive cosmology case Λ > 0, the
+nonlinear stability of slowly rotating Kerr-Newman-de Sitter black holes was proved by Hintz [39].
+1.2. Related works. Besides the above references for charged black holes, we also mention the closely
+related results on Einstein vacuum equations. Nonlinear stability problems for solutions of Einstein vacuum
+equations have attracted a large amount of interest. The nonlinear stability of Minkowski spacetime was first
+proved by Christodoulou-Klainerman [11] and later an alternative proof was given by Lindblad-Rodnianski
+[58–60] using wave coordinates. For the case Λ > 0, see [26] for the nonlinear stability of de Sitter spacetime.
+For the Schwarzschild metric, the mode stability was studied in [4, 56, 72, 83, 88]. Dafermos-Holzegel-
+Rodnianski [14] proved the linear stability of the Schwarzschild metric in a double null gauge by recon-
+structing a perturbation from a certain decoupled quantity.
+Later, the linear stability of Schwarzschild
+metric in (generalized) harmonic gauge was studied in [47–50].
+For progress in the nonlinear stability,
+Klainerman-Szeftel [55] proved the nonlinear stability of the Schwarzschild metric under axially symmetric
+and polarized perturbations. Recently, Dafermos-Holzegel-Rodnianski-Taylor [15] proved the nonlinear sta-
+bility of Schwarzschild spacetime under a restrictive symmetry class, which excludes rotating Kerr solutions
+as final state of the evolution.
+For the Kerr metric, the mode stability was studied in [74, 76, 85]. Dafermos-Holzegel-Rodnianski [13]
+studied the Teukolsky equation on Kerr spacetime for |a| ≪ m. Andersson-B¨ackdahl-Blue-Ma [1] proved
+the linear stability of slowly rotating Kerr metric for initial data with strong decay using Newman-Penrose
+formalism. H¨afner-Hintz-Vasy [36] proved the linear stability of slowly rotating Kerr spacetime using wave
+map gauge. Recently, the nonlinear stability of slowly rotating Kerr spacetime was proved by a series of work
+of Klainerman-Szeftel and Giorgi-Klainerman-Szeftel [31,32,52–54]. For large a case, Dafermos-Rodnianski-
+Shlapentokh-Rothman [16] established the decay for solutions of the wave equation on full subextremal Kerr
+exterior spacetimes. Shlapentokh-Rothman-Costa [75] proved the boundedness and decay for the Teukolsky
+equation on Kerr in the full subextremal range |a| < m. Anderson-H¨afner-Whiting [2] gave the mode analysis
+for the linearized equations on subextremal Kerr metric.
+In the case of the Einstein equations with a positive cosmological constant, the global nonlinear stability of
+slowly rotating Kerr-de Sitter spacetime was proved by Hintz-Vasy [43], followed by the work of Fang [22,23].
+1.3. Main ideas of the proof. The proof of Theorem 1.1 builds on the framework developed in [36] for
+the study of the Einstein vacuum equations. The presence of electromagnetic term in the right hand side of
+Einstein equations adds new difficulties to the analysis of the problem.
+(1) In addition to the diffeomorphism invariance, which for the Einstein vacuum equations is eliminated
+by imposing a wave map gauge in [36], one needs to fix a gauge (generalized Lorenz gauge) for the
+electromagnetic 4-potential A in order to express the Einstein-Maxwell system (1.1) as a quasilin-
+ear system of wave equations, called the gauge-fixed Einstein-Maxwell system. Correspondingly,
+the linearization of the gauge-fixed Einstein-Maxwell system gives rise to a linear system of wave
+equations.
+(2) We need to show the (generalized) mode stability for the linearization of (1.1) around a RN solution:
+all generalized mode solutions are sums of linearized KN solutions and pure gauge solutions. In this
+paper, we give a self-contained proof in full detail of this generalized mode stability, completing and
+extending work by Kodama and Ishibashi [57].
+
+LINEAR STABILITY OF KERR-NEWMAN FAMILY
+5
+(3) The fact that the KN metric is not Ricci-flat creates the major difficulty in the mode analysis of
+the wave type operator δgbGgbδ∗
+gb (where (δ∗
+gbu)αβ = 1
+2(∇αuβ + ∇βuα) is the symmetric gradient
+operator and (δgbh)β = −∇αhαβ is the negative divergence operator, and Ggb = Id − 1
+2gbtrgb is the
+trace reversal operator) acting on 1-forms. We note that the operator δgbGgbδ∗
+gb (or its modifications)
+occurs as both the gauge propagation operator and the gauge potentials operator for the Einstein
+equations. To deal with this issue, we restrict to the weakly charged case, where we are able to
+exploit a perturbation argument to extend the invertibility of relevant operators on Schwarzschild
+metric to weakly charged Reissner-Nordstr¨om spacetime.
+Concretely, we first employ the generalized wave map gauge
+�ΥE(g; gb) := ΥE(g; gb) − θ(g; gb) = 0,
+ΥE(g; gb) = gµνgκλ (Γ(g)µ
+κλ − Γ(gb)µ
+κλ)
+(1.2)
+where θ(g; gb) is linear in g and θ(gb; gb) = 0. By introducing this type of gauge condition with a
+suitable choice of θ, the gauge potential operator is given by ˜δgbGgbδ∗
+gb where ˜δgb is a zero order
+modification of δg. In contrast to δgbGgbδ∗
+gb which is the gauge potential operator corresponding
+to the standard wave map gauge ΥE(g; gb) = 0, the operator ˜δgbGgbδ∗
+gb is invertible (on suitable
+function spaces) in the Schwarzschild case b = (m, 0, 0), and thus a perturbation argument enables
+us to extend this invertibility to the weakly charged and slowly rotating KN metrics with sufficiently
+small charge. Moreover, the generalized wave map gauge allows us to exclude a pure gauge solution
+which has no geometric meaning.
+Next, we implement the constraint damping, which was first discussed in [34] and played a central
+role both in numerical work [71] and the stability proofs [36, 39, 42, 43]. The reason we introduce
+the constraint damping, i.e., use ˜δ∗
+gb in the construction of linearized gauge-fixed Einstein-Maxwell
+equations, is that we need to use the invertibility of the corresponding gauge propagation operator
+δgbGgb ˜δ∗
+gb to exclude the generalized modes, which grows linearly in tb,∗ and whose leading term
+is given by linearized (in ( ˙m, 0, ˙Q)) KN solutions, to the linearized gauge-fixed Einstein-Maxwell
+equations Lb(˙g, ˙A) = 0. We also note that we use a perturbation argument to prove the invertibility
+of the gauge propagation operator δgbGgb ˜δ∗
+gb on the weakly charged and slowly rotating KN metrics
+gb.
+1.3.1. Gauge fixing. In this paper, we introduce the generalized wave map gauge and generalized Lorenz
+gauge.
+Concretely, one fixes a background metric g0, and requires that the identity map (M, g) → (M, g0) be a
+wave map. This is equivalent to
+ΥE(g; g0) = gµνgκλ �
+Γ(g)µ
+κλ − Γ(g0)µ
+κλ
+�
+= 0.
+We define the generalized wave map gauge as follows
+�ΥE(g; g0) := ΥE(g; g0) − θ(g; g0) = 0
+where θ(g; g0) is linear in g, contains no derivatives of g and θ(g0, g0) = 0. Then we consider the gauge-fixed
+Einstein equations
+P E(g, A) := Ric(g) − 2T (g, dA) − �δ∗
+g �Υ(g; g0) = 0.
+At the linearized level, we take gb, around which we linearize the equations, as the background metric, and
+thus obtain the linearized gauge-fixed Einstein equations
+LE
+b (˙g, ˙A) := DgbP E(˙g, ˙A) = −1
+2□gb ˙g + l.o.t = 0
+which is a system of linear wave equations.
+As for the Maxwell equations, we introduce the generalized Lorenz gauge. We fix a background metric
+g0, and then a (modified) Lorenz gauge reads
+ΥM(g, A; g0) = trgδ∗
+g0A = 0.
+We fix a background 1-form A0 and define the generalized Lorenz gauge as
+�ΥM(g, A; g0, A0) := ΥM(g, A; g0) − ΥM(g, A0; g0),
+
+6
+LILI HE
+and then obtain the gauge-fixed Maxwell equations
+P M(g, A) := δgdA − d�ΥM(g, A; g0, A0) = 0.
+Again, at the linearized level, we take (gb, Ab), around which we linearize the equations, as the background
+metric and 1-form, and thus have
+LM
+b (˙g, ˙A) := DgbP M(˙g, ˙A) = −□gb ˙A + l.o.t = 0
+which is a system of linear wave equations.
+We see that a solution of the ungauged equations satisfying the gauge conditions gives rise to a solution
+of the gauge-fixed equations. As for the other direction, suppose that (g, A) solves the gauge-fixed Einstein-
+Maxwell system
+�
+P E(g, A), P M(g, A)
+�
+= 0,
+and satisfies
+(�ΥE(g; gb), �ΥM(g, A; gb, Ab) = 0
+at Σ0
+and
+(L∂t �ΥE(g; gb), L∂t �ΥM(g, A; gb, Ab) = 0
+at Σ0
+where Σ0 = {t = 0} is the spacelike Cauchy hypersurface. Applying δg to
+P M(g, A) = δgdA − d�ΥM(g, A; gb, Ab) = 0
+gives
+−δd�ΥM(g, A; gb, Ab) = □g �ΥM(g, A; gb, Ab) = 0.
+and thus according to the vanishing initial data of �ΥM(g, A; gb, Ab), we obtain that �ΥM(g, A; gb, Ab) = 0.
+Applying δgGg to P E(g, A) = Ric(g) − 2T (g, dA) − �δ∗
+g �Υ(g; g0) = 0 and using the second Bianchi identity
+and the fact δgT = 0 yields
+−δgGg�δ∗
+g �ΥE(g; gb) = 0
+which is a wave-type equation on the 1-form �ΥE(g; gb) and thus �ΥE(g; gb) = 0. We conclude that (g, A)
+satisfies the gauge conditions, and thus solves the Einstein-Maxwell system. Correspondingly, this direction
+holds on the linearized level as well.
+1.3.2. Fredholm framework setup. According to the above discussion, it suffices to consider the equation
+Lb(˙g, ˙A) := (2LE
+b , LM
+b )(˙g, ˙A) = 0. A simple linear theory presented in [43] reduces the initial value problem
+for Lb(˙g, ˙A) = 0 to the following inhomogeneous problem.
+Concretely, let b = (m, a, Q) be close to b0 = (m0, 0, Q0) with |Q0| ≪ m0. Fix a function tb,∗ which
+equals t + r(m0,0,Q0),∗ near event horizon and t − r(m,0,Q),∗ near null infinity I +, where rb,∗ is the tortoise
+coordinate. Then we consider the following inhomogeneous equation
+Lb(˙g, ˙A) = (2LE
+b (˙g, ˙A), LM
+b (˙g, ˙A)) = f
+where f has compact support in t∗ and f decays in r at the rate r−2−α. Our strategy is to take Fourier
+transform of the inhomogeneous equation Lb(˙g, ˙A) = f in tb,∗. We define the Fourier transform of (˙g, ˙A) as
+(ˆ˙g(σ), ˆ˙A(σ)) :=
+�
+R
+eitb,∗σ(˙g, ˙A)(tb,∗) dtb,∗
+(1.3)
+and likewise for f.
+According to a crude energy estimate in Proposition 5.27, we have the following integral representation
+(˙g(tb,∗),
+˙A(tb,∗)) = 1
+2π
+�
+Im σ=M+1
+e−iσtb,∗�
+Lb(σ)−1 ˆf(σ) dσ,
+M ≫ 1.
+(1.4)
+We expect to shift contour of the integration to Im σ = c for any c > 0. To this end, we need to understand
+the location of the poles of �
+Lb(σ)−1 and expect invertibility of �
+Lb(σ)−1 when |Re σ| ≫ 1. Concretely, we
+need to establish the uniform Fredholm estimates (down to σ = 0) and high energy estimates. The uniform
+Fredholm estimates allow us to perform the perturbation argument (which is used a lot in this paper) while
+the high energy estimates imply the invertibility of �
+Lb(σ)−1 when |Re σ| ≫ 1. We point out that the whole
+section §5 is devoted to using microlocal tools to obtain the aforementioned uniform Fredholm estimates
+and high energy estimates. In brief, the proof of uniform Fredholm estimates uses the non-elliptic Fredholm
+
+LINEAR STABILITY OF KERR-NEWMAN FAMILY
+7
+framework of [77], radial point estimates at the event horizon [77], scattering radial point estimates at infinity
+( [65,79] for non-zero σ and [81] for σ near zero), propagation of singularities estimates [18], and (for σ = 0)
+elliptic b-theory [64]. As for the proof of high energy estimates, in addition to the aforementioned estimates
+at a semiclassical version, it uses estimates at normally hyperbolic trapping [20, 40, 86] and high energy
+estimates at infinity [79,82].
+1.3.3. Mode stability of the gauge-fixed Einstein-Maxwell operator for C−1 ≤ |σ| ≤ C, Im σ ≥ 0 with C > 0.
+In view of the high energy estimates, we are left with the analysis of �
+Lb(σ) for σ in a bounded region in the
+closed upper half plane. For the analysis in the region C−1 ≤ |σ| ≤ C, Im σ ≥ 0 with C > 0, we prove that
+�
+Lb0(σ), with b0 = (m0, 0, Q0) and Q0 ≪ m0 being a Reissner-Nordstr¨om parameter, is invertible and then
+use a perturbation argument (in this compact (in σ) region) to extend the invertibility to �
+Lb(σ) for nearby
+b.
+For the proof of invertibility of �
+Lb0(σ), we make use of the geometric mode stability result of Reissner-
+Nordstr¨om black holes and the invertibility of the gauge progation operator and gauge potential operator.
+Concretely, suppose that �
+Lb0(˙g, ˙A) = 0. Then (˙g, ˙A) = e−itb0,∗σ(˙g, ˙A) is a mode solution of Lb0(˙g, ˙A) = 0.
+According to the discussion in §1.3.1, we see that the solution satisfies
+δgb0 d
+�
+D(gb0 ,Ab0) �ΥM(˙g, ˙A; gb0, Ab0)
+�
+= 0.
+According to the mode stability of □gb0 ,0, we see that D(gb0 ,Ab0 ) �ΥM(˙g, ˙A; gb0, Ab0) = 0. Therefore, (˙g, ˙A)
+solves the Maxwell equations. By the discussion in §1.3.1 again, we obtain that
+δgb0 Ggb0 ˜δ∗
+gb0
+�
+Dgb0 �ΥE(˙g; gb0)
+�
+= 0.
+Using the mode stability of the gauge propagation operator (this is where we need the smallness of the
+charge), we see that Dgb0 �ΥE(˙g; gb0) = 0. Therefore, (˙g, ˙A) is a mode solution of the ungauged linearized
+Einstein-Maxwell system. Using the geometric mode stability of Reissner-Nordstr¨om solutions, we conclude
+that (˙g, ˙A) is a pure gauge mode solution, that is,
+(˙g, ˙A) =
+�
+2δ∗
+gω, Lω♯A + dφ
+�
+where ω is a 1-form and φ is a scalar function, both of which are modes. Plugging them back into the
+linearized gauge condition, and with our choice of the generalized gauge condition, we have
+˜δgb0 Ggb0 δ∗
+gb0 ω = 0,
+δgb0 d
+�
+Lω♯A + dφ
+�
+= 0.
+First by the mode stability of the gauge potential operator (this is again where we need the smallness of the
+charge) and next the mode stability of □gb0 ,0, we see that (ω, φ) = 0. This proves the injectivity of �
+Lb0(σ).
+Since �
+Lb0(σ) has index 0, it follows that �
+Lb0(σ) is invertible.
+1.3.4. The analysis of �
+Lb(σ)−1 near σ = 0. According to Theorem 10.1, the space Kb of zero energy modes
+of Lb is 8-dimensional; it is the sum of a 5-dimensional space of linearized Kerr-Newman solutions corrected
+by addition of pure gauge solutions to arrange the linearized gauge conditions, and a 3-dimensional space
+of pure gauge solutions (2δ∗
+gbωb, Lω♯
+bAb + dφb) with ωb asymptotic to a spatial translation. According to
+Proposition 10.5, the space �Kb of generalized zero energy modes (with o(1) decay as r → ∞ for fixed tb,∗,
+but allowing for polynomial growth in tb,∗) of Lb is the sum of Kb and a 3-dimensional space of pure gauge
+solutions (2δ∗
+gb ˆωb, Lˆω♯
+bAb + dˆφb) with ˆωb asymptotic to a Lorentz boost and the coefficient of tb,∗ being a
+stationary pure gauge solution (2δ∗
+gbωb, Lω♯
+bAb + dφb).
+Therefore, �
+Lb(σ) is more singular when acting on the 3-dimensional space of stationary pure gauge so-
+lutions because of the existence of the generalized zero energy solutions of Lb(˙g, ˙A) = 0 with linear growth
+in tb,∗ and leading order terms given as a stationary pure gauge solution. Motivated by this fact, we write
+�
+Lb(σ) (actually after multiplied by an operator which is invertible near σ = 0) as a 3 × 3 block matrix.
+
+8
+LILI HE
+The basic mechanism underlying the analysis of �
+Lb(σ)−1 near σ = 0, is the formal expansion of the
+resolvent near σ = 0
+�
+Lb(σ)−1f = �
+Lb(0)−1f + (�
+Lb(σ)−1 − �
+Lb(0)−1)f
+= u0 + σ�
+Lb(σ)−1f1,
+where we use the formal resolvent identity �
+Lb(σ)−1 − �
+Lb(0)−1 = −�
+Lb(σ)−1(�
+Lb(σ) − �
+Lb(0))�
+Lb(0)−1 and thus
+obtain
+u0 = �
+Lb(0)−1f,
+f1 = −σ−1��
+Lb(σ) − �
+Lb(0)
+�
+u0.
+The first term u0 is σ-independent. As for the second term σ�
+Lb(σ)−1f1, we gain a factor of σ (i.e. more
+regularity in σ and thus more decay in tb,∗ after taking inverse Fourier transform). However, since �
+Lb(0)−1
+loses two orders of decay in r while σ−1(�
+Lb(σ) − �
+Lb(0)) = 2iρ(ρ∂ρ − 1) + ρ2C∞ with ρ = 1/r only gains
+back one order decay in r, it follows that f1 has one order of decay less than f. We expect to do the above
+expansion iteratively
+�
+Lb(σ)−1f = u0 + σu1 + · · · + σkuk
+where
+uk = �
+Lb(0)−1fk,
+fk+1 = −σ−1��
+Lb(σ) − �
+Lb(0)
+�
+uk
+as often as possible. However, this iteration requires that uk = �
+Lb(0)−1fk has a certain asymptotic behavior
+or decay rate as r → ∞ such that σ−1(�
+Lb(σ) − �
+Lb(0))uk lies in the domain of �
+Lb(σ)−1. That is, we have to
+stop the iteration once uk = �
+Lb(0)−1fk does not satisfy this requirement and thus we obtain an error term
+σkL(σ)−1fk in the expansion.
+Specifically, a detailed analysis in §11.2 implies that �
+Lb(σ)−1 (up to a multiplication of an operator
+invertible near σ = 0) is equal to
+
+
+
+L00(b, σ)
+σ2 �L01(b, σ)
+σ�L02(b, σ)
+σ�L10(b, σ)
+σ2 �L11(b, σ)
+σ2 �L12(b, σ)
+σ�L20(b, σ)
+σ2 �L21(b, σ)
+σ�L22(b, σ)
+
+
+
+where L00(b, σ), ˜L11(b, σ), ˜L22(b, σ) are invertible near σ = 0. Therefore, one obtain
+�
+Lb(σ)−1 = σ−2P2(b) + σ−1P1(b) + R1(b, σ) + R2(b, σ),
+R1(b, σ) ∼ |σ|α−1,
+R2(b, σ) ∼ |σ|α
+where P1(b), P2(b) are independent of σ and of finite rank with explicit range: linearized KN solutions and
+pure gauge solutions. The range of R1(b, σ) consists of pure gauge solutions and this allow us to rewrite
+the inverse Fourier transform of R1(b, σ)f as a pure gauge solution plus a remainder term decaying faster in
+time tb,∗ (in fact at the rate t−1−α
+b,∗
+).
+1.4. Outline. The paper is organized as follows.
+• In §2, we introduce the b- and scattering geometric structures on manifolds with boundaries or
+corners (in this paper, we work on the compactification of the spatial slice Σ of the spacetime M),
+and corresponding function spaces.
+• In §3, we discuss the initial value problems formalism for Einstein-Maxwell equations.
+We also
+introduce our choice of gauge, generalized wave map gauge and generalized Lorenz gauge, and then
+derive the gauge-fixed Einstein-Maxwell system. We include the discussion at both nonlinear and
+linearized levels.
+• In §4, we realize the family of slowly rotating KN metric as a smooth family of stationary metrics on
+a fixed manifold M. We also introduce several vector bundles over M and discuss how to construct
+the gauged Cauchy data for the gauge-fixed Einstein-Maxwell system from any given initial data
+(h, k, E, B) satisfying the constraint equations.
+• In §5, we set up the general Fredholm framework, based on tools from microlocal analysis. Concretely,
+we prove the uniform Fredholm estimates and high energy estimates for the relevant wave type
+operators (acting on both scalar functions and tensor bundles). We also establish a crude energy
+estimate which gives an exponentially growing bound on the energy (over a spacelike hypersurface
+terminating at null infinity) of solutions to the wave equations on slowly rotating Kerr-Newman
+metric.
+
+LINEAR STABILITY OF KERR-NEWMAN FAMILY
+9
+• In §6, we introduce the spherical harmonic decompositions, which will be used in the proof of the
+geometric mode stability of Reissner-Nordstr¨om spacetime in §7.
+• In §7, we prove the version of the geometric mode stability of Reissner-Nordstr¨om spacetime used
+in the subsequent sections.
+• In §8, we study the modes of the scalar wave operator on RN and slowly rotating KN spacetimes.
+• In §9, we analyze the modes on various function spaces (which correspond to different decay rate at
+spatial infinity) of the gauge propagation operator Pb,γ = δgbGgb�δ∗
+gb,γ and the gauge potential wave
+operator Wb,γ = �δgb,γGgbδ∗
+gb, both of which are wave-type operators acting on 1-form. From this
+section on, we are restricted in the small charge case.
+• In §10, we prove that the linearized gauge-fixed Einstein-Maxwell system Lgb0 ,γ (linearized around
+weakly charged RN metrics and 4-electromagnetic potentials (gb0, Ab0)) has no non-zero modes in
+the closed upper half plane. Moreover, we describe the space of zero modes and generalized zero
+modes (with linearly growth in tb,∗) of Lb,γ for both RN case and slowly rotating KN case with small
+charge.
+• In §11, we prove the mode stability of the linearized gauge-fixed Einstein-Maxwell operator Lb,γ(σ)
+for b = (m, a, Q) near b0 = (m0, 0, Q0), |Q0| ≪ m0 and Im σ ≥ 0, σ ̸= 0 (i.e. the invertibility of the
+operator �
+Lb,γ(σ) on suitable function spaces for Im σ ≥ 0, σ ̸= 0), and also provide a description of
+the structure of the resolvent �
+Lb,γ(σ)−1 near σ = 0.
+• In §12, we study the higher regularity of the resolvent ( �
+Lb,γ(σ))−1 of the linearized gauge-fixed
+Einstein-Maxwell operator.
+• In §13, we put all the previous sections together to study the asymptotic behavior of the solutions
+(˙g, ˙A) to the inhomogeneous linearized gauge-fixed Einstein-Maxwell equations Lb,γ(˙g, ˙A) = f.
+• In §14, we reduce this initial value problem for Einstein-Maxwell equations to the inhomogeneous
+problem studied in the previous section.
+• In Appendix §A, we provide a detailed calculation of the linearized Einstein-Maxwell system around
+RN black holes, which is needed in the proof of the geometric mode stability of Reissner-Nordstr¨om
+spacetime in §7. In Appendix B, we present the calculation of the subprincipal operator of the wave
+operator acting on tensor bundles at trapping. In Appendix C, we include the computation of the
+subprincipal operator of the wave operator acting on tensor bundles at radial points at event horizon,
+which is needed in the radial point estimates at event horizons.
+1.5. List of notations. We give a list of notation used frequently throughout the present paper, together
+with a reference to the first appearance:
+Aℓ
+weighted L∞ conormal function space, see (2.4)
+Ab0
+Reissner-Nordstr¨om electromagnetic 4-potentials with parameters b0, see (4.23)
+Ab
+Kerr-Newman electromagnetic 4-potentials with parameters b, see (4.23)
+b0
+fixed Reissner-Nordstr¨om parameters, see (4.1)
+b
+Kerr-Newman parameters, see (4.13)
+δg
+negative divergence, (δgu)µ1...µN = −∇λuλµ1...µN
+δ∗
+g
+symmetric gradient, (δ∗
+gu)µν = 1
+2(uµ;ν + uν;µ)
+˜δg,γ
+modification of the negative divergence, see (4.62)
+˜δ∗
+g,γ
+modification of the symmetric gradient, see (4.62)
+gb0
+Reissner-Nordstr¨om metrics with parameter b0, see (4.23)
+gb
+Kerr-Newman metrics with parameters b, see (4.23)
+γ0
+map assigning to a function on M its Cauchy data at Σ0, see (4.85)
+¯Hs,ℓ
+b
+weighted b-Sobolev space of extendible distributions, see (2.8)
+˙Hs,ℓ
+b
+weighted b-Sobolev space of supported distributions, see (2.9)
+¯Hs,ℓ
+b,h
+semiclassical weighted b-Sobolev space of extendible distributions, see (2.11)
+˙Hs,ℓ
+b,h
+semiclassical weighted b-Sobolev space of supported distributions, see (2.11)
+ib
+map constructing Cauchy data from geometric initial data, see Proposition 4.14
+Kb
+the space of zero energy modes of Lb,γ, see (10.8)
+�Kb
+the space of generalized zero energy modes of Lb,γ, see Proposition 10.5
+
+10
+LILI HE
+K∗
+b
+the space of dual zero energy modes of L∗
+b,γ, see (10.9)
+�K∗
+b
+the space of dual generalized zero energy modes of L∗
+b,γ, see Proposition 10.5
+LV
+Lie derivative along the vector field V
+�LT V
+equal to LV T , see Definition 3.9
+Lb,γ
+linearized gauge-fixed Einstein-Maxwell equations (2LE
+b,γ, LM
+b,γ)
+LE
+b,γ
+linearized gauge-fixed Einstein equations, see (4.63)
+LM
+b,γ
+linearized gauge-fixed Maxwell equations, see (4.63)
+M
+the static region of a fixed reissner-Nordstr¨o spacetime, see (4.5)
+M
+extended manifold where gb is defined, see (4.8)
+Pb,γ
+gauge propagation operator for the generalized wave map gauge 1-form �ΥE, see (4.66)
+Σ0
+a spacelike Cauchy hpersurface of M, see (3.10)
+t
+static time coordinate, see (4.2), or Boyer-Lindquist coordinate, see (4.14)
+t0
+incoming Eddington-Finkelstein coordinate on a fixed Reissner-Nordstr¨om manifold, see (4.6)
+tχ0
+a time function interpolating t0 near the even thorizon and t near r = ∞, see (4.9)
+tb,∗
+a time function which is smooth across the even thorizon and transverse to null infinity near r = ∞,
+see (4.29)
+t∗
+equal to tb0,∗, see (4.10)
+t
+a timelike function with respect to Kerr-Newman metric gb, which equals to t near r = ∞, see (4.15)
+θ
+gauge source function for the generalized wave map gauge, see (3.17)
+Ub0
+parameter space for slowly rotating Kerr-Newman black holes, see Lemma 4.1
+Ub0,m
+parameter space for slowly rotating Kerr-Newman black holes allowing for magnetic charges, see
+(4.24)
+ΥE
+wave map gauge 1-form, see (3.15)
+ΥM
+modified Lorenz gauge function, see (3.19)
+�ΥE
+generalized wave map gauge 1-form, see (3.17)
+�ΥM
+generalized Lorenz gauge function, see (3.20)
+Wb,γ
+gauge potential operator for Einstein equations, see (4.66)
+X
+the spatial slice of the static region M, see (4.5)
+X
+the spatial slice of the extended manifold M, see (4.8)
+¯X
+compactification of X at r = ∞, see (4.11)
+∂− ¯X
+an artificial boundary of ¯X, see (4.11)
+∂+ ¯X
+the boundary at spatial infinity r = ∞ of ¯X, see (4.11)
+�•(σ)
+Fourier transform eitb,∗σ • e−itb,∗σ of • with respect to the time function tb,∗; in this paper, • =
+□gb,0, Pb,γ, Wb,γ, Lb,γ
+Acknowledgements. The author would like to thank her advisor, Hans Lindblad, for his constant support
+and encouragement. The author is also grateful to Peter Hintz and Andr´as Vasy for helpful discussions.
+2. b- and scattering geometry
+In this section, we will discuss the b- and scattering geometric structures on manifolds with boundaries or
+corners, and corresponding function spaces. Here, we follow the discussion in [36, §2]. In §2.1, we recall the
+notions of b- and scattering vector bundles. We refer the readers to [64, §2] and [66, §2] for a more detailed
+exposition. In §2.2, we recall the notion of radial compactification. In §2.3, we define the relevant Sobolev,
+conormal and polyhomogeneous functions defined on manifolds with boundaries or corners.
+Let ¯X be a compact n-dimensional manifold with boundary ∂ ¯X ̸= ∅, and let ρ ∈ C∞(X) denote a
+boundary defining function: ∂ ¯X = ρ−1(0), dρ ̸= 0 on ∂ ¯X.
+2.1. b- and scattering vector bundles. We define the Lie algebras of b-vector fields and scattering vector
+fields by
+Vb( ¯X) = {V ∈ V( ¯X): V is tangent to ∂ ¯X},
+Vsc( ¯X) = ρVb( ¯X).
+(2.1)
+In local coordinates x ≥ 0, y ∈ Rn−1 on ¯X, with x = 0 locally defining the boundary of ¯X, elements of
+Vb( ¯X) are of the form a(x, y)x∂x + �n−1
+i=1 bi(x, y)∂yi, with a, bi ∈ C∞( ¯
+X), while elements of Vsc( ¯X) are of
+the form a(x, y)x2∂x + �n−1
+i=1 bi(x, y)x∂yi.
+
+LINEAR STABILITY OF KERR-NEWMAN FAMILY
+11
+Correspondingly, one can define the b-tangent bundle, resp. scattering tangent bundle
+bT ¯X → ¯X,
+resp.
+scT ¯X → ¯X,
+with local frames given by {x∂x, ∂yi}, resp. {x2∂x, x∂yi}, such that Vb( ¯X) = C∞( ¯X; bT ¯X) and Vsc( ¯X) =
+C∞( ¯X; scT ¯X). Therefore, for example, x∂x is a smooth, non-vanishing section of bT ¯X down to ∂ ¯X. Over
+the interior X, these bundles are naturally isomorphic to T X, but the maps bT ¯X → T X and scT ¯X → T X
+fail to be injective over ∂ ¯X. We denote by Diffm
+b ( ¯X), resp. Diffm
+sc(X) the space of m-th order b-, resp.
+scattering differential operators, consisting of linear combinations of up to m-fold products of elements of
+Vb(X), resp. Vsc(X).
+The dual bundles bT ∗ ¯X → ¯X (b-cotangent bundle), resp. scT ∗ ¯X → ¯X (scattering cotangent bundle) have
+local frames
+dx
+x , dyi,
+resp.
+dx
+x2 , dyi
+x ,
+which are smooth down to ∂ ¯X as sections of these bundles (despite their being singular as standard covectors,
+i.e. elements of T ∗ ¯X). A scattering metric is then a section g ∈ C∞( ¯X; S2 scT ∗ ¯X) which is a non-degenerate
+quadratic form on each scattering tangent space scT ∗
+p ¯X, p ∈ ¯X; b-metrics are defined analogously as a section
+g ∈ C∞( ¯X; S2 bT ∗ ¯X)
+2.2. Radial compactification. These aforementioned b- and scattering structures arise naturally on com-
+pactifications of non-compact manifolds (see [66, §1]), the simplest example being the radial compactification
+of Rn, defined by
+Rn :=
+�
+Rn ⊔ ([0, 1)ρ × Sn−1)
+�
+/ ∼
+(2.2)
+where the relation ∼ identifies a point in Rn \ {0}, expressed in polar coordinates as rω, r > 0, ω ∈ Sn−1,
+with the point (ρ, ω) where
+ρ = r−1.
+This has a natural smooth structure, with smoothness near ∂Rn = ρ−1(0) meaning smoothness in (ρ, ω). In
+polar coordinates in r > 1, the space of b-vector fields is then locally spanned over C∞(Rn) by ρ∂ρ = −r∂r
+and V(Sn−1); scattering vector fields are spanned by ρ2∂ρ = −∂r and ρV(Sn−1). Using standard coordinates
+x1, . . . , xn on Rn, scattering vector fields on Rn are precisely those of the form
+n
+�
+i=1
+ai∂xi,
+ai ∈ C∞(Rn);
+this involves the statement that {∂x1, . . . , ∂xn}, which is a frame of T Rn, extends by continuity to a smooth
+frame of scT Rn down to ∂Rn. Thus, the space of scattering vector fields on Rn is generated over C∞(Rn) by
+constant coefficient (translation-invariant) vector fields on Rn. On the other hand, Vb(Rn) is spanned over
+C∞(Rn) by vector fields on Rn with coefficients which are linear functions, i.e. by ∂x1, . . . , ∂xn, and xi∂xj,
+1 ≤ i, j ≤ n.
+On the dual side, scT ∗Rn is spanned by dxi, 1 ≤ i ≤ n, down to ∂Rn. Therefore, a scattering metric
+g ∈ C∞(Rn, S2 scT ∗Rn) is a non-degenerate linear combination of dxi ⊗s dxj = 1
+2(dxi ⊗ dxj + dxj ⊗ dxi)
+with C∞(Rn) coefficients. In particular, the Euclidean metric
+(dx1)2 + · · · + (dxn)2 ∈ C∞(Rn; S2 scT ∗Rn)
+is a Riemannian scattering metric.
+In this manuscript, we are particularly interested in the behavior of the Hamiltonian vector field associated
+to a smooth function p ∈ C∞(scT ∗ ¯X) where ¯X is a manifold with boundaries. Concretely, by extension from
+T ∗X, one can define Hamilton vector fields Hp of smooth functions p ∈ C∞(scT ∗ ¯X). In fact Hp ∈ Vsc(scT ∗ ¯X)
+is a scattering vector field on scT ∗ ¯X, which is a manifold with boundary scT ∗
+∂ ¯
+X ¯X. For us, the main example
+will be the Hamilton vector field HG where G(x, ξ) := |ξ|2
+g(x)−1 and g ∈ C∞( ¯X; S2 scT ∗ ¯X) is a scattering
+metric.
+To give a uniform discussion of the behavior of the Hamiltonian vector field Hp ∈ Vsc(scT ∗ ¯X) near the
+boundary of ∂ ¯X and near the infinity in the fibers of the scattering cotangent bundle scT ∗ ¯X, we consider
+the compactified scattering cotangent bundle scT ∗ ¯X as our basic microlocal space. We note that scT ∗ ¯X is a
+compact manifold with corners obtained by the radial compactification of the fibers of scT ∗ ¯X (see Figure 1).
+Let ρtotal = xρξ, where x, resp. ρξ is the boundary defining function of the boundary of ¯X, resp. the fiber
+
+12
+LILI HE
+infinity, be the total boundary defining function of scT ∗ ¯X. Then we have Vsc(scT ∗ ¯X) = ρtotalVb(scT ∗ ¯X), and
+thus the rescaled Hamiltonian vector field ρ−1
+totalHp, where p ∈ C∞(scT ∗ ¯X), is a b-vector field on scT ∗ ¯X.
+scT ∗
+∂ ¯
+X ¯X = {x = 0}
+scS∗ ¯X = {ρξ = 0}
+scS∗ ¯X = {ρξ = 0}
+0 section
+Figure 1. The microlocal space scT ∗ ¯X.
+2.3. Function spaces. We next introduce Sobolev spaces corresponding to b- and scattering structures. As
+an integration measure on ¯X, let us fix a scattering density, i.e. a positive section of scΩ1 ¯X = |Λn scT ∗ ¯X|,
+which in local coordinates x ≥ 0, y ∈ Rn−1 takes the form a(x, y)| dx
+x2
+dy
+xn−1 | with 0 < a ∈ C∞( ¯X). (On Rn,
+one can take |dx1 · · · dxn|.) This provides us with a space L2( ¯X); the norm depends on the choice of density,
+but all choices lead to equivalent norms. Working with a b-density on the other hand would give a different
+space, namely a weighted version of L2(X); we therefore stress that even for b-Sobolev spaces, we work with
+a scattering density. Thus, for k ∈ N, we define
+Hk
+• (X) := {u ∈ L2(X): V1 · · · Vju ∈ L2(X) ∀ V1, . . . , Vj ∈ V•(X), 0 ≤ j ≤ k},
+• = b, sc,
+called b- or scattering Sobolev space. Using a finite spanning set in V•(X), one can give this the structure of
+a Hilbert space; Hs
+•(X) for general s ∈ R is then defined by duality and interpolation. If ℓ ∈ R, we denote
+weighted Sobolev spaces by
+Hs,ℓ
+• (X) = ρℓHs
+•(X) = {ρℓu: u ∈ Hs
+•(X)}.
+For example, Hs,ℓ
+sc (R) ∼= ⟨x⟩−ℓHs(Rn) is the standard weighted Sobolev space on Rn. The space of weighted
+(L2-)conormal functions on ¯X is
+H∞,ℓ
+b
+(X) =
+�
+s∈R
+¯Hs,ℓ
+b (X).
+Dually, we define
+H−∞,ℓ
+b
+(X) =
+�
+s∈R
+¯Hs,ℓ
+b (X).
+Note that ¯Hs,ℓ
+b (X) ⊂ C−∞( ¯
+X) := ˙C∞( ¯X)∗ (where ˙C∞( ¯X) ⊂ C∞(X) is the subspace of functions vanishing
+to infinite order at ∂ ¯X) consists of tempered distributions. (In particular, they are extendible distributions
+at ∂X in the sense of [46, Appendix B].) We furthermore introduce the notation
+Hs,ℓ+
+b
+( ¯X) :=
+�
+ǫ>0
+Hs,ℓ+ǫ
+b
+( ¯X),
+Hs,ℓ−
+b
+( ¯X) :=
+�
+ǫ>0
+Hs,ℓ−ǫ
+b
+( ¯X)
+(2.3)
+for s ∈ R ∪ {±∞}.
+A space closely related to H∞,ℓ
+b
+( ¯X) is
+Aℓ( ¯
+X) := {u ∈ ρℓL∞( ¯X): Diffb( ¯X)u ⊂ ρℓL∞( ¯X)},
+{ρ = 0} = ∂ ¯X
+(2.4)
+consisting of weighted L∞-conormal functions. For ¯X = R3, we have the inclusions
+H∞,ℓ
+b
+(R3) ⊂ Aℓ+3/2(R3),
+Aℓ(R3) ⊂ H∞,ℓ−3/2−
+b
+(R3),
+
+LINEAR STABILITY OF KERR-NEWMAN FAMILY
+13
+by Sobolev embedding. (The shift 3
+2 in the weight is due to our defining b-Sobolev spaces with respect to
+scattering densities instead of b-densities. Indeed, for s > 3
+2, we have
+Hs,ℓ
+b (R3; |dx1 dx2 dx3|) = Hs,ℓ+3/2
+b
+(R3; ⟨r⟩−3|dx1 dx2 dx3|) ֒→ ⟨r⟩−ℓ−3/2L∞(R3),
+(2.5)
+with the second density here being a b-density on R3.) We define Aℓ+( ¯X) and Aℓ−( ¯X) analogously to (2.3).
+These notions extend readily to sections of rank k vector bundles E → X: for instance, in a local trivialization
+of E, an element of Hs,ℓ
+• (X, E) is simply a k-tuple of elements of Hs,ℓ
+• (X).
+We next turn to the notion of E-smoothness (see [64, §5]), where E ⊂ C × N is an index set; the latter
+means that E is a discrete subset of C × N satisfying that (z, k) ∈ E implies (z, j) ∈ E for 0 ≤ j ≤ k and
+(z − i, k) ∈ E, and that any sequence (zj, kj) ∈ E with |zj| + kj → ∞ satisfies Im zj → −∞. The space
+AE
+phg( ¯X) ⊂ A−∞( ¯X) =
+�
+ℓ∈R
+Aℓ( ¯X)
+then consists of all u for which
+�
+(z,j)∈E
+Im z≥−N
+(ρDρ − z)u ∈ AN( ¯X)
+∀ N ∈ R,
+where we define Dρ = i−1∂ρ with respect to any fixed collar neighborhood of ∂ ¯X. Equivalently, there exist
+a(z,j) ∈ C∞(∂ ¯X), (z, j) ∈ E, such that
+u −
+�
+(z,j)∈E
+Im z≥−N
+ρiz(log ρ)ja(z,j) ∈ AN( ¯X)
+∀ N ∈ R.
+(2.6)
+We say that u ∈ A−∞(X) is polyhomogeneous if it is E-smooth for some index set E.
+Suppose now ¯X′ is a compact manifold with boundary, and let ¯X ⊂ ¯X′ be a submanifold with boundary.
+Suppose that the boundaries of ¯X decomposes into two non-empty sets
+∂ ¯X = ∂− ¯X ⊔ ∂+ ¯X,
+∂− ¯X = ∂ ¯X \ ∂ ¯X′,
+∂+X = ∂ ¯X′;
+(2.7)
+we consider ∂+ ¯X to be a boundary ‘at infinity’, while ∂− ¯X is an interior, ‘artificial’ boundary. Concretely,
+this means that we define (by a slight abuse of notation)
+Vb( ¯
+X) := {V |X : V ∈ Vb( ¯X′)},
+Vsc(X) := {V |X : V ∈ Vsc( ¯
+X′)};
+these vector fields are b or scattering at infinity, but are unrestricted at ∂− ¯X. A typical example is ¯X′ = Rn
+and ¯X = {r ≥ 1} ⊂ ¯X′, in which case ∂− ¯X = {r = 1}, while ∂+ ¯X = ∂X′ is the boundary (at infinity) of
+Rn. See Figure 2.
+∂− ¯X
+∂+ ¯X = ∂ ¯X′
+Figure 2. A typical example of the setting (2.7):
+¯X (dark gray) is a submanifold of ¯X′
+(the union of the dark and light gray regions) with two boundary components ∂+ ¯X = ∂ ¯X′
+and ∂− ¯X ⊂ ( ¯X′)◦. We then consider function spaces such as ¯Hs,ℓ
+b ( ¯X), which measures b-
+regularity of degree s at ∂+ ¯X (with decay rate ℓ), and standard regularity (regularity with
+respect to incomplete vector fields) at ∂− ¯X.
+Then there are now two natural classes of Sobolev spaces: those consisting of extendible distributions,
+¯Hs,ℓ
+• ( ¯
+X) := {u| ¯
+X◦ : u ∈ Hs,ℓ
+• ( ¯X′)},
+• = b, sc,
+(2.8)
+
+14
+LILI HE
+and those consisting of supported distributions,
+˙Hs,ℓ
+• ( ¯X) := {u: u ∈ Hs,ℓ
+• ( ¯X′), supp u ⊂ ¯X}.
+(2.9)
+Away from ∂− ¯X, these are the same as the standard spaces Hs,ℓ
+• ( ¯X); thus, the extendibility/support con-
+siderations arise only at ∂− ¯X, not at ∂+ ¯X.
+If ¯X is the ‘spatial part’ of a stationary spacetime M = Rt × ¯X with projection π ¯
+X : M → ¯X, and F( ¯X)
+denotes a space of distributions on ¯X such as F( ¯
+X) = ¯H∞,ℓ
+b
+( ¯X) or ˙H−∞,ℓ
+b
+( ¯X) in the setting (2.7), we will
+be interested not only in zero modes F( ¯X) ∼= π∗¯
+XF( ¯X), i.e. t-independent distributions, but also generalized
+zero modes,
+Poly(t)F( ¯X) :=
+�
+k∈N
+Polyk(t)F( ¯X),
+Polyk(t)F( ¯
+X) :=
+
+
+
+k
+�
+j=0
+tjaj : aj ∈ F( ¯X)
+
+
+ .
+(2.10)
+Here, we do not require ¯X to be spacelike or dt to be timelike.
+For the high energy estimates, we will work with semiclassical b-Sobolev spaces. Thus, if ¯X is a manifold
+with boundary, we define Hs,ℓ
+b,h( ¯X) = Hs,ℓ
+b ( ¯X) as a set, but with norm depending on the semiclassical
+parameter h ∈ (0, 1]: if V1, . . . , VN ∈ Vb( ¯
+X) spans Vb( ¯X) over C∞( ¯X), we let
+∥u∥2
+Hk
+b,h(X) :=
+�
+1≤i1,...,ij≤N
+0≤j≤k
+∥(hVi1) · · · (hVij)u∥2
+L2( ¯
+X),
+∥u∥Hk,ℓ
+b,h( ¯
+X) := ∥ρ−ℓu∥Hk
+b,h( ¯
+X),
+(2.11)
+for k ∈ N; for k ∈ R, we define the norm Hk
+b,h( ¯X) by duality and interpolation. (Alternatively, one can
+define Hs,ℓ
+b,h( ¯X) using semiclassical b-pseudodifferential operators, see [42, Appendix A].) On manifolds ¯X
+as in (2.7), one can also define semiclassical spaces ¯Hs,ℓ
+b,h(X) and
+˙Hs,ℓ
+b,h(X) of extendible and supported
+distributions analogously to (2.8)–(2.9).
+3. Initial value problems formalism of Einstein-Maxwell equations
+In this section, we shall discuss the initial value problems formalism for Einstein-Maxwell equations. As
+a general reference, we refer the readers to [9, §6.10]. In §3.1, we reduce the Einstein-Maxwell equations for
+(g, F) in the form (3.1)–(3.2) to the study of the equations for (g, A) of the form (3.9). In §3.2, we formulate
+the initial value problems for Einstein-Maxwell equations and then write the Einstein-Maxwell equations
+with respect to a generalized wave map gauge and Lorenz gauge. In §3.3, we discuss the linearization around
+a special solution of the Einstein-Maxwell equations, as well as the linearization of the gauge-fixed Einstein-
+Maxwell equations.
+3.1. Einstein-Maxwell equations. On a time-oriented 4-dimensional Lorentzian spacetime (M, g) with
+signature (−, +, +, +), the Einstein-Maxwell system, for the metric g and the electromagnetic 2-form F,
+reads
+Ric(g)µν = 2T (g, F)µν,
+(3.1)
+dF = 0,
+δgF = 0.
+(3.2)
+where T (g, F) := FµαF α
+ν − 1
+4gµνFαβF αβ is the energy-momentum tensor associated with the electromagnetic
+2-form F and δgF = −∇αFαβ is the negative divergence. A direct calculation implies that if F solves (3.2),
+then we have δgT (g, F) = 0.
+Assuming now that (g, F) is a smooth solution of (3.1)–(3.2), let i: Σ0 ֒→ M be a spacelike hypersurface
+with future timelike unit normal field N, and let h = g|Σ0 be the induced metric on Σ0 with respect ot g,
+which is thus a Riemannian metric. The electromagnetic initial data on Σ0 are the magnetic field B and
+electric field E. The 1-form B is the form induced on Σ0 by the electromagnetic field F, while E is the
+1-form on Σ0 relative to the future timelike unit normal N to Σ0 in the spacetime metric g. Specifically, the
+electric and magnetic fields E, B, as measured by observers with 4-velocity N, are defined by
+E := −i∗ιNF = ⋆hi∗ ⋆g F,
+B := i∗ιN ⋆g F = ⋆hi∗F
+(3.3)
+where ⋆ is the Hodge star operator.
+
+LINEAR STABILITY OF KERR-NEWMAN FAMILY
+15
+We compute for Tg,F ≡ T (g, F)
+Tg,F(N, N) = 1
+2(|E|2
+h + |B|2
+h),
+Tg,F (N, ·) = ⋆h(B ∧ E)
+on
+T Σ0,
+(3.4)
+where Tg,F (N, N) is the energy density of the electromagnetic field as measured by an observer with velocity
+N.
+We now define the following rotation of F, E, B:
+Fθ := cos(θ)F + sin(θ) ⋆g F,
+(3.5)
+(Eθ, Bθ) := (cos(θ)E − sin(θ)B, sin(θ)E + cos(θ)B).
+(3.6)
+Then Eθ and Bθ are the electric and magnetic field, respectively, corresponding to Fθ. Now we establish the
+following lemma.
+Lemma 3.1. If (g, F) solves the Einstein-Maxwell equations (3.1)–(3.2), then so does (g, Fθ) for all θ ∈ R,
+with Fθ defined in (3.5).
+Proof. According to the facts that ⋆g⋆g = −Id and δg = ⋆gd⋆g when acting on 2-forms, the equations (3.2)
+are equivalent to dF = 0, d ⋆g F = 0. Then it follows that
+dFθ = cos(θ)dF + sin(θ)d ⋆g F = 0,
+δgFθ = ⋆g(cos(θ)d ⋆g F − sin(θ)dF) = 0.
+Since the energy momentum tensor T (g, F) can be rewritten as
+T (g, F)µν = 1
+2gλσ�
+FµλFνσ + (⋆gF)µλ(⋆gF)νσ
+�
+,
+it follows that T (g, Fθ)µν = T (g, F)µν.
+This proves that (g, Fθ) solves the Einstein-Maxwell equations
+(3.1)–(3.2).
+□
+Let us now assume for simplicity that
+M ∼= Rt × Ir × S2,
+Σ0 = {t = 0} ⊂ M,
+(3.7)
+with I ⊂ R a non-empty open interval, and assume that Σ0 is spacelike. If S = {t = t0, r = r0} ⊂ M (with
+t0 ∈ R, r0 ∈ I) is a 2-sphere, we can define the electric and magnetic charges, associated with F solving
+(3.2), by
+Qe(g, F) := 1
+4π
+�
+S
+⋆gF,
+Qm(F) := 1
+4π
+�
+S
+F.
+(3.8)
+By Stokes’ theorem and the equations (3.2) dF = 0, d ⋆g F = 0, Qe and Qm are independent of the specific
+choice of S. In particular, we can take S ⊂ Σ0, and use (3.3) and ⋆h⋆h = Id when acting on 1-form to find
+that
+Qe(g, F) = Qe(h, E) := 1
+4π
+�
+S
+⋆hE = 1
+4π
+�
+S
+⟨E, ν⟩dσ,
+Qm(F) = Qm(B) := 1
+4π
+�
+S
+⋆hB = 1
+4π
+�
+S
+⟨B, ν⟩dσ,
+with ν the outward pointing unit normal to S (with respect to h) and dσ the volume element.
+Since Ir × S2 is not a simply connected domain, dF = 0 does not imply that F = dA for some 1-form
+A globally. In fact, the de Rham cohomology group H2
+dR(Ir × S2, R) ∼= R and F �→ Qm(F) induces an
+isomorphism of the de Rham cohomology group H2
+dR(Ir × S2, R). Therefore, the vanishing of Qm(F) is
+equivalent to the condition that F = dA for some 1-form A. Now we will show that the vanishing of Qm(F)
+can always be arranged by the rotation transformation (3.5).
+Lemma 3.2. In the setting (3.7), and given any (g, F) solving (3.2), there exists θ ∈ R such that Qm(Fθ) =
+0, with Fθ defined in (3.5); for such θ, we have
+Qe(g, Fθ)2 = Qm(F)2 + Qe(g, F)2.
+Proof. With Fθ = cos(θ)F + sin(θ) ⋆g F, we have Qm(Fθ) = cos(θ)Qm(F) + sin(θ)Qe(g, F). Choosing θ ∈ R
+such that (cos θ, sin θ) ⊥ (Qm(F), Qe(g, F)) finishes the proof.
+□
+
+16
+LILI HE
+In view of Lemma 3.2, we may assume Qm(F) = 0, equivalently F = dA for some 1-form A. This reduces
+the study of the Einstein-Maxwell equations on M in the form (3.1)–(3.2) to the study of the following
+system in (g, A)
+Ric(g) = 2T (g, dA),
+δgdA = 0,
+(3.9)
+which we continue to call Einstein-Maxwell equations. In this work, we will study the Einstein-Maxwell
+equations of the above form (3.9).
+3.2. Initial value problems for the nonlinear system. Initial data for the Einstein-Maxwell equations
+(3.9) are 5-tuples
+(Σ0, h, k, E, B),
+(3.10)
+where Σ0 is a smooth 3-manifold with Riemannian metric h, k is a symmetric 2-tensor, and E, B are 1-forms
+on Σ0. In the initial value problem, one seeks a solution (M, g, A) to (3.9) with data (Σ0, h, k, E, B) in the
+sense that Σ0 ֒→ M is a spacelike embedding hypersurface with respect to g, and h, k are respectively the
+induced metric and second fundamental form of Σ0 with respect to g, and F = dA induces the given fields
+E, B on Σ0 via (3.3). We define the initial data map
+τ : (g, F) �→ (h, k, E, B)
+(3.11)
+if (g, F) induces the data (h, k, E, B) at Σ0 in the above sense. We note that the data (h, k, E, B) cannot be
+prescribed freely and they must satisfy the following constraint equations, which are obtained by evaluating
+Gg(Ric(g) − 2T (g, F)) on (N, N) where Gg = Id − 1
+2gtrg and N is future timelike unit normal to Σ0, and
+on (N, X) with X ∈ T Σ0, and pulling back dF = 0, d ⋆g F = 0 to Σ0, and finally adding the constraint that
+the magnetic charge Qm(F) = Qm(B) = 0 vanishes.
+Rh − |k|2
+h + (trhk)2 = 2(|E|2
+h + |B|2
+h),
+δhk + dtrhk = 2 ⋆h (B ∧ E);
+(3.12)
+δhE = 0,
+δhB = 0;
+(3.13)
+Qm(B) = 1
+4π
+�
+S
+⋆hB = 0.
+(3.14)
+The constraints (3.12)–(3.13) are necessary and sufficient conditions for the local solvability of the Einstein-
+Maxwell equations (see [9, §6.10]).
+Remark 3.3. In the setting Σ0 ∼= Ir × S2 as in (3.7), by the discussion around (3.5) and Lemma 3.2, the
+constraint equations are invariant upon replacing (E, B) by (Eθ, Bθ), and for suitable θ, we indeed have
+Qm(Bθ) = 0, thus we can solve the Einstein-Maxwell system for (g, A) with initial data (Eθ, Bθ), and then
+F := (dA)−θ yields a solution of (3.1)–(3.2) with initial data (E, B) for F.
+Under the additional assumption
+�
+S ⋆hB = 0, one can solve the Einstein-Maxwell equations of the form
+(3.9) for (g, A), with electromagnetic tensor then given by F = dA. Due to the diffeomorphism invariance
+of Einstein-Maxwell equations, one has the gauge freedom and thus can fix a gauge. Here, we employ the
+generalized wave map gauge. Concretely, we fix a Lorentzian background metric g0 and define the gauge
+1-form
+ΥE(g; g0) := g(g0)−1δgGgg0.
+(3.15)
+In local coordinates, the above gauge 1-form can be written as
+ΥE(g; g0)α = gακgµν(Γ(g)κ
+µν − Γ(g0)κ
+µν).
+(3.16)
+We note that ΥE(g) ≡ 0 if and only if the identity map Id: (M, g) → (M, g0) is a wave map, and thus
+ΥE(g) ≡ 0 gives rise to the wave map gauge (see [17]). The generalized wave map gauge is defined as
+�ΥE(g; g0) = ΥE(g; g0) − θ(g; g0) = 0
+(3.17)
+where θ is a 1-form linear in g, satisfying that θ(g0; g0) = 0, and independent of the derivatives of g. We also
+point out that if the background metric g0 is Minkowski metric, one obtains the generalized wave coordinates
+for θ(g; g0) ̸= 0 (see [25]) and wave coordinates for θ(g; g0) = 0 (see [9, §6]), respectively.
+The system (3.9) has an additional gauge freedom, namely, Einstein-Maxwell equations are invariant
+under the gauge transformation
+(g, A) �→ (φ∗g, φ∗A + da).
+(3.18)
+
+LINEAR STABILITY OF KERR-NEWMAN FAMILY
+17
+for any diffeomorphism φ : M → M and smooth function a ∈ C∞(M). A typical method of fixing a gauge
+for A is by demanding δgA = 0, called Lorenz gauge; it is in fact convenient to note δg = −trgδ∗
+g and to
+introduce the gauge function
+ΥM(g, A; g0) := trgδ∗
+g0A,
+(3.19)
+where g0 is a fixed background metric as above; thus, ΥM(g, A) equals −δgA up to terms of order 0 in A.
+The generalized Lorenz gauge is defined as
+�ΥM(g, A; g0, A0) = ΥM(g, A; g0) − ΥM(g, A0; g0) = 0
+(3.20)
+where A0 is a fixed 1-form.
+Remark 3.4. We claim that any solution (g, A) to the Einstein-Maxwell equations can be transformed into
+another solution (φ∗g, φ∗A + da) which satisfies the generalized wave map gauge and generalized Lorenz
+gauge conditions. Concretely, we define new coordinates (yα) by solving
+�
+□gyα − θ(g; g0)β∇βy = 0
+(yα, dyα)|Σ0 = (xα, dxα)|Σ0.
+Then the pull back (φ∗g, φ∗A) of (g, A) with respect to the diffeomorphism φ(y) = x satisfies the generalized
+wave map gauge condition and is also a solution to Einstein-Maxwell equations. Then with (φ∗g, φ∗A), one
+continues to solve the equation ΥM(φ∗g, φ∗A + da; g0) = ΥM(φ∗g, A0; g0), which is a linear scalar wave
+equation for a, and then (φ∗g, φ∗A + da) satisfies the generalized Lorenz gauge condition and solves the
+Einstein-Maxwell equations.
+We now consider the gauge-fixed system
+P E(g, A) := Ric(g) − ˜δ∗
+g �ΥE(g; g0) − 2T (g, dA) = 0
+(3.21)
+P M(g, A) := δgdA − d�ΥM(g, A; g0, A0) = 0
+(3.22)
+where ˜δ∗
+g is zero order modification of δ∗
+g, i.e., ˜δ∗
+g = δ∗
+g + B with B ∈ C∞(M; Hom(S2T ∗M, T ∗M)), and
+(δ∗
+gu)µν = 1
+2(uµ;ν + uν;µ) is the symmetric gradient. Now, (3.21)–(3.22) is a quasilinear hyperbolic system
+for (g, A) (which is principally scalar if one multiplies the first equation by 2).
+In order to relate the initial value problem for the Einstein-Maxwell system for (g, A) to that of the
+quasilinear hyperbolic system (3.21)–(3.22), we need the following lemma.
+Lemma 3.5. Suppose (g, A) solves the gauge-fixed Einstein-Maxwell equation
+P(g, A) = (P E(g, A), P M(g, A)) = 0
+and attains the given initial data (g0, g1, A0, A1) at the initial hypersurface Σ0 = {t = 0} which satisfy
+the constraint equations (3.12)–(3.14). If (g, A) satisfies the generalized wave map gauge and Lorenz gauge
+conditions initially at Σ0, that is ,
+�ΥE(g; g0) = 0,
+�ΥM(g, A; g0, A0) = 0
+at
+Σ0 = {t = 0},
+(3.23)
+then (g, A) solves the Einstein-Maxwell equations with the same initial data.
+Proof. We first prove that if (g, A) solves P(g, A) = 0 and satisfies the constraint equations and the gauge
+conditions at Σ0, then
+L∂t �ΥE(g; g0) = 0,
+L∂t �ΥM(g, A; g0, A0) = 0
+at
+Σ0.
+(3.24)
+Since the constraint equation (3.12) is equivalent to evaluating Gg(Ricg − 2T (g, dA)) on (N, N) and (N, X)
+on Σ0 where N is the future timelike unit normal to Σ0 and X ∈ T Σ0, then evaluating GgP M(g, A) on
+(N, X) and (N, N) and using the fact that �ΥE(g, A; g0) = 0 at Σ0 yield L∂t �ΥE(g, A; g0) = 0 at Σ0. As for
+the Maxwell part, the constraint equation implies that the evaluation of δgdA on N is zero at Σ0 and thus
+we have L∂t �ΥM(g, A; g0, A0) = 0 at Σ0.
+Applying the negative divergence operator δg on the equation P M(g, A) = 0 yields
+δgd�ΥM(g, A; g0, A0) = 0.
+
+18
+LILI HE
+Then we obtain a linear hyperbolic system for �ΥM(g, A; g0, A0) with vanishing initial data.
+Therefore,
+�ΥM(g, A; g0, A0) = 0 on M, which implies that (g, A) solve the Maxwell part δgdA = 0 of the Einstein-
+Maxwell equations. According to the second Bianchi identity (δgGgRic(g) = 0) and the fact that δgdA = 0
+(and thus δgGgT (g, dA) = δgT (g, dA) = 0), applying first the trace reversal operator Gg = Id − 1
+2gtrg and
+then the negative divergence operator δg on P E(g, A) = 0 yields
+δgGg˜δ∗
+g �ΥE(g; g0) = 0.
+We again obtain a linear hyperbolic system for �ΥE(g; g0) with vanishing initial data, and thus �ΥE(g; g0) = 0
+on M, which implies that (g, A) solves Einstein part Ric(g)−2T (g, dA) = 0 of the Einstein-Maxwell equations.
+This completes the proof.
+□
+Owing to Lemma 3.5, in order to solve the initial value problem for Einstein-Maxwell equations, it suffices
+to construct the gauged initial data satisfying the conditions (3.23) and solve the gauge-fixed Einstein-
+Maxwell equations (3.21)–(3.22) with the constructed gauged initial data. We postpone the construction of
+the gauged initial data to §4.4.
+3.3. Initial value problems for the linearized system. For our stability arguments, we are interested
+in understanding long time behavior of a general solution to the linearize Einstein-Maxwell equations.
+Thus, suppose (g, F) is a solution to the Einstein-Maxwell system (3.1)–(3.2) on a spacetime M = R×Σ0.
+Then around (g, F), the linearized Einstein-Maxwell equations are defined as
+DgRic(˙g) − 2Dg,FT (˙g, ˙F) := d
+ds
+���
+s=0
+�
+(Ric − 2T )(g + s˙g, F + s ˙F)
+�
+= 0,
+d ˙F = 0,
+Dg,F (δ(·)(·))(˙g, ˙F) := d
+ds(δg+s ˙g(F + s ˙F))
+���
+s=0 = 0.
+(3.25)
+Suppose Σ0 is spacelike for g. We denote by (˙h, ˙k, ˙E, ˙B) the linearization of the initial data map τ around
+(g, F). We note that (˙h, ˙k, ˙E, ˙B) are the initial data we impose in the initial value problem for the linearized
+Einstein-Maxwell equations (3.25). As in the case of the non-linear Einstein-Maxwell equations where the
+initial data should satisfy the constraint equations, the initial data (˙h, ˙k, ˙E, ˙B) for the linearized Einstein-
+Maxwell equations are required to satisfy the linearized constraint equations (see [9, §7.2]), which is defined
+as the linearization of (3.12)–(3.13) around the data (h, k, E, B) induced by (g, F).
+Motivated from (3.5) and (3.6), we induce the following transformation
+˙Fθ := cos(θ) ˙F + sin(θ) d
+ds
+���
+s=0
+�
+⋆g+s ˙g (F + s ˙F)
+�
+,
+(3.26)
+( ˙Eθ, ˙Bθ) := (cos(θ) ˙E − sin(θ) ˙B, sin(θ) ˙E + cos(θ) ˙B).
+(3.27)
+Then we have
+(h, k, ˙Eθ, ˙Bθ) = D(g,Fθ)τ(˙g, ˙Fθ).
+Now we prove the analogue of Lemma 3.1 at the linearized level.
+Lemma 3.6. If (˙g, ˙F) solves the linearized Einstein-Maxwell equations (3.25) linearized around (g, F), then
+(˙g, ˙Fθ + cFθ+ π
+2 ) solves (3.25) linearized around (g, Fθ) for any c ∈ R.
+Proof. Since dF = 0, d
+�
+d
+ds
+���
+s=0
+�
+⋆g+s ˙g (F + s ˙F)
+��
+= 0, it follows that
+d( ˙Fθ + cFθ+ π
+2 ) = cos(θ)d ˙F + sin(θ)d
+� d
+ds
+���
+s=0
+�
+⋆g+s ˙g (F + s ˙F)
+��
++ cdFθ+ π
+2 = 0.
+We next calculate
+d
+ds
+���
+s=0
+�
+⋆g+s ˙g
+�
+Fθ + s( ˙Fθ + cFθ+ π
+2 )
+��
+= d
+ds
+���
+s=0
+�
+⋆g+s ˙g
+�
+Fθ + s ˙Fθ
+��
++ c ⋆g Fθ+ π
+2
+= cos(θ) d
+ds
+���
+s=0
+�
+⋆g+s ˙g (F + s ˙F)
+�
+− sin(θ) ˙F + c ⋆g Fθ+ π
+2
+and this implies
+d
+ds
+���
+s=0
+�
+δg+s ˙g
+�
+Fθ + s( ˙Fθ + cFθ+ π
+2 )
+��
+= 0.
+
+LINEAR STABILITY OF KERR-NEWMAN FAMILY
+19
+According to the calculation for T (g, F) in the proof of Lemma 3.1, with (F + s ˙F)θ = cos(θ)(F + s ˙F) +
+sin(θ) ⋆g+s ˙g (F + s ˙F) we have
+T (g + s˙g, F + s ˙F) = T (g + s˙g, (F + s ˙F)θ) = T (g + s˙g, Fθ + s ˙Fθ) + O(s2)
+and thus
+D(g,Fθ)T (˙g, ˙Fθ + cFθ+ π
+2 ) = D(g,F )T (˙g, ˙F) + c
+2gλσ((Fθ)µλ(Fθ+ π
+2 )νσ + (Fθ+ π
+2 )µλ(Fθ)νσ)
++ c
+2gλσ((⋆gFθ)µλ(⋆gFθ+ π
+2 )νσ + (⋆gFθ+ π
+2 )µλ(⋆gFθ)νσ)
+= D(g,F )T (˙g, ˙F).
+This implies that (˙g, ˙Fθ + cFθ+ π
+2 ) solves the linearization of the equation Ric − 2T = 0 around (g, Fθ).
+□
+If M ∼= R × Σ0, we can define linearized charges as follows
+˙Qe(˙g, ˙F) := 1
+4π
+�
+S
+d
+ds
+���
+s=0 ⋆g+s ˙g (F + s ˙F),
+˙Qm( ˙F) := 1
+4π
+�
+S
+d
+ds
+���
+s=0(F + s ˙F) = 1
+4π
+�
+S
+˙F.
+(3.28)
+If (˙g, ˙F) solves the linearized Einstein-Maxwell equations, by Stokes’ theorem and the linearized equations
+(3.25) again, ˙Qe and ˙Qm are independent of the specific choice of S. In particular, we can take S ⊂ Σ0, and
+use the linearization of (3.3) around (g, F) to find that
+˙Qe(˙g, ˙F) = ˙Qe(˙h, ˙E) := 1
+4π
+�
+S
+d
+ds
+���
+s=0 ⋆hs Es,
+˙Qm( ˙F) = Qm( ˙B) := 1
+4π
+�
+S
+d
+ds
+���
+s=0 ⋆hs Bs.
+We note that the analogue of Lemma 3.2 holds for the linearized charges.
+Lemma 3.7. In the setting (3.7), and given any (˙g, ˙F) solving the linearized Einstein-Maxwell equations
+around (g, F), there exist c, θ ∈ R such that Qm(Fθ) = 0 and ˙Qm( ˙Fθ + cFθ+ π
+2 ) = 0, with Fθ+ π
+2 and ˙Fθ
+defined in (3.5) and (3.26), respectively.
+Proof. First, according to Lemma 3.2, there exists θ ∈ R such that Qm(Fθ) = 0. Since
+Q2
+m(Fθ+ π
+2 ) = Qe(g, Fθ)2 = Qm(F)2 + Qe(g, F)2 ̸= 0,
+we can choose
+c = −
+˙Qm( ˙Fθ)
+Qm(Fθ+ π
+2 ),
+with which we have ˙Qm( ˙Fθ + cFθ+ π
+2 ) = ˙Qm( ˙Fθ) + cQm(Fθ+ π
+2 ) = 0. This finishes the proof.
+□
+Therefore, in addition to the linearized constraint equation, we may assume that ˙Qm( ˙F) = 0 which is
+an analogue of condition (3.14) and can always be arranged by Lemma 3.7. Concretely, without loss of
+generality, suppose that F satisfies Qm(F) = 0. Then for any (˙h, ˙k, ˙E, ˙B) satisfying the linear constraint
+equations, by choosing a suitable c ∈ R such that ˙Qm( ˙Bθ) − cQm(E) = 0, the condition (3.14) is arranged.
+In the context of the present paper, given initial data satisfying the linearized constraints and having
+vanishing linearized magnetic charge, we solve the following linearized Einstein-Maxwell system
+L (˙g, ˙A) =
+�
+L1(˙g, ˙A), L2(˙g, ˙A)
+�
+= 0,
+(3.29)
+where
+L1(˙g, ˙A) = DgRic(˙g) − 2D(g,dA)T (˙g, ˙A),
+L2(˙g, ˙A) = D(g,A)(δ(·)d(·))(˙g, ˙A),
+(3.30)
+By choosing (g, A), around which we linearize the Einstein-Maxwell equations, as the background Lorentzian
+metric and 1-form in the gauge 1-form ΥE, gauge source θ and gauge function ΥM, one can consider the
+initial value problem for the corresponding linearized gauge-fixed Einstein-Maxwell equations
+LE(˙g, ˙A) := Dg(Ric)(˙g) − ˜δ∗
+gDg
+�
+ΥE − θ
+�
+(˙g; g) − 2D(g,dA)T (˙g, d ˙A) = 0,
+LM(˙g, ˙A) := D(g,A)(δ(·)d(·))(˙g, ˙A) − d
+�
+D(g,A)ΥM(˙g, ˙A; g) − DgΥM(˙g, A; g)
+�
+= 0.
+(3.31)
+
+20
+LILI HE
+We recall from [84, §7.5] and [33, §3] that
+(DgRic)(˙g) = −1
+2□g ˙g − δ∗
+gδgGg ˙g + Rg(˙g),
+(3.32)
+where □g = ∇α∇α is the tensor wave operator, Rg(˙g)µν = R(g)κ
+λ
+µν ˙gκλ + 1
+2(Ric(g)κ
+µ ˙gνκ + Ric(g)κ
+ν ˙gµκ), and
+DgΥE(˙g; g) = −δgGg ˙g.
+(3.33)
+Since ˜δ∗
+g is a zero order modification of δ∗
+g and D(g,dA)T (˙g, ˙A) is of order 0 in ˙g and order 1 in A, it follows
+that the first equation in (3.31) has the principal part − 1
+2□g ˙g. We define the linearized generalized wave
+map gauge as
+Dg �ΥE(˙g; g) = Dg(ΥE − θ)(˙g; g) = 0.
+(3.34)
+Analogously, we define the linearized generalized Lorenz gauge as
+D(g,A) �ΥM(˙g, ˙A; g, A) = D(g,A)ΥM(˙g, ˙A; g) − DgΥM(˙g, A; g) = −δg ˙A = 0,
+(3.35)
+and the second equation in (3.31) also has principal part (δgd+dδg) ˙A, and only involves up to first derivatives
+of ˙g. Thus, the linearized system (3.31) is linear hyperbolic system for (˙g, ˙A), whose initial value problem
+we are able to solve.
+As discussed in the non-linear case, in order to relate the initial value problem for the linearized Einstein-
+Maxwell system for (˙g, ˙A) to that of the linear hyperbolic system (3.31), we need the an analogue of Lemma
+3.5 at the linearized level.
+Lemma 3.8. Suppose (˙g, ˙A) solves the linearized gauge-fixed Einstein-Maxwell equation (3.31)
+L(˙g, ˙A) = (LE(˙g, ˙A), LM(˙g, ˙A)) = 0
+and attains the given initial data (˙g0, ˙g1, ˙A0, ˙A1) at the initial hypersurface Σ0 = {t = 0} which satisfy the
+linearized constraint equations. If (˙g, ˙A) satisfies the linearized generalized wave map gauge and Lorenz gauge
+conditions initially at Σ0, that is,
+Dg �ΥE(˙g; g) = 0,
+δg ˙A = 0
+at
+Σ0,
+(3.36)
+then (˙g, ˙A) solves the Einstein-Maxwell equations with the same initial data.
+Proof. The proof is completely analogous to that of Lemma 3.5.
+□
+Therefore, in order to solve the initial value problem for linearized Einstein-Maxwell equations, we again
+need to construct the gauged initial data satisfying the conditions (3.36). We postpone the construction of
+the gauged initial data to §4.4.
+Finally, we discuss the pure gauge solution generated from the invariance of Einstein-Maxwell system under
+the gauge transformation introduced in the non-linear setting (g, A) �→ (φ∗g, φ∗A + da). Linearizing around
+(φ, a) = (Id, 0) the gauge transformation on a solution (˙g, ˙A) of (3.29) yields the following transformation
+acting on (˙g, ˙A)
+(˙g, ˙A) �→ (˙g + LV g, ˙A + LV A + da),
+(3.37)
+and the linearized Einstein-Maxwell system is invariant under the above transformation (3.37).Equivalently,
+C∞(M; T ∗M) × C∞(M) ∋ (ω, a) acts on (˙g, ˙A) by the same formula via the identification V = ω♯. We note
+that Lω♯g = 2δ∗
+gω. For notational convenience, we make the following definition:
+Definition 3.9. For a vector field V and a tensor T , define ˜LT V := LV T .
+According to Remark 3.4 in the non-linear setting, any solution (g, A) can be transformed into another
+one satisfying the generalized wave map and Lorenz gauge conditions. Here we state its analogue in the
+linear case.
+Remark 3.10. Any solution (˙g, ˙A) to the Einstein-Maxwell equations can be transformed into another solution
+(˙g + LV g, ˙A + LV A + da) which satisfies the linearized generalized wave map gauge and Lorenz gauge
+conditions. Concretely, we first solve define new coordinates (yα) by solving the equation
+Dg �ΥE(˙g + LV g; g) = 0
+(3.38)
+
+LINEAR STABILITY OF KERR-NEWMAN FAMILY
+21
+for the vector field V . The equation (3.38) is a wave equation since its principal part is given by −□gV ♭.
+Then we solve
+δg( ˙A + LV A + da) = 0,
+(3.39)
+whose principal part is −□ga, for the scalar function a. If one chooses trivial Cauchy data for V and a on
+Σ0, then (˙g + LV g, ˙A + LV A + da) solves (3.29), satisfies the linearized generalized wave map and Lorenz
+gauge conditions, and attains the same initial data (˙g0, ˙g1, ˙A0, ˙A1) as (˙g, ˙A).
+4. Kerr-Newman black holes
+In §4.1, we introduce the Reissner-Nordstr¨om (RN) family of subextremal, non-rotating black holes, which
+is parametrized by (m0, 0, Q0) ∈ R × R3 × R with m0 > 0, |Q0| < m0. In §4.2, we introduce the Kerr-
+Newman (KN) family of black holes with parameters (m, a, Q) ∈ R× R3 × R close to (m0, 0, Q0) and realize
+this family of slowly rotating KN metric as a smooth family of stationary metrics on a fixed manifold M. In
+§4.3, we introduce several vector bundles over M and analyze the structure of differential operators, which
+will be used in the subsequent sections, near infinity. In §4.4, we will discuss how to construct the gauged
+Cauchy data for the gauge-fixed Einstein-Maxwell system from the given initial data (h, k, E, B) and also
+consider its linearized version.
+4.1. The Reissner-Nordstr¨om family. Fix a set of parameters
+b0 = (m0, 0, Q0) ∈ R × R3 × R,
+m0 > 0, |Q0| < m0,
+(4.1)
+where the parameter m0 > 0 denote the mass and Q0 denotes the charge, the Reissner-Nordstr¨om (RN)
+solution to the Einstein-Maxwell equations is then described by the spherically symmetric and static RN
+metric
+gb0 = −µb0(r) dt2 + µb0(r)−1 dr2 + r2/g,
+(4.2)
+where /g is the round metric on S2 and
+µb0(r) = 1 − 2m0
+r
++ Q2
+0
+r2 ,
+(4.3)
+and the RN 4-potential for the electromagnetic field
+˘Ab0 = Q0
+r dt,
+˘Fb0 = d ˘Ab0 = −Q0
+r2 dr ∧ dt.
+(4.4)
+(We reserve the name Ab0 for the 4-potential that we eventually use, see (4.23).) When |Q0| < m0, the
+expression µb0 has two distinct positive roots m0 −
+�
+m2
+0 − Q2
+0 = rc < rb0 = m0 +
+�
+m2
+0 − Q2
+0. There is a
+Cauchy horizon at r = rc and an event horizon H at rb0. The expressions (4.2) and (4.4) are valid in the
+static region, which sometimes is also called exterior region or domain of outer communication
+M = Rt × X,
+X = (rb0, ∞)r × S2.
+(4.5)
+The singularity of the expression (4.2) at r = rb0 is a coordinate singularity and can be removed by change
+of coordinates. We define a new new coordinate system (t0, r, θ, ϕ) which is called incoming Eddington-
+Finkelstein coordinates and make use of the function t0
+t0 = t + rb0,∗,
+drb0,∗
+dr
+=
+1
+µb0
+,
+(4.6)
+where rb0,∗ is called the tortoise coordinate or Regge-Wheeler radial coordinate. In this coordinate system,
+the RN metric is given by
+gb0 = −µb0(r)dt2
+0 + 2 dt0dr + r2/g.
+(4.7)
+Now the expression (4.7) for gb0 extends analytically beyond the event horizon H = {r = rb0}, and thus gb0
+is smooth and non-degenerate metric on the extended manifold
+M = Rt0 × X ⊃ M,
+X = [r−, ∞)r × S2 ⊃ X,
+(4.8)
+where r− ∈ (rc, rb0). At the same time, we can take the RN 4-potential to be Q0r−1 dt0, which differs from
+˘Ab0 by an exact form d(
+�
+Q0r−1µ−1 dr).
+
+22
+LILI HE
+Since dt is timelike when r is sufficiently large, to capture this useful structure, we introduce another
+coordinate
+tχ0 := t +
+�
+χ0(r)
+µb0(r) dr,
+(4.9)
+where χ0(r) ∈ C∞(R) is smooth, is identically 1 for r ≤ 4rb0/3, and vanishes for r ≥ 3rb0/2. Therefore, tχ0
+is a smooth interpolation between rb0 near event horizon and t near infinity r = ∞ with a suitable choice of
+the constant of integration.
+In order to describe the waves near the future event horizon and future null infinity (and in between), we
+introduce the following function
+t∗ = tb0,∗ := t + rb0,∗χ0(r) − rb0,∗(1 − χ0(r)),
+(4.10)
+where χ0(r) is defined as in (4.9); it smoothly interpolates between t+rb0,∗ near the event horizon and t−rb0,∗
+near null infinity. Later on, we will study forcing problems for wave equations of the type □g(m0,0,Q0)u = f
+on the manifold
+Rt∗ × X,
+X = [r−, ∞)r × S2,
+where f is supported in t∗ ≥ 0.
+Recall the definition of R3 from (2.2), now we compactify X in a similar manner
+¯X :=
+�
+X ⊔ ([0, r−1
+b0 )ρ × S2)
+�
+/ ∼
+(4.11)
+where the relation ∼ identifies a point (r, ω) ∈ X = [r−, ∞)r × S2 with the point (ρ, ω) where ρ = r−1.
+Namely, we add the boundary {ρ = 0} ∼= S2 at infinity. Thus, ¯X = {r ≥ r−} ⊂ R3, and we let ∂− ¯X =
+r−1(r−), ∂+ ¯X = ∂R3 = {ρ = 0} ⊂ ¯X. We note that Rt∗ × ∂ ¯X has two components,
+Σfin := Rt∗ × ∂− ¯X
+(4.12)
+(which is a spacelike hypersurface inside the black hole) and Rt∗ ×∂+X (which is future null infinity, typically
+denoted I +). See Figure 3 for the level sets of t, t0, tχ0, t∗.
+t
+Σfin
+t0
+tχ0
+t∗
+i+
+i−
+i0
+I +
+I −
+H−
+H+
+Figure 3.
+Penrose diagram of the Reissner-Nordstr¨om metric (including future/past null
+infinity I ±, the future/past event horizon H±, spacelike infinity i0, and future/past timelike
+infinity i±), together with the level sets of the static time function t, of the null coordinate
+t0 from (4.6), of the coordinate tχ0 from (4.9), and of the function t∗ from (4.10). Also
+shown are the boundaries Σfin = Rt∗ × ∂− ¯X and Rt∗ × ∂+ ¯X (future null infinity I +).
+
+LINEAR STABILITY OF KERR-NEWMAN FAMILY
+23
+4.2. The slowly rotating Kerr-Newman family. Let b0 = (m0, 0, Q0) be a parameter of a RN black
+hole. Consider black hole parameters
+b = (m, a, Q) ∈ R × R3 × R,
+(4.13)
+with a ∈ R3 denoting the angular momentum. If a = |a| ̸= 0, choose adapted polar coordinates (θ, ϕ) on
+S2 such that ˆa = a/|a| is defined by θ = 0, and the vector field ∂ϕ generates counterclockwise rotation
+around a/|a|, with R3 carrying the standard orientation. For a = 0, adapted polar coordinates are simply
+any polar coordinates. A very convenient coordinate system for the Kerr-Newman metric is Boyer-Lindquist
+coordinates (t, r, θ, ϕ) (see [5]), in which the Kerr-Newman metric takes the form
+gBL
+b
+= −∆b
+ρ2
+b
+(dt − a sin2 θ dϕ)2 + ρ2
+b
+�dr2
+∆b
++ dθ2�
++ sin2 θ
+̺2
+b
+�
+a dt − (r2 + a2)dϕ
+�2,
+(gBL
+b )−1 = −
+1
+∆bρ2
+b
+�
+(r2 + a2)∂t + a∂ϕ
+�2 + ∆b
+ρ2
+b
+∂2
+r + 1
+ρ2
+b
+∂2
+θ +
+1
+ρ2
+b sin2 θ (∂ϕ + a sin2 θ ∂t)2,
+∆b = r2 − 2mr + Q2 + a2,
+ρ2
+b = r2 + a2 cos2 θ,
+a = |a|.
+(4.14)
+In the Boyer-Lindquist coordinates, the KN 4-potential reads
+˘Ab = Qr
+ρ2
+b
+(dt − a sin2 θ dϕ)
+and the corresponding electromagnetic 2-form is
+˘Fb = − Q
+ρ4
+b
+�
+(r2 − a2 cos2 θ)dr ∧ (dt − a sin2 θ dϕ)
+�
++ Qr
+ρ4
+b
+�
+2a cosθ sin θ dθ ∧ (a dt − (r2 + a2) dϕ)
+�
+.
+We note that (gBL
+b , ˘Ab) is a solution to the Einstein-Maxwell equations.
+In the initial value problem for the Einstein-Maxwell equations, one places asymptotically flat initial data
+on a Cauchy hypersurface
+Σ0 ⊂ M,
+which is spacelike and equal to t−1(0) where r is large. We introduce the following coordinates
+t = t +
+�
+χ1(r)
+�r2 + a2
+∆b
+�
+1 −
+∆b
+r2 + a2
+�
+dr,
+ϕ1 = ϕ +
+�
+χ1(r) a
+∆b
+dr
+(4.15)
+where χ1(r) is a nonnegative smooth function such that χ1(r) = 1 for r ≤ 3m and χ1(r) = 0 for r ≥ 4m. In
+the new coordinates (t, r, θ, ϕ1), gBL
+b
+and (gBL
+b )−1 are smooth at event horizon, and furthermore
+(gBL
+b )−1(dt, dt) = −1 + χ2 − 1
+ρ2
+b
+�(r2 + a2)2
+∆b
+− r2 + a2�
+=
+
+
+
+
+
+−1,
+r ≤ 3m
+≤ −1,
+3m ≤ r ≤ 4m
+(gBL
+b )−1(dt, dt),
+r ≥ 4m
+.
+By choosing a proper integration constant we have t = t for r ≥ 4m. Then we impose asymptotically flat
+initial data on the Cauchy hypersurface
+Σ0 := {t = 0} ⊂ M.
+(4.16)
+In the present paper, we will focus on parameters (m, a, Q) close to (m0, 0, Q0); in particular, we are
+looking at slowly rotating (|a| ≪ m + |Q|) Kerr-Newman black holes. As in the RN case, the singularity of
+the expression (4.14) of the KN metric at
+r = rb := m +
+�
+m2 − (|Q|2 + |a|2).
+(4.17)
+is a coordinate singularity and can be removed by introducing new coordinates. We let
+tb,χ = t +
+� r2 + a2
+∆b
+χ(r) dr,
+ϕb,χ = ϕ +
+�
+a
+∆b
+χ(r) dr.
+(4.18)
+
+24
+LILI HE
+where χ(r) ∈ C∞(R) is identically 1 near r ≤ m +
+�
+m2 − Q2. Then the metric gBL
+b
+and its inverse take
+the form
+gBL
+b
+= −∆b
+ρ2
+b
+(dtb,χ − a sin2 θ dϕb,χ)2 + 2χ(dtb,χ − a sin2 θ dϕb,χ)dr
++ ρ2
+b
+∆b
+(1 − χ2)dr2 + ρ2
+b dθ2 + sin2 θ
+ρ2
+b
+�
+a dtb,χ − (r2 + a2)dϕb,χ
+�2,
+(gBL
+b )−1 = ρ−2
+b
+�
+∆b∂2
+r + χ2 − 1
+∆b
+�
+(r2 + a2)∂tχ + a∂ϕb,χ
+�2
++ ∂2
+θ +
+1
+sin2 θ
+�
+∂ϕb,χ + a sin2 θ∂tχ
+�2
++ 2χ
+�
+(r2 + a2)∂tχ + a∂ϕb,χ
+�
+∂r
+�
+,
+(4.19)
+which is smooth and non-degenerate on the manifold Mb = Rtb,χ ×[r−, ∞)r ×S2
+θ,ϕb,χ with r− defined in §4.1.
+Correspondingly, we have
+˘Ab = Qr
+ρ2
+b
+(dtb,χ − a sin2 θ dϕb,χ) − χQr
+∆b
+dr.
+Since the last term is an exact 1-form which makes no contribution to the electromagnetic 2-form, it follows
+that one can define the 4-potential on Mb as
+ABL
+b
+:= Qr
+ρ2
+b
+(dtb,χ − a sin2 θ dϕb,χ).
+(4.20)
+Smooth coordinates on S2 near the poles θ = 0, π are x = sin θ cos ϕb,χ and y = sin θ sin ϕb,χ. Then one finds
+that sin2 θ = x2 + y2 and
+∂ϕb,χ = −y∂x + x∂y.
+∂θ = cos θ cos ϕb,χ∂x + cos θ sin ϕb,χ∂y
+and thus the smoothness of (gBL
+b )−1 near the poles θ = 0, π (i.e, x = y = 0) follows from writing ∂2
+θ +
+sin−2 θ∂2
+ϕb,χ as
+∂2
+θ +
+1
+sin2 θ∂2
+ϕb,χ = (1 − x2 − y2)
+�
+∂2
+x + ∂2
+y
+�
+− (x∂x + y∂y)2,
+(4.21)
+which is smooth near x = y = 0. Since the volume form is given by
+dvolgBL
+b
+= (r2 + a2(1 − x2 − y2))(1 − x2 − y2)−1/2 dtb,χ dr dx dy
+and thus smooth at the poles θ = 0, π, we conclude that gBL
+b
+indeed extends smoothly and non-degenerately
+to the poles θ = 0, π.
+For b = (m, a, Q) close to (m0, 0, Q0), we take χ = χ0 as defined in (4.9) and choose suitable constants
+of integration such that tb,χ0 ≡ t, ϕb,χ0 ≡ ϕ for r ≥ 3rb0/2. Then we define the following diffeomorphism
+Φb : M = Rtχ0 × X = Rtχ0 × [r−, ∞)r × S2
+θ,ϕ → Mb = Rtb,χ0 × [r−, ∞)r × S2
+θ,ϕb,χ0,
+(tb,χ0, r, θ, ϕb,χ0)(Φb(p)) = (tχ0, r, θ, ϕ)(p)
+(4.22)
+where tχ0 is defined in (4.9) and (θ, ϕ) denote polar coordinates on S2 adapted to a. Using this family of
+diffeomorphisms Φb, we define
+gb = (Φb)∗(gBL
+b ) ∈ C∞(M; S2T ∗M),
+Ab = (Φb)∗(ABL
+b ) ∈ C∞(M; T ∗M)
+(4.23)
+which is a family of smooth and stationary metric and 1-form on M, respectively. For b = b0 = (m0, 0, Q0),
+this pullback indeed gives rise to the RN black hole (gb0, Ab0). Following the argument in [42, Proposition 3.5],
+one can prove that (gb, Ab) depends smoothly on b when b is close to b0.
+Lemma 4.1. Let Ub0 is a sufficiently small neighborhood of b0 = (m0, 0, Q0) and let M = Rtχ0 × X with
+X = {p ∈ R2 | |p| ≥ r−}. Then (gb, Ab) ∈ C∞(M; S2T ∗M ⊕ T ∗M) depends smoothly on b ∈ Ub0.
+Proof. Given (m, a, Q) with a = |a| ̸= 0, let us denote the spherical coordinate system with north pole θ = 0
+at a/|a| by (θb, ϕb), so the pullbacks of the functions ∆b, ρ2
+b to M are simply obtained by replacing θ by θb.
+Then, if (r, θb, ϕb) are the polar coordinates of a point p ∈ X, we have
+r = |p|,
+a cosθb = ⟨a, p
+|p|⟩,
+a2 sin2 θb = |a|2 − a2 cos2 θb
+
+LINEAR STABILITY OF KERR-NEWMAN FAMILY
+25
+where | · | and ⟨·, ·⟩ denote the Euclidean norm and inner product on X ⊂ R3, respectively. Since r ̸= 0, this
+shows that r, a cos θb and a2 sin2 θb, and thus the pullbacks of ∆b and ρb are smooth (in b) families of smooth
+functions on M. Then the smoothness of gb and Ab in b follows from that of a sin2 θbdϕb = (ar−2∂ϕb)♭ (using
+the musical isomorphism on Euclidean R3), which we will show subsequently.
+We now prove the smooth dependence of the inverse metric g−1
+b : in view of the expression (4.19) and the
+discussion around (4.21), all we need to show is that the vector fields ∂tχ0 , ∂r, a∂ϕb and a sin θb∂θb depend
+smoothly on a.
+For ∂tχ0 and for the radial vector field ∂r = |p|−1p∂p, which do not depend on b, the
+smoothness is clear. Further, we have
+a∂ϕb = ∇a×p
+at
+p ∈ X,
+and the expression on the right-hand side is smooth in a. Indeed, if a = a⃗e3 := (0, 0, a), both sides equal
+a(x∂y − y∂x) on R3
+x,y,z, and if a ∈ R3 is any given vector and R ∈ SO(R3) is a rotation with Ra = a⃗e3,
+a = |a|, then (R∗(a∂ϕb))|p = (a∂ϕa⃗e3,)|R(p) which we just observed to be equal to ∇a ⃗e3×R(p) = R∗∇a×p.
+In a similar manner, we have
+a sin θb∂θb = |p|−1∇p×(p×a)
+at
+p ∈ X,
+and again the expression on the right-hand side is smooth in a. This finishes the proof of the Lemma.
+□
+Remark 4.2. We consider the enlarged parameter space
+Ub0,m = {(m, a, Qe, Qm)} ⊂ R × R3 × R × R;
+(4.24)
+we take Ub0,m to be a small neighborhood of the parameters
+b0,m = (m0, 0, Q0,e, Q0,m)
+of a RN spacetime with magnetic charge. We define the map
+β : Ub0,m ∋ (m, a, Qe, Qm) �→ (m, a,
+�
+Q2e + Q2m) ∈ Ub0
+and then for bm = (m, a, Qe, Qm) ∈ Bb0,m, we define
+gbm := gβ(bm),
+Fbm := QeF(m,a,1) + Qm ⋆gbm F(m,a,1).
+(4.25)
+In view of Lemma 3.1, we see that (gbm, Fbm) is a solution of the Einstein–Maxwell system (3.1)–(3.2). Since
+the charge always appears in the second power in the KN metric, one concludes that gbm and Fbm depend
+smoothly on bm ∈ Bb0,m.
+Remark 4.3. The diffeomorphism used to realize the family of KN black holes as a smooth family of black
+holes on a fixed manifold is not unique, and in fact another presentation is more convenient for calculations
+(especially near event horizon) later on. Namely, we also consider coordinates tb,1, ϕb,1 (that is, take χ ≡ 1
+in the definitions of tb,χ and ϕb,χ) and use the following diffeomorphism
+Φ0
+b : M = Rt0 × [r−, ∞)r × S2
+θ,ϕ → Mb = Rtb,1 × [r−, ∞)r × S2
+θ,ϕb,1,
+(tb,1, r, θ, ϕb,χ0)(Φ0
+b(p)) = (t0, r, θ, ϕ)(p)
+(4.26)
+to define
+g0
+b = (Φ0
+b)∗(gBL
+b ) ∈ C∞(M; S2T ∗M),
+A0
+b = (Φ0
+b)∗(ABL
+b ) ∈ C∞(M; T ∗M)
+(4.27)
+Again, (g0
+b, A0
+b) depends smoothly on b when b is close to b0; for b = b0 = (m0, 0, Q0), the above pullback
+(4.27) produces the RN black hole (gb0, Ab0). For b = (m, 0, Q), we have
+g0
+(m,0,Q) = −µ(m,0,Q)dt2
+0 + 2dt0 dr + r2/g,
+A0
+(m,0,Q) = Q
+r dt0,
+µ(m,0,Q) = 1 − 2m
+r
++ Q
+r2 .
+(4.28)
+For b = (m, a, Q) close to b0 = (m0, 0, Q0), we also introduce the following function
+tb,∗ := t + rb0,∗χ0(r) − r(m,0,Q),∗(1 − χ0(r)),
+(4.29)
+which generalizes t∗ = tb0,∗ defined in (4.10); it equals t − r(m,0,Q),∗ for r ≥ 3rb0/2. In particular, in the
+region r ≥ 3rb0/2 where tχ0 = t, we have for b = (m, 0, Q)
+g(m,0,Q) = −µ(m,0,Q) dt2 + µ−1
+(m,0,Q) dr2 + r2/g = −µ(m,0,Q) dt2
+b,∗ − 2dtb,∗ dr + r2/g,
+A(m,0,Q) = Q
+r dt = Q
+r dtb,∗ + Q
+r∆b
+dr,
+
+26
+LILI HE
+and then dtb,∗ is null for large r.
+Given the smooth families (gb, Ab), now we can define linearized KN solutions:
+Definition 4.4. For ˙b ∈ R × R3 × R, we define the linearized KN solutions as
+˙gb(˙b) = Dbgb(˙b) = d
+ds
+���
+s=0gb+s˙b,
+˙Ab(˙b) = DbAb(˙b) = d
+ds
+���
+s=0Ab+s˙b.
+(4.30)
+Thus, (˙gb(˙b), ˙Ab(˙b)) is a solution of the linearized Einstein-Maxwell system (3.29) linearized around
+(gb, Ab).
+We can also linearize the family (g0
+b, A0
+b) in the parameter b to obtain another version of linearized KN
+solutions
+˙g0
+b(˙b) = Dbg0
+b(˙b) = d
+ds
+���
+s=0g0
+b+s˙b,
+˙A0
+b(˙b) = DbA0
+b(˙b) = d
+ds
+���
+s=0A0
+b+s˙b.
+(4.31)
+We record the particular case
+˙g0
+(m0,0,Q0)( ˙m, 0, ˙Q) =
+�2 ˙m
+r
+− 2Q0 ˙Q
+r2
+�
+dt2
+0,
+˙A0
+(m0,0,Q0)( ˙m, 0, ˙Q) =
+˙Q
+r dt0
+˙g0
+(m0,0,Q0)(0, ˙a, 0) = ((−4m0
+r
++ 2Q0
+r2 )dt0 − 2dr) sin2 θ dϕ,
+˙A0
+(m0,0,Q0)(0, ˙a, 0) = −Q
+r sin2 θdϕ,
+(4.32)
+where in the second line |˙a| = 1, and (θ, ϕ) are spherical coordinates adapted to ˙a.
+Remark 4.5. Since gb = ((Φ0
+b)−1 ◦ Φb)∗g0
+b, the linearized KN solution (˙gb(˙b), ˙Ab(˙b)) can be obtained from
+(˙g0
+b(˙b), ˙g0
+b(˙b)) by subtracting off a Lie derivative of (g0
+b, A0
+b) along a vector field generating the flow (Φ0
+b)−1◦Φb.
+More precisely, in the coordinates (t∗, r, θ, ϕ) on M, we have
+(Φ0
+b)−1 ◦ Φb : (t∗, r, θ, ϕ) �→
+�
+t∗ +
+� �r2 + a2
+∆b
+− r2
+∆b0
+�
+(1 − χ0) dr, r, θ, ϕ +
+�
+a
+∆b
+(1 − χ0) dr
+�
+.
+(4.33a)
+In particular, for ˙b = ( ˙m, ˙a, ˙Q), the two versions of the linearization of the KN solution at gb0 are related by
+˙gb0(˙b) = ˙g0
+b0(˙b) − LV (˙b)g0
+b0,
+˙gb0(˙b) = ˙g0
+b0(˙b) − LV (˙b)A0
+b0,
+V (˙b) = d
+ds
+����
+s=0
+((Φ0
+b0+s˙b)−1 ◦ Φb0+s˙b)
+=
+�
+˙m
+�� r
+r0
+2r3
+∆2
+b0
+(1 − χ0) dr
+�
+− ˙Q
+�� r
+r0
+2Q0r2
+∆2
+b0
+(1 − χ0) dr
+��
+∂t∗ + ˙a
+�� r
+r0
+1 − χ0
+∆b0
+dr
+�
+∂ϕ.
+(4.33b)
+where r0 in the definition of V (˙b), can be chosen arbitrarily.
+4.3. Stationarity, vector bundles, and geometric operators. Now we introduce some basics of the
+vector bundles and geometric operators (see [36, §3.3]). In the notation (4.8), denote the projection to the
+spatial manifold by
+πX : M → X;
+this is independent of the choice of time function. Suppose that E1 → X is a vector bundle; then differenti-
+ation along ∂t = ∂t0 = ∂t∗ is a well-defined operation on sections of the pullback bundle π∗
+XE1. The tangent
+bundle of M is an important example of such a pullback bundle, as
+T M ∼= π∗
+X(Tt−1
+0
+(0)M) ∼= π∗
+X(Tt−1
+∗
+(0)M),
+likewise for the cotangent bundle and other tensor bundles.
+Let E2 → X be another vector bundle, and suppose that �L(0) ∈ Diff(X; E1, E2) is a differential operator;
+fixing t = t∗ + F, F ∈ C∞(X), with dt ̸= 0 everywhere, we can then define its stationary extension by
+assigning to u ∈ C∞(M; π∗
+XE1) the section (Lu)(t, −) := �L(0)(u(t, −)) of π∗
+XE2; this extension does depend
+on the choice of t. The action of L on stationary functions on the other hand is independent of the choice of
+t since
+Lπ∗
+X = π∗
+X �L(0).
+(4.34)
+Via stationary extension, one can consider Diffb( ¯X; E) (for E → ¯X a smooth vector bundle down to ∂+ ¯X)
+to be a subalgebra of Diff(M; π∗
+XE); likewise Diffsc( ¯X; E) ֒→ Diff(M; π∗
+XE).
+
+LINEAR STABILITY OF KERR-NEWMAN FAMILY
+27
+Conversely, if L ∈ Diff(M; π∗
+XE1, π∗
+XE2) is stationary, i.e. commutes with ∂t, there exists a unique (in-
+dependent of the choice of t) operator �L(0) ∈ Diff(X; E1, E2) such that the relation (4.34) holds. More
+generally, we can consider the formal conjugation of L by the Fourier transform in t,
+�L(σ) := eiσtLe−iσt ∈ Diff(X; E1, E2),
+where we identify the stationary operator eiσtLe−iσt with an operator on X. Switching from t to another
+time function, t + F ′, F ′ ∈ C∞(X), amounts to conjugating �L(σ) by eiσF ′.
+In order to describe the uniform behavior of geometric operators at ∂+X concisely, we need to define
+a suitable extension of T ∗M to ‘infinity’. To accomplish this, note that the product decomposition (4.8)
+induces a splitting
+T ∗M ∼= T ∗Rt0 ⊕ T ∗X = π∗
+T (T ∗Rt0) ⊕ π∗
+X(T ∗X),
+where πT : M0 → Rt0 is the projection. We therefore define the extended scattering cotangent bundle of ¯X
+by
+�
+scT ∗ ¯X := Rdt0 ⊕ scT ∗ ¯X.
+(4.35)
+At this point, dt0 is merely a name for the basis of a trivial real rank 1 line bundle over ¯X; considering the
+pullback bundle π∗
+X �
+scT ∗ ¯X → M, we identify it with the differential of t0 ∈ C∞(M ◦), giving an isomorphism
+(π∗
+X �
+scT ∗ ¯X)|M ∼= T ∗M.
+(4.36)
+Smooth sections of �
+scT ∗ ¯X → ¯X are linear combinations, with C∞( ¯X) coefficients, of dt0 and the 1-forms dxi,
+where (x1, x2, x3) are standard coordinates on X ⊂ R3. One can switch to another time function in (4.35),
+say, t∗, by writing dt∗ = dt0 + (dt∗ − dt0), with the second term being a smooth scattering 1-form on ¯X;
+likewise for tχ0 and also for the static time t in r > rb0.
+For a stationary metric g on M, there exists a unique g′ ∈ C∞(X; S2 �
+scT ∗ ¯X) such that π∗
+Xg′ = g, namely
+g′ is the restriction (as a section of S2 �
+scT ∗ ¯X) of g to any transversal of πX, such as level sets of t0, tχ0, t∗.
+Identifying g with g′ and applying this to the Kerr-Newman family, we then have
+gb, g0
+b ∈ C∞( ¯X; S2 �
+scT ∗ ¯X);
+(4.37)
+they are non-degenerate down to ∂+X. Moreover, we have
+gb − g ∈ ρC∞,
+g := −dt2
+χ0 + dr2 + r2/g,
+g(m,a,Q) − g(m,0,Q) ∈ ρ2C∞( ¯X; S2 �
+scT ∗ ¯X),
+(4.38)
+i.e. a KN metric equals the Minkowski metric g to leading order, and is a O(ρ2) perturbation of the RN
+metric of the same mass and charge.
+We proceed to discuss basic geometric operators on Kerr-Newman spacetimes. We write
+(δ∗
+gω)µν = 1
+2(∇νωµ + ∇µων),
+(δgh)α = −gµν∇µhνα,
+Gg = Id − 1
+2g trg,
+(4.39)
+and furthermore denote by
+□g,0, □g,1, □g,2,
+(4.40)
+the wave operator gµν∇µ∇ν on scalar functions, 1-forms, and symmetric 2-tensors, respectively. When the
+bundle is clear from the context, we shall simply write □g. We use the splitting
+�
+scT ∗ ¯X = ⟨dt⟩ ⊕ scT ∗ ¯X,
+(4.41)
+S2 �
+scT ∗ ¯X = ⟨dt2⟩ ⊕ (2dt ⊗s scT ∗ ¯X) ⊕ S2 scT ∗ ¯X.
+(4.42)
+and work in the trivialization of scT ∗ ¯X, resp. S2 scT ∗ ¯X given by dxi, 1 ≤ i ≤ 3, resp. 2dxi ⊗s dxj, 1 ≤ i ≤
+j ≤ 3.
+Proposition 4.6. Working in the splittings (4.41) and (4.42) and the trivializations of scT ∗ ¯X and S2 scT ∗ ¯X,
+we write the operators □gb,2, □gb,1, □gb,0 as
+□gb,0 = g−1
+b (dtχ0, dtχ0)∂2
+tχ0 + �
+□gb,0(0) + Q1
+b,0∂tχ0 ,
+(4.43)
+□gb,1 = □gb,0 ⊗ Id4×4 + Q1
+b,1∂tχ0 + Q0
+b,1,
+(4.44)
+□gb,2 = □gb,0 ⊗ Id10×10 + Q1
+b,2∂tχ0 + Q0
+b,2.
+(4.45)
+
+28
+LILI HE
+Then we have
+�
+□gb,0(0) ∈ ρ2Diff2
+b( ¯X),
+Q1
+b,0 ∈ ρ3Diff1
+b( ¯X),
+(4.46)
+Q1
+b,1 ∈ ρ2Diff0
+b( ¯X; �
+scT ∗ ¯X),
+Q0
+b,1 ∈ ρ3Diff1
+b( ¯
+X; �
+scT ∗ ¯X),
+(4.47)
+Q1
+b,2 ∈ ρ2Diff0
+b( ¯X; S2 �
+scT ∗ ¯X),
+Q0
+b,2 ∈ ρ3Diff1
+b( ¯
+X; S2 �
+scT ∗ ¯X).
+(4.48)
+Moreover,
+�
+□gb,j(0) − �
+□g,j(0) ∈ ρ3Diff2
+b( ¯X),
+g = −dt2
+χ0 + dx2,
+j = 0, 1, 2.
+(4.49)
+Since the metric components are smooth away from ∂+ ¯X, it suffices to analyze □g,•, • = 0, 1, 2 near spatial
+infinity ∂+ ¯X where tχ0 ≡ t is the static time coordinate. Proposition 4.6 is a consequence of the following
+lemma (by taking h = dx2).
+Lemma 4.7. Suppose g is a stationary Lorentzian metric on M for which
+g(∂t, ∂t) ∈ −1 + ρC∞( ¯X),
+g(∂t, −) ∈ ρ2C∞( ¯X; scT ∗ ¯X),
+g|scT ¯
+X×scT ¯
+X ∈ h + ρC∞( ¯X; S2 �
+scT ∗ ¯X),
+(4.50)
+where h ∈ C∞( ¯X; S2 �
+scT ∗ ¯X) is Riemannian. Then the operator □g,2, □g,1, □g,0 satisfy the statements in
+Proposition 4.6 with Q0
+b,j ∈ ρ3Diff1
+b( ¯X) replaced by Q0
+b,j ∈ ρ2Diff1
+b( ¯X) for j = 1, 2.
+Moreover, �
+□g,j(0) mod ρ3Diff2
+b( ¯X) only depends on h for j = 0, 1, 2.
+Proof. We introduce coordinates
+x = (x0, x1, x2, x3),
+where x0 = t, and x1, x2, x3 are standard coordinates on R3. We use Latin letters i, j, k, · · · for indices from
+1 to 3, and Greek letters α, β, µ, ν, · · · for indices from 0 to 3. We write Ok := ρkC∞( ¯X) and write f = Ok
+in the spirit of O-notation instead of f ∈ Ok when f is a smooth function.
+We first compute the Levi-Civita connection of g. Using the facts that g is stationary, ∂i ∈ ρVb(X) and
+g−1(dt, dt) ∈ −1 + ρC∞( ¯X),
+g−1(dt, −) ∈ ρ2C∞( ¯X; scT ¯X),
+g−1|scT ∗ ¯
+X×scT ∗ ¯
+X ∈ h−1 + ρC∞( ¯X; S2 scT ¯X),
+we find that
+Γ0
+00 = O4,
+Γi
+00 = O2,
+Γ0
+0i = O2,
+Γj
+0i = O3,
+Γ0
+ij = O3,
+Γk
+ij = Γk
+ij(h) + O2,
+Γk
+ij(h) = O1.
+(4.51)
+Since □g,0 = gαβ∂α∂β − gαβΓµ
+αβ∂µ where gαβ is the inverse to g, we see that
+□g,0 = gαβ∂α∂β + f µ∂µ,
+f 0 = O3, f i = O1,
+and thus we can write
+□g,0 = g00∂2
+t + Q∂t + �
+□g,0(0)
+with
+Q ∈ ρ3Diff1
+b( ¯X), �
+□g,0(0) ∈ ρ2Diff2
+b( ¯X).
+Moreover, �
+□g,0(0) mod ρ3Diff2
+b( ¯X) only depends on h.
+Next, we calculate □g,1. Since
+(□g,1u)α = □g,0uα − 2gµνΓκ
+µα∂νuκ + gµνΓκ
+µνΓβ
+καuβ + gµνΓκ
+µαΓβ
+κνuβ − gµν(∂µΓκ
+να)uκ,
+we can write
+□g,1 = □g,0 ⊗ Id4×4 + Q1
+g,1∂t + Q0
+g,1
+where Q1
+g,1 ∈ ρ2Diff0
+b( ¯X; �
+scT ∗ ¯X) and Q0
+g,1 ∈ ρ2Diff1
+b( ¯
+X; �
+scT ∗ ¯X). Moreover, Q0
+g,1 ∈ ρ3Diff1
+b( ¯X; �
+scT ∗ ¯X) if
+Γk
+ij(h) vanishes, and �
+□g,1(0) mod ρ3Diff2
+b( ¯X; �
+scT ∗ ¯X) only depends on h.
+Finally, we calculate □g,2. Since
+(□g,2h)αβ = □g,0hαβ − 2gµνΓκ
+µα∂νhκβ − 2gµνΓκ
+µβ∂νhκα + S0
+g,2
+where S0
+g,2 ∈ Diff0
+b( ¯X; S2 �
+scT ∗ ¯X) whose elements are linear combinations of terms of the types gΓ · Γ and
+g∂Γ, it follows that we can write
+□g,2 = □g,0 ⊗ Id10×10 + Q1
+g,2∂t + Q0
+g,2
+
+LINEAR STABILITY OF KERR-NEWMAN FAMILY
+29
+where Q1
+g,2 ∈ ρ2Diff0
+b( ¯
+X; S2 �
+scT ∗ ¯X) and Q0
+g,2 ∈ ρ2Diff1
+b( ¯X; S2 �
+scT ∗ ¯X). Again, Q0
+g,2 ∈ ρ3Diff1
+b( ¯X; S2 �
+scT ∗ ¯X) if
+Γk
+ij(h) vanishes, and �
+□g,2(0) mod ρ3Diff2
+b( ¯X; S2 �
+scT ∗ ¯X) only depends on h.
+□
+We next discuss the curvature tensor and other geometric operators associated to gb.
+Proposition 4.8. Let g be a metric of the form (4.50) and ˜g = −dt2 + h where h ∈ C∞( ¯
+X; S2 �
+scT ∗ ¯X) is
+Riemannian. Then we have
+Riem(g) − Riem(˜g) ∈ ρ3C∞� ¯X; �
+scT ¯X ⊗ ( �
+scT ∗ ¯X)3�
+,
+Ric(g) − Ric(˜g) ∈ ρ3C∞� ¯X; S2 �
+scT ∗ ¯X
+�
+,
+(4.52)
+δg = −ιdt♯
+χ0∂tχ0 + �δg(0),
+�δg(0) ∈ ρDiff1
+b( ¯X),
+�δg(0) − �δ˜g(0) ∈ ρ2Diff1
+b( ¯X),
+(4.53)
+δ∗
+g = dtχ0 ⊗s ∂tχ0 + �δ∗g(0),
+�δ∗g(0) ∈ ρDiff1
+b( ¯X),
+�δ∗g(0) − �δ∗
+˜g(0) ∈ ρ2Diff0
+b( ¯X).
+(4.54)
+In particular, these hold for g = gb and ˜g = g = −dt2
+χ0 + dx2.
+Proof. Since
+Γ0
+00(˜g) = Γi
+00(˜g) = Γ0
+0i(˜g) = Γj
+0i(˜g) = Γ0
+ij(˜g) = 0,
+Γk
+ij(˜g) = O1,
+Γµ
+αβ(g) = Γµ
+αβ(˜g) + O2,
+and g, ˜g are stationary, the proposition follows from the formulas
+Riem(g)ν
+µαβ = ∂αΓν
+βµ(g) − ∂βΓν
+αµ(g) + Γν
+ακ(g)Γκ
+βµ(g) − Γν
+βκ(g)Γκ
+αµ(g),
+δgu = −gµν∂µuν + gµνΓκ
+µν(g)uκ,
+(δgh)α = −gµν∂µhνα + gµνΓκ
+µν(g)hκα + gµνΓκ
+µα(g)hνκ,
+(δ∗
+gu)αβ = 1
+2(∂αuβ + ∂βuα) − Γκ
+αβ(g)uκ.
+□
+In the context of (4.38), it is useful to record the following strengthening of the leading order control:
+Lemma 4.9. If g1, g2 are two metrics of the form (4.50) and so that in addition g1−g2 ∈ ρ2C∞(X; S2 �
+scT ∗ ¯X),
+then
+�
+□g1,j(0) − �
+□g2,j(0) ∈ ρ4Diff2
+b( ¯
+X),
+j = 0, 1, 2,
+(4.55)
+Riem(g2) − Riem(g1) ∈ ρ4C∞� ¯X; �
+scT ¯X ⊗ ( �
+scT ∗ ¯X)3�
+,
+Ric(g2) − Ric(g1) ∈ ρ4C∞� ¯X; S2 �
+scT ∗ ¯X
+�
+,
+(4.56)
+�
+δg1(0) − �
+δg2(0) ∈ ρ3Diff1
+b( ¯X),
+(4.57)
+�
+δ∗g1(0) − �
+δ∗g2(0) ∈ ρ3C∞( ¯X; Hom( �
+scT ∗ ¯X, S2 �
+scT ∗ ¯X)),
+(4.58)
+Proof. The proof follows from the facts that Γµ
+αβ(g1) = O1 and Γµ
+αβ(g2) − Γµ
+αβ(g1) = O3.
+□
+4.3.1. Relevant geometric operators in linearized gauge-fixed Einstein-Maxwell equations. We now study the
+linearized gauge-fixed Einstein-Maxwell operator arising from the generalized wave map and Lorenz gauge.
+Definition 4.10. Let B ∈ C∞(M; Hom(T ∗M, S2T ∗M)), F ∈ C∞(M; Hom(S2T ∗M, T ∗M)) and g be a
+Lorentzian metric. Then we define the modified symmetric gradient �δ∗
+g,B and modified negative divergence
+�δg,F as follows.
+�δ∗
+g,B = δ∗
+g + Bg,
+�δg,F = δg + Fg.
+(4.59)
+In this paper, we will use Bg, Fg of the following form
+Bg = B(g; c, γ) := 2γc ⊗s (•) − 1
+2γg−1(c, •)g,
+(4.60)
+Fg = F(g; c, γ) := 2γιg
+c(•) − 1
+2γctrg(•).
+(4.61)
+where γ ∈ R is a sufficiently small constant to be determined later on, c ∈ C∞
+c (X, �
+scT ∗ ¯X) is a stationary
+1-form with compact support and ιg
+c(h) = gµνcνhµα. With the above choices of Bg and Fg in (4.60) and
+(4.61), we find that �δ∗
+g,B and �δg,F are formally adjoint to each other. From now on, we denote
+�δ∗
+g,γ := �δ∗
+g,B = δ∗
+g + Bg,
+�δg,γ := �δg,F = δg + Fg.
+(4.62)
+
+30
+LILI HE
+By choosing ˜δ∗
+gb = �δ∗
+gb,γ and θ(g; gb) = F(gb; c, γ)[Ggbg], we obtain the linearized gauge-fixed Einstein-
+Maxwell system Lb,γ(˙g, ˙A) = (2LE
+b,γ(˙g, ˙A), LM
+b,γ(˙g, ˙A)) = 0 around (gb, Ab)
+LE
+b,γ(˙g, ˙A) := Dgb(Ric)(˙g) − ˜δ∗
+gb,γDgb �ΥE(˙g; gb) − 2D(gb,dAb)T (˙g, d ˙A),
+�ΥE = ΥE − θ,
+LM
+b,γ(˙g, ˙A) := D(gb,Ab)(δ(·)d(·))(˙g, ˙A) − d
+�
+D(gb,Ab) �ΥM(˙g, ˙A; gb, Ab)
+�
+.
+(4.63)
+According to the calculation (3.32)–(3.35), (A.1) and (A.4), we find that
+LE
+b,γ(˙g, ˙A) = −1
+2□gb,2 ˙g + (˜δ∗
+gb,γ − δ∗
+gb)δgbGgb ˙g + δ∗
+gb(˜δgb,γ − δgb)Ggb ˙g
++ (˜δ∗
+gb,γ − δ∗
+gb)(˜δgb,γ − δgb)Ggb ˙g + Rgb(˙g)µν − 2Dgb,dAbT (˙g, d ˙A),
+LM
+b,γ(˙g, ˙A) = (dδgb + δgbd) ˙A − (δgbGgb ˙g)κ(dAb)κµ + 1
+2(∇ν ˙gκ
+µ − ∇κ ˙gν
+µ)(dAb)νκ + ˙gνα∇α(dAb)νµ
+(4.64)
+where dδgb + δgbd = −□gb,1 + Ric(gb) and
+(Rgb ˙g)µν = Riem(gb)α
+β
+µν ˙gαβ + 1
+2
+�
+Ric(gb) κ
+µ ˙gνκ + Ric(gb) κ
+ν ˙gµκ
+�
+,
+Dgb,dAbT (˙g, d ˙A) =
+�
+(gb)αβ(d ˙A)µα(dAb)νβ + (gb)αβ(dAb)µα(d ˙A)νβ
+�
+− 1
+2(gb)µν(d ˙A)αβ(dAb)αβ
+− ˙gαβ(dAb)µα(dAb)νβ − 1
+4
+�
+˙gµν(dAb)αβ(dAb)αβ−2(gb)µν ˙gκα(dAb)αβ(dAb) β
+κ
+�
+.
+(4.65)
+We note that the first two terms of the second line of LE
+b,γ(˙g, ˙A), the last two terms of Dgb,dAbT (˙g, d ˙A) and
+the last term of LM
+b,γ(˙g, ˙A) contain no derivatives of (˙g, ˙A).
+For later use, we also define the gauge propagation operator Pb,γ and the gauge potential wave operator
+Wb,γ for the gauge 1-form �ΥE(˙g; gb):
+Pb,γ := δgbGgb ˜δ∗
+gb,γ = −1
+2□gb,1 − 1
+2Ric(gb) + δgbGgb(˜δ∗
+gb,γ − δ∗
+gb),
+Wb,γ := ˜δgb,γGgbδ∗
+gb = −1
+2□gb,1 − 1
+2Ric(gb) + (˜δgb,γ − δgb)Ggbδ∗
+gb.
+(4.66)
+Then we have
+Lemma 4.11. Let Lb,γ = (2LE
+b,γ, LM
+b,γ), Pb,γ, Wb,γ be defined as above. Then we have
+Pb,γ = −1
+2□gb,1 + P 1∂tχ0 + P 0,
+�
+Pb,γ(0) = −1
+2
+�
+□g,1(0) + P 0,
+(4.67)
+Wb,γ = −1
+2□gb,1 + W 1∂tχ0 + W 0,
+�
+Wb,γ(0) = −1
+2
+�
+□g,1(0) + W 0,
+(4.68)
+where
+P 1, W 1 ∈ ρ∞Diff0
+b( ¯X; �
+scT ∗ ¯X),
+P 0, W 0 ∈ ρ4Diff1
+b( ¯X; �
+scT ∗ ¯X).
+(4.69)
+We also have
+Lb,γ = −□gb,0 ⊗ Id14×14 + L1∂tχ0 + L0,
+�
+Lb,γ(0) = − �
+□g,0(0) ⊗ Id14×14 + L0,
+(4.70)
+where
+L1 ∈ ρ2Diff0
+b( ¯
+X; S2 �
+scT ∗ ¯X ⊕ �
+scT ∗ ¯X),
+L0 ∈ ρ3Diff1
+b( ¯X; S2 �
+scT ∗ ¯X ⊕ �
+scT ∗ ¯X).
+(4.71)
+Moreover, we have
+− �
+Lb,γ(0) − �
+□g,0(0) ⊗ Id14×14 ∈ ρ3Diff2
+b( ¯X; S2 �
+scT ∗ ¯X ⊕ �
+scT ∗ ¯X),
+g = −dt2
+χ0 + dx2.
+(4.72)
+Proof. Since ˜δ∗
+gb,γ − δ∗
+gb and ˜δgb,γ − δgb are compactly supported and Ric(gb) = O4, then the conclusions
+(4.67)–(4.69) follow.
+According to (4.65) and using the facts that Riem(gb) = O3, dAb = O2, we see that Rgb ∈ ρ3Diff0
+b and
+D(gb,dAb)T = T 1∂tχ0 + T 0 with
+T 1 ∈ ρ2Diff0
+b( ¯X; S2 �
+scT ∗ ¯X ⊕ �
+scT ∗ ¯X),
+T 0 ∈ ρ3Diff1
+b( ¯X; S2 �
+scT ∗ ¯X ⊕ �
+scT ∗ ¯X),
+and thus
+2LE
+b,γ(˙g, ˙A) = −□gb,2 ˙g + LE,1∂tχ0 + LE,0
+
+LINEAR STABILITY OF KERR-NEWMAN FAMILY
+31
+with
+LE,1 ∈ ρ2Diff0
+b( ¯X; S2 �
+scT ∗ ¯X ⊕ �
+scT ∗ ¯X),
+LE,0 ∈ ρ3Diff1
+b( ¯X; S2 �
+scT ∗ ¯X ⊕ �
+scT ∗ ¯X).
+According to the expression for LE
+b,γ(˙g, ˙A) in (4.64) and using Proposition 4.8, we find that
+Lb,γ(˙g, ˙A) = −□gb,1 ˙A + LM,1∂tχ0 + LM,0
+where
+LM,1 ∈ ρ2Diff0
+b( ¯X; S2 �
+scT ∗ ¯X ⊕ �
+scT ∗ ¯X),
+LM,0 ∈ ρ3Diff1
+b( ¯X; S2 �
+scT ∗ ¯X ⊕ �
+scT ∗ ¯X).
+Combining the above conclusions for LE
+b,γ and LM
+b,γ and using the description of □gb,1, □gb,2 in Proposition
+4.6, we obtain (4.70)–(4.72).
+□
+Now we let
+�
+Lb,γ(σ) = eitb,∗σLb,γe−itb,∗σ,
+�
+Pb,γ(σ) = eitb,∗σPb,γe−itb,∗σ,
+�
+Wb,γ(σ) = eitb,∗σWb,γe−itb,∗σ.
+(4.73)
+In order to study the property of �
+Lb,γ(σ), �
+Pb,γ(σ), �
+Wb,γ(σ), we first express Lb,γ, Pb,γ, Wb,γ in the coordinates
+(tb,∗, xi), i = 1, 2, 3.
+Lemma 4.12. In the coordinates (tb,∗, xi), i = 1, 2, 3, the operator □gb,0 take the form
+□gb,0 = Q2∂2
+tb,∗ + Q1∂tb,∗ + �
+□gb,0(0)
+(4.74)
+where Q2 ∈ ρ2C∞( ¯X), Q1 ∈ 2ρ2∂ρ − 2ρ + ρ3Diff1
+b( ¯
+X) and �
+□gb,0(0) ∈ ρ2Diff2
+b( ¯X).
+Moreover, the operators Pb,γ, Wb,γ, Lb,γ have the form
+Pb,γ = −1
+2□gb,0 ⊗ Id4×4 + ˜P 1∂tb,∗ + ˜P 0,
+(4.75)
+Wb,γ = −1
+2□gb,0 ⊗ Id4×4 + ˜W 1∂tb,∗ + ˜W 0,
+(4.76)
+Lb,γ = −□gb,0 ⊗ Id14×14 + ˜L1∂tb,∗ + ˜L0,
+(4.77)
+where
+˜P 1, ˜W 1, ˜L1 ∈ ρ2Diff0
+b( ¯X),
+˜P 0, ˜
+W 0, ˜L0 ∈ ρ3Diff0
+b( ¯X).
+Proof. With parameter b = (m, a, Q), near ∂+ ¯X, we have tχ0 = t and tb,∗ = t − r(m,0,Q),∗. Then changing
+from (tχ0, r) coordinates to (tb,∗, r) coordinates transforms ∂tχ0 , ∂r into
+∂tb,∗,
+∂r −
+1
+µ(m,0,Q)
+∂tb,∗
+near
+∂+ ¯X,
+µ(m,0,Q) = 1 − 2m
+r
++ Q2
+r2 .
+Since gb − g(m,0,Q) ∈ ρ2C∞( ¯X; S2 �
+scT ∗ ¯X), using the facts that in the coordinate (t, xi),
+Γκ
+αβ(g(m,0,Q)) = O2,
+Γκ
+αβ(gb) − Γκ
+αβ(g(m,0,Q)) ∈ O3,
+it follows that
+□gb,0 − □g(m,0,Q),0 = ρ2C∞( ¯X)∂2
+tχ0 + ρ3Diff1
+b( ¯X)∂tχ0 + ρ4Diff2
+b( ¯
+X)
+and thus
+□gb,0 − □g(m,0,Q),0 = ρ2C∞( ¯X)∂2
+tb,∗ + ρ3Diff1
+b( ¯
+X)∂tb,∗ + ρ4Diff2
+b( ¯
+X).
+Then it remains to calculate □g(m,0,Q),0 in (tb,∗, r) coordinates. Since near ∂+ ¯X
+g(m,0,Q) = −µ(m,0,Q)dt2
+b,∗ − 2dtb,∗dr + r2/g,
+we have for ρ = 1/r
+□g(m,0,Q),0 = −2∂r∂tb,∗ + µ(m,0,Q)∂2
+r + 2r − 2m
+r2
+∂r − 2
+r ∂tb,∗ + 1
+r2 /∆
+= 2ρ2∂ρ∂tb,∗ − 2ρ∂tb,∗ + ρ2Diff2
+b( ¯
+X)
+near
+∂+ ¯X.
+This completes the proof for □gb,0. Then the proof for Pb,γ, Wb,γ, Lb,γ follows directly from Proposition 4.6
+and Lemma 4.11.
+□
+Then the following proposition is a direct result of Lemma 4.12.
+
+32
+LILI HE
+Proposition 4.13. The operator □gb,0 take the form
+�
+□gb,0(σ) = −2iσρ(ρ∂ρ − 1) + �
+□gb,0(0) + σQ + σ2V
+(4.78)
+where V ∈ ρ2C∞( ¯X), Q ∈ ρ3Diff1
+b( ¯X) and �
+□gb,0(0) ∈ ρ2Diff2
+b( ¯X).
+Moreover, the operators , Pb,γ, Wb,γ, Lb,γ have the form
+Pb,γ = iσρ(ρ∂ρ − 1) ⊗ Id4×4 + �
+Pb,γ(0) + σQP + σ2VP ,
+(4.79)
+Wb,γ = iσρ(ρ∂ρ − 1) ⊗ Id4×4 + �
+Wb,γ(0) + σQW + σ2VW ,
+(4.80)
+Lb,γ = 2iσρ(ρ∂ρ − 1) ⊗ Id14×14 + �
+Lb,γ(0) + σQL + σ2VL,
+(4.81)
+where
+VP , VW , VL ∈ ρ2Diff0
+b( ¯X),
+QP , QW , QL ∈ ρ2Diff1
+sc( ¯X),
+�
+Pb,γ(0), �
+Wb,γ(0), �
+Lb,γ(0) ∈ ρ2Diff2
+b( ¯X).
+4.4. Constructing gauged initial data. According to Lemmas 3.5 and 3.8, we now show how to construct
+gauged initial data for the gauge-fixed Einstein-Maxwell system (3.22) from initial data given on Σ0 which
+are close to the data (hb, kb, Eb, Bb) induced by the KN black holes with parameter b.
+We use the generalized wave map gauge for the Einstein part of the system,
+�ΥE(g; gb) = ΥE(g; gb) − θ(g; gb) = gg−1
+b δgGggb − θ(g; gb) = 0
+(4.82)
+i.e. letting the background metric in (3.15) be gb, and the generalized Lorenz gauge for the Maxwell part,
+�ΥM(g, A; gb, Ab) = ΥM(g, A; gb) − ΥM(g, Ab; gb) = trgδ∗
+gbA − trgδ∗
+gbAb = 0.
+(4.83)
+We point out that for a general treatment, we will construct the gauged initial data for any θ(g; gb) satisfying
+that θ(gb; gb) = 0 and θ(g; gb) contains no derivatives of g, without being restricted to the specific θ(g; gb)
+we chose in §4.3.
+In order to capture the vanishing magnetic charge, we consider for s > 2, 0 < α < 1 the subspace of the
+initial data for ungauged Einstein-Maxwell equations
+Zs,α :=
+�
+(h, k, E, B) | d ⋆h B = 0,
+�
+S2 ⋆hB = 0,
+(h − hb, k − kb, E − Eb, B − Bb) ∈ ¯Hs+1,−1/2+α
+b
+(¯Σ0; S2
+>0
+scT ∗¯Σ0)
+× ¯Hs,1/2+α
+b
+(¯Σ0; S2scT ∗ ¯Σ0) × ¯Hs,1/2+α
+b
+(¯Σ0; scT ∗¯Σ0) × ¯Hs,1/2+α
+b
+(¯Σ0; scT ∗¯Σ0)
+�
+,
+(4.84)
+where S2
+>0
+scT ∗¯Σ0 denotes the fiber bundle of positive definite inner products on scT ¯Σ0. We remark that the
+definition of the space Zs,α incorporates (3.14) and one of the constraint equations (3.13), which is necessary
+since we study the Einstein-Maxwell system using the potential A rather than the electromagnetic 2-form
+F.
+For the initial value problems of the gauge-fixed Einstein-Maxwell system P(g, A) = 0, which is a quasi-
+linear hyperbolic system, the initial data induced by (g, A) at the spacelike hypersurface Σ0 = {t = 0} is
+defined as
+γ0(g, A) := (g|Σ0, L∂tg|Σ0, A|Σ0, L∂tA|Σ0).
+(4.85)
+In particular, we have γ0(gb, Ab) = (gb|Σ0, 0, Ab|Σ0, 0). Then we define the subspace of the initial data for
+the gauge-fixed Einstein-Maxwell equations
+Y s,α :=
+�
+(g0, g1, A0, A1) | (g0, g1, A0, A1) − γ0(gb, Ab) ∈ ¯Hs+1,−1/2+α
+b
+(¯Σ0; S2 �
+scT ∗¯Σ0)
+× ¯Hs,1/2+α
+b
+(¯Σ0; S2 �
+scT ∗ ¯Σ0) × ¯Hs,−1/2+α
+b
+(¯Σ0; �
+scT ∗¯Σ0) × ¯Hs−1,1/2+α
+b
+(¯Σ0; �
+scT ∗Σ0)
+�
+.
+(4.86)
+We now prove:
+Proposition 4.14. There exist a neighborhood
+(hb, kb, Eb, Bb) ∈ U ⊂ Z2,α
+of KN initial data and a smooth map
+ib : U ∩ Zs,α → Y s,α
+
+LINEAR STABILITY OF KERR-NEWMAN FAMILY
+33
+for all s ≥ 2, 0 < α < 1, such that for (h, k, E, B) ∈ U, the sections
+(g0, g1, A0, A1) = ib(h, k, E, B)
+induce the data (h, k, E, B) on Σ0 = {t = 0}, and they satisfy the gauge conditions (4.82) and (4.83) in the
+sense that for any section (g, A) of S2T ∗M ⊕ T ∗M near Σ0 with γ0(g, A) = (g0, g1, A0, A1), it induces the
+initial data (h, k, E, B) at Σ0 (i.e., τ((g, dA)) = (h, k, E, B)) and the conditions (4.82) and (4.83) hold at
+Σ0.
+Moreover, for exact KN data with parameter b, we have ib(hb, kb, Eb, Bb) = γ0(gb, Ab) = (gb|Σ0, 0, Ab|Σ0, 0).
+Proof. We will start with the metric components (g0, g1) of the map ib. We write
+gb = −N 2
+b (dt)2 + gb,ij(dxi + Xi
+bdt)(dxj + Xj
+b dt)
+where Nb ∈ 1 + ρC∞(Σ0) and Xb ∈ ρ2C∞(¯Σ0; scT ¯Σ0). Then we define the component g0 of ib(h, k, E, B) as
+g0 := g|{t=0} = −N 2
+b (dt)2 + hij(dxi + Xi
+bdt)(dxj + Xj
+b dt).
+Moreover, g0 = gb|Σ0 if h = hb and g0 satisfies the estimate
+∥g0 − gb|Σ0∥ ¯
+Hs+1,−1/2+α
+b
+(¯Σ0;S2 �
+scT ∗ ¯Σ0) ≲ ∥h − hb∥ ¯
+Hs+1,−1/2+α
+b
+(¯Σ0;S2
+>0
+scT ∗ ¯Σ0).
+We next define g1 = L∂tg|Σ0. Let ∇g0 be the Levi-Civita connection of g0 and let n resp. nb be the future
+timelike unit normal to Σ0 with respect to g0 resp. gb|Σ0. Then we have
+n =
+−∇g0t
+�
+−g0(∇g0t, ∇g0t)
+= (1 + N −2
+b
+h(Xb, Xb))−1/2( 1
+Nb
+, −Xi
+b
+Nb
+),
+n − nb ∈ ¯Hs+1,1/2+α
+b
+(¯Σ0; �
+scT ¯Σ0).
+Since we require
+kij = IIg(∂i, ∂j)|Σ0 = −⟨∇g
+∂in, ∂j⟩ = −1
+2Lngij = −1
+2nα∂αgij − 1
+2giα∂jnα − 1
+2gjα∂inα,
+where the Greek index α ranges over 0, 1, 2, 3 with x0 = t, we must define
+(g1)ij = ∂tgij|Σ0 = −2(1 + N −2
+b
+h(Xb, Xb))1/2Nbkij + LXbhij
+Now the initial data for the remaining components of g1, i.e. (g1)0α = ∂tg0α|Σ0, are fixed by the generalized
+wave map gauge condition. More precisely, since θ(g; gb) does not contain the derivatives of g, it follows that
+0 = �ΥE(g; gb)i = giκgµν(Γ(g)κ
+µν − Γ(gb)κ
+µν) − θ(g; gb)i
+= gµν∂µgiν − 1
+2gµν∂igµν − giκgµνΓ(g0)κ
+µν − θ(g; gb)i
+at
+Σ0
+fixes ∂tg0i|Σ0 and
+0 = �ΥE(g; gb)0 = g0κgµν(Γ(g)κ
+µν − Γ(gb)κ
+µν) − θ(g; gb)0
+= gµν∂µg0ν − 1
+2gµν∂0gµν − g0κgµνΓ(gb)κ
+µν − θ(g; gb)0
+at
+Σ0
+fixes ∂tg00|Σ0. This explicit construction of g1 implies that g1 = ∂tgb|Σ0 = 0 if (h, k) = (hb, kb), and
+∥g1∥ ¯
+Hs,1/2+α
+b
+(¯Σ0;S2 �
+scT ∗ ¯Σ0) ≲ ∥h − hb∥ ¯
+Hs+1,−1/2+α
+b
+(¯Σ0;S2
+>0
+scT ∗ ¯Σ0) + ∥k − kb∥ ¯
+Hs,1/2+α
+b
+(¯Σ0;S2scT ∗ ¯Σ0).
+So we finish the construction of the initial data (g0, g1) which induces the data (h, k) on Σ0 and satisfies the
+generalized wave map gauge condition at Σ0.
+We next turn to the construction of the components (A0, A1) of the map ib. Here we closely follow the
+construction in [39, §3.5]. Motivated by the relation F = dA, we first construct a bounded linear map
+A♯ :
+�
+u ∈ ¯Hs,ℓ
+b (¯Σ0; Λ2scT ∗¯Σ0): du = 0,
+�
+S2 u = 0
+�
+→ ¯Hs,ℓ−1
+b
+(¯Σ0; scT ∗ ¯Σ0)
+(4.87)
+such that d ◦ A♯ = Id with ℓ > −1/2. Using the structure ¯Σ0 ∼= J × S2, J = [0, 1/r−]ρ=1/r, of ¯Σ0, we define
+π: ¯Σ0 → S2 to be the projection onto the second factor, and j : S2 ֒→ ¯Σ0 be the embedding ω �→ (1/r−, ω).
+Then the map K : ¯Hσ,ℓ
+b (¯Σ0; ΛscT ∗ ¯Σ0) → ¯Hσ,ℓ−1
+b
+(¯Σ0; ΛscT ∗Σ0), σ ≥ 0, ℓ > −1/2, defined by linear extension
+from
+K(f(r, ω)dr ∧ π∗u)(r, ω) := (π∗u)(r, ω)
+� r
+r−
+f(s, ω) ds,
+K(f(r, ω)π∗u) := 0,
+
+34
+LILI HE
+for f ∈ ¯Hσ,1/2+α
+b
+(¯Σ0) and u ∈ C∞(S2; ΛT ∗S2), satisfies Id − π∗j∗ = dK + Kd. Therefore, for closed u as
+in (4.87), we have u = dKu + π∗(j∗u). According to Hodge decomposition theory and the fact that the de
+Rham cohomology group H2
+dR(S2) ∼= R, we see that u′ := j∗u ∈ Hs(S2; Λ2T ∗S2) can be written uniquely
+as u′ = /∗u0 + /du′′ where u0 ∈ R and u′′ ∈ Hs+1(S2; T ∗S2). Here, /d, /δ, /∗ denote the exterior differential,
+codifferential and Hodge star on S2. Since K(f(r, ω)π∗u) = 0 and
+�
+S2 u = 0, it follows that
+0 =
+�
+S2 π∗(j∗u) =
+�
+S2 u′ =
+�
+S2 u0 dVol/g = 4πu0,
+and thus u = dA♯u with A♯u := Ku + π∗u′′, finishing the construction of the map (4.87).
+Returning to the construction of the map ib, we define
+A0 = (ι∂tAb) dt + A♯
+0,
+A♯
+0 := i∗Ab + A♯(⋆hB − ⋆hbBb) ∈ i∗Ab + ¯Hs,−1/2+α
+b
+(¯Σ0; scT ∗¯Σ0),
+and thus dA♯
+0 = i∗(⋆hB), and A0 = Ab if (h, B) = (hb, Bb). We make the ansatz
+A1 = a1 dt + A♯
+1
+with a1 ∈ ¯Hs−1,ℓ1
+b
+(¯Σ0) and A♯
+1 ∈ ¯Hs−1,ℓ
+b
+(¯Σ0; scT ∗¯Σ0) to be determined. Recall the future timelike unit
+normal n to Σ0 with respect to g0 as defined above. The requirement that −i∗ιndA = E at Σ0 then reads
+E = n0(d(ι∂tAb) − A♯
+1) − ιnk∂kdA♯
+0. Then we must define
+A♯
+1 := d(ι∂tAb) − (E + ιnk∂kdA♯
+0)
+n0
+∈ ¯Hs−1,1/2+α
+b
+(¯Σ0; scT ∗ ¯Σ0)
+so in particular A♯
+1 = 0 if (h, k, E, B) = (hb, kb, Eb, Bb).
+Lastly, the generalized Lorenz gauge condition
+0 = �ΥM(g, A; gb, Ab) = trg0δ∗
+gb(A − Ab)
+= 1
+2gµν
+0
+�
+∂µ(Aν − Ab,ν) + ∂µ(Aν − Ab,ν) − 2Γκ
+µν(g0)(Aκ − Ab,κ)
+�
+at
+Σ0,
+where gµν
+0
+is the inverse g0, fixes a1 = ∂tA0 ∈ ¯Hs−1,1/2+α
+b
+(¯Σ0). In particular, we have a1 = 0 if the data
+(h, k, E, B) are induced by (gb, Ab). Moreover, we have
+∥A0 − Ab|Σ0∥ ¯
+Hs,−1/2+α
+b
+(¯Σ0; �
+scT ∗ ¯Σ0) + ∥A1∥ ¯
+Hs−1,1/2+α
+b
+(¯Σ0; �
+scT ∗ ¯Σ0)
+≲ ∥h − hb∥ ¯
+Hs,−1/2+α
+b
+(¯Σ0;S2
+>0
+scT ∗ ¯Σ0) + ∥B − Bb∥ ¯
+Hs,1/2+α
+b
+(¯Σ0;scT ∗ ¯Σ0) + ∥E − Eb∥ ¯
+Hs,1/2+α
+b
+(¯Σ0;scT ∗ ¯Σ0)
+Also, according to the above explicit construction of the map ib, we see that the map ib is smooth. This
+completes the proof.
+□
+We can use the map ib to construct gauged initial data for the initial value problem for the linearized
+Einstein–Maxwell system (3.29), linearized around (gb, Ab) and with initial data (˙h, ˙k, ˙E, ˙B). Correspond-
+ingly, we consider for s > 2, 0 < α < 1 the subspace of the initial data for ungauged linearized Einstein-
+Maxwell equations around (gb, Ab)
+˙Zs,α :=
+�
+(˙h, ˙k, ˙E, ˙B) | D(hb,Bb)(⋆(•)(•))(˙h, ˙B) = 0,
+�
+S2
+d
+ds|s=0 ⋆hb+s˙h (Bb + s ˙B) = 0,
+(˙h, ˙k, ˙E, ˙B) ∈ ¯Hs+1,−1/2+α
+b
+(¯Σ0; S2
+>0
+scT ∗¯Σ0)
+× ¯Hs,1/2+α
+b
+(¯Σ0; S2scT ∗ ¯Σ0) × ¯Hs,1/2+α
+b
+(¯Σ0; scT ∗¯Σ0) × ¯Hs,1/2+α
+b
+(¯Σ0; scT ∗¯Σ0)
+�
+,
+(4.88)
+where S2
+>0
+scT ∗¯Σ0 denotes the fiber bundle of positive definite inner products on scT ¯Σ0. We again remark
+that the definition of the space Zs,α incorporates the vanishing of the linearized magnetic charge and the
+linearization of one of the constraint equations (3.13).
+We also define the subspace of the initial data for the gauge-fixed linearized Einstein-Maxwell equations
+˙Y s,α :=
+�
+(˙g0, ˙g1, ˙A0, ˙A1) | (˙g0, ˙g1, ˙A0, ˙A1) ∈ ¯Hs+1,−1/2+α
+b
+(¯Σ0; S2 �
+scT ∗ ¯Σ0)
+× ¯Hs,1/2+α
+b
+(¯Σ0; S2 �
+scT ∗ ¯Σ0) × ¯Hs,−1/2+α
+b
+(¯Σ0; �
+scT ∗¯Σ0) × ¯Hs−1,1/2+α
+b
+(¯Σ0; �
+scT ∗Σ0)
+�
+.
+(4.89)
+Then we have
+
+LINEAR STABILITY OF KERR-NEWMAN FAMILY
+35
+Corollary 4.15. For all s ≥ 2, 0 < α < 1, the map
+D(hb,kb,Eb,Bb)ib : ˙Zs,α → ˙Y s,α
+satisfies that for (˙h, ˙k, ˙E, ˙B) ∈ ˙Zs,α, the sections
+(˙g0, ˙g1, ˙A0, ˙A1) = D(hb,kb,Eb,Bb)ib(˙h, ˙k, ˙E, ˙B)
+induce the data (˙h, ˙k, ˙E, ˙B) on Σ0 = {t = 0}, and they satisfy the linearized gauge conditions Dgb �ΥE(˙g) = 0
+and D(gb,Ab) �ΥM(˙g, ˙A; gb, Ab) = −δgb ˙A = 0 in the sense that for any section (˙g, ˙A) of S2T ∗M ⊕ T ∗M near
+Σ0 with γ0(˙g, ˙A) = (˙g0, ˙g1, ˙A0, ˙A1), it induces the initial data (˙h, ˙k, ˙E, ˙B) at Σ0 and the linearized gauge
+conditions hold at Σ0.
+Proof. We define
+h(s) = hb + s˙h, k(s) = kb + s˙k, E(s) = Eb + s ˙E, B(s) = Bb + s ˙B
+and
+⋆h+s˙hB(s) = ⋆hB + sDhb,Bb(⋆(·)(·))(˙h, ˙B)
+for s close to 0; thus h′(0) = ˙h etc., and d(s) := (h(s), k(s), E(s), B(s)) ∈ U ⊂ Z2,α. Then it follows that
+ib(d(s)) is well-defined and smooth in s, and
+(˙g0, ˙g1, ˙A0, ˙A1) = d
+dsib(d(s))
+��
+s=0 = D(hb,kb,Eb,Bb)ib(˙h, ˙k, ˙E, ˙B)
+gives initial data satisfying the linearized gauge conditions
+Dgb �ΥE(˙g) = 0,
+D(gb,Ab) �ΥM(˙g, ˙A; gb, Ab) = −δgb ˙A = 0,
+(4.90)
+for ˙g and ˙A with γ0(˙g, ˙A) = (˙g0, ˙g1, ˙A0, ˙A1).
+□
+5. Analysis of the linearized gauge-fixed Einstein-Maxwell operator
+Throughout this section, we continue using the notation from §4.
+In §5.1, we will discuss the characteristic set and the global dynamics of the Hamiltonian flow of the
+principal symbol p(σ) of the operator �
+□gb(σ) := eitb,∗σ□gbe−itb,∗σ on full subextremal KN spacetimes (see
+Figure 6 for an illustration). Concretely, first we will show that the characteristic set of p(σ) has two parts,
+one of which lies inside the ergoregion while the other is at the spatial infinity ∂+ ¯X. Then we prove that
+there exist radial points, to which the nearby integral curves of the Hamiltonian vector field Hp(σ) converge
+in either forward direction or backward direction, at event horizon and spatial infinity ∂+ ¯X. Finally, we
+describe the global dynamics of the Hamiltonian flow of the principal symbol p(σ).
+In §5.2, we combine the global dynamics of the Hamiltonian flow of the principal symbol p(σ) established
+in §5.1 with the elliptic estimate, propagation of singularities estimate, the radial point estimate at event
+horizon, the scattering radial point estimate at spatial infinity ∂+ ¯X and hyperbolic estimate to prove the
+uniform Fredholm estimates for the operators �
+□gb(σ), �
+Pb,γ(σ), �
+Wb,γ(σ), �
+Lb,γ(σ) for bounded σ in the closed
+upper half plane.
+In§5.3, we will give a detailed description of kernel of the following wave operators: �
+□gb(σ) on scalar
+functions, �
+Pb,γ(σ), �
+Wb,γ(σ) on scattering 1-forms and the linearized gauge-fixed Einstein-Maxwell operator
+�
+Lb,γ(σ).
+In §5.4, we will analyze the characteristic set and the global dynamics of the Hamiltonian flow of the
+semiclassical principal symbol ph,z of the semiclassical rescaled operator h2�
+□gb(h−1z) on full subextremal
+KN spacetimes for z ∈ R\0 (see Figure 10 for an illustration). Specifically, we first show that the characteristic
+set of ph,z can be split into a disjoint union of two components. Then we prove that there exist radial points
+at event horizon and spatial infinity ∂+ ¯X. Next, we discuss the trapped set associated to the Hamiltonian
+flow of the semiclassical symbol ph,z and prove that it is normally hyperbolic. Finally, we describe the global
+dynamics of the Hamiltonian flow of the semiclassical principal symbol ph,z.
+In §5.5, we combine the global dynamics of the Hamiltonian flow of the semiclassical principal symbol ph,z
+established in §5.4 with the elliptic estimate, propagation of singularities estimate, the radial point estimate
+at event horizon, the scattering radial point estimate at spatial infinity ∂+ ¯X and hyperbolic estimate, all of
+which are in the semiclassical version, together with the estimate at normally hyperbolic trapping to establish
+
+36
+LILI HE
+the high energy estimates for the operators �
+□gb(σ), �
+Pb,γ(σ), �
+Wb,γ(σ), �
+Lb,γ(σ) for |Re σ| ≫ 1, Im σ ≤ C in the
+closed upper half plane. This implies the invertibility of these operators for |Re σ| ≫ 1, Im σ ≤ C in the
+closed upper half plane.
+In §5.6, we shall establish the energy estimates for the solutions to various wave type equations on slowly
+rotating Kerr-Newman metrics.
+Fix KN black hole parameters b = (m, a, Q) close to b0 = (m0, a0, Q0) and let
+g = gb
+Recall that with a = |a|
+g−1 = ρ−2
+b
+�
+∆b∂2
+r + χ2 − 1
+∆b
+�
+(r2 + a2)∂tχ + a∂ϕ
+�2
++ ∂2
+θ +
+1
+sin2 θ
+�
+∂ϕ + a sin2 θ∂tχ
+�2
++ 2χ
+�
+(r2 + a2)∂tχ + a∂ϕ
+�
+∂r
+�
+.
+We define
+�
+□g(σ) := eitb,∗σ□gbe−itb,∗σ.
+Let scT ∗ ¯X be the compact manifold with the corners of codimension two obtained by radial compactification
+of the fibers scT ∗ ¯X (whose detail will be discussed later on). Then its boundary consists of three hypersurfaces
+∂scT ∗ ¯X = scS∗ ¯X ∪ scT ∗∂− ¯
+X ¯X ∪ scT ∗∂+ ¯
+X ¯X
+where scS∗ ¯X = (scT ∗ ¯X \ 0)/R+ is called fiber infinity.
+5.1. Microlocal geometry and dynamics of Kerr-Newman spacetimes. Now, we are ready to discuss
+the microlocal geometry of the KN metric g = gb. Away from ∂+ ¯X, we note that scT ∗ ¯X is isomorphic to
+T ∗ ¯X and write the covectors as
+ξrdr + ξθdθ + ξϕdϕ.
+Then the principal symbol of �
+□g(σ) is given by
+p = σ2(�
+□g(σ)) = −ρ−2
+b
+�
+∆bξ2
+r + ξ2
+θ +
+� a2
+∆b
+(χ2 − 1) +
+1
+sin2 θ
+�
+ξ2
+ϕ + 2χaξϕξr
+�
+.
+Near ∂+ ¯X, we write the scattering covectors as
+ζρ
+dρ
+ρ2 + ζθ
+dθ
+ρ + ζϕ
+dϕ
+ρ ,
+ρ = 1
+r .
+Then the scattering principal symbol (see [65]) of �
+□g(σ) at ∂+ ¯X is given by
+2σζρ − ζ2
+ρ − ζ2
+θ −
+1
+sin2 θζ2
+ϕ = −(ζρ − σ)2 − ζ2
+θ −
+1
+sin2 θζ2
+ϕ + σ2
+To handle the poles {θ = 0} and {θ = π}, where the spherical coordinates break down, we introduce new
+coordinates
+x1 = sin θ cos ϕ,
+x2 = sin θ sin ϕ,
+and let ξ1 and ξ2 be dual variables to x1 and x2 respectively. Correspondingly, we have
+ξθ = (x1ξ1 + x2ξ2) cot θ,
+ξϕ = x1ξ2 − x2ξ1.
+and note that ξϕ is a smooth function in (x1, x2, ξ1, ξ2). Since
+ξ2
+θ +
+1
+sin2 θξ2
+ϕ = 1 − x2
+1 − x2
+2
+x2
+1 + x2
+2
+(x1ξ1 + x2ξ2)2 +
+1
+x2
+1 + x2
+2
+(x1ξ2 − x2ξ1)2 = ξ2
+1 + ξ2
+2 − (x1ξ1 + x2ξ2)2
+is a smooth function in (r, x1, x2, ξr, ξ1, ξ2) near the poles, so is p. Therefore, we can perform all symbol
+calculations away from these two poles and extend the results to the poles. From now on, we do not emphasize
+the analysis near these two poles.
+
+LINEAR STABILITY OF KERR-NEWMAN FAMILY
+37
+5.1.1. Characteristic set. We first study the characteristic set. Away from ∂+ ¯X, since
+p = −ρ−2
+b
+�
+∆b(ξr + aχ
+∆b
+ξϕ)2 + ξ2
+θ + ∆b − a2 sin2 θ
+∆b sin2 θ
+ξ2
+ϕ
+�
+,
+it follows that the characteristic set {p = 0, ξ ̸= 0} of p satisfies
+{p = 0, ξ = (ξr, ξθ, ξϕ) ̸= 0} ⊂ {∆b − a2 sin2 θ ≤ 0, ξ ̸= 0} ⊂ scT ∗ ¯X \ 0.
+Since χ = 1 in the region ∆b − a2 sin2 θ ≤ 0, the principal symbol can be written as
+p = −ρ−2
+b
+�
+∆bξ2
+r + 2aξrξϕ + ˜p
+�
+where
+˜p = ξ2
+θ +
+1
+sin2 θξ2
+ϕ,
+and thus ξr ̸= 0 at the characteristic set of p. As a consequence, in the fiber radial compactification scT ∗ ¯X
+of scT ∗ ¯X, we use the following coordinates near the fiber infinity scT ∗ ¯X,
+ρξ =
+1
+|ξr| ∈ [0, ∞),
+ˆξ = (ˆξθ, ˆξϕ) = ρξ(ξθ, ξϕ).
+(5.1)
+We note that ρξ is a boundary defining function of scT ∗ ¯X, i.e., ρξ = 0 at ∂scT ∗ ¯X = scS∗ ¯X, and ˆξ is a
+coordinate system on scS∗ ¯X.
+Since p is a homogeneous polynomial of order 2 in ξ = (ξr, ξθ, ξϕ), the rescaled symbol ρ2
+ξp and the
+rescaled Hamiltonian vector field ρξHp extend smoothly to scT ∗ ¯X.
+Therefore, we study the flow of the
+rescaled Hamiltonian vector field ρξHp on the characteristic set {ρ2
+ξp = 0} ⊂ scT ∗ ¯X \ 0. We now split the
+characteristic set {ρ2
+ξp = 0} ⊂ scT ∗ ¯X \ 0 into two components
+Σ± := {ρ2
+ξp = 0} ∩ {±ξr > 0},
+∂Σ± = Σ± ∩ scS∗ ¯X.
+Near ∂+ ¯X, we have χ = 0 and thus
+g−1 = ρ−2
+b
+�
+∆b∂2
+r − 1
+∆b
+�
+(r2 + a2)∂tχ + a∂ϕ
+�2
++ ∂2
+θ +
+1
+sin2 θ
+�
+∂ϕ + a sin2 θ∂tχ
+�2�
+.
+Then using tχ = t and tb,∗ = t − r(m,a0,Q),∗ near ∂+ ¯X, we calculate
+�
+□g(σ) = −2iσρ2∂ρ + (ρ2∂ρ)2 + ρ2∂2
+θ +
+ρ2
+sin2 θ ∂2
+ϕ + ρDiff2
+sc( ¯X),
+ρ = 1
+r .
+Therefore, the scattering principal symbol of �
+□g(σ) at ∂+ ¯X is given by
+psc(σ) = σsc(�
+□g(σ)) = −(ζρ − σ)2 − ˜p + σ2,
+˜p = ζ2
+θ +
+1
+sin2 θζ2
+ϕ,
+and the scattering characteristic set is the surface
+Σsc(σ) = {psc(σ) = 0} ⊂ scT ∗∂+ ¯
+X ¯X.
+We note that Σsc(σ) is the scattering zero section o∂+ ¯
+X ⊂ scT ∗∂+ ¯
+X ¯X for Im σ > 0 and σ = 0.
+5.1.2. The radial points at event horizon. We now analyze the behavior of the Hamiltonian flow exp(sρξHp)
+on the characteristic set Σ = Σ+ ∪ Σ− ⊂ scT ∗ ¯X \ 0, that is, we consider the portion whose projection to the
+base space (r, θ, ϕ) is away from ∂+ ¯X. Near Σ = Σ+ ∪ Σ−, we see that χ = 1 and thus
+p = −ρ−2
+b
+�
+∆bξ2
+r + 2aξrξϕ + ˜p
+�
+where
+˜p = ξ2
+θ +
+1
+sin2 θξ2
+ϕ.
+Then we calculate
+Hp = −ρ−2
+b
+�
+(2∆bξr + 2aξϕ)∂r + 2aξr∂ϕ − ∂r∆b
+∂r ξ2
+r∂ξr + H˜p
+�
+− ρ−2
+b pHρ2
+b.
+
+38
+LILI HE
+In the coordinates (5.1) near the fiber infinity scS∗ ¯X, we compute
+ρξHp = −ρ−2
+b
+�
+(2∆b(sgn ξr) + 2aˆξϕ)∂r + 2a(sgn ξr)∂ϕ − ∂∆b
+∂r (sgn ξr)ξr∂ξr + ρξH˜p
+�
+− ρ−2
+b (ρξp)Hρ2
+b
+= −ρ−2
+b
+�
+(2∆b(sgn ξr) + 2aˆξϕ)∂r + 2a(sgn ξr)∂ϕ + 2ˆξθ∂θ +
+2
+sin2 θ
+ˆξϕ∂ϕ
+�
+− ρ−2
+b
+∂∆b
+∂r (sgn ξr)
+�
+ρξ∂ρξ + ˆξθ∂ˆξθ + ˆξϕ∂ˆξϕ
+�
+− ρ−2
+b
+2 cos θ
+sin3 θ
+ˆξ2
+ϕ∂ˆξθ − ρ−2
+b (ρ2
+ξp)ρ−1
+ξ Hρ2
+b.
+(5.2)
+We let
+Λ± = {∆b = 0, (ξθ, ξϕ) = 0, ±ξr > 0}
+and
+L± := ∂Λ± = Λ± ∩ scS∗ ¯X = {∆b = 0, ρξ = ± 1
+ξr
+= 0, ˆξ = 0}.
+Now we describe the property of the set L±. More specifically, L± are radial sources/sinks in the following
+sense (see [21, Definition E.50]):
+Definition 5.1. Let κ : scT ∗ ¯X \ 0 → scS∗ ¯X = (scT ∗ ¯X \ 0)/R+ be the natural projection map, i.e., for
+each (r, θ, ϕ, ξ) ∈ scT ∗ ¯X \ 0, the ray (r, θ, ϕ, sξ) converges to κ((r, θ, ϕ, ξ)) as s → ∞. Then for the rescaled
+Hamiltonian flow
+φs := exp(sρξHp) : scT ∗ ¯X → scT ∗ ¯X,
+we say that a nonempty compact φs-invariant set
+L ⊂ {ρ2
+ξp = 0} ∩ scS∗ ¯X
+is a radial source for p, if there exists a neighborhood U ⊂ scT ∗ ¯X of L such that uniformly in (r, θ, ϕ, ξ) ∈
+U ∩ scT ∗ ¯X, one have
+κ(φs(r, θ, ϕ, ξ)) → L
+as
+s → −∞
+and
+|φs(r, θ, ϕ, ξ)| ≥ BeB|s||ξ|,
+s ≤ 0
+for some B > 0. Here, | · | denotes a norm on the fibers of scT ∗ ¯X.
+A radial sink for p is be definition a radial source for −p.
+Lemma 5.2. L± are invariant under the Hamiltonian flow exp(sρξHp) on the characteristic set Σ. More-
+over, L+ is a radial sink and L− is a radial source for the flow exp(sρξHp), see Figure 4.
+L+
+L−
+Figure 4. The flow of exp(sρξHp) near L±. It is projected to the (r, ξr) coordinates and
+drawn in a fiber-radially compactified view. The horizontal coordinate is r; the dashed line
+is the event horizon {r = rb}. The vertical coordinate is ξr/⟨ξr⟩, so the top and bottom lines
+correspond to the fiber infinity {ρξ = 0}. The midline {ξr = 0} lies outside the characteristic
+set {ρ2
+ξp = 0}.
+Proof. Since {ρ2
+ξ ˜p = 0} ⇔ {ˆξ = (ˆξθ, ˆξϕ) = 0}, we rewrite
+L± = {∆b = 0, ρξ = ± 1
+ξr
+= 0, ρ2
+ξ ˜p = 0}.
+
+LINEAR STABILITY OF KERR-NEWMAN FAMILY
+39
+Using the coordinates (5.1) and expression (5.2), we find that
+ρξHpρξ = −ρ−2
+b
+�∂∆b
+∂r + 2rρ2
+ξp
+�
+(sgn ξr)ρξ,
+ρξHp ˆξϕ = −ρ−2
+b
+�∂∆b
+∂r + 2rρ2
+ξp
+�
+(sgn ξr)ˆξϕ,
+ρξHp∆b = −2ρ−2
+b
+∂∆b
+∂r
+�
+(sgn ξr)∆b + aˆξϕ
+�
+,
+ρξHp(ρ2
+ξ ˜p) = −ρ−2
+b
+�
+2∂∆b
+∂r + 4rρ2
+ξp
+�
+(sgn ξr)ρ2
+ξ ˜p − 2ρ−2
+b a2 sin(2θ)(ρ2
+ξp)ˆξθ.
+(5.3)
+This implies that ρξHp is tangent to L±, so L± is invariant under the flow exp(sρξHp). Moreover, in a
+neighborhood of L± = {∆b = 0, ρξ = ±ξ−1
+r
+= 0, ρ2
+ξ ˜p = 0} ⊂ {ρ2
+ξp = 0}, we have
+±ρξHpρ2
+ξ ≤ −C0ρ2
+ξ,
+±ρξHp ˆξ2
+ϕ ≤ −C0 ˆξ2
+ϕ,
+±ρξHp∆2
+b ≤ −C0∆2
+b + C1 ˆξ2
+ϕ,
+±ρξHp(ρ2
+ξ ˜p) ≤ −C0ρ2
+ξ ˜p + C1
+�
+∆2
+b + ˆξ2
+ϕ
+�
+.
+(5.4)
+where C0, C1 > 0. It follows from (5.4) that in a sufficiently small neighborhood of L±
+± ρξHpf ≤ −Cf,
+f = ρ2
+ξ + ˆξ2
+ϕ + C2∆2
+b + C3ρ2
+ξ ˜p ≥ 0.
+(5.5)
+for some C, C2, C3 > 0. This implies that for (r, θ, ϕ, ξ) in a sufficiently small neighborhood of L±, we have
+f(φs((r, θ, ϕ, ξ))) ≤ e−C|s|f((r, θ, ϕ, ξ))
+for
+± s > 0.
+Therefore, we have uniformly in (r, θ, ϕ, ξ) in a neighborhood of L±
+φs((r, θ, ϕ, ξ)) → L±
+as
+s → ±∞.
+Moreover, according to the first inequality in (5.4), we have
+ρξ(φs((r, θ, ϕ, ξ))) ≤ e−C0|s|ρξ((r, θ, ϕ, ξ))
+for
+± s ≥ 0.
+This proves that L+ is a radial sink and L− is a radial source.
+□
+Finally, we consider the imaginary part of the operator �
+□g(σ) at L±, which is needed to calculate the
+threshold quantity for the order of the Sobolev spaces (here the differential order) in the radial point estimates.
+Recall that near L±, the inverse metric G takes the form
+g−1 = ρ−2
+b
+�
+a2 sin2 θ∂2
+tχ + 2(r2 + a2)∂tχ∂r + 2a∂tχ∂ϕ + ∆b∂2
+r + 2a∂r∂ϕ + ∂2
+θ +
+1
+sin2 θ ∂2
+ϕ
+�
+and tχ = tb,∗. A direct calculation shows that
+�
+□g(σ)∗ = �
+□g(¯σ).
+where �
+□g(σ)∗ denotes the formal adjoint of �
+□g(σ) defined by ⟨�
+□gb(σ)u, , v⟩ = ⟨u, �
+□g(σ)∗v⟩ with respect to
+L2( ¯X;
+�
+| det g|drdθdϕ) for u, v ∈ C∞
+c (X). We first compute
+p1 := σ1
+�
+Im �
+□g(σ)
+�
+= σ1
+� �
+□g(σ) − �
+□g(σ)∗
+2i
+�
+= ρ−2
+b Im σ
+�
+2(r2 + a2)ξr + 2aξϕ
+�
+Let β0 ∈ C∞(L±) be a positive function defined as
+β0 = ∓ρξHpρξ
+ρξ
+|L±.
+Then we define ˜β ∈ C∞(L±) as
+˜β = ∓ρξp1
+β0
+|L±
+Let
+βsup = sup ˜β,
+βinf = inf ˜β.
+
+40
+LILI HE
+If ˜β is a constant along L±, we may write
+β = βsup = βinf.
+We note that βsup and βinf are the relevant quantities in the radial point estimates. More specifically, in our
+setting where we consider the operator �
+□g(σ) with Im σ ≥ 0, the threshold regularity condition in the high
+regularity radial estimate is given by
+s > 1
+2 + βsup,
+(5.6)
+while the low regularity radial estimate requires
+s < 1
+2 + βinf.
+(5.7)
+Using the first equality in (5.3), we find for the operator �
+□g(σ) that
+˜β = −Im σ r2
+b + a2
+rb − m ,
+and thus
+β = βsup = βinf = −Im σ r2
+b + a2
+rb − m ≤ 0
+for
+Im σ ≥ 0.
+(5.8)
+5.1.3. The radial points at spatial infinity ∂+ ¯X. We now turn to the analysis near ∂+ ¯X. Recall that for the
+operator �
+□g(σ) with Im σ ≥ 0, the scattering characteristic set is the surface
+Σsc(σ) = {(ρ = 1
+r = 0, θ, ϕ, ζ)) : psc(σ) = 0} ⊂ scT ∗∂+ ¯
+X ¯X.
+where
+psc(σ) = −(ζρ − σ)2 − ˜p + σ2,
+˜p = ζ2
+θ +
+1
+sin2 θζ2
+ϕ.
+Using the identification of T ∗ ¯X and scT ∗ ¯X in ρ > 0
+ζρ = ρ2ξρ = −ξr,
+(ζθ, ζϕ) = ρ(ξθ, ξϕ),
+we find that the scattering Hamiltonian vector field associated to p in ρ > 0 is given by
+scHp = ρ2 ∂p
+∂ζρ
+� ∂
+∂ρ + 2ζρ
+ρ
+∂
+∂ζρ
++
+� �
+µ=θ,ϕ
+ζµ
+ρ
+∂
+∂ζµ
+��
+−
+�∂p
+∂ρ + 2ζρ
+ρ
+∂p
+∂ζρ
++
+� �
+µ=θ,ϕ
+ζµ
+ρ
+∂p
+∂ζµ
+��
+ρ2 ∂
+∂ζρ
++
+�
+µ=θ,ϕ
+ρ
+� ∂p
+∂ζµ
+∂
+∂µ − ∂p
+∂µ
+∂
+∂ζµ
+�
+= ρ
+� ∂p
+∂ζρ
+�
+ρ ∂
+∂ρ + (
+�
+µ=θ,ϕ
+ζµ
+∂
+∂ζµ
+)
+�
+−
+�
+ρ∂p
+∂ρ + (
+�
+µ=θ,ϕ
+ζµ
+∂p
+∂ζµ
+)
+� ∂
+∂ζρ
++
+�
+µ=θ,ϕ
+� ∂p
+∂ζµ
+∂
+∂µ − ∂p
+∂µ
+∂
+∂ζµ
+��
+.
+By introducing the following coordinates in the fiber radial compactification scT ∗ ¯X of scT ∗ ¯X
+ρζ = (1 + ζ2
+r + ζ2
+θ +
+1
+sin2 θζ2
+ϕ)− 1
+2 ,
+ˆζ = (ˆζρ, ˆζθ, ˆζϕ) = ρζ(ζρ, ζθ, ζϕ),
+the rescaled scattering Hamiltonian vector field ρζρ−1scHp can be extended to an element in Vb(scT ∗ ¯X)
+taking the following form
+scH2,0
+p
+:= ρ2−1
+ζ
+ρ−0−1scHp
+= ρζ
+� ∂p
+∂ζρ
+�
+ρ ∂
+∂ρ + (
+�
+µ=θ,ϕ
+ζµ
+∂
+∂ζµ
+)
+�
+−
+�
+ρ∂p
+∂ρ + (
+�
+µ=θ,ϕ
+ζµ
+∂p
+∂ζµ
+)
+� ∂
+∂ζρ
++
+�
+µ=θ,ϕ
+� ∂p
+∂ζµ
+∂
+∂µ − ∂p
+∂µ
+∂
+∂ζµ
+��
+.
+We now discuss the integral curves of scH2,0
+psc(σ) on the scattering characteristic set Σsc(σ) = {psc(σ) =
+0} ⊂ scT ∗∂+ ¯
+X ¯X for σ ∈ R \ 0 (see [63]). Since Σsc(σ) has no intersection with the fiber infinity ∂scT ∗∂+ ¯
+X ¯X,
+we can drop the factor ρζ which is non-zero on Σsc(σ). Therefore, with psc(σ) = −(ζρ − σ)2 − ˜p + σ2, we
+write
+scH2,0
+psc(σ) = (−2ζρ + 2σ)ρ∂ρ + 2˜p∂ζρ + (−2ζρ + 2σ)
+�
+µ=θ,ϕ
+ζµ∂ζµ − H˜p
+on
+Σsc(σ)
+
+LINEAR STABILITY OF KERR-NEWMAN FAMILY
+41
+where
+H˜p = 2ζθ∂θ + 2ζϕ
+sin2 θ∂ϕ + cos θ
+sin2 θ ζ2
+ϕ∂ζθ.
+Lemma 5.3. For 0 ̸= σ ∈ R, on the characteristic variety
+Σsc(σ) = {psc(σ) = −(ζρ − σ)2 − ˜p + σ2 = 0} ⊂ scT ∗∂+ ¯
+X ¯X,
+˜p = ζ2
+θ +
+1
+sin2 θζ2
+ϕ,
+(5.9)
+there are two invariant submanifolds under the Hamiltonian flow exp(sscH2,0
+psc(σ)), one of which is the zero
+section R(0) = {ρ = 0, ζρ = 0, (ζθ, ζϕ) = 0} and the other is R(σ) = {ρ = 0, ζρ = 2σ, (ζθ, ζϕ) = 0}.
+For (θ0, ϕ0, ζ0
+ρ, ζ0
+θ, ζ0
+ϕ) ∈ Σsc(σ) \ (R(0) ∪ R(σ)), the integral curves exp(sscH2,0
+psc(σ))(θ0, ϕ0, ζ0
+ρ, ζ0
+θ, ζ0
+ϕ) take
+the following form
+ζρ(s) = σ + |σ| sin(s + s0),
+ζρ(0) = ζ0
+ρ,
+(ζθ, ζϕ) = |σ| cos(s + s0)(˜ζθ, ˜ζϕ),
+|σ| cos(s0) = ˜p(θ0, ϕ0, ζ0
+θ, ζ0
+ϕ)
+1
+2 ,
+(θ, ϕ, ˜ζθ, ˜ζϕ) = exp((s + s0)H− 1
+2 ˜p)(θ0, ϕ0, ˜ζ0
+θ, ˜ζ0
+ϕ),
+(˜ζ0
+θ, ˜ζ0
+ϕ) = ˜p(θ0, ϕ0, ζ0
+θ, ζ0
+ϕ)− 1
+2 · (ζ0
+θ, ζ0
+ϕ)
+(5.10)
+where s0 ∈ (− π
+2 , π
+2 ), s ∈ (− π
+2 − s0, π
+2 − s0) and
+ds
+ds′ = 2˜p(θ, ϕ, ζθ, ζϕ)
+1
+2 where
+d
+ds′ := scH2,0
+psc(σ).
+This implies that for σ > 0, the zero section is a radial source and the nonzero one is a radial sink, while
+for σ < 0, the zero section is a radial sink and the nonzero one is a radial source, see Figure 5.
+(ζθ, ζϕ)
+ζρ
+R(σ)
+R(0)
+(ζθ, ζϕ)
+ζρ
+R(σ)
+R(0)
+Figure 5.
+The integral curves of scH2,0
+psc(σ) in ζρ and (ζθ, ζϕ) coordinates. The left hand
+side is the case σ > 0 and the right hand side is the case σ < 0.
+Proof. At ∂ ¯X, the rescaled scattering Hamiltonian vector field is
+scH2,0
+psc(σ) = (−2ζρ + 2σ)ρ∂ρ + 2˜p∂ζρ + (−2ζρ + 2σ)
+�
+µ=θ,ϕ
+ζµ∂ζµ − H˜p,
+Introducing the polar coordinates with respect to the radial variable in (ζθ, ζϕ)
+(˜ζθ, ˜ζϕ) = ˜p(θ, ϕ, ζθ, ζϕ)− 1
+2 · (ζθ, ζϕ),
+|(ζθ, ζϕ)| = ˜p(θ, ϕ, ζθ, ζϕ)
+1
+2 .
+Let
+d
+ds′ := scH2,0
+psc(σ). Then we have
+d
+ds′ ζρ = 2˜p(θ, ϕ, ζθ, ζϕ),
+d
+ds′ ˜p(θ, ϕ, ζθ, ζϕ)1/2 = −2(ζρ − σ)˜p(θ, ϕ, ζθ, ζϕ)
+1
+2 ,
+d
+ds′ (˜ζθ, ˜ζϕ) = ˜p− 1
+2 (∂˜p
+∂θ, ∂˜p
+∂ϕ)(θ, ϕ, ζθ, ζϕ),
+d
+ds′ (θ, ϕ) = −( ∂˜p
+∂ζθ
+, ∂˜p
+∂ζϕ
+)(θ, ϕ, ζθ, ζϕ).
+This implies that
+d
+ds′ is tangent to R(0) and R(σ), so they are invariant under the Hamiltonian flow
+exp(sscH2,0
+psc(σ)).
+When ˜p(θ, ϕ, ζθ, ζϕ) ̸= 0, we introduce a new parameter s satisfying
+ds
+ds′ = 2˜p(θ, ϕ, ζθ, ζϕ)
+1
+2 and obtain
+d
+ds(ζρ − σ) = ˜p(θ, ϕ, ζθ, ζϕ)
+1
+2 ,
+d
+ds ˜p(θ, ϕ, ζθ, ζϕ)
+1
+2 = −(ζρ − σ),
+d
+ds(˜ζθ, ˜ζϕ) = 1
+2(∂˜p
+∂θ, ∂˜p
+∂ϕ)(θ, ϕ, ˜ζθ, ˜ζϕ),
+d
+ds(θ, ϕ) = −1
+2( ∂˜p
+∂˜ζθ
+, ∂˜p
+∂˜ζϕ
+)(θ, ϕ, ˜ζθ, ˜ζϕ).
+(5.11)
+
+42
+LILI HE
+Integrating the above system with respect to the new parameter s gives (5.10).
+□
+For the propagation of singularities estimates at these radial points R(σ) and R(0), there is a threshold
+quantity for the order of the Sobolev spaces (here the scattering decay order). Let β0 be a positive function
+defined at the radial points as
+scH2,0
+psc(σ)ρ = ±β0ρ.
+Here and in what follows, we use + for radial sources and − for radial sinks. We next define ˜β at the radial
+points as
+σsc(ρ−1Im �
+□g(σ)) = σsc(
+�
+□g(σ) − �
+□g(σ)∗
+2iρ
+) = ±β0 ˜β
+Let
+βsup = sup ˜β,
+βinf = inf ˜β
+where the supremum and infimum are taken over the radial points set. If ˜β is a constant along the radial
+points set, we may write
+β = βsup = βinf.
+We note that βsup and βinf are the relevant quantities in the radial point estimates. More specifically, in our
+setting where we consider the operator �
+□g(σ) with σ ∈ R \ 0, the threshold scattering decay condition in the
+high scattering decay radial estimate is given by
+r > −1
+2 − βinf,
+(5.12)
+while the low scattering decay radial estimate requires
+r < −1
+2 − βsup.
+(5.13)
+In our setting, since �
+□g(σ) = �
+□g(σ)∗ for σ ∈ R \ 0, it follows that
+β = βsup = βinf = 0
+on
+R(σ)
+and
+R(0).
+(5.14)
+5.1.4. Global dynamics of the Hamiltonian flow. Now we study the global behavior of the flow of ρξHp (resp.,
+scH2,0
+psc(σ)) away from infinity ∂+ ¯X = {ρ = 1
+r = 0} (resp., at ∂+ ¯X).
+Proposition 5.4. The characteristic set of �
+□g(σ) is a disjoint union Σ ∪ Σsc(σ) (see Figure 6).
+(1) Let (r, θ, ϕ, ξ) ∈ Σ± and put φ(s) = exp(sρξHp)(r, θ, ϕ, ξ). Then φ(s) → L± as s → ±∞. Moreover,
+if (r, θ, ϕ, ξ) /∈ Λ± = {∆b = 0, (ξθ, ξϕ) = 0, ±ξr > 0}, then φ(s) crosses ∂− ¯X into the inward direction
+of deceasing r at some s0 with ±s0 ≤ 0.
+(2) For σ ∈ R \ 0, let (ρ = 0, θ, ϕ, ζ) ∈ Σsc(σ) and put φ(s) = exp(sscH2,0
+psc(σ))(θ, ϕ, ζ). Then either
+φ(s) → R( σ
+2 + |σ|
+2 ) = {ρ = 0, ζρ = σ + |σ|, ˜p = 0} as s → ∞, or φ(s) → R( σ
+2 − |σ|
+2 ) = {ρ = 0, ζρ =
+σ − |σ|, ˜p = 0} as s → −∞. Moreover, if (ρ = 0, θ, ϕ, ζ) /∈ R(0) ∪ R(σ), then φ(s) → R( σ
+2 ± |σ|
+2 ) as
+s → ±∞.
+Proof. We only discuss the case Σ+ as the other case Σ− can be handled in a similar manner. Let (r, θ, ϕ, ξ) ∈
+Σ+ and put φ(s) = exp(sρξHp)(r, θ, ϕ, ξ). Since Σ+ is invariant under the flow exp(sρξHp), it follows that
+φ(s) ∈ Σ± for all s for which it is well-defined. Put
+f(s) =
+�
+ρ2
+ξ + (∆b + 2asgn(ξr)ˆξϕ)2 + ρ2
+ξ ˜p
+�
+(φ(s)),
+˙f(s) = ρξHpf(φ(s)).
+Using the calculation in (5.3), we find that
+˙f(s) ≤ −2
+�
+ρ−2
+b
+∂∆b
+∂r f
+�
+(φ(s)) ≤ −C0f(s)
+on
+Σ+ ⊂ {r ≥ r− | ∆b − a2 sin2 θ ≤ 0}
+for some C0 > 0. This gives
+f(s) ≤ e−C0sf(0) = e−C0sf((r, θ, ϕ, ξ))
+for
+s ≥ 0
+and thus
+φ(s) → L+
+as
+s → ∞.
+
+LINEAR STABILITY OF KERR-NEWMAN FAMILY
+43
+Σsc(σ)
+R(σ)
+R(0)
+L+
+L−
+Σ+
+Σ−
+σ > 0
+Σsc(σ)
+R(σ)
+R(0)
+L+
+L−
+Σ+
+Σ−
+σ < 0
+Σsc(σ)
+L+
+L−
+Σ+
+Σ−
+Im σ > 0 or σ = 0
+Figure 6. The flow of ρξHp (resp.
+scH1,0
+psc(σ)) away from infinity ∂+ ¯X = {ρ =
+1
+r = 0}
+(resp., at ∂+ ¯X), which is projected to the coordinates (r, ξr) (resp. (ρ, ζρ)) and drawn in
+a fiber-radially compactified view. The shaded region is the characteristic set Σ = Σ+ ∪
+Σ− = {ρ2
+ξp = 0} away from ∂+ ¯X, while the red region is the scattering characteristic set
+Σsc(σ) = {psc(σ) = 0} at ∂+ ¯X. The horizontal coordinate is ρ and the rightmost vertical
+line corresponds to ∂+ ¯X. The dotted line is the even horizon r = rb and the vertical line
+in the middle is {r = rergo} where rergo is the equatorial radius (i.e. greatest radius) of the
+ergosphere. The vertical coordinate is ξr/⟨ξr⟩ (resp. ζρ/⟨ζρ⟩), so the top and bottom lines
+stand for the fiber infinity. The midline {ξr = 0} (resp. ζρ = 0) does not intersect Σ, but
+intersects Σsc(σ).
+Using the last expression in (5.3), we have
+ρξHp(ρ2
+ξ ˜p)(φ(s)) = −2
+�
+ρ−2
+b
+∂∆b
+∂r ρ2
+ξ ˜p
+�
+(φ(s)) ≤ −C0(ρ2
+ξ ˜p)(φ(s)).
+for some C0 > 0. Therefore
+(ρ2
+ξ ˜p)(φ(s)) ≥ e−C0s(ρ2
+ξ ˜p)(r, θ, ϕ, ξ)
+for s ≤ 0 as long as the flow remains in the region Σ+ ⊂ {r ≥ r− | ∆b − a2 sin2 θ ≤ 0}. Since on Σ+
+2a2 + 1
+2
+ˆξ2
+ϕ ≥ −2aˆξϕ = ∆b + ρ2
+ξ ˜p ≥ ∆b + 1
+2ρ2
+ξ ˜p + 1
+2
+ˆξ2
+ϕ
+where we use ∆b +2aˆξϕ +ρ2
+ξ ˜p = 0 on Σ+ in the second equality and ρ2
+ξ ˜p ≥ ˆξ2
+ϕ in the last inequality, it follows
+that
+2a2 − ∆b ≥ 1
+2ρ2
+ξ ˜p
+on
+Σ+.
+Since
+{ρ2
+ξ ˜p = 0} ∩ Σ+ = Λ+,
+then if (r, θ, ϕ, ξ) ∈ Σ+ \ Λ+, we have
+(2a2 − ∆b)(φ(s)) ≥ 1
+2ρ2
+ξ ˜p(φ(s)) ≥ Ce−C0s,
+C, C0 > 0
+for s ≤ 0 as long as the flow remains in the region Σ+ ⊂ {r ≥ r− | ∆b − a2 sin2 θ ≤ 0}. As a consequence,
+φ(s) cross ∂− ¯X into the inward direction of deceasing r at some s0 with s0 ≤ 0. This proves the part (i).
+The proof of part (2) follows from Lemma 5.3.
+□
+5.1.5. Second microlocalized scattering algebra. To do the analysis at the scattering zero section at ∂+ ¯X, we
+introduce the second microlocalized ¯X by blowing up the zero section o∂+ ¯
+X at ∂+ ¯X (i.e. replacing o∂+ ¯
+X by
+its inward pointing spherical normal bundle and thus obtaining a smooth manifold with corners), see Figure
+7.
+
+44
+LILI HE
+scT ∗∂− ¯
+X ¯X
+scT ∗∂+ ¯
+X ¯X
+scS∗ ¯X
+scS∗ ¯X
+¯X × {0}
+Σsc(σ)
+R(0)
+R(σ)
+σ > 0
+scT ∗∂− ¯
+X ¯X
+scT ∗∂+ ¯
+X ¯X
+scS∗ ¯X
+scS∗ ¯X
+¯X × {0}
+R(σ)
+bT ∗
+∂+ ¯
+X ¯X
+σ > 0
+Figure 7. Second microlocalized
+¯X.
+The left hand side is the fiber-compactification
+scT ∗ ¯X of the scattering cotangent bundle scT ∗ ¯X, and the right hand side is its blow-up
+[scT ∗ ¯X; o∂+ ¯
+X] at the scattering zero section at the boundary ∂+ ¯X. Also shown is the scat-
+tering characteristic set Σsc(σ) at ∂+ ¯X of �
+□g(σ) for σ > 0. The interior of the front face of
+the blow-up, shown by curved arcs, can be identified with bT ∗
+∂+ ¯
+X ¯X.
+On the other hand, from an analytically better behaved, but geometrically equivalent perspective, one
+takes bT ∗ ¯X and blows up its corner bS∗
+∂+ ¯
+X ¯X, i.e. the fiber infinity at ∂+ ¯X, see Figure 8. These two spaces
+[scT ∗ ¯X; o∂+ ¯
+X] and [bT ∗ ¯X; bS∗
+∂+ ¯
+X ¯X] are naturally the same, see [80, Lemma 5.1].
+bT ∗∂− ¯
+X ¯X
+bT ∗∂+ ¯
+X ¯X
+bS∗ ¯X
+bS∗ ¯X
+¯X × {0}
+bT ∗∂− ¯
+X ¯X
+bS∗ ¯X
+bS∗ ¯X
+¯X × {0}
+bT ∗∂+ ¯
+X ¯X
+ff=[scT ∗∂+ ¯
+X ¯X; o∂+ ¯
+X]
+ff=[scT ∗∂+ ¯
+X ¯X; o∂+ ¯
+X]
+Figure 8. The left hand side is the fiber-compactification bT ∗ ¯X of the b-cotangent bundle
+bT ∗ ¯X, and the right hand side is its blow-up [bT ∗ ¯X; bS∗
+∂+ ¯
+X ¯X] at the fiber infinity at the
+boundary ∂+ ¯X. The interior of the front face of the blow-up, shown by curved arcs, can be
+identified with [scT ∗
+∂+ ¯
+X ¯X; o∂+ ¯
+X].
+In order to actually do analysis, one needs to introduce the pseudodifferential operator space Ψs,r,l
+sc,b( ¯X).
+The three orders of Ψs,r,l
+sc,b ( ¯X) are the sc-diffferential order s, the sc-decay order r and b-decay order l. The
+operators Ψs,r,l
+sc,b( ¯
+X) arise from the quantization of the symbols in the following class
+Ss,r,l( ¯X) := ρ−l
+b ρ−r
+sc ρ−s
+∞ S0,0,0([bT ∗ ¯X; bS∗
+∂+ ¯
+X ¯X]) = ρ−l
+b ρ−r
+sc ρ−s
+∞ S0,0(bT ∗ ¯X)
+where ρ∞, resp. ρsc, resp. ρb correspond to the boundary defining functions of the three boundary hypersur-
+faces of [bT ∗ ¯X; bS∗
+∂+ ¯
+X ¯X] (or equivalently [scT ∗ ¯X; o∂+ ¯
+X]): the lift of b-fiber infinity (or equivalently the lift of
+sc-fiber infinity), the new front face of [bT ∗ ¯X; bS∗
+∂+ ¯
+X ¯X] (or the lift of scT ∗
+∂+ ¯
+X ¯X in [scT ∗ ¯X; o∂+ ¯
+X]), and the lift
+of ∂+ ¯X in [bT ∗ ¯X; bS∗
+∂+ ¯
+X ¯X] (or the new front face of [scT ∗ ¯X; o∂+ ¯
+X]), respectively. Here, the symbol spaces
+S0,0(bT ∗ ¯X) (resp. S0,0,0([bT ∗ ¯X; bS∗
+∂+ ¯
+X ¯X])) consist of L∞(bT ∗ ¯X) (resp. L∞([bT ∗ ¯X; bS∗
+∂+ ¯
+X ¯X])) functions
+which are smooth away from the boundary hypersurfaces bS∗ ¯X ∪ bT ∗∂+ ¯
+X ¯X (resp. bS∗ ¯X ∪ bT ∗∂+ ¯
+X ¯X ∪ ff),
+and they remain so under iterated applications of C∞ b-vector fields on bT ∗ ¯X (resp. [bT ∗ ¯X; bS∗ ¯X]), namely,
+vector fields tangent to the aforementioned boundary hypersurfaces. In particular, we have
+Ψs,s+l,l
+sc,b
+( ¯X) = Ψs,l
+b ( ¯X).
+
+LINEAR STABILITY OF KERR-NEWMAN FAMILY
+45
+We refer the reader to [80, §5] for a more detailed description of the operators Ψs,r,l
+sc,b( ¯X) and the symbols
+Ss,r,l( ¯X).
+Correspondingly, one can define the second microlocal Sobolev space Hs,r,l
+sc,b as follows. Fix an elliptic
+operator A ∈ Ψs,r,l
+sc,b( ¯
+X), let
+Hs,r,l
+sc,b ( ¯X) = {u ∈ C−∞( ¯
+X) | Au ∈ L2}.
+Therefore, we have
+Hs,s+l,l
+sc,b
+( ¯X) = Hs,l
+b ( ¯X).
+5.2. Uniform Fredholm estimates. In this section, we prove the uniform Fredholm estimates first for
+operators �
+□gb(σ) acting on scalar functions, then for �
+Pb,γ(σ), �
+Wb,γ(σ) acting on scattering 1-forms and the
+linearized gauge-fixed Einstein-Maxwell operator �
+Lb,γ(σ), as well as their formal adjoints for σ ∈ C, |σ| ≤ C.
+To this end, we combine the global dynamics of of the Hamiltonian flow of the principal symbol pσ established
+in 5.4 with the elliptic estimate, propagation of singularities estimate, the radial point estimate at event
+horizon, the scattering radial point estimate at spatial infinity ∂+ ¯X and hyperbolic estimate.
+5.2.1. Uniform Fredholm estimates for scalar wave operators. We first prove uniform Fredholm estimates for
+the operator �
+□gb(σ) acting on scalar functions, as well as its formal adjoint with respect to volume density
+L2( ¯X;
+�
+det |gb|drdθdϕ) for σ ∈ C, |σ| ≤ C.
+B+
+B−
+B0
+E0
+Σsc(σ)
+R(σ)
+Bσ
+R(0)
+L+
+L−
+suppχ
+B′
++
+E′
++
+B′
+−
+E′
+−
+L+
+L−
+Σsc(σ)
+R(σ)
+R(0)
+B′
+0
+B′
+σ
+E′
+σ
+suppχ
+supp(1 − χ)
+Figure 9. A phase space picture of the proof of the estimate (5.23), on the left, and (5.24),
+on the right. The coordinates and notations L±, R(0), R(σ) are the same as in Figure 6.
+For (5.23), we use hyperbolic estimates to control u via χu; χ is controlled (modulo elliptic
+estimates) by B±, B0, Bσ; B0 is controlled by E0 using low b-decay radial point estimates
+and E0 is again controlled by Bσ; finally B±, Bσ are controlled using high regularity radial
+points estimates at L± and high sc-decay radial point estimates at R(σ). For (5.24), we
+use hyperbolic estimates to bound (1 − χ)u; χ is controlled (modulo elliptic estimates) by
+B′
+±, B′
+0, B′
+σ; B′
+± is controlled by E′
+± using low regularity radial point estimates and E′
+± is
+controlled by 1 − χ; B′
+σ is controlled by E′
+σ using low sc-decay radial points estimates and
+E′
+σ is controlled by B′
+0; finally B′
+0 is controlled using high b-decay radial points estimates
+at R(0).
+Theorem 5.5. Let b0 = (m0, a0, Q0) with |Q0| + |a0| < m0.
+Then there exists ǫ > 0 such that for
+b = (m, a, Q) with |b − b0| < ǫ and for s > s0 > 1
+2, ℓ − 1 ≤ ℓ0 < ℓ < − 1
+2 with s + ℓ > s0 + ℓ0 > − 1
+2 and
+ℓ ̸= − 3
+2, the following holds.
+
+46
+LILI HE
+(1) For any fixed C1 > 0, there exist C > 0 independent of b, such that for σ ∈ C, Im σ ≥ 0, C−1
+1
+≤ |σ| ≤
+C1, we have
+∥u∥ ¯
+Hs,ℓ
+b
+( ¯
+X) ≤ C
+�
+∥�
+□gb(σ)u∥ ¯
+Hs,ℓ+1
+b
+( ¯
+X) + ∥u∥ ¯
+Hs0,ℓ0
+b
+( ¯
+X)
+�
+,
+(5.15)
+∥u∥ ˙H−s,−ℓ−1
+b
+( ¯
+X) ≤ C
+�
+∥�
+□gb(σ)∗u∥ ˙H−s,−ℓ
+b
+( ¯
+X) + ∥u∥ ˙H−N,−ℓ−3
+b
+( ¯
+X)
+�
+;
+(5.16)
+∥u∥ ¯
+Hs,ℓ
+b
+( ¯
+X) ≤ C
+�
+∥�
+□gb(σ)u∥ ¯
+Hs−1,ℓ+2
+b
+( ¯
+X) + ∥u∥ ¯
+Hs0,ℓ0
+b
+( ¯
+X)
+�
+,
+(5.17)
+∥u∥ ˙H−s+1,−ℓ−2
+b
+( ¯
+X) ≤ C
+�
+∥�
+□gb(σ)∗u∥ ˙H−s,−ℓ
+b
+( ¯
+X) + ∥u∥ ˙H−N,−ℓ−3
+b
+( ¯
+X)
+�
+(5.18)
+where C only depends on b0, s, s0, ℓ, δ, C1. Therefore, the operators
+�
+□gb(σ) : {u ∈ ¯Hs,ℓ
+b ( ¯X) | �
+□gb(σ)u ∈ ¯Hs,ℓ+1
+b
+( ¯X)} → ¯Hs,ℓ+1
+b
+( ¯X)
+(5.19)
+�
+□gb(σ) : {u ∈ ¯Hs,ℓ
+b ( ¯X) | �
+□gb(σ)u ∈ ¯Hs−1,ℓ+2
+b
+( ¯X)} → ¯Hs−1,ℓ+2
+b
+( ¯X)
+(5.20)
+are Fredholm for Im σ ≥ 0, σ ̸= 0.
+(2) Let ℓ ∈ (− 3
+2, − 1
+2). For any fixed C1 > 0, there exist C > 0 independent of b, such that for σ ∈
+C, Im σ ≥ 0, |σ| ≤ C1, we have
+∥u∥ ¯
+Hs,ℓ
+b
+( ¯
+X) ≤ C
+�
+∥�
+□gb(σ)u∥ ¯
+Hs−1,ℓ+2
+b
+( ¯
+X) + ∥u∥ ¯
+Hs0,ℓ0
+b
+( ¯
+X)
+�
+,
+∥u∥ ˙H−s+1,−ℓ−2
+b
+( ¯
+X) ≤ C
+�
+∥�
+□gb(σ)∗u∥ ˙H−s+1,−ℓ−2
+b
+( ¯
+X) + ∥u∥ ˙H−N,−ℓ−3
+b
+( ¯
+X)
+�
+(5.21)
+where C only depends on b0, s, s0, ℓ, δ, C1. Therefore, the operator
+�
+□gb(0) : {u ∈ ¯Hs,ℓ
+b ( ¯X) | �
+□gb(0)u ∈ ¯Hs−1,ℓ+2
+b
+( ¯X)} → ¯Hs−1,ℓ+2
+b
+( ¯X)
+(5.22)
+is Fredholm.
+Proof. We first prove the statement (1).
+We claim that it suffices to prove the following estimates for
+s > s0 > 1
+2, r > r0 > − 1
+2, ℓ < − 1
+2, ℓ ̸= − 3
+2
+∥u∥ ¯
+Hs,r,ℓ
+sc,b ( ¯
+X) ≤ C
+�
+∥�
+□gb(σ)u∥ ¯
+Hs−1,r+1,ℓ+1
+sc,b
+( ¯
+X) + ∥u∥ ¯
+Hs0,r0,ℓ−1
+sc,b
+( ¯
+X)
+�
+(5.23)
+and
+∥u∥ ˙H−s+1,−r−1,−ℓ−1
+sc,b
+( ¯
+X) ≤ C
+�
+∥�
+□gb(σ)∗u∥ ˙H−s,−r,−ℓ
+sc,b
+( ¯
+X) + ∥u∥ ˙H−N,−N,−ℓ−3
+sc,b
+( ¯
+X)
+�
+(5.24)
+Letting r = s + ℓ > − 1
+2 and r0 = s0 + ℓ0 > − 1
+2, we have
+∥u∥ ¯
+Hs,s+ℓ,ℓ
+sc,b
+( ¯
+X) ≤ C
+�
+∥�
+□gb(σ)u∥ ¯
+Hs−1,s+ℓ+1,ℓ+1
+sc,b
+( ¯
+X) + ∥u∥ ¯
+Hs0,s0+ℓ0,ℓ0
+sc,b
+( ¯
+X)
+�
+and
+∥u∥ ˙H−s+1,−s−ℓ−1,−ℓ−1
+sc,b
+( ¯
+X) ≤ C
+�
+∥�
+□gb(σ)∗u∥ ˙H−s,−s−ℓ,−ℓ
+sc,b
+( ¯
+X) + ∥u∥ ˙H−N,−N,−ℓ−3
+sc,b
+( ¯
+X)
+�
+.
+Then using the facts ¯Hs,s+ℓ,ℓ
+sc,b
+= ¯Hs,l
+b , ˙Hs,s+ℓ,ℓ
+sc,b
+= ˙Hs,l
+b ,
+¯Hs,ℓ+1
+b
+= ¯Hs,s+ℓ+1,ℓ+1
+sc,b
+⊂ ¯Hs−1,s+ℓ+1,ℓ+1
+sc,b
+,
+¯Hs−1,ℓ+2
+b
+= ¯Hs−1,s+ℓ+1,ℓ+2
+sc,b
+⊂ ¯Hs−1,s+ℓ+1,ℓ+1
+sc,b
+and
+˙H−s,−ℓ−1
+b
+= ˙H−s,−s−ℓ−1,−ℓ−1
+sc,b
+⊃ ˙H−s+1,−s−ℓ−1,−ℓ−1
+sc,b
+,
+˙H−s+1,−ℓ−2
+b
+= ˙H−s+1,−s−ℓ−1,−ℓ−2
+sc,b
+⊃ ˙H−s+1,−s−ℓ−1,−ℓ−1
+sc,b
+,
+we obtain (5.15), (5.16), (5.17) and (5.18). Now we will show (5.23) and (5.24). Here we only discuss the
+case Re σ > 0 (see Figure 9 for a phase space illustration of the proof of estimates (5.23) and (5.24)), as the
+case Re σ ≤ 0 can be handled in a similar manner.
+
+LINEAR STABILITY OF KERR-NEWMAN FAMILY
+47
+• Proof of the estimate (5.23). For s > s0 >
+1
+2 ≥
+1
+2 − Im (σ) r2
+b +a2
+r2
+b−m (as calculated (5.8)), using the
+high regularity radial point estimate (see [77, Proposition 2.3], [78, Proposition 5.27]), there exist
+B±, G±, S± ∈ Ψ0( ¯X) microlocally supported near L± with L± ⊂ ell(B±), L± ⊂ ell(S±) and satisfy-
+ing that all the forward (backward) null characteristics from WF(B±) tend to L±, while remaining
+in the elliptic set of G±, such that
+∥B±u∥Hs,r,ℓ
+sc,b ( ¯
+X) ≤ C∥G±�
+□gb(σ)u∥Hs−1,r+1,ℓ+1
+sc,b
+( ¯
+X) + C∥S±u∥Hs0,r0,ℓ0
+sc,b
+( ¯
+X)
+(5.25)
+where r0 < r, ℓ0 < ℓ and the sc-decay order r and b-decay order ℓ are actually irrelevant here.
+For r > r0 > − 1
+2 (as calculated in (5.14)), using the high sc-decay radial point estimates at the
+non-zero scattering section R(σ) (see [79, §4]. The proof follows from a positive commutator estimate
+i(�
+□gb(σ)∗A − A�
+□gb(σ)) = Im �
+□gb(σ)A + AIm �
+□gb(σ) + i[Re �
+□gb(σ), A]
+where
+Re �
+□gb(σ) =
+�
+□gb(σ) + �
+□gb(σ)∗
+2
+,
+Im �
+□gb(σ) =
+�
+□gb(σ) − �
+□gb(σ)∗
+2i
+.
+Let A ∈ Ψ2s−1,2r+1,2l+1
+sc,b
+( ¯X) with principal symbol
+a = χ0(ζ2
+θ +
+1
+sin2 θζ2
+ϕ)χ1((ζρ − 2Re z)2)ρ−2r+1(ζ2
+θ +
+1
+sin2 θζ2
+ϕ + ζ2
+ρ)s− 1
+2
+where χ0, χ1 are identically 0 near 0 and have compact support sufficiently close to 0, and χ1
+has relatively large support such that suppχ0 ∩ suppχ′
+1 is disjoint from the characteristic set of
+Re Ph(z).
+Then using Lemma 4.1 and the calculation before Lemma 4.8 in [79], it follows that
+(Im �
+□gb(σ)A+AIm �
+□gb(σ)+i[Re �
+□gb(σ), A]) ∈ Ψ2s,2r,2l
+sc,b
+( ¯X) whose principal symbol is positive definite
+elliptic near R(σ) if r > − 1
+2. This, together with a regularization argument, proves high sc-decay
+radial point estimates at R(σ)), there exist Bσ, Gσ, Sσ ∈ Ψ0,0
+sc ( ¯X) microlocally supported near R(σ)
+with R(σ) ⊂ ellsc(Bσ), R(σ) ⊂ ellsc(Sσ) and satisfying that all the forward null characteristics from
+WFsc(Bσ) tend to R(σ), while remaining in the scattering elliptic set of Gσ, such that
+∥Bσu∥Hs,r,ℓ
+sc,b ( ¯
+X) ≤ C∥Gσ �
+□gb(σ)u∥Hs−1,r+1,ℓ+1
+sc,b
+( ¯
+X) + C∥Sσu∥Hs0,r0,ℓ0
+sc,b
+( ¯
+X)
+(5.26)
+where s0 < s, ℓ0 < ℓ and the differential order s and b-decay order ℓ are actually irrelevant here.
+For ℓ < − 1
+2 (as calculated in (5.14)), using the low b-decay radial point estimates at the zero
+scattering section R(0) (see [79, §4]), there exist B0, G0 ∈ Ψ0,0,0
+sc,b ( ¯X) microlocally supported near
+R(0) (in fact near the blown up R(0) in the second microlocalized space introduced in §5.1.5) with
+R(0) ⊂ ellsc,b(B0) and E0 ∈ Ψ0,0,0
+sc,b ( ¯X), WFsc,b(E0)∩R(0) = ∅, and satisfying that all the forward null
+characteristics from WFsc,b(B0)\R(0) enter ellsc,b(E0), while remaining in the second microlocalized
+scattering elliptic set of G0, such that for any sufficiently large N > 0
+∥B0u∥Hs,r,ℓ
+sc,b ( ¯
+X) ≤ Ch−1∥G0�
+□gb(σ)u∥Hs−1,r+1,ℓ+1
+sc,b
+( ¯
+X) + C∥E0u∥Hs,r,ℓ
+sc,b ( ¯
+X) + C∥u∥ ¯
+H−N,−N,ℓ
+sc,b
+( ¯
+X)
+(5.27)
+where the differential order s is actually irrelevant here.
+Let r− < r0 < rb and χ ∈ C∞
+c ( ¯X) with χ = 1 near r ≥ rb and suppχ ⊂ {r ≥ r0}. By part
+(1) in Proposition 5.4, propagation of singularities estimates and elliptic estimates (Concretely, if
+(r, θ, ϕ, ξ)( or (ρ, θ, ϕ, ζ)) ∈ WF(χ)∩(Σ∪Σsc)c, we use elliptic estimates. If (r, θ, ϕ, ξ)( or (ρ, θ, ϕ, ξ)) ∈
+WF(χ) ∩ (Σ ∪ Σsc), then there exists s ∈ R with exp(sscH2,0
+ph,z)(ρ, θ, ϕ, ξ) ∈ ellh(B±) ∪ ellh(B1) ∪
+ellh(B0) ∪ ellh(Bz), and thus we use propagation of singularities estimates. We point out that in
+the set {Re psc(σ) = 0} ∩ ∂+ ¯X, the scattering symbol has a nonnegative imaginary part, so one can
+propagate estimates towards R(0). Finally, by using a pseudodifferential partition of unity, χ can be
+written as s sum of operators falling into the above two case), we have
+∥χu∥Hs,r,ℓ
+sc,b ( ¯
+X) ≤ C∥�
+□gb(σ)u∥ ¯
+Hs−1,r+1,ℓ+1
+sc,b
+( ¯
+X) + C∥B+u∥Hs,r,ℓ
+sc,b ( ¯
+X) + C∥B−u∥Hs,r,ℓ
+sc,b ( ¯
+X)
++ C∥B0u∥Hs,r,ℓ
+sc,b ( ¯
+X) + C∥Bσu∥Hs,r,ℓ
+sc,b ( ¯
+X) + C∥u∥ ¯
+H−N,−N,ℓ
+sc,b
+( ¯
+X).
+(5.28)
+By the same reasoning as above in the control of χu, it follows that
+∥E0u∥Hs,r,ℓ
+sc,b ( ¯
+X) ≤ C∥�
+□gb(σ)u∥ ¯
+Hs−1,r+1,ℓ+1
+sc,b
+( ¯
+X) + C∥Bσu∥Hs,r,ℓ
+sc,b ( ¯
+X) + C∥u∥ ¯
+H−N,−N,ℓ
+sc,b
+( ¯
+X).
+(5.29)
+
+48
+LILI HE
+Putting all the above estimates together yields for s > s0 > 1
+2, r > r0 > − 1
+2, ℓ < − 1
+2 and for any
+sufficiently large N > 0
+∥χu∥Hs,r,ℓ
+sc,b ( ¯
+X) ≤ C∥�
+□gb(σ)u∥ ¯
+Hs−1,r+1,ℓ+1
+sc,b
+( ¯
+X) + C∥u∥ ¯
+Hs0,r0,−N
+sc,b
+( ¯
+X) + C∥u∥ ¯
+H−N,−N,ℓ
+sc,b
+( ¯
+X).
+(5.30)
+Since
+p = −ρ−2
+b
+�
+∆b
+�
+ξr + aξϕ
+∆b
+�2 − a2ξ2
+ϕ
+∆b
++ ˜p
+�
+where
+˜p = ξ2
+θ +
+1
+sin2 θ ξ2
+ϕ
+is hyperbolic with respect to r in the region {r− ≤ r < rb} (see [21, definition E.55]), using the
+hyperbolic estimates (see [21, Theorem E.56]), we have for all s, r, ℓ ∈ R
+∥(1 − χ)u∥ ¯
+Hs,r,ℓ
+sc,b ( ¯
+X) ≤ C∥�
+□gb(σ)u∥ ¯
+Hs−1,r+1,ℓ+1
+sc,b
+( ¯
+X) + ∥χu∥ ¯
+Hs,r,ℓ
+sc,b ( ¯
+X)
+(5.31)
+where the sc-decay order r and b-decay order ℓ are actually relevant here. Combining (5.30) with
+(5.31) yields for s > s0 > 1
+2, r > r0 > − 1
+2, ℓ < − 1
+2 and for any sufficiently large N > 0
+∥u∥ ¯
+Hs,r,ℓ
+sc,b ( ¯
+X) ≤ C∥�
+□gb(σ)u∥ ¯
+Hs−1,r+1,ℓ+1
+sc,b
+( ¯
+X) + C∥u∥ ¯
+Hs0,r0,−N
+sc,b
+( ¯
+X) + C∥u∥ ¯
+H−N,−N,ℓ
+sc,b
+( ¯
+X).
+(5.32)
+Since near ∂+ ¯X
+�
+□gb(σ) − N(�
+□gb(σ)) ∈ ρ2Diff2
+b( ¯X),
+N(�
+□gb(σ)) = −2iσρ(ρ∂ρ − 1),
+it follows that from [79, Lemma 4.13 and Proposition 4.16] that for ℓ < − 1
+2
+∥u∥ ¯
+H−N,−N,ℓ
+sc,b
+( ¯
+X) ≤ ∥u∥ ¯
+H−N−ℓ,−N,ℓ
+sc,b
+( ¯
+X) = ∥u∥ ¯
+H−N−ℓ,ℓ
+b
+( ¯
+X) ≤ C∥N(�
+□gb(σ))u∥ ¯
+H−N−ℓ,ℓ+1
+b
+( ¯
+X)
+≤ C∥�
+□gb(σ)u∥ ¯
+H−N−ℓ,ℓ+1
+b
+( ¯
+X) + C∥u∥ ¯
+H−N−ℓ+2,ℓ−1
+b
+( ¯
+X)
+= C∥�
+□gb(σ)u∥ ¯
+H−N−ℓ,−N+1,ℓ+1
+sc,b
+( ¯
+X) + C∥u∥ ¯
+H−N−ℓ+2,−N+1,ℓ−1
+sc,b
+( ¯
+X),
+and thus for s > s0 > 1
+2, r > r0 > − 1
+2, ℓ < − 1
+2
+∥u∥ ¯
+Hs,r,ℓ
+sc,b ( ¯
+X) ≤ C∥�
+□gb(σ)u∥ ¯
+Hs−1,r+1,ℓ+1
+sc,b
+( ¯
+X) + C∥u∥ ¯
+Hs0,r0,ℓ−1
+sc,b
+( ¯
+X).
+(5.33)
+This finishes the proof of (5.23).
+• Proof of the estimate (5.24). For s >
+1
+2 ≥
+1
+2 − Im (σ) r2
+b +a2
+r2
+b−m , we have 1 − s ≤
+1
+2 − Im (¯σ) r2
+b +a2
+r2
+b−m .
+Then using the low regularity radial point estimates (see [77, Proposition 2.4], [78, Proposition
+5.27]), there exist B′
+±, G′
+± ∈ Ψ0( ¯
+X) microlocally supported near L± with L± ⊂ ell(B′
+±) and E′
+± ∈
+Ψ0( ¯X), WF(E′
+±) ∩ L± = ∅, and satisfying that all the backward (forward) null characteristics from
+WF(B′
+±) \ L± reach ell(E′
+±), while remaining in the elliptic set of G′
+±, such that for any N ∈ R
+∥B′
+±u∥H1−s,−r−1,−ℓ−1
+sc,b
+( ¯
+X)
+≤ C∥G′�
+□gb(σ)∗u∥H−s,−r,−ℓ
+sc,b
+( ¯
+X) + C∥E′
+±u∥H1−s,−r−1,−ℓ−1
+sc,b
+( ¯
+X) + C∥u∥ ˙H−N,−N,−N
+sc,b
+( ¯
+X)
+(5.34)
+where the sc-decay order r and b-decay order ℓ are actually irrelevant here.
+For r > − 1
+2, we have −r−1 < − 1
+2. Then using the low sc-decay radial point estimates at the non-
+zero scattering section R(σ) (see [79, §4] and the above discussion about the proof of high sc-decay
+radial point estimates at R(σ). We note that the only difference is that now the term involving χ′
+0
+does not have the correct sign and needs to be treated as an error term E′
+σu), there exist B′
+σ, G′
+σ ∈
+Ψ0,0
+sc ( ¯X) microlocally supported near R(σ) with R(σ) ⊂ ellsc(Bσ) and E′
+σ ∈ Ψ0,0
+sc ( ¯X), WFsc(E′
+σ) ∩
+R(σ) = ∅, and satisfying that all the backward null characteristics from WFsc(B′
+σ) \ R(σ) reach
+ellsc,h(E′
+σ), while remaining in the scattering elliptic set of G′
+σ, such that
+∥B′
+σu∥H1−s,−r−1,−ℓ−1
+sc,b
+( ¯
+X)
+≤ C∥G′
+σ �
+□gb(σ)∗u∥H−s,−r,−ℓ
+sc,b
+( ¯
+X) + C∥E′
+σu∥H1−s,−r−1,−ℓ−1
+sc,b
+( ¯
+X) + C∥u∥ ˙H−N,−N,−N
+sc,b
+( ¯
+X)
+(5.35)
+where the differential order s and b-decay order ℓ are actually irrelevant here.
+
+LINEAR STABILITY OF KERR-NEWMAN FAMILY
+49
+For ℓ < − 1
+2, we have −ℓ − 1 > − 1
+2. Then using the high b-decay radial point estimates at the
+zero scattering section R(0) (see [79, §4]), there exist B′
+0, G′
+0 ∈ Ψ0,0,0
+sc,b ( ¯X) microlocally supported
+near R(0) (in fact near the blown up R(0) in the second microlocalized space introduced in §5.1.5)
+with R(0) ⊂ ellsc,b(B′
+0), and satisfying all the backward null characteristics from WFsc,b(B′
+0) tend
+to R(0), while remaining in the second microlocalized scattering elliptic set of G′
+0, such that for any
+N ∈ R
+∥B′
+0u∥H1−s,−r−1,−ℓ−1
+sc,b
+( ¯
+X) ≤ C∥G′
+0�
+□gb(σ)∗u∥H−s,−r,−ℓ
+sc,b
+( ¯
+X) + C∥u∥ ˙H−N,−N,−ℓ−1
+sc,b
+( ¯
+X)
+(5.36)
+where the differential order s is actually irrelevant here.
+By the same reasoning as in the proof of the estimate (5.23), we have
+∥χu∥H1−s,−r−1,−ℓ−1
+sc,b
+( ¯
+X)
+≤ C∥�
+□gb(σ)∗u∥ ˙H−s,−r,−ℓ
+sc,b
+( ¯
+X) + C∥B+u∥H1−s,−r−1,−ℓ−1
+sc,b
+( ¯
+X)
++ C∥B′
+−u∥H1−s,−r−1,−ℓ−1
+sc,b
+( ¯
+X) + C∥B′
+0u∥H1−s,−r−1,−ℓ−1
+sc,b
+( ¯
+X)
++ C∥B′
+σu∥H1−s,−r−1,−ℓ−1
+sc,b
+( ¯
+X) + C∥u∥ ˙H−N,−N,−ℓ−1
+sc,b
+( ¯
+X),
+(5.37)
+and
+∥E′
+±u∥H1−s,−r−1,−ℓ−1
+sc,b,h
+( ¯
+X) + ∥E′
+σu∥H1−s,−r−1,−ℓ−1
+sc,b,h
+( ¯
+X)
+≤ C∥�
+□gb(σ)∗u∥ ˙H−s,−r,−ℓ
+sc,b,h
+( ¯
+X) + C∥(1 − χ)u∥ ˙H1−s,−r−1,−ℓ−1
+sc,b
+( ¯
+X)
++ C∥B′
+0u∥H1−s,−r−1,−ℓ−1
+sc,b
+( ¯
+X) + C∥u∥ ˙H−N,−N,−ℓ−1
+sc,b
+( ¯
+X).
+(5.38)
+Putting all the above estimates together yields for s > 1
+2, r > − 1
+2, ℓ < − 1
+2
+∥χu∥H1−s,−r−1,−ℓ−1
+sc,b
+( ¯
+X) ≤ C∥�
+□gb(σ)∗u∥ ˙H−s,−r,−ℓ
+sc,b
+( ¯
+X) + C∥(1 − χ)u∥ ˙H1−s,−r−1,−ℓ−1
+sc,b
+( ¯
+X)
++ C∥u∥ ˙H−N,−N,−ℓ−1
+sc,b
+( ¯
+X),
+(5.39)
+Again using the semiclassical hyperbolic estimates (see [21, Theorem E.56]), we have for all s, r, ℓ ∈ R
+∥(1 − χ)u∥ ˙H1−s,−r−1,−ℓ−1
+sc,b,h
+( ¯
+X) ≤ C∥�
+□gb(σ)∗u∥ ˙H−s,−r,−ℓ
+sc,b
+( ¯
+X)
+(5.40)
+where the sc-decay order r and b-decay order ℓ are actually relevant here. Combining (5.39) with
+(5.40) yields for s > 1
+2, r > − 1
+2, ℓ < − 1
+2
+∥u∥ ˙H1−s,−r−1,−ℓ−1
+sc,b
+( ¯
+X) ≤ C∥�
+□gb(σ)∗u∥ ˙H−s,−r,−ℓ
+sc,b
+( ¯
+X) + C∥u∥ ˙H−N,−N,−ℓ−1
+sc,b
+( ¯
+X).
+(5.41)
+By the same argument as in the proof of the estimate (5.23), we have for s > 1
+2, r > − 1
+2, ℓ < − 1
+2
+∥u∥ ˙H1−s,−r−1,−ℓ−1
+sc,b
+( ¯
+X) ≤ C∥�
+□gb(σ)∗u∥ ˙H−s,−r,−ℓ
+sc,b
+( ¯
+X) + C∥u∥ ˙H−N,−N,−ℓ−2
+sc,b
+( ¯
+X).
+(5.42)
+This proves the estimate (5.24).
+• Fredholm property of �
+□gb(σ) for Im σ ≥ 0, σ ̸= 0. We only discuss the proof of (5.19) in detail as
+(5.20) can be handled in a completely analogous manner. We define the space
+X(σ) := {u ∈ ¯Hs,ℓ
+b ( ¯X) | �
+□gb(σ)u ∈ ¯Hs,ℓ+1
+b
+( ¯X)}.
+We note that X(σ) is a Hilbert space endowed with the norm ∥u∥X (σ) := ∥u∥ ¯
+Hs,ℓ
+b + ∥�
+□gb(σ)u∥ ¯
+Hs,ℓ+1
+b
+.
+Let {uj} is a bounded sequence in ker �
+□gb(σ) ⊂ X(σ). Since ¯Hs,ℓ
+b
+→ ¯Hs0,ℓ0
+b
+is a compact embedding,
+by passing to a subsequence we may assume that uj converges in ¯Hs0,ℓ0
+b
+. By (5.15), we see that
+uj is a Cauchy sequence in ¯Hs,ℓ
+b . This implies that ker �
+□gb(σ) is finite dimensional. We then show
+that ranX (σ)�
+□gb(σ) is closed in ¯Hs,ℓ+1
+b
+. Let {fj} ⊂ ranX (σ)�
+□gb(σ) be a convergent sequence with
+fj → f∞ ∈ ¯Hs,ℓ+1
+b
+. We may write fj = �
+□gb(σ)uj where uj ∈ X(σ) is in the orthogonal complement
+of ker �
+□gb(σ). We claim that {uj} is bounded in X(σ). (Otherwise, suppose that {uj} is unbounded
+in X(σ). Passing to a subsequence, we may assume that ∥uj∥X (σ) → ∞. Let ˜uj = uj/∥uj∥X (σ)
+and ˜fj = �
+□gb(σ)˜uj = fj/∥uj∥X (σ). Then ∥˜uj∥X (σ) = 1 and ∥ ˜fj∥ ¯
+Hs,ℓ+1
+b
+→ 0. By the argument
+
+50
+LILI HE
+as above, passing to a subsequence, we have that ˜uj → ˜u∞ ∈ X(σ) and thus �
+□gb(σ)˜u∞ = 0. On
+the other hand, since ˜u∞ is in the orthogonal complement of ker �
+□gb(σ), we have �
+□gb(σ)˜u∞ ̸= 0;
+this gives a contradiction.) Then argue as above, by passing to a subsequence we may assume that
+uj → u∞ ∈ X(σ). It follows that �
+□gb(σ)uj → �
+□gb(σ)u∞ ∈ ¯Hs,ℓ+1
+b
+and thus �
+□gb(σ)u∞ = f∞. This
+proves that f∞ ∈ ranX (σ)�
+□gb(σ) and thus ranX (σ)�
+□gb(σ) is closed. Finally, it remains to show that
+ranX (σ)�
+□gb(σ) has finite codimension. Proceeding as in the proof of the finite dimension property of
+ker �
+□gb(σ) and using (5.16), we obtain that ker �
+□gb(σ)∗ ⊂ ˙H−s,−ℓ−1
+b
+is also finite dimensional. Let
+f ∈ ¯Hs,ℓ+1
+b
+be such that ⟨f, v⟩ = 0 for all v ∈ ker �
+□gb(σ)∗. We claim that there exists u ∈ ¯Hs,ℓ
+b
+such
+that �
+□gb(σ)u = f. (This implies that ranX (σ)�
+□gb(σ) has finite codimension.) We define
+Y(σ) := {v ∈ ˙H−s,−ℓ−1
+b
+( ¯X) | �
+□gb(σ)∗v ∈ ˙H−s,−ℓ
+b
+( ¯X)}.
+Let V be a complementary space of ker �
+□gb(σ)∗ in ˙H−s,−ℓ−1
+b
+. Then there exists a constant C2 such
+that
+∥v∥ ˙H−s,−ℓ−1
+b
+≤ C2∥�
+□gb(σ)∗v∥ ˙H−s,−ℓ
+b
+,
+v ∈ Y(σ) ∩ V.
+If this were not true, we could find a sequence vj ⊂ Y(σ) ∩ V such that
+∥vj∥ ˙H−s,−ℓ−1
+b
+= 1,
+∥vj∥ ˙H−s,−ℓ−1
+b
+≥ j∥�
+□gb(σ)vj∥ ˙H−s,−ℓ
+b
+.
+By (5.16) and the compact embedding ˙H−s,−ℓ−1
+b
+→ ˙H−N,−ℓ−3
+b
+, passing to a subsequence we have
+vj → v∞ with ∥v∞∥ ˙H−s,−ℓ−1
+b
+= 1 and v∞ ∈ V ∩ ker �
+□gb(σ)∗; this gives a contradiction. If f ∈ ¯Hs,ℓ+1
+b
+satisfies ⟨f, v⟩ = 0 for all v ∈ ker �
+□gb(σ)∗, we have
+|⟨f, v⟩| ≤ C2∥f∥ ¯
+Hs,ℓ+1
+b
+∥�
+□gb(σ)∗v∥ ˙H−s,−ℓ
+b
+,
+v ∈ Y(σ) ∩ V.
+Since any v ∈ Y(σ) can be written as v = v1 + v2 with v1 ∈ ker �
+□gb(σ)∗, v2 ∈ Y(σ) ∩ V , it follows
+that
+|⟨f, v⟩| = |⟨f, v2⟩| ≤ C2∥f∥ ¯
+Hs,ℓ+1
+b
+∥�
+□gb(σ)∗v2∥ ˙H−s,−ℓ
+b
+= C2∥f∥ ¯
+Hs,ℓ+1
+b
+∥�
+□gb(σ)∗v∥ ˙H−s,−ℓ
+b
+,
+v ∈ Y(σ).
+By Hahn-Banach Theorem, the anti-linear form �
+□gb(σ)∗v → ⟨f, v⟩ with v ∈ Y(σ) can be extended
+to a continuous anti-linear form on ˙H−s,−ℓ
+b
+. Since ( ˙H−s,−ℓ
+b
+)∗ = ¯Hs,ℓ
+b , there exists u ∈ ¯Hs,ℓ
+b
+such that
+⟨u, �
+□gb(σ)∗v⟩ = ⟨f, v⟩ for v ∈ Y(σ). In particular, for v ∈ C∞
+c (X◦)
+⟨�
+□gb(σ)u, v⟩ = ⟨u, �
+□gb(σ)∗v⟩ = ⟨f, v⟩.
+This proves that �
+□gb(σ)u = f.
+We now prove statement (2). Combining the estimates near spatial infinity ∂+ ¯X (see [81, Propo-
+sition 5.3]) with the radial point estimates at event horizon, propagation of singularity estimates,
+elliptic estimates and hyperbolic estimates as in the proof of statement (1), we obtain (5.21). The
+Fredholm property follows as in the nonzero σ case.
+□
+5.2.2. Uniform Fredholm estimates for tensor wave operators. Now we prove an analogy to Theorem 5.5
+for the operators �
+Pb,γ(σ), �
+Wb,γ(σ) and �
+Lb,γ(σ) acting on bundles.
+We note that the formal adjoint of
+Pb,γ, Wb,γ, Lb,γ with respect to the natural inner product induced by gb and the volume form dvolgb is
+Wb,γ,
+Pb,γ,
+�Ggb
+0
+0
+4
+�
+Lb,γ
+�Ggb
+0
+0
+1
+4
+�
+,
+respectively (see (9.4) and (9.5) for the derivation of the adjoint of Lb,γ).
+Theorem 5.6. Let b0 = (m0, a0, Q0) with |Q0|+|a0| < m0 and |a0| ≪ |m0|+|Q0|. Then there exists ǫ > 0
+such that for b = (m, a, Q) with |b − b0| < ǫ and for s > s0 > 2 (resp. s > s0 > 3), ℓ − 1 ≤ ℓ0 < ℓ < − 1
+2
+with s + ℓ > s0 + ℓ0 > − 1
+2 and ℓ ̸= − 3
+2, the conclusions in Theorem 5.5 hold for �
+Pb,γ(σ) and �
+Wb,γ(σ) (resp.
+�
+Lb,γ(σ)).
+
+LINEAR STABILITY OF KERR-NEWMAN FAMILY
+51
+Proof. The proof is analogous to that of Theorem 5.5, except for the computation of threshold regularity
+in the radial point estimate at event horizon and the threshold decay rate in the radial point estimate at
+spatial infinity ∂+ ¯X.
+As for the calculation of threshold regularity in the radial point estimate at event horizon, the Reissner-
+Nordstr¨om metric case gb = g(m0,0,Q0) was done in appendix C. According to (C.21) and (5.8), for Im σ ≥ 0,
+the threshold regularity is 3
+2 − 2γ
+κ for �
+Pb0,γ(σ) and 3
+2 + γ
+2κ for �
+Wb0,γ(σ) acting on 1-forms, and is 5
+2 − 2γ
+κ for
+�
+Lb,γ(σ) acting on S2 �
+scT ∗ ¯X ⊕ �
+scT ∗ ¯X. This implies that the threshold regularity for nearby Kerr-Newman
+metrics is close to 3
+2 for �
+Pb,γ(σ) and �
+Wb,γ(σ), and close to 5
+2 for �
+Lb,γ(σ) as γ is a sufficiently small constant.
+For 0 ̸= σ ∈ R, the radial point estimate at R(0) and R(σ) for �
+Pb,γ(σ), �
+Wb,γ(σ), �
+Lb,γ(σ) requires the
+computation of a threshold decay rate relative to L2( ¯X). Concretely, the threshold − 1
+2 from [79, Theorems 1.1
+and 1.3] is modified by the subprincipal symbol σsc( 1
+2iρ(�•(σ) − �•(σ)∗))|R(0),R(σ) where • = Pb,γ, Wb,γ, Lb,γ.
+Working in the trivialization of S2 �
+scT ∗ ¯X ⊕ �
+scT ∗ ¯X given in terms of the differentials of standard coordinates
+t, x1, x2, x3, according to Proposition 4.6 and Lemma 4.11, we see that
+σsc( 1
+2iρ(�•(σ) − �•(σ)∗))|R(0),R(σ) = σsc( 1
+2iρ( �
+□gb,0(σ) − �
+□gb,0(σ)∗) ⊗ Id4×4)|R(0),R(σ) = 0,
+• = P, W,
+σsc( 1
+2iρ( �
+Lb,γ(σ) − �
+Lb,γ(σ)∗))|R(0),R(σ) = σsc( 1
+2iρ( �
+□gb,0(σ) − �
+□gb,0(σ)∗) ⊗ Id14×14)|R(0),R(σ) = 0.
+Therefore, the threshold decay rate is still − 1
+2.
+□
+5.3. Description of the kernel of the wave type operators. In this section, we give a detailed descrip-
+tion of kernel of the following wave operators: �
+□gb(σ) on scalar functions, �
+Pb,γ(σ), �
+Wb,γ(σ) on scattering
+1-forms and the linearized gauge-fixed Einstein-Maxwell operator �
+Lb,γ(σ).
+Proposition 5.7. Let b = (m, a, Q) with |Q| + |a| < |m| and a = |a| ≪ |m| + |Q|. Suppose u ∈ ¯Hs,ℓ
+b ( ¯X; C)
+with s > 1
+2.
+(1) If �
+□gb(0)u = 0 and u ∈ ¯Hs,ℓ
+b ( ¯X; C) with ℓ ∈ R, then u ∈ ¯H∞,ℓ
+b
+( ¯X; C) and has a polyhomogeneous
+expansion with index set contained in {(z, k) | z ∈ iZ, k ∈ N}. In particular, if − 3
+2 < ℓ < − 1
+2, then
+u ∈ A1( ¯X; C). More precisely, there exists u0 ∈ C∞(∂+ ¯X; C) such that u − u0
+r ∈ A2−( ¯X; C).
+(2) If �
+□gb(0)∗u = 0 and u ∈ ˙H−∞,ℓ
+b
+( ¯
+X; C) with ℓ ∈ R, then u ∈ ˙H
+1
+2 −,ℓ
+b
+( ¯X; C). Near ∂+ ¯X, u has the
+same polyhomegeneous expansion as in part (1).
+(3) If σ ̸= 0 and u ∈ ker �
+□gb(σ) ∩ ¯Hs,ℓ
+b ( ¯X; C) for some ℓ ∈ R with s + ℓ > − 1
+2, then u ∈ ρC∞(∂+ ¯X; C) +
+A2−( ¯X; C).
+The above statements also hold for �
+Pb,γ(σ), �
+Wb,γ acting on ω ∈ ¯Hs,ℓ
+b ( ¯X; �
+scT ∗ ¯X) with s > 2 (for statement
+(2), ω ∈ ˙H
+− 1
+2 −C(γ,a),ℓ
+b
+) and �
+Lb,γ(σ) acting on (˙g, ˙A) ∈ ¯Hs,ℓ
+b ( ¯X; �
+scT ∗ ¯X ⊕ S2 �
+scT ∗ ¯X) with s > 3 (for statement
+(2), ω ∈ ˙H
+− 3
+2 −C(γ,a),ℓ
+b
+) where C(γ, a) > 0 is a sufficiently small constant depending on γ, a.
+Proof. We only discuss the proof for the operator �
+□gb(σ) in detail here as the other operators �
+Pb,γ(σ), �
+Wb,γ,
+�
+Lb,γ(σ) can be handled similarly.
+• Proof of statement (1). According to the high regularity radial point estimate (since s > 1
+2 and 1
+2
+is the threshold regularity at the radial points at event horizon), 5.4, propagation of singularity
+estimate and elliptic estimate for b-pseudodifferential operators, we conclude that u ∈ H∞,ℓ
+b
+( ¯X; C).
+To prove the polyhomogeneous expansion of u, we exploit a typical normal operator argument
+(see [64, §5]). Concretely, since
+�
+□gb(0) = (ρ2∂ρ)2 − 2ρ3∂ρ + ρ2(∂2
+θ +
+1
+sin2 θ∂2
+ϕ) + ρ3Diff2
+b = ρ2�
+ρ∂ρ(ρ∂ρ − 1) + /∆
+�
++ ρ3Diff2
+b ∈ ρ2Diff2
+b,
+the normal operator N(�
+□gb(0)) of �
+□gb(0) is defined by factoring out an overall weight (here is ρ2)
+and freezing the coefficients at the boundary ρ = 0. Therefore, we have
+N(�
+□gb(0)) = ρ2�
+ρ∂ρ(ρ∂ρ − 1) + /∆
+�
+.
+
+52
+LILI HE
+Let χ(ρ) ∈ C∞
+c (R) be identically 1 near 0 and vanishing when ρ = 1/r ≥ 1/3m. Then we have
+ρ−2N(�
+□gb(0))(χu) = ρ−2�
+N(�
+□gb(0)) − �
+□gb(0)
+�
+(χu) + ρ−2[�
+□gb(0), χ]u := f ∈ ¯H∞,ℓ+1
+b
+( ¯X; C).
+(5.43)
+For simplicity of notation, we denote ρ−2N(�
+□gb(0)) by N. Using the Mellin transform in ρ, defined
+by (we drop the dependence on the angular variables on ∂+ ¯X from the notation)
+�
+χu(ξ) :=
+� ∞
+0
+ρ−iξ(χu)(ρ) dρ
+ρ ,
+the equation N(χu) = f becomes
+�
+N(ξ)
+�
+�
+χu
+�
+(ξ) =
+�
+iξ(iξ − 1) + /∆
+�
+(�
+χu)(ξ) = ˆf(ξ).
+At this point, we only know that �
+χu(ξ) is holomorphic and Schwartz in Re ξ (with Im ξ fixed) in
+the region Im ξ > −ℓ − 3
+2 and so is ˆf(ξ) in Im ξ > −ℓ − 5
+2. Therefore, we have the following inverse
+Mellin transform
+χu(ρ) = 1
+2π
+�
+Im ξ=−ℓ− 3
+2 +ǫ
+ρiξ �
+χu(ξ) dξ,
+ǫ > 0.
+We denote by Y m
+l
+with l ∈ N, m ∈ Z, |m| ≤ l the spherical harmonic function of degree l and
+order m which satisfies /∆Y m
+l
+= −l(l + 1)Y m
+l . We denote by Sl = {Y m
+l
+: |m| ≤ l} the space of degree
+l spherical harmonic functions. Then one can conclude that {Sl : l ∈ N} form an orthogonal basis
+of L2(S2). Therefore, one can expand (�
+χu)(ξ) and ˆf(ξ) in terms of spherical harmonics Y m
+l , i.e., we
+write (�
+χu)(ξ) = � um
+l (ξ)Y m
+l , ˆf = � f m
+l (ξ)Y m
+l . Restricted to Sl, the inverse
+�
+N(ξ)−1 =
+�
+iξ(iξ − 1) + /∆
+�−1
+=
+�
+iξ(iξ − 1) − l(l + 1))−1
+is meromorphic in ξ with simple poles given by ξ = −i(l+1), il. Then in the inverse Mellin transform
+χu(ρ) = 1
+2π
+�
+Im ξ=−ℓ− 3
+2 +ǫ
+ρiξ( �
+N(ξ))−1 ˆf(ξ) dξ,
+ǫ > 0,
+−ℓ − 3
+2 + ǫ /∈ Z,
+we can shift the contour of integration through the pole at ξ = ik to Im ξ = −ℓ − 5
+2 + ǫ where k ∈ Z
+and k ∈ (−ℓ − 3
+2 + ǫ, −ℓ − 5
+2 + ǫ), and the Residue Theorem gives
+χu(ρ) = ρ−kuk + ˜uk,
+˜uk = 1
+2π
+�
+Im ξ=−ℓ− 5
+2 +ǫ
+ρiξ( �
+N(ξ))−1 ˆf(ξ) dξ ∈ Aℓ+ 5
+2 −ǫ( ¯X; C)
+(5.44)
+where
+uk =
+��
+−k≤m≤k
+−1
+2k+1f m
+k (ik)Y m
+k ,
+k ≥ 0
+�
+k+1≤m≤−k−1
+1
+2k+1f m
+−k−1(−ik − i)Y m
+−k−1,
+k ≤ −1 .
+To deduce the full polyhomogeneous expansion for χu, we plug the above partial expansion (5.44)
+into the equation (5.43) to obtain
+N ˜uk = N(χu) = f1 + f2,
+f1 ∈ ρ−k+1C∞(∂+ ¯X; C)
+and
+f2 ∈ Aℓ+ 7
+2 −ǫ( ¯X, C).
+Now we solve the following two equations N ˜uj
+k = fj, j = 1, 2. First, using the Mellin transform as
+before, we see that
+˜u2
+k = ρ−k+1uk+1 + ˜uk
+where
+uk+1 ∈
+�
+Sk−1,
+k − 1 ≥ 0
+S−k,
+k − 1 ≤ −1
+and
+˜uk+1 ∈ Aℓ+ 7
+2 −ǫ( ¯X; C).
+As for u1
+k, we can solve N ˜u1
+k = f1 more directly. Since any function in C∞(∂+ ¯X) can be expressed
+as a linear combination of Sl, it suffices to solve N ˜u1
+k = f1 for f1 ∈ ρ−kSl. More explicitly,
+u1
+k =
+�
+(k(k + 1) − l(l + 1))−1f1,
+k(k + 1) ̸= l(l + 1)
+−(2k + 1)−1f1 ln ρ,
+k(k + 1) = l(l + 1) .
+Therefore, we have
+χu = ρ−kuk + u1
+k + ρ−k+1uk+1 + ˜uk+1.
+
+LINEAR STABILITY OF KERR-NEWMAN FAMILY
+53
+Plugging the partial expansion into the equation (5.43) and proceeding as above iteratively give the
+full polyhomogeneous expansion.
+• Proof of statement (2). Near ∂+ ¯X, according to the elliptic estimate for b-pseudodifferential opera-
+tor, we conclude that u ∈ ¯H∞,ℓ
+b
+there. Away form spatial infinity ∂+ ¯X, since u is 0 in the region
+r < rb, by Proposition 5.4, propagation of singularity estimate and elliptic estimate we see that u is
+smooth away from the radial points at event horizon. Finally, using the low regularity radial point
+estimate yields u ∈ ˙H
+1
+2 −,ℓ
+b
+( ¯X; C). The polyhomogeneous expansion of u near ∂+ ¯X can be obtained
+as in the proof of statement (1) by exploiting the normal operator argument.
+• Proof of statement (3). Since s + ℓ > − 1
+2, according to the high sc-decay estimate at the radial
+point R(σ), propagation of singularity estimate and elliptic estimate, we see that away from the
+radial point R(0), u is in ¯H∞,ℓ
+b
+( ¯X; C).
+Near R(0) the low b-decay radial point estimate gives
+u ∈ ¯H
+∞,min{− 1
+2 −,ℓ}
+b
+( ¯X; C) at R(0). Therefore, we have u ∈ ¯H
+∞,min{− 1
+2 −,ℓ}
+b
+( ¯
+X; C). Noe since σ ̸=
+0, we have �
+□gb(σ) = −2iσρ(ρ∂ρ − 1) + ρ2Diff2
+b, and thus N(�
+□gb(σ)) = −2iσρ(ρ∂ρ − 1).
+Then
+using the typical normal operator argument as in the proof of statement (1), we obtain that u =
+ρC∞(∂+ ¯X; ¯X) + A2−( ¯
+X; C).
+• Proof of �
+Pb,γ(σ), �
+Wb,γ and �
+Lb,γ(σ). Near the spatial infinity, we work in the standard coordinate
+trivialization of �
+scT ∗ ¯X for �
+Pb,γ(σ), �
+Wb,γ (which is given in terms of the differentials dt, dx1, dx2, dx3
+of the standard coordinates) and S2 �
+scT ∗ ¯X ⊕ �
+scT ∗ ¯X for �
+Lb,γ(σ) (which is given in terms of the
+differentials dt2, dtdxi, dxidxj, dt, dxi1 ≤ j ≤ i ≤ 3 of the standard coordinates).
+Then according to Propositions 4.6 and 4.13, the normal operator of 2 �
+Pb,γ(σ), 2�
+Wb,γ (resp.
+�
+Lb,γ(σ)) is, −ρ2�
+ρ∂ρ(ρ∂ρ − 1) + /∆
+�
+for σ = 0 and 2iσρ(ρ∂ρ − 1) for σ ̸= 0, tensored with 4 × 4
+identity matrix (reps. 14 × 14 identity matrix). Therefore, the proof for these operators follows in a
+completely analogous manner as before.
+□
+5.4. Semiclassical behavior of Kerr-Newman spacetimes. Recall that
+�
+□g(σ) := eitb,∗σ□gbe−itb,∗σ.
+where the inverse metric g−1 is given by
+g−1 = ρ−2
+b
+�
+∆b∂2
+r + χ2 − 1
+∆b
+�
+(r2 + a2)∂tχ + a∂ϕ
+�2
++ ∂2
+θ +
+1
+sin2 θ
+�
+∂ϕ + a sin2 θ∂tχ
+�2
++ 2χ
+�
+(r2 + a2)∂tχ + a∂ϕ
+�
+∂r
+�
+.
+(5.45)
+We introduce the semiclassically rescaled version of the operator �
+□g(σ):
+h2 �
+□g(h−1z),
+h = 1
+|σ|
+and
+z = Re σ
+|σ| + iIm σ
+|σ| .
+(5.46)
+When Im σ is bounded, we write z = zR + ihzI and parametrize h2 �
+□g(h−1z) by zR and zI = h−1Im z rather
+than Re z and Im z. This still gives a family of operators in Diff2
+sc,h which depends smoothly on zR, zI, and its
+semiclassical principal symbol does not depend on zI. Therefore, when analyzing the semiclassical principal
+symbol of h2 �
+□g(h−1z), it suffices to consider the case z ∈ R.
+For z ∈ R, the semiclassical principal symbol of h2 �
+□g(h−1z) is given by
+ph,z(ξ) := σh(h2 �
+□g(h−1z)) = −g−1(−zdtb,∗ + ξ, −zdtb,∗ + ξ),
+ξ ∈ ∂⊥
+tb,∗ = scT ∗ ¯X.
+(5.47)
+As before, away from ∂+ ¯X, we write the scattering covectors as
+ξ := ξrdr + ξθdθ + ξϕdϕ.
+While near ∂+ ¯X, we write them as
+ξ := ξρ
+dρ
+ρ2 + ξθ
+dθ
+ρ + ξϕ
+dϕ
+ρ ,
+ρ = 1
+r .
+
+54
+LILI HE
+5.4.1. Characteristic set. We first study the characteristic set
+Σh := {⟨ξ⟩−2ph,z(ξ) = 0} ⊂ scT ∗ ¯X,
+⟨ξ⟩ =
+�
+1 + ξ2r + ξ2
+θ +
+1
+sin2 θξ2ϕ.
+We will show that for z ∈ R \ 0, it splits into two components. To achieve this, we make use of an auxiliary
+covector dt which is timelike everywhere on ¯X.
+Lemma 5.8 (Splitting of the characteristic set). There exist closed z-dependent sets Σ±
+h ⊂ scT ∗ ¯X such that
+{⟨ξ⟩−2ph,z(ξ) = 0} = Σ+
+h ⊔ Σ−
+h ,
+z ∈ R \ 0.
+(5.48)
+Moreover,
+± ⟨ξ⟩−1g−1(−zdtb,∗ + ξ, dt) > 0
+on
+Σ±
+h
+for
+z ∈ R \ 0.
+(5.49)
+Finally,
+(Σ±
+h ∩ scS∗ ¯X) ∩ {∆b − a2 sin2 θ > 0} = ∅
+for
+z ∈ R;
+(5.50)
+Σ±
+h ∩ {∆b − a2 sin2 θ > 0} = ∅
+for
+∓ z > 0.
+(5.51)
+Proof. We put
+Σ±
+h := {⟨ξ⟩−2ph,z(ξ) = 0} ∩ {±⟨ξ⟩−1g−1(−zdtb,∗ + ξ, dt) ≥ 0}.
+It is clear that Σ±
+h are closed and their union is the characteristic set. In order to show that Σ+
+h ∩ Σ−
+h = ∅
+and (5.49) for z ∈ R \ 0, we need
+{⟨ξ⟩−2ph,z(ξ) = 0} ∩ {⟨ξ⟩−1g−1(−zdtb,∗ + ξ, dt) = 0} = ∅
+for
+z ∈ R \ 0.
+On the fiber infinity scS∗ ¯X, ⟨ξ⟩−1g−1(ξ, dt) = 0 implies that ⟨ξ⟩−1ξ is spacelike or 0 as dt is timelike.
+If the intersection were not empty at scS∗ ¯X, ⟨ξ⟩−1ξ must be 0, which is impossible. Also, the fact that
+⟨ξ⟩−2ph,z(ξ) = ⟨ξ⟩−2p(ξ) = 0 gives (5.50).
+In the interior scT ∗ ¯X, if the intersection were not empty, since dt is timelike, g−1(−zdtb,∗ + ξ, dt) = 0
+implies that −zdtb,∗ + ξ must be spacelike or 0. Then the equality g−1(−zdtb,∗ + ξ, −zdtb,∗ + ξ) = 0 forces
+−zdtb,∗ + ξ to be 0, which cannot happen for ξ ∈ ∂⊥
+tb,∗ and z ∈ R \ 0.
+On the region {∆b − a2 sin2 θ > 0}, we see that g−1(ξ, ξ) > 0 for 0 ̸= ξ ∈ scT ∗ ¯X.
+For any p ∈
+{∆b − a2 sin2 θ > 0}, let {dt, X1, X2, X3} be an orthogonal basis of T ∗
+p ¯
+M with G|p(Xi, Xi) = 1. We write
+−zdtb,∗ + ξ = (a0 − z)dt +
+3
+�
+i=1
+aiXi,
+and then g−1(−zdtb,∗+ξ, −zdtb,∗+ξ) = 0 gives (a0−z)2G(dt, dt)+�3
+i=1 a2
+i = 0. Since −zdtb,∗+ξ+zdt ∈ T ∗
+p ¯X,
+it follows that g−1(−zdtb,∗ + ξ + zdt, −zdtb,∗ + ξ + zdt) = a2
+0G(dt, dt) + �3
+i=1 a2
+i > 0. Therefore, we have
+z2 − 2a0z > 0. Then
+±g−1(−zdtb,∗ + ξ, dt) = ±(a0 − z)g−1(dt, dt) = ±(a0 − z
+2 − z
+2)g−1(dt, dt) > 0
+for
+± z > 0.
+This finishes the proof of (5.51).
+□
+5.4.2. The radial points at event horizon. Now in the fiber radial compactification scT ∗ ¯X of scT ∗ ¯X, we use
+the following coordinates near the fiber infinity scS∗ ¯X,
+ρξ = (ξ2
+r + ξ2
+θ +
+1
+sin2 θξ2
+ϕ)− 1
+2 ∈ [0, ∞),
+ˆξ = (ˆξr, ˆξθ, ˆξϕ) = ρξ(ξr, ξθ, ξϕ).
+(5.52)
+We note that ρξ is a boundary defining function of scT ∗ ¯X, i.e., ρξ = 0 at scS∗ ¯X, and
+ˆξ ∈ scS∗ ¯X = {ˆξ2
+r + ˆξ2
+θ +
+1
+sin2 θ
+ˆξ2
+ϕ = 1}.
+We now analyze the behavior of the Hamiltonian flow exp(sρξHph,z) on the characteristic set Σh = Σ+
+h ∪Σ−
+h ,
+concentrating on the behavior away from ∂+ ¯X. With tb,∗ = tχ + c(r), we write
+ph,z = −ρ−2
+b
+�
+∆b
+�
+ξr − zc′(r) + χ
+�
+aξϕ − (r2 + a2)z
+�
+∆b
+�2 − 1
+∆b
+�
+aξϕ − (r2 + a2)z
+�2 + ˜ph,z
+�
+(5.53)
+
+LINEAR STABILITY OF KERR-NEWMAN FAMILY
+55
+where
+˜ph,z = ξ2
+θ +
+1
+sin2 θ(ξϕ − a sin2 θz)2.
+(5.54)
+Then we calculate
+Hph,z = −ρ−2
+b
+��
+2∆b
+�
+ξr − zc′(r)
+�
++ 2χ
+�
+aξϕ − (r2 + a2)z
+��
+∂r +
+�
+2aχ2 − 1
+∆b
+�
+aξϕ − (r2 + a2)z
+�
++ 2χa
+�
+ξr − c′(r)z
+��
+∂ϕ
+− ∂(−ρ2
+bph,z)
+∂r
+∂ξr + H˜ph,z
+�
+− ρ−2
+b ph,zHρ2
+b
+where
+∂(−ρ2
+bph,z)
+∂r
+= 2∆b
+�
+ξr − zc′(r) + χ
+�
+aξϕ − (r2 + a2)z
+�
+∆b
+�
+∂r
+�
+ξr − zc′(r) + χ
+�
+aξϕ − (r2 + a2)z
+�
+∆b
+�
++ ∂∆b
+∂r
+�
+ξr − zc′(r) + χ
+�
+aξϕ − (r2 + a2)z
+�
+∆b
+�2 − ∂r
+� 1
+∆b
+�
+aξϕ − (r2 + a2)z
+�2�
+In the coordinates (5.52) near the fiber infinity scS∗ ¯X, we compute
+ρξHph,z = −ρ−2
+b
+��
+2∆b
+�ˆξr − ρξzc′(r)
+�
+− 2ρξχ(r2 + a2)z + 2χaˆξϕ
+�
+∂r
++
+�
+2aχ2 − 1
+∆b
+�
+aˆξϕ − ρξ(r2 + a2)z
+�
++ 2χa
+�ˆξr − ρξc′(r)z
+��
+∂ϕ
+− ρξ
+∂(−ρ2
+bph,z)
+∂r
+∂ξr + ρξH˜ph,z
+�
+− ρ−2
+b (ρ2
+ξph,z)ρ−1
+ξ Hρ2
+b.
+(5.55)
+We let
+L± := {ρξ = 0, ˆξr = ±1, (ˆξθ, ˆξϕ) = 0, ∆b = 0} ⊂
+�
+Σ±
+h ∩ scS∗ ¯X
+�
+.
+Now we describe the property of the set L±.
+Lemma 5.9. L± are invariant under the Hamiltonian flow exp(sρξHph,z) on the characteristic set Σh.
+Moreover, L+ is a radial sink and L− is a radial source for the flow exp(sρξHph,z).
+Proof. We rewrite
+L± = {∆b = 0, ρξ = 0, ˆξr = ±1, ρ2
+ξ ˜ph,z = 0}.
+Using the coordinates (5.52), the expression (5.55) and the facts that c(r) = 0, χ = 1 near L±, we find that
+near L±
+ρξHph,zρξ = −ρ−2
+b
+�∂∆b
+∂r
+ˆξ3
+r − 4rρξ ˆξ2
+rz + H˜ph,zρξ
+�
+ρξ − ρ−2
+b (ρ2
+ξph,z)ρ−1
+ξ Hρ2
+bρξ,
+ρξHph,z ˆξϕ = (Hph,zρξ)ˆξϕ,
+ρξHph,z∆b = −2ρ−2
+b
+∂∆b
+∂r
+�
+∆b ˆξr − ρξ(r2 + a2)z + aˆξϕ
+�
+,
+ρξHph,z(ρ2
+ξ ˜ph,z) = 2(Hph,zρξ)ρ2
+ξ ˜ph,z − ρ−2
+b (ρ2
+ξph,z)ρξHρ2
+b ˜ph,z.
+(5.56)
+This implies that ρξHp is tangent to L±, so L± is invariant under the flow exp(sρξHph,z). Moreover, in a
+neighborhood of L± = {∆b = 0, ρξ = 0, ˆξr = ±1, ρ2
+ξ ˜ph,z = 0} ⊂ {ρ2
+ξph,z = 0}, we have
+±ρξHpρ2
+ξ ≤ −C0ρ2
+ξ,
+±ρξHp ˆξ2
+ϕ ≤ −C0ˆξ2
+ϕ,
+±ρξHp∆2
+b ≤ −C0∆2
+b + C1(ρ2
+ξ + ˆξ2
+ϕ)
+±ρξHp(ρ2
+ξ ˜p) ≤ −C0ρ2
+ξ ˜p + C1(ρ2
+ξ + ∆2
+b + ˆξ2
+ϕ)
+(5.57)
+where C0, C1 > 0. It follows from (5.4) that in a sufficiently small neighborhood of L±
+± ρξHpf ≤ −Cf,
+f = ρ2
+ξ + ˆξ2
+ϕ + C2∆2
+b + C3ρ2
+ξ ˜p ≥ 0.
+(5.58)
+for some C, C2, C3 > 0. Proceeding as in the proof of Lemma 5.2 finishes the proof.
+□
+
+56
+LILI HE
+Finally, since Hph,zρξ = Hpρξ and
+ρξσh
+�
+h−1Im h2�
+□gb(h−1z)
+�
+= Im (h−1z)ρ−2
+b
+�
+2(r2 + a2)ˆξr + 2aˆξϕ)
+�
+= ρξσ1
+�
+Im �
+□gb(h−1z)
+�
+at
+L±,
+(5.59)
+the calculation of the threshold regularity at L± in the microlocal case applies here as well.
+5.4.3. The radial points at spatial infinity ∂+ ¯X. We now turn to the analysis near ∂+ ¯X. Recall that near
+∂+ ¯X, the semiclassical principal symbol of h−2 �
+□g(h−1z) for z ∈ R is given by
+ph,z(ζ) = −g−1(−zdtb,∗ + ξ, −zdtb,∗ + ξ)
+= −(ξρ − z)2 + z2 − ˜p + ρ
+�
+0≤j,k,m≤2
+0≤j+k+m≤2
+ajkm(ρ, θ, ϕ)ξj
+ρξk
+θ ξm
+ϕ ,
+˜p =
+�
+ξ2
+θ +
+1
+sin2 θξ2
+ϕ
+�
+(5.60)
+where ξ = ξρ
+dρ
+ρ2 + ξθ dθ
+ρ + ξϕ
+dϕ
+ρ and ajkm ∈ A0( ¯X). Then the scattering Hamiltonian vector field associated
+to ph,z in ρ > 0 is given by
+scHph,z = ρ
+�∂ph,z
+∂ξρ
+�
+ρ ∂
+∂ρ + (
+�
+µ=θ,ϕ
+ξµ
+∂
+∂ξµ
+)
+�
+−
+�
+ρ∂ph,z
+∂ρ
++ (
+�
+µ=θ,ϕ
+ξµ
+∂ph,z
+∂ξµ
+)
+� ∂
+∂ξρ
++
+�
+µ=θ,ϕ
+�∂ph,z
+∂ξµ
+∂
+∂µ − ∂ph,z
+∂µ
+∂
+∂ξµ
+��
+.
+We introduce the rescaled scattering Hamiltonian vector field (here we drop the factor ⟨ρξ⟩ again)
+scH2,0
+ph,z := ρ−1scHph,z
+= (−2ξρ + 2z)ρ∂ρ + 2˜p∂ξρ + (−2ξρ + 2z)
+�
+µ=θ,ϕ
+ξµ∂ξµ − H˜p + ρV,
+V ∈ Vb(scT ∗ ¯X).
+(5.61)
+We now discuss the integral curves of scH2,0
+ph,z on the semiclassical characteristic set Σh for z ∈ R \ 0 near
+∂+ ¯X.
+Lemma 5.10. For z ∈ R\0, we put φs(ρ, θ, ϕ, ξ) = exp(sscH2,0
+ph,z)(ρ, θ, ϕ, ξ). Then there exists a neighborhood
+V± ⊂ scT ∗ ¯X of
+R(z
+2 ± |z|
+2 ) = {ρ = 0, ξ = (z ± |z|)dρ
+ρ2 }
+such that uniformly in (ρ, θ, ϕ, ξ) ∈ V± ∩ Σh, one has
+φs(ρ, θ, ϕ, ξ) → R(z
+2 ± |z|
+2 )
+as
+s → ±∞.
+This implies that R( z
+2 + |z|
+2 ) is a radial sink while R( z
+2 − |z|
+2 ) is a radial source.
+Proof. Recall that
+R(z
+2 ± |z|
+2 ) = {ρ = 0, ˜p = 0, ξρ = z ± |z|}.
+Using the expression (5.61), we find that in a neighborhood U 2ǫ
+± = {(ξρ − (z ± |z|))2 + ρ2 + ˜p < 2ǫ} of
+R( z
+2 ± |z|
+2 ) where ǫ > 0 is a sufficiently small constant,
+±scH2,0
+ph,zρ2 ≤ −C0ρ2,
+±scH2,0
+ph,z ˜p ≤ −C0˜p
+(5.62)
+for some C0 > 0. According to the expression (5.60) of ph,z, there exists ǫ′ > 0 such that
+{ρ2 + ˜p < ǫ′} ∩ Σh ⊂ U ǫ
++ ∪ U ǫ
+−
+where U ǫ
+± are neighborhoods of R( z
+2± |z|
+2 ) satisfying U ǫ
++∩U ǫ
+− = ∅. Then for (ρ, θ, ϕ, ξ) ∈ U ǫ
+±∩{ρ2+˜p < ǫ′}∩Σh,
+since ph,z is a constant along the integral curves of scH2,0
+ph,z, we have
+φs(ρ, θ, ϕ, ξ) ⊂ U ǫ
+± ∩ Σh
+for
+± s ≥ 0.
+(5.63)
+Indeed, if (5.63) failed, we could find ±s0 > 0 such that
+φs0(ρ, θ, ϕ, ξ) ∈ U 2ǫ
+±
+and
+± scH2,0
+ph,z(ρ2 + ˜p)(φs0(ρ, θ, ϕ, ξ)) ≥ 0
+
+LINEAR STABILITY OF KERR-NEWMAN FAMILY
+57
+which contradicts (5.62). Proceeding as in the proof of Lemma 5.2, it follows that uniformly in (ρ, θ, ϕ, ξ) ∈
+U ǫ
+± ∩ Σh, one has
+φs(ρ, θ, ϕ, ξ) → {ρ = 0, ˜p = 0} ∩ (U ǫ
+± ∩ Σh) = R(z
+2 ± |z|
+2 )
+as
+s → ±∞.
+□
+Finally, since ph,z at ∂+ ¯X equals to psc(z), the calculation of the threshold scattering decay order at the
+radial points in the microlocal case applies here as well.
+5.4.4. Structure of the trapped sets. We now study the structure of the trapped sets for KN spacetime with
+|a|2 + Q2 < m2.
+First, we define and investigate the incoming and outgoing tails and the trapped set
+associated to the Hamiltonian flow exp(sscH2,0
+ph,z) where ph,z is the semiclassical symbol given by
+ph,z = −ρ−2
+b
+�
+∆b
+�
+ξr − zc′(r) + χ
+�
+aξϕ − (r2 + a2)z
+�
+∆b
+�2 − 1
+∆b
+�
+aξϕ − (r2 + a2)z
+�2 + ˜ph,z
+�
+where
+˜ph,z = ξ2
+θ +
+1
+sin2 θ(ξϕ − a sin2 θz)2.
+To define them, we need an escape function.
+Proposition 5.11. Let f(r) ∈ C∞([r−, ∞)) be defined as
+f(r) = ∆b(r)
+r4
+.
+(5.64)
+Then there exists δ0 = δ0(m, a, Q) > 0 such that for (r, θ, ϕ, ξ) ∈ scT ∗ ¯X \ (L± ∪ ∂+ ¯X) and z ∈ R \ 0
+f(r) < δ0,
+⟨ξ⟩−2ph,z(r, θ, ϕ, ξ) = 0,
+⟨ξ⟩−1Hph,zf(r, θ, ϕ, ξ) = 0 =⇒ (⟨ξ⟩−1Hph,z)2f(r, θ, ϕ, ξ) < 0. (5.65)
+Proof. We first show (5.65) on the fiber infinity scS∗ ¯X where ⟨ξ⟩−1 = 0. According to (5.50) in Lemma 5.8,
+we know that
+{⟨ξ⟩−2ph,z = 0} ∩ scS∗ ¯X ⊂ {r ≥ r− | ∆b ≤ a2 sin2 θ} ⊂ {r− ≤ r ≤ m +
+�
+m2 − Q2}.
+A direct calculation gives
+f ′(r) = r∂r∆b(r) − 4∆b(r)
+r5
+= −2r2 − 6mr + 4a2 + 4Q2
+r5
+> 0
+when
+r ∈ [r−, m +
+�
+m2 − Q2].
+Therefore, ⟨ξ⟩−1Hph,zf = 0 implies ⟨ξ⟩−1Hph,zr = 0, that is,
+⟨ξ⟩−1Hph,zr = −ρ−2
+b ⟨ξ⟩−1�
+2∆b
+�
+ξr − zc′(r)
+�
+− 2χ(r2 + a2)z + 2χaξϕ
+�
+= 0
+on
+scS∗ ¯X ∩ Σh.
+This is equivalent to
+∆b ˆξr = −χaˆξϕ
+where ˆξ is defined as in (5.52). Plugging this back to ⟨ξ⟩−2ph,z = 0 yields
+⟨ξ⟩−2ph,z = −ρ−2
+b
+�
+− ∆b ˆξ2
+r + χ2 − 1
+∆b
+a2ˆξ2
+ϕ + ˆξ2
+θ +
+1
+sin2 θ
+ˆξ2
+ϕ
+�
+= 0.
+Since χ = 1 on [r−, rb], it follows that
+{⟨ξ⟩−2ph,z = 0, ⟨ξ⟩−1Hph,zf = 0} ∩ scS∗ ¯X = L±.
+On {⟨ξ⟩−2ph,z = 0, ⟨ξ⟩−1Hph,zf = 0, r > rb} ∩ scS∗ ¯X, we calculate
+(⟨ξ⟩−1Hph,z)2f = −2ρ−2
+b f ′(r)∆b⟨ξ⟩−2Hph,zξr = −2f ′(r)
+ρ4
+b∆b
+∂∆b
+∂r a2 ˆξ2
+ϕ < 0
+where we use the fact that ⟨ξ⟩−1Hph,z⟨ξ⟩−1 = 0 on scS∗ ¯X.
+We next prove that in the region r− ≤ r ≤ rb where f(r) ≤ 0, ⟨ξ⟩−2ph,z and ⟨ξ⟩−1Hph,zf cannot vanish
+at the same time in the interior scT ∗ ¯X (i.e. away from the fiber infinity). Suppose that Hph,zf = 0 and
+ph,z = 0 in r− ≤ r ≤ rb. Then we have
+Hph,zf = −ρ−2
+b f ′(r)
+�
+2∆b
+�
+ξr − zc′(r)
+�
+− 2χ(r2 + a2)z + 2χaξϕ
+�
+= 0.
+
+58
+LILI HE
+Since f ′(r) > 0 when r ∈ [r−, rb], Hph,zf = 0 implies
+∆b
+�
+ξr − zc′(r)
+�
+= χ(r2 + a2)z − χaξϕ.
+Plugging this into ph,z = 0 and using the fact that χ = 1 in [r−, rb] yield
+−ρ−2
+b
+�
+− ∆b
+�
+ξr − zc′(r)
+�2 + ˜ph,z
+�
+= 0
+where
+˜ph,z = ξ2
+θ +
+1
+sin2 θ(ξϕ − a sin2 θz)2,
+and thus
+∆b(ξr − c′(r)z) = 0,
+ξθ = 0,
+ξϕ = a sin2 θz
+in
+r ∈ [r−, rb].
+Therefore, for z ∈ R \ 0
+Hph,zf = ρ−2
+b f ′(r)(2r2z + 2a2 cos2 θz) ̸= 0
+in
+r ∈ [r−, rb],
+which is a contradiction.
+We now turn to the region r > rb where f > 0. We return to the analysis of
+f ′(r) = −2r2 − 6mr + 4a2 + 4Q2
+r5
+.
+It is clear that f ′(r) < 0 when r > 3m and f ′(r) > 0 when rb = m+
+�
+m2 − a2 − Q2 ≤ r <
+3m+√
+9m2−8a2−8Q2
+2
+,
+so in the region f(r) < δ with δ = f(3m), we have f ′(r) ̸= 0. Therefore, in the region f(r) < δ = f(3m),
+Hph,zf = 0 is equivalent to Hph,zr = 0, which is given by
+∆b
+�
+ξr − c′(r)z
+�
+− χ
+�
+(r2 + a2)z − aξϕ
+�
+= 0.
+Under the conditions that ph,z = 0 and Hph,zf = 0, we calculate
+H2
+ph,zf = −2ρ−2
+b f ′(r)∆bHph,zξr = −2f ′(r)
+ρ4
+b∆b
+(aξϕ − (r2 + a2)z)
+�∂∆b
+∂r (aξϕ − (r2 + a2)z) + 4zr∆b
+�
+.
+Next, we let
+A = aξϕ − (r2 + a2)z,
+B = ξϕ − a sin2 θz.
+Then we have z = −ρ−2
+b (A − aB) and
+H2
+ph,zf = −2f ′(r)
+ρ6
+b∆b
+A
+�∂∆b
+∂r Aρ2
+b − 4r∆b(A − aB)
+�
+= −2f ′(r)
+ρ6
+b∆b
+��∂∆b
+∂r ρ2
+b − 4r∆b
+�
+A2 + 4ar∆bAB
+�
+.
+Using ph,z = 0 and Hph,zf = 0, we obtain
+ph,z = −ρ−2
+b
+�
+− 1
+∆b
+A2 + ξ2
+θ +
+B2
+sin2 θ
+�
+= 0
+and thus
+A2 ≥ ∆bB2
+(5.66)
+where we use the fact that B = ξϕ = 0 when sin θ = 0. We also note that ph,z = 0 and z ∈ R \ 0 imply
+A ̸= 0 since otherwise B = 0 and z = 0. We now calculate
+�∂∆b
+∂r ρ2
+b − 4r∆b
+�
+A2 + 4ar∆bAB < −2r(r2 − 3mr + (a2 + 2Q2))A2 + 4|a|r∆b|AB|
+< −2r(r2 − 3mr + (a2 + 2Q2))A2 + 4mr∆1/2
+b
+A2
+< −2r(r2 − 5mr + (a2 + 2Q2))A2
+where we use |a| < m and A2 ≥ ∆bB2 in the second step and ∆b < r2 for r ≥ rb in the third step. As a
+consequence, H2
+ph,zf < 0 when r > 5m.
+We now assume that rb < r ≤ 5m and f(r) < δ′. Then we have
+∆b(r) = r4f(r) ≤ (5m)4f(r) < (5m)4δ′
+∂r∆b = 2r − 2m > 2
+�
+m2 − a2 − Q2.
+By choosing δ′ > 0 sufficiently small such that
+2m
+�
+m2 − a2 − Q2 − 4(5m)4δ′ − 4m(5m)2√
+δ′ ≥ 0,
+
+LINEAR STABILITY OF KERR-NEWMAN FAMILY
+59
+we conclude that on {rb < r ≤ 5m} ∩ {f(r) < δ′}
+r5f ′(r) = r∂r∆b − 4∆b > 2m
+�
+m2 − a2 − Q2 − 4(5m)4δ′ ≥ 0,
+�∂∆b
+∂r ρ2
+b − 4r∆b
+�
+A2 + 4ar∆bAB ≥ r
+�∂∆b
+∂r r − 4∆b − 4m∆1/2
+b
+�
+A2
+> m
+�
+2m
+�
+m2 − a2 − Q2 − 4(5m)4δ′ − 4m(5m)2√
+δ′�
+A2 ≥ 0,
+and thus H2
+ph,zf < 0. Then letting δ0 = min{f(5m), δ′} finishes the proof.
+□
+We now define the outgoing/incoming tails and the trapped set (see [19, Definition 2.1], [21, Definition
+6.1]). Recall that
+scH2,0
+ph,z = ρ−1⟨ξ⟩−1scHph,z,
+ρ = r−1.
+Definition 5.12. Let f be the function defined as in Proposition 5.11. Let (r, θ, ϕ, ξ) ∈ scT ∗ ¯X and φ(s) =
+exp(sscH2,0
+ph,z)(r, θ, ϕ, ξ) be a maximally extended integral curve with domain of definition I ⊂ R on Σh =
+{⟨ξ⟩−2ph,z = 0} ⊂ scT ∗ ¯X. Let s− = inf I, s+ = sup I. We say that φ(s) is trapped as s → ±∞, if there
+exists δ > 0 and T > 0 such that f(φ(s)) > δ for all ±s ≥ T (as a consequence, s± = ±∞). We define the
+incoming tail Γ− and the outgoing tail Γ+ as
+Γ∓ = {(r, θ, ϕ, ξ) ∈ scT ∗ ¯X | φ(s) = exp(sscH2,0
+ph,z)(r, θ, ϕ, ξ) is trapped as s → ±∞}.
+Then we define the trapped set as K := Γ− ∩ Γ+.
+It follows from the definition that Γ±, K are invariant under the flow exp(sscH2,0
+ph,z). We next study the
+properties of Γ± and K. We need the following lemmas.
+Lemma 5.13. For z ∈ R \ 0, we have
+± ⟨ξ⟩−1Hph,zr > 0
+on
+(Σ±
+h ∩ {r− ≤ r ≤ rb}) \ L±.
+(5.67)
+Proof. We first consider the case r− ≤ r < rb and finite ξ. Since χ = 1, c′(r) = 0 on {r− ≤ r < rb}, we write
+ph,z = 0 as
+ph,z = −ρ−2
+b
+�
+∆bξ2
+r − 2(r2 + a2)ξrz + 2aξrξϕ + ˜ph,z
+�
+= 0
+where
+˜ph,z = ξ2
+θ +
+1
+sin2 θ(ξϕ − a sin2 θz)2.
+This is a quadratic equation in ξr with positive discriminant on {r− ≤ r < rb} since ∆b < 0 there. As
+a consequence, it has two distinct roots ξ−
+r
+< 0 < ξ+
+r and ±∂ξrph,z > 0 at ξ±
+r . By Lemma 5.8, Σ±
+h is
+characterized by the conditions ±g−1(−zdtχ + ξ, dt) > 0 (because tb,∗ = tχ in the region r− ≤ r ≤ rb),
+that is, with t = tχ + b(r) (according to the definition of t in (4.15), we see that b′(r) ≤ 0 in the region
+r− ≤ r ≤ rb), we have
+± ξr > ±b′(r)
+�
+(r2 + a2)z − aξϕ
+�
++ a
+�
+a sin2 θz − ξϕ
+�
+r2 + a2 + b′(r)∆b(r)
+.
+(5.68)
+If g−1(−zdtχ + ξ, dt) = 0, then −zdtχ + ξ is a nonzero spacelike vector and thus ph,z(ξ) = −g−1(−zdtχ +
+ξ, −zdtχ + ξ) < 0. That is, substituting the right-hand side of (5.68) into ph,z yields a negative number.
+Therefore,
+(r, θ, ϕ, ξ±
+r , ξθ, ξϕ) ∈ Σ±
+h
+and
+± ⟨ξ⟩−1Hph,zr = ±⟨ξ⟩−1∂ξrph,z > 0
+at
+ξ±
+r .
+In case of fiber infinity scS∗ ¯X ∩ {r− ≤ r < rb}, using the coordinates (5.52), we write
+ρ2
+ξph,z = −ρ−2
+b
+�
+∆b ˆξ2
+r + 2aˆξr ˆξϕ + ˆξ2
+θ +
+1
+sin2 θ
+ˆξ2
+ϕ
+�
+and
+ρξg−1(−zdtχ + ξ, dt) = ρ−2
+b
+��
+r2 + a2 + b′(r)∆b(r)
+�ˆξr +
+�
+a(1 + b′(r))
+�ˆξϕ
+�
+.
+Then the proof proceeds as in the finite ξ case.
+It remains to consider the case r = rb where ∆b = 0. For finite ξ, we write
+ph,z = −ρ−2
+b
+�
+− 2(r2 + a2)ξrz + 2aξrξϕ + ˜ph,z
+�
+= 0
+where
+˜ph,z = ξ2
+θ +
+1
+sin2 θ(ξϕ − a sin2 θz)2.
+
+60
+LILI HE
+Since aξϕ − (r2 + a2)z ̸= 0 on Σh (otherwise a2 sin2 θ = r2 + a2 at rb, which is impossible), the right-hand
+side of ph,z is a linear equation in ξr with nonzero leading coefficient and thus has only one root. Again, by
+Lemma 5.8, Σ±
+h is characterized by the conditions ±g−1(−zdtχ + ξ, dt) > 0, i.e.,
+± ξr > ±−
+�
+a(1 + b′(r))
+�
+ξϕ + z
+�
+a2 sin2 θ + (r2 + a2)b′(r)
+�
+r2 + a2
+.
+(5.69)
+Similarly, substituting the right-hand side of (5.69) into ph,z yields a negative number. Therefore, we have
+either
+(r, θ, ϕ, ξr, ξθ, ξϕ) ∈ Σ+
+h
+and
+⟨ξ⟩−1Hph,zr = ⟨ξ⟩−1∂ξrph,z > 0
+at
+ξr,
+or
+(r, θ, ϕ, ξr, ξθ, ξϕ) ∈ Σ−
+h
+and
+⟨ξ⟩−1Hph,zr = ⟨ξ⟩−1∂ξrph,z < 0
+at
+ξr.
+In case of fiber infinity scS∗ ¯X ∩ {r = rb}, using the coordinates (5.52), we write
+ρ2
+ξph,z = −ρ−2
+b
+�
+2aˆξr ˆξϕ + ˆξ2
+θ +
+1
+sin2 θ
+ˆξ2
+ϕ
+�
+.
+Since ˆξϕ ̸= 0 on ({⟨ξ⟩−2ph,z = 0, r = rb} ∩ scS∗ ¯X) \ L±, the right-hand side of ρξph,z s a linear equation in
+ˆξr with nonzero leading coefficient and thus has only one root. Then the proof proceeds as in the finite ξ
+case.
+□
+Lemma 5.14. Let z ∈ R\0. Let δ0 > 0 and f(r) be defined as in Proposition 5.11. Let (r, θ, ϕ, ξ) ∈ scT ∗ ¯X \
+(L± ∩ ∂+ ¯X) and φ(s) = exp(sscH2,0
+ph,z)(r, θ, ϕ, ξ) be a maximally extended flow on Σh = {⟨ξ⟩−2ph,z = 0}.
+(1) If f(φ)(s0) < δ0 and ±∂sf(φ(s)) ≤ 0 for some ±s0 ≥ 0, then f(φ(s)) < δ0 and ±∂sf(φ(s)) < 0 for
+±s > s0 and φ(s) is not trapped as ±s → ∞.
+(2) If φ(s) is not trapped as ±s → ∞, then either f(φ(s)) < δ0 and ±∂sf(φ(s)) ≤ 0 for all sufficiently
+large ±s, or φ(s) → L± (i.e. f(s) → 0) as ±s → ∞.
+Moreover,
+(3) If φ(s) is not trapped as s → ∞, then for φ(s) ⊂ Σ−
+h , φ(s) crosses ∂− ¯X into the inward direction of
+decreasing r at some finite time s0 < ∞ or ρ(φ(s)) → 0 as s → ∞, while for φ(s) ⊂ Σ+
+h , φ(s) → L+
+or ρ(φ(s)) → 0 as s → ∞.
+(4) If φ(s) is not trapped as s → −∞, then for φ(s) ⊂ Σ−
+h , φ(s) → L− or ρ(φ(s)) → 0 as s → −∞,
+while for φ(s) ⊂ Σ+
+h , φ(s) crosses ∂− ¯X into the inward direction of decreasing r at some finite time
+s0 > −∞ or ρ(φ(s)) → 0 as s → −∞.
+Proof. We only prove the case ∂sf(φ(s)) ≤ 0 as the other case −∂sf(φ(s)) ≤ 0 can be handled similarly. We
+put
+f(s) = f(φ(s)),
+˙f(s) = ∂sf(φ(s)) = scH2,0
+ph,zf(φ(s)),
+¨f(s) = ∂2
+sf(φ(s)) = (scH2,0
+ph,z)2f(φ(s)).
+Let s > s0. If f(s) attains its minimum of on the interval [s0, s] at some point s1 ∈ (s0, s), then
+f(s1) < δ0,
+˙f(s1) = 0,
+¨f(s1) ≥ 0,
+which contradicts (5.65). If f(s) attains its minimum value at s0, then ˙f(s0) ≥ 0, and thus ˙f(s0) = 0.
+Again, it follows from (5.65) that s0 is a strict local maximum point, which is a contradiction. Therefore,
+f(s) must attain its minimum value at s and ˙f(s) < 0 (otherwise ˙f(s) = 0 and then by (5.65), s is a strict
+local maximum point, which is a contradiction), showing that
+f(s) < δ0,
+˙f(s) < 0
+for all
+s > s0.
+Suppose φ(s) is trapped as s → ∞. Then there exists δ > 0 and T > 0 such that f(s) > δ for all s ≥ T .
+Since {f(r) ≥ δ} ∩ scT ∗ ¯X is a compact set, we can take a sequence sj → ∞ such that φ(sj) converges to
+some (r∞, θ∞, ϕ∞, ξ∞) ∈ {δ ≤ f(r) ≤ f(s0) < δ0}. Since f(s) is non increasing and bounded for all s ≥ s0,
+lims→∞ f(s) exists. Since ¨f(s),
+...
+f (s) are bounded for all s ≥ max{s0, T }, by Barb˘alat’s Lemma [3], we have
+scH1,0
+ph,zf((r∞, θ∞, ϕ∞, ξ∞)) = (scH1,0
+ph,z)2f((r∞, θ∞, ϕ∞, ξ∞)) = 0. This contradicts (5.65).
+We now prove the statement (2), (3) and (4). We only prove the case φ(s) is not trapped as s → ∞ in detail
+as the other case can be handled similarly. If φ(s) ⊂ {f(r) ≤ 0} for all s ≥ 0, then φ(s) ⊂ {r− ≤ r ≤ rb}
+since φ(0) /∈ ∂+ ¯X. For the case φ(s) ⊂ Σ−
+h , since φ(0) /∈ L−, it follows from Lemma 5.13 that φ(s) crosses
+∂− ¯X into the inward direction of decreasing r at finite time and thus there must exist s0 > 0 such that
+
+LINEAR STABILITY OF KERR-NEWMAN FAMILY
+61
+f(s0) < δ0, ˙f(s0) < 0. Proceeding as in the proof of (1), we see that f(s) < δ0, ˙f(s) ≤ 0 for all s ≥ s0. While
+for the case φ(s) ⊂ Σ+
+h , we claim that φ(s) → L+ as s → ∞. Otherwise, there exists a sufficiently small
+neighborhood U+ of L+ such that φ(s) ⊂ Σ+
+h \ U+ for all s > 0. By Lemma 5.13, there exists δ > 0 such
+that
+⟨ξ⟩−1Hph,zr ≥ δ
+on
+(Σ+
+h ∩ {r− ≤ r ≤ rb + δ}) \ U+
+(5.70)
+(Otherwise, if (5.70) does not hold for any positive δ, then we can find a sequence {(rj, θj, ϕj, ξj)} such that
+r− ≤ rj ≤ rb + 1
+j and ⟨ξ⟩−1Hph,zr((rj, θj, ϕj, ξj)) < 1
+j . Let j → ∞, and then we obtain a limit point
+(r∞, θ∞, ϕ∞, ξ∞) ∈ (Σ+
+h ∩ {r− ≤ r ≤ rb}) \ L+
+such that ⟨ξ⟩−1Hph,zr((r∞, θ∞, ϕ∞, ξ∞)) ≤ 0, which contradicts Lemma 5.13). It follows from (5.70) that
+φ(s) ⊂ {r ≥ rb + δ} for all large enough s, which contradicts the assumption that φ(s) ⊂ {r− ≤ r ≤ rb}.
+On the other hand, suppose there exists s0 ≥ 0 such that f(s0) > 0. Since φ(s) is not trapped as s → ∞,
+we can find s2 > s1 > 0 such that f(s2) < f(s1) < δ0. Then arguing as in the proof of (1), we see that f(s)
+attains its minimum value on in the interval [s1, s2] at the point s2, and thus ˙f(s2) ≤ 0. Proceeding again
+as in the proof of (1), we see that f(s) < δ0, ˙f(s) ≤ 0 for all s ≥ s2.
+Also, since φ(s) is not trapped as s → ∞, it follows that either ρ(φ(s)) → 0 as s → ∞, or φ(s) enters
+the region {r ≤ rb + δ} for any δ > 0 in finite time. Using (5.70), we find that φ(s) → L+ as s → ∞ if
+φ(s) ⊂ Σ+
+h . If φ(s) ⊂ Σ−
+h , suppose that φ(s) never crosses ∂− ¯X for s ≥ 0. Then φ(s) is well defined for all
+s ≥ 0. Since (ρ, θ, ϕ, ξ) /∈ L−, then there exists a neighborhood V− of L− such that (ρ, θ, ϕ, ξ) /∈ V−. Since
+L− is a radial source, there exists a neighborhood U− of L− and T > 0 such that
+φ(s)(U−) ⊂ V−
+for all
+s ≤ −T.
+Since (ρ, θ, ϕ, ξ) /∈ V−, it follows that φ(s) /∈ U− for all s ≥ T . Proceeding as in the proof of (5.70) yields
+−⟨ξ⟩−1Hph,zr ≥ δ
+on
+(Σ−
+h ∩ {r− ≤ r ≤ rb + δ}) \ U−
+for some δ > 0. This forces r(φ(s)) > rb + δ for all s > T and thus ρ(φ(s)) → 0 as s → ∞.
+□
+According to Lemma 5.14, it is clear that K ⊂ {f(r) ≥ δ0} ⊂ {r > rb}. Therefore, we now restrict our
+analysis in the region r > rb. Recall the calculation in the proof of Proposition 5.11
+∆b > 0, Hph,zr = 0 =⇒ H2
+ph,zr = 2ρ−4
+b ∆b∂r
+�(aξϕ − (r2 + a2)z)2
+∆b(r)
+�
+We define
+F(r) := (aξϕ − (r2 + a2)z)2
+∆b(r)
+.
+(5.71)
+The key property of F(r) is given by
+Proposition 5.15. Let z ∈ R \ 0. For each r ∈ (rb, ∞), we have
+ph,z = 0, ∂rF(r) = 0 =⇒ ∂2
+rF(r) > 0.
+(5.72)
+Moreover, for each fixed ξϕ, there exists a unique rξϕ ∈ (rb, ∞) such that ∂rF(rξϕ) = 0.
+Proof. Since ph,z = 0, by the discussion around (5.66), we see that aξϕ − (r2 + a2)z ̸= 0. Then
+∂rF(r) = −(aξϕ − (r2 + a2)z)
+∆2
+b
+�∂∆b
+∂r (aξϕ − (r2 + a2)z) + 4zr∆b
+�
+= 0
+implies
+∂2
+rF(r) = −(aξϕ − (r2 + a2)z)
+∆2
+b
+�∂2∆b
+∂r2 (aξϕ − (r2 + a2)z) + 2zr∂∆b
+∂r + 4z∆b
+�
+= −
+8z2r
+∆b(∂r∆b)2
+�
+2∂2∆b
+∂r2 r∆b − r(∂r∆b)2 − 2∆b∂r∆b
+�
+=
+32z2r
+∆b(∂r∆b)2
+�
+r3 − 3mr2 + 3m2r − m(a2 + Q2)
+�
+> 0
+on
+rb < r < ∞.
+Since F(r) → ∞ as r → rb and r → ∞, F(r) must attains its global minimum value at rξϕ ∈ (rb, ∞) and
+∂rF(rξϕ) = 0. Moreover, the existence of the critical point of F is unique (otherwise, suppose that there are
+
+62
+LILI HE
+two points r1, r2 ∈ (rb, ∞) with r1 < r2 such that ∂rF(r1) = ∂rF(r2) = 0. Then let D = {r1 < r < r2 |
+∂rF(r) > 0}. Since ∂rF(r1) = 0, ∂2
+rF(r1) > 0, it follows that D is nonempty and we let d := sup D. Since
+∂rF(r2) = 0, ∂2
+rF(r2) > 0, we see that d ∈ (r1, r2). By continuity, we have ∂rF(d) = 0. By the definition of
+d, we have ∂2
+rF(d) ≤ 0, which is a contradiction).
+□
+Let (r0, θ0, ϕ0, ξ0) ∈ Σh and (r(s), θ(s), ϕ(s), ξ(s)) = exp(sHph,z)((r0, θ0, ϕ0, ξ0)). Then we define
+Φ0(r) := Φ(r; θ0, ϕ0, ξ0
+θ, ξ0
+ϕ) = F(r; ξ0
+ϕ) − ˜ph,z((θ0, ϕ0, ξ0
+θ, ξ0
+ϕ)).
+(5.73)
+Since ˜ph,z and ξϕ are constants along the Hamiltonian flows exp(sHph,z) on Σh, it follows that (r(s), ξr(s))
+is a Hamiltonian flow trajectory of
+H0(r, ξr) := ∆b
+�
+ξr − zc′(r) + χ(aξ0
+ϕ − (r2 + a2)z)
+∆b
+�2 − Φ0(r)
+In particular, (r(s), ξr(s)) solves the equation H0(r, ξr) = 0.
+Proposition 5.16. In the region r > rb, the outgoing tail Γ+ and incoming tail Γ− are given by
+Γ± =
+�
+(r, θ, ϕ, ξ)|ph,z = 0, ξr − zc′(r) + χ(aξϕ − (r2 + a2)z)
+∆b
+= ∓sgn(r − rξϕ)
+�
+Φ(r; θ, ϕ, ξθ, ξϕ)
+∆b
+�
+(5.74)
+where
+Φ(r; θ, ϕ, ξθ, ξϕ) = F(r; ξϕ) − ˜ph,z(θ, ϕ, ξθ, ξϕ)
+and rξϕ is the only solution to the equations Φ(r; θ, ϕ, ξθ, ξϕ) = 0 and ∂rΦ(r; θ, ϕ, ξθ, ξϕ) = 0. Moreover, Γ±
+are smooth codimension 1 submanifolds of Σh intersecting transversely, and their intersection is the trapped
+set
+K =
+�
+(r, θ, ϕ, ξ)|ph,z = 0, r = rξϕ, ξr − zc′(r) + χ(aξϕ − (r2 + a2)z)
+∆b
+= 0
+�
+,
+(5.75)
+which is a smooth codimension 2 submanifold of Σh
+Proof. Let (r0, θ0, ϕ0, ξ0) ∈ Σh and (r(s), θ(s), ϕ(s), ξ(s)) = exp(sHph,z)((r0, θ0, ϕ0, ξ0)). By Proposition
+5.15, ∂rΦ0(rξ0ϕ) = 0. If Φ0(rξ0ϕ) ̸= 0, we have |Φ0(r(s))|+|∂rΦ0(r(s))| > 0. Suppose that (r(s), θ(s), ϕ(s), ξ(s))
+is trapped as s → ∞. Since (r(s), ξr(s)) solves the equation H0(r, ξr) = 0, arguing as in the proof of the
+statement (1) of Lemma 5.14, we can find a sequence sj → ∞ such that (r(sj), ξr(sj)) converges to some
+(r∞, (ξr)∞) which satisfies H0(r∞, (ξr)∞) = 0, HH0r((r∞, (ξr)∞)) = HH0ξr((r∞, (ξr)∞)) = 0. This contra-
+dicts the fact that the Hamiltonian vector field associated to H0 is nonzero on the set H0 = 0. This proves
+that (r(s), θ(s), ϕ(s), ξ(s)) is not trapped as s → ∞. Similarly, (r(s), θ(s), ϕ(s), ξ(s)) escapes as s → −∞ as
+well. As a result, (r0, θ0, ϕ0, ξ0) /∈ Γ±.
+On the other hand, we consider the case Φ0(rξ0ϕ) = 0. We note that the set of solutions to the equation
+H0(r, ξr) = 0 is given by the union Γ+,0 ∪ Γ−,0 where
+Γ±,0 =
+�
+ξr − zc′(r) + χ(aξ0
+ϕ − (r2 + a2)z)
+∆b
+= ∓sgn (r − rξ0ϕ)
+�
+Φ(r; θ0, ϕ0, ξ0
+θ, ξ0ϕ)
+∆b
+�
+.
+First, (r0, θ0, ϕ0, ξ0) ∈ Σh is equivalent to H0(r0, ξ0
+r) = 0, i.e., (r0, θ0, ϕ0, ξ0) ∈ Σh implies (r0, ξ0
+r) ∈
+Γ+,0 ∪ Γ−,0. Next, if (r0, ξ0
+r) ∈ Γ∓,0, we see that (r(s), θ(s), ϕ(s), ξ(s)) is trapped as s → ±∞. Therefore,
+we conclude that (r(s), θ(s), ϕ(s), ξ(s)) is trapped as s → ±∞ if and only if (r0, ξ0
+r) ∈ Γ∓,0. Then we obtain
+K = Γ+ ∩ Γ− =
+�
+(r, θ, ϕ, ξ)|ph,z = 0, r = rξϕ, ξr − zc′(r) + χ(aξϕ − (r2 + a2)z)
+∆b
+= 0
+�
+=
+�
+(r, θ, ϕ, ξ)|ph,z = 0, ∂rF = 0, Hph,zr = 0
+�
+.
+We note that d(Hph,zr) ∈ {dr, dξr, dξϕ} and the coefficient of dξr is 1. In view of (5.72), d(∂rF) ∈ {dr, dξϕ}
+and the coefficient of dr is positive on K. Since Hph,zr = 0 and ∂rF = 0 imply ∂ξrph,z = 0 and ∂rph,z = 0
+respectively on K, we have dph,z ∈ {dθ, dξθ, dξϕ}.
+Therefore, dph,z, d(∂rF) and d(Hph,zr) are linearly
+independent on K and K is a smooth codimension 2 submanifold of Σh.
+For any (r0, θ0, ϕ0, ξ0) ∈ K, since Φ0(r) = 1
+2|∂2
+rΦ0(rξ0ϕ)|(r − rξ0ϕ)2 + O((r − rξ0ϕ)3), the defining functions
+of Γ±,0 are smooth functions in r, ξr and thus Γ±,0 is a smooth codimension 1 submanifold of T ∗(rb, ∞). If
+
+LINEAR STABILITY OF KERR-NEWMAN FAMILY
+63
+(r, θ, ϕ, ξ) ∈ Γ±, then its projection onto (θ, ϕ, ξθ, ξφ) must lie in the set of projection of K onto (θ, ϕ, ξθ, ξφ).
+Therefore, Γ± are smooth codimension 1 submanifolds of Σh intersecting transversely at K.
+□
+Since T K is in the kernel of d(Hph,zr) and d(∂rF), we see that the symplectic complement of T K in Σh
+is (T K)⊥ = {HHph,z r, H∂rF }. By a direct calculation, we see that H∂rF (Hp,zr) = −2∆b∂2
+rF ̸= 0 on K,
+which implies T K ∩(T K)⊥ = 0 and thus K is symplectic. Now we verify that the trapped set K is normally
+hyperbolic whose definition will be given below (see [21, §6.3]).
+Definition 5.17. Let K be given as above, i.e., K is symplectic. Let φs = exp(sHph,z) : scT ∗ ¯X → scT ∗ ¯X
+be the Hamiltonian flow and ϕ± be the defining functions of Γ±. We say K is normally hyperbolic if there
+exist C, ν > 0 such that for all (r, θ, ϕ, ξ) ∈ K
+d(ϕ± ◦ φ±s)(r, θ, ϕ, ξ)
+dϕ±(r, θ, ϕ, ξ)
+≤ Ce−νs,
+s ≥ 0
+(5.76)
+Define the minimal expansion rate νmin > 0 as the supremum of all values of ν for which there exists a
+constant C such that (5.76) holds.
+Proposition 5.18. Let K be defined as in (5.75). Then K is normally hyperbolic.
+Proof. It suffices to prove that the minimal expansion νmin rate defined above is positive. By (5.74), we have
+Γ± = {ϕ± = ξr − zc′(r) + χ(aξϕ − (r2 + a2)z)
+∆b
+± sgn (r − rξϕ)
+�
+Φ(r; θ, ϕ, ξθ, ξϕ)
+∆b
+= 0}
+where (θ, ϕ, ξθ, ξϕ) ∈ K. That is ϕ± are defining function of Γ±. Since ph,z = ρ−2
+b ∆bϕ+ϕ−, it follows that
+Hph,zϕ± = ∓ρ−2
+b ∆b(Hϕ+ϕ−)ϕ± + (Hρ−2
+b
+∆bϕ±)ϕ∓ϕ± := ∓ν±ϕ±
+and ν+|K = ν−|K = ρ−2
+b ∆b(Hϕ+ϕ−)|K = ν. Since
+∂ξr(Hph,zϕ±) = ∂ξr(∓ν±ϕ±) = ∓
+�
+(∂ξrν±)ϕ± + ν±
+�
+,
+using Proposition 5.15 we find that
+ν = −∂ξr(Hph,zϕ+)|K = ρ−2
+b
+�
+2∆b∂2rF(rξϕ) > 0.
+For (r, θ, ϕ, ξ) ∈ K, we calculate
+∂sd(ϕ± ◦ φ±s) = ∓d((ν±ϕ±) ◦ φ±s) = ∓(ν± ◦ φ±s)d(ϕ± ◦ φ±s).
+Therefore, for all T > 0 and (r, θ, ϕ, ξ) ∈ K, we have
+d(ϕ± ◦ φ±T )(r, θ, ϕ, ξ)
+dϕ±(r, θ, ϕ, ξ)
+= e−⟨ν⟩±
+T T ,
+⟨ν⟩±
+T = 1
+T
+� T
+0
+ν ◦ φ±s(r, θ, ϕ, ξ) ds.
+As a result, the minimal expansion rate is given by
+νmin = min{ lim inf
+T →∞
+inf
+(r,θ,ϕ,ξ)∈K⟨ν⟩+
+T ,
+lim inf
+T →∞
+inf
+(r,θ,ϕ,ξ)∈K⟨ν⟩−
+T } > 0.
+(5.77)
+□
+Finally, we further determine the position of the tapped set K.
+Lemma 5.19. For z ∈ R \ 0, let Σ±
+h be defined as in Lemma 5.8. Then K ⊂ Σ±
+h for ±z > 0. Moreover,
+Γ+ ∪ Γ− ⊂ Σ±
+h for ±z > 0.
+Proof. First, according to (5.51) in Lemma 5.8, we have
+Σ∓
+h ⊂ {r ≥ r− | ∆b ≤ a2 sin2 θ} ⊂ {r− ≤ r ≤ m +
+�
+m2 − Q2}
+for
+± z > 0
+and thus
+K ∩ {r > m +
+�
+m2 − Q2} ⊂ Σ±
+h
+for
+± z > 0.
+Since c′(r) = 0, χ = 1 on r− ≤ r ≤ m +
+�
+m2 − Q2, then for
+(r, θ, ϕ, ξ) ∈ K ∩ {r− ≤ r ≤ m +
+�
+m2 − Q2} = K ∩ {rb < r ≤ m +
+�
+m2 − Q2},
+
+64
+LILI HE
+we have
+aξϕ − (r2 + a2)z = −4zr∆b
+∂r∆b
+and
+ξr = 4zr
+∂r∆b
+,
+and thus
+±g−1(−zdtb,∗ + ξ, dt) = ±ρ−2
+b
+�
+ξr(r2 + a2 + b′(r)∆b) + b′(r)(aξϕ − (r2 + a2)z) + (aξϕ − a2 sin2 θz)
+�
+= ±
+4zr
+ρ2
+b∂r∆b
+�
+r2 + a2 − ∆b
+�
+± z = ±
+4zr
+ρ2
+b∂r∆b
+�
+2mr − Q2�
+± z > 0
+for
+± z > 0.
+This implies that
+K ∩ {r− ≤ r ≤ m +
+�
+m2 − Q2} ⊂ Σ±
+h
+for
+± z > 0,
+and thus proves K ⊂ Σ±
+h for ±z > 0.
+In order to prove Γ+ ∪ Γ− ⊂ Σ±
+h for ±z > 0, it suffices to show that if (r, θ, ϕ, ξ) ∈ Γ±, then φ(s) =
+exp(sscH2,0
+ph,z)(r, θ, ϕ, ξ) → K as s → ∓∞. We only prove the case of (r, θ, ϕ, ξ) ∈ Γ− as the other case can
+be handled in a similar manner. Suppose that φ(s) ↛ K as s → ∞. Then there exists a sequence sj → ∞
+and a neighborhood U of K such that φ(sj) /∈ U for all j. Since φ(s) is trapped as s → ∞ (i.e., there exist
+δ, T > 0 such that f(φ(s)) > δ for all s > T ), passing to a subsequence we have
+φ(sj) → (r∞, θ∞, ϕ∞, ξ∞)
+for some
+(r∞, θ∞, ϕ∞, ξ∞) /∈ K = Γ+ ∩ Γ−.
+By Lemma 5.14, we find that Γ− is closed and thus (r∞, θ∞, ϕ∞, ξ∞) ∈ Γ−. It follows that (r∞, θ∞, ϕ∞, ξ∞) /∈
+Γ+, which means that φ(s) is not trapped as s → −∞. Then there exists T1 > 0 such that f(φ∞(−T1)) < δ
+where φ∞(s) = exp(sscH2,0
+ph,z)(r∞, θ∞, ϕ∞, ξ∞). Since φ(sj) → (r∞, θ∞, ϕ∞, ξ∞), it follows that
+exp(−T1
+scH2,0
+ph,z)(φ(sj)) → φ∞(−T1).
+Therefore, f(φ(sj − T1)) < δ for all sufficiently large j. But this contradicts the fact that φ(s) is trapped as
+s → ∞.
+□
+5.4.5. Global dynamics of the Hamiltonian flow. Now we shall analyze the global behavior of the flow of
+scH2,0
+ph,z := ρ−1⟨ξ⟩−1scHph,z on the semiclassical characteristic set Σh = {⟨ξ⟩−2ph,z = 0}.
+We need the
+following technical lemma.
+Lemma 5.20. For z ∈ R \ 0, let (ρ, θ, ϕ, ξ) ∈ Σh and we put φ(s) = exp(sscH2,0
+ph,z)(ρ, θ, ϕ, ξ). If ρ(φ(s)) → 0
+as s → ±∞, then one has
+φ(s) → R(z
+2 ± |z|
+2 ) = {ρ = 0, ξ = (z ± |z|)dρ
+ρ2 }
+as
+s → ±∞.
+Proof. Recall that
+⟨ξ⟩scH2,0
+ph,z = (−2ξρ + 2z)ρ∂ρ + 2˜p∂ξρ + (−2ξρ + 2z)
+�
+µ=θ,ϕ
+ξµ∂ξµ − H˜p + ρV,
+V ∈ Vb(scT ∗ ¯X).
+Since ph,z = 0 along the integral curves of scH2,0
+ph,z, we calculate
+⟨ξ⟩scH2,0
+ph,z
+�ξρ − z
+ρ
+�
+= ⟨ξ⟩2(ξρ − z)2 + 2˜p + ρa1
+ρ
+= ⟨ξ⟩z2 + ρa2
+ρ
+,
+a1, a2 ∈ A0( ¯X).
+Since ρ(φ(s)) → 0 as s → ±∞, then for any ǫ > 0, there exists T > 0 such that ρ(φ(s)) < ǫ for all
+±s > T . Picking ǫ sufficiently small, we see that scH2,0
+ph,z
+�
+ξρ−z
+ρ
+�
+≥ c1 > 0 for all ±s > T . This implies that
+(ξρ − z)/ρ → ±∞ as s → ±∞; in particular ±(ξρ − z) ≥ 0 for all ±s > T1 where T1 > 0 is a sufficiently
+large constant.
+According to Lemma 5.10, there exist neighborhoods V ǫ′
+± = {(ξρ − (z ± (sgn z)z)))2 + ˜p + ρ2 ≤ ǫ′} of
+R( z
+2 ± |z|
+2 ) such that once φ(s) enters V ǫ′
+± at some time ±s0 > 0, it will converge to R( z
+2 ± |z|
+2 ) as ±s → ∞.
+Suppose that φ(s) /∈ V ǫ′
+± for all ±s ≥ 0. Since ph,z = 0 along the integral curves of scH2,0
+ph,z, it follows that
+|ξρ − z| ≤ δ1
+˜p > δ2
+for all
+± s > T2
+(5.78)
+for some δ1, δ2 > 0, T2 > T1. Then for all ±s > T2, we have
+scH2,0
+ph,z(ξρ − z) = ⟨ξ⟩−1(2˜p + ρa2) > c2 > 0,
+a2 ∈ A0( ¯X).
+
+LINEAR STABILITY OF KERR-NEWMAN FAMILY
+65
+which contradicts (5.78). Therefore, φ(s) must enter V ǫ′
+± at some time ±s0 > 0 and this finishes the proof
+of the lemma.
+□
+Now we are at the position to describe the the global behavior of the flow of scH2,0
+ph,z := ρ−1⟨ξ⟩−1scHph,z
+on the semiclassical characteristic set Σh = {⟨ξ⟩−2ph,z = 0}.
+L+
+L−
+R(z)
+R(0)
+K
+Γ−
+Γ+
+Σ−
+h
+Σ+
+h
+z > 0
+L+
+L−
+R(z)
+R(0)
+K
+Γ+
+Γ−
+Σ−
+h
+Σ+
+h
+z < 0
+Figure 10. The flow of exp(sscH2,0
+ph,z), which is projected to the coordinates (r, ξr) and
+drawn in a fiber-radially compactified view. The shaded region is the semiclassical char-
+acteristic set Σh = Σ+
+h ⊔ Σ−
+h = {⟨ξ⟩−2ph,z = 0}. The horizontal coordinate is r and the
+rightmost vertical line corresponds to ∂+ ¯X. The dotted line is the even horizon r = rb and
+the vertical line in the middle is {r = rergo} where rergo is the equatorial radius (i.e. greatest
+radius) of the ergosphere. The vertical coordinate is ξr/⟨ξr⟩, so the top and bottom lines
+stand for the fiber infinity.
+Proposition 5.21. Let ±z > 0. Let φ(s) = exp(sscH2,0
+ph,z)(ρ, θ, ϕ, ξ) be the maximally extended integral curve
+of scH2,0
+ph,z on Σh with the domain of definition s ∈ I ⊂ R. Let s− = inf I, s+ = sup I. (see Figure 10).
+(1) Let z > 0.
+(i) If φ(s) ⊂ Σ−
+h , then either φ(s) ⊂ L−, or φ(s) → L− as s → s− = −∞ and φ(s) crosses ∂− ¯X
+into the inward direction of deceasing r at finite time s+ < ∞.
+(ii) If φ(s) ⊂ Σ+
+h , then either φ(s) ⊂ L+∪R(z)∪R(0)∪K, or φ(s) → L+∪R(z)∩K as s → s+ = ∞
+and φ(s) → R(0) ∪ K as s → s− = −∞ or φ(s) crosses ∂− ¯X into the inward direction of
+deceasing r at finite time s− > −∞. Moreover, if (ρ, θ, ϕ, ξ) /∈ K, then φ(s) cannot converge to
+K in both the forward and backward direction.
+(2) Let z < 0.
+(i) If φ(s) ⊂ Σ+
+h , then either φ(s) ⊂ L+, or φ(s) → L+ as s → s+ = ∞ and φ(s) crosses ∂− ¯X into
+the inward direction of deceasing r at finite time s− > −∞.
+(ii) If φ(s) ⊂ Σ−
+h , then either φ(s) ⊂ L−∪R(z)∪R(0)∪K, or φ(s) → L−∪R(z)∪K as s → s− = −∞
+and φ(s) → R(0)∪K as s → s+ = ∞ or φ(s) crosses ∂− ¯X into the inward direction of deceasing
+r at finite time s+ < ∞. Moreover, if (ρ, θ, ϕ, ξ) /∈ K, then φ(s) cannot converge to K in both
+the forward and backward direction.
+Proof. We only consider the case z > 0 as the other case z < 0 can be proved similarly. First, we note that
+L+, L−, R(0), R(z), K are invariant under the flow φ(s) and if (ρ, θ, ϕ, ξ) ∈ Γ±, then φ(s) → K as ∓s → ∞.
+If (ρ, θ, ϕ, ξ) /∈ L+ ∪ L− ∪ R(0) ∪ R(z) ∪ K, then either (ρ, θ, ϕ, ξ) /∈ Γ− or (ρ, θ, ϕ, ξ) /∈ Γ+.
+
+66
+LILI HE
+If (ρ, θ, ϕ, ξ) /∈ Γ−, by Lemma 5.13, for φ(s) ⊂ Σ+
+h , either φ(s) → L+ or ρ(φ(s)) → 0 as s → ∞, while
+for φ(s) ⊂ Σ−
+h , φ(s) crosses ∂− ¯X into the inward direction of deceasing r at some finite time s0 < ∞. If
+ρ(φ(s)) → 0 as s → ∞, according to Lemma 5.20, we have φ(s) → R(z) as s → ∞.
+If (ρ, θ, ϕ, ξ) /∈ Γ+, by Lemma 5.13, for φ(s) ⊂ Σ+
+h , either ρ(φ(s)) → 0 as s → ∞ or φ(s) crosses ∂− ¯X
+into the inward direction of deceasing r at some finite time s0 > −∞, while for φ(s) ⊂ Σ−
+h , φ(s) → L− as
+s → −∞. If ρ(φ(s)) → 0 as s → −∞, according to Lemma 5.20, we have φ(s) → R(0) as s → −∞.
+□
+5.5. High energy estimates. In this section, we prove the high energy estimates first for operators
+h2�
+□gb(h−1z) acting on scalar functions, then for h2 �
+Pb,γ(h−1z), h2�
+Wb,γ(h−1z) acting on scattering 1-forms
+and the linearized gauge-fixed Einstein-Maxwell operator h2 �
+Lb,γ(h−1z), as well as their formal adjoints. To
+this end, we combine the global dynamics of of the Hamiltonian flow of the semiclassical principal symbol ph,z
+established in 5.21 with the elliptic estimate, propagation of singularities estimate, the radial point estimate
+at event horizon, the scattering radial point estimate at spatial infinity ∂+ ¯X and hyperbolic estimate, all of
+which are in the semiclassical version, together with the estimate at normally hyperbolic trapping.
+5.5.1. High energy estimates for scalar wave operators. We first prove high energy estimates for the oper-
+ator h2�
+□gb(h−1z) acting on scalar functions, as well as its formal adjoint with respect to volume density
+L2( ¯X;
+�
+det |gb|drdθdϕ).
+B+
+B−
+B0
+E0
+Bz
+B1
+B2
+B2
+L+
+L−
+R(z)
+R(0)
+K
+Γ−
+Γ+
+suppχ
+B′
++
+E′
++
+B′
+−
+E′
+−
+B′
+0
+B′
+z
+E′
+z
+B′
+1
+B′
+2
+B′
+2
+L+
+L−
+R(z)
+R(0)
+K
+Γ−
+Γ+
+suppχ
+supp(1 − χ)
+Figure 11. A phase space picture of the proof of the estimate (5.81), on the left, and (5.82),
+on the right. The coordinates and notations L±, R(0), R(z), Γ±, K are the same as in Figure
+10. For (5.81), we use semiclassical hyperbolic estimates to control u via χu; χ is controlled
+(modulo elliptic estimates) by B±, B1, B0, Bz; B0 is controlled by E0 using low b-decay
+radial point estimates and E0 is again controlled by B+, B1, Bz; B1 is controlled by B2 by
+using normally hyperbolic trapping estimate and B2 is controlled by B+, Bz; finally B±, Bz
+are controlled using high regularity radial points estimates at L± and high sc-decay radial
+point estimates at R(z). For (5.82), we use semiclassical hyperbolic estimates to bound
+(1 − χ)u; χ is controlled (modulo elliptic estimates) by B′
+±, B′
+1, B′
+0, B′
+z; B′
+± is controlled
+by E′
+± using low regularity radial point estimates and E′
+± is controlled by B′
+1, B′
+0, 1 − χ;
+B′
+z is controlled by E′
+z using low sc-decay radial points estimates and E′
+z is controlled by
+B′
+0, B′
+1, 1−χ; B′
+1 is controlled by B′
+2 by using normally hyperbolic trapping estimate and B′
+2
+is controlled by B′
++, 1−χ; finally B′
+0 is controlled using high b-decay radial points estimates
+at R(0).
+
+LINEAR STABILITY OF KERR-NEWMAN FAMILY
+67
+Theorem 5.22. Let b0 = (m0, a0, Q0) with |Q0| + |a0| < m0.
+Then there exists ǫ > 0 such that for
+b = (m, a, Q) with |b − b0| < ǫ and for s > 1
+2, ℓ < − 1
+2 with s + ℓ > − 1
+2, the following holds. For any fixed
+C1 > 0, there exist C2 > 1, C > 0 which are independent of b, such that for σ ∈ C, Im σ ∈ [0, C1], |Re σ| > C2,
+we have
+∥u∥ ¯
+Hs,ℓ
+b,h( ¯
+X) ≤ C∥�
+□gb(σ)u∥ ¯
+Hs,ℓ+1
+b,h
+( ¯
+X),
+∥u∥ ˙H−s,−ℓ−1
+b,h
+( ¯
+X) ≤ C∥�
+□gb(σ)∗u∥ ˙H−s,−ℓ
+b,h
+( ¯
+X);
+(5.79)
+∥u∥ ¯
+Hs,ℓ
+b,h( ¯
+X) ≤ C∥�
+□gb(σ)u∥ ¯
+Hs−1,ℓ+2
+b,h
+( ¯
+X),
+∥u∥ ˙H−s+1,−ℓ−2
+b,h
+( ¯
+X) ≤ C∥�
+□gb(σ)∗u∥ ˙H−s,−ℓ
+b,h
+( ¯
+X),
+(5.80)
+where h = |σ|−1 and C only depends on b0, s, ℓ, C1.
+Therefore,
+�
+□gb : {u ∈ ¯Hs,ℓ
+b,h( ¯X) | ¯Hs,ℓ+1
+b,h
+( ¯X)} → ¯Hs,ℓ+1
+b,h
+( ¯X)
+and
+�
+□gb : {u ∈ ¯Hs,ℓ
+b,h( ¯X) | ¯Hs−1,ℓ+2
+b,h
+( ¯X)} → ¯Hs−1,ℓ+2
+b,h
+( ¯
+X)
+are invertible for σ ∈ C, Im σ ∈ [0, C1], |Re σ| > C2.
+Proof. For the simplicity of notations, we let Ph(z) := h2�
+□gb(h−1z). We first claim that it suffices to prove
+the following estimates for s > 1
+2, r > − 1
+2, ℓ < − 1
+2
+∥u∥ ¯
+Hs,r,ℓ
+sc,b,h( ¯
+X) ≤ Ch−2∥Ph(z)u∥ ¯
+Hs−1,r+1,ℓ+1
+sc,b,h
+( ¯
+X)
+(5.81)
+and
+∥u∥ ˙H−s+1,−r−1,−ℓ−1
+sc,b,h
+( ¯
+X) ≤ Ch−2∥Ph(z)∗u∥ ˙H−s,−r,−ℓ
+sc,b,h
+( ¯
+X)
+(5.82)
+Since ¯Hs,s+ℓ,ℓ
+sc,b,h
+= ¯Hs,l
+b,h and ˙Hs,s+ℓ,ℓ
+sc,b,h
+= ˙Hs,l
+b,h, letting r = s + ℓ > − 1
+2 we have
+∥u∥ ¯
+Hs,s+ℓ,ℓ
+sc,b,h ( ¯
+X) ≤ Ch−2∥Ph(z)u∥ ¯
+Hs−1,s+ℓ+1,ℓ+1
+sc,b,h
+( ¯
+X)
+and
+∥u∥ ˙H−s+1,−s−r−1,−ℓ−1
+sc,b,h
+( ¯
+X) ≤ Ch−2∥Ph(z)∗u∥ ˙H−s,−s−r,−ℓ
+sc,b,h
+( ¯
+X).
+Then using the facts that
+¯Hs,ℓ+1
+b,h
+= ¯Hs,s+ℓ+1,ℓ+1
+sc,b,h
+⊂ ¯Hs−1,s+ℓ+1,ℓ+1
+sc,b,h
+,
+¯Hs−1,ℓ+2
+b,h
+= ¯Hs−1,s+ℓ+1,ℓ+2
+sc,b,h
+⊂ ¯Hs−1,s+ℓ+1,ℓ+1
+sc,b,h
+,
+˙H−s,−ℓ−1
+b,h
+= ˙H−s,−s−ℓ−1,−ℓ−1
+sc,b,h
+⊃ ˙H−s+1,−s−ℓ−1,−ℓ−1
+sc,b,h
+,
+˙H−s+1,−ℓ−2
+b,h
+= ˙H−s+1,−s−ℓ−1,−ℓ−2
+sc,b,h
+⊃ ˙H−s+1,−s−ℓ−1,−ℓ−1
+sc,b,h
+,
+we obtain (5.79) and (5.80). Now we will show (5.81) and (5.82). Here we only discuss the case Re z > 0
+(see Figure 11 for a phase space illustration of the proof of estimates (5.81) and (5.82)), as the case Re z < 0
+can be handled in a similar manner.
+• Proof of the estimate (5.81). For s > s0 >
+1
+2 ≥
+1
+2 − Im (h−1z) r2
+b+a2
+r2
+b−m (as calculated in (5.59) and
+(5.8)), using the semiclassical high regularity radial point estimate (see [77, Proposition 2.10], [78,
+Proposition 5.27]), there exist B±, G±, S± ∈ Ψ0
+h( ¯X) microlocally supported near L± with L± ⊂
+ellh(B±), L± ⊂ ellh(S±) and satisfying that all the forward (backward) null characteristics from
+WFh(B±) tend to L±, while remaining in the elliptic set of G±, such that
+∥B±u∥Hs,r,ℓ
+sc,b,h( ¯
+X) ≤ Ch−1∥G±Ph(z)u∥Hs−1,r+1,ℓ+1
+sc,b,h
+( ¯
+X) + Chs−s0∥S±u∥Hs0,r0,ℓ0
+sc,b,h
+( ¯
+X)
+(5.83)
+where r0 < r, ℓ0 < ℓ and the sc-decay order r and b-decay order ℓ are actually irrelevant here.
+For r > r0 > − 1
+2 (as calculated in (5.14)), using the semiclassical high sc-decay radial point esti-
+mates at the non-zero scattering section R(z) (see [79]. The proof follows from a positive commutator
+estimate
+i(Ph(z)∗A − APh(z)) = Im Ph(z)A + AIm Ph(z) + i[Re Ph(z), A]
+where
+Re Ph(z) = Ph(z) + Ph(z)∗
+2
+,
+Im Ph(z) = Ph(z) − Ph(z)∗
+2i
+.
+Let A ∈ Ψ2s−1,2r+1,2l+1
+sc,b,h
+( ¯X) with semiclassical principal symbol
+a = χ0(ξ2
+θ +
+1
+sin2 θξ2
+ϕ)χ1((ξρ − 2Re z)2)ρ−2r+1(ξ2
+θ +
+1
+sin2 θξ2
+ϕ + ξ2
+ρ)s− 1
+2
+
+68
+LILI HE
+where χ0, χ1 are identically 0 near 0 and have compact support sufficiently close to 0, and χ1
+has relatively large support such that suppχ0 ∩ suppχ′
+1 is disjoint from the characteristic set of
+Re Ph(z).
+Then using Lemma 4.1 and the calculation before Lemma 4.8 in [79], it follows that
+h−1(Im Ph(z)A + AIm Ph(z) + i[Re Ph(z), A]) ∈ Ψ2s,2r,2l
+sc,b,h ( ¯X) whose semiclassical principal symbol
+is positive definite elliptic near R(z) if r > − 1
+2. This, together with a regularization argument,
+proves semiclassical high sc-decay radial point estimates at R(z)), there exist Bz, Gz, Sz ∈ Ψ0,0
+sc,h( ¯X)
+microlocally supported near R(z) with R(z) ⊂ ellsc,h(Bz), R(z) ⊂ ellsc,h(Sz) and satisfying that all
+the forward null characteristics from WFsc,h(Bz) tend to R(z), while remaining in the scattering
+elliptic set of Gz, such that
+∥Bzu∥Hs,r,ℓ
+sc,b,h( ¯
+X) ≤ Ch−1∥GzPh(z)u∥Hs−1,r+1,ℓ+1
+sc,b,h
+( ¯
+X) + Chs−s0∥Szu∥Hs0,r0,ℓ0
+sc,b,h
+( ¯
+X)
+(5.84)
+where s0 < s, ℓ0 < ℓ and the differential order s and b-decay order ℓ are actually irrelevant here.
+For ℓ < − 1
+2 (as calculated in (5.14)), using the semiclassical low b-decay radial point estimates
+at the zero scattering section R(0) (see [79, §4–5]), there exist B0, G0 ∈ Ψ0,0,0
+sc,b,h( ¯X) microlocally
+supported near R(0) (in fact near the blown up R(0) in the second microlocalized space introduced
+in §5.1.5) with R(0) ⊂ ellsc,b,h(B0) and E0 ∈ Ψ0,0,0
+sc,b,h( ¯X), WFsc,b,h(E0) ∩ R(0) = ∅, and satisfying
+that all the forward null characteristics from WFsc,b,h(B0)\ R(0) enter ellsc,b,h(E0), while remaining
+in the second microlocalized scattering elliptic set of G0, such that for any sufficiently large N > 0
+∥B0u∥Hs,r,ℓ
+sc,b,h( ¯
+X) ≤ Ch−1∥G0Ph(z)u∥Hs−1,r+1,ℓ+1
+sc,b,h
+( ¯
+X) + C∥E0u∥Hs,r,ℓ
+sc,b,h( ¯
+X) + ChN∥u∥ ¯
+H−N,−N,ℓ
+sc,b,h
+( ¯
+X)
+(5.85)
+where the differential order s is actually irrelevant here.
+Since at the tapped set K
+σh(h−1Im Ph(z)) = σh(Ph(z) − Ph(z)∗
+2ih
+) = σh(Ph(z) − Ph(¯z)
+2ih
+)
+= −ρ−2
+b Im σ
+�
+−
+�
+ξr − c′(r)Re z + χ(aξϕ − (r2 + a2)Re z)
+∆b
+��
+c′(r) + χ(r2 + a2)
+∆b
+�
++ 2
+∆b
+�
+aξϕ − (r2 + a2)Re z
+��
+r2 + a2�
+− 2a
+�
+ξϕ − a sin2 θRe z
+��
+= −ρ−2
+b Im σ
+� 2
+∆b
+�
+aξϕ − (r2 + a2)Re z
+��
+r2 + a2�
+− 2a
+�
+ξϕ − a sin2 θRe z
+��
+> 0
+where we use ξr − c′(r)Re z + χ(aξϕ−(r2+a2)Re z)
+∆b
+= 0 at K in the last equality and
+aξϕ − (r2 + a2)Re z = −−4r∆bRe z
+∂r∆b
+< 0,
+(aξϕ − (r2 + a2)Re z)2 ≥ ∆b(ξϕ − a sin2 θRe z)2
+at K,
+which follow from Proposition 5.15 and (5.66), in the last step, it follows that
+σh(h−1Im − Ph(z)) = σh((−Ph(z)) − (−Ph(z))∗
+2ih
+) < 0 < νmin
+2
+at
+K.
+Using the hyperbolic trapping estimates (see [20] and [40, Theorem 4.7]), there exist B1, G1 ∈
+Ψ0
+h( ¯X) microlocally supported near K with K ⊂ ellh(B1) and B2 ∈ Ψ0
+h( ¯X), WFh(B2) ∩ Γ− = ∅,
+and satisfying that WFh(B1) ∪ WFh(B2) is contained in the elliptic set of G1, such that for any
+N, s′, r′, ℓ′ ∈ R
+∥B1u∥Hs,r,ℓ
+sc,b,h( ¯
+X) ≤ Ch−2∥G1Ph(z)u∥Hs−1,r+1,ℓ+1
+sc,b,h
+( ¯
+X) + C∥B2u∥Hs,r,ℓ
+,hsc,b( ¯
+X) + ChN∥u∥ ¯
+Hs′,r′,ℓ′
+sc,b,h ( ¯
+X)
+(5.86)
+where the differential order s′, the sc-decay order r′ and b-decay order ℓ′ are actually irrelevant here.
+Let r− < r0 < rb and χ ∈ C∞
+c ( ¯X) with χ = 1 near r ≥ rb and suppχ ⊂ {r ≥ r0}.
+By
+part (1) in Proposition 5.21, semiclassical propagation of singularities estimates and semiclassical
+elliptic estimates (Concretely, when (ρ, θ, ϕ, ξ) ∈ WFh(χ) ∩ {⟨ξ⟩−2ph,z(ξ) ̸= 0}, we use semiclassical
+elliptic estimates.
+If (ρ, θ, ϕ, ξ) ∈ WFh(χ) ∩ {⟨ξ⟩−2ph,z(ξ) = 0}, then there exists s ∈ R with
+exp(sscH2,0
+ph,z)(ρ, θ, ϕ, ξ) ∈ ellh(B±) ∪ ellh(B1) ∪ ellh(B0) ∪ ellh(Bz), and thus we use semiclassical
+propagation of singularities estimates. We point out that in the set {Re ph(z) = 0} ∩ ∂+ ¯X, the
+semiclassical symbol has a nonnegative imaginary part, so one can propagate estimates towards
+
+LINEAR STABILITY OF KERR-NEWMAN FAMILY
+69
+R(0). Finally, by using a pseudodifferential partition of unity, χ can be written as s sum of operators
+falling into the above two case), we have
+∥χu∥Hs,r,ℓ
+sc,b,h( ¯
+X) ≤ Ch−1∥Ph(z)u∥ ¯
+Hs−1,r+1,ℓ+1
+sc,b,h
+( ¯
+X) + C∥B+u∥Hs,r,ℓ
+sc,b,h( ¯
+X) + C∥B−u∥Hs,r,ℓ
+sc,b,h( ¯
+X)
++ C∥B1u∥Hs,r,ℓ
+sc,b,h( ¯
+X) + C∥B0u∥Hs,r,ℓ
+sc,b,h( ¯
+X) + C∥Bzu∥Hs,r,ℓ
+sc,b,h( ¯
+X) + ChN∥u∥ ¯
+H−N,−N,ℓ
+sc,b,h
+( ¯
+X).
+(5.87)
+By the same reasoning as above in the control of χu, it follows that
+∥E0u∥Hs,r,ℓ
+sc,b,h( ¯
+X) ≤ Ch−1∥Ph(z)u∥ ¯
+Hs−1,r+1,ℓ+1
+sc,b,h
+( ¯
+X) + C∥B+u∥Hs,r,ℓ
+sc,b,h( ¯
+X) + C∥B1u∥Hs,r,ℓ
+sc,b,h( ¯
+X)
++ C∥Bzu∥Hs,r,ℓ
+sc,b,h( ¯
+X) + ChN∥u∥ ¯
+H−N,−N,ℓ
+sc,b,h
+( ¯
+X)
+(5.88)
+and
+∥B2u∥Hs,r,ℓ
+sc,b,h( ¯
+X) ≤ Ch−1∥Ph(z)u∥ ¯
+Hs−1,r+1,ℓ+1
+sc,b,h
+( ¯
+X) + C∥B+u∥Hs,r,ℓ
+sc,b,h( ¯
+X)
++ C∥Bzu∥Hs,r,ℓ
+sc,b,h( ¯
+X) + ChN∥u∥ ¯
+H−N,−N,ℓ
+sc,b,h
+( ¯
+X).
+(5.89)
+Putting all the above estimates together yields for s > s0 > 1
+2, r > r0 > − 1
+2, ℓ < − 1
+2
+∥χu∥Hs,r,ℓ
+sc,b,h( ¯
+X) ≤ Ch−2∥Ph(z)u∥ ¯
+Hs−1,r+1,ℓ+1
+sc,b,h
+( ¯
+X) + Chs−s0∥u∥ ¯
+Hs0,r0,ℓ
+sc,b,h ( ¯
+X).
+(5.90)
+Since
+ph,z = −ρ−2
+b
+�
+∆b
+�
+ξr +
+�
+aξϕ − (r2 + a2)z
+�
+∆b
+�2 − 1
+∆b
+�
+aξϕ − (r2 + a2)z
+�2 + ˜ph,z
+�
+where
+˜ph,z = ξ2
+θ +
+1
+sin2 θ(ξϕ − a sin2 θz)2
+is semiclassical hyperbolic with respect to r in the region {r− ≤ r < rb} (see [21, definition E.55]),
+using the semiclassical hyperbolic estimates (see [21, Theorem E.57]), we have for all s, r, ℓ ∈ R
+∥(1 − χ)u∥ ¯
+Hs,r,ℓ
+sc,b,h( ¯
+X) ≤ Ch−1∥Ph(z)u∥ ¯
+Hs−1,r+1,ℓ+1
+sc,b,h
+( ¯
+X) + ∥χu∥ ¯
+Hs,r,ℓ
+sc,b,h( ¯
+X)
+(5.91)
+where the sc-decay order r and b-decay order ℓ are actually relevant here. Combining (5.90) with
+(5.91) yields for s > s0 > 1
+2, r > r0 > − 1
+2, ℓ < − 1
+2
+∥u∥ ¯
+Hs,r,ℓ
+sc,b,h( ¯
+X) ≤ Ch−2∥Ph(z)u∥ ¯
+Hs−1,r+1,ℓ+1
+sc,b,h
+( ¯
+X) + Chs−s0∥u∥ ¯
+Hs0,r0,ℓ
+sc,b,h ( ¯
+X).
+(5.92)
+Taking h small enough, the term Chs−s0∥u∥ ¯
+Hs0,r0,ℓ
+h,sc,b ( ¯
+X) can be absorbed into the left-hand side and
+this proves the estimate (5.81).
+• Proof of the estimate (5.82). For s > 1
+2 ≥ 1
+2 − Im (h−1z) r2
+b+a2
+r2
+b−m , we have 1 − s ≤ 1
+2 − Im (h−1z) r2
+b +a2
+r2
+b−m .
+Then using the semiclassical low regularity radial point estimates (see [77, Proposition 2.11], [78,
+Proposition 5.27]), there exist B′
+±, G′
+± ∈ Ψ0
+h( ¯X) microlocally supported near L± with L± ⊂ ellh(B′
+±)
+and E′
+± ∈ Ψ0
+h( ¯X), WFh(E′
+±) ∩ L± = ∅, and satisfying that all the backward (forward) null charac-
+teristics from WFh(B′
+±) \ L± reach ellh(E′
+±), while remaining in the elliptic set of G′
+±, such that for
+any N ∈ R
+∥B′
+±u∥H1−s,−r−1,−ℓ−1
+sc,b,h
+( ¯
+X)
+≤ Ch−1∥G′Ph(z)∗u∥H−s,−r,−ℓ
+sc,b,h
+( ¯
+X) + C∥E′
+±u∥H1−s,−r−1,−ℓ−1
+sc,b,h
+( ¯
+X) + ChN∥u∥ ˙H−N,−N,−N
+sc,b,h
+( ¯
+X)
+(5.93)
+where the sc-decay order r and b-decay order ℓ are actually irrelevant here.
+For r > − 1
+2, we have −r − 1 < − 1
+2.
+Then using the semiclassical low sc-decay radial point
+estimates at the non-zero scattering section R(z) (see [79] and the above discussion about the proof
+of high sc-decay radial point estimates at R(z).
+We note that the only difference is that now
+the term involving χ′
+0 does not have the correct sign and needs to be treated as an error term
+E′
+zu), there exist B′
+z, G′
+z ∈ Ψ0,0
+sc,h( ¯X) microlocally supported near R(z) with R(z) ⊂ ellsc,h(Bz) and
+
+70
+LILI HE
+E′
+z ∈ Ψ0,0
+sc,h( ¯X), WFsc,h(E′
+z)∩R(z) = ∅, and satisfying that all the backward null characteristics from
+WFsc,h(B′
+z) \ R(z) reach ellsc,h(E′
+z), while remaining in the scattering elliptic set of G′
+z, such that
+∥B′
+zu∥H1−s,−r−1,−ℓ−1
+sc,b,h
+( ¯
+X)
+≤ Ch−1∥G′
+zPh(z)∗u∥H−s,−r,−ℓ
+sc,b,h
+( ¯
+X) + C∥E′
+zu∥H1−s,−r−1,−ℓ−1
+sc,b,h
+( ¯
+X) + ChN∥u∥ ˙H−N,−N,−N
+sc,b,h
+( ¯
+X)
+(5.94)
+where the differential order s and b-decay order ℓ are actually irrelevant here.
+For ℓ < − 1
+2, we have −ℓ − 1 > − 1
+2.
+Then using the semiclassical high b-decay radial point
+estimates at the zero scattering section R(0) (see [79, §4–5]), there exist B′
+0, G′
+0 ∈ Ψ0,0,0
+sc,b,h( ¯X) mi-
+crolocally supported near R(0) (in fact near the blown up R(0) in the second microlocalized space
+introduced in §5.1.5) with R(0) ⊂ ellsc,b,h(B′
+0), and satisfying all the backward null characteristics
+from WFsc,b,h(B′
+0) tend to R(0), while remaining in the second microlocalized scattering elliptic set
+of G′
+0, such that for any N ∈ R
+∥B′
+0u∥H1−s,−r−1,−ℓ−1
+sc,b,h
+( ¯
+X) ≤ Ch−1∥G′
+0Ph(z)∗u∥H−s,−r,−ℓ
+sc,b,h
+( ¯
+X) + ChN∥u∥ ˙H−N,−N,−ℓ−1
+sc,b,h
+( ¯
+X)
+(5.95)
+where the differential order s is actually irrelevant here.
+We again calculate
+σh(h−1Im Ph(z)∗) = σh(Ph(¯z) − Ph(z)
+2ih
+) < 0 < νmin
+2
+at
+K.
+Then using the hyperbolic trapping estimates (see [20] and [40, Theorem 4.7]), there exist B′
+1, G′
+1, ∈
+Ψ0
+h( ¯X) microlocally supported near K with K ⊂ ellh(B′
+1) and B′
+2 ∈ Ψ0
+h( ¯X), WFh(B′
+2) ∩ Γ+ = ∅,
+and satisfying that WFh(B′
+1) ∪ WFh(B′
+2) is contained in the elliptic set of G′
+1, one has for any
+N, s′, r′, ℓ′ ∈ R
+∥B′
+1u∥H1−s,−r−1,−ℓ−1
+sc,b,h
+( ¯
+X)
+≤ Ch−2∥G′
+1Ph(z)∗u∥H−s,−r,−ℓ
+sc,b,h
+( ¯
+X) + C∥B′
+2u∥H1−s,−r−1,−ℓ−1
+sc,b,h
+( ¯
+X) + ChN∥u∥ ˙Hs′,r′,ℓ′
+sc,b,h ( ¯
+X)
+(5.96)
+where the differential order s′, the sc-decay order r′ and b-decay order ℓ′ are actually irrelevant here.
+By the same reasoning as in the proof of the estimate (5.81), we have
+∥χu∥H1−s,−r−1,−ℓ−1
+sc,b,h
+( ¯
+X)
+≤ Ch−1∥Ph(z)∗u∥ ˙H−s,−r,−ℓ
+sc,b,h
+( ¯
+X) + C∥B+u∥H1−s,−r−1,−ℓ−1
+sc,b,h
+( ¯
+X)
++ C∥B′
+−u∥H1−s,−r−1,−ℓ−1
+sc,b,h
+( ¯
+X) + C∥B′
+1u∥H1−s,−r−1,−ℓ−1
+sc,b,h
+( ¯
+X)
++ C∥B′
+0u∥H1−s,−r−1,−ℓ−1
+sc,b,h
+( ¯
+X) + C∥B′
+zu∥H1−s,−r−1,−ℓ−1
+sc,b,h
+( ¯
+X) + ChN∥u∥ ˙H−N,−N,−ℓ−1
+sc,b,h
+( ¯
+X),
+(5.97)
+∥E′
+±u∥H1−s,−r−1,−ℓ−1
+sc,b,h
+( ¯
+X) + ∥E′
+zu∥H1−s,−r−1,−ℓ−1
+sc,b,h
+( ¯
+X)
+≤ Ch−1∥Ph(z)∗u∥ ˙H−s,−r,−ℓ
+sc,b,h
+( ¯
+X) + C∥(1 − χ)u∥ ˙H1−s,−r−1,−ℓ−1
+sc,b,h
+( ¯
+X)
++ C∥B′
+1u∥H1−s,−r−1,−ℓ−1
+sc,b,h
+( ¯
+X) + C∥B′
+0u∥H1−s,−r−1,−ℓ−1
+sc,b,h
+( ¯
+X) + ChN∥u∥ ˙H−N,−N,−ℓ−1
+sc,b,h
+( ¯
+X)
+(5.98)
+and
+∥B′
+2u∥H1−s,−r−1,−ℓ−1
+sc,b,h
+( ¯
+X) ≤ Ch−1∥Ph(z)∗u∥ ˙H−s,−r,−ℓ
+sc,b,h
+( ¯
+X) + C∥(1 − χ)u∥ ˙H1−s,−r−1,−ℓ−1
+sc,b,h
+( ¯
+X)
++ C∥B′
+0u∥H1−s,−r−1,−ℓ−1
+sc,b,h
+( ¯
+X) + ChN∥u∥ ˙H−N,−N,−ℓ−1
+sc,b,h
+( ¯
+X).
+(5.99)
+Putting all the above estimates together yields for s > 1
+2, r > − 1
+2, ℓ < − 1
+2
+∥χu∥H1−s,−r−1,−ℓ−1
+sc,b,h
+( ¯
+X) ≤ Ch−2∥Ph(z)∗u∥ ˙H−s,−r,−ℓ
+sc,b,h
+( ¯
+X) + C∥(1 − χ)u∥ ˙H1−s,−r−1,−ℓ−1
+sc,b,h
+( ¯
+X)
++ ChN∥u∥ ˙H−N,−N,−ℓ−1
+sc,b,h
+( ¯
+X),
+(5.100)
+Again using the semiclassical hyperbolic estimates (see [21, Theorem E.57]), we have for all s, r, ℓ ∈ R
+∥(1 − χ)u∥ ˙H1−s,−r−1,−ℓ−1
+sc,b,h
+( ¯
+X) ≤ Ch−1∥Ph(z)∗u∥ ˙H−s,−r,−ℓ
+sc,b
+( ¯
+X)
+(5.101)
+
+LINEAR STABILITY OF KERR-NEWMAN FAMILY
+71
+where the sc-decay order r and b-decay order ℓ are actually relevant here. Combining (5.100) with
+(5.101) yields for s > 1
+2, r > − 1
+2, ℓ < − 1
+2
+∥u∥ ˙H1−s,−r−1,−ℓ−1
+sc,b,h
+( ¯
+X) ≤ Ch−2∥Ph(z)∗u∥ ˙H−s,−r,−ℓ
+sc,b,h
+( ¯
+X) + ChN∥u∥ ˙H−N,−N,−ℓ−1
+sc,b,h
+( ¯
+X).
+(5.102)
+Taking h small enough, the term ChN∥u∥ ˙H−N,−N,ℓ
+h,sc,b
+( ¯
+X) can be absorbed into the left-hand side and
+this proves the estimate (5.82).
+• Invertibility of �
+□gb(σ) for 0 ≤ Im σ ≤ C1, Re σ ≥ C2. We only discuss the proof of the invertibility
+of the operator �
+□gb : {u ∈ ¯Hs,ℓ
+b,h( ¯X) | ¯Hs,ℓ+1
+b,h
+( ¯
+X)} → ¯Hs,ℓ+1
+b,h
+( ¯X) in detail as the operator �
+□gb : {u ∈
+¯Hs,ℓ
+b,h( ¯X) | ¯Hs−1,ℓ+2
+b,h
+( ¯X)} → ¯Hs−1,ℓ+2
+b,h
+( ¯X) can be handled in a completely analogous manner. We
+define the space
+X(σ) := {u ∈ ¯Hs,ℓ
+b ( ¯X) | �
+□gb(σ)u ∈ ¯Hs,ℓ+1
+b
+( ¯X)},
+which is a Hilbert space endowed with the norm ∥u∥X (σ) := ∥u∥ ¯
+Hs,ℓ
+b
++ ∥�
+□gb(σ)u∥ ¯
+Hs,ℓ+1
+b
+. First, the
+injectivity of �
+□gb(σ) : X(σ) → ¯Hs,ℓ
+b
+immediately follows from (5.79). As for the surjectivity, we need
+to show that for any f ∈ ¯Hs,ℓ+1
+b
+, there exists u ∈ ¯Hs,ℓ
+b
+such that �
+□gb(σ)u = f. We define
+Y(σ) := {v ∈ ˙H−s,−ℓ−1
+b
+( ¯X) | �
+□gb(σ)∗v ∈ ˙H−s,−ℓ
+b
+( ¯X)}.
+According to (5.79), we have
+|⟨f, v⟩| ≤ C∥f∥ ¯
+Hs,ℓ+1
+b
+∥�
+□gb(σ)∗v∥ ˙H−s,−ℓ
+b
+,
+v ∈ Y(σ).
+By Hahn-Banach Theorem, the anti-linear form �
+□gb(σ)∗v → ⟨f, v⟩ with v ∈ Y(σ) can be extended
+to a continuous anti-linear form on ˙H−s,−ℓ
+b
+. Since ( ˙H−s,−ℓ
+b
+)∗ = ¯Hs,ℓ
+b , there exists u ∈ ¯Hs,ℓ
+b
+such that
+⟨u, �
+□gb(σ)∗v⟩ = ⟨f, v⟩ for v ∈ Y(σ). In particular, for v ∈ C∞
+c (X◦)
+⟨�
+□gb(σ)u, v⟩ = ⟨u, �
+□gb(σ)∗v⟩ = ⟨f, v⟩.
+This proves that �
+□gb(σ)u = f.
+□
+Now we further determine the index of the Fredholm operators �
+□gb(σ) on suitable function spaces for
+σ ∈ C, Im σ ≥ 0.
+Lemma 5.23. Let s > 1
+2, ℓ < − 1
+2, s + ℓ > − 1
+2. Then the following holds.
+(1) For σ ∈ C, Im σ ≥ 0, σ ̸= 0, the operators
+�
+□gb(σ) : {u ∈ ¯Hs,ℓ
+b ( ¯X) | �
+□gb(σ)u ∈ ¯Hs−1,ℓ+2
+b
+( ¯X)} → ¯Hs−1,ℓ+2
+b
+( ¯X)
+(5.103)
+�
+□gb(σ) : {u ∈ ¯Hs,ℓ
+b ( ¯X) | �
+□gb(σ)u ∈ ¯Hs,ℓ+1
+b
+( ¯X)} → ¯Hs,ℓ+1
+b
+( ¯X)
+(5.104)
+are Fredholm of index 0.
+(2) Let − 3
+2 < ℓ < − 1
+2. Then
+�
+□gb(0) : {u ∈ ¯Hs,ℓ
+b ( ¯X) | �
+□gb(0)u ∈ ¯Hs−1,ℓ+2
+b
+( ¯X)} → ¯Hs−1,ℓ+2
+b
+( ¯X)
+(5.105)
+is Fredholm of index 0.
+Proof. We now prove that (5.103) and (5.105) have index 0, and the index 0 property of (5.104) follows in
+a similar manner.
+• Proof of the index 0 property of (5.103). In view of the high energy estimates in Theorem 5.22,
+�
+□gb(σ) : {u ∈ ¯Hs,ℓ
+b ( ¯X) | �
+□gb(σ)u ∈ ¯Hs−1,ℓ+2
+b
+( ¯X)} → ¯Hs−1,ℓ+2
+b
+( ¯X)
+is invertible and thus has index 0 for fixed Im σ = C ≥ 0 when |Re σ| ≫ 1. We claim that the index
+of �
+□gb(σ) with Im σ ≥ 0, σ ̸= 0 is a constant on the horizontal line Im σ = C and thus is 0 for all σ
+with Im σ ≥ 0, σ ̸= 0. Let
+X(σ) := {u ∈ ¯Hs,ℓ
+b ( ¯X) : �
+□gb(σ)u ∈ ¯Hs−1,ℓ+2
+b
+( ¯X)}.
+
+72
+LILI HE
+For any fixed σ0 ̸= 0 with Im σ0 = C ≥ 0, without loss of generality, we may assume Re σ0 ≥ 0.
+First, there exists a C′ such that �
+□gb(σ) is invertible and thus has index 0 for σ ∈ [C′ + iC, ∞ + iC).
+We next prove that for each σ′ ∈ [σ0 = Re σ0 + iC, C′ + iC], there exists ǫ > 0 such that the index
+of �
+□gb(σ) : X(σ) → ¯Hs−1,ℓ+2
+b
+( ¯X) is a constant for σ with Im σ ≥ 0, σ ̸= 0, |σ − σ′| < ǫ. Then by
+compactness of [σ0 = Re σ0 + iC, C′ + iC] we conclude that the index of �
+□gb(σ0) is the same as that
+of �
+□gb(C′ + iC) and thus is 0. Suppose �
+□gb(σ′) has kernel and cokernel of dimension k+ and k−,
+respectively, then we establish a Grushin problem [21, Appendix C.1]. More specifically, we define
+the following operator
+L(σ) =
+��
+□gb(σ)
+L−
+L+
+0
+�
+: X(σ) ⊕ Ck− → ¯Hs−1,ℓ+2
+b
+( ¯X) ⊕ Ck+
+where L+, L− are defined as follows. If ker �
+□gb(σ′) is spanned by xj ∈ X(σ′), j = 1, · · · , k+, then
+by Hahn-Banach Theorem we can find x∗
+j ∈ ˙H−s,−ℓ
+b
+( ¯X) with ∥x∗
+j∥ ≤ 1 such that ⟨xi, x∗
+j⟩ = δij. It
+follows that
+L+ : ¯Hs,ℓ
+b ( ¯X) → Ck+,
+L+(f) := (⟨f, x∗
+1⟩, · · · , ⟨f, x∗
+k+⟩)
+restricts to an isomorphism ker �
+□gb(σ′) → Ck+. As for the construction of L−, we choose yj ∈
+¯Hs−1,ℓ+2
+b
+( ¯X) with ∥yj∥ ≤ 1, j = 1, · · · , k− so that yj+ran�
+□gb(σ′) form a basis of ¯Hs−1,ℓ+2
+b
+/ran�
+□gb(σ′).
+Then we define
+L− : Ck− → ¯Hs−1,ℓ+2
+b
+( ¯X),
+L−((a1, · · · , ak−)) :=
+k−
+�
+j=1
+ajyj.
+The operator L− has maximal rank and ranL− ∩ ran�
+□gb(σ′) = ∅. The above construction of L+, L−
+implies L(σ′) has a trivial kernel and onto, and thus is invertible. Since �
+□gb(σ) satisfy the uniform
+Fredholm estimates (5.17) for all σ ∈ C, Im σ ∈ [0, 2C] satisfying |σ0|/2 ≤ |σ| ≤ 2(C + C′), so do
+L(σ). That is, there exists a constant ˜C (independent of b and σ) such that
+∥(u, u−)∥ ¯
+Hs,ℓ
+b
+( ¯
+X)⊕Ck− ≤ ˜C
+�
+∥L(σ)(u, u−)∥ ¯
+Hs−1,ℓ+2
+b
+( ¯
+X)⊕Ck+ + ∥(u, u−)∥ ¯
+Hs0,ℓ0
+b
+( ¯
+X)⊕Ck−
+�
+for σ ∈ C, Im σ ∈ [0, 2C] satisfying |σ0|/2 ≤ |σ| ≤ 2(C + C′), and s0 < s, ℓ0 < ℓ. Then following
+the perturbation arguments in [77, §2.7], we prove the invertibility of L(σ) when σ is close to σ′.
+Concretely, suppose there exists a sequence δj → δ′ such that L(δj) is not invertible, so either
+ker L(δj) on ¯Hs,ℓ
+b ( ¯X) ⊕ Ck− or ker L(δj)∗ on ˙H−s+1,−ℓ−2
+b
+( ¯X) ⊕ Ck+ is non-trivial. By passing to a
+subsequence, we may assume that the former possibility holds for all j, as the case of the adjoint
+is completely analogous. Now, if ∥(uj, u−
+j )∥ ¯
+Hs,ℓ
+b
+( ¯
+X)⊕Ck− = 1 and L(δj)(uj, u−
+j ) = 0, then the above
+uniform Fredholm estimate gives ˜C−1 ≤ ∥(uj, u−
+j )∥ ¯
+Hs0,ℓ0
+b
+( ¯
+X)⊕Ck− . Using the sequential compactness
+of the unit ball in ¯Hs,ℓ
+b ( ¯X) ⊕ Ck− in the weak topology, and the compactness of the inclusion
+¯Hs,ℓ
+b ( ¯
+X) ⊕ Ck− → ¯Hs0,ℓ0
+b
+( ¯X) ⊕ Ck−, s0 < s, ℓ0 < ℓ, it follows that (uj, u−
+j ) has a weakly convergent
+subsequence in ¯Hs,ℓ
+b ( ¯X) ⊕ Ck− to some (u0, u+
+0 ) ∈ ¯Hs,ℓ
+b ( ¯X) ⊕ Ck− which is norm-convergent in
+¯Hs0,ℓ0
+b
+( ¯X) ⊕ Ck−. We also have
+L(δ′)(u0, u+
+0 ) = L(δ′)
+�
+(u0, u+
+0 ) − (uj, u−
+j )
+�
++ (L(δ′) − L(δj))(uj, u−
+j ) → 0
+in
+¯Hs0−2,ℓ0
+b
+( ¯X) ⊕ Ck−
+since L(δj) → L(δ′) as bounded operators in L( ¯Hs0,ℓ0
+b
+( ¯X)⊕Ck−, ¯Hs0−2,ℓ0
+b
+( ¯X)⊕Ck−) and (uj, u−
+j ) →
+(u0, u−
+0 ) in ¯Hs0,ℓ0
+b
+( ¯X) ⊕ Ck−. Therefore, L(δ′)(u0, u−
+0 ) = 0. Since ˜C−1 ≤ ∥(u, u−)∥ ¯
+Hs0,ℓ0
+b
+( ¯
+X)⊕Ck− ,
+(u0, u−
+0 ) ̸= 0 and thus ker L(σ′) is non-trivial, which leads to a contradiction. This proves that L(σ)
+is invertible for σ close to σ′, which implies �
+□gb(σ) has index k+ − k− (see [21, Theorem C.4]) for
+for σ close to σ′.
+• Proof of 0 index property of (5.105). Suppose that − 3
+2 < ℓ < − 1
+2. Then using the uniform Fredholm
+estimates (5.21) (up to σ = 0) and the argument as above yields the index 0 property of (5.105).
+□
+
+LINEAR STABILITY OF KERR-NEWMAN FAMILY
+73
+5.5.2. High energy estimates for tensor wave operators. Now we prove an analogy to Theorem 5.22 for the
+semicalssical operators h2 �
+Pb,γ(h−1z), h2 �
+Wb,γ(h−1z) and h2 �
+Lb,γ(h−1z) acting on bundles.
+Theorem 5.24. Let b0 = (m0, a0, Q0) with |Q0| + |a0| < m0 and |a0| ≪ |m0| + |Q0|. Then there exists
+ǫ > 0 such that for b = (m, a, Q) with |b − b0| < ǫ and for s > s0 > 2 (resp. s > s0 > 3), ℓ < − 1
+2 with
+s+ℓ > − 1
+2, the conclusions in Theorem 5.22 hold for h2 �
+Pb,γ(h−1z) and h2 �
+Wb,γ(h−1z) (resp. h2 �
+Lb,γ(h−1z)).
+Proof. The proof is analogous to that of Theorem 5.22, except for the computation of threshold regularity
+in the radial point estimate at event horizon and the threshold decay rate in the radial point estimate at
+spatial infinity ∂+ ¯X, and the verification of
+σ1,h
+� 1
+2ih
+�
+h2�•(h−1z) − (h2�•(h−1z))∗��
+< νmin
+2
+,
+• = Pb,γ, Wb,γ, Lb,γ
+(5.106)
+at trapping K.
+The computation of threshold regularity in the radial point estimate at event horizon and the threshold
+decay rate in the radial point estimate at spatial infinity ∂+ ¯X is the same as that in the fixed σ case in
+Theorem 5.6. The verification of (5.106) at trapping K follows from the discussion in appendix B (see
+Theorems B.6, B.7 and Remark B.8) and the relevant calculation in the proof of Theorem 5.22.
+□
+Correspondingly, we have the following analogy to Lemma 5.23 for �
+Pb,γ, �
+Wb,γ and �
+Lb,γ.
+Lemma 5.25. Let b = (m, a, Q) with |Q| + |a| < m and |a| ≪ |m| + |Q|. Let s > 2(resp. s > 3), ℓ < − 1
+2
+and s + ℓ > − 1
+2. Then the conclusions in Lemma 5.23 also hold for �
+Pb,γ(σ) and �
+Wb,γ(σ) (resp. �
+Lb,γ(σ)).
+Proof. The proof is completely analogous to that of Lemma 5.23.
+□
+5.6. Energy estimates. In this section, we shall establish the energy estimates for the solutions to various
+wave type equations on slowly rotating Kerr-Newman metrics.
+First, we introduce another modification of the Boyer-Lindquist coordinates, which is defined by
+¯t = t + Ft(r),
+¯ϕ = ϕ + Fϕ(r).
+In the new coordinates (¯t, r, θ, ¯ϕ), the Kerr-Newman metric takes the following form
+gb = −∆b
+ρ2
+b
+�
+d¯t − a sin2 θd ¯ϕ +
+�
+a sin2 θF ′
+ϕ(r) − F ′
+t(r)
+�
+dr
+�2
++ sin2 θ
+ρ2
+b
+�
+ad¯t − (r2 + a2)d ¯ϕ +
+�
+(r2 + a2)F ′
+ϕ(r) − aF ′
+t(r)
+�
+dr
+�2
++ ρ2
+b(dr2
+∆b
++ dθ2)
+(5.107)
+and its inverse is
+g−1
+b
+= −
+1
+ρ2
+b∆b
+�
+(r2 + a2)∂¯t + a∂ ¯ϕ
+�2
++
+1
+ρ2
+b sin2 θ
+�
+∂ ¯ϕ + a sin2 θ∂¯t
+�2
++ 1
+ρ2
+b
+∂2
+θ + ∆b
+ρ2
+b
+�
+∂r + F ′
+t(r)∂¯t + F ′
+ϕ(r)∂ ¯ϕ
+�2
+.
+(5.108)
+In order for gb, g−1
+b
+to be smooth at event horizon, Ft(r) and Fϕ(r) are required to satisfy that F ′
+t(r) =
+±(r2 + a2)/∆b, F ′
+ϕ(r) = ±a/∆b (up to a smooth modification) at event horizon. Now we shall prove
+Lemma 5.26. There exist Ft(r), Fϕ(r) such that they satisfy the following conditions
+(1) F ′
+ϕ(r) =
+a
+∆b + ˜Fϕ(r) where ˜Fϕ(r) is smooth on [r−, ∞)r, and Fϕ(r) = 0 near r = ∞.
+(2) F ′
+t(r) = r2+a2
+∆b
++ ˜Ft(r) where ˜Ft(r) is smooth on [r−, ∞), and Ft(r) = −rb,∗ + O(r−1) near r = ∞
+where drb,∗
+dr
+=
+1
+∆b .
+(3) The hypersurfaces ¯t = const are spacelike, i.e., d¯t is timelike. More precisely,
+g−1
+b (d¯t, d¯t) ≤ −m2ρ−2
+b
+< 0.
+With this choice of Ft(r), Fϕ(r), gb in the coordinates (¯t, r, θ, ¯ϕ) is smooth and non degenerate on the extended
+manifold R¯t × [r−, ∞)r × S2.
+
+74
+LILI HE
+Proof. Let χ(r) ∈ C∞
+c (R) such that χ(r) = 1 when r ≤ 3m and χ = 0 when r ≥ 4m. We first define
+Fϕ(r) = χ
+�
+a/∆b dr which satisfies statement (1). Next, we expect
+F ′
+t(r) = r2 + a2
+∆b
++ µ1(r)
+near
+r = rb;
+F ′
+t(r) = −r2 + a2
+∆b
++ µ2(r)
+near
+r = ∞
+(5.109)
+where µ1(r) is smooth near event horizon and µ2(r) ∼ r−2 near r = ∞. We let
+g−1
+b (d¯t, d¯t) = − 1
+ρ2
+b
+�(r2 + a2)2
+∆b
+− ∆bF ′
+t(r)2 − a2 sin2 θ
+�
+= −m2
+ρ2
+b
+< 0,
+In order to arrange (5.109), we need
+F ′
+t(r) =
+
+
+
+
+
+
+
+��
+r2+a2
+∆b
+�2
+− m2+a2 sin2 θ
+∆b
+= r2+a2
+∆b
+�
+1 − ∆b(m2+a2 sin2 θ)
+(r2+a2)2
+,
+r is near rb
+−
+��
+r2+a2
+∆b
+�2
+− 1+a2 sin2 θ
+∆b
+= − r2+a2
+∆b
+�
+1 − ∆b(m2+a2 sin2 θ)
+(r2+a2)2
+,
+r is near ∞
+.
+Therefore, we define
+F ′
+t(r) = χ(r)
+�r2 + a2
+∆b
+�
+1 − ∆b(m2 + a2 sin2 θ)
+(r2 + a2)2
+�
++ (1 − χ(r))
+�
+− r2 + a2
+∆b
+�
+1 − ∆b(m2 + a2 sin2 θ)
+(r2 + a2)2
+�
+.
+With this choice of F ′
+t(r), we have g−1
+b (d¯t, d¯t) = −m2ρ−2
+b
+for r ≤ 3m, r ≥ 4m and g−1
+b (d¯t, d¯t) ≤ −m2ρ−2
+b
+for 3m ≤ r ≤ 4m, and thus the requirement statement (3) is satisfied. By choosing a proper integration
+constant we have ¯t = rb,∗ + O(r−1) near r = ∞, and thus the requirement statement (2) is satisfied
+□
+We now establish the energy estimates for the solutions to the equation □gbu = f on the extended Kerr-
+Newman manifold M = R¯t × [r−, ∞)r × S2. We denote by Σ¯t0 the spacelike hypersurfaces {¯t = ¯t0} ∩ M.
+Here, the integral on M is taken with respect to the volume form dV = dvolgb = ρ2
+b sin θd¯tdrdθdϕ.
+We begin with the energy-momentum tensor
+Tαβ[g] = ∂αu∂βu − 1
+2gαβ∂γu∂γu
+Its contraction with respect to a vector field X is denoted by
+Pα[g, X] = Tαβ[g]Xβ
+and its divergence is
+∇αPα[g, X] = □gu · Xu + 1
+2Tαβ[g]παβ
+X
+and παβ
+X is the deformation tensor of X, which is given in terms of the Lie derivative by
+πX
+αβ = ∇αXβ + ∇βXα = (LXg)αβ.
+Since Σ¯t is “asymptotically null”, we shall define the (degenerate) energy at Σ¯t0 as
+E[u](¯t0) :=
+� ∞
+r−
+�
+S2
+1
+r2 |∂¯tu(¯t0, ·)|2 + |∂ru(¯t0, ·)|2 + 1
+r2
+�
+|∂θu(¯t0, ·)|2 +
+1
+sin2 θ|∂ϕu(¯t0, ·)|2�
+r2 sin θ dr dθ dϕ.
+Now we establish the energy estimate.
+Proposition 5.27. Let b = (m, a, Q) with |a| ≪ |Q|+m. Let u be the solution to the following initial value
+problem
+□gbu = f,
+u|¯t=0 = u0,
+∂¯tu|¯t=0 = u1
+(5.110)
+where rf ∈ L1([0, ∞)¯t; L2(Σ¯t)) and u0
+r , u1
+r , ∂u0 ∈ L2(Σ0; r2 sin θ dr dθ dϕ). Then there exists a constant C(b)
+which depends on the parameter b = (m, a, Q) such that
+E[u](¯t) ≤ C(b)
+�
+∥u0
+r ∥L2(Σ0) + ∥∂u0∥L2(Σ0) + ∥u1
+r ∥L2(Σ0) + ∥rf∥2
+L1([0,¯t);L2(Σ¯t))
+�
+eC(b)¯t.
+(5.111)
+where the integral is taken with respect the volume form r2 dr dθ dϕ and ∂ ∈ {∂r, 1
+r ∂θ,
+1
+r sin θ∂ϕ}.
+Moreover, the above statements also hold for the wave type operators Pb,γ, Wb,γ and Lb,γ.
+
+LINEAR STABILITY OF KERR-NEWMAN FAMILY
+75
+Proof. We only do the analysis for the case a = 0 here and the same argument will also work for small a
+since all the calculation below near event horizon depends continuously on b = (m, a, Q) (away from the
+event the horizon, we make use of the timelike property of d¯t and ∂¯t). Therefore, we let b = (m, 0, Q) and
+then we have
+g = gb = −∆b
+r2 d¯t2 + 2∆b
+r2 F ′
+t(r)d¯tdr +
+� r2
+∆b
+− ∆b
+r2 F ′
+t(r)2�
+dr2 + r2dθ2 + r2 sin2 θdϕ2,
+g−1 = g−1
+b
+= −
+� r2
+∆b
+− ∆b
+r2 F ′
+t(r)2�
+∂2¯t + 2∆b
+r2 F ′
+t(r)∂¯t∂r + ∆b
+r2 ∂2
+r + 1
+r2 ∂2
+θ +
+1
+r2 sin2 θ∂2
+ϕ.
+By a density statement, we may assume that u is supported away from spatial infinity.
+• Killing multiplier: X = ∂¯t. We compute
+g−1(d¯t, P[g, ∂¯t]) =
+�∆b
+r2 F ′
+t(r)2 − r2
+∆b
+�
+T (∂¯t, ∂¯t) + ∆b
+r2 F ′
+t(r)T (∂r, ∂¯t)
+= −1
+2
+�� r2
+∆b
+− ∆b
+r2 F ′
+t(r)2�
+|∂¯tu|2 + ∆b
+r2 |∂ru|2 + |∂θu|2 +
+1
+sin2 θ|∂ϕ|2
+�
+∼ −
+� 1
+r2 |∂¯tu|2 + |∂ru|2 + 1
+r2 |∂θu|2 +
+1
+r2 sin2 θ|∂ϕu|2�
+on
+[r−, ∞),
+g−1(dr, P[g, ∂¯t]) = ∆b
+r2 F ′
+t(r)T (∂¯t, ∂¯t) + ∆b
+r2 T (∂r, ∂¯t)
+= ∆b(r−)
+r2
+−
+F ′
+t(r−)|∂¯tu|2 + ∆b(r−)
+r2
+−
+∂¯tu∂ru
+at
+r = r−.
+(5.112)
+We see that g−1(d¯t, P[g, ∂¯t]) gives control of | 1
+r∂¯tu| and angular derivative but not ∂ru near event
+horizon because ∆b(r) is nonpositive at and beyond event horizon. We also note that the first term
+of g−1(dr, P[g, ∂¯t]) is positive if r− is sufficiently close to event horizon r = rb while the second term
+does not have a definite sign. So we need to introduce a suitable multiplier which allows us to deal
+with the above issues at event horizon.
+• Energy estimates near event horizon. Following [62], We introduce X = b(r)∂r which is expressed in
+terms of the coordinates (¯t, r, θ, ϕ) with b(r) = −χ(r) where χ(r) = 1 when r ≤ 3m and χ = 0 when
+r ≥ 4m. First we compute the boundary terms at the spacelike hypersurfaces ¯t = const
+g−1(d¯t, P[g, X]) =
+�∆b
+r2 F ′
+t(r)2 − r2
+∆b
+�
+T (∂¯t, b(r)∂r) + ∆b
+r2 F ′
+t(r)T (∂r, b(r)∂r)
+= −χ(r)
+�
+−
+�∆b
+r2 F ′
+t(r)2 − r2
+∆b
+�
+∂¯tu∂ru + ∆b
+r2 F ′
+t(r)|∂ru|2
+�
+.
+Since ∆b
+r2 F ′
+t(r) ∼ 1 near event horizon r = rb, it follows that on r ∈ [r−, ∞)
+g−1(d¯t, P[g, C∂¯t + X]) ∼ −E[u](¯t)
+(5.113)
+provided C is large enough. At the hypersurface r = r−, we have
+g−1(dr, P[g, X]) = ∆b
+r2 F ′
+t(r)T (∂¯t, b(r)∂r) + ∆b
+r2 T (∂r, b(r)∂r)
+= −1
+2χ(r)
+�� r2
+∆b
+− ∆b
+r2 F ′
+t(r)2�
+|∂¯tu|2 + ∆b
+r2 |∂ru|2 − 1
+r2 |∂2
+θu|2 −
+1
+r2 sin2 θ |∂ϕu|2
+�
+.
+Therefore, combining the calculation of g−1(dr, P[g, X]) with (5.112) yields
+g−1(dr, C∂¯t + X)
+=
+�
+C ∆b(r−)
+r2
+−
+F ′
+t(r−) + 1
+2
+� r2
+∆b
+− ∆b
+r2 F ′
+t(r)2�
+− ǫ
+�
+|∂¯tu|2 + ǫ(∂¯tu + C∆b(r−)
+2ǫr2
+−
+∂ru
+�2
++
+�
+− ∆b(r−)
+2r2
+−
+− C2∆2
+b(r−)
+4ǫr4
+−
+�
+|∂ru|2 +
+1
+2r2
+−
+|∂2
+θu|2 +
+1
+2r2
+− sin2 θ|∂ϕu|2 ≥ 0
+at
+r = r−.
+(5.114)
+provided C is sufficiently large, ǫ is small enough and r− is sufficiently close to rb.
+
+76
+LILI HE
+Then we integrate ∇αPα[g, C∂¯t + X] = □gu · (∂¯tu + u) + 1
+2Tαβ[g]παβ
+X over the region
+D = {0 ≤ ¯t ≤ ¯t1, r ≥ r−}
+with respect to the volume form dVgb = r2 sin θd¯tdrdθdϕ and use Divergence Theorem
+�
+D
+∇αPα dV =
+�
+∂D
+ιP dV
+to obtain�
+D
+□gu · (C∂¯tu + Xu) r2 sin θd¯tdrdθdϕ + 1
+2
+�
+D
+Tαβ[g]παβ
+X r2 sin θd¯tdrdθdϕ
+=
+� ∞
+r−
+�
+S2 g−1(d¯t, P[g, C∂¯t + X]) r2 sin θdrdθdϕ|
+¯t=¯t1
+¯t=0
+−
+� ¯t1
+0
+�
+S2 g−1(dr, P[g, C∂¯t + X])r2 sin θd¯tdθdϕ|r=r−.
+Since g−1(dr, P[g, C∂¯t + X])|r=r− ≥ 0, and Tαβ[g]παβ
+X is quadratic in ∂u, we find that
+sup
+0≤¯t≤¯t1
+E[u](¯t) ≤ C′�
+∥∂u0∥L2(Σ0) + ∥u1
+r ∥L2(Σ0) + ∥rf∥2
+L1([0,¯t);L2(Σ¯t))
+�
++ C′
+� ¯t1
+0
+E[u](¯t) d¯t.
+where ∂ ∈ {∂r, 1
+r∂θ,
+1
+r sin θ∂ϕ} and L2(Σ0) = L2(Σ0; r2 sin θ dr dθ dϕ).
+As for the L2 norm of u, we have
+∥u
+r ∥L2(Σ¯t) ≲ ∥u0
+r ∥L2(Σ0) +
+� ¯t1
+0
+∥∂¯tu
+r ∥L2(Σ¯t) d¯t
+where L2(Σ¯t) = L2(Σ¯t; r2 sin θ dr dθ dϕ). Therefore, we have
+sup
+0≤¯t≤¯t1
+E[u](¯t) ≤ C
+�
+∥u0
+r ∥L2(Σ0) + ∥∂u0∥L2(Σ0) + ∥u1
+r ∥L2(Σ0) + ∥rf∥2
+L1([0,¯t);L2(Σ¯t))
+�
++ C
+� ¯t1
+0
+E[u](¯t) d¯t.
+Finally by Gr¨onwall inequality we obtain the exponential growth of the energy
+E[u](¯t) ≤ C
+�
+∥u0
+r ∥H1(Σ0) + ∥∂u0∥L2(Σ0) + ∥u1
+r ∥L2(Σ0) + ∥rf∥2
+L1([0,¯t);L2(Σ¯t))
+�
+eC¯t.
+• Proof for Pb,γ, Wb,γ and Lb,γ. According to Lemma 4.12, −2Pb,γ, −2Wb,γ (resp.
+−Lb,γ) are □gb
+tensored with 4 × 4 (resp. 14 × 14) identity matrix plus a a matrix whose entries are first order
+differential operators with coefficients decaying like r−2, and then the above arguments in the proof
+for □gb still apply here.
+□
+We next show that if Im σ is sufficiently large, then �
+□gb(σ), �
+Pb,γ(σ), �
+Wb,γ(σ), �
+Lb,γ(σ) are invertible on
+suitable function spaces.
+Theorem 5.28. Let b0 = (m0, 0, Q0) with |Q0| < m0. Then there exists ǫ > 0 such that for b = (m, Q, a)
+with |b − b0| < ǫ and for s > 1
+2, ℓ < − 1
+2 with s + ℓ > − 1
+2, the operators
+�
+□gb(σ) : {u ∈ ¯Hs,ℓ
+b ( ¯X) | �
+□gb(σ)u ∈ ¯Hs,ℓ+1
+b
+( ¯X)} → ¯Hs,ℓ+1
+b
+( ¯X)
+(5.115)
+�
+□gb(σ) : {u ∈ ¯Hs,ℓ
+b ( ¯X) | �
+□gb(σ)u ∈ ¯Hs−1,ℓ+2
+b
+( ¯X)} → ¯Hs−1,ℓ+2
+b
+( ¯X)
+(5.116)
+are invertible for Im σ ≥ C′(b0) where C′(b0) > 0 is a sufficiently large constant only depending on b0.
+Moreover, the above statements also hold for the wave type operators Pb,γ, Wb,γ and Lb,γ.
+
+LINEAR STABILITY OF KERR-NEWMAN FAMILY
+77
+Proof. According to Proposition 5.27, there exist ǫ > 0, ˜C(b0) > 0 such that for |b − b0| < ǫ, the solution u
+to the initial value problem (5.110) satisfies
+E[u](¯t) ≤
+�
+˜C(b0)E[u](0) + ˜C(b0)
+� � ¯t
+0
+∥rf∥L2(Σ¯t) d¯t
+�2�
+e
+˜
+C(b0)¯t.
+We now show that if Im σ > ˜C(b0), then (5.115) and (5.116) are invertible. According to Lemma 5.23,
+it suffices to prove that (5.115) and (5.116) are injective. Assume the contrary, by Proposition 5.7, there
+exists a nontrivial solution w = ρC∞(∂+ ¯X) + A2−( ¯
+X) to �
+□gb(σ)w = 0. Then, u(tb,∗, r, θ, ϕ) = e−itb,∗σw
+is a solution to □gbu = 0. Let χ(¯t) ∈ C∞(R) be such that suppχ ⊂ [1, ∞) and supp(1 − χ) ⊂ (−∞, 2].
+Then χu is the unique solution which is supported in (1, ∞) to the inhomogeneous equation □˜u = [□gb, χ]u.
+According to Lemma 4.11 and ¯t − tb,∗ = O( 1
+r), we have
+□˜u = [□gb, χ]u ∼
+1
+r3−
+and thus χ(¯t)u satisfies E(χu) ≲ e ˜
+C(b0)¯t. But a direct calculation implies E(χu) ∼ eIm σtb,∗ ∼ eIm σ¯t, which
+leads to a contradiction.
+The proof for �
+Pb,γ(σ), �
+Wb,γ(σ), �
+Lb,γ(σ) is completely analogous.
+□
+6. Spherical harmonic decompositions
+In this section, we shall introduce the spherical harmonic decompositions, which will be used in the proof
+of the geometric mode stability of Reissner-Nordstr¨om spacetime (see §7).
+Let /g be the standard metric on the unit sphere S2. We denote the operators on S2 using a slash, e.g.
+/tr = tr/g, /δ = δ/g, /δ
+∗ = δ∗
+/g, etc. We denote by Y m
+l
+with l ∈ N, m ∈ Z, |m| ≤ l the spherical harmonic function
+of degree l and order m which satisfies /∆HY m
+l
+= l(l+1)Y m
+l
+where /∆H = /d/δ+/δ/d is the Hodge Laplacian. We
+denote by Sl = {Y m
+l
+: |m| ≤ l} the space of degree l spherical harmonic functions. Then one can conclude
+that {Sl : l ∈ N} form an orthogonal basis of L2(S2).
+According to Hodge decomposition Theorem and the fact that the de Rham cohomology group H1
+dR(S2) =
+0, we see that any 1-form ω ∈ C∞(S2, T ∗S2) can be uniquely decomposed as ω = /dφ + /δη where φ ∈
+C∞(S2), η ∈ C∞(S2; Λ2T ∗S2). Since /δ = −/∗/d/∗ where /∗ is the Hodge star operator, we can rewrite ω =
+/dφ + /∗/dψ where φ, ψ ∈ C∞(S2). Therefore, one can conclude that an orthogonal basis of L2(S2, T ∗S2) is
+given by {/dSl, Vl = /∗/dSl : l ∈ N} where Sl, Vl are the eigen-1-forms of the Hodge Laplacian ∆H with
+eigenvalue l(l + 1). We note that /δVl = 0 and /dS0 = V0 = 0. In other words, a spectral decomposition of
+the negative tensor Laplacian − /∆ = − /tr /∇a /∇b satisfying − /∆ = /∆H − 1 is given by the scalar part
+/dSl ∈ ker(− /∆ − (l(l + 1) − 1))
+with
+l ≥ 1
+(6.1)
+and the vector part
+Vl ∈ ker(− /∆ − (l(l + 1) − 1))
+with
+l ≥ 1.
+(6.2)
+As for the symmetric 2-tensors h ∈ C∞(S2; S2T ∗S2), we have the transverse-traceless decomposition [87]
+h = 1
+2(/trh)/g + /δ
+∗
+0ω + hT T where /δ
+∗
+0 = /δ
+∗ + 1
+2/g/δ is the traceless symmetric gradient, ω ∈ C∞(S2; T ∗S2) and
+hT T is both traceless and divergence free. According to [37], we know that hT T = 0. We then decompose
+ω further into eigen-1-forms (we also call them spherical harmonic 1-forms) of − /∆ of scalar type /dSl and
+vector type Vl, and /trh into scalar spherical harmonic functions Sl. In the spherical coordinates (θ, ϕ), since
+/dY 0
+1 = − sin θdθ,
+/dY 1
+1 = cos θ cos ϕdθ − sin θ sin ϕdϕ,
+/dY −1
+1
+= cos θ sin ϕdθ + sin θ cos ϕdϕ,
+we see that /dS1 is conformal killing and thus /δ
+∗
+0/dS1 = 0. We also have
+/∗/dY 0
+1 = − sin2 θdϕ,
+/∗/dY 1
+1 = sin θ cos θ cos ϕdϕ + sin ϕdθ,
+/∗/dY −1
+1
+= sin θ cos θ sin ϕdϕ − cos ϕdθ.
+(6.3)
+and find that V♯
+1 are rotations, and thus killing, i.e. /δ
+∗V1 = 0. Then any symmetric 2-tensors on sphere
+can be decomposed into the following spherical harmonic symmetric 2-tensors of the scalar type
+Sl/g (l ≥ 0),
+/δ
+∗
+0/dSl (l ≥ 2)
+(6.4)
+and the vector type
+/δ
+∗
+0Vl = /δ
+∗Vl (l ≥ 2).
+(6.5)
+
+78
+LILI HE
+We claim that the following geometric operators on sphere S2
+/g, /tr, /d, /δ, /δ
+∗, − /∆ and their compositions
+preserve the the scalar and vector type of the spherical harmonics. More precisely, these geometric operators
+map the scalar type functions/1-forms/symmetric 2-tensors built out of a particular S ∈ Sl into another scalar
+type tensor with the same S, likewise for the vector type 1-forms/symmetric 2-tensors. This is obviously
+true for /g acting on functions, /tr on symmetric 2-tensors, /d on functions, /δ on 1-forms, and − /∆ on functions
+and 1-forms. For the verification of /δ, /δ
+∗, − /∆, we make use of the identities /δ/δ
+∗ω = − 1
+2 /∆ω + 1
+2/d/δω − 1
+2ω and
+− /∆/δ
+∗ω = /δ
+∗(− /∆ω) − 3/δ
+∗ω − 2/g/δω to obtain
+/δ(S/g) = −/dS, /δ/δ
+∗
+0/dS = l(l + 1) − 2
+2
+/dS, /δ/δ
+∗/∗/dS = l(l + 1) − 2
+2
+/∗/dS; /δ
+∗/dS = /δ
+∗
+0/dS−l(l + 1)
+2
+S/g;
+− /∆(S/g) = l(l + 1)S/g,
+− /∆(/δ
+∗
+0/dS) = (l(l + 1) − 4) (/δ
+∗
+0/dS),
+− /∆(δ∗/∗/dS) = (l(l + 1) − 4) (δ∗/∗/dS).
+(6.6)
+We now decompose the spacetime M into the aspherical ˚
+X = Rt∗ × [r−, ∞)r and spherical part S2, i.e.
+M = ˚
+X × S2,
+˚π : M → ˚
+X,
+/π : M → S2.
+(6.7)
+We then call the functions aspherical if they are only functions of (t∗, r), and spherical if they are only
+functions on S2. Alternatively, we can identify the aspherical and spherical functions as subspaces of C∞(M)
+as follows:
+C∞( ˚
+X) ∼= ˚π∗C∞( ˚
+X) ⊂ C∞(M),
+C∞(S2) ∼= /π∗C∞(S2) ⊂ C∞(M).
+(6.8)
+Then functions on M, say L2(M), can be decomposed into an infinite sum of products of aspherical functions
+and spherical harmonic functions.
+We can also split the cotangent bundle T ∗M into aspherical and spherical part as follows:
+T ∗M = T ∗
+AS ⊕ T ∗
+S
+where
+T ∗
+AS = ˚π∗T ∗ ˚
+X
+and
+T ∗
+S = /π∗T ∗S2.
+(6.9)
+We can further write ω ∈ T ∗
+S as an infinite sum of the spherical harmonic 1-forms /dSl, Vl of − /∆. Therefore,
+a 1-form ω ∈ C∞(M; T ∗M) can be expressed as an infinite sum of products of aspherical functions/1-forms
+on ˚
+X and spherical harmonic functions/1-forms.
+The above splitting of T ∗M induces a natural splitting of the symmetric 2-tensors,into aspherical part
+S2T ∗
+AS, mixed part T ∗
+AS ⊗ T ∗
+S and spherical part S2T ∗
+S as follows:
+S2T ∗M = S2T ∗
+AS ⊕ (T ∗
+AS ⊗ T ∗
+S) ⊕ S2T ∗
+S
+(6.10)
+where we identify a ⊗ b ∈ T ∗
+AS ⊗ T ∗
+S as 2a ⊗s b = a ⊗ b + b ⊗ a. Therefore, a symmetric 2-tensor h ∈
+C∞(M; S2T ∗M) can be expressed as an infinite sum of products of aspherical functions/1-forms/symmetric
+2-tensors on ˚
+X and spherical harmonic functions/1-forms/symmetric 2-tensors.
+The above decomposition for functions/1-form/symmetric 2-tensors on M implies that the perturbation
+(˙g, ˙A) can be divided into two categories: the first consists of scalar perturbations (also called even parity
+modes in [72] and closed portions in [49]):
+scalar l ≥ 2 :
+�
+˙g = �fS + (f ⊗s /dS) + (HLS/g + HT /δ
+∗
+0/dS)
+˙A = �KS + K/dS
+with
+S ∈ Sl
+scalar l = 1 :
+�
+˙g = �fS + (f ⊗s /dS) + HLS/g
+˙A = �KS + K/dS
+with
+S ∈ S1
+scalar l = 0 :
+�
+˙g = �f + HL/g
+˙A = �K
+(6.11)
+where
+HL, HT , K ∈ C∞( ˚
+X),
+f, �K ∈ C∞( ˚
+X; T ∗ ˚
+X),
+�f ∈ C∞( ˚
+X; S2T ∗ ˚
+X).
+
+LINEAR STABILITY OF KERR-NEWMAN FAMILY
+79
+The second comprises of the vector perturbations (also called odd parity modes in [72] and co-closed portions
+in [49]):
+vector l ≥ 2 :
+�
+˙g = f ⊗s V + HT /δ
+∗V
+˙A = KV
+with
+V ∈ Vl
+vector l = 1 :
+�
+˙g = f ⊗s V
+˙A = KV
+with
+V ∈ V1
+(6.12)
+where HT , K ∈ C∞( ˚
+X) and f ∈ C∞( ˚
+X; T ∗ ˚
+X).
+We decompose the Reissner-Nordstr¨om metric g into aspherical and spherical part as in (6.7), which takes
+the form in static coordinates (t, r, ω) with ω ∈ S2 as follows
+g = ˚g + r2/g
+(6.13)
+where ˚g is a Lorentzian metric on ˚
+X = Rt∗ × [r−, ∞) and /g denote the standard metric on the unit sphere
+S2. In static coordinates (t, r), ˚g takes the form
+˚g = −µb0dt2 + µ−1
+b0 dr2
+with
+µb0 = 1 − 2m
+r
++ Q2
+r2 ,
+(6.14)
+and it can be extended beyond the event horizon by introducing (t∗, r) coordinates. We denote the operators
+associated with ˚g and /g by rings and slashes respectively. Here we use the Einstein summation convention
+with Greek indices µ, ν, . . . for coordinates on M, lower case Roman indices i, j, k, m, n for coordinates on
+˚
+X and a, b, c, d, e for coordinates on S2. Indices are raised and lowered using the Reissner-Nordstr¨om metric
+g, except that we write /gab = (/g−1)ab for the inverse metric to /g on S2.
+7. Mode stability of the Reissner-Nordstr¨om spacetime
+In this section we shall characterize the non-decaying (generalized) modes of the linearized Einstein-
+Maxwell system around the Reissner-Nordstr¨om spacetime (M, gb0, Ab0) with subextremal charge, following
+the method of Kodama-Ishibashi [56,57] for the study of perturbations of black holes without (or with) charge
+in high dimensions. Throughout this section, we drop the subscript b0, i.e., we take (g, A) = (gb0, Ab0) and
+m = m0, Q = Q0. Here we adopt some notations from [36,39,57].
+Theorem 7.1. Let σ ∈ C with Im σ ≥ 0, suppose (˙g, ˙A) is an outgoing mode solution to the linearized
+Einstein-Maxwell system, i.e., (˙g, ˙A) = (e−iσt∗ ˙g0, e−iσt∗ ˙A0) with (˙g0, ˙A0) ∈ ¯H∞,ℓ
+b
+( ¯X; S2 �
+scT ∗ ¯X ⊕ �
+scT ∗ ¯X) for
+some ℓ ∈ R solves
+L1(˙g, ˙A) = −1
+2□g ˙g − δ∗
+gδgGg ˙g + Rg ˙g − 2D(g,dA)T (˙g, d ˙A) = 0
+L2(˙g, ˙A) = D(g,A) (δgdA) (˙g, ˙A) = 0.
+(7.1)
+Then there exist parameters ˙m ∈ R, ˙a ∈ R3, ˙Q ∈ R and an outgoing 1-form ω and a function φ on ¯
+M, i.e.
+(ω, φ) ∈ e−iσt∗ ¯H∞,ℓ′
+b
+( ¯X; �
+scT ∗ ¯X ⊕ C) for some ℓ′ ∈ R, such that
+(˙g, ˙A) =
+�
+˙g(m,0,Q)( ˙m, ˙a, ˙Q) + 2δ∗
+gω,
+˙A(m,0,Q)( ˙m, ˙a, ˙Q) + Lω#A + dφ
+�
+.
+(7.2)
+More specifically
+(1) If σ ̸= 0, then
+(˙g, ˙A) =
+�
+2δ∗
+gω, Lω♯A + dφ
+�
+where (ω, φ) ∈ e−iσt∗ ¯H∞,ℓ′
+b
+( ¯X; �
+scT ∗ ¯X ⊕ C) for some ℓ′ ∈ R.
+(2) If σ = 0 and − 3
+2 < ℓ < − 1
+2, we decompose the perturbation (˙g, ˙A) into spherical harmonics whose
+detail is given in §6. Then
+(i) If (˙g, ˙A) is of scalar type with l ≥ 1 or vector type l ≥ 2, then
+(˙g, ˙A) =
+�
+2δ∗
+gω, Lω♯A + dφ
+�
+where (ω, φ) ∈ ¯H∞,ℓ−1
+b
+( ¯X; �
+scT ∗ ¯X ⊕ C) is of the same type as (˙g, ˙A).
+
+80
+LILI HE
+(ii) If (˙g, ˙A) is of scalar type with l = 0, i.e. spherically symmetric, then
+(˙g, ˙A) =
+�
+˙g(m,0,Q)( ˙m, 0, ˙Q) + 2δ∗
+gω,
+˙A(m,0,Q)(0, 0, ˙Q) + Lω♯A + dφ
+�
+where (ω, φ) ∈ ¯H∞,ℓ−1
+b
+( ¯X; �
+scT ∗ ¯X ⊕ C) is spherically symmetric.
+(iii) If (˙g, ˙A) is of vector type l = 1, then
+(˙g, ˙A) =
+�
+˙g(m,0,Q)(0, ˙a, 0) + 2δ∗
+gω,
+˙A(m,0,Q)(0, ˙a, 0) + Lω♯A + dφ
+�
+where (ω, φ) ∈ ¯H∞,ℓ−1
+b
+( ¯X; �
+scT ∗ ¯X ⊕ C) is of vector type with l = 1.
+The statement of stationary perturbation (˙g, ˙A) (σ = 0) of scalar type with l = 0, 1 can be extended
+to the following cases:
+(iv) If (˙g, ˙A) ∈ ¯H∞,ℓ
+b
+( ¯X; S2 �
+scT ∗ ¯X ⊕ �
+scT ∗ ¯X) with − 5
+2 < ℓ < − 3
+2 is a stationary perturbation of scalar
+type l = 1, then
+(˙g, ˙A) =
+�
+2δ∗
+gω, Lω♯A + dφ
+�
+where (ω, φ) ∈ ¯H∞,ℓ−1
+b
+( ¯X; �
+scT ∗ ¯X ⊕ C) is of the scalar type l = 1.
+(v) If (˙g, ˙A) ∈ Poly(t∗)k ¯H∞,ℓ
+b
+( ¯
+X; S2 �
+scT ∗ ¯X ⊕ �
+scT ∗ ¯X) is of scalar type l = 0, then
+(˙g, ˙A) =
+�
+˙g(m,0,Q)( ˙m, 0, ˙Q) + 2δ∗
+gω,
+˙A(m,0,Q)(0, 0, ˙Q) + Lω♯A + dφ
+�
+where (ω, φ) ∈ Poly(t∗)k+1 ¯H∞,ℓ′
+b
+( ¯
+X; �
+scT ∗ ¯X ⊕ C) is of scalar type l = 0 for some ℓ′ < ℓ.
+Remark 7.2. Our discussion corresponds to the case n = 2, K = 1, κ2 = 2, E0 = r−2Q, Q2 = Q2, Λ = 0 (and
+thus λ = 0), in [57]. We note that [57] discusses the scalar and vector type perturbations for non-stationary
+modes (σ ̸= 0) with l ≥ 2 in detail, but the corresponding stationary perturbations (σ = 0) are only treated
+in the uncharged case in [56]. [57] also studies the scalar (Appendix D) and vector (§4) perturbations for
+modes l = 1 (which is called exceptional modes there), however, they do not give a full description of the
+scalar l = 0 (spherically symmetric) perturbations.
+7.1. Scalar type perturbations. We shall consider the scalar type perturbations of modes l ≥ 2, l = 1
+and l = 0 separately. Recall that we can write the perturbations under the splitting (6.9) and (6.10) as in
+(6.11). We further introduce a rescaled version of the traceless part of the spherical harmonic symmetric
+2-tensor /δ
+∗
+0/dSl which is built from the spherical harmonic function S ∈ Sl with eigenvalues k2 = l(l + 1):
+/HkS = k−2/δ
+∗
+0/dS
+where
+S ∈ Sl with l ̸= 0.
+7.1.1. Modes with l ≥ 2. Suppose (˙g, ˙A) is the scalar perturbations of the following form
+˙g =
+
+
+�fS
+− r
+kf ⊗ /dS
+2r2(HLS/g + HT /HkS)
+
+ ,
+˙A =
+�
+�KS
+− r
+kK/dS
+�
+with
+S ∈ Sl (l ≥ 2).
+(7.3)
+We notice that the pure gauge solutions take the form
+δ ˙g = 2δ∗
+gω =
+
+
+2˚δ∗T
+− r
+k(− k
+r T + r˚
+dr−1L)/dS
+2r2(r−1ιdrT + k
+2rL)S/g − k
+r L /HkS
+
+ ,
+δ ˙A = Lω♯A + dφ =
+�
+(˚dιAT + ιT ˚dA + ˚dP)S
+− r
+k(− k
+r ιAT − k
+r P)/dS
+�
+(7.4)
+with
+ω =
+�
+T S
+− r
+kL/dS
+�
+,
+φ = PS
+(7.5)
+where T ∈ C∞( ˚
+X; T ∗ ˚
+X), L, P ∈ C∞( ˚
+X). When adding (δ ˙g, δ ˙A) to (˙g, ˙A), the quantities ˜f, f, HL, HT , ˜K, K
+change by
+δ �f = 2˚δ∗T,
+δf = −k
+r T + r˚
+d(r−1L),
+δHL = r−1ιdrT + k
+2rL,
+δHT = −k
+r L;
+δ �K = ˚
+dιAT + ιT ˚
+dA + ˚dP,
+δK = −k
+r (ιAT + P).
+(7.6)
+
+LINEAR STABILITY OF KERR-NEWMAN FAMILY
+81
+We define X := r
+k(f + r
+k ˚
+dHT ), then δX = r
+k(δf + r
+k ˚dδHT ) = −T because δ commutes with any geometric
+operators we use in this manuscript. As a consequence, the following quantities
+�F := �f + 2˚δ∗X ∈ C∞( ˚
+X; S2T ∗ ˚
+X),
+J := HL + 1
+2HT + r−1ιdrX ∈ C∞( ˚
+X; T ∗ ˚
+X),
+N := �K + ˚
+d
+� r
+k K
+�
++ ιX˚dA ∈ C∞( ˚
+X; T ∗ ˚
+X)
+(7.7)
+are gauge-invariant, that is, δ �F = 0, δJ = δN = 0. Conversely, if �F = 0, J = N = 0, one can verify (˙g, ˙A) is
+a pure gauge solution
+(˙g, ˙A) = (2δ∗ω, Lω♯A + dφ) with ω =
+� −XS
+r2
+k2 HT /dS
+�
+,
+φ = − r
+kK + ιAX.
+(7.8)
+If (˙g, ˙A) are outgoing mode solutions with frequency σ ̸= 0, then ω and φ are modes of the same type, with
+an extra factor r2 and r respectively relative to ˙g and ˙A. If σ = 0, (ω, φ) grow at most by a factor r more
+than (˙g, ˙A).
+Since the linearized Einstein-Maxwell system is gauge-invariant, we can express it in terms of the gauge-
+invariant quantities defined above.
+Concretely, one can choose a gauge, i.e., add a pure gauge solution
+(δ ˙g, δ ˙A) to (˙g, ˙A) for suitable ω = ω(T, L), φ = φ(P) such that the non gauge-invariant quantities ˜f etc.
+take a simple form in the new gauge. More specifically, we adds (δ ˙g, δ ˙A) built from T = X, L = r
+kHT , P =
+r
+kK − ιAX, then X + δX = T − T = 0, HT + δHT = k
+r L − k
+r L = 0 which implies f + δf = 0, �f + δ �f =
+˜F, HL+δHL = J, and K +δK = K − k
+r ιAX− k
+r ( r
+kK −ιAX) = 0 which implies �K +δ �K = N. To summarize,
+if we replace (˙g, ˙A) by (˙g + δ ˙g, ˙A + δ ˙A), in the new gauge we have ( �f, f, HL, HT , �K, K) = ( �F , 0, J, 0, N, 0).
+Therefore, using the detailed calculation in §6 and §A.1 we can write the linearized Einstein-Maxwell system,
+acting on the new (˙g, ˙A)
+˙g =
+
+
+�FS
+0
+2r2JS/g
+
+ ,
+˙A =
+�
+NS
+0
+�
+in terms of the gauge-invariant quantities �F, J, N.
+Now we express 2L1(˙g, ˙A) = 0 and L2(˙g, ˙A) = 0 in the form of (7.3) in terms of �f E, f E, HE
+L , HE
+T and
+�KE, KE respectively. Then the linearized Einstein-Maxwell system reads
+�f E =
+�
+−˚□ − 2˚δ∗˚δ − ˚δ∗˚
+d˚tr
+�
+�F + 2r−1 �
+2˚δ∗ιdr �F − (∂ir)˚∇i �F
+�
+− 4˚δ∗˚dJ − 8r−1(dr) ⊗s ˚dJ
++ (−µ′′
+b0 + k2r−2) �F + (−2Q2r−4 + µ′′
+b0)˚g˚tr �F − 4Qr−2˚g˚∗˚dN = 0,
+(7.9a)
+− r
+kf E = −˚δ �F − 2˚dJ − r˚
+d(r−1˚tr �F) + 4Qr−2˚∗N = 0,
+(7.9b)
+2r2HE
+L = −˚□(2r2J) − 2rιdr˚δ �F + 2ιdrιdr �F − rιdr˚d˚tr �F + (rµ′
+b0 + k2
+2 + 2Q2r−2)˚tr �F
++ (2k2 − 4Q2r−2)J + 4Q˚∗˚dN = 0,
+(7.9c)
+2r2HE
+T = −k2˚tr �F = 0,
+(7.9d)
+�KE = r−2˚δ(r2˚dN) + k2r−2N + 1
+2Qr−2˚∗˚
+d˚tr �F − 2Qr−2˚∗˚
+dJ = 0,
+(7.9e)
+− r
+kKE = −˚δN = 0.
+(7.9f)
+By (7.9d), all terms containing ˚tr �F vanish, then plugging ˚δ �F = −2˚dJ + 4Qr−2˚∗N into (7.9a) yields the can-
+cellation of the term 4˚δ∗˚
+dJ and thus one obtains a wave equation of �F (coupled to J, N only via subprincipal
+terms). Moreover, by (7.9f), one adds the zero term ˚d˚δN to (7.9e) and then can obtain a wave equation for
+N (coupled to J via subprincipal terms). In conclusion, we can rewrite equation (7.9a), (7.9c) and (7.9e) as
+
+82
+LILI HE
+a principally scalar system of wave equations for ( �F, J, N)
+− ˚□B − DB = 0
+where
+B =
+
+
+�F
+J
+N
+
+
+(7.10)
+where D is a first order stationary differential operator acting on C∞( ˚
+X; S2T ∗ ˚
+X ⊕ R ⊕ T ∗ ˚
+X). When (˙g, ˙A)
+and thus B are smooth modes, this system of equations become a system of ODEs on the one dimensional
+space t−1
+∗ (0) with a regular singular point at the event horizon r = rb0 whose solution is given by Frobenius
+series, which means that the vanishing of B in the static region r > rb0 implies the vanishing of B on ˚
+X. As
+a result, it suffices to prove B = 0 in the static region r > rb0. To achieve this, we can work in the static
+coordinates (t, r).
+First, since ˚δN = ˚∗˚d˚∗N = 0, we have ˚
+d˚∗N = 0 and then ˚∗N = ˚dN for some N ∈ C∞( ˚
+X) according to
+H1
+dR( ˚
+X) = 0, which implies N = ˚∗˚dN. Returning to the equation (7.9e), we find
+r2˚∗ �KE = ˚d(−r2˚□N) + k2˚dN − 2Q˚dJ = ˚d
+�
+−r2˚□N + k2N − 2QJ
+�
+= 0
+(7.11)
+and thus
+−r2˚□N + k2N − 2QJ = C
+for some C ∈ R.
+Replacing N by N − C
+k2 , we still have N = ˚∗˚
+dN and meanwhile
+− ˚□N + k2r−2N − 2Qr−2J = 0.
+(7.12)
+We note that since N, and thus ˚∗N is a mode, N can also be chosen as a mode. More specifically, suppose
+˚∗N = e−it∗σ(N0dt∗ + N1dr) where N0, N1 are smooth functions of r and σ ̸= 0, we let N = e−it∗σN0 with
+N0 = iσ−1N0. Then N1 = ∂rN0 = iσ−1∂rN0 will be satisfied automatically because ˚
+d˚∗N = e−it∗σ(−∂rN0 −
+iσN1)dt∗ ∧ dr = 0.
+Next, according to the second Bianchi identity δgGgRic(g) = 0 (which holds for any metric g), one
+concludes for all (g, F)
+δgGg(Ric(g) − 2T (g, F)) = −2δgT (g, F) = −2gαβ(δgF)αFνβ − gακgβλFαβ(∇νFκλ + 2∇κFλν).
+Since dFνκλ = ∇νFκλ − ∇λFνκ + ∇κFλν and gακgβλFαβ(∇λFνκ + ∇κFλν) = 0, it follows that
+δgGg(Ric(g) − 2T (g, F)) = −2δgT (g, F) = −2gαβ(δgF)αFνβ − gακgβλFαβ(dF)νκλ.
+(7.13)
+Linearizing the above equation around (g, F) = (gb0, dAb0), if (˙g, ˙A) is a solution to the Maxwell part of
+the linearized Einstein-Maxwell system, i.e., L2(˙g, ˙A) = 0, we find by using Ric(g) − 2T (g, dA) = 0, δgdA =
+0, d2A = 0 that
+δgGgL1(˙g, d ˙A) = 0,
+i.e.,
+δgGg
+
+
+�f ES
+− r
+kf E ⊗ /dS
+2r2(HE
+L S/g + HE
+T /HkS)
+
+ = 0,
+(7.14)
+which gives the following system of equations for ( �f E, f E, HE
+L , HE
+T )
+r−2˚δ(r2 �f E) + 1
+2
+˚
+d˚tr �f E − k
+r f E + 2r−2˚d(r2HE
+L ) = 0,
+1
+2
+˚tr �f E −
+1
+kr2˚δ(r3f E) + k2 − 2
+k2
+HE
+T = 0.
+We define
+�E = Gg �f E − 2HE
+L˚g,
+(7.15)
+it is clear that �E = 0 implies ( �f E, HE
+L ) = (h˚g, 0) for some h ∈ C∞( ˚
+X). If in addition equations (7.9b) and
+(7.9d) hold (i.e. f E = 0 and HE
+T = 0), one obtains ˚tr �f E = 0,˚δ(r2 �E) = 0, and thus ( �f E, HE
+L ) = (0, 0). We
+further write ˚δ(r2 �E) = 0 in (t, r) coordinates and find its dr component
+− µ′
+b0
+2µ2
+b0
+�Ett + µ′
+b0∂t �Etr + µ−1/2
+b0
+∂r(µ3/2
+b0 �Err) = 0.
+Then the vanishing of �Etr and �Err implies �Ett = 0 because µ−2
+b0 µ′
+b0 = µ−2
+b0 (2mr−2 − 2Q2r−3) ̸= 0 in the
+static region r > rb0 = m +
+�
+m2 − Q2 > Q2/m in the subextremal charge case m > Q. In summary,
+
+LINEAR STABILITY OF KERR-NEWMAN FAMILY
+83
+( �f E, f E, HE
+L , HE
+T ) = 0 is equivalent to (f E, HE
+T , �Etr, �Err) = 0. So our goal is to prove ( �F, J, N) = 0, provided
+that (f E, HE
+T , �Etr, �Err, �KE, KE) = 0.
+Following [57, equation (5.27)] or [39, equation (5.37)], we write
+�F − 2J˚g =
+
+
+µb0X
+−µ−1
+b0 Z
+−µ−1
+b0 Y
+
+
+(7.16)
+in the splitting (A.12). From now on, we will use the fact ˚tr �F = 0 which is implied by HE
+T = 0 without
+specified till the end of this subsubsection. One can recover �F, J as
+�F =
+
+
+µb0
+2 (X − Y )
+−µ−1
+b0 Z
+1
+2µb0 (X − Y )
+
+ ,
+J = X + Y
+4
+.
+(7.17)
+Then the equation (7.9b) f E = 0 implies ˚δ( �F − 2J˚g) = 4Qr−2˚g˚dN, which we express in terms of X, Y, Z, N
+as
+∂tX + ∂rZ = 4Qr−2∂tN,
+− µ′
+b0
+2µb0
+(X − Y ) − µ−2
+b0 ∂tZ + ∂rY = 4Qr−2∂rN.
+(7.18)
+The equation �Etr = �f E
+tr = 0 reads (we use the equation (7.18) during the calculation)
+− ∂t∂rX − ∂t∂rY + µ′
+b0
+2µb0
+∂tX + ( µ′
+b0
+2µb0
+− 2
+r )∂tY −
+k2
+r2µb0
+Z = 0.
+(7.19)
+Finally the equation µb0 �Err = 1
+2(µ−1
+b0 �f E
+tt + µb0 �f E
+rr) − 2HE
+L = 0 becomes (we use equation (7.18) during the
+calculation)
+−µ−1
+b0 ∂2
+t X −µ−1
+b0 ∂2
+t Y + µ′
+b0
+2 ∂rX +(µ′
+b0
+2 + 2µb0
+r
+)∂rY −
+4
+rµb0
+∂tZ − Q2
+r4 X −
+�k2 − 2
+r2
++ 3Q2
+r2
+�
+Y + 4k2Q
+r4
+N = 0.
+(7.20)
+Therefore, it suffices to prove (X, Y, Z, N) = 0 provided that they satisfy the equations (7.12) and (7.18)–
+(7.20).
+We now discuss the case that (˙g, ˙A) is a mode solution with σ ̸= 0. After taking Fourier transform with
+respect to t, setting ∂t(X, Y, Z, N) = −iσ(X, Y, Z, N) and solving for (X′, Y ′, Z′) = ∂r(X, Y, Z), we obtain
+from equations (7.18) and (7.19)
+
+
+X′
+Y ′
+Z′
+iσ
+
+ = T
+
+
+X
+Y
+Z
+iσ
+
+ + f
+where
+T =
+
+
+
+
+0
+µ′
+b0
+µb0 − 2
+r
+k2
+r2µb0 − σ2
+µ2
+b0
+µ′
+b0
+2µb0
+−
+µ′
+b0
+2µb0
+σ2
+µ2
+b0
+1
+0
+0
+
+
+
+ ,
+f = 4Q
+r2
+
+
+−N ′
+N ′
+−N
+
+ ,
+while the equation (7.20) implies the following linear constraint on X, Y, Z
+γ
+
+
+X
+Y
+Z
+iσ
+
+ = h
+(7.21)
+where h = − 4Q
+r2 (
+2µb0
+r N ′ + k2
+r2 N) and
+γ = ( σ2
+µb0
+− Q2
+r4 + (µ′
+b0)2
+4µb0
++ µ′
+b0
+r , σ2
+µb0
+− k2 − 2
+r2
+− 3Q2
+r4
++ (µ′
+b0)2
+4µb0
+− 2µ′
+b0
+r
+, − 2σ2
+rµb0
++ k2µ′
+b0
+2r2µb0
+).
+Following [57] and [39], one can find a linear combination Φ of X, Y, Z
+iσ which satisfies a second order
+ODE. Concretely, let
+Φ :=
+2Z
+iσ − r(X + Y )
+H
+with
+m := k2 − 2,
+x := 2m
+r ,
+z := Q2
+r2 ,
+H := m + 3x − 4z.
+(7.22)
+Then the second order ODE for Φ is (see [57, equation (5.41)], [39, equation (5.49)])
+(µb0∂r)2Φ + (σ2 − VΦ)Φ = FΦN
+(7.23)
+
+84
+LILI HE
+where VΦ and FΦ are (see [57, equations (5.43), (5.44), (C.1)], [39, equation (B.1)])
+VΦ =
+µb0
+r2H2
+�
+9x3 − 9(6z − m)x2 + (72z2 − 8(4m − 3)z + 3m2)x − 32z3 + 24mz(z + 1) + m2(m + 2)
+�
+FΦ = 8Qµb0
+r3H2
+�
+−3x2 + (2z + 6)x + m(m + 4)
+�
+.
+(7.24)
+Conversely, X, Y, Z
+iσ can be expressed in terms of Φ and N as
+X = (σ2r
+µb0
+− PX0
+2rH2 )Φ + PX1
+2H Φ′ − 2QPXN
+r2H2 N − 8Qµb0
+rH
+N ′
+Y = (−σ2r
+µb0
+− PY0
+2rH2 )Φ + PY1
+2H Φ′ − 2QPY N
+r2H2 N + 8Qµb0
+rH
+N ′
+Z
+iσ = PZ
+2H Φ − rµb0Φ′ − 8Qµb0
+rH
+N ′
+(7.25)
+where PX0, PX1, PXN , PY0, PY1, PY N , PZ are functions of r (see [57, equations (5.46), (C.4)–(C.8)] and [39,
+equations (B.2)])
+PX0 = 27x3 − 24(5z − m)x2 +
+�
+152z2 − 2(35m − 12)z + 3m(3m + 2)
+�
+x
+− 64z3 + 48mz2 − 8m(m − 2)z + 2m2(m + 2),
+PX1 = 9x2 − (16z − 5m + 6)x + 8z2 − 6mz − 4m,
+PXN = −18x2 + 4(8z − m + 6)x − 16z2 + 4(m − 4)z + 2m(m + 6),
+PY0 = 9x3 − 6(10z − m)x2 +
+�
+120z2 − 2(11m − 12)z + 3m(m + 2)
+�
+x
+− 64z3 + 16(m − 4)z2 − 8m(m + 2)z,
+PY1 = 3x2 − (12z + m + 6)x + 8z2 + 2(m + 8)z,
+PY N = −6x2 + 4(6z − m)x − 16z2 + 4(m − 4)z − 2m(m + 2),
+PZ = 3x2 + (−2z + 3m)x − (4m + 8)z − 2m.
+(7.26)
+Therefore it suffices to prove (Φ, N) = 0. Recall that N satisfies the wave equation (7.12) and since N is
+also a mode, we can rewrite it as
+(µb0∂r)2N + (σ2 − µb0k2
+r2
+)N = −2QJ
+r2
+= −µb0Q(X + Y )
+2r2
+.
+(7.27)
+Using (7.25) and (7.26), we find
+X + Y = −( PX0
+2rH2 + PY0
+2rH2 )Φ + (PX1
+2H + PY1
+2H )Φ′ − (2QPXN
+r2H2
++ 2QPY N
+r2H2 )N
+= −(H
+r − PZ
+rH )Φ − 2µb0Φ′ − 16Qµb0
+r2H
+N
+and thus (see [57, equation (5.49)] and [39, eqaution (5.52)])
+(µb0∂r)2N + (σ2 − VN )N = FN0Φ + FN1Φ′
+(7.28)
+with
+VN = µb0k2
+r2
++ 8Q2µ2
+b0
+r4H
+,
+FN0 = Qµb0
+2r2 (H
+r − PZ
+rH ),
+FN1 = Qµ2
+b0
+r2
+.
+(7.29)
+Now we have two coupled second order ODEs (7.23) and (7.28)for Φ and N and in fact we can transform
+them into two decoupled second order ODEs by introducing suitable linear combinations of Φ and N (see [57,
+equations (5.56)–(5.64)] and [39, equations (5.55)–(5.59)]). Concretely, we set
+Ψ± = a±Φ + b±N
+(7.30)
+where a±, b± are smooth on ˚
+X (which extends a little bit beyond the event horizon r = rb0)
+(a+, b+) = (Qm
+2˜c + Q
+2r, 1),
+(a−, b−) = ( ˜c
+6m − 2Q2
+3mr, − 4Q
+3m),
+˜c = 3m +
+�
+9m2 + 4Q2m.
+(7.31)
+
+LINEAR STABILITY OF KERR-NEWMAN FAMILY
+85
+Since
+�����
+Qm
+2˜c + Q
+2r
+1
+˜c
+6m − 2Q2
+3mr
+− 4Q
+3m
+����� = − 2Q2m
+3m˜c −
+˜c
+6m < 0, we can recover Φ, N from Ψ±. Next the decoupled equations
+for Φ± are given as
+(µb0∂r)2Ψ± + (σ2 − V±)Ψ± = 0
+(7.32)
+where
+V± = a±FΦ + b±VN
+b±
+.
+(7.33)
+Introducing the tortoise coordinate r∗ =
+�
+µ−1
+b0 dr, we see that the equations for Ψ± are reduced to an
+eigenvalue problem of the type AΨ = σ2Ψ on the region r∗ ∈ (−∞, ∞) where A = −∂2
+r∗ + V (r) is a self-
+adjoint operator. As explained in [57, §6], we expect V (r) to be non-negative on r ∈ [rb0, ∞) and then
+naturally so is A.
+Lemma 7.3. Given r ∈ [rb0, ∞), we have V+ ≥ 0 for l ≥ 1, i.e., k2 ≥ 2 and m ≥ 0.
+Proof. Since rb0 = m +
+�
+m2 − Q2 and H = m + 2
+r(3m − 2Q2
+r ), in the subextremal case Q < m, we have
+r ≥ rb0 > m > Q > 2Q2
+3m and thus H(r) > 0. We first analyze
+FΦ = 8Qµb0
+r3H2
+�
+−3x2 + (2z + 6)x + m(m + 4)
+�
+,
+since x = 2m
+r
+∈ [0, 2), z ≥ 0, m ≥ 0, we have −3x2 + (2z + 6)x + m(m + 4) ≥ m(m + 4) ≥ 0 and thus FΦ ≥ 0.
+Next it is clear from (7.29) that VN ≥ 0. Therefore V+ ≥ 0.
+□
+However, the positivity of V− fails. To deal with this issue, we follow [57, §6] and introduce the procedure
+S-deformation of V−. More specifically, suppose Ψ is compactly supported in r∗, we have
+(Ψ, AΨ)L2(dr∗) =
+�
+|∂r∗Ψ|2 + V (r)|Ψ|2 dr∗ =
+�
+| �DΨ|2 + �V (r)|Ψ|2 dr∗
+where �D = ∂r∗ + S = µb0∂r + S and �V = V + µb0∂rS − S2. We apply the S-deformation to V− with
+S = S− := µb0∂rH−
+H−
+= µb0∂r(log H−) where H− = k2 − 2 + ˜c
+r > 0 on [rb0, ∞) for k2 ≥ 2(l ≥ 1)
+and then
+�V− = k2(k2 − 2)µb0
+r2H−
+≥ 0 on [rb0, ∞)
+for
+k2 ≥ 2(l ≥ 1).
+(7.34)
+Namely, with H− and V− (which are also smooth near r = rb0) defined above, we write �D = ∂r∗ + S− =
+e−
+�
+S−dr∗∂r∗e
+�
+S−dr∗ = H−1
+− ∂r∗H− = H−µb0∂rH− and find that D∗ which is the adjoint of D with respect
+to dr∗ = µ−1
+b0 dr can be expressed as −H−∂r∗H−1
+−
+= −H−µb0∂rH−1
+− . As a consequence, we have
+�D∗ �DΨ− + �V−Ψ− = −H−µb0∂r
+�
+H−2
+− µb0∂r(H−Ψ)
+�
++ �V−Ψ
+= −(µb0∂r)2Ψ − ΨH−µb0∂r(H−2
+− µb0∂rH−) + (V + H−1
+− (µb0∂r)2H− − 2H−2
+− (µb0∂rH−)2)Ψ
+= −(µb0∂r)2Ψ + V Ψ.
+(7.35)
+Now we are at the position to show that Ψ± = 0 on r > rb0, and thus from the previous discussion
+the mode solution (˙g, ˙A) with σ ̸= 0 is a pure gauge solution. To achieve this, we first need to discuss the
+behavior of Ψ± near r = rb0 and r = ∞, i.e., r∗ = ∓∞. Since H, a±, b± are bounded away from zero, we
+only need to analyze the contributions of X, Y, Z, N. Note that �F, J are smooth on ˚
+X (which extends a
+little bit beyond the event horizon). Therefore, the contribution of J = (X + Y )/4 to Φ is smooth on ˚
+X. As
+for the contribution from Z, we notice that ∂t, ∂r∗, which are expressed as ∂t, µb0∂r in the static coordinates
+(t, r), are smooth vector fields on ˚
+X, and thus Z = µb0 �Ftr = − �F(∂t, ∂r∗) is smooth on ˚
+X as well. As a
+result, Φ = e−iσt∗C∞([rb0, ∞)r), which is written in the static coordinates as
+Φ = (t, r) = e−iσtΘ(r)
+where
+Θ(r) ∈ e±iσr∗C∞([rb0, ∞))
+when
+r∗ → ±∞.
+which implies the exponential decay of Φ as r∗ → −∞ when Im σ > 0. Also, when r∗ → ∞, since ˙g = e−iσt∗ ˙g0
+with ˙g0 ∈ ¯H∞,ℓ
+b
+( ¯
+X; S2 �
+scT ∗ ¯X), we still obtain the rapid decay of Φ when r∗ → ∞. As for N, from the previous
+
+86
+LILI HE
+discussion, we know N is a mode, that is, N = e−iσt∗N0 with N0 ∈ C∞([rb0, ∞)r) ∩ ¯H∞,ℓ
+b
+( ¯
+X) as well, and
+thus decays rapidly when r∗ → ±∞.
+Consequently, Ψ± decays rapidly as well when r∗ → ±∞ when
+Im σ > 0.
+Then when Im σ > 0, by pairing (−∂2
+r∗ + V+ − σ2)Ψ+ = 0 and (− �D∗ �D + �V− − σ2)Ψ− = 0 with Ψ+ and
+Ψ− respectively with respect to the L2(Rr∗; dr∗) and then integrating by parts (whose validity is guaranteed
+by the rapid decay of Ψ± when r∗ → ±∞ ), one obtains
+0 = ∥∂r∗Ψ+∥L2 + ∥V 1/2
++
+Ψ+∥L2 − σ2∥Ψ+∥L2,
+0 = ∥ �DΨ−∥L2 + ∥�V 1/2
+−
+Ψ−∥L2 − σ2∥Ψ−∥L2.
+When Re σ = 0 and thus σ2 < 0, we have a sum of squares and thus Ψ± = 0. When Re σ ̸= 0, the imaginary
+part −2iRe σIm σ∥Ψ±∥L2 = 0 and thus Ψ± = 0 on (rb0, ∞).
+When σ ∈ R\{0}, we still have eiσtΦ, eiσtN ∈ e±iσr∗
+�
+C∞([rb0, ∞))∩ ¯H∞,ℓ
+b
+( ¯X)
+�
+when r∗ → ±∞ for some
+ℓ ∈ R, and thus so do Ψ±. First, by normal operator arguments as in the proof of Proposition 5.7, we find
+Ψ+ = e−iσteiσr∗(a + A1−) with a ∈ C when r∗ → ∞. Next, a standard boundary pairing argument, see the
+proof of Theorem 8.1 below for the detail (starting at (8.8)), allows us to conclude limr∗→±∞ e∓iσr∗Ψ+ = 0.
+Then an indicial root argument implies Ψ+ ∈ e−iσteiσr∗A∞, i.e., Ψ+ vanishes to infinite order at r∗ = ∞.
+Lastly, the unique continuation at infinity ( [45, Theorem 17.2.8]) implies Ψ+ = 0 on (rb0, ∞). The proof for
+Ψ− proceeds similarly.
+Next we treat the stationary mode solutions, i.e., σ = 0.
+We still assume l ≥ 2. Now (˙g, ˙A) and thus
+�F, J, N and X, Y, Z are stationary.
+We first analyze the nature of N defined as ˚∗N = ˚
+dN. Let ˚∗N = N0dt∗ + N1dr where N0, N1 are smooth
+functions of r only, and then ˚d˚∗N = 0 gives ∂rN0 = 0 and thus N0 is a constant. Therefore, we have the
+unique (up to additive constants) N = N0t∗ +
+� r
+r− N1(s)ds, which is a generalized mode with frequency
+σ = 0 and grows at most linearly in t∗. Since X, Y, Z are stationary, equation (7.19) implies
+Z = 0,
+(7.36)
+which in turn implies ∂tN = 0 by using equation (7.18). Therefore, we conclude N0 = 0 and N is stationary
+as well.
+Remark 7.4. Alternatively, using N ∈ ¯Hℓ,∞
+b
+( ¯X; �
+scT ∗ ¯X) ⊂ Aℓ+3/2 ⊂ A0+ (because −3/2 < ℓ < −1/2) when
+σ = 0, it also follows that N0 = c = 0.
+As for Φ, we notice that Z/(iσ) is not well-defined when σ = 0 and we need to do some remedies. Using
+(7.21), Z/(iσ) can be expressed as a linear combination of X, Y, N, N ′ and thus we can rewrite Ψ± as linear
+combinations of X, Y, N, N ′ whose coefficients depend on σ
+Ψ±(σ) = CX±(σ)X + CY ±(σ)Y + CN±(σ)N + CN ′±(σ)N ′
+(7.37)
+where C•±(σ) are rational functions of r depending on σ ∈ C. To make the dependence on σ explicit, we
+write (see [39, equations (5.65), (B.3)])
+CX± = a±
+PX
+3 �H
+,
+CY ± = a±
+PY
+3 �H
+,
+CN+ = 1 + µb0(m + 2)PN
+˜cr �H
+,
+CN− = b−
+�
+1 + 6a−µb0(m + 2)x
+r �
+H
+�
+CN ′+ = 2µ2
+b0PN
+˜c �H
+,
+CN ′− = 4b−µ2
+b0(˜c − 4rz)
+r ˜H
+(7.38)
+where �H(σ) = H(k2µ′
+b0 − 4rσ2) and PX, PY , PN are smooth on ˚
+X
+PX = (9x − 36z − 3m − 18)x + 6(4z + m + 8)z,
+PY = −3(9x − 16z + 5m − 6)x − 6(4z − 3m)z + 12m,
+PN = −8(˜c + mr)z,
+(7.39)
+Since µ′
+b0 = 2r−2(m − r−1Q2) > 0 in the region r ≥ rb0 > m > m−1Q2 and thus �H(0) = Hk2µ′
+b0 > 0 there
+as well, we conclude that Ψ±(σ) exist down to σ = 0. We define Ψ± := Ψ±(0). Conversely, using the first
+
+LINEAR STABILITY OF KERR-NEWMAN FAMILY
+87
+two equations in (7.25) with σ = 0, one can recover X, Y in terms of Φ, N, N ′ and thus Ψ±, N, N ′. One can
+also check that Ψ± satisfy the equation (7.32) with σ = 0 (because Ψ±(σ) is holomorphic near σ = 0), i.e.,
+(µb0∂r)2Ψ± − V±Ψ± = 0
+(7.40)
+with V± defined as before in (7.33), (7.31), (7.29) and (7.24).
+Since (˙g, ˙A) ∈ ¯H∞,ℓ
+b
+( ¯X; S2 �
+scT ∗ ¯X ⊕ �
+scT ∗ ¯X) ⊂ Aℓ+3/2, we have X, Y, N ′ ∈ Aℓ+3/2( ¯X) and N ∈ Aℓ+1/2.
+Let ρ = 1/r, we notice that H = m + 6mρ − 4Q2ρ2 and k2µ′
+b0 = 2k2mρ2 − 2k2Q2ρ3, so �H(0) ∈ 2k2mmρ2 +
+ρ3C∞( ¯X). In view of the expression in (7.39), we find PY ∈ C∞( ¯X) and PX, PN ∈ ρC∞( ¯X), and then
+CY ± ∈ ρ−2C∞( ¯X), CX±, CN ′± ∈ ρ−1C∞( ¯X) and CN± ∈ C∞( ¯X).
+As a consequence, we have Ψ± ∈
+Aℓ−1/2 ⊂ A−2+ a priori because −3/2 < ℓ < −1/2. Now we are ready to describe the asymptotic behavior
+of Ψ± near r = rb0 and r = ∞. We only present the proof of Ψ+ because the proof of Ψ− follows analogously.
+Since FΦ, VN , a±, b± are smooth across the event horizon, we have µb0(∂rΨ+)Ψ+|r=rb0 = 0. This implies
+that boundary term at r = rb0 arising in the integration by parts we will do later on vanishes. Near r = ∞,
+owing to (7.24), (7.29) (7.31) and (7.33), we conclude FΦ ∈ ρ3C∞( ¯X), VN ∈
+k2
+r2 + ρ4C∞( ¯X) and then
+V± ∈ k2
+r2 + ρ3C∞( ¯X), which implies V± = k2
+r2
+∗ + A3− as a function of r∗ near r∗ = ∞. We set x = 1/r∗ for
+the moment,
+(µb0∂r)2Ψ+ − V+Ψ+ = ∂2
+r∗Ψ+ − V+(r∗)Ψ = x4∂xΨ+ + 2x3∂xΨ+ − V+(1/x) = 0,
+which implies x = 0(i.e. r∗ = ∞) is a regular singular point and its indicial equation is λ(λ−1)+2λ−k2 = 0.
+The roots to the indicial equation are λ± = (−1±
+√
+1 + 4k2)/2. Since k2 ≥ 6, we see that λ+ ≥ 2 corresponds
+to one solution Ψ+ with decay ∼ r−λ+
+∗
+near r∗ = ∞, while λ− ≤ −3 implies another linearly independent
+solution Ψ+ with growth ∼ r−λ−
+∗
+near r∗ = ∞. Since we have Ψ+ ∈ A−2+ a priori, this excludes the second
+solution with growth. Namely, Ψ+ ∼ r−λ+ which means that Ψ+ ∼ r−2
+∗
+near r∗ = ∞. With this decay rate
+near r∗ = ∞, we can justify the L2(Rr∗; dr∗) pairing between ((µb0∂r)2 − V+)Ψ+ and Ψ+ and also find that
+the boundary term at r∗ = ∞ vanishes when integrating by parts. Therefore we proceed in the same way
+as in the proof of the case Im σ > 0 (use the L2(Rr∗; dr∗) pairing and then integrate by parts) and conclude
+that Ψ± = 0 on (rb0, ∞). Therefore, (˙g, ˙A) is a pure gauge solution.
+7.1.2. Modes with l = 1. We now let k2 = l(l + 1) = 2. Our goal is again to prove that an outgoing mode
+solution (˙g, ˙A) is a pure gauge solution, with the gauge potential an outgoing mode as well. The scalar
+perturbations with l = 1 take the form
+˙g =
+
+
+�fS
+− r
+kf ⊗ /dS
+2r2HLS/g
+
+ ,
+˙A =
+�
+�KS
+− r
+kK/dS
+�
+with
+S ∈ Sl (l = 1).
+(7.41)
+where the term HT is absent because /H1S = 0. Now we follow the notations and expressions in the case
+l ≥ 2 with HT set to be 0. First, the pure gauge solutions take the form
+δ ˙g = 2δ∗
+gω =
+
+
+2˚δ∗T
+− r
+k(− k
+r T + r˚
+dr−1L)/dS
+2r2(r−1ιdrT + k
+2rL)S/g
+
+ ,
+δ ˙A = Lω♯A + dφ =
+�
+(˚dιAT + ιT ˚dA + ˚dP)S
+− r
+k(− k
+r ιAT − k
+r P)/dS
+�
+(7.42)
+with
+ω =
+�
+T S
+− r
+kL/dS
+�
+,
+φ = PS
+(7.43)
+where T ∈ C∞( ˚
+X; T ∗ ˚
+X), L, P ∈ C∞( ˚
+X). When adding (δ ˙g, δ ˙A) to (˙g, ˙A), the quantities ˜f, f, HL, ˜K, K
+change by
+δ �f = 2˚δ∗T,
+δf = −k
+r T + r˚
+d(r−1L),
+δHL = r−1ιdrT + k
+2r L,
+δ �K = ˚
+dιAT + ιT ˚
+dA + ˚dP,
+δK = −k
+r (ιAT + P).
+(7.44)
+We define X := r
+kf, then δX = r
+kδf = −T + r2
+k ˚d(r−1L). We observe that for any given L, one can choose
+T = X + r2
+k
+˚
+d(r−1L),
+P = −ιAT + r
+kK
+(7.45)
+
+88
+LILI HE
+which implies X + δX = 0, K + δK = 0. If (˙g, ˙A) are outgoing mode solutions and L is a mode, then T, P
+are modes of the same type which grows at most by the factor r relative to ˙g, ˙A and L. We again define
+�F := �f + 2˚δ∗X ∈ C∞( ˚
+X; S2T ∗ ˚
+X),
+J := HL + r−1ιdrX ∈ C∞( ˚
+X; T ∗ ˚
+X),
+N := �K + ˚d
+� r
+kK
+�
++ ιX˚dA ∈ C∞( ˚
+X; T ∗ ˚
+X).
+(7.46)
+However, �F, J, N are not gauge-invariant in this case. In fact
+δ �F = 2
+k
+˚δ∗�
+r2˚d(r−1L)
+�
+,
+δJ = k
+2rL + r
+k ιdr˚d(r−1L),
+δN = r2
+k ι˚
+d(r−1L)˚dA.
+(7.47)
+For any fixed L, we pick T, P as explained above and then can work in the gauge X = 0, K = 0, which
+implies
+( �f, f, HL, �K, K) = ( �F, 0, J, N, 0),
+and thus the linearized Einstein-Maxwell system 2L1(˙g, ˙A) = 0, L2(˙g, ˙A) = 0 is again given by (7.9a)–(7.9f)
+with the equation (7.9d) for HE
+T absent. Conversely, in the gauge X = 0, K = 0, if ( �F, J, N) = 0, we have
+(˙g, ˙A) = 0, which implies the original perturbations (˙g, ˙A) take the form
+(˙g, ˙A) = (2δ∗
+gω, Lω♯A + dφ) with ω = −
+��
+X + r2
+k ˚d(r−1L)
+�
+S
+− r
+kL˚dS
+�
+,
+φ = ιA(X + r2
+k
+˚
+d(r−1L)) − r
+k K. (7.48)
+If (˙g, ˙A) are outgoing mode solutions and L is a mode , then ω and φ are modes of the same type which
+grow at most by a factor r more than (˙g, ˙A) and L.
+Now we exploit this extra gauge freedom resulting from L to simplify the structure of the linearized
+Einstein-Maxwell system. Concretely, we define HE
+T := − k2
+2r2 ˚tr �F and we shall arrange HE
+T + δHE
+T = 0 by
+choosing L properly and thus the absent equation (7.9d) still holds in the case l = 1. In order to achieve
+this, we should choose L such that
+δHE
+T = − k
+r2 ˚tr˚δ∗�
+r2˚d(r−1L)
+�
+= k
+r2˚δ
+�
+r2˚d(r−1L)
+�
+= −k˚□˚g(r−1L) − 2kr−1∂i(r)∂i(r−1L)
+= −k˚□˚g(r−1L) + kgabΓi
+ab∂i(r−1L) = −k□g(r−1L) = −HE
+T .
+(7.49)
+We first consider the non-stationary mode solutions with Im σ ≥ 0, σ ̸= 0. We again define N as ˚∗N =
+˚dN and X, Y, Z as in (7.16). As explained in the case l ≥ 2, N, X, Y, Z are modes with the same frequency
+σ. We shall find a mode L satisfying the above equation (7.49). Writing HE
+T = e−iσt∗HE
+T,0(r), then the
+equation (7.49) for L = e−iσt∗ ˆL(r) becomes
+�
+□g(σ)(r−1 ˆL) = HE
+T,0
+k
+(7.50)
+which has a spherically symmetric solution since by Theorem 8.1 �
+□g(σ) : ¯H∞,ℓ
+b
+( ¯X) → ¯H∞,ℓ+1
+b
+( ¯X) is invertible
+for any ℓ < −1/2. Therefore, as discussed around (7.45), we add a pure gauge solution using this L and
+the corresponding T, P as in (7.45) , the linearized Einstein-Maxwell system is again given by (7.9a)–(7.9e).
+This allows us to repeat the proof of the case of non-stationary mode solutions with l ≥ 2 to conclude that
+in the case l = 1, (˙g, ˙A) is a gauge solution as well. More precisely, (˙g, ˙A) take the form as in (7.48).
+Next we discuss the stationary case σ = 0. Suppose (˙g, ˙A) ∈ ¯H∞,ℓ
+b
+( ¯X; S2 �
+scT ∗ ¯X ⊕ �
+scT ∗ ¯X) where we now
+assume −5/2 < ℓ < −1/2, then X = rf/k ∈ ¯H∞,ℓ−1
+b
+( ¯X; �
+scT ∗ ¯X) and thus ˚δ∗X ∈ ¯H∞,ℓ
+b
+( ¯X; S2 �
+scT ∗ ¯X)
+(because ˚δ∗ acts as ρDiff1
+b).
+Therefore, the right-handed side of (7.49) is in ¯H∞,ℓ+2
+b
+( ¯X).
+According to
+Theorem 8.1,�
+□g(0) : ¯H∞,ℓ
+b
+( ¯
+X) → ¯H∞,ℓ+2
+b
+( ¯X) is surjective for −5/2 < ℓ < −1/2,and then we can solve
+(7.49) for r−1L ∈ ¯H∞,ℓ
+b
+( ¯X) and thus L ∈ ¯H∞,ℓ−1
+b
+( ¯X). Again, according to (7.45), by adding a pure gauge
+solution (2˚δ∗
+gω, Lω♯A + dφ) with (ω, φ) ∈ ¯H∞,ℓ−1
+b
+( ¯X; �
+scT ∗ ¯X ⊕ R), the linearized Einstein-Maxwell system is
+again given by (7.9a)–(7.9e).
+Next, we need to modify the argument in the case l ≥ 2, σ = 0 a little bit. We again write
+Ψ±(σ) = CX±(σ)X + CY ±(σ)Y + CN±(σ)N + CN ′±(σ)N ′
+(7.51)
+
+LINEAR STABILITY OF KERR-NEWMAN FAMILY
+89
+where C•±(σ) are rational functions of r depending on σ ∈ C (obtained by letting k2 = 2, m = 0 in (7.38))
+CX± = a±
+PX
+3 �H
+,
+CY ± = a±
+PY
+3 �H
+,
+CN+ = 1 + 2µb0PN
+˜cr �H
+,
+CN− = b−
+�
+1 + 12a−µb0x
+r �H
+�
+CN ′+ = 2µ2
+b0PN
+˜c �H
+,
+CN ′− = 4b−µ2
+b0(˜c − 4rz)
+r ˜H
+(7.52)
+where �H(σ) = H(2µ′
+b0 −4rσ2) with H(r) = 6mr−1 −4Q2r−2, (a+, b+) = ( Q
+2r, 1), (a−, b−) = (1− 2Q2
+3mr, − 4Q
+3m),
+˜c = 6m and PX, PY , PN are smooth on ˚
+X(we again let k2 = 2, m = 0 in (7.39))
+PX = (9x − 36z − 18)x + 6(4z + 8)z,
+PY = −3(9x − 16z − 6)x − 24z2,
+PN = −8˜cz,
+(7.53)
+Since H(r) = 2r−1(3m − 2Q2r−1) > 0 and µ′
+b0 = 2r−2(m − r−1Q2) > 0 in the region r ≥ rb0 > m >
+m−1Q2 > (2/3)m−1Q2, we have �H(0) = 2Hµ′
+b0 > 0 there as well.
+We again define Ψ± := Ψ±(0).
+Conversely, using the the first two equations in (7.25) with σ = 0 and m = 0, one can recover X, Y in terms
+of Φ, N, N ′ and thus Ψ±, N, N ′ as well. Again, Ψ±(0) satisfy
+(µb0∂r)2Ψ± − V±Ψ± = 0
+(7.54)
+with V± defined as before in (7.33) (7.31), (7.29) and (7.24) with m = 0, k2 = 2.
+Since (˙g, ˙A) ∈ ¯H∞,ℓ
+b
+( ¯X; S2 �
+scT ∗ ¯X ⊕ �
+scT ∗ ¯X) ⊂ Aℓ+3/2( ¯X), we have X, Y, N ′ ∈ Aℓ+3/2( ¯X) and N ∈
+Aℓ+1/2( ¯X).
+Let ρ = 1/r, we notice that H = 6mρ − 4Q2ρ2 and k2µ′
+b0 = 4mρ2 − 4Q2ρ3, so �H(0) ∈
+24m2ρ3 + ρ4C∞( ¯X). In view of the expression in (7.53), we find PX, PY ∈ ρC∞( ¯X) and PN ∈ ρ2C∞( ¯X),
+and then CX+, CY +, CN ′+ ∈ ρ−1C∞( ¯X) and CN+ ∈ C∞( ¯X). As a consequence, we have
+Ψ+ ∈ Aℓ+1/2( ¯
+X) ⊂ A−2+( ¯X) a priori if
+− 5/2 < ℓ < −1/2.
+Now we are ready to describe the asymptotic behavior of Ψ+ near r = rb0 and r = ∞. Since FΦ, VN , a±, b±
+are smooth across the event horizon, we have µb0(∂rΨ+)Ψ+|r=rb0 = 0. This implies that boundary term
+at r = rb0 arising in the integration by parts we will do later on vanishes. Near r = ∞, owing to (7.24),
+(7.29) (7.31) and (7.33) with m = 0, we conclude FΦ ∈ 8Q
+3mρ2 + ρ3C∞( ¯X), VN ∈
+2
+r2 + ρ3C∞( ¯X) and then
+V+ ∈
+2
+r2 + ρ3C∞( ¯X), which implies V+ =
+2
+r2
+∗ + A3− as a function of r∗ near r∗ = ∞. We set x = 1/r∗ for
+the moment,
+(µb0∂r)2Ψ+ − V+Ψ+ = ∂2
+r∗Ψ+ − V+(r∗)Ψ = x4∂xΨ+ + 2x3∂xΨ+ − V+(1/x) = 0,
+which implies x = 0(i.e. r∗ = ∞) is a regular singular point and its indicial equation is λ(λ−1)+2λ−2 = 0.
+The roots to the indicial equation are λ+ = 1, λ− = −2. Again we see that λ+ = 1 corresponds to one
+solution Ψ+ with decay ∼ r−1
+∗
+near r∗ = ∞, while λ− ≤ −2 implies another linearly independent solution
+Ψ+ with growth ∼ r2
+∗ near r∗ = ∞. Since we have Ψ+ ∈ A−2+ a priori, this excludes the second solution with
+growth. Namely, Ψ+ ∼ r−1
+∗
+near r∗ = ∞. With this decay rate near r∗ = ∞, we can justify the L2(Rr∗; dr∗)
+pairing between ((µb0∂r)2 − V+)Ψ+ and Ψ+ and also find that the boundary term at r∗ = ∞ vanishes when
+integrating by parts. Therefore we proceed in the same way as in the proof of the case Im σ > 0 (use the
+L2(Rr∗; dr∗) pairing and then integrate by parts) and conclude that Ψ+ = 0 on (rb0, ∞) and thus on ˚
+X.
+As for Ψ−, recall the S-deformation we discussed around (7.35) and we can rewrite the equation ((µb0∂r)2−
+V−)Ψ− = 0 as
+�D∗ �DΨ− = −H−∂r
+�
+H−2
+− µb0∂r(H−Ψ−)
+�
+= 0
+with
+H−(r) = ˜c
+r = 6m
+r
+> 0 on r ∈ [rb0, ∞)
+because �V− =
+k2(k2−2)µb0
+r2H−
+= 0 if l = 1. Solving the above equation for Ψ− yields
+Ψ− = c1
+H−
+� H2
+−
+µb0
+dr + c2
+H−
+= d1 ln
+�����
+r − rH
+r − (m −
+�
+m2 − Q2)
+����� + d2r.
+(7.55)
+
+90
+LILI HE
+where c1, c2, d1, d2 ∈ R. Since Ψ− is smooth across the event horizon r = rb0, we conclude d1 = 0 and thus
+Ψ− = d2r. Using (Ψ+, Ψ−) = (0, d2r), (7.30) and (7.31), we have (Ψ, N) = (d2r, −d2Q/2). Exploiting the
+first two equations in (7.25) we obtain (X, Y ) = (−d2, −d2). As discussed previously, the equation (7.19)
+implies Z = 0 in the stationary perturbation. Finally using (7.17) and ˚∗N = ˚dN we see
+�F = 0,
+J = −d2
+2 ,
+N = 0
+Since J ∈ ¯H∞,ℓ
+b
+( ¯X) and 0 ̸= d2 ∈ ¯H
+∞,− 3
+2 −
+b
+( ¯X), we have d2 = 0 if ℓ ≥ −3/2 and thus we are done. If
+−5/2 < ℓ < −3/2, d2 may be non-zero and then in the gauge X = 0
+˙g =
+
+
+0
+0
+−2r2 d2
+2 S/g
+
+ ,
+˙A = 0.
+This implies in the gauge X = 0, (˙g, ˙A) is a gauge solution with T = 0, L = −d2r/k, P = 0. Therefore,
+according to the discussion around (7.48), the original perturbations (˙g, ˙A) = (2δ∗
+gω, Lω♯A+dφ) with (ω, φ) ∈
+¯H∞,ℓ−1
+b
+( ¯X; �
+scT ∗ ¯X ⊕ R) for −5/2 < ℓ < −1/2.
+7.1.3. Modes with l = 0. As stated in [36,39], the linearization of the simple proof of the Birkhoff’ theorem
+[73] can be extended to the spherically symmetric Einstein-Maxwell system. Following the argument in [73],
+we work in the incoming Eddington-Finkelstein coordinates (t0 = t + r∗, r) where the Reissner-Nordstr¨om
+metric and the electromagnetic 4-potential take the form
+g = −µb0dt2
+0 + 2dt0dr + r2/g,
+A = Q
+r dt0
+(7.56)
+Remark 7.5. The electromagnetic 4-potential defined above differs from (A.17) by an exact 1-form f(r)dr.
+We note that adding an exact 1-form to A changes neither the electromagnetic field nor the linearized
+Einstein-Maxwell system because the linearized system only involves dA.
+Representing the Reissner-Nordstr¨om metric and the electromagnetic 4-potential in the way (7.56), we can
+linearize (gb, Ab) with respect to the parameters (m, 0, Q) in the direction ( ˙m, 0, ˙Q) to obtain the following
+linearized metric and electromagnetic 4-potential
+g′ = (2 ˙m
+r
+− 2Q ˙Q
+r2
+)dt2
+0,
+A′ =
+˙Q
+r dt0.
+(7.57)
+We note that (g′, A′) differs from (˙g(m,0,Q)( ˙m, 0, ˙Q), ˙A(m,0,Q)( ˙m, 0, ˙Q)) by a pure gauge term.
+With this we now turn to the analysis of the spherically symmetric modes. Since there is no spherical com-
+ponent in 1-forms and no aspherical-spherical mixed component in symmetric 2-tensors for spherically sym-
+metric perturbations, we use the following splitting of 1-forms and symmetric 2-tensors respectively:⟨dt0⟩ ⊕
+⟨dr⟩ and ⟨dt2
+0⟩⊕⟨2dt0dr⟩⊕⟨dr2⟩⊕⟨/g⟩. Under this splitting, we write the scalar perturbations of modes l = 0
+(spherically symmetric modes) in the following form
+˙g =
+
+
+
+
+˙X
+˙Y
+˙Z
+2r2HL
+
+
+
+ ,
+˙A =
+� ˙A0
+˙A1
+�
+where ˙X, ˙Y , ˙Z, HL are smooth function on Rt0 × [r−, ∞). To describe the pure gauge perturbations, we use
+the following lemma
+Lemma 7.6. Let (g, A) be as defined in (7.56) , under the splitting ⟨∂t0⟩⊕⟨∂r⟩ of T M and ⟨dv2⟩⊕⟨2dt0dr⟩⊕
+⟨dr2⟩ of S2T ∗M, the map L(•)g : T M → S2T ∗M takes the form
+L(•)g =
+
+
+
+
+−2µb0∂t0
+2∂t0 − µ′
+b0
+∂t0 − µb0∂r
+∂r
+2∂r
+0
+0
+2r
+
+
+
+ ;
+(7.58)
+
+LINEAR STABILITY OF KERR-NEWMAN FAMILY
+91
+the map L(•)A between sections of ⟨∂t0⟩ ⊕ ⟨∂r⟩ and ⟨dt0⟩ ⊕ ⟨dr⟩ is given by
+L(•)A =
+�
+Qr−1∂t0
+−Qr−2
+Qr−1∂r
+0
+�
+,
+(7.59)
+and as a mapping between the sections of the trivial line bundle R and ⟨dt0⟩ ⊕ ⟨dr⟩ , d =
+�∂t0
+∂r
+�
+.
+Proof. It follows from the formula (LW g)αβ = W γ∂γgαβ +(∂αW γ)gγβ +(∂βW γ)gαγ, (LW A)α = W γ∂γAα +
+(∂αW γ)Aγ.
+□
+Therefore, adding a pure gauge solution (LW g, LW A) to (˙g, ˙A) with W := Z1∂t0 − rHL∂r and Z1 :=
+(−
+� r
+3m ˙Z(t0, s)ds)/2, the dr2 and spherical components of ˙g vanish. Next, by adding another pure gauge
+solution (0, dφ) with φ = − � r
+3m(A1 + Qs−1∂rZ1)(t0, s)ds, the dr-component of ˙A can be eliminated.
+Suppose the original perturbations (˙g, ˙A) ∈ e−iσt∗ ¯H∞,ℓ
+b
+( ¯X; S2 �
+scT ∗ ¯X ⊕ �
+scT ∗ ¯X) are outgoing modes, then
+so are ˙X, ˙Y , ˙Z, HL. If σ = 0, it is obvious that (ω = W ♭, φ) ∈ ¯H∞,ℓ−1
+b
+( ¯
+M; �
+scT ∗ ¯X ⊕ R) are still stationary
+modes. If σ ̸= 0, since t∗ = t0 + f(r) where f is smooth on [r−, ∞) and ∼ −2r∗ near infinity, we find that
+(ω = W ♭, φ) ∈ e−iσt∗ ¯H∞,ℓ′
+b
+( ¯X; �
+scT ∗ ¯X ⊕ R) for some ℓ′ < ℓ is also outgoing modes. Further, if (˙g, ˙A) ∈
+Poly(t∗)k ¯H∞,ℓ
+b
+( ¯X; S2 �
+scT ∗ ¯X ⊕ �
+scT ∗ ¯X) are the generalized stationary modes, the integration in the definition
+of W and φ implies (ω = W ♭, φ) ∈ Poly(t∗)k ¯H∞,ℓ′
+b
+( ¯X; �
+scT ∗ ¯X ⊕ R) for some ℓ′ < ℓ. To summarize, by
+adding pure gauge solutions which are outgoing modes(or generalized stationary modes), the perturbations
+are reduced to the following form
+˙g = ˙Xdt2
+0 + 2 ˙Y dt0dr,
+˙A = ˙A0dt0.
+Next, we analyze the linearized Einstein-Maxwell system for spherically symmetric modes. Concretely, for
+(ˇg, ˇA) of the general form ˇg = Xdt2
+0+2Y dt0dr+r2/g and ˇA = A0dt0 where X, Y, A0 are functions of (t0, r), the
+dr2-component of Ric(ˇg) − 2T (ˇg, d ˇA) = 0 implies 2∂rY/(rY ) = 0. Linearizing it around (g, A) in (7.56), we
+obtain the corresponding equation for (˙g, ˙A): ∂r ˙Y = 0, so ˙Y = ˙Y (t0). If ˙Y is an outgoing mode with σ ̸= 0,
+we must have ˙Y = ˙Y (t0) = ce−iσt0 which contradicts the definition of outgoing modes unless ˙Y = 0. If ˙Y
+is a stationary mode, then ˙Y = c. However the assumption ˙Y ∈ ¯H∞,ℓ
+b
+( ¯X) ⊂ A0+( ¯X) for −3/2 < ℓ < −1/2
+require ˙Y = 0. Lastly, if ˙Y is a generalized zero mode, we then have ˙Y = ˙Y (t0) = Poly(t0)k. Letting
+W1 = f∂t0 with f(t0) ∈ Poly(t0)k+1 satisfying ∂t0f(t0) = − ˙Y (t0), and adding the pure gauge solution
+(LW1g, LW1A = 0) to (˙g, ˙A), dvdr-component of ˙g again vanishes. Therefore, we now have
+˙g = ˙Xdt2
+0,
+˙A = ˙A0dt0.
+(7.60)
+Again, for (ˇg, ˇA) the Maxwell equation becomes
+δˇgd ˇA =
+�
+− 1
+Y ∂t0∂rA0 + X
+Y 2 ∂2
+rA0 + 2X
+rY 2 ∂rA0 − X
+Y 3 ∂rY ∂rA0 + 1
+Y 2 ∂rA0∂t0Y
+�
+dv
++
+� 1
+Y ∂2
+rA0 + 2
+rY ∂rA0 − 1
+Y 2 ∂rY ∂rA0
+�
+dr.
+(7.61)
+Linearizing around (g, A) (i.e. X = −µb0, Y = 1, A0 = r−1Q) and using the previously obtained fact ˙Y = 0,
+the dr component tells ∂2
+r ˙A0 + 2r−1∂r ˙A0 = 0, so ˙A0 = c1(t0)r−1 + c2(t0). Since c2(t0)dt0 is an exact 1-form
+and thus a pure gauge solution, we may assume ˙A0 = c1(t0)r−1. Then the equation for dt0-component gives
+−∂t0∂r ˙A0 − µb0∂2
+r ˙A0 − 2µb0r−1∂r ˙A0 = r−2∂t0c1(t0) = 0 and thus c1(t0) is a constant that we denote by ˙Q.
+That is, ˙A0 = r−1 ˙Q.
+Next, the spherical component of Ric(ˇg) − 2T (ˇg, d ˇA) = 0 is given by
+1 + X
+Y 2 + r
+Y 2 ∂rX − rX
+Y 3 ∂rY − r2
+Y 2 (∂rA0)2 = 0.
+Linearizing around (g, A) (i.e.
+X = −µb0, Y = 1, A0 = r−1Q) and using the previously obtained facts
+˙Y = 0, ˙A0 = r−1 ˙Q, we find ˙X + r∂r ˙X − 2r−2Q ˙Q = 0 and thus
+˙X = −2Q ˙Q
+r2
++ c(t0)
+r
+.
+
+92
+LILI HE
+Lastly, the dt2
+0-component of Ric(ˇg) − 2T (ˇg, d ˇA) = 0 is
+− X
+Y 2 ∂t0∂rY + X
+2Y 2 ∂2
+rX + X
+Y 3 ∂t0Y ∂rY − X
+2Y 3 ∂rX∂rY − 2X
+rY 2 ∂t0Y + X
+rY 2 ∂rX + 1
+rY ∂t0X + X
+Y 2 (∂rA0)2 = 0.
+Again linearizing around (g, A) (i.e. X = −µb0, Y = 1, A0 = r−1Q) and using the previously obtained facts
+˙Y = 0, ˙A0 = r−1 ˙Q, ˙X = − 2Q ˙Q
+r2
++ c(v)
+r , we obtain r−2∂vc(v) = 0 and thus c(v) is a constant which we denote
+by 2 ˙m. Therefore, we conclude that up to pure gauge solutions
+˙g = (2 ˙m
+r
+− 2Q ˙Q
+r2
+)dt2
+0 = g′,
+˙A0 =
+˙Q
+r dt0 = A′.
+(7.62)
+7.2. Vector type perturbations. We shall consider the vector type perturbations of modes l ≥ 2 and
+l = 1 separately. Recall that we can write the perturbations under the splitting (6.9) and (6.10) as in (6.12).
+We denote by V ∈ Vl the spherical harmonic 1-form with eigenvalues k2 = l(l + 1) − 1.
+7.2.1. Modes with l ≥ 2. Suppose (˙g, ˙A) is the vector perturbations of the following form
+˙g =
+
+
+0
+rf ⊗ V
+− 2
+kr2HT /δ
+∗V
+
+ ,
+˙A =
+� 0
+rKV
+�
+with
+V ∈ Vl (l ≥ 2).
+(7.63)
+We notice that the pure gauge solutions take the form
+δ ˙g = 2δ∗
+gω =
+
+
+0
+r2˚d(r−1L)/dS
+− 2
+kr2(−kr−1L)/δ
+∗V
+
+ ,
+δ ˙A = Lω♯A = 0
+(7.64)
+with
+ω =
+�
+0
+rLV
+�
+(7.65)
+where L ∈ C∞( ˚
+X). When adding (δ ˙g, δ ˙A) to (˙g, ˙A), the quantities f, HT , K change by
+δf = r˚
+d(r−1L),
+δHT = −kr−1L,
+δK = 0.
+(7.66)
+Then we have the following gauge-invariant quantities
+J := f + r
+k
+˚dHT ∈ C∞( ˚
+X; T ∗ ˚
+X),
+K ∈ C∞( ˚
+X)
+(7.67)
+We also note that if J = 0, K = 0, (˙g, ˙A) is a pure gauge solution
+(˙g, ˙A) = (2δ∗ω, Lω♯A) with ω =
+�
+0
+− r2
+k HT V
+�
+(7.68)
+If (˙g, ˙A) are outgoing mode solutions, then ω is a mode of the same type, which grows by a factor r more
+than (˙g, ˙A).
+Again we can express the linearized Einstein-Maxwell system in terms of the gauge-invariant quantities
+J, K defined above. More specifically, we adds (δ ˙g, δ ˙A) built from L = r
+kHT , then HT + δHT = 0, and thus
+(f, HT , K) = (J, 0, K). Therefore, using the detailed calculation in §6 and §A.1 we can write the linearized
+Einstein-Maxwell system, acting on the new (˙g, ˙A)
+˙g =
+
+
+0
+rJ ⊗ V
+0
+
+ ,
+˙A =
+�
+0
+rKV
+�
+in terms of the gauge-invariant quantities J, K.
+
+LINEAR STABILITY OF KERR-NEWMAN FAMILY
+93
+Now we express 2L1(˙g, ˙A) = 0 and L2(˙g, ˙A) = 0 in the form of (7.3) in terms of f E, HE
+T and KE
+respectively. Then the linearized Einstein-Maxwell system reads
+rf E = r−2˚δ(r4˚
+d(r−1J)) + k2 − 1
+r
+J − 4Qr−2˚∗˚d(rK) = 0,
+(7.69a)
+−2r2
+k HE
+T = −2˚δ(rJ) = 0,
+(7.69b)
+rKE = −˚□(rK) + k2 + 1
+r
+K + Q˚∗˚d(r−1J) = 0.
+(7.69c)
+By (7.69b), one adds the zero term ˚
+d˚δ(rJ) to (7.69a) and then can obtain a wave equation for J (coupled
+to K via subprincipal terms) as (˚δ˚
+d + ˚d˚δ)J = (−˚□ − 1
+2µ′′
+b0)J. In conclusion, we can rewrite equation (7.69a)
+and (7.69c) as a principally scalar system of wave equations for (J, K)
+− ˚□B − DB = 0
+where
+B =
+�
+J
+N
+�
+(7.70)
+where D is a first order stationary differential operator acting on C∞( ˚
+X; T ∗ ˚
+X ⊕ T ∗ ˚
+X). As discussed in the
+scalar perturbations, it suffices to prove B = 0 in the static region r > rb0. To achieve this, we again work
+in the static coordinates (t, r).
+First, by (7.69b) we have ˚
+d˚∗(rJ) = 0 (because ˚δ = ˚∗˚d˚∗), and then ˚∗rJ = ˚d(rΦ) for some function Φ on
+˚
+X. Applying ˚∗r2 on both sides of equation (7.69a) yields
+˚
+d
+�
+r4˚δ(r−2˚
+d(rΦ)) + (k2 − 1)rΦ − 4QrK
+�
+= 0
+which implies r4˚δ(r−2˚d(rΦ)) + (k2 − 1)rΦ − 4QrK = c for some constant c. Replacing Φ by Φ −
+c
+(k2−1)r, we
+obtain r4˚δ(r−2˚d(rΦ)) + (k2 − 1)rΦ − 4QrK = 0. and thus
+r˚δ(r−2˚
+d(rΦ)) + (k2 − 1)
+r2
+Φ − 4Q
+r2 K = 0.
+(7.71)
+We can also rewrite it as
+− ˚□Φ + 1
+r2
+�
+k2 + 1 − 6m
+r
++ 4Q2
+r2
+�
+Φ − 4Q
+r2 K = 0.
+(7.72)
+Next, equation (7.69c) can be rewritten as (where we use equation (7.71))
+− ˚□(rK) + 1
+r2
+�
+k2 + 1 + 4Q2
+r2
+�
+(rK) − (k2 − 1) Q
+r3 Φ = 0
+(7.73)
+We note that (7.72) and (7.73) recover [57, equations (4.28) and (4.30)] with Ω = rΦ, A = −rK, mV =
+k2 − 1, κ2 = 2, q = Q, n = 2, K = 1, ¯τ = 0, J = 0 there. Following [57, equations(4.34) and (4.35)], we
+introduce the following combinations
+Ψ± = a±Φ + b±rK
+(7.74)
+with
+(a+, b+) = (
+Q(k2 − 1)
+3m +
+�
+9m2 + 4(k2 − 1)Q2 , −1),
+(a−, b−) = (1,
+4Q
+3m +
+�
+9m2 + 4(k2 − 1)Q2 ).
+(7.75)
+When expressed in terms of Ψ±, equations (7.72) and (7.73) are transformed into two decoupled wave
+equations (see [57, equations (4.37)–(4.39)])
+˚□Ψ± − V±
+µb0
+Ψ± = 0
+(7.76)
+where
+V± = µb0
+r2
+�
+k2 + 1 + 4Q2
+r2
+− 3m
+r
+±
+�
+9m2 + 4(k2 − 1)Q2
+r
+�
+.
+(7.77)
+Suppose (˙g, ˙A) ∈ e−iσt∗ ¯H∞,ℓ
+b
+( ¯X; S2 �
+scT ∗ ¯X ⊕ �
+scT ∗ ¯X) are modes, then (J, K) ∈ e−iσt∗ ¯H∞,ℓ−1
+b
+( ¯X; �
+scT ∗ ¯X ⊕ R)
+for σ ̸= 0, while (J, K) ∈ ¯H∞,ℓ
+b
+( ¯X; �
+scT ∗ ¯X ⊕ R) for σ = 0. As discussed around (7.12) and (7.36) and using
+
+94
+LILI HE
+(7.73) in the σ = 0 case, we conclude Φ, rK ∈ e−iσt∗ ¯H∞,ℓ−1
+b
+( ¯X) for Im σ ≥ 0, and so are Ψ±. Therefore in
+the static coordinates (t, r), equations (7.76) take the form
+(µb0∂r)2Ψ± − (V± − σ2)Ψ± = 0.
+We observe that when r > rb0 = m +
+�
+m2 − Q2, V+ > 0 and
+r2µ−1
+b0 V− > (r2µ−1
+b0 V−)(rb0) = k2 − 1 + (m −
+�
+9m2 + 4(k2 − 1)Q2)/rb0
+= r−1
+b0
+�
+k2m + (k2 − 1)
+�
+m2 − Q2 −
+�
+(4k2 + 5)Q2 + 9(m2 − Q2)
+�
+> r−1
+b0
+�
+k2m −
+�
+4k2 + 5Q + (k2 − 4)
+�
+m2 − Q2
+�
+> 0
+where we use k2 ≥ 5, m > Q in the last inequality, so V− > 0 in the static region as well. We also notice the
+asymptotic behavior of V± is V± = (k2+1)ρ2+ρ3C∞( ¯X) where ρ = 1/r. Then we can proceed as in the scalar
+perturbations l ≥ 2 case to conclude that Ψ± = 0 and thus (J, K) = 0. More specifically, when Im σ > 0,
+we pair (µb0∂r)2Ψ± − (V± − σ2)Ψ± with Ψ± with respect to L2(Rr∗; dr∗) and integrate by parts, then the
+positivity of V± allows us to obtain Ψ± = 0. When σ ∈ R \ {0}, we use the boundary paring argument.
+Lastly, when σ = 0, we first obtain the a priori decay rate Ψ± ∈ ¯H∞,ℓ−1
+b
+( ¯X) ⊂ Aℓ+1/2( ¯X) ⊂ A−1+( ¯X)
+because −3/2 < ℓ < −1/2. Next we use the Frobenius method (i.e. analyze the indicial equation) and the
+decay rate of V± = (k2 +1)ρ2 +ρ3C∞( ¯
+X) to conclude that Ψ± ∼ r−2 near r = ∞. So the conclusion Ψ± = 0
+again follows from the L2 pairing and integration by parts.
+7.2.2. Modes with l = 1. We now let k2 = l(l + 1) − 1 = 1. Suppose (˙g, ˙A) is the vector perturbations of the
+following form
+˙g =
+
+
+0
+rf ⊗ V
+0
+
+ ,
+˙A =
+�
+0
+rKV
+�
+with
+V ∈ Vl (l = 1).
+(7.78)
+We notice that the pure gauge solutions take the form
+δ ˙g = 2δ∗
+gω =
+
+
+0
+r2˚d(r−1L)/dS
+0
+
+ ,
+δ ˙A = Lω♯A = 0
+(7.79)
+with
+ω =
+�
+0
+rLV
+�
+(7.80)
+where L ∈ C∞( ˚
+X). When adding (δ ˙g, δ ˙A) to (˙g, ˙A), the quantities f, HT , K change by
+δf = r˚
+d(r−1L),
+δK = 0.
+(7.81)
+Then we define the following gauge-invariant quantities
+˚d(r−1f) ∈ C∞( ˚
+X; Λ2T ∗ ˚
+X),
+K ∈ C∞( ˚
+X)
+(7.82)
+If ˚d(r−1f) = 0, K = 0, since ˚
+X is contractible, we can find L ∈ C∞( ˚
+X) such that r−1f = ˚
+d(r−1L), which
+implies
+(˙g, ˙A) = (2δ∗ω, Lω♯A) with ω =
+� 0
+rLV
+�
+(7.83)
+is a pure gauge solution. As explained around (7.12) and (7.36) again, we have L ∈ e−iσt∗ ¯H∞,ℓ
+b
+( ¯X) for
+σ ̸= 0, while L ∈ ¯H∞,ℓ−1
+b
+( ¯X) for σ = 0.
+Again we can express the linearized Einstein-Maxwell system in terms of the gauge-invariant quantities
+˚d(r−1f), K as follows
+rf E = r−2˚δ(r4˚d(r−1f)) − 4Qr−2˚∗˚d(rK) = 0,
+(7.84a)
+rKE = −˚□(rK) + 2
+r K + Q˚∗˚d(r−1f) = 0.
+(7.84b)
+Before solving the above system, we first describe the linearized Kerr-Newman solution in which we lin-
+earize the Kerr-Newman solution family (gb, Ab) around Reissner-Nordstr¨om solution (gb0, Ab0) with respect
+
+LINEAR STABILITY OF KERR-NEWMAN FAMILY
+95
+to the angular momentum parameter a in the direction ˙a, which is parallel to the rotation axis of V. We see
+that in the static coordinates
+g′
+(m,0,Q)(0, ˙a, 0) = 2(µb0 − 1) sin2 θdtdϕ,
+A′
+(m,0,Q)(0, ˙a, 0) = −Q
+r sin2 θdϕ
+(7.85)
+where |˙a| = 1 and (θ, ϕ) are the spherical coordinates adapted to ˙a, i.e, ˙a is defined by θ = 0. We note
+that (7.85) is of the form (7.78) with f = (−2mr−2 + Q2r−3)dt, K = −Qr−2 and V = sin2 θdφ. Then the
+associated gauge-invariant quantities are ˚∗˚
+d(r−1f) = 6mr−4 − 4Q2r−5, K = −Qr−2.
+Now we turn to solving the equations (7.84a) and (7.84b). First (7.84a) implies ˚d
+�
+r4˚∗˚
+d(r−1f)−4QrK
+�
+= 0
+and thus
+r4˚∗˚
+d(r−1f) − 4QrK = c
+(7.86)
+for some constant c. Plugging this into the equation (7.84b) yields
+(˚□ − V )(rK) = cQ
+r4
+where
+V = 2
+r2 + 4Q2
+r4 .
+Since V > 0 and the right-handed side is stationary, as discussed previously, rK = 0, c = 0 is the only mode
+solution with frequency σ ̸= 0 to the above equation, and thus ˚d(r−1f) = 0 as well. As for the case σ = 0,
+since V = 2ρ2 + 4Q2ρ4 where ρ = 1/r, using the Frobenius method and analyzing the indicial equation, it
+follows that for each c ∈ R, there is at most one solution K ∈ ¯H∞,ℓ
+b
+( ¯X) for −3/2 < ℓ < −1/2. On the other
+hand, one can verify
+K = − cQ
+6mr2 ∈ ¯H∞,ℓ
+b
+( ¯
+X)
+is indeed a solution. Returning to (7.86) we find
+˚∗˚d(r−1f) =
+c
+6m(6m
+r4 − 4Q2
+r5 ).
+Therefore, replacing (˙g, ˙A) by (˙g, ˙A) −
+c
+6m(g′, A′), we may assume c = 0 and thus K = 0, ˚d(r−1f) = 0, which
+implies (˙g, ˙A) is a pure gauge solution.
+This finishes the proof of Theorem 7.1.
+8. Mode analysis of the scalar wave operator
+In this section we shall discuss the modes of the scalar wave operator on slowly rotating KN spacetimes
+with subextremal charge. It not only occurs as the gauge propagation operator acting on the gauge function
+D(gb,Ab) �ΥM(˙g, ˙A; gb, Ab) = −δgb ˙A, but an operator on a scalar function which generate the pure gauge
+solutions to the Maxwell equation. We define the Fourier-transformed scalar wave operator on slowly rotating
+Kerr-Newman spacetime gb with parameters b = (m, a, Q) near b0 = (m0, 0, Q0) as
+�
+□gb(σ) := eiσtb,∗□gbe−iσtb,∗
+where tb,∗ = χ0(r)(t + rb0,∗) + (1 − χ0(r))(t − r(m,0,Q),∗) is defined as in (4.29).
+For ease of notation, we drop C in the notations of the functions space ¯Hs,ℓ
+b ( ¯X; C), ¯Hs,ℓ
+b ( ¯X; C), A( ¯X; C)
+and C∞( ¯X; C) if it is clear from the context.
+Theorem 8.1. Let b0 = (m0, 0, Q0) and g = gb0 be the RN metric with subextremal charge. There exists
+ǫ > 0 such that for b = (m, a, Q) with |b − b0| < ǫ, the following holds
+(1) If Im σ ≥ 0 and σ ̸= 0,
+�
+□gb(σ) : {u ∈ ¯Hs,ℓ
+b ( ¯X; C) : �
+□g(σ)u ∈ ¯Hs,ℓ+1
+b
+( ¯X; C)} → ¯Hs,ℓ+1
+b
+( ¯X; C)
+(8.1)
+is invertible for s > 1
+2, ℓ < − 1
+2, s + ℓ > − 1
+2.
+(2) If s > 1
+2 and − 3
+2 < ℓ < − 1
+2, then stationary operator
+�
+□gb(0) : {u ∈ ¯Hs,ℓ
+b ( ¯X; C) : �
+□g(0)u ∈ ¯Hs−1,ℓ+2
+b
+( ¯X; C)} → ¯Hs−1,ℓ+2
+b
+( ¯X; C)
+(8.2)
+is invertible.
+Proof. The proof closely follows [36, Theorem 6.1]. We first prove (8.1) and (8.2) for the RN metric g and
+then use a perturbation argument to extend them to the slowly rotating KN metric gb with b near b0.
+
+96
+LILI HE
+• Injectivity of �
+□g(0). Suppose u ∈ ¯Hs,ℓ
+b ( ¯X) with s > 1
+2, ℓ ∈ (− 3
+2, − 1
+2) and
+�
+□g(0)u = □gu = 1
+r2 ∂rµb0r2∂ru + 1
+r2 /∆u = 0,
+µb0 = 1 − 2m0
+r
++ Q2
+0
+r2 .
+Then by Proposition 5.7, u ∈ ρC∞( ¯X) + A2−( ¯X) where ρ = r−1, which implies |u|, |r∂ru|, | /∇u| ≲
+r−1. As a consequence, the boundary term at r = ∞ in the following integration by parts
+0 = −
+�
+S2
+� ∞
+rb0
+□gu · u r2drd/g =
+�
+S2
+� ∞
+rb0
+�
+µb0(r)|r∂ru|2 + | /∇u|2�
+drd/g
+(8.3)
+vanishes, and the boundary term at r = rb0 vanishes since u is smooth there and µb0(rb0) = 0.
+Then u is a constant in r ≥ rb0 and thus vanishes there since limr→∞ u = 0. Since u vanishes to
+infinite order at r = rb0 and smooth in r ≤ rb0, we shall prove that u = 0 in r ≤ rb0 as well by
+following the arguments in [89, Lemma 1]. Concretely, we first compute the following energy identity
+in r− ≤ r < rb0
+∂r
+�
+|r − rb0|−N�
+−µb0(r)|∂ru|2 + 1
+r2 |/du|2
+/g + |u|2��
++ 2|r − rb0|−N 1
+r2 /δ
+�
+Re (∂ru/du)
+�
+= N|r − rb0|−N−1�
+−µb0(r)|∂ru|2 + 1
+r2 |/du|2
+/g + |u|2�
+− |r − rb0|−N�
+2Re (∂ru · �
+□g(0)u) − R(u, du)
+�
+(8.4)
+where
+R(u, du) =
+�
+4r−1µb0(r) − µ′
+b0(r)
+�
+|∂ru|2 + 2Re (u∂ru) − 2r−3|/du|2
+/g
+is a quadratic form in u and du independent of N. As a result, R(u, du) can be controlled by the first
+term on the right-hand side of the above energy identity (8.4) when N is sufficiently large. Then for
+any fixed r0 ∈ [r−, rb0) we apply Stokes’ Theorem in [r0, rb0 − ǫ] × S2. For N large enough we have
+�
+S2(−µb0(r)|∂ru|2 + |/du|2 + |u|2)|r=r0 d/g
+≤ (rb0 − r0)Nǫ−N
+�
+S2(−µb0(r)|∂ru|2 + |/du|2 + |u|2)|r=rb0−ǫ d/g ≤ CKǫ−N+K
+(8.5)
+for any K as ǫ → 0+ since u is smooth and vanishes to infinite order at r = rb0. By choosing K > N
+and letting ǫ → 0+ we conclude that u = 0 at r = r0.
+• Surjectivity of �
+□g(0). The index 0 property of �
+□g(0) established in Lemma 5.23 directly gives the
+surjectivity. But here we present an alternative proof of the surjectivity. To prove the surjectivity
+of �
+□g(0), we note that it is equivalent to proving the injectivity of the adjoint �
+□g(0)∗. Suppose
+v ∈ ˙H−s+1,−ℓ−2
+b
+( ¯X) with s > 1
+2, ℓ ∈ (− 3
+2, − 1
+2) satisfies �
+□g(0)∗v = �
+□g(0)v = 0. By Proposition 5.7,
+v is smooth in r− ≤ r < rb0 and r > rb0, and u ∈ ρC∞( ¯X) + A2−( ¯X) near r = ∞. First we have
+v = 0 in r < rb0 since �
+□g(0)∗ is a hyperbolic operator there with r being the timelike function,
+see [21, Theorem E.56]. Next, according to [35, Theorem 6.3], v is conormal at r = rb0. Since
+the normal operator of µb0 �
+□g(0)∗ is (µb0∂r)2 whose boundary spectrum consists of {(0, 0), (0, 1)}, it
+follows that v = H(r − rb0)(v0 + v1 log µb0 + ˜v) where v0, v1 ∈ C∞(S2) and ˜v ∈ A1−([rb0, ∞)) which
+means that ˜v is conormal at r = rb0 and ˜v ∼ (r − rb0)1− as r → r+
+b0. Since �
+□g(0)H(r − rb0) =
+r−2∂r(r2µb0(r)δ(r − rb0)) = 0 and �
+□g(0)(H(r − rb0) log µb0) = µ′
+b0(r)δ(r − rb0) + 2Q2
+0r−4H(r − rb0),
+we have
+0 = �
+□g(0)v = µ′
+b0(r)δ(r − rb0)v1 + R
+where
+R = H(r − rb0)
+r2
+� /∆v0 + log(r − rb0) /∆v1 + 2Q2
+0
+r2 v1
+�
++ H(r − rb0)�
+□g(0)˜v ∈ A0−.
+However, δ(r − rb0) /∈ A0−, and this implies v1 = 0. Since v1 = 0 implies µb0∂rv → 0 as r → r+
+b0, we
+find that the integration by parts (8.3) still works here and thus gives v = 0 in r > rb0. Then we
+can conclude that v is supported at r = rb0, and consequently v /∈ L2 [44, Theorem 7.1.27] unless
+v = 0. However, the radial point estimates at the event horizon imply that v ∈ H1/2− near r = rb0.
+Therefore, v must be 0.
+
+LINEAR STABILITY OF KERR-NEWMAN FAMILY
+97
+• Invertibility of �
+□g(σ) for σ ∈ R \ {0}. According to Lemma 5.23, it suffices to prove the injectivity
+of �
+□g(σ) for non-zero real σ.
+We first illustrate the relationship between �
+□g(σ) and �
+□g(σ) =
+eiσt□ge−iσt, that is, u ∈ ker �
+□g(σ) gives rise to an outgoing solution
+�
+□g˜u = 0
+where
+˜u = eiσ(t−tb0,∗)u.
+(8.6)
+We next prove that ˜u = 0 and thus u = 0 in r ≥ rb0 by a boundary pairing argument (see [66, §2.3] [41,
+§3.2] [36, §7]), and a unique continuation theorem at infinity. Concretely, let f(r) ∈ C∞([rb0, ∞))
+be a nonnegative function with f(r) = r − rb0 for rb0 ≤ r < 3m0 and f(r) = r−1 for r > 4m0. Fix
+a cutoff χ ∈ C∞([0, ∞)) which is 0 on [0, 1/2] and 1 on [1, ∞). For small ǫ > 0, we define
+χǫ(r) := χ(f
+ǫ ) ∈ C∞
+c ((rb0, ∞)).
+(8.7)
+We see that χǫ = 1 when r − rb0 ≥ ǫ, r ≤ ǫ−1, and r − rb0 ≥ ǫ/2, r ≤ 2ǫ−1 on suppχǫ. We then
+calculate
+0 = lim
+ǫ→0
+� �
+S2
+� ∞
+rb0
+χǫ˜u · �
+□g(σ)˜u r2drd/g −
+�
+S2
+� ∞
+rb0
+χǫ �
+□g(σ)˜u · ˜u r2drd/g
+�
+= lim
+ǫ→0
+�
+S2
+� ∞
+rb0
+[�
+□g(σ), χǫ]˜u · ˜u r2drd/g
+= lim
+ǫ→0
+��
+S2
+� ∞
+rb0
+2µb0∂rχǫ · ∂r˜u · ˜u r2drd/g −
+�
+S2
+� ∞
+rb0
+µb0∂rχǫ · ∂r|˜u|2 r2drd/g
+�
+.
+(8.8)
+where ∂rχǫ may be nonzero only near the event horizon r = rb0 and near infinity r = ∞. Near r = rb0,
+we have ˜u = e−iσr∗u where we write r∗ = rb0,∗ for brevity of notation, ∂rχǫ = ǫ−1χ′(
+r−rb0
+ǫ
+) →
+δ(r − rb0) as ǫ → 0, and thus
+lim
+ǫ→0
+��
+S2
+� ∞
+rb0
+2µb0∂rχǫ · ∂r˜u · ˜u r2drd/g −
+�
+S2
+� ∞
+rb0
+µb0∂rχǫ · ∂r|˜u|2 r2drd/g
+�
+=
+�
+S2
+�
+− 2iσr2|u|2 + 2µb0r2∂ru · u − r2µb0∂r|u|2�
+|r=rb0 d/g
+=
+�
+S2
+�
+− 2iσr2|u|2�
+|r=rb0 d/g
+(8.9)
+where we use the smoothness of u at r = rb0 in the last step. While near r = ∞, we have r2∂rχǫ =
+−ǫ−1χ′( ρ
+ǫ ) → −δ(ρ) with ρ = r−1 as ǫ → 0, ˜u = eiσr∗u where u = ρu+(ω) + e(ρ, ω) with u+(ω) ∈
+C∞(S2) and e(ρ, ω) ∈ A2−( ¯X) (see Proposition 5.7), and thus
+lim
+ǫ→0
+��
+S2
+� ∞
+rb0
+2µb0∂rχǫ · ∂r˜u · ˜u r2drd/g −
+�
+S2
+� ∞
+rb0
+µb0∂rχǫ · ∂r|˜u|2 r2drd/g
+�
+=
+�
+S2
+�
+− 2iσr2|u|2 − 2µb0r2∂ru · u + r2µb0∂r|u|2�
+|r=∞ d/g
+=
+�
+S2
+�
+− 2iσr2|u|2�
+|r=∞ d/g =
+�
+S2−2iσ|u+|2 d/g
+(8.10)
+Putting (8.9) and (8.10) together yields
+0 = −2iσ
+��
+S2r2|u|2|r=rb0 d/g +
+�
+S2|u+|2 d/g
+�
+,
+(8.11)
+and this implies u|r=rb0 = 0 and u+(ω) = 0. This proves that the leading order term of u at infinity
+vanishes and thus u ∈ A2−( ¯X). For the next step, we rewrite �
+□g(σ)u = 0 near infinity as follows
+2iσ∂r(ru) = −1
+r ∂rr2µb0∂ru − 1
+r /∆u = ρDiff2
+bu.
+Then integrating both sides from infinity to r and using ru|r=∞ = 0 yield u ∈ A3−( ¯X). Proceeding
+iteratively, we find that u ∈ A∞, which means that u vanishes to infinite order at r = ∞. Then
+using the unique continuation theorem at infinity [45, Theorem 17.2.8] for �
+□g(σ)˜u = 0 gives ˜u = 0
+
+98
+LILI HE
+and thus u = 0 in r > rb0. By Proposition 5.7 again, u is smooth and vanishes to infinite order at
+r = rb0. Therefore, we follow the arguments in [89, Lemma 1] again to prove u = 0 in r < rb0. Since
+we are working in the region near the r = rb0 where tb0,∗ = t0, it would be more convenient to use
+the coordinates (tb0, r, ω) with ω ∈ S2, and then we write
+�
+□g(σ)u = 1
+r2 ∂rr2µb0∂ru + 1
+r2 /∆u − 2iσ
+r ∂rru.
+(8.12)
+We can again obtain the energy identity as in (8.4) with R(u, du) replaced by
+R(u, du) − 2Re
+�
+2iσr−1∂ru · ∂r(ru)
+�
+which can still be controlled when N is sufficiently large. Proceeding in the same way as in (8.5), it
+follows that u = 0 in r < rb0 as well.
+• Invertibility of �
+□g(σ) for Im σ > 0. Again according to Lemma 5.23, it suffices to prove the injectivity
+of �
+□g(σ) for σ ∈ C, Im σ> 0. We again use relationship between �
+□g(σ) and �
+□g(σ) as illustrated in
+(8.6). Now ˜u decays exponentially when rb0,∗ → ±∞. So we can apply integration by parts to
+conclude
+0 = −
+�
+S2
+� ∞
+rb0
+�
+□g(σ)˜u · ˜u r2dr/g =
+�
+S2
+� ∞
+rb0
+�
+µb0(r)|r∂r ˜u|2 + | /∇˜u|2 − σ2µ−1
+b0 (r)|˜u|2�
+drd/g.
+Then by taking imaginary part for Re σ ̸= 0, while using −σ2 > 0 for Re σ = 0, it follows that ˜u = 0
+and thus u = 0 in r > rb0. The proof of u = 0 in r < rb0 is the same as that in the non-zero real σ
+case.
+• Proof for �
+□gb(σ). Finally, we consider the slowly rotating KN metric gb, with b near b0. By Theorem
+5.28 and high energy estimate 5.22, �
+□gb(σ) is invertible for |σ| large, say |σ| ≥ C, when |b − b0| < ǫ1,
+so we only need to consider the bounded region|σ| ≤ C. Near σ = 0, the invertibility of �
+□gb(σ) for
+b close to b0 follows from the perturbation argument as above and the fact that �
+□g(0) is invertible.
+For σ ∈ C, Im σ ≥ 0, bounded away from 0 and infinity, we prove the theorem by perturbation
+arguments (starting with the invertibility of �
+□g(σ)) in a compact set of Im σ ≥ 0.
+Concretely,
+suppose there exist sequences δj → 0, bj → b0 such that ker �
+□gbj (σj) is non-trivial. Then we can
+find uj ∈ ¯Hs,ℓ
+b ( ¯X) ∩ ker �
+□gbj (σj) with ∥uj∥ ¯
+Hs,ℓ
+b
+( ¯
+X) = 1. By the uniform Fredholm estimate 5.5, we
+have 1 ≤ C′∥uj∥ ¯
+Hs0,l0
+b
+( ¯
+X). Therefore, there exists a subsequence uj → u weakly in ¯Hs,ℓ
+b ( ¯X) which
+is norm convergent in ¯Hs0,ℓ0
+b
+( ¯
+X) with the limit u being non-zero and satisfying �
+□g(0)u = 0. But
+this contradicts the invertibility of �
+□g(0). So this proves the injectivity. The surjectivity which is
+equivalent to the injectivity of the adjoint �
+□gb(σ)∗ can be proved in a similar manner. This proves
+that for s > 1
+2, ℓ ∈ (− 3
+2, 1
+2) and Im σ ≥ 0, |σ| < c, |b − b0| < ǫ2 for some c, ǫ2 > 0,
+�
+□gb(σ) : {u ∈ ¯Hs,ℓ
+b ( ¯X) : �
+□gb(σ)u ∈ ¯Hs−1,ℓ+2
+b
+( ¯X)} → ¯Hs−1,ℓ+2
+b
+( ¯X)
+is invertible. By Proposition 5.7, this implies that �
+□gb(σ) acting on the function spaces in (8.1) is
+injective for Im σ ≥ 0, 0 < |σ| < c, and the invertibility follows from the fact that �
+□gb(σ) has index
+0. This proves the theorem for Im σ ≥ 0, 0 ≤ |σ| < c and |b − b0| < ǫ2. For Im σ ≥ 0, c ≤ |σ| ≤ C,
+the fact that �
+□g(σ) is invertible and a perturbation argument as above imply the invertibility of
+�
+□gb(σ) for (b, σ) in an open neighborhood of (b0, σ). Then the compactness of the region c ≤ |σ| ≤ C
+implies that there exists an ǫ3 > 0 (depending on c, C) such that �
+□gb(σ) in (8.1) is invertible for
+Im σ ≥ 0, c ≤ |σ| ≤ C and |b − b0| < ǫ3.
+Therefore, the theorem holds for |b − b0| < ǫ where
+ǫ = min{ǫ1, ǫ2, ǫ3}.
+□
+8.1. Growing zero modes. For later use, we now discuss the explicit form of the scalar functions in
+ker �
+□gb(0) which are allowed to have more growth at infinity.
+
+LINEAR STABILITY OF KERR-NEWMAN FAMILY
+99
+Proposition 8.2. For b = (m, a, Q) near b0 = (m0, 0, Q0), we have
+ker �
+□gb(0) ∩ ¯H∞,−3/2−
+b
+( ¯
+X) = ⟨ub,s0⟩,
+ub,s0 = 1.
+(8.13)
+ker �
+□gb(0)∗ ∩ ˙H−∞,−3/2−
+b
+( ¯
+X) = ⟨u∗
+b,s0⟩,
+u∗
+b,s0 = H(r − rb).
+(8.14)
+The following spaces
+ker �
+□gb(0) ∩ ¯H∞,−5/2−
+b
+( ¯X) = ⟨ub,s0⟩ ⊕ {ub,s1(S) : S ∈ S1},
+(8.15)
+ker �
+□gb(0)∗ ∩ ˙H−∞,−5/2−
+b
+( ¯X) = ⟨u∗
+b,s0⟩ ⊕ {u∗
+b,s1(S) : S ∈ S1},
+(8.16)
+are 4-dimensional. Moreover, the maps b �→ ub,s1(S) and b �→ u∗
+b,s1(S) can be chosen to be continuous in b
+with values in the respective spaces. More explicitly, we have
+ub=(m,0,Q),s1(S) = (r − m)S,
+ub=(m,0,Q),s1(S) = (r − m)H(r − rb)S.
+(8.17)
+Remark 8.3. We use the notation ub,s0, ub,s1 because they are of scalar type l = 0 and l = 1 respectively
+when b = (m, 0, Q) is a parameter of RN metric.
+Proof of Proposition 8.2. We first prove (8.13). Let u ∈ ker �
+□gb(0) ∩ ¯H∞,−3/2−
+b
+( ¯X). By the normal operator
+argument as in Proposition 5.7, u must have the form u = u0 + ˜u where u0 ∈ C and ˜u ∈ A1−( ¯X) ⊂
+¯H∞,−1/2−
+b
+( ¯X). Then we have
+0 = �
+□gb(0)u = �
+□gb(0)u0 + �
+□gb(0)˜u = �
+□gb(0)˜u.
+In view of Theorem 8.1, ˜u must be 0, and thus u = u0 is a constant.
+Conversely, we certainly have
+�
+□gb(0)1 = 0. This proves (8.13).
+The proof of (8.14) is analogous. Let u ∈ ker �
+□gb(0) ∩ ˙H−∞,−3/2−
+b
+( ¯X). By the normal operator argument
+again, we conclude that u = χu0 + ˜u where χ is a smooth cutoff with χ = 0 for r ≤ 3m0, χ = 1 with
+r ≥ 4m0, and u0 ∈ C, ˜u ∈ ˙H∞,−1/2−
+b
+( ¯X). We calculate
+−�
+□gb(0)∗χu0 ∈ ˙H−∞,∞
+b
+( ¯X) ⊂ ˙H−∞,3/2−
+b
+( ¯X).
+Owing to Theorem 8.1, there exists a unique ˜u ∈ ˙H−∞,−1/2−
+b
+( ¯X) satisfying �
+□gb(0)∗˜u = −�
+□gb(0)∗χu0. This
+proves that the space in (8.14) is at most 1-dimensional. Conversely, a direct calculation implies
+�
+□gb(0)H(r − rb) = ρ−2
+b ∂r(∆bδ(r − rb)) = 0
+since ∆b|r=rb = 0. This proves (8.14).
+As for (8.15), since ¯H∞,−5/2−
+b
+( ¯
+X) ⊂ A−1−( ¯X), in the normal operator argument as in Proposition 5.7,
+we shift the contour of integration through the pole i with the space of resonant states given by rS with
+S ∈ S1. That is to say, u must take the form u = χrS + ˜u where χ is the smooth cutoff as defined above
+and S ∈ S1, ˜u ∈ ¯H∞,−3/2−
+b
+( ¯X).
+Since S1 is 3-dimensional, we find that the space in (8.15) is at most
+4-dimensional. Now we prove that it is indeed 4-dimensional. We compute
+− �
+□gb(0)χrS = −χ�
+□g(0)rS + [χ, �
+□g(0)]rS + (�
+□g(0) − �
+□gb(0))χrS = 0 + ¯H∞,∞
+b
+( ¯X) + ¯H∞,1/2−
+b
+( ¯X) (8.18)
+where we use �
+□g(0) − �
+□gb(0) ∈ ρ3Diff2
+b for the third summand. Since ker �
+□gb(0) ∩ ˙H−∞,−1/2−
+b
+( ¯X) = 0, there
+exists ˜u ∈ ¯H∞,−3/2−
+b
+( ¯X), which is unique modulo a multiple of ub,s0, satisfying the above equation (8.18).
+Concretely, let v ∈ C∞
+c (X◦) such that ⟨ub,s0, v⟩ = 1. Then there exists a unique ˜u ∈ ann(v) satisfying the
+equation (8.18). Therefore, χrS + ˜u, where ˜u is uniquely determined by S and b, indeed gives rise to an
+element in the space (8.15) with leading order term rS at infinity. This proves (8.15). The proof of (8.16) is
+completely analogous.
+Now we turn to discussing the continuous dependence of ub,s1(S), u∗
+b,s1(S) on b. Suppose bj → b and
+ubj(S) = χrS + ˜ubj(S) ∈ ker �
+□gbj ∩ ¯H∞,−5/2−
+b
+( ¯X) where ˜ubj(S) is in the annihilator of v ∈ C∞
+c (X◦) which
+satisfies ⟨ub,s0, v⟩ ̸= 0. Let u = χrS + ˜u(S) ∈ ker �
+□gb(0) and ej(S) = ubj(S) − u(S) = ˜ubj(S) − ˜u(S), and we
+find
+�
+□gb(0)ej(S) = (�
+□gb(0) − �
+□gbj (0))(˜ubj(S) + χrS).
+
+100
+LILI HE
+Since {˜ubj(S)} is uniformly bounded in ann(v) and �
+□gb(0)− �
+□gbj (0) ∈ (b−bj)ρ3Diff2
+b, we see that �
+□gb(0)ej(S) ∈
+(b − bj) ¯H∞,1/2−
+b
+( ¯X). As �
+□gb(0)|ann(v) : annv → ¯H∞,1/2−
+b
+( ¯X) is invertible, it follows that ej(S) is of size
+O ¯
+H∞,−3/2−
+b
+( ¯
+X)(b − bj). This completes the proof of the continuity.
+It remains to derive the explicit expressions in (8.17). We first consider the case b = b0, i.e. gb = g. Using
+/∆S = −2S, it follow that
+�
+□g(0)((r − m0)S) = 1
+r2
+�
+∂r(r2 − 2m0r + Q2
+0)S − 2(r − m0)S
+�
+= 0.
+�
+□g(0)∗((r − m0)H(r − rb0)S) = H(r − rb0)�
+□g(0)((r − m0)S) + 2µb0δ(r − rb0)S
++ r−2∂r(r2µb0δ(r − rb0))(r − m0)S = 0
+(8.19)
+For the general RN metric gb with parameter b = (m, 0, Q), we notice that the expression of gb is the same
+as that of g with (m0, 0, Q0) replaced by (m, 0, Q). Therefore, the above calculation still follows and gives
+the explicit form of u(m,0,Q),s1(S) and u∗
+(m,0,Q),s1(S).
+□
+9. Mode analysis of the 1-form wave-type operator
+In this section we shall analyze the modes of the gauge propagation operator Pb,γ = δgbGgb�δ∗
+gb,γ and the
+gauge potential wave operator Wb,γ = �δgb,γGgbδ∗
+gb, both of which are wave-type operators acting on 1-form.
+The former one occurs as the (Einstein part) of gauge propagation operator acting on the gauge 1-form
+Dgb �ΥE(˙g; gb) = ˜δgb,γGgb ˙g, while the latter one acts on the gauge potentials (which generate the pure gauge
+solutions). We also note that from this section on, the smallness of the charge Q is required, i.e., we are
+restricted in the weakly charged RN and KN (slowly rotating) metric. Throughout this section, the bundle
+we mostly deal with is scattering 1-form, so we shall drop “ �
+scT ∗ ¯X” in ¯Hs,ℓ
+b ( ¯X; �
+scT ∗ ¯X), ˙Hs,ℓ
+b ( ¯
+X; �
+scT ∗ ¯X) and
+A( ¯X, �
+scT ∗ ¯X) when it is clear from the context.
+We first recall the linearized gauge-fixed Einstein-Maxwell system
+Lb,γ = (2LE
+b,γ(˙g, ˙A), LM
+b,γ(˙g, ˙A)) = 0
+where
+LE
+b,γ(˙g, ˙A) = DgbRic(˙g) − 2D(gb,dAb)T (˙g, d ˙A) + �δ∗
+gb,γ�δgb,γGgb ˙g = 0,
+(9.1)
+LM
+b,γ(˙g, ˙A) = D(gb,Ab)(δgdA)(˙g, ˙A) + dδgb ˙A = 0.
+(9.2)
+We note that for (gb, Ab) satisfying the Einstein-Maxwell system, (2δ∗
+gbω, Lω♯Ab+dφ) solves the linearized
+gauge-fixed Einstein-Maxwell system (9.1) and (9.2) provided that the 1-form ω and the scalar function φ
+satisfy
+�δgb,γGgbδ∗
+gbω = 0,
+δgbdφ = −δgb(Lω♯Ab).
+(9.3)
+Therefore, zero mode solutions to the above wave-type system of equations (9.3) give rise to pure gauge zero
+mode solutions to L(gb,Ab),γ.
+Dually, using (A.1), (A.2), (A.4) and the following calculation for (DgbRic)∗ : S2 �
+scT ∗ ¯X → S2 �
+scT ∗ ¯X,
+(D(gb,dAb)T )∗ : S2 �
+scT ∗ ¯X → �
+scT ∗ ¯X ⊕ S2 �
+scT ∗ ¯X and (D(gb,Ab)(δgdA))∗ : �
+scT ∗ ¯X → �
+scT ∗ ¯X ⊕ S2 �
+scT ∗ ¯X
+(DgbRic)∗(˙g) = GgbDgbRic(Ggb ˙g)
+(D(gb,dAb)T )∗(˙g) = (−D(gb,Ab)(δgdA)(˙g, 0), D(gb,dAb)T (˙g, 0) − 1
+2(gb)µν ˙gαβF γ
+α Fβγ + 1
+2trgb ˙gFµαF α
+ν ),
+= (−D(gb,Ab)(δgdA)(Ggb ˙g, 0), GgbD(gb,dAb)T (Ggb ˙g, 0))
+(D(gb,Ab)(δgdA))∗( ˙A) =
+�
+δgbd ˙A, −D(gb,dAb)T (0, d ˙A)
+�
+=
+�
+dδgb ˙A, −GgbD(gb,dAb)T (0, d ˙A)
+�
+,
+we conclude
+L∗
+(gb,Ab),γ(˙g, ˙A) = (2L∗E
+(gb,Ab),γ(˙g, ˙A), 4L∗M
+(gb,Ab),γ(˙g, ˙A))
+
+LINEAR STABILITY OF KERR-NEWMAN FAMILY
+101
+with
+L∗E
+(gb,Ab),γ(˙g, ˙A) = Ggb
+�
+DgbRic − 2D(gb,dAb)T + �δ∗
+gb,γ�δgb,γGgb
+�
+(Ggb ˙g, 1
+4d ˙A),
+(9.4)
+L∗M
+(gb,Ab),γ(˙g, ˙A) =
+�
+D(gb,Ab)(δgdA) + dδgb
+�
+(Ggb ˙g, 1
+4
+˙A).
+(9.5)
+Therefore, we have L∗
+(gb,Ab),γ(2Ggbδ∗
+gbω∗, 4Lω∗♯Ab + dφ∗) = 0 provided that
+�δgb,γGgbδ∗
+gbω∗ = 0,
+δgbdφ∗ = −4δgb(Lω∗♯Ab).
+(9.6)
+Such dual pure gauge zero modes (2Ggbδ∗
+gbω∗, 4Lω∗♯Ab + dφ∗) give rise to zero mode solutions to
+L∗
+(gb,Ab),γ(˙g, ˙A) = 0.
+Remark 9.1. The reason we introduce the constraint damping, i.e., replace δ∗
+gb by δ∗
+gb,γ in this manuscript, is
+that we need to use the invertibility of the corresponding gauge propagation operator δgbGgb ˜δ∗
+gb to exclude the
+generalized modes, which grows linearly in tb,∗ and whose leading term is given by linearized (in ( ˙m, 0, ˙Q)) KN
+solutions, to the linearized gauge-fixed Einstein-Maxwell equations. We also note that we use a perturbation
+argument to prove the invertibility of the gauge propagation operator δgbGgb ˜δ∗
+gb on the weakly charged and
+slowly rotating KN metrics gb. Moreover, we use the generalized linearized gauge condition for the Einstein
+part �δgb,γGgb ˙g = 0 in this paper to exclude the zero mode of no geometric significance, see [36, Remark
+10.13].
+Let ˜χ(r) ∈ C∞
+c ((r−, 3m0)) such that ˜χ = 1 near r = 2m0 (and thus near the event horizon of gb with b
+near (m0, 0, Q0) where |Q0| ≪ m0). We use the following
+c = ˜χ(r)(dt0 − bdr),
+t0 = t + r(m0,0,Q0),∗
+(9.7)
+with b to be determined later, to define �δ∗
+gb,γ, �δgb,γ, L(gb,Ab),γ, L∗
+(gb,Ab),γ. As discussed in (9.3), we then define
+gauge propagation operator
+Pb,γ := δgbGgb�δ∗
+gb,γ
+(9.8)
+and gauge potential wave operator
+Wb,γ := �δgb,γGgbδ∗
+gb.
+(9.9)
+Remark 9.2. We notice that Pb,γ and Wb,γ are formally dual to each other. However, on the level of function
+spaces, they are not because we need to work with the extendible b-Sobolev spaces for both when analyzing
+the modes of L(gb,Ab),γ. We also note that the dual pure gauge zero mode solutions to L∗
+(gb,Ab),γ(˙g, ˙A) = 0
+(as discussed in (9.6)) are closely related to the zero mode solutions to Wb,γω∗ = 0.
+In order to study the modes of Pb,γ and Wb,γ for b = (m, a, Q) near b0 = (m0, 0, Q0), we define
+�
+Pb,γ(σ) := eiσtb,∗Pb,γe−iσtb,∗,
+�
+Wb,γ(σ) := eiσtb,∗Wb,γe−iσtb,∗
+(9.10)
+where tb,∗ = (t + rb0,∗)χ0(r) + (t − r(m,0,Q),∗)(1 − χ0(r)) is defined as in (4.29).
+We first record the following result about Schwarzschild metric which we rely on in the analysis of �
+Pb,γ(σ)
+and �
+Wb,γ(σ) on weakly charged and slowly rotating KN metrics.
+Proposition 9.3 ( [36, Theorem 7.1]). Let (g, M(m0,0,0)) be a Schwarzschild spacetime with mass m0 and
+P0 = δgGgδ∗
+g be the wave operator acting on 1-form. Define �
+P0(σ) := eiσt0,∗P0e−iσt0,∗ with t0,∗ = (t +
+r0,∗)χ0(r) + (t − r0,∗)(1 − χ0(r)) and r0,∗ = r + 2m0 log(r − 2m0). Then we have
+(1) If Im σ ≥ 0 and σ ̸= 0, then
+�
+P0(σ) : {ω ∈ ¯Hs,ℓ
+b ( ¯X; �
+scT ∗ ¯X) : �
+P0(σ)ω ∈ ¯Hs,ℓ+1
+b
+( ¯X; �
+scT ∗ ¯X)} → ¯Hs,ℓ+1
+b
+( ¯X; �
+scT ∗ ¯X)
+(9.11)
+is invertible for s > 3
+2, ℓ < − 1
+2, s + ℓ > − 1
+2.
+(2) If s > 3
+2 and − 3
+2 < ℓ < − 1
+2, then stationary operator
+�
+P0(0) : {ω ∈ ¯Hs,ℓ
+b ( ¯X; �
+scT ∗ ¯X) : �
+P0(0)ω ∈ ¯Hs−1,ℓ+2
+b
+( ¯X; �
+scT ∗ ¯X)} → ¯Hs−1,ℓ+2
+b
+( ¯X; �
+scT ∗ ¯X)
+(9.12)
+has 1-dimensional kernel and cokernel.
+
+102
+LILI HE
+More explicitly, with v = t + r0,∗ = t + r + 2m0 log(r − 2m0) we have
+ker �
+P0(0) ∩ ¯H∞,−1/2−
+b
+( ¯X; �
+scT ∗ ¯X) = ⟨ωs0⟩,
+ωs0 = r−1(dv − dr)
+ker �
+P0(0)∗ ∩ ˙H−∞,−1/2−
+b
+( ¯X; �
+scT ∗ ¯X) = ⟨ω∗
+s0⟩,
+ω∗
+s0 = δ(r − 2m0)dr
+.
+(9.13)
+Remark 9.4. According to (9.13), �
+P0(0) restricted to 1-forms of scalar type l = 1 (and vector type l = 1
+respectively) is invertible. Since �
+Wb0,γ(0) also maps 1-forms of scalar type l = 1 (and vector type l = 1
+respectively) to the same type and is a small perturbation of �
+P0(0) for b0 = (m0, 0, Q0) with |Q0| ≪ m0
+and γ ≪ 1, it follows that �
+Wb0,γ(0)−1 restricted to 1-forms of scalar type l = 1 (and 1-forms of vector type
+l = 1 respectively) exists with norm O(1).
+9.1. Mode stability of the gauge propagation operator. We first record the following two propositions
+about Schwarzschild metric which are needed in the course of the proof of mode stability of the gauge
+propagation operator Pb,γ for the weakly charged RN metric and also slowly rotating and weakly charged
+KN metric.
+Proposition 9.5 ( [36, Proposition 10.3]). Let (g, M(m0,0,0)) be the Schwarzschild spacetime with mass m0
+and P0,γ = δgGg˜δ∗
+g,γ where ˜δ∗
+g,γ is defined as in (4.62) and (4.60) with c = ˜χ(r)(dv − bdr), v = t + r0,∗ =
+t + r + 2m0 log(r − 2m0) and b ̸= 1. Then there exists γ0 > 0 such that for fixed γ with 0 < |γ| < γ0, the
+following holds for �
+P0,γ(σ) := eiσt0,∗P0,γe−iσt0,∗ with t0,∗ = (t + r0,∗)χ0(r) + (t − r0,∗)(1 − χ0(r)).
+�
+P0,γ(0) : {ω ∈ ¯Hs,ℓ
+b ( ¯X; �
+scT ∗ ¯X) : �
+P0,γ(0)ω ∈ ¯Hs−1,ℓ+2
+b
+( ¯X; �
+scT ∗ ¯X)} → ¯Hs−1,ℓ+2
+b
+( ¯X; �
+scT ∗ ¯X)
+(9.14)
+with s > 2, − 3
+2 < ℓ < − 1
+2 is invertible.
+Proposition 9.6 ( [36, Proposition 10.12]). Let �
+P0,γ(σ) be defined as in Proposition 9.5 with b > 1, then
+we have for Im σ ≥ 0, σ ̸= 0 and s > 2, ℓ < − 1
+2, s + ℓ > − 1
+2
+�
+P0,γ(σ) : {ω ∈ ¯Hs,ℓ
+b ( ¯X; �
+scT ∗ ¯X) : �
+P0,γ(σ)ω ∈ ¯Hs,ℓ+1
+b
+( ¯X; �
+scT ∗ ¯X)} → ¯Hs,ℓ+1
+b
+( ¯X; �
+scT ∗ ¯X)
+(9.15)
+is invertible.
+Remark 9.7. We note that δ∗
+g,γ defined in [36, §10] is slightly different from that in this manuscript ((4.62)
+and (4.60)) in the way that the coefficient of the second term in B in [36] (which is denoted by E there) is
+−1 instead of −1/2. However, this difference has no influence on the conclusions stated in the above two
+propositions. In particular, it does not affect the calculation in [36, equation (10.5)].
+Next we shall prove the analogues of the above two propositions for RN and slowly rotating KN metric
+with small charge by a perturbation argument.
+We define c as in (9.7) with b > 1. With γ0 as in Propositions 9.5 and 9.6, we fix γ ∈ (0, γ0).
+Theorem 9.8. Let b0 = (m0, 0, Q0) and b > 1. Fix γ ∈ (0, γ0), there exists a small constant C(γ) > 0 such
+that for the weakly charged RN metric gb0 with |Q0| < C(γ), the following holds.
+(1) If Im σ ≥ 0 and σ ̸= 0,
+�
+Pb0,γ(σ) : {ω ∈ ¯Hs,ℓ
+b ( ¯X; �
+scT ∗ ¯X) : �
+Pb0,γ(σ)ω ∈ ¯Hs,ℓ+1
+b
+( ¯X; �
+scT ∗ ¯X)} → ¯Hs,ℓ+1
+b
+( ¯X; �
+scT ∗ ¯X)
+(9.16)
+is invertible when s > 2, ℓ < − 1
+2, s + ℓ > − 1
+2.
+(2) If s > 2 and − 3
+2 < ℓ < − 1
+2, then stationary operator
+�
+Pb0,γ(0) : {ω ∈ ¯Hs,ℓ
+b ( ¯X; �
+scT ∗ ¯X) : �
+Pb0,γ(0)ω ∈ ¯Hs−1,ℓ+2
+b
+( ¯X; �
+scT ∗ ¯X)} → ¯Hs−1,ℓ+2
+b
+( ¯X; �
+scT ∗ ¯X)
+(9.17)
+is invertible.
+Both statements also hold for the weakly charged and slowly rotating KN metric gb with b = (m, a, Q) near
+b0 = (m0, 0, Q0).
+Proof. We note that a perturbation argument as in the final step in the proof of Theorem 8.1 allows us
+to extend Propositions 9.5 and 9.6 to weakly charged RN metrics, which however is a smooth family of
+Lorentzian metric on the fixed manifold M(m0,0,0) and is a pull back of the RN metric gb0 defined in (4.7).
+
+LINEAR STABILITY OF KERR-NEWMAN FAMILY
+103
+We need to prove that Propositions 9.5 and 9.6 in fact hold for gb0 with sufficiently small charge as well.
+Specifically, let (g, M(m0,0,0)) be the Schwarzschild spacetime
+g = −(1 − 2m0
+r
+)dv2 + 2dvdr + r2/g,
+M(m0,0,0) = Rv × [r−, ∞) × S2
+where r− ∈ (0, 2m0) is a fixed number and v = t + r0,∗ = t + r + 2m0 log(r − 2m0). We introduce another
+coordinates for (g, M(m0,0,0)): (˜tχ0, r, θ, φ) where ˜tχ0 = t +
+� r
+4m0 s−2(s2 − 2m0s)χ0(s) ds. Let gb0 be the RN
+metric as defined in (4.7). We then define ˜gb0 = (Ψb0)∗gb0 as the pull back of gb0 under a diffeomorphism
+Ψb0
+Ψb0 : M(m,0,0) → M,
+(tχ0, r, θ, φ)(Ψb0(p)) = (˜tχ0, r, θ, φ)(p).
+Then we have
+gb0 = −µb0dt2
+χ0 + 2χ0dtχ0dr + 1 − χ2
+0
+µb0
+dr2 + r2/g,
+(9.18)
+˜gb0 = −µb0d˜t2
+χ0 + 2χ0d˜tχ0dr + 1 − χ2
+0
+µb0
+dr2 + r2/g.
+(9.19)
+We now define
+P˜gb0 ,γ := δ˜gb0 G˜gb0 δ∗
+˜gb0 ,γ
+with c = ˜χ(r)(dv − bdr) and v = t + r∗ in the definition of δ∗
+˜gb0 ,γ. In view of (9.18) and (9.19), we find that
+Pb0,γ can be obtained by replacing ∂˜tχ in P˜gb0 ,γ by ∂tχ. Since
+�
+Pb0,γ(σ) = eiσtb0,∗Pb0,γe−iσtb0,∗,
+tb0,∗ = (t + rb0,∗)χ0(r) + (t − rb0,∗)(1 − χ0(r)),
+�
+P˜gb0 ,γ(σ) = eiσ˜tb0,∗P˜gb0 ,γe−iσ˜tb0,∗,
+˜tb0,∗ = (t + r0,∗)χ0(r) + (t − rb0,∗)(1 − χ0(r)),
+we rewrite
+�
+Pb0,γ(σ) = eiσ(tχ0 +F1(r))Pb0,γe−iσ(tχ0 +F1(r)),
+�
+P˜gb0 ,γ(σ) = eiσ(˜tχ0 +F2(r))P˜gb0 ,γe−iσ(˜tχ0 +F2(r)).
+Therefore
+�
+Pb0,γ(σ) = eiσ(F1(r)−F2(r)) �
+P˜gb0 ,γ(σ)e−iσ(F1(r)−F2(r))
+where F1(r) − F2(r) ∈ C∞
+c ((2m0, ∞)). As discussed at the beginning, Propositions 9.5 and 9.6 hold for
+�
+P˜gb0 ,γ(σ), and thus for �
+Pb0,γ(σ) as well with |Q0| < C(γ) where C(γ) is a sufficiently small constant.
+Finally it remains to analyze the weakly charged and slowly rotating KN case. It follows from a pertur-
+bation argument as in the final step in the proof of Theorem 8.1.
+□
+9.2. Mode stability of the gauge potential wave operator. Now we turn to the analysis of the gauge
+potential wave operator Wb,γ = �δgb,γGgbδ∗
+gb. We will first prove the analogues of Propositions 9.5 and 9.6 for
+W0,γ. We note that the proofs closely follow [36, Proposition 10.3, 10.12]. However, we present the proofs
+in detail for completeness here. Then proceeding as in Theorem 9.8, we can obtain its counterpart for Wb,γ.
+Proposition 9.9. Let (g, M(m0,0,0)) be the Schwarzschild spacetime with mass m0 and W0,γ = ˜δg,γGgδ∗
+g
+where ˜δg,γ is defined as in (4.62) and (4.61) with c = ˜χ(r)(dv − bdr), v = t + r0,∗ = t + r + 2m0 log(r − 2m0)
+and b ̸= 1.
+Then there exists γ0 > 0 such that for fixed γ with 0 < |γ| < γ0, the following holds for
+�
+W0,γ(σ) := eiσt0,∗W0,γe−iσt0,∗ with t0,∗ = (t + r0,∗)χ0(r) + (t − r0,∗)(1 − χ0(r)).
+�
+W0,γ(0) : {ω ∈ ¯Hs,ℓ
+b ( ¯X; �
+scT ∗ ¯X) : �
+W0,γ(0)ω ∈ ¯Hs−1,ℓ+2
+b
+( ¯X)} → ¯Hs−1,ℓ+2
+b
+( ¯X)
+(9.20)
+with s > 2, − 3
+2 < ℓ < − 1
+2 is invertible.
+Proof. Fix − 3
+2 < ℓ < − 1
+2, s > 2 and let
+X s,ℓ := {u ∈ ¯Hs,ℓ
+b ( ¯X) : �
+W0,γ(0)u ∈ ¯Hs−1,ℓ+2
+b
+( ¯X)}.
+In view of Lemma 5.25, we know that �
+W0,γ(0) : X s,ℓ → ¯Hs−1,ℓ+2
+b
+( ¯X) is Fredholm of index 0 when γ is
+sufficiently small. As �
+W0,γ1(0) − �
+W0,γ2(0) is a first-order differential operator with compactly supported
+coefficients, the space X s,ℓ does not depend on γ.
+
+104
+LILI HE
+We split the domain X s,ℓ and the target space ¯Hs,ℓ
+b ( ¯
+X) as follows
+X s,ℓ = K⊥ ⊕ K,
+K = ker �
+P0(0) ∩ X s,ℓ = ⟨ωs0⟩
+¯Hs,ℓ
+b ( ¯X) = R ⊕ R⊥,
+R = ranX s,ℓ �
+P0(0) = ann ω∗
+s0
+where ann means annihilator, K⊥ is a complementary subspace to K inside X s,ℓ while R⊥ = ⟨η⟩ is com-
+plementary to R inside ¯Hs−1,ℓ+2
+b
+(in other word, R⊥ is the cokernel of �
+P0(0)), and ω0, ω∗
+0 are defined as
+in (9.13). We may choose η ∈ C∞
+c (X; �
+scT ∗ ¯X) satisfying ⟨η, ω∗
+s0⟩ = 1. Then the operator �
+W0(0) can be
+rewritten as
+�
+W0,γ(0) =
+�W00 + γW♭
+00
+γW♭
+01
+γW♭
+10
+γW♭
+11
+�
+.
+(9.21)
+We notice that W00 = �
+P0(0)|K⊥ : K⊥ → R is invertible, and so is W00 + γW♭
+00 provided that γ is small
+enough. Moreover, if we identify K ∼= C via cωs0 �→ c and R⊥ ∼= C via cη �→ ⟨cη, ω∗
+s0⟩ = c, correspondingly,
+W♭
+11 can be identified as an endmorphism of C, and thus is simply a number which can be computed as
+follows.
+W♭
+11 = γ−1⟨
+��
+W0,γ(0) − �
+P0(0)
+�
+ωs0, ω∗
+s0⟩
+= γ−1⟨
+�
+FgGgδ∗
+g
+�
+ωs0, ω∗
+s0⟩ = γ−1⟨δ∗
+gωs0, GgBgω∗
+s0⟩
+= ⟨δ∗
+gωs0,
+�
+2c ⊗s ω∗
+s0 − 1
+2g−1(c, ω∗
+s0)g
+�
+⟩
+= 4π(b − 1)
+(9.22)
+where we use g = −(1 − 2m0
+r )dv2 + 2dvdr + r2/g, 2c ⊗s ω∗
+s0 = δ(r − 2m0)˜χ(r)(2dvdr − 2bdr2), g−1(c, ω∗
+s0) =
+δ(r − 2m0)˜χ(r)(1 − b(1 − 2m0
+r )) and δ∗
+gωs0 = − 2m2
+0
+r4 dv2 − 2( 1
+2r2 + m0
+r3 )dvdr + 1
+r2 dr2 − (1 − 2m0
+r )/g in the last
+step.
+Now suppose (ω0, ω1) ∈ (K⊥ ⊕ K) ∩ ker �
+W0,γ(0), then we have
+ω0 = −γ(W00 + γW♭
+00)−1W♭
+01ω1
+and thus
+�
+W♭
+11 − γW♭
+10(W00 + γW♭
+00)−1W♭
+01
+�
+ω1 = 0.
+if γ is nonzero. When b ̸= 1, W♭
+11 is invertible, and thus so is W♭
+11 − γW♭
+10(W00 + γW♭
+00)−1W♭
+01 if γ is
+sufficiently small. So we must have ω1 = 0 and thus ω0 = 0. This proves the injectivity of �
+W0,γ(0). The
+surjectivity follows from the fact that �
+W0,γ(0) is Fredholm of index 0.
+□
+As explained in (9.3) and (9.6), Wb,γ acts as the gauge potential wave operator. In order to obtain the
+mode stability of Lgb,γ(˙g, ˙A) = 0 for Im σ ≥ 0, σ ̸= 0, we need to prove that for the operator Wb,γ. In other
+words, we need to show that for suitable choices of b and small γ, Wb,γ has no modes with σ ̸= 0 in the
+closed upper half plane. We first prove this for Schwarzschild metric, and then the RN and slowly rotating
+KN metric with small charge follows from a perturbation argument as in Theorem 9.8.
+Proposition 9.10. Let �
+W0,γ(σ) be defined as in Proposition 9.9 with b > 1, then there exists ˜γ0 > 0 such
+that for 0 < γ < ˜γ0, Im σ ≥ 0, σ ̸= 0 and s > 2, ℓ < − 1
+2, s + ℓ > − 1
+2
+�
+W0,γ(σ) : {ω ∈ ¯Hs,ℓ
+b ( ¯X; �
+scT ∗ ¯X) : �
+W0,γ(σ)ω ∈ ¯Hs,ℓ+1
+b
+( ¯X; �
+scT ∗ ¯X)} → ¯Hs,ℓ+1
+b
+( ¯X; �
+scT ∗ ¯X)
+(9.23)
+is invertible.
+In the course of the proof of Proposition 9.10, we closely follow the arguments in [36, §10.2]. As explained
+there, the proof consists of two steps:
+(1) We show in Lemma 9.12 that for b > 1 and for all sufficiently small γ, �
+W0,γ(σ) has no modes in a
+fixed neighborhood of σ = 0 in the closed upper half plane.
+(2) In the proof of Proposition 9.10, we proceed as in the final step in the proof of Theorem 8.1 to prove
+the mode stability of �
+W0,γ(σ) in the closed upper half plane.
+
+LINEAR STABILITY OF KERR-NEWMAN FAMILY
+105
+Fix s > 2, − 3
+2 < ℓ < − 1
+2, the domains
+X s,ℓ(σ) := {ω ∈ ¯Hs,ℓ
+b ( ¯X) : �
+W0,γ(σ)ω ∈ ¯Hs−1,ℓ+2
+b
+( ¯X)}
+(9.24)
+of the operators �
+W0,γ(σ) : X s,ℓ(σ) → ¯Hs−1,ℓ+2
+b
+now depend on σ (but still independent of γ). As a conse-
+quence, we need the following technical lemma which allows us to pass to operators with fixed domain and
+target space.
+Lemma 9.11. Let s > 2, − 3
+2 < ℓ < − 1
+2 and b ̸= 1 and fix γ1 satisfying 0 < |γ1| < γ0 where γ0 is as in Propo-
+sition 9.9. Then �
+W0,γ1(σ) : X s,ℓ(σ) → ¯Hs−1,ℓ+2
+b
+( ¯X) is invertible for σ ∈ C with Im σ ≥ 0, |σ| < c where c is
+some small constant. Moreover, �
+W0,γ1(σ)−1 is continuous in σ with values in Lweak( ¯Hs−1,ℓ+2
+b
+( ¯X), ¯Hs,ℓ
+b ( ¯X))
+(the space of linear bounded operators equipped with weak operator topology), while continuous in σ with
+values in Lop( ¯Hs−1+ǫ,ℓ+2+ǫ
+b
+( ¯X), ¯Hs−ǫ,ℓ−ǫ
+b
+( ¯X)) (norm topology) for any ǫ > 0.
+Proof. By Theorem 5.6, we have the uniform Fredholm estimates for ω ∈ X s,ℓ(σ)
+∥ω∥ ¯
+Hs,ℓ
+b
+( ¯
+X) ≤ C
+�
+∥�
+W0,γ1(σ)ω∥ ¯
+Hs−1,ℓ+2
+b
+( ¯
+X) + ∥ω∥ ¯
+Hs0,ℓ0
+b
+( ¯
+X)
+�
+(9.25)
+for Im σ ≥ 0 with |σ| being small and s0 < s, ℓ0 < ℓ. We now follow the arguments in [79, Propsition 4.4] to
+show that the compact error term ∥ω∥ ¯
+Hs0,ℓ0
+b
+( ¯
+X) can be dropped (while we may need to take larger constant
+C) if |σ| is sufficiently small. Concretely, suppose for the sake of contradiction that there exists a sequence
+δj → 0 and a sequence ωj with ∥ωj∥ ¯
+Hs,ℓ
+b
+( ¯
+X) = 1 and �
+W0,γ1(σj)ωj ∈ ¯Hs−1,ℓ+2
+b
+( ¯X) such that 1 = ∥ωj∥ ¯
+Hs,ℓ
+b
+( ¯
+X) ≥
+j∥�
+W0,γ1(σj)ωj∥ ¯
+Hs−1,ℓ+2
+b
+( ¯
+X), then �
+W0,γ1(σj)ωj → 0 in ¯Hs−1,ℓ+2
+b
+( ¯X) and lim infj→∞ ∥ωj∥ ¯
+Hs0,ℓ0
+b
+( ¯
+X) ≥ C−1 > 0.
+As a consequence, there exists a subsequence ωj → ω weakly in ¯Hs,ℓ
+b ( ¯X) and strongly in ¯Hs0,ℓ0
+b
+( ¯X) with the
+limit ω being non-zero. Moreover,
+�
+W0,γ1(σj)ωj − �
+W0,γ1(0)ω =
+��
+W0,γ1(σj) − �
+W0,γ1(0)
+�
+ωj + �
+W0,γ1(0)
+�
+ωj − ω
+�
+→ 0
+in
+¯Hs0−2,ℓ0
+b
+( ¯X)
+since �
+W0,γ1(σj) → �
+W0,γ1(0) as bounded operators in L( ¯Hs0,ℓ0
+b
+( ¯X), ¯Hs0−2,ℓ0
+b
+( ¯X)) and ωj → ω in ¯Hs0,ℓ0
+b
+( ¯X).
+Therefore, �
+W0,γ1(0)ω = 0. But this contradicts Proposition 9.9, and thus proves the injectivity of �
+W0,γ1(σ)
+for σ with small |σ| in the closed upper half plane. Since �
+W0,γ1(σ) has index 0, it follows that �
+W0,γ1(σ)−1
+exists with a uniform bound in a small neighborhood of σ = 0 in the closed upper half plane.
+Next we prove the sequential continuity of �
+W0,γ1(σ)−1 in σ (which is equivalent to continuity). Suppose
+δj → δ, one need to show �
+W0,γ1(σj)−1 → �
+W0,γ1(σ)−1 in weak operator topology of L( ¯Hs−1,ℓ+2
+b
+( ¯X), ¯Hs,ℓ
+b ( ¯X)).
+Suppose {fj} is a sequence in ¯Hs−1,ℓ+2
+b
+which is norm converging to f, let ωj = �
+W0,γ1(σj)−1fj and we shall
+show that ωj converges weakly to �
+W0,γ1(σ)−1f in ¯Hs,ℓ
+b ( ¯X). In view of the uniform bound of �
+W0,γ1(σj)−1
+proved in the first step above, we see that ωj is uniformly bounded in ¯Hs,ℓ
+b ( ¯X). Therefore, there exists a
+subsequence ωjk → ω weakly in ¯Hs,ℓ
+b ( ¯X), then fjk = �
+W0,γ1(σjk)ωjk → �
+W0,γ1(σ)ω weakly in ¯Hs−2,ℓ
+b
+( ¯X).
+Meanwhile by assumption fjk → f strongly in ¯Hs−1,ℓ+2
+b
+( ¯
+X) and thus �
+W0,γ1(σ)ω = f. This implies that ω
+is uniquely determined by f and independent of the subsequence. Namely, every subsequence of ωj has a
+subsequence converging weakly to the same ω. This means that the full sequence ωj → ω = �
+W0,γ1(σj)−1f
+weakly in ¯Hs,ℓ
+b ( ¯
+X) and thus proves the continuity in the weak operator topology (Otherwise, one can find
+g ∈
+˙H−s,−ℓ
+b
+( ¯
+X)), δ > 0 and a subsequence {ωjk} ⊂ {wj} such that |g(ωjk) − g(ω)| ≥ δ.
+However, by
+assumption {ωjk} should have a subsequence converging weakly to ω, which is impossible).
+As for the continuity in the norm topology, we shall prove it by contradiction. Suppose there exists a
+δ > 0 and a sequence σj → σ such that ∥�
+W0,γ1(σj)−1 − �
+W0,γ1(σ)−1∥L( ¯
+Hs−1+ǫ,ℓ+2+ǫ
+b
+( ¯
+X), ¯
+Hs−ǫ,ℓ−ǫ
+b
+( ¯
+X)) ≥ δ, then
+one can find a bounded sequence fj ∈ ¯Hs−1+ǫ,ℓ+2+ǫ
+b
+such that
+∥�
+W0,γ1(σj)−1fj − �
+W0,γ1(σ)−1fj∥ ¯
+Hs−ǫ,ℓ−ǫ
+b
+( ¯
+X) ≥ δ.
+(9.26)
+Passing to a subsequence, we may assume that fj → f weakly in ¯Hs−1+ǫ,ℓ+2+ǫ
+b
+( ¯X) and strongly in ¯Hs−1,ℓ+2
+b
+( ¯X).
+Then it follows that �
+W0,γ1(σ)−1fj → �
+W0,γ1(σ)−1f in ¯Hs,ℓ
+b ( ¯X). However, by the continuity in the weak op-
+erator topology, we see that �
+W0,γ1(σj)−1fj → �
+W0,γ1(σ)−1f weakly in ¯Hs,ℓ
+b ( ¯X) and thus (passing to a
+
+106
+LILI HE
+subsequence) strongly in ¯Hs−ǫ,ℓ−ǫ
+b
+( ¯X), which contradicts (9.26). This finishes the proof of the continuity in
+the norm topology.
+□
+With the help of the invertibility of �
+W0,γ1(σ), the analysis of �
+W0,γ(σ) can be converted to that of another
+operator �
+W0,γ(σ)�
+W0,γ1(σ)−1 : ¯Hs−1,ℓ+2
+b
+( ¯X) → ¯Hs−1,ℓ+2
+b
+( ¯X) with fixed domain and target space independent
+of σ. This will enable us to prove the invertibility of �
+W0,γ(σ) for all sufficiently small γ > 0 in a fixed small
+neighborhood of σ = 0 in the closed upper half plane.
+Lemma 9.12. Let s > 2, − 3
+2 < ℓ < − 1
+2 and b > 1. Then there exists C0 > 0 such that for all γ > 0 and
+σ ∈ C, Im σ ≥ 0 with γ + |σ| < C0, the operator �
+W0,γ1(σ) : X s,ℓ(σ) → ¯Hs−1,ℓ+2
+b
+( ¯X) is invertible.
+Proof. Let
+�
+W0,γ(σ)�
+W0,γ1(σ)−1 : ¯Hs−1,ℓ+2
+b
+( ¯X) → ¯Hs−1,ℓ+2
+b
+( ¯X),
+then it suffices to prove the injectivity of �
+W0,γ(σ)�
+W0,γ1(σ)−1 provided that b > 1 and γ > 0, Im σ ≥ 0 with
+γ + |σ| < C0 where C0 is a small constant. To this end, we split the domains and the target spaces as follows
+domain: ¯Hs−1,ℓ+2
+b
+( ¯X) = K⊥ ⊕ K,
+K = ker �
+P0(0)�
+W0,γ1(0)−1 ∩ ¯Hs−1,ℓ+2
+b
+( ¯X) = ⟨˜ω⟩ with ˜ω = �
+W0,γ1(0)ωs0
+target: ¯Hs−1,ℓ+2
+b
+( ¯X) = R ⊕ R⊥,
+R = ranX s,ℓ(0)�
+P0(0)
+where K⊥ and R⊥ = ⟨η⟩ are the complementary subspaces to K and R respectively inside ¯Hs−1,ℓ+2
+b
+( ¯X), and
+ωs0 is defined as in (9.13). We notice that
+˜ω = �
+W0,γ1(0)ωs0 =
+��
+W0,γ1(0) − �
+P0(0)
+�
+ωs0 = �
+FgGgδgωs0 ∈ C∞
+c (X◦)
+and we can identify R⊥ ∼= C via cη �→ ⟨cη, ω∗
+s0⟩ where ω∗
+s0 is defined as in (9.13). Under these splittings,
+�
+W0,γ(σ)�
+W0,γ1(σ)−1 takes the form
+�
+W0,γ(σ)�
+W0,γ1(σ)−1 =
+�W00(γ, σ)
+W01(γ, σ)
+W10(γ, σ)
+W11(γ, σ)
+�
+.
+(9.27)
+We first list several basic facts: W01(0, 0), W10(0, 0), W11(0, 0) are 0, and W00(0, 0) is invertible.
+Since
+�
+W0,γ(σ)�
+W0,γ1(σ)−1 ∈ L( ¯Hs−1,ℓ+2
+b
+( ¯X), ¯Hs−1,ℓ+2
+b
+( ¯X)) is Fredholm of index 0 (see Lemma 5.25) for γ + |σ|
+small and the three components (0, 1), (1, 0), (1, 1) are compact operators (because they all have finite rank),
+we can conclude that W00(γ, σ) is Fredholm of index 0 as well for γ + |σ| small.
+Suppose ω = (ω0, ω1) ∈ (K⊥ ⊕ K) ∩ ker �
+W0,γ(σ)�
+W0,γ1(σ)−1, we have W00(γ, σ)ω0 + W01(γ, σ)ω1 = 0.
+Assume for the moment that W00(γ, σ) is invertible for γ + |σ| small, then
+ω0 = −W00(γ, σ)−1W01(γ, σ)ω1
+(9.28)
+and thus
+�
+W11(γ, σ) − W10(γ, σ)W00(γ, σ)−1W01(γ, σ)
+�
+ω1 = 0.
+(9.29)
+We want to show that ω1 = 0 and thus ω0 = 0 for γ+|σ|. small. Formally speaking, since W01(0, 0), W10(0, 0)
+are 0, W01(γ, σ), W10(γ, σ) are of size O(γ + |σ|). Therefore, it suffices to show that W11(γ, σ) is nonzero
+modulo O(γ2 + |σ|2) for γ + |σ| small. To make these arguments rigorous, we need to prove the injectivity of
+W00(γ, σ) (and thus the invertibility in view of its 0 index property) for γ +|σ| small and the differentiability
+of W01, W10 at (γ, σ) = (0, 0), and calculate the Taylor expansion of W11(γ, σ) at (0, 0) up to the first order.
+• Invertibility of W00. We shall prove that for γ + |σ| small, there exists a uniform constant C > 0
+such that
+∥W00(γ, σ)−1∥R→K⊥ ≤ C.
+(9.30)
+The proof is similar to that in the first step of the proof of Lemma 9.11. Specifically, we define
+Π : ¯Hs−1,ℓ+2
+b
+( ¯X) → R to be the projection onto R.
+Suppose (9.30) is not true, one can find
+a sequence (γj, δj) → (0, 0) and a sequence {fj} ⊂ K⊥ with ∥fj∥ ¯
+Hs−1,ℓ+2
+b
+( ¯
+X) = 1 such that for
+ωj = �
+W0,γ1(σj)−1fj
+1 = ∥fj∥ ¯
+Hs−1,ℓ+2
+b
+( ¯
+X) ≥ j∥Π�
+W0,γj(σj)ωj∥ ¯
+Hs−1,ℓ+2
+b
+( ¯
+X)
+
+LINEAR STABILITY OF KERR-NEWMAN FAMILY
+107
+which implies Π�
+W0,γj(σj)ωj → 0 in ¯Hs−1,ℓ+2
+b
+( ¯X). In view of Theorem 5.6, we have the following
+uniform Fredholm estimates for γ + |σ| small
+∥ωj∥ ¯
+Hs,ℓ
+b
+( ¯
+X) ≤ ˜C
+�
+∥Π�
+W0,γj(σj)ωj∥ ¯
+Hs−1,ℓ+2
+b
+( ¯
+X) + ∥(I − Π)�
+W0,γj(σj)ωj∥ ¯
+Hs−1,ℓ+2
+b
+( ¯
+X)
++ ∥ωj∥ ¯
+Hs0,ℓ0
+b
+( ¯
+X)
+�
+.
+(9.31)
+Using the identification R⊥ ∼= C introduced previously, we calculate
+∥(I − Π)�
+W0,γj(σj)ωj∥ ¯
+Hs−1,ℓ+2
+b
+( ¯
+X) ∼ |⟨�
+W0,γj(σj)ωj, ω∗
+s0⟩| = |⟨ωj, �
+W0,γj(σj)∗ω∗
+s0⟩|
+= |⟨ωj,
+�
+�
+W0,γj(σj)∗ − �
+P0(0)∗�
+ω∗
+s0⟩| → 0
+as
+(γj, σj) → (0, 0)
+(9.32)
+where we use ω∗
+s0 ∈
+˙H
+− 1
+2 −,∞
+b
+( ¯
+X) in the first equality and �
+P0,γj(σj)∗ − �
+P0(0)∗ → 0 as bounded
+operators L( ˙H1−s,−ℓ−2
+b
+( ¯X), ˙H−s,−ℓ−1
+b
+( ¯X)) in the last step.
+Since ωj = �
+W0,γ1(σj)−1fj, we have
+∥ωj∥ ¯
+Hs,ℓ
+b
+( ¯
+X) ∼ 1 and thus lim infj→∞ ∥ωj∥ ¯
+Hs0,ℓ0
+b
+( ¯
+X) ≥ ˜C−1 lim infj→∞ ∥ωj∥ ¯
+Hs,ℓ
+b
+( ¯
+X) > 0. As a re-
+sult, there exists a subsequence (not shown in the notation) ωj → ω ̸= 0 weakly in ¯Hs,ℓ
+b ( ¯X).
+Then �
+W0,γj(σj)ωj → �
+W0(0)ω = �
+P0(0)ω weakly in ¯Hs−2,ℓ
+b
+( ¯X). Since �
+W0,γj(σj)ωj → 0 strongly in
+¯Hs−1,ℓ+2
+b
+( ¯X), we obtain �
+P0(0)ω = 0.
+Meanwhile, since ∥fj∥ ¯
+Hs−1,ℓ+2
+b
+( ¯
+X) = 1, passing to a subsequence, we see that fj → f weakly
+in ¯Hs−1,ℓ+2
+b
+( ¯X) and strongly in ¯Hs−1−ǫ,ℓ+2−ǫ
+b
+( ¯X). Owing to Lemma 9.11 (the continuity in weak
+operator topology), ωj = �
+W0,γ1(σj)−1fj → �
+W0,γ1(0)−1f weakly in ¯Hs−ǫ,ℓ−ǫ
+b
+. Therefore, we obtain
+0 ̸= ω = �
+W0,γ1(0)−1f, i.e., 0 ̸= f = �
+W0,γ1(0)ω ∈ K. It is clear that one can find g ∈ ˙H1−s,ℓ−2
+b
+( ¯X)
+such that g(K⊥) = 0 while g( ˜w) = 1 (K = ⟨˜ω⟩), but this contradicts the fact that fj → f weakly,
+and thus proves (9.30). From (9.30), we conclude that for γ + |σ| small, W00 is injective and then
+the invertibility follows from the fact that W00 has index 0.
+• Differentiability of W01, W10. We define
+W1(γ, σ) := W01(γ, σ) ⊕ W11(γ, σ) : K = ⟨˜ω⟩ → ¯Hs−1,ℓ+2
+b
+( ¯X).
+(9.33)
+We will show that W1(γ, σ) is continuous for γ + |σ| small and differentiable at (0, 0).
+We write
+�
+W0,γ(σ)�
+W0,γ1(σ)−1 =
+�
+�
+W0,γ(σ) − �
+W0,γ1(σ) + �
+W0,γ1(σ)
+�
+�
+W0,γ1(σ)−1
+=
+�
+�
+W0,γ(σ) − �
+W0,γ1(σ)
+�
+�
+W0,γ1(σ)−1 + I.
+(9.34)
+Then it suffices to analyze the first term on the right hand side of (9.34).
+For the continuity,
+since �
+W0,γ1(σ)−1˜ω ∈ ¯Hs,ℓ
+b ( ¯X) depending on σ continuously for any s > 2, − 3
+2 < ℓ < − 1
+2 (because
+˜ω ∈ C∞
+c (X◦) and �
+W0,γ1(σ)−1 is continuous in σ with value in Lop( ¯Hs−1+ǫ,ℓ+2+ǫ
+b
+( ¯X), ¯Hs−ǫ,ℓ−ǫ
+b
+( ¯X)))
+and �
+W0,γ(σ)−�
+W0,γ1(σ) ∈ Diff1
+b with C∞
+c (X) coefficients and depends on γ, σ smoothly, the continuity
+of �
+W0,γ(σ)�
+W0,γ1(σ)−1˜ω ∈ C∞
+c (X) in (γ, σ) follows
+Now we turn to the analysis of the differentiability at (0, 0). Since �
+W0,γ(σ) − �
+W0,γ1(σ) are poly-
+nomials in γ, σ with coefficients being first order differential operators, it is certainly differentiable
+at (0, 0). So it suffices to prove that �
+W0,γ1(σ)−1˜ω is differentiable at σ = 0. Now we claim that if
+�
+W0,γ1(0)−1ω ∈ ρC∞( ¯X) + ¯H∞,1/2−
+b
+( ¯X), then the following resolvent identity holds
+�
+�
+W0,γ1(σ)−1 − �
+W0,γ1(0)−1�
+ω = �
+W0,γ1(σ)−1�
+�
+W0,γ1(0) − �
+W0,γ1(σ)
+�
+�
+W0,γ1(0)−1ω.
+(9.35)
+The proof of the above resolvent identity closely follows [81, §6] and uses a regularization argument.
+Concretely, let χǫ(ρ) = χ(ǫ−1ρ) where χ is a nonnegative smooth cutoff such that χ = 1 on [1, ∞)
+
+108
+LILI HE
+and χ = 0 on (−∞, 1/2]. With the limit being strong operator limits, we have
+�
+W0,γ1(σ)−1 − �
+W0,γ1(0)−1
+= lim
+ǫ→0
+�
+�
+W0,γ1(σ)−1χǫ(ρ)�
+W0,γ1(0)�
+W0,γ1(0)−1 − �
+W0,γ1(σ)−1�
+W0,γ1(σ)χǫ(ρ)�
+W0,γ1(0)−1�
+= lim
+ǫ→0
+�
+W0,γ1(σ)−1�
+[χǫ(ρ), �
+W0,γ1(0)] + �
+W0,γ1(0)χǫ(ρ) − �
+W0,γ1(σ)χǫ(ρ)
+�
+�
+W0,γ1(0)−1
+= �
+W0,γ1(σ)−1 lim
+ǫ→0
+��
+�
+W0,γ1(0) − �
+W0,γ1(σ)
+�
+χǫ(ρ)
+�
+�
+W0,γ1(0)−1
++ �
+W0,γ1(σ)−1 lim
+ǫ→0
+�
+[χǫ(ρ), �
+W0,γ1(0)]
+�
+�
+W0,γ1(0)−1 := I + II.
+As for I, since �
+W0,γ1(0)−1ω ∈ ρC∞( ¯X) + ¯H∞,1/2−
+b
+( ¯X) which is annihilated modulo ¯H∞,3/2−
+b
+( ¯X) by
+the normal operator −iσρ(ρ∂ρ − 1) of �
+W0,γ1(0) − �
+W0,γ1(σ), it follows that
+��
+�
+W0,γ1(0) − �
+W0,γ1(σ)
+�
+χǫ(ρ)
+�
+�
+W0,γ1(0)−1ω
+= χǫ(ρ)
+�
+�
+W0,γ1(0) − �
+W0,γ1(σ)
+�
+�
+W0,γ1(0)−1ω + ǫ−1χ′(ǫ−1ρ)
+�
+ρ3C∞( ¯X) + ¯H∞,5/2−
+b
+( ¯X)
+�
+∈ χǫ(ρ) ¯H∞,1/2−
+b
+( ¯X) + ǫ−1χ′(ǫ−1ρ)
+�
+ρ3C∞( ¯X) + ¯H∞,5/2−
+b
+( ¯X)
+�
+.
+Since the ¯H∞,1/2−
+b
+( ¯
+X) norm of the second term ǫχ′(ǫ−1ρ)
+�
+ρC∞( ¯X) + ¯H∞,1/2−
+b
+( ¯X)
+�
+goes to 0 as
+ǫ → 0, we arrive at
+I = �
+W0,γ1(σ)−1�
+�
+W0,γ1(0) − �
+W0,γ1(σ)
+�
+�
+W0,γ1(0)−1ω.
+For II, since �
+W0,γ1(0) ∈ ρ2Diff2
+b, it follows that [χǫ(ρ), �
+W0,γ1(0)] is uniform (in ǫ) in ρ2Diff1
+b and
+[χǫ(ρ), �
+W0,γ1(0)] goes to 0 in the strong operator topology L( ¯Hs,ℓ
+b ( ¯X), ¯Hs−1,ℓ+2
+b
+( ¯X)), and thus II = 0.
+This proves (9.35).
+Since �
+W0,γ1(0)−1˜ω = ωs0 = r−1(dv − dr) ∈ ρC∞( ¯X) + ¯H∞,1/2−
+b
+( ¯X), applying (9.35) yields
+�
+�
+W0,γ1(σ)−1 − �
+W0,γ1(0)−1�
+˜ω = �
+W0,γ1(σ)−1�
+�
+W0,γ1(0) − �
+W0,γ1(σ)
+�
+�
+W0,γ1(0)−1˜ω.
+(9.36)
+Let
+˜ω(σ) :=
+�
+�
+W0,γ1(0) − �
+W0,γ1(σ)
+�
+�
+W0,γ1(0)−1˜ω ∈ ¯H∞,3/2−
+b
+( ¯X),
+(9.37)
+and write
+˜ω(σ) = −σ(∂σ �
+W0,γ1(0)ωs0) + O ¯
+H∞,3/2−
+b
+( ¯
+X)(|σ|2)
+(9.38)
+where the notation O ¯
+H∞,3/2−
+b
+( ¯
+X)(|σ|2) means that the term has norm in ¯H∞,3/2−
+b
+( ¯X) of size O(|σ|2).
+Using the fact that �
+W0,γ1(σ)−1 is continuous in σ with value in Lop( ¯Hs−1+ǫ,ℓ+2+ǫ
+b
+( ¯X), ¯Hs−ǫ,ℓ−ǫ
+b
+( ¯X)),
+it follows that
+�
+�
+W0,γ1(σ)−1 − �
+W0,γ1(0)−1�
+˜ω = −σ�
+W0,γ1(0)−1∂σ �
+W0,γ1(0)ωs0 + o ¯
+Hs,ℓ
+b
+( ¯
+X)(|σ|)
+(9.39)
+for any s > 2, − 3
+2 < ℓ < − 1
+2. This proves the differentiability of W1 and thus W01 at (γ, σ) = (0, 0).
+As for W10, we proceed in a similar manner. We define
+W2(γ, σ) := W10(γ, σ) ⊕ W11(γ, σ) : ¯Hs−1,ℓ+2
+b
+( ¯X) → R⊥ ∼= C
+ω �→ ⟨�
+W0,γ(σ)�
+W0,γ1(σ)−1ω, ω∗
+s0⟩
+(9.40)
+We will show that W2(γ, σ) is continuous for γ + |σ| small and differentiable at (0, 0).
+As ω∗
+s0 = δ(r − 2m0)dr ∈ ˙H−1/2−,∞
+b
+( ¯X), we can rewrite
+⟨�
+W0,γ(σ)�
+W0,γ1(σ)−1ω, ω∗
+s0⟩ = ⟨�
+W0,γ1(σ)−1ω, �
+W0,γ(σ)∗ω∗
+s0⟩
+= ⟨�
+W0,γ1(σ)−1ω,
+��
+W0,γ(σ)∗ − �
+W0(0)∗�
+ω∗
+s0⟩.
+(9.41)
+
+LINEAR STABILITY OF KERR-NEWMAN FAMILY
+109
+Since �
+W0,γ1(σ)−1 is continuous in σ with value in Lop( ¯Hs−1,ℓ+2
+b
+( ¯X), ¯Hs−ǫ,ℓ−ǫ
+b
+( ¯X)), i.e.,
+�
+W0,γ1(σ)−1 = �
+W0,γ1(0)−1 + oL( ¯
+Hs−1,ℓ+2
+b
+( ¯
+X), ¯
+Hs−ǫ,ℓ−ǫ
+b
+( ¯
+X))(1),
+and �
+W0,γ(σ)∗ − �
+W0(0)∗ is a polynomial in γ, σ with coefficients in L( ˙H1−s,−ℓ−2
+b
+( ¯X), ˙H−s,−ℓ−1
+b
+( ¯X))
+�
+W0,γ(σ)∗ − �
+W0(0)∗
+= σ
+�
+∂σ(�
+W0,γ(σ)∗)|(0,0)
+�
++ γ
+�
+∂γ(�
+W0,γ(σ)∗)|(0,0)
+�
++ OL( ˙H1−s,−ℓ−2
+b
+( ¯
+X), ˙H−s,−ℓ−1
+b
+( ¯
+X))((γ + |σ|)2),
+it follows that
+⟨�
+W0,γ(σ)�
+W0,γ1(σ)−1ω, ω∗
+s0⟩
+= σ⟨�
+W0,γ1(0)−1ω, ∂σ(�
+W0,γ(σ)∗)|(0,0)ω∗
+s0⟩ + γ⟨�
+W0,γ1(0)−1ω, ∂γ(�
+W0,γ(σ)∗)|(0,0)ω∗
+s0⟩
++ o(γ + |σ|) · ∥ω∥ ¯
+Hs−1,ℓ+2
+b
+( ¯
+X).
+(9.42)
+This proves the differentiability of W10 at (0, 0).
+• Taylor expansion of W11. Since W11 : K = ⟨˜ω⟩ → R⊥ ∼= C is given by ⟨�
+W0,γ(σ)�
+W0,γ1(σ)−1˜ω, ω∗
+s0⟩,
+by (9.42) we find
+W11(γ, σ) = σ⟨∂σ �
+W0,γ(σ)|(0,0)ωs0, ω∗
+s0⟩ + γ⟨∂γ �
+W0,γ(σ)|(0,0)ωs0, ω∗
+s0⟩ + o(γ + |σ|).
+(9.43)
+Since ∂γ �
+W0,γ(σ)|(0,0) = γ−1��
+W0,γ(0) − �
+P0(0)
+�
+, using the calculation of W♭
+11 in (9.22) gives
+⟨∂γ �
+W0,γ(σ)|(0,0)ωs0, ω∗
+s0⟩ = 4π(b − 1).
+(9.44)
+We notice that ∂σ �
+W0,γ(σ)|(0,0) = ∂σ �
+P0(σ)|σ=0 = −i[P0, t0,∗], so we have
+⟨∂σ �
+P0,γ(σ)|(0,0)ωs0, ω∗
+s0⟩ = −i⟨[P0, t0,∗]ωs0, ω∗
+s0⟩ = −i⟨[P0, v]ωs0, ω∗
+s0⟩ = −2πi
+(9.45)
+where we use
+[P0, v]ωs0 = −(Ggδ∗
+gωs0)αβ(∇αv) + δgGg(∇αv ⊗s ωs0)
+=
+�
+(− 1
+2r2 + m0
+r3 )dv − 1
+r2 dr
+�
++
+� 1
+2r2 dv + 1
+r2 dr
+�
+= m0
+r3 dv.
+Therefore, we arrive at
+W11(γ, σ) = −2πiσ + 4π(b − 1)γ + o(γ + |σ|).
+(9.46)
+• Final step. Owing to the previous discussion starting at (9.28), in order to prove that �
+W0,γ(σ)�
+W0,γ1(σ)−1
+is injective, it suffices to prove the injectivity of the following operator
+W11(γ, σ) − W10(γ, σ)W00(γ, σ)−1W01(γ, σ) : K → R⊥.
+(9.47)
+Combining the differentiability of W01, W10 at (0, 0) and (9.46) together yields
+W11(γ, σ) − W10(γ, σ)W00(γ, σ)−1W01(γ, σ) = −2πiσ + 4π(b − 1)γ + o(γ + |σ|)
+which is nonzero for γ > 0, Im σ ≥ 0 with γ + |σ| small if we set b > 1, and thus is injective. Finally,
+the invertibility follows from the fact that �
+W0,γ(σ)�
+W0,γ1(σ)−1 has index 0 for γ + |σ| small.
+□
+Now we are at the position to prove Proposition 9.10.
+Proof of Proposition 9.10. The proof proceed in a similar fashion to that in the last step in the proof of
+Theorem 8.1. Specifically, by Lemma 9.12, there exists γ′
+0 > 0, C′ > 0 such that for 0 < γ < γ′
+0, Im σ ≥
+0, |σ| < C′ and s > 2, ℓ ∈ (− 3
+2, 1
+2),
+�
+W0,γ(σ) : {ω ∈ ¯Hs,ℓ
+b ( ¯X) : �
+W0,γ(σ)u ∈ ¯Hs−1,ℓ+2
+b
+( ¯X)} → ¯Hs−1,ℓ+2
+b
+( ¯X)
+is invertible. By Proposition 5.7, this implies for 0 < γ < γ′
+0, Im σ ≥ 0, |σ| < C′ and s > 2, ℓ < − 1
+2, s+ℓ > − 1
+2,
+�
+W0,γ(σ) : {ω ∈ ¯Hs,ℓ
+b ( ¯X) : �
+W0,γ(σ)u ∈ ¯Hs,ℓ+1
+b
+( ¯X)} → ¯Hs,ℓ+1
+b
+( ¯X)
+is injective as well, and the invertibility follows from the fact that �
+P0,γ(σ) has index 0 (see Lemma 5.25).
+
+110
+LILI HE
+Next, according to the high energy estimates (Theorem 5.24) and energy estimate (Proposition 5.27 and
+Theorem 5.28), there exists γ′′
+0 > 0, C′′ > 0 such that for 0 < γ < γ′′
+0 and Im σ ≥ 0, |σ| > C′′, �
+W0,γ(σ)
+defined as in (9.23) is invertible.
+For σ with Im σ ≥ 0, C′ ≤ |σ| ≤ C′′, the fact that �
+W0,0(σ) = �
+P0(σ) (see Proposition 9.3) is invertible and
+a perturbation argument (as in the final step in the proof of Theorem 8.1) give the invertibility of �
+W0,γ(σ) for
+(γ, σ) in an open neighborhood of (0, σ). Then the compactness of the region C′ ≤ |σ| ≤ C′′ implies that there
+exists an γ′′′
+0 > 0 (depending on C′, C′′) such that �
+W0,γ(σ) in (9.23) is invertible for Im σ ≥ 0, C′ ≤ |σ| ≤ C′′
+and 0 < γ < γ′′′
+0 . Therefore, the proposition holds for 0 < γ < ˜γ0 where ˜γ0 = min{γ′
+0, γ′′
+0 , γ′′′
+0 }.
+□
+Finally, having Propositions 9.9 and 9.10 at our disposal, we are able to establish the following theorem
+for �
+Wb,γ(σ).
+Theorem 9.13. Let b0 = (m0, 0, Q0) and b > 2. Fix γ ∈ (0, ˜γ0), there exists a small constant C(γ) > 0
+such that for the weakly charged RN metric gb0 with |Q0| < C(γ), the following holds for
+(1) If Im σ ≥ 0 and σ ̸= 0,
+�
+Wb0,γ(σ) : {ω ∈ ¯Hs,ℓ
+b ( ¯X; �
+scT ∗ ¯X) : �
+Wb0,γ(σ)ω ∈ ¯Hs,ℓ+1
+b
+( ¯X; �
+scT ∗ ¯X)} → ¯Hs,ℓ+1
+b
+( ¯X; �
+scT ∗ ¯X)
+(9.48)
+is invertible when s > 2, ℓ < − 1
+2, s + ℓ > − 1
+2.
+(2) If s > 2 and − 3
+2 < ℓ < − 1
+2, then stationary operator
+�
+Wb0,γ(0) : {ω ∈ ¯Hs,ℓ
+b ( ¯X; �
+scT ∗ ¯X) : �
+Wb0,γ(0)ω ∈ ¯Hs−1,ℓ+2
+b
+( ¯X; �
+scT ∗ ¯X)} → ¯Hs−1,ℓ+2
+b
+( ¯X; �
+scT ∗ ¯X)
+(9.49)
+is invertible.
+Both statements also hold for the weakly charged and slowly rotating Kerr Newman metric gb with b =
+(m, a, Q) near b0 = (m0, 0, Q0).
+Proof. The proof is completely analogous to that of Theorem 9.8.
+□
+9.3. Growing zero modes of the gauge potential wave operator. As discussed around (9.3) and (9.6),
+the zero mode solutions to Wb,γω = 0 on extendible and supported b-Sobolev spaces are related to the pure
+gauge zero mode solutions of Lgb,γ(˙g, ˙A) = 0 and dual pure gauge zero mode solutions of L∗
+gb,γ(˙g, ˙A) = 0,
+respectively. In this section we will discuss the zero mode solutions to Wb,γω = 0 which are allowed to
+grow at infinity and are asymptotic to translations and rotations. We will see in the next section that these
+growing zero modes may generate (dual) pure gauge zero modes solutions to L(∗)
+gb,γ(˙g, ˙A) = 0, which have the
+expected decay rate at infinity (more precisely, lie in the space ¯H∞,ℓ
+b
+( ¯X) or ˙H−∞,ℓ
+b
+( ¯X) with − 3
+2 < ℓ < − 1
+2).
+Throughout this subsection, we focus on the RN metric and slowly rotating metric with sufficiently small
+charge. Namely, for the parameter b0 = (m0, 0, Q0) of the RN metric, we assume that
+Q0 ≪ m0.
+We first discuss the growing zero modes which are asymptotic to translations.
+Proposition 9.14. For b = (m, a, Q) near b0 = (m0, 0, Q0), we obtain
+ker �
+Wb,γ(0) ∩ ¯H∞,−3/2−
+b
+( ¯X) = ⟨ωb,s0⟩ ⊕ {ωb,s1(S) : S ∈ S1},
+ωb,s0 = ∂♭
+t,
+(9.50)
+ker �
+Wb,γ(0) ∩ ˙H−∞,−3/2−
+b
+( ¯X) = ⟨ω∗
+b,s0⟩ ⊕ {ω∗
+b,s1(S) : S ∈ S1},
+(9.51)
+where ♭ denotes the musical isomorphism V ♭ := gb(V, •) and ωb,s1(S) and ω∗
+b,s1(S) depend on S ∈ S1 linearly.
+They satisfy
+δ∗
+gbωb,s1 ∈ ¯H∞,1/2−
+b
+( ¯X; S2 �
+scT ∗ ¯X),
+Ggbδ∗
+gbω∗
+b,• ∈ ˙H−∞,1/2−
+b
+( ¯X; S2 �
+scT ∗ ¯X)
+(9.52)
+Moreover, the maps b �→ ωb,s0, ωb,s1(S) and b �→ ω∗
+b,s0, ω∗
+b,s1(S) can be chosen to be continuous in b with values
+in the respective spaces.
+For later use, we further determine the leading term of ωb0,s1(S) and ω∗
+b0,s1(S)
+ωb0,s1(S) = dub0,s1(S)+O ¯
+H∞,−1/2−
+b
+( ¯
+X)(γ+|Q0|2),
+ω∗
+b0,s1(S) = du∗
+b0,s1(S)+O ˙H−∞,−1/2−
+b
+( ¯
+X)(γ+|Q0|2) (9.53)
+where ub0,s1(S) = (r − m0)S and u∗
+b0,s1(S) = (r − m0)H(r − rb0)S are defined as in (8.17).
+
+LINEAR STABILITY OF KERR-NEWMAN FAMILY
+111
+Proof. The proof exploits a normal operator argument as in Propositions 5.7 and 8.2. In particular, the
+normal operator of −2�
+Wb,γ(0) is equal to �
+□g,1(0) which is the Euclidean Laplacian ∆R3 = ρ2(ρ∂ρ(ρ∂ρ−1)+ /∆)
+tensored with 4 × 4 identity matrix when working in the standard coordinate trivialization of �
+scT ∗ ¯X and
+annihilates the following 1-forms
+dt, dx1, dx2, dx3.
+(9.54)
+We note that dt is of scalar type l = 0 while dxi = Sdr + rdS with S = xi/r (i = 1, 2, 3) is of scalar type
+l = 1.
+We first prove (9.50). Let u ∈ ker �
+Wb,γ(0) ∩ ˙H−∞,−3/2−
+b
+( ¯X). By the normal operator argument as in
+Proposition 5.7 ( where we shift the contour of integration through the pole 0 with the space of resonant states
+given by S0), ω must take the form ω = χω0+˜ω where ω0 is in the 4-dimensional space span{dt, dx1, dx2, dx3},
+˜ω ∈ ¯H∞,−1/2−
+b
+( ¯X) and χ is a smooth cutoff with χ = 0 for r ≤ 3m0, χ = 1 with r ≥ 4m0. This proves that
+the space in (9.50) is at most 4-dimensional. Now we prove that it is indeed 4-dimensional. We compute for
+ω0 ∈ {dt, dx1, dx2, dx3}
+−�
+Wb,γ(0)χω0 = −(�
+Wb,γ(0) − □g,1)χω0 − [□g,1, χ]ω0 ∈ ¯H∞,3/2−
+b
+( ¯
+X) + ¯H∞,∞
+b
+( ¯X) = ¯H∞,3/2−
+b
+( ¯X).
+In view of Theorem 9.13, there exists a unique ˜ω ∈ ¯H∞,−1/2−
+b
+( ¯X) satisfying �
+Wb,γ(0)˜ω = −�
+Wb,γ(0)χω0.
+Therefore, χω0+˜ω gives rise to an element in the space (9.50) which we denote by ωb,s0 if ω0 = dt and ωb,s1(S)
+if ω0 = d(rS) ∈ {dx1, dx2, dx3}. The explicit expression for ωb,s0 follows from the direction calculation
+�
+Wb,γ∂♭
+t = δgb,γGgδ∗
+g∂♭
+t = 0 since ∂t is killing (i.e. δ∗
+g∂♭
+t = 0). As for the symmetric gradient, we calculate
+δ∗
+gbωb,s1 = δgb ˜ω + (δ∗
+gb − δ∗
+g)χω0 + [δ∗
+g, χ]ω0 ∈ ¯H∞,1/2−
+b
+( ¯X) + ¯H∞,∞
+b
+( ¯X) = ¯H∞,1/2−
+b
+( ¯
+X)
+where we use δ∗
+gb − δ∗
+g ∈ ρ2Diff1
+b, δ∗
+gb ∈ ρDiff1
+b and δ∗
+gω0 = 0.
+Now let us turn to the continuous dependence of ωb,s1(S) on b. Suppose bj → b and ωbj,s1(S) = χω0+˜ωbj(S)
+where ω0 = d(rS) ∈ {dx1, dx2, dx3} and ˜ωbj(S) is uniquely determined by bj and S. Let ωb,s1(S) = χω0+˜ωb(S)
+and ej,s1(S) = ωbj,s1(S) − ωb,s1(S) = ˜ωbj,s1(S) − ˜ωb,s1(S), we find
+�
+Wb,γ(0)ej,s1(S) = (�
+Wb,γ(0) − �
+Wbj,γ(0))(˜ωbj,s1(S) + χω0).
+Since {˜ωbj,s1(S)} is uniformly bounded in ¯H∞,−1/2−
+b
+( ¯X) and �
+Wb,γ(0) − �
+Wbj,γ(0) ∈ (b − bj)ρ3Diff2
+b, we see
+that �
+Wb,γ(0)ej,s1(S) ∈ (b − bj) ¯H∞,3/2−
+b
+, and thus ej,s1(S) ∈ ¯H∞,−1/2−
+b
+( ¯X) is of size O(b − bj). This proves
+the continuous dependence of ωb,s1(S) on b.
+The proof for the dual zero modes ω∗
+b,s0 and ω∗
+b,s1(S )proceeds in an analogous manner (we use Theorem
+9.8 because �
+Wb,γ(0) acting on supported b-Sobolev space is identical to �
+Pb,γ(0)∗).
+It remains to derive the expressions in (9.53). Since
+�
+Wb0,γ(0) = δgb0 Ggb0 δ∗
+gb0 + Fgb0 Ggb0 δ∗
+gb0 = 1
+2(dδgb0 + δgb0 d) − Ric(gb0) + Fgb0 Ggb0 δ∗
+gb0 ,
+we have
+�
+Wb0,γ(0)(ωb0,s1(S) − dub,s1(S)) = Ric(g) β
+α (dub0,s1(S))β − Fgb0 Ggb0 δ∗
+gb0 dub0,s1(S)
+= O ¯
+H∞,5/2−
+b
+(|Q0|2) + O ¯
+H∞,∞
+b
+( ¯
+X)(γ).
+(9.55)
+Since �
+Wb0,γ(0)(ωb0,s1(S) − dub,s1(S)) is of scalar type l = 1 and �
+Wb0,γ(0)−1 restricted in scalar type l =
+1 1-form spaces exists with norm of size O(1) (see Remark 9.4), it follows that ωb0,s1(S) − dub,s1(S) =
+O ¯
+H∞,−1/2−
+b
+( ¯
+X)(γ + |Q0|2).
+□
+Next we turn to the growing zero modes asymptotic to rotations.
+Proposition 9.15. For b = (m, a, Q) near b0 = (m0, 0, Q0), there exists continuous (in b) families
+b �→ ωb,v1(V) ∈ ker �
+Wb,γ(0) ∩ ¯H∞,−5/2−
+b
+( ¯X),
+(9.56)
+b �→ ω∗
+b,v1(V) ∈ ker �
+Wb,γ(0) ∩ ˙H−∞,−5/2−
+b
+( ¯X),
+(9.57)
+depending linearly on V ∈ V1, such that
+δ∗
+gbωb,v1(V) ∈ ¯H∞,1/2−
+b
+( ¯X; S2 �
+scT ∗ ¯X),
+Ggbδ∗
+gbω∗
+b,v1(V) ∈ ˙H−∞,1/2−
+b
+( ¯X; S2 �
+scT ∗ ¯X).
+(9.58)
+
+112
+LILI HE
+More explicitly, we have
+ωb0,v1(V) = r2V,
+ω∗
+b0,v1(V) = r2H(r − rb0)V + O ˙H−∞,−1/2−
+b
+( ¯
+X)(γ)
+(9.59)
+Proof. We first notice that, with respect to RN metric, r2V ∈ ¯H∞,−5/2−
+b
+( ¯X) is dual to the rotation vector
+field (see the calculation in (6.3)), and thus killing. Namely, we have δ∗
+gb0 r2V = 0 and thus �
+Wb0,γ(0)r2V =
+δgb0,γGgb0 δ∗
+gb0 r2V = 0. As for KN metric gb with b = (m, a, Q) near b0 = (m0, 0, Q0), a direct calculation
+implies
+−�
+Wb,γ(0)χr2V = −
+�
+�
+Wb,γ(0)−�
+Wb′,γ(0)
+�
+χr2V−[�
+Wb′,γ(0), χ]r2V ∈ ¯H∞,3/2−
+b
+( ¯X)+ ¯H∞,∞
+b
+( ¯X) = ¯H∞,3/2−
+b
+( ¯X).
+where b′ = (m, 0, Q) and we use �
+Wb,γ(0)−�
+Wb′,γ(0) ∈ ρ4Diff2
+b. In view of Theorem 9.13, there exists a unique
+˜ωb(V) ∈ ¯H∞,−1/2−
+b
+( ¯X) satisfying �
+Wb,γ(0)˜ωb(V) = −�
+Wb,γ(0)χr2V. We then define ωb,v1(V) := χr2V + ˜ωb(V).
+For b = (m, a, Q), the symmetric gradient δgbωb,v1(V) is given by
+δ∗
+gbωb,v1(V) = δ∗
+gb ˜ωb(V) + (δ∗
+gb−δ∗
+g(m,0,Q))χr2V + [δ∗
+g(m,0,Q), χ]r2V ∈ ¯H∞,1/2−
+b
+( ¯
+X) + ¯H∞,∞
+b
+( ¯X) = ¯H∞,1/2−
+b
+( ¯X)
+where we use δ∗
+gb − δ∗
+g(m,0,Q) ∈ ρ3Diff1
+b, δ∗
+gb ∈ ρDiff1
+b and δ∗
+g(m,0,Q)r2V = 0.
+The proof of the continuous dependence of ωb,v1 on b proceeds as in Proposition 9.14. Suppose bj =
+(mj, aj, Qj) → b = (m, a, Q) and let ωbj,v1(V) = χr2V + ˜ωbj(V), ωb,v1(V) = χr2V + ˜ωb(V) and ej(V) =
+ωbj,v1(V) − ωb,v1(V) = ˜ωbj(V) − ˜ωb(V), we find
+�
+Wb,γ(0)ej(V) = (�
+Wb,γ(0) − �
+Wbj,γ(0))(˜ωbj(V) + χr2V)
+= |b − bj|ρ3Diff2
+b ˜ωbj(V) +
+���
+Wb,γ(0) − �
+Wb′,γ(0)
+�
+−
+��
+Wbj,γ(0) − �
+Wb′
+j,γ(0)
+��
+χr2V
++
+�
+�
+[Wb′,γ(0) − �
+Wb′
+j,γ(0), χ]r2V
+�
+= |b − bj|ρ3Diff2
+b ˜ωbj(V) + |b − bj|ρ4Diff2
+bχr2V + |b − bj|ρ∞Diff1
+br2V
+∈ |b − bj|
+�
+¯H∞,5/2−
+b
+( ¯X) + ¯H∞,3/2−
+b
+( ¯
+X) + ¯H∞,∞
+b
+( ¯X)
+�
+⊂ |b − bj| ¯H∞,3/2−
+b
+( ¯X)
+where b′
+j = (mj, 0, Qj) and b′ = (m, 0, Q). As a consequence, ej(V) ∈ ¯H∞,−1/2−
+b
+( ¯X) is of size O(b − bj).
+This proves the continuous dependence of ωb,v1(V) on b.
+The statements for dual zero modes ω∗
+b,v1(V) can be proved in a similar way (we use Theorem 9.8 because
+�
+Wb,γ(0) acting on supported b-Sobolev space is identical to �
+Pb,γ(0)∗). Here, we present the detail of the
+derivation of the expression for ω∗
+b0,v1(V) in (9.59). First, we calculate
+δgb0 Ggb0 δ∗
+gb0 r2H(r − rb0)V = δgb0
+�
+(r2δ(r − rb0)dr) ⊗s V
+�
+= − 1
+2r2 ∂r(r4δ(r − rb0)µb0)V = 0.
+It follows that
+�
+Wb0,γ(0)(ω∗
+b0,v1(V) − r2H(r − rb0)V) = −Fgb0
+�
+(r2δ(r − rb0)dr) ⊗s V
+�
+= O ¯
+H−∞,∞
+b
+( ¯
+X)(γ).
+Since �
+Wb0,γ(0)(ωb0,v1(V) − r2H(r − rb0)V) is of vector type l = 1 and �
+Wb0,γ(0)−1 restricted in vector type
+l = 1 1-form spaces exists with norm of size O(1) (see Remark 9.4), we conclude that
+�
+Wb0,γ(0)(ωb0,v1(V) − r2H(r − rb0)V) = O ¯
+H−∞,−1/2−
+b
+( ¯
+X)(γ).
+□
+9.4. Generalized zero modes of the gauge potential wave operator. Apart from the growing modes,
+we also need to take the generalized zero modes into consideration (the zero modes which allow for polynomial
+growth in tb,∗). Before the discussion about the generalized zero modes, we establish the following technical
+lemma about another family of growing zero modes.
+Lemma 9.16. For b = (m, a, Q) near b0 = (m0, 0, Q0), there exists continuous (in b) families
+b �→ ω(1)
+b,s1(S) ∈ ker �
+Wb,γ(0) ∩ ¯H∞,−5/2−
+b
+( ¯X),
+(9.60)
+b �→ ω(1)∗
+b,s1 (S) ∈ ker �
+Wb,γ(0) ∩ ˙H−∞,−5/2−
+b
+( ¯X),
+(9.61)
+
+LINEAR STABILITY OF KERR-NEWMAN FAMILY
+113
+depending linearly on S ∈ S1, such that
+δ∗
+gbω(1)
+b,s1(S) = χdt ⊗s d(rS) + ¯H∞,−1/2−
+b
+( ¯
+X; S2 �
+scT ∗ ¯X)
+δ∗
+gbω(1)∗
+b,s1 (S) = χdt ⊗s d(rS) + ˙H∞,−1/2−
+b
+( ¯
+X; S2 �
+scT ∗ ¯X)
+(9.62)
+Moreover, the leading terms of ω(1)
+b,s1(S), ω(1)∗
+b,s1 (S)are given by
+ω(1)
+b,s1(S) − χrSdt ∈ ¯H∞,−3/2−
+b
+( ¯X),
+ω(1)∗
+b,s1 (S) − χrSdt ∈ ˙H−∞,−3/2−
+b
+( ¯
+X)
+(9.63)
+where χ is a smooth cutoff satisfying χ = 1 for r ≥ 4m0 and χ = 0 for r ≤ 3m0.
+For later use, we further determine the leading term of ω(1)
+b0,s1(S)
+ω(1)
+b0,s1(S) = rS(µb0dt0 − dr) + O ¯
+H∞,−1/2−
+b
+( ¯
+X)(γ + |Q0|2).
+(9.64)
+Proof. The proof uses the fact that the normal operator of −2�
+Wb,γ(0) is equal to �
+□g,1(0), which is the
+Euclidean Laplacian ∆R3 = ρ2(ρ∂ρ(ρ∂ρ − 1) + /∆) tensored with 4 × 4 identity matrix when working in the
+standard coordinate trivialization of �
+scT ∗ ¯X and annihilates the following 1-form xidt = rSdt (i = 1, 2.3)
+where S = xi/r ∈ S1. Concretely, we have
+−�
+Wb,γ(0)χrSdt = −
+�
+�
+Wb,γ(0) − □g,1
+�
+χrSdt − [□g,1, χ]rSdt ∈ ¯H∞,1/2−
+b
+( ¯X) + ¯H∞,∞
+b
+( ¯X) = ¯H∞,1/2−
+b
+( ¯X).
+Since ker �
+Wb,γ(0)∗ ∩ ˙H−∞,−1/2+
+b
+( ¯X) = {0} by Theorem 9.13, the above equation is solvable. More pre-
+cisely, there exists a unique ˜ω(1)
+b (S) ∈ ¯H∞,−3/2−
+b
+( ¯X) which is in ann{f1, · · · , f4}, where {f1, · · · , f4} ⊂
+C∞
+c (X◦) is a set of linearly independent functionals on �
+Wb0,γ(0)∩ ¯H∞,−3/2−
+b
+( ¯X), such that �
+Wb,γ(0)˜ω(1)
+b (S) =
+−�
+Wb,γ(0)χrSdt. We then define ω(1)
+b,s1(S) := χrSdt + ˜ω(1)
+b (S) . As for the symmetric gradient δgbω(1)
+b,s1(S), we
+find
+δ∗
+gbω(1)
+b,s1(S) = χδ∗
+grSdt + [δ∗
+g, χ]rSdt + (δ∗
+gb−δ∗
+g)χrSdt + δ∗
+gb ˜ω(1)
+b (S)
+= χdt ⊗s d(rS) + ¯H∞,∞
+b
+( ¯
+X) + ¯H∞,−1/2−
+b
+( ¯X) + ¯H∞,−1/2−
+b
+( ¯X)
+= χdt ⊗s d(rS) + ¯H∞,−1/2−
+b
+( ¯X)
+where we use δ∗
+gb − δ∗
+g ∈ ρ2Diff1
+b and δ∗
+gb ∈ ρDiff1
+b.
+As for the continuous dependence of ω(1)
+b,s1(S) on b, suppose bj → b and let ω(1)
+bj,s1(S) = χrSdt + ˜ω(1)
+bj (S),
+ω(1)
+b,s1(S) = χrSdt + ˜ω(1)
+b (S) and ej(S) = ω(1)
+bj,s1(S) − ω(1)
+b,s1(S) = ˜ω(1)
+bj (S) − ˜ω(1)
+b (S), we find
+�
+Wb,γ(0)ej(S) = (�
+Wb,γ(0) − �
+Wbj,γ(0))(˜ω(1)
+bj (S) + χrSdt) ∈ |b − bj| ¯H∞,1/2−
+b
+( ¯X).
+As �
+Wb,γ(0)|ann{f1,··· ,f4} : ann{f1, · · · , f4} →
+¯H∞,1/2−
+b
+( ¯
+X) is invertible, it follows that ej(S) is of size
+O ¯
+H∞,−3/2−
+b
+( ¯
+X)(b − bj). This completes the proof of the continuity.
+The proofs for dual zero modes ˆω(1)∗
+b,s1 (S) are analogous where we use
+ker �
+Wb,γ(0)∗ ∩ ¯H∞,−1/2+
+b
+( ¯X) = ker �
+Pb,γ(0) ∩ ¯H∞,−1/2+
+b
+( ¯X) = {0}.
+It remains to derive the expressions in (9.64). Since
+δgb0 Ggb0 δgb0
+�
+rS(µb0dt0 − dr)
+�
+= Q2
+0
+r3 S(µb0dt0 − dr) ∈ Q2
+0 ¯H∞,3/2−
+b
+( ¯X),
+we have
+�
+Wb0,γ(0)
+�
+ω(1)
+b0,s1(S) − rS(µb0dt0 − dr)
+�
+= O ¯
+H∞,3/2−
+b
+(|Q0|2) + O ¯
+H∞,∞
+b
+( ¯
+X)(γ).
+Since the right-hand side of the above equation is of scalar type l = 1 and �
+Wb0,γ(0)−1 restricted in scalar type
+l = 1 1-form spaces exists with norm of size O(1) (see Remark 9.4), it follows that ω(1)
+b0,s1(S)−rS(µb0dt0−dr) =
+O ¯
+H∞,−1/2−
+b
+( ¯
+X)(γ + |Q0|2).
+□
+
+114
+LILI HE
+Now we are ready to discuss (dual) generalized zero modes which have less decay rate in r at infinity
+for fixed tb,∗ and grow linearly in tb,∗. In the course of the proof, we will find that they are asymptotic to
+Lorentz boosts.
+Proposition 9.17. For b = (m, a, Q) near b0 = (m0, 0, Q0), there exists continuous (in b) families
+b �→ ˆωb,s1(S) ∈ ker Wb,γ ∩ Poly1(tb,∗) ¯H∞,−5/2−
+b
+( ¯X),
+(9.65)
+b �→ ˆω∗
+b,s1(S) ∈ ker Wb,γ ∩ Poly1(tb,∗) ˙H−∞,−5/2−
+b
+( ¯X),
+(9.66)
+depending linearly on S ∈ S1, such that
+δ∗
+gb ˆωb,s1(S) ∈ Poly1(tb,∗) ¯H∞,−1/2−
+b
+( ¯X; S2 �
+scT ∗ ¯X),
+(9.67)
+Ggbδ∗
+gb ˆω∗
+b,s1(S) ∈ Poly1(tb,∗) ˙H−∞,−1/2−
+b
+( ¯X; S2 �
+scT ∗ ¯X).
+(9.68)
+Moreover, the leading terms of ˆωb,s1(S) are given by
+ˆω(1)
+b,s1(S) − (tdxi − xidt) ∈ Poly1(tb,∗) ¯H∞,−3/2−
+b
+( ¯X),
+r ≫ 1
+(9.69)
+where S = xi/r.
+For later use, we further determine the leading term of ˆωb0,s1(S)
+ˆωb0,s1(S) = (t0 − r)ωb0,s1(S) − rS(µb0dt0 − dr) + m0S(dt0 + dr)
++ d
+�
+2m0
+�
+2m0 + (m0 − r) log( r
+m0
+)
+�
+S
+�
++ O ¯
+H∞,−1/2−
+b
+( ¯
+X)(γ + |Q0|2).
+(9.70)
+Proof. We make the ansatz
+ˆωb,s1(S) = tχ0ωb,s1(S) + ˇωb,s1(S)
+(9.71)
+with ˇωb,s1(S) ∈ ¯H∞,−5/2−
+b
+( ¯X) to be determined. Our aim is to find
+ˆωb,s1 ∈ ker Wb,γ ∩Poly1(tb,∗) ¯H∞,5/2−
+b
+( ¯X)
+such that δ∗
+gb ˆωb,s1(S) ∈ Poly1(tb,∗) ¯H∞,−1/2−
+b
+( ¯X). To this end, since
+tχ0 − tb∗ ∈ A−1( ¯X)
+and
+δ∗
+gbωb,s1 ∈ ¯H∞,1/2−
+b
+( ¯
+X),
+it suffices to solve the following two equations for ˇωb,s1(S) ∈ ¯H∞,−5/2−
+b
+( ¯X)
+[δ∗
+gb, tχ0]ωb,s1(S) + δ∗
+gb ˇωb,s1(S) ∈ ¯H∞,−1/2−
+b
+( ¯
+X),
+(9.72)
+�
+Wb,γ(0)ˇωb,s1(S) = −[Wb,γ, tχ0]ωb,s1(S).
+(9.73)
+We first analyze (9.72). Since
+[δ∗
+gb, tχ0]ωb,s1(S) = dtχ0 ⊗s ωb,s1(S),
+we further make the ansatz
+ˇωb,s1(S) = −ω(1)
+b,s1(S) + ˜ωb,s1(S) ∈ ¯H∞,−5/2−
+b
+( ¯X).
+(9.74)
+with ˜ωb,s1(S) ∈ ¯H∞,−3/2−
+b
+( ¯X) to be determined. Therefore, using ωb,s1(S) = χd(rS) + ¯H∞,−1/2−
+b
+( ¯X) and
+(9.62), it follows that
+[δ∗
+gb, tχ0]ωb,s1(S) + δ∗
+gb ˇωb,s1 ∈ ¯H∞,−1/2−
+b
+( ¯X)
+and the requirement (9.72) is satisfied with the choice of ˇωb,s1(S) defined as in (9.74).
+Next we solve (9.73) to determine ˜ωb,s1(S) in (9.74). We rewrite (9.73) as
+�
+Wb,γ(0)˜ωb,s1(S) = �
+Wb,γ(0)ω(1)
+b,s1(S) − [Wb,γ, tχ0]ωb,s1(S)
+= ˜δgb,γGgb
+�
+δ∗
+gbω(1)
+b,s1(S) − [δ∗
+gb, tχ0]ωb,s1(S)
+�
+− [˜δgb,γ, tχ0]Ggbδ∗
+gbωb,s1(S) ∈ ¯H∞,1/2−
+b
+( ¯X)
+(9.75)
+where we use ˜δgb,γ ∈ ρDiff1
+b, δ∗
+gbω(1)
+b,s1(S) − [δ∗
+gb, t|chi0]ωb,s1(S) ∈ ¯H∞,−1/2−
+b
+( ¯X) and
+δ∗
+gbωb,s1(S) ∈ ¯H∞,1/2−
+b
+( ¯X),
+[˜δgb,γ, tχ0] ∈ A0( ¯X).
+
+LINEAR STABILITY OF KERR-NEWMAN FAMILY
+115
+Since ker �
+Wb,γ(0)∗∩ ˙H−∞,−1/2+
+b
+( ¯X) = {0}, we can solve the above equation (9.75) for ˜ωb,s1(S) ∈ ¯H∞,−3/2−
+b
+( ¯X),
+which is unique modulo ker �
+Wb,γ(0) ∩ ¯H∞,−3/2−
+b
+( ¯X). We then define
+ˆω(1)
+b,s1(S) := tχ0ωb,s1(S) − ω(1)
+b,s1(S) + ˜ωb,s1(S)
+where ˜ωb,s1(S) is in the orthogonal complement
+�
+ker �
+Wb,γ(0)∩ ¯H∞,−3/2−
+b
+( ¯X)
+�⊥ and thus uniquely determined
+by S and b. As a result, we have for r ≫ 1
+ˆωb,s1(S) = tωb,s1(S) − χrSdt + ¯H∞,−3/2
+b
+( ¯X) = td(rS) − rSdt + Poly1(tb,∗) ¯H∞,−3/2−
+b
+( ¯X).
+Finally, the continuity in b follows in the same way as in the proof of Lemma 9.16.
+The proof for ˆω∗
+b,s1(S) is completely analogous.
+It remains to derive the expressions in (9.70). For the simplicity of calculation, now we let ˜t = t0 − r and
+make the ansatz
+ˆωb0,s1(S) = ˜tωb0,s1(S) − ω(1)
+b0,s1 + ˜ωb0,s1(S).
+Our aim is to solve the following equation for ˜ωb0,s1(S) ∈ ¯H−3/2−
+b
+( ¯X)
+�
+Wb0,γ(0)˜ωb0,s1(S) = −[Wb0,γ, ˜t ]ωb0,s1(S) = 1
+2[□gb0 , ˜t ]ωb0,s1(S) + O ¯
+H∞,∞
+b
+( ¯
+X)(γ)
+= S
+�
+(m0
+r2 − Q2
+0
+r3 )dt0 + Q2
+0
+r3 dr
+�
++ m0
+r (1 + m0
+r
+− Q2
+0
+r2 )/dS + O ¯
+H∞,3/2−
+b
+( ¯
+X)(γ + |Q0|2)
+(9.76)
+where we use [□gb0 , ˜t ] = □gb0 ,0˜t+ 2∇α˜t∇α ∈ ρ2Diff1
+b and (9.53) in the last step. A direct calculation implies
+�
+Wb0,γ(0)m0(dt0 + dr) = δgb0 Ggb0 δ∗
+gb0 m0(dt0 + dr) + O ¯
+H∞,∞
+b
+( ¯
+X)(γ)
+= S
+�
+(m0
+r2 + m0Q2
+0
+r4
+)dt0+(3m0
+r2
+− 2m2
+0
+r3 + m0Q2
+0
+r4
+)dr
+�
++ m0
+r (−2+ 2m0
+r
+− 2Q2
+0
+r2 )/dS +O ¯
+H∞,∞
+b
+( ¯
+X)(γ).
+Then
+�
+Wb0,γ(0)
+�
+˜ωb0,s1(S) − m0(dt0 + dr)
+�
+= S(−3m0
+r2
++ 2m2
+0
+r3 )dr + m0
+r (3 − m0
+r )/dS + O ¯
+H∞,3/2−
+b
+( ¯
+X)(γ + |Q0|2)
+= d
+�
+(3m0
+r
+− m2
+0
+r2 )S
+�
++ O ¯
+H∞,3/2−
+b
+( ¯
+X)(γ + |Q0|2)
+Meanwhile, with u = 2m0(2m0 + (m0 − r) log(r/m0)), we have
+�
+Wb0,γ(0)du = 1
+2dδgb0 du − Ric(g0) β
+α (du)β + Fgb0 Ggb0 du
+= d
+�
+(3m0
+r
+− m2
+0
+r2 )S
+�
++ O ¯
+H∞,5/2−
+b
+( ¯
+X)(γ + |Q0|2).
+Therefore, we obtain
+�
+Wb0,γ(0)
+�
+˜ωb0,s1(S) − m0(dt0 + dr) − du
+�
+= O ¯
+H∞,3/2−
+b
+( ¯
+X)(|Q0|2 + γ).
+Since the right-hand side of the above equation is of scalar type l = 1 and �
+Wb0,γ(0)−1 restricted in scalar type
+l = 1 1-form spaces exists with norm of size O(1) (see Remark 9.4), it follows that ˜ωb0,s1(S)−m0(dt0 +dr)−
+du = O ¯
+H∞,−1/2−
+b
+( ¯
+X)(γ + |Q0|2). Therefore, (9.70) follows from the combination of this fact and (9.64).
+□
+10. Modes of the linearized gauge-fixed Einstein-Maxwell system
+In this section, we will discuss the modes of the linearized gauge-fixed Einstein-Maxwell system
+L(gb,Ab),γ(˙g, ˙A) = (2LE
+(gb,Ab),γ(˙g, ˙A), LM
+(gb,Ab),γ(˙g, ˙A)) = 0
+(10.1)
+where
+LE
+(gb,Ab),γ(˙g, ˙A) = DgbRic(˙g) − 2D(gb,dAb)T (˙g, d ˙A) + �δ∗
+gb,γ�δgb,γGgb ˙g = 0,
+(10.2)
+LM
+(gb,Ab),γ(˙g, ˙A) = D(gb,Ab)(δgdA)(˙g, ˙A) + dδgb ˙A = 0.
+(10.3)
+
+116
+LILI HE
+on the RN metric and 4-electromagnetic potential, i.e., the case b = b0 = (m0, 0, Q0), |Q0| ≪ m0.
+For the brevity of notation, we rewrite the Lie derivative of a 1-form ω with respect to the vector field X
+as
+�LωX := LXω
+and put
+Lb,γ := L(gb,Ab),γ,
+LE
+b,γ := LE
+(gb,Ab),γ,
+LM
+b,γ := LM
+(gb,Ab),γ.
+(10.4)
+For the various spaces in this section, for instance, ¯Hs,ℓ
+b ( ¯X; E), ˙Hs,ℓ
+b ( ¯X; E), A( ¯
+X; E), etc., where E → ¯X is
+a vector bundle over ¯X, we will drop the notation E of the bundle if it it clear from the context.
+We shall prove that Lb0,γ has no non-zero modes in the closed upper half plane for RN metric and 4-
+electromagnetic potential (gb0, Ab0) (we note that the same result for slowly rotating KN is nontrivial and will
+be discussed in detail in next section). Moreover, we will describe the space of zero modes and generalized
+zero modes (with linearly growth in tb,∗) of Lb,γ for both RN case and slowly rotating KN case.
+10.1. Modes in the closed upper half plane. Let (gb, Ab) be the KN metric and 4-electromagnetic
+potential with b = (m, a, Q) near b0 = (m0, 0, Q0) and the linearized gauge-fixed Einstein-Maxwell operator
+Lb,γ : ¯Hs,ℓ
+b ( ¯X; S2 �
+scT ∗ ¯X ⊕ �
+scT ∗ ¯X) → ¯Hs,ℓ
+b ( ¯X; S2 �
+scT ∗ ¯X ⊕ �
+scT ∗ ¯X) be defined as in (10.4), (10.1), (10.2)
+and (10.3). In the definition of ˜δ∗
+gb,γ, ˜δgb,γ(see (4.62), (4.60) and (4.61)), c is defined as in (9.7) and γ ∈
+(0, min{γ0, ˜γ0}) where γ0, ˜γ0 are as in Theorem 9.8 and 9.13 respectively. We also note that we consider the
+weakly charged case, namely, |Q0| ≪ m0 (more precisely, |Q0| ≤ C(γ) for some small constant C(γ)). As in
+the previous sections, we define the Fourier-transformed operator
+�
+Lb,γ(σ) := eiσtb,∗Lb,γe−iσtb,∗
+(10.5)
+using the function tb,∗ = χ0(r)(t + rb0,∗) + (1 − χ0(r))(t − r(m,0,Q),∗) defined in (4.29).
+Theorem 10.1. Let (gb0, Ab0) be the RN metric and 4-electromagnetic potential where b0 = (m0, 0, Q0)
+with |Q0| ≪ m0. Then the Fourier-transformed operator �
+Lb0,γ(σ) satisfies the following statements:
+(1) If Im σ ≥ 0 and σ ̸= 0,
+�
+Lb0,γ(σ) : {(˙g, ˙A) ∈ ¯Hs,ℓ
+b ( ¯X; S2 �
+scT ∗ ¯X ⊕ �
+scT ∗ ¯X) : �
+Lb0,γ(σ)(˙g, ˙A) ∈ ¯Hs,ℓ+1
+b
+( ¯X; S2 �
+scT ∗ ¯X ⊕ �
+scT ∗ ¯X)}
+→ ¯Hs,ℓ+1
+b
+( ¯X; S2 �
+scT ∗ ¯X ⊕ �
+scT ∗ ¯X)
+(10.6)
+is invertible for s > 3, ℓ < − 1
+2 and s + ℓ > − 1
+2.
+(2) If s > 3 and − 3
+2 < ℓ < − 1
+2, then stationary operator
+�
+Lb0,γ(0) : {(˙g, ˙A) ∈ ¯Hs,ℓ
+b ( ¯X; S2 �
+scT ∗ ¯X ⊕ �
+scT ∗ ¯X) : �
+Lb0,γ(0)(˙g, ˙A) ∈ ¯Hs−1,ℓ+2
+b
+( ¯X; S2 �
+scT ∗ ¯X ⊕ �
+scT ∗ ¯X)}
+→ ¯Hs−1,ℓ+2
+b
+( ¯X; S2 �
+scT ∗ ¯X ⊕ �
+scT ∗ ¯X)
+(10.7)
+has 8-dimensional kernel and cokernel.
+The second statement also holds for the weakly charged and slowly rotating KN metric and 4-electromagnetic
+potential (gb, Ab) with b = (m, a, Q) near b0 = (m0, 0, Q0). Concretely, we have
+Kb := ker �
+Lb,γ(0) ∩ ¯H∞,−1/2−
+b
+( ¯X) = {(˙gb,s0, ˙Ab,s0)} ⊕ {(˙gb,s1(S), ˙Ab,s1(S)) : S ∈ S1}
+(10.8)
+⊕ {(˙gb,v1(V), ˙Ab,v1(V)) : V ∈ V1},
+K∗
+b := ker �
+Lb,γ(0)∗ ∩ ˙H−∞,−1/2−
+b
+( ¯X) = {(˙g∗
+b,s0, ˙A∗
+b,s0)} ⊕ {(˙g∗
+b,s1(S), ˙A∗
+b,s1(S)) : S ∈ S1}
+(10.9)
+⊕ {(˙g∗
+b,v1(V), ˙A∗
+b,v1(V)) : V ∈ V1},
+where
+(˙gb,s0, ˙Ab,s0) = (˙gb( ˙m, 0, ˙Q) + 2δgb ˜ωb,s0,
+˙Ab(0, 0, ˙Q) + �LAb ˜ω♯
+b,s0 + dφb,s0),
+˙m, ˙Q ∈ R,
+(10.10)
+(˙gb,s1(S), ˙Ab,s1(S)) = (2δ∗
+gbωb,s1(S), �LAb(ωb,s1(S))♯ + dφb,s1(S)),
+(10.11)
+(˙gb,v1(V), ˙A∗
+b,v1(V)) = (˙gb(0, ˙a, 0) + 2δ∗
+gb ˜ωb,v1(V),
+˙Ab(0, ˙a, 0) + �LAb(˜ωb,v1(V))♯ + dφb,v1(V)),
+(10.12)
+
+LINEAR STABILITY OF KERR-NEWMAN FAMILY
+117
+and
+(˙g∗
+b,s0, ˙A∗
+b,s0) = c1(2Ggbδ∗
+gbω∗
+b,s0, 4 �LAb(ω∗
+b,s0)♯ + dφ∗
+b,s0) + c2(0, δ(r − rb)dr),
+c1, c2 ∈ R,
+(10.13)
+(˙g∗
+b,s1(S), ˙A∗
+b,s1(S)) = (2Ggbδ∗
+gbω∗
+b,s1(S), 4 �LAb(ω∗
+b,s1(S))♯ + dφ∗
+b,s1(S)),
+(10.14)
+(˙g∗
+b,v1(V), ˙A∗
+b,v1(V)) = (2Ggbδ∗
+gbω∗
+b,v1(V), 4 �LAb(ω∗
+b,v1(V))♯ + dφ∗
+b,v1(V))
+(10.15)
+with ωb,s1(S), ω∗
+b,s0 and ω∗
+b,s1(S), ω∗
+b,v1(V) defined as in (9.50), (9.51) and (9.57).
+Moreover, we have
+(˙gb,s1(S),
+˙Ab,s1(S)),
+(˙gb,v1(V),
+˙Ab,v1(V)) ∈ ¯H∞,1/2−
+b
+( ¯X),
+and the maps b �→ (˙gb,•(•),
+˙Ab,•(•)) and b �→ (˙g∗
+b,•(•),
+˙A∗
+b,•(•)) can be chosen to be continuous in b with
+values in the respective spaces.
+For later use, we further determine the leading term of (˙gb0,v1(V),
+˙Ab0,v1(V))
+˙gb0,v1(V) =
+�
+(−4m0
+r
++ 2Q2
+0
+r2 )dt0 +
+4m0
+r − (m0 −
+�
+m2
+0 − Q2
+0)
+dr
+�
+⊗s V + O ¯
+H∞,1/2−
+b
+( ¯
+X)(γ),
+˙Ab0,v1(V) = O ¯
+H∞,1/2−
+b
+( ¯
+X)(|Q0|).
+(10.16)
+Remark 10.2. In the course of the proof of the above theorem, we will find that the zero modes in Kb satisfy
+both the linearized Einstein-Maxwell system and the linearized generalized wave map gauge and Lorenz
+gauge conditions.
+Proof of Theorem 10.1. We first analyze the RN case b = b0 and write (g, A) = (gb0, Ab0).
+Suppose
+�
+Lb0,γ(σ)(˙g, ˙A) = 0. Then by Proposition 5.7, (˙g, ˙A) ∈ ¯H∞,−1/2−
+b
+( ¯X). In view of (10.3), applying δg to
+LM
+b0,γe−iσtb0,∗(˙g, ˙A) = 0 yields
+δgd(δge−iσtb0,∗ ˙A) = 0.
+Since ˙A ∈ ¯H∞,−1/2−
+b
+( ¯X; �
+scT ∗ ¯X), we obtain that δg(e−iσtb0,∗ ˙A) is a mode. More precisely, we have
+δg(e−iσtb0,∗ ˙A) ∈
+�
+e−iσtb0,∗ ¯H∞,−1/2−
+b
+( ¯X; C)
+if
+σ ̸= 0,
+¯H∞,1/2−
+b
+( ¯
+X; C)
+if
+σ = 0.
+By Theorem 8.1, it follows that
+δg(e−iσtb0,∗ ˙A) = 0,
+(10.17)
+and thus e−iσtb0,∗(˙g, ˙A) is a mode solution to the Maxwell part of the linearized Einstein-Maxwell system.
+Applying δgGg to LE
+b0,γe−iσtb0,∗(˙g, ˙A) = 0 and using the linearized second Bianchi identity (see the discussion
+around (7.13) and (7.14)) give
+δgGg˜δ∗
+g,γ(˜δg,γGge−iσtb0,∗ ˙g) = Pb0,γ(˜δg,γGge−iσtb0,∗ ˙g) = 0.
+Again, since ˜δg,γGge−iσtb0,∗ ˙g is a mode in the same way as δg(e−iσtb0,∗ ˙A), it follows from Theorem 9.8 that
+˜δg,γGge−iσtb0,∗ ˙g = 0,
+(10.18)
+and thus e−iσtb0,∗(˙g, ˙A) is a mode solution to the linearized Einstein-Maxwell system.
+• The case Im σ ≥ 0, σ ̸= 0. Then we apply the mode stability result Theorem 7.1 and find that
+e−iσtb0,∗(˙g, ˙A) = (2δ∗
+gω, Lω♯A + dφ)
+(10.19)
+where (ω, φ) = e−iσtb0,∗(ω0, φ0) ∈ e−iσtb0,∗ ¯H∞,ℓ′
+b
+( ¯X; �
+scT ∗ ¯X ⊕ C) with ℓ′ < −1/2. Plugging (10.19)
+into the equation (10.18), we find
+�
+Wb0,γ(σ)ω0 = 0,
+and thus ω0 = 0 by Theorem 9.13. We then plug (10.19) with ω = 0 into the equation (10.17) and
+obtain
+�
+□g(σ)φ0 = 0.
+Therefore, we have φ0 = 0 by Theorem 8.1. This proves the injectivity of �
+Lb0,γ(σ) for Im σ ≥ 0, σ ̸= 0,
+and the surjectivity follows from the fact that �
+Lb0,γ(σ) is Fredholm of index 0.
+
+118
+LILI HE
+• Scalar type l ≥ 2 or vector type l ≥ 2 zero modes. We apply the mode stability result Theorem 7.1
+and find that
+(˙g, ˙A) = (2δ∗
+gω, Lω♯A + dφ)
+(10.20)
+where (ω, φ) ∈ ¯H∞,−3/2−
+b
+( ¯X; �
+scT ∗ ¯X ⊕ C). Plugging (10.20) into the equation (10.18), we find
+�
+Wb0,γ(0)ω = 0.
+Since ω ∈ ¯H∞,−3/2−
+b
+( ¯X) is of scalar or vector type l ≥ 2, it follows that ω = 0 by Theorem 9.13 and
+Proposition 9.14. We then plug (10.20) with ω = 0 into the equation (10.17) and obtain
+�
+□g(0)φ = 0.
+Again, since φ ∈ ¯H∞,−3/2−
+b
+( ¯X) with vector or scalar type l ≥ 2, we conclude that φ = 0 by Theorem
+8.1 and Proposition 8.2. This proves the injectivity of �
+Lb0,γ(0) when restricted on scalar or vector
+type l ≥ 2 modes.
+• Scalar type l = 1 zero modes. As in the previous case (scalar or vector l ≥ 2 case), we find that
+(˙g, ˙A) = (2δ∗
+gω, Lω♯A + dφ) where (ω, φ) ∈ ¯H∞,−3/2−
+b
+( ¯X; �
+scT ∗ ¯X ⊕ C) and satisfies first
+�
+Wb0,γ(0)ω = 0.
+Since ω ∈ ¯H∞,−3/2−
+b
+( ¯
+X) is of scalar type l = 1, in view of Proposition 9.14 we see that
+ω = ωb0,s1(S)
+for some
+S ∈ S1
+(10.21)
+where ωb0,s1(S) is defined as in (9.50). We then plug ˙A = Lω♯A + dφ and (10.21) into the equation
+(10.17) and obtain
+−�
+□g(0)φ = −δg �LA(ωb0,s1(S))♯ ∈ Q0 ¯H∞,3/2−
+b
+( ¯X)
+where we use δg ∈ ρDiff1
+b and L(·)♯A ∈ ρ2Diff1
+b (see (A.24)). Owing to Theorem 8.1 and the fact
+that ker �
+□g(0) ∩ ¯H∞,−3/2−
+b
+( ¯X) restricted to scalar type l = 1 spaces is 0, there exists a unique
+φ ∈ ¯H∞,−1/2−
+b
+( ¯X), which we denote by φb0,s1(S), such that the above equation holds. Therefore,
+the scalar type l = 1 nullspace of �
+Lb0,γ(0) is 3-dimensional.
+We also note that φb0,s1(S) is of
+size O ¯
+H∞,−1/2−
+b
+( ¯
+X)(|Q0|) and thus
+˙Ab0,s1(S) = O ¯
+H∞,1/2−
+b
+( ¯
+X)(|Q0|). In view of (9.52), we see that
+˙gb0,s1(S) = 2δ∗
+gωb0,s1(S) ∈ ¯H∞,1/2−
+b
+( ¯X; S2 �
+scT ∗ ¯X).
+• Scalar type l = 0 zero modes. If (˙g, ˙A) is of scalar type l = 0, by Theorem 7.1 we have
+(˙g, ˙A) = (˙gb0( ˙m, 0, ˙Q) + 2δ∗
+gω,
+˙Ab0(0, 0, ˙Q) + Lω♯A + dφ)
+(10.22)
+where (ω, φ) ∈ ¯H∞,−3/2−
+b
+( ¯X; �
+scT ∗ ¯X ⊕ C). Plugging (10.22) into (10.18), it follows that
+2�
+Wb0,γ(0)ω = −˜δg,γGg ˙gb0( ˙m, 0, ˙Q) ∈ ¯H∞,1/2−
+b
+( ¯X).
+In view of Theorem 9.13, ker �
+Wb0,γ(0)∗ ∩ ˙H−∞,−1/2+
+b
+( ¯X; �
+scT ∗ ¯X) = {0}. Therefore, we can solve the
+above equation for ω ∈ ¯H∞,−3/2−
+b
+( ¯
+X) which is of scalar type l = 0 and unique modulo ⟨∂♭
+t⟩. However,
+since δ∗
+g∂♭
+t = 0 and L∂tA = 0, a multiple of ∂♭
+t does not change ˙g and the equation for φ below. This
+implies ω is unique and we denote it by ˜ωb0,s0. Then we plug ˙A = ˙Ab0(0, 0, ˙Q) + Lω♯A + dφ and
+ω = ˜ωb0,s0 into the equation (10.17) and obtain
+−�
+□g(0)φ = −δg ˙Ab0(0, 0, ˙Q) − δgL˜ω♯
+b0,s0 A ∈ ¯H∞,1/2−
+b
+( ¯X)
+By Theorem 8.1, we have ker �
+□g(0)∗ ∩ ˙H−∞,−1/2+
+b
+( ¯X; C) = {0}. Therefore, we can solve the above
+equation for φ ∈ ¯H∞,−3/2−
+b
+( ¯X; C) which is of scalar type l = 0 and unique modulo ⟨1⟩. Again,
+adding a constant to φ make no change to ˙A, which means that φ is unique. We then denote this
+unique φ by φb0,s0. This proves that the scalar type l = 0 kernel of �
+Lb0,γ(0) is 2-dimensional.
+
+LINEAR STABILITY OF KERR-NEWMAN FAMILY
+119
+• Vector type l = 1 zero modes. If (˙g, ˙A) is of vector type l = 1, by Theorem 7.1 we have
+(˙g, ˙A) = (˙gb0(0, ˙a, 0) + 2δ∗
+gω,
+˙Ab0(0, ˙a, 0) + Lω♯A + dφ)
+(10.23)
+where (ω, φ) ∈ ¯H∞,−3/2−
+b
+( ¯X; �
+scT ∗ ¯X ⊕ C). Plugging (10.23) into (10.18), it follows that
+2�
+Wb0,γ(0)ω = −˜δg,γGg ˙gb0(0, ˙a, 0) ∈ ¯H∞,3/2−
+b
+( ¯X).
+In view of Theorem 9.13, there exists a unique ω ∈ ¯H∞,−1/2−
+b
+( ¯X; �
+scT ∗ ¯X) of vector type l = 1
+satisfying the above equation.
+We then denote this unique ω by ˜ωb0,v1(V).
+Therefore, we have
+˙gb0,v(V) ∈ ¯H∞,1/2−
+b
+( ¯X). Plugging ˙A = ˙Ab0(0, ˙a, 0) + Lω♯A + dφ and ω = ˜ωb0,v1 into the equation
+(10.17), we obtain that
+−�
+□g(0)φ = −δg ˙Ab0(0, ˙a, 0) − δg �LA(˜ωb0,v1(V))♯ ∈ Q0 ¯H∞,3/2−
+b
+( ¯X)
+By Theorem 8.1, there exists a unique φ ∈ ¯H∞,−1/2−
+b
+of vector type l = 1 satisfying the above
+equation.
+We then denote this unique φ by φb0,v1(V).
+Therefore, the vector l = 1 kernel of
+�
+Lb0,γ(0) is 3-dimensional. We also note that φb0,v1 is of size O ¯
+H∞,−1/2−
+b
+( ¯
+X;C)(|Q0|), and thus ˙Ab0,v1 =
+O ¯
+H∞,1/2−
+b
+( ¯
+X;C)(|Q0|).
+Now we shall derive the expression for (˙gb0,v1(V), ˙Ab0,v1(V)) in (10.16). We write
+˙gb0,v1(V) = ˙g0
+b0(0, ˙a, 0) + 2δ∗
+gω0.
+When rescaling to |˙a| = 1 and letting (θ, ϕ) be the spherical coordinates adapted to ˙a, we can rewrite
+˙gb0,v1(V) = (2(µb0 − 1)dt0 − 2dr) ⊗s V + 2δ∗
+gω0
+with
+V = sin2 θdϕ.
+Plugging ˙gb0,v1(V) into (10.18) yields
+2�
+Wb0,γ(0)ω0 = −˜δg,γGg
+�
+(2(µb0 − 1)dt0 − 2dr) ⊗s V
+�
+= −2r−1V + O ¯
+H∞,∞
+b
+( ¯
+X)(γ).
+We also find that
+δ∗
+gf(r)V =
+�
+1 +
+2m0
+r − (m0 −
+�
+m2
+0 − Q2
+0)
+�
+dr ⊗s V,
+δgGgδ∗
+gf(r)V = −r−1V
+with
+f(r) = (
+2m0
+m0 −
+�
+m2
+0 − Q2
+0
+− 1)r +
+2m0
+(m0 −
+�
+m2
+0 − Q2
+0)2 r2 ln(1 − m0 −
+�
+m2
+0 − Q2
+0
+r
+).
+Then it follows that
+2�
+Wb0,γ(0)(ω0 − f(r)V) = O ¯
+H∞,∞
+b
+( ¯
+X)(γ),
+and thus ω0 − f(r)V = O ¯
+H∞,−1/2−
+b
+( ¯
+X)(γ) (according to Remark 9.4, �
+Wb0,γ(0)−1 restricted in vector
+type l = 1 1-form spaces exists with norm of size O(1) ). This implies
+˙gb0,v1(V)= (2(µb0 − 1)dt0 − 2dr) ⊗s V + 2δ∗
+gf(r)V+O ¯
+H∞,1/2−
+b
+( ¯
+X)(γ)
+=
+�
+(−4m0
+r
++ 2Q2
+0
+r2 )dt0 +
+4m0
+r − (m0 −
+�
+m2
+0 − Q2
+0)
+dr
+�
+⊗s V + O ¯
+H∞,1/2−
+b
+( ¯
+X)(γ).
+(10.24)
+Therefore, ˙gb0,v1(V) ∈ ¯H∞,1/2−
+b
+( ¯X; S2 �
+scT ∗ ¯X).
+• Dual zero modes for RN and KN case. Combining the discussion around (9.4), (9.5) and (9.6) with
+the growing dual zero modes established in §9.3, we see that
+(2Ggbδ∗
+gbω∗
+b,•(•), 4 �LAb(ω∗
+b,•(•))♯ + dφ∗
+b,•(•)) ∈ ker �
+Lb,γ(0)∗
+provided that
+− �
+□gb(0)φ∗
+b,•(•) = −4δgb( �LAb(ω∗
+b,•(•))♯).
+(10.25)
+For ω∗
+b,s•(•), in view of (9.51), δgb ∈ ρDiff1
+b and L(•)♯Ab ∈ ρ2Diff1
+b, the right-hand side of (10.25) is
+in ˙H−∞,3/2−
+b
+( ¯X). For the case φ∗
+b,s0, by Theorem 8.1, there exists a unique φ∗
+b,s0 ∈ ˙H−∞,−1/2−
+b
+( ¯X)
+such that (10.25) holds.
+We note that adding a multiple of H(r − rb0) to φ∗
+b,s0 still gives rise
+
+120
+LILI HE
+to A∗
+b,s0 ∈
+˙H−∞,1/2−
+b
+( ¯X).
+For the case φ∗
+b,s1(S), S ∈ S1, by Theorem 8.1 again, there exists a
+unique φ∗
+b,s1(S) ∈
+˙H−∞,−1/2−
+b
+( ¯X) such that (10.25) holds.
+Therefore, with the above choice of
+φ∗
+b,•(•), we have
+˙A∗
+b,s•(•) ∈
+˙H−∞,1/2−
+b
+( ¯X).
+As for ωb,v1(V), according to (9.57), now the right-
+hand side of (10.25) is in ˙H−∞,1/2−
+b
+( ¯X). Since ker �
+□gb(0) ∩ ¯H∞,−1/2+
+b
+( ¯X) = {0} by Theorem 8.1,
+(10.25) is solvable for φ∗
+b,v1(V). More precisely, there exists a unique φ∗
+b,v1(V) ∈
+˙H−∞,−3/2−
+b
+( ¯X),
+which lies in ann(v) with v ∈ C∞
+c (X) and satisfying ⟨v, H(r − rb)⟩ ̸= 0, such that (10.25) holds,
+and thus ˙A∗
+b,v1 ∈ ˙H−∞,−1/2−
+b
+( ¯X; �
+scT ∗ ¯X). We also note that by (9.52) and (9.58), Ggbδ∗
+gbω∗
+b,•(•) ∈
+˙H−∞,1/2−
+b
+( ¯X; S2 �
+scT ∗ ¯X). As a consequence,
+(2Ggbδ∗
+gbω∗
+b,•(•), 4 �LAb(ω∗
+b,•(•))♯ + dφ∗
+b,•(•))
+and
+(0, δ(r − rb)dr)
+indeed span a 8-dimension subspace of K∗
+b.
+• Zero modes for KN case. First, the above construction of the dual zero modes of �
+Lb,γ(0)∗ and the
+index 0 property of �
+Lb,γ(0) imply that Kb is at least 8-dimensional.
+Next, we shall prove by a
+contradiction argument that it is at most 8-dimensional for b near b0, and thus must be 8-dimensional.
+Suppose {h1, · · · , h8} is a basis of Kb0 and choose h♭
+1, · · · , h♭
+8 ∈ C∞
+c (X◦; S2 �
+scT ∗ ¯X ⊕ �
+scT ∗ ¯X) such that
+⟨hi, h♭
+j⟩ = δij. (Consider the map Φ : C∞
+c (X◦) → C8 defined by Φ(h♭) = (⟨h1, h♭⟩, · · · , ⟨h8, h♭⟩),
+and note that Φ is surjective.
+Otherwise, there would exist some a = (a1, · · · , a8) ̸= 0 ∈ C8
+such that a · Φ(h♭) = ⟨�8
+i=1 aihi, h♭⟩ = 0 for all h♭ ∈ C∞
+c (X◦), which implies ⟨�8
+i=1 aihi, h♭⟩ = 0
+all h♭ ∈ ˙H−∞,−ℓ
+b
+( ¯X) since C∞
+c (X◦) is dense in ˙H−∞,−ℓ
+b
+( ¯X). However, this is impossible because
+�8
+i=1 aihi ̸= 0). Assume for the sake of contradiction that there exists a sequence bj → 0 such
+that for all j, the dimension of Kb is greater than 8.
+Then one can find uj ∈ Kbj such that
+uj ∈ ann{h♭
+1, · · · , h♭
+8} with ∥uj∥ ¯
+H∞,−1/2−
+b
+( ¯
+X) = 1. As in the proof of the first step of Lemma 9.12
+(using the Fredholm estimates for �
+Lb,γ(0)), there exists a subsequence uj → u ̸= 0 weakly in
+¯H∞,−1/2−
+b
+( ¯X) and then u ∈ ker �
+Lb0,γ(0). Therefore, one can pick h♭ ∈ span{h♭
+1, · · · , h♭
+8} such that
+⟨u, h♭⟩ = 1, but 0 = ⟨uj, h♭⟩ → ⟨u, h♭⟩ = 1, which leads to a contradiction.
+Getting the explicit expressions for (˙gb,•(•), ˙Ab,•(•)) as in (10.10), (10.11) and (10.12) requires a di-
+rect argument. First, by (9.52), we indeed have (˙gb,s1(S), ˙Ab,s1(S)) = (2δgbωb,s1(S), �LAb(ωb,s1(S))♯ +
+dφb,s1(S)) ∈ ker �
+Lb,γ ∩ ¯H∞,1/2−
+b
+( ¯X) for suitable φb,s1.
+It remains to construct (˙gb,s0, ˙Ab,s0) and
+(˙gb,v1(V), ˙Ab,v1(V)) which extend (˙gb0,s0, ˙Ab0,s0) and (˙gb0,v1(V), ˙Ab0,v1(V)), respectively. We make
+the ansatz
+(˙gb,s0, ˙Ab,s0) = (˙gb( ˙m, 0, ˙Q) + 2δ∗
+gbω,
+˙Ab(0, 0, ˙Q) + Lω♯Ab + dφ)
+(10.26)
+with ω, φ to be found. Since (˙gb,s0, ˙Ab,s0) of the form in the above ansatz satisfies the linearized
+Einstein-Maxwell system, we have (˙gb,s0, ˙Ab,s0) ∈ ker �
+Lb,γ(0) provided that the generalized linearized
+gauge conditions are satisfied, namely,
+˜δgb,γGgb ˙gb,s0 = 0
+and
+δgbd ˙Ab,s0 = 0.
+(10.27)
+Then as in the proof of scalar type l = 0 zero modes of �
+Lb0,γ(0), we can find a unique ωb,s0 ∈
+¯H∞,−3/2−
+b
+( ¯X; �
+scT ∗ ¯X), which lies in ann{f1, · · · , f4} with {f1, · · · , f4} ⊂ C∞
+c (X) and being a set of
+linearly independent functionals on ker �
+Wb0,γ(0)∩ ¯H∞,−3/2−
+b
+( ¯X; �
+scT ∗ ¯X), and φb,s0 ∈ ¯H∞,−3/2−
+b
+( ¯X; C),
+which lies in ann(v) with v ∈ C∞
+c (X) and satisfying ⟨1, v⟩ ̸= 0, such that (10.27) holds. The con-
+struction of (˙gb,v1(V), ˙Ab,v1(V)) is analogous. We make the ansatz
+(˙gb,v1(V), ˙Ab,v1(V)) = (˙gb(0, ˙a, 0) + 2δ∗
+gbω,
+˙Ab(0, ˙a, 0) + Lω♯Ab + dφ)
+(10.28)
+where V ∈ V1 is dual to the rotation around the axis ˙a and ω, φ are to be found. Then by using
+the discussion above and proceeding as in the proof of vector type l = 1 zero modes of �
+Lb0,γ(0), the
+conclusion (10.12) follows and (˙gb,v1(V), ˙Ab,v1(V)) ∈ ¯H∞,1/2−
+b
+( ¯X).
+• Continuity. The continuity (in b) of the zero modes and dual zero modes follows from the explicit
+construction above (see the proof of continuity in §9.3) and that of ωb,s1(S) and ω∗
+b,•(•).
+
+LINEAR STABILITY OF KERR-NEWMAN FAMILY
+121
+□
+10.2. Generalized zero modes: linear growth. Now we turn to discussing the generalized zero mode
+solutions of Lb,γ(˙g, ˙A) = 0. Suppose (˙g, ˙A) = (�k
+j=1 tj
+b,∗ ˙gj, �k
+j=1 tj
+b,∗ ˙Aj) and Lb,γ(˙g, ˙A) = 0. Due to the
+fact that [Lb,γ, ∂tb,∗] = 0, we find that ∂j
+tb,∗(˙g, ˙A) ∈ ker Lb,γ for 0 ≤ j ≤ k. Taking j = k, we obtain that
+the leading order term (˙gk, ˙Ak) of (˙g, ˙A) lies in the ker �
+Lb,γ(0). Next we further determine what the leading
+order term (˙gk, ˙Ak) should be.
+Lemma 10.3. Let k ≥ 1. Then there does not exist (˙g, ˙A) = (�k
+j=1 tj
+b,∗ ˙gj, �k
+j=1 tj
+b,∗ ˙Aj), with 0 ̸= (˙gk, ˙Ak) ∈
+{(˙gb,v1(V), ˙Ab,v1(V)) : V ∈ V1} and (˙gj, ˙Aj) ∈ ¯H∞,−1/2−
+b
+( ¯X; S2 �
+scT ∗ ¯X ⊕ �
+scT ∗ ¯X), such that Lb,γ(˙g, ˙A) = 0.
+Proof. According to the discussion above, it suffices to consider the case k = 1. Assume
+(˙g, ˙A) = (tb,∗ ˙gb,v1(V) + ˙g0, tb,∗ ˙Ab,v1(V) + ˙A0)
+with V ̸= 0. Then we need to show that there is no (˙g0, ˙A0) ∈ ¯H∞,−1/2−
+b
+( ¯X) such that the following equation
+holds
+�
+Lb,γ(0)(˙g0, ˙A0) = −[Lb,γ, tb,∗](˙gb,v1(V), ˙Ab,v1(V)) ∈ ¯H∞,3/2−
+b
+( ¯X)
+where we use [Lb,γ, tb,∗] ∈ ρDiff1
+b and (˙gb,v1(V), ˙Ab,v1(V) ∈ ¯H∞,1/2−
+b
+( ¯X). To this end, we need to verify that
+the pairing of the right-hand side with (˙g∗
+b,v1(V), ˙A∗
+b,v1(V)) is non-zero. We note that it suffices to check the
+RN case b = b0 because the general KN case with b near b0 follows directly by the continuity (in b). Using
+the expression (10.16) for (˙gb0,v1(V), ˙Ab0,v1(V)), we find that
+⟨[Lb0,γ, tb0,∗](˙gb0,v1(V), ˙Ab0,v1(V)), (˙g∗
+b0,v1(V′), ˙A∗
+b0,v1(V′))⟩
+= ⟨[LE
+b0,γ, tb0,∗](˙gb0,v1(V), 0), (˙g∗
+b0,v1(V′), 0)⟩ + O(|Q0|)
+= ⟨[LE
+b0,γ, tb0,∗](˙gb0,v1(V), 0), (2Ggδ∗
+g(r2H(r − rb0)V′), 0)⟩ + O(|Q0| + γ)
+(10.29)
+where we use (9.59) in the last step. Since Ggb0 δgb0 (r2H(r − rb0)V′) = r2δ(r − rb0)dr ⊗s V′ is supported at
+r = rb0 where tb0,∗ = t0, we can replace tb0,∗ by t0 in the subsequent calculation. We then compute
+⟨[LE
+b0,γ, t0](˙gb0,v1(V), 0), 2r2δ(r − rb0)dr ⊗s V′⟩
+= ⟨[LE
+b0,γ, t0](4m0r−1(dr − dt0) ⊗s V, 0), 2r2δ(r − rb0)dr ⊗s V′⟩ + O(|Q0|2 + γ)
+= −⟨[□gb0 ,2, t0](4m0r−1(dr − dt0) ⊗s V), 2r2δ(r − rb0)dr ⊗s V′⟩ + O(|Q0|2 + γ)
+= ⟨8m0r−2�
+(dr − (1 + m0
+r )dt0)
+�
+⊗s V, 2r2δ(r − rb0)dr ⊗s V′⟩ + O(|Q0|2 + γ)
+= −12m0⟨V, V ′⟩L2(S2;T ∗S2) + O(|Q0|2 + γ).
+(10.30)
+where we use (10.16) in the first step, (10.2) and (A.2) in the second step and [□gb0 ,2, t0]h = h□gb0 t0 +
+2(∇αt0)∇αh in the third step. Therefore, the pairing is indeed non-zero if V = V′ and |Q0|+ γ is sufficiently
+small.
+□
+We now exclude the generalized zero modes whose leading order term is (˙gb,s0, ˙Ab,s0).
+Lemma 10.4. Let k ≥ 1. Then there does not exist (˙g, ˙A) = (�k
+j=1 tj
+b,∗ ˙gj, �k
+j=1 tj
+b,∗ ˙Aj), with 0 ̸= (˙gk, ˙Ak) =
+(˙gb,s0, ˙Ab,s0) and (˙gj, ˙Aj) ∈ ¯H∞,−1/2−
+b
+( ¯
+X; S2 �
+scT ∗ ¯X ⊕ �
+scT ∗ ¯X), such that Lb,γ(˙g, ˙A) = 0.
+Proof. As discussed in the proof of Lemma 10.3, it suffices to prove the statement for the case k = 1. We
+first consider the case b = b0. Suppose
+(˙g, ˙A) = (tb0,∗ ˙gb0,s0 + ˙g0, tb0,∗ ˙Ab0,s0 + ˙A0) ∈ ker Lb0,γ
+with (˙g0, ˙A0) ∈ ¯H∞,−1/2−
+b
+( ¯X). Applying δgb0 to LM
+b0,γ(˙g, ˙A) = 0 gives δgb0 ˙A ∈ ker □gb0 . As pointed out in
+Remark 10.2, we have δgb0 ˙Ab0,s0 = 0. This implies
+ker �
+□gb0 (0) ∋ δgb0 ˙A = [δgb0 , tb0,∗] ˙Ab0,s0 + δgb0 ˙A0 ∈ ¯H∞,−1/2−
+b
+( ¯X; C).
+Therefore, δgb0 ˙A = 0 by Theorem 8.1. That is, (˙g, ˙A) satisfies the Maxwell part of the linearized Einstein-
+Maxwell system. Then as in the proof of Theorem 10.1, we apply δgb0 Ggb0 to LE
+b0,γ(˙g, ˙A) = 0 and use the
+
+122
+LILI HE
+linearized second Bianchi identity to obtain ˜δgb0 ,γGgb0 ˙g ∈ ker Pb0,γ. Since ˜δgb0 ,γGgb0 ˙gb0,s0 = 0 (see Remark
+10.2), it follows that
+ker �
+Pb0,γ(0) ∋ ˜δgb0 ,γGgb0 ˙g = [˜δgb0 ,γGgb0 , tb0,∗]˙gb0,s0 + ˜δgb0 ,γGgb0 ˙g0 ∈ ¯H∞,−1/2−
+b
+( ¯X; �
+scT ∗ ¯X),
+and thus ˜δgb0 ,γGgb0 ˙g = 0 by Theorem 9.8. As a consequence, (˙g, ˙A) satisfies the linearized Einstein-Maxwell
+system. By the statement (v) in mode stability Theorem 7.1, we have
+(˙g, ˙A) = (˙gb0( ˙m, 0, ˙Q) + 2δ∗
+gb0 ω,
+˙Ab0(0, 0, ˙Q) + Lω♯Ab0 + dφ)
+where (ω, φ) ∈ Poly2(tb0,∗) ¯H∞,−3/2−
+b
+( ¯X; �
+scT ∗ ¯X ⊕ C). To make the calculation simpler, we work in (t0 = t +
+rb0,∗, r, θ, ϕ) coordinates instead. Since ˙g is linear in t0, it follows that the coefficient of t2
+0 in ω and φ must be a
+multiple of ∂♭
+t0 and 1, respectively. Then equating the coefficient of t0 of the equality ˙g = ˙gb0( ˙m, 0, ˙Q)+2δ∗
+gb0 ω
+yields
+(2 ˙m
+r
+− 2Q0 ˙Q
+r2
+)dt2
+0 = ˙g0
+b0( ˙m, 0, ˙Q) = c(dt0) ⊗s ∂♭
+t0 + δ∗
+gb0 ˜ω
+(10.31)
+where c ∈ R and ˜ω ∈ ¯H∞,ℓ′
+b
+( ¯X; �
+scT ∗ ¯X) for some ℓ′ ∈ R is of scalar type l = 0. Assume ˜ω = f(r)dt0 + g(r)dr
+and let S = c(dt0) ⊗s ∂♭
+t0 + δ∗
+gb0 ˜ω. A direct calculation implies 0 = Srr = g′(r) and thus g(r) = m for some
+m ∈ R. Then 0 = St0r = c
+2 + 1
+2f ′(r) + m
+2 µ′
+b0 implies f(r) = −cr + 2mm0
+r
+− mQ2
+0
+r2
++ n for some n ∈ R. Next
+0 = Sθθ = r(f(r)+mµb0) = −cr2 +(m+n)r gives c = 0, m+n = 0, and thus ˜ω = −mµb0dt0 +mdr. Finally,
+2 ˙m
+r − 2Q0 ˙Q
+r2
+= St0t0 = m
+2 µb0µ′
+b0 − m
+2 µb0µ′
+b0 = 0 implies that ˙m = Q0 ˙Q = 0 and thus the leading term gb0,s0
+of ˙g is 0. If ˙Q ̸= 0, Q0 = 0, equating the coefficient of t0 of the equality ˙A = ˙Ab0(0, 0, ˙Q) + dφ yields
+˙Q
+r dt0 = cdt0 + d˜φ
+(10.32)
+where c ∈ R and ˜φ ∈ ¯H∞,ℓ′
+b
+( ¯X; C) for some ℓ′ ∈ R is of scalar type l = 0. This implies that ˙Q = 0 and thus
+the leading term Ab0,s0 of ˙A is 0. This proves the lemma for the RN case b = b0.
+This non-existence result for b = b0 in turn means that the following equation
+�
+Lb0,γ(0)(˙g0, ˙A0) = −[Lb0,γ, tb0,∗](˙gb0,s0, ˙Ab0,s0) ∈ ¯H∞,3/2−
+b
+( ¯X)
+cannot be solved for (˙g0, ˙A0) ∈ ¯H∞,−1/2−
+b
+( ¯X; S2 �
+scT ∗ ¯X⊕ �
+scT ∗ ¯X) (The right-hand side is in the stated Sobolev
+space because by Proposition 5.7, (˙gb0,s0, ˙Ab0,s0) = ρC∞( ¯X) + ¯H∞,1/2−
+b
+( ¯X) whose leading term ρC∞( ¯
+X) is
+annihilated by the normal operator 2ρ(ρ∂ρ − 1) of −[Lb0,γ, tb0,∗]). Suppose {(˙g1
+b,s0, ˙A1
+b,s0), (˙g2
+b,s0, ˙A2
+b,s0)} is a
+basis of {(˙gb,s0, ˙Ab,s0)} and {(˙g∗1
+b,s0, ˙A∗1
+b,s0), (˙g∗2
+b,s0, ˙A∗2
+b,s0)} is a basis of {(˙g∗
+b,s0, ˙A∗
+b,s0)}. In terms of pairing, we
+have
+⟨[Lb0,γ, tb0,∗](˙gi
+b0,s0, ˙Ai
+b0,s0), (˙g∗j
+b0,s0, ˙A∗j
+b0,s0)⟩,
+1 ≤ i, j ≤ 2
+is non-degenerate. Due to the continuity of (˙gb,s0, ˙Ab,s0) and (˙g∗
+b,s0, ˙A∗
+b,s0) in b, this pairing is still non-zero
+for KN case with b near b0. This finishes the proof of the lemma.
+□
+However, there do exist generalized zero modes (growing linearly in tb,∗) whose leading order term is given
+by (˙gb,s1(S), ˙Ab,s1(S)) for S ∈ S1.
+Proposition 10.5. Let b0 = (m0, 0, Q0) with |Q0| ≪ m0. Then the following space
+�Kb0 := ker Lb0,γ ∩ Poly1(tb0,∗) ¯H∞,−1/2−
+b
+( ¯X; S2 �
+scT ∗ ¯X ⊕ �
+scT ∗ ¯X).
+(10.33)
+is 11-dimensional. The 3-dimensional quotient space �Kb0/Kb0 is spanned by the following continuous (in b)
+families
+{(ˆgb0,s1(S), ˆAb0,s1(S)) : S ∈ S1}.
+(10.34)
+with
+(ˆgb0,s1(S), ˆAb0,s1(S)) = (2δ∗
+gb0 ˆωb0,s1(S), �LAb0 (ˆωb0,s1(S))♯ + dˆφb0,s1(S))
+(10.35)
+where ˆωb0,s1(S) is defined as in (9.65) and ˆφb0,s0(S) ∈ Poly1(tb0,∗) ¯H∞,−3/2−
+b
+( ¯X; C) depends linearly in S ∈ S1.
+The generalized zero modes (ˆgb0,s1(S), ˆAb0,s1(S)) extend to continuous families in b
+b → (ˆgb,s1(S), ˆAb,s1(S)) ∈ ker Lb,γ ∩ Poly1(tb,∗) ¯H∞,−1/2−
+b
+( ¯X; S2 �
+scT ∗ ¯X ⊕ �
+scT ∗ ¯X)
+(10.36)
+
+LINEAR STABILITY OF KERR-NEWMAN FAMILY
+123
+with
+(ˆgb,s1(S), ˆAb,s1(S)) = (2δ∗
+gb ˆωb,s1(S), �LAb(ˆωb,s1(S))♯ + dˆφb,s1(S))
+(10.37)
+where ˆωb,s1(S) is defined as in (9.65) and ˆφb,s0(S) ∈ Poly1(tb,∗) ¯H∞,−3/2−
+b
+( ¯
+X; C) depends linearly in S ∈
+S1. Moreover, (ˆgb,s1(S), ˆAb,s1(S)) satisfies both the linearized Einstein-Maxwell system and the linearized
+generalized wave map gauge and generalized Lorenz gauge conditions.
+Meanwhile, there exists continuous families (in b)
+b → (ˆg∗
+b,s1(S), ˆA∗
+b,s1(S)) ∈ ker L∗
+b,γ ∩ Poly1(tb,∗) ˙H−∞,−1/2−
+b
+( ¯X; S2 �
+scT ∗ ¯X ⊕ �
+scT ∗ ¯X)
+(10.38)
+with
+(ˆg∗
+b,s1(S), ˆA∗
+b,s1(S)) = (2Ggbδ∗
+gb ˆω∗
+b,s1(S), 4 �LAb0 (ˆω∗
+b,s1(S))♯ + dˆφ∗
+b,s1(S))
+(10.39)
+where ˆω∗
+b,s1(S) is defined as in (9.65) and ˆφ∗
+b,s0(S) ∈ Poly1(tb,∗) ˙H∞,−3/2−
+b
+( ¯X; C).
+Proof. Let (˙g, ˙A) = (tb,∗ ˙g1 + ˙g0, tb,∗ ˙A1 + ˙A0) ∈ ker Lb,γ with (˙gj, ˙Aj) ∈ ¯H∞,−1/2−
+b
+( ¯X).
+• Generalized zero modes in the RN case b = b0. In view of Lemma 10.3 and 10.4, (˙g1, ˙A1) must take
+the form (2δgb0 ωb0,s1(S), �LAb0 (ωb0,s1(S))♯ + dφb0,s1(S)) for some S ∈ S1. By the argument at the
+beginning of the proof of Lemma 10.4, we see that (˙g, ˙A) satisfies the generalized linearized gauge
+condition, that is,
+˜δgb0 Ggb0 ˙g = 0,
+δgb0 ˙A = 0.
+(10.40)
+Therefore, (˙g, ˙A) solves the linearized Einstein-Maxwell system. Since
+˙g = 2δ∗
+gb0(tb0,∗ωb0,s1(S)) − 2[δ∗
+gb0 , tb0,∗]ωb0,s1(S) + ˙g0 = 2δ∗
+gb0 (tb0,∗ωb0,s1(S)) + ˙g′
+0,
+˙A = �LAb0 (tb0,∗ωb0,s1(S))♯ + d
+�
+tb0,∗φb0,s1(S)
+�
+−
+�
+ιωb0,s1(S)♯Ab0 + φb0,s1(S)
+�
+dtb0,∗ + ˙A0
+= �LAb0 (tb0,∗ωb0,s1(S))♯ + d
+�
+tb0,∗φb0,s1(S)
+�
++ ˙A′
+0,
+(10.41)
+with
+˙g′
+0 = ˙g0 − 2[δ∗
+gb0 , tb0,∗]ωb0,s1(S) ∈ ¯H∞,−3/2−
+b
+( ¯X),
+˙A′
+0 = ˙A0 −
+�
+ιωb0,s1 (S)♯Ab0 + φb0,s1(S)
+�
+dtb0,∗ ∈ ¯H∞,−1/2−
+b
+( ¯X).
+where we use [δ∗
+gb0 , tb0,∗] ∈ A0( ¯X), ωb0,s1(S) ∈
+¯H∞,−3/2−
+b
+( ¯X) and φb0,s1(S) ∈
+¯H∞,−1/2−
+b
+( ¯X), it
+follows that (˙g′
+0, ˙A′
+0) solves the linearized Einstein-Maxwell system as well. By the statement (iv) in
+Theorem 7.1, we have
+(˙g′
+0, ˙A′
+0) = (2δ∗
+gb0 ω, Lω♯Ab0 + dφ)
+(10.42)
+with (ω, φ) ∈ ¯H∞,−5/2−
+b
+( ¯X; �
+scT ∗ ¯X ⊕ C). Plugging (10.41) and (10.42) into (10.40) yields
+�
+Wb0,γ(0)ω = −[Wb0,γ, tb0,∗]ωb0,s1(S) ∈ ¯H∞,−1/2−
+b
+( ¯
+X),
+(10.43)
+− �
+□gb0 (0)φ = −δgb0
+�
+Lω♯Ab0 + (ιωb0,s1 (S)♯Ab0)dtb0,∗
+�
+(10.44)
++ [□gb0 , tb0,∗]φb0,s1(S) − [δgb0 , tb0,∗] �LAb0 (ωb0,s1(S))♯
+We first consider (10.43). Since ker �
+Wb0,γ(0)∗ ∩ ˙H−∞,1/2+
+b
+( ¯X) = {0} by Theorem 9.13, it follows
+that (10.43) can be solved for ω ∈ ¯H∞,−5/2−
+b
+( ¯X). According to Proposition 9.17, we know that
+ˆωb0,s1(S) = tb0,∗ωb0,s1(S) + ˇωb0,s1(S) ∈ ker Wb0,γ
+with ˇωb0,s1(S) ∈ ¯H∞,−5/2−
+b
+( ¯X). Therefore,
+�
+Wb0,γ(0)
+�
+ω − ˇωb0,s1(S)
+�
+= −Wb0,γ ˆωb0,s1(S) = 0
+
+124
+LILI HE
+and ω − ˇωb0,s1(S) ∈ span{ω(1)
+b0,s1(S) : S ∈ S1} (In fact ker �
+Wb0,γ(0) ∩ ¯H∞,−5/2−
+b
+( ¯X) restricted to the
+scalar type l = 1 1-form is equal to span{ωb0,s1(S), ω(1)
+b0,s1(S) : S ∈ S1}. Since (˙g, ˙A) is in the quotient
+space �Kb0/Kb0, we can exclude ωb0,s1(S)). Using (9.67), we rewrite
+˙g = 2δ∗
+gb0 ˆωb0,s1(S) + 2δ∗
+gb0
+�
+ω − ˇωb0,s1(S)
+�
+= Poly1(tb0,∗) ¯H∞,−1/2−
+b
+( ¯X) + 2δ∗
+gb0
+�
+ω − ˇωb0,s1(S)
+�
+.
+Since we require ˙g ∈ Poly1(tb0,∗) ¯H∞,−1/2−
+b
+( ¯X), it follows that δ∗
+gb0
+�
+ω − ˇωb0,s1(S)
+�
+must lie in
+¯H∞,−1/2−
+b
+( ¯X). Owing to (9.62), ω − ˇωb0,s1(S) must be 0. That it, ˙g = 2δgb0 ˆωb0,s1(S).
+Next we turn to (10.44). Since ω = ˇωb0,s1(S) ∈ ¯H∞,−5/2
+b
+( ¯X) and φb0,s1 ∈ ¯H∞,−1/2−
+b
+( ¯X), the right-
+hand side of (10.44) lies in ¯H∞,1/2−
+b
+( ¯X). By Theorem 8.1 (ker �
+□gb0 (0)∗ ∩ ˙H−∞,−1/2+
+b
+( ¯X) = {0})
+and Proposition 8.2 (ker �
+□gb0 (0) ∩ ¯H∞,−3/2−
+b
+( ¯X) restricted to scalar type l = 1 is 0), there exists a
+unique φ ∈ ¯H∞,−3/2−
+b
+( ¯X) which we denote by ˇφb0,s1(S) satisfying (10.44). This proves that �Kb0/Kb0
+is at most 3-dimensional.
+Using ˙g = 2δ∗
+gb0 ˆωb0,s1(S) ∈ Poly1(tb0,∗) ¯H∞,−1/2
+b
+( ¯X) (see (9.68)) and rewriting
+˙A = �LAb0 (ˆωb0,s1(S))♯ + d
+�
+tb0,∗φb0,s1(S) + ˇφb0,s1(S)
+�
+=
+�
+�LAb0 (ˇωb0,s1(S))♯ + φb0,s1dtb0,∗ + ι(ωb0,s1(S))♯Ab0dtb0,∗ + dˇφb0,s1(S)
+�
++ tb0,∗
+�
+�LAb0 (ωb0,s1(S))♯ + dφb0,s1(S)
+�
+∈ Poly1(tb0,∗) ¯H∞,−1/2
+b
+( ¯X),
+(10.45)
+we conclude that (˙g, ˙A) = (ˆgb0,s1(S), ˆAb0,s1(S)) = (2δ∗
+gb0 ˆωb0,s1(S), �LAb0 (ˆωb0,s1(S))♯ + d(ˆωb0,s1(S)))
+with ˆφb0,s1(S) = tb0,∗φb0,s1(S) + ˇωb0,s1(S) ∈ Poly1(tb0,∗) ¯H∞,−3/2
+b
+( ¯X) indeed lies in �Kb0.
+• Generalized zero modes in the KN case. For b near b0, we make the ansatz
+(˙g, ˙A) = (2δ∗
+gb ˆωb,s1(S), �LAb(ˆωb,s1(S))♯ + d(tb,∗φb,s1(S)) + dφ).
+with φ ∈ ¯H∞,−3/2
+b
+( ¯X) to be determined. By the arguments in the proof of RN case above (the last two
+paragraphs), there exists ˇφb,s1(S) ∈ ¯H∞,−3/2
+b
+( ¯X) such that with ˆφb,s1(S) = tb,∗φb,s1(S) + ˇφb,s1(S) ∈
+Poly1(tb,∗) ¯H∞,−3/2
+b
+( ¯X),
+(ˆgb,s1(S), ˆAb,s1(S))
+= (2δ∗
+gb ˆωb,s1(S), �LAb(ˆωb,s1(S))♯ + d(ˆφb,s1(S))) ∈ ker Lb,γ ∩ Poly1(tb,∗) ¯H∞,−1/2−
+b
+( ¯X)
+extends (ˆgb0,s1(S), ˆAb0,s1(S)) and is continuous in b.
+• Generalized dual zero modes for RN and KN cases. Combining the discussion around (9.4), (9.5)
+and (9.6) with the generalized dual zero mode ˆω∗
+b,s1(S) = tb,∗ω∗
+b,s1(S) + ˇω∗
+b,s1(S) of Wb,γ established
+in Proposition 9.17, we see that
+(ˆg∗
+b,s1(S), ˆA∗
+b,s1(S)) = (2Ggbδ∗
+gb ˆω∗
+b,s1(S), 4 �LAb(ˆω∗
+b,s1(S))♯ + d(tb,∗φ∗
+b,s1(S)) + dˇφ∗
+b,s1(S)) ∈ ker L∗
+b,γ
+provided that
+−�
+□gb(0)ˇφ∗
+b,s1(S) = −4δgb
+�
+�LAb(ˇω∗
+b,s1(S))♯ + (ιω∗
+b,s1 (S)♯Ab)dtb,∗
+�
++ [□gb, tb,∗]φ∗
+b,s1(S) − 4[δgb, tb,∗] �LAb(ω∗
+b,s1(S))♯ ∈ ˙H−∞,1/2−
+b
+( ¯X).
+In view of Theorem 8.1 and Proposition 8.2, ker �
+□gb(0) ∩ ¯H∞,−1/2+
+b
+( ¯X) = {0}, and thus there exists
+a unique ˇφ∗
+b,s1(S) ∈ ˙H−∞,−3/2−
+b
+( ¯X) (up to addition of a multiple of H(r − rb))) such that (10.46)
+holds. We put ˆφ∗
+b,s1(S) = tb,∗φ∗
+b,s1 + ˇφ∗
+b,s1(S) and find that ˆAb,s1(S) ∈ poly1(tb,∗) ˙H−∞,−1/2−
+b
+( ¯X)
+by the argument in (10.45). We also recall that Ggbδ∗
+gb ˆω∗
+b,s1(S) ∈ Poly1(tb,∗) ˙H−∞,−1/2−
+b
+( ¯X) (see
+(9.68)). As a consequence, the continuous (in b) families (ˆg∗
+b,s1(S), ˆA∗
+b,s1(S)) indeed lie in ker L∗
+b,γ ∩
+poly1(tb,∗) ˙H−∞,−1/2−
+b
+( ¯X).
+
+LINEAR STABILITY OF KERR-NEWMAN FAMILY
+125
+□
+Let
+(ˇgb,s1(S), ˇAb,s1(S)) := (ˆgb,s1(S) − tb,∗ ˙gb,s1(S), ˆAb,s1(S) − tb,∗ ˙Ab,s1(S)) ∈ ¯H∞,−1/2−
+b
+( ¯X),
+(10.46)
+(ˇg∗
+b,s1(S), ˇA∗
+b,s1(S)) := (ˆg∗
+b,s1(S) − tb,∗ ˙g∗
+b,s1(S), ˆA∗
+b,s1(S) − tb,∗ ˙A∗
+b,s1(S)) ∈ ˙H−∞,−1/2−
+b
+( ¯X).
+(10.47)
+be the coefficients of t0
+b,∗ of the generalized (dual) zero modes constructed above. For later use, we determine
+the leading order behavior of them.
+Lemma 10.6. For S ∈ S1, we have
+(ˇgb,s1(S),
+ˇAb,s1(S)) ∈ ρC∞( ¯X) + ¯H∞,1/2−
+b
+( ¯X),
+(ˇg∗
+b,s1(S),
+ˇA∗
+b,s1(S)) ∈ ρC∞( ¯X) + ˙H−3/2−C(a,γ),1/2−
+b
+( ¯
+X)
+(10.48)
+where C(a, γ) > 0 is a sufficiently small constant depending on a, γ.
+Proof. Arguing as in the proof of Proposition 5.7, we again exploit the normal operator argument. First, since
+(˙gb,s1(S),
+˙Ab,s1(S)) ∈ ¯H∞,1/2−
+b
+( ¯X), according to Proposition 5.7 (where we shift the contour of integration
+through the pole ξ = −2i and thus the space of resonant states is S1), it follows that
+(˙gb,s1(S),
+˙Ab,s1(S)) ∈ ρ2Ω1 + ¯H∞,3/2−
+b
+( ¯X)
+where
+Ω1 ∈ span{Sdt2, Sdt ⊗s dxi, Sdxi ⊗s dxj, Sdt, Sdxi | S ∈ S1, 1 ≤ i, j ≤ 3}.
+Since
+− �
+Lb,γ(0)(ˇgb,s1(S), ˇAb,s1(S)) = [Lb,γ, tb,∗](˙gb,s1(S),
+˙Ab,s1(S)) ∈ −2ρ3Ω1 + ¯H∞,5/2−
+b
+( ¯X)
+where we use the fact [Lb,γ, tb,∗] ∈ −2ρ(ρ∂ρ − 1) + ρ2Diff1
+b, it follows that we can solve the above equation
+for (ˇgb,s1(S), ˇAb,s1(S)) by first solving away the leading term −2ρ3Ω1 and then the error term which lies in
+¯H∞,5/2−
+b
+( ¯X).
+For the leading term 2ρ3Ω1, proceeding as in the proof of Proposition 5.7 (here we have k = −1, l = 1),
+we need to solve the following equation
+�
+ρ∂ρ(ρ∂ρ − 1) + /∆
+�
+(˙g1, ˙A1) = −2ρΩ1.
+Since now k(k + 1) = (−1) · 0 ̸= 1 · 2 = l(l + 1), we conclude that (˙g1, ˙A2) ∈ (−1/2) · (−2ρΩ1) = ρΩ1.
+For the error term, we need to solve the following equation
+�
+ρ∂ρ(ρ∂ρ − 1) + /∆
+�
+(˙g2, ˙A2) = f ∈ ¯H∞,5/2−
+b
+( ¯X),
+(˙g2, ˙A2) ∈ ¯H∞,−1/2
+b
+( ¯X).
+Arguing as in the proof of Proposition 5.7, we see that (˙g2, ˙A2) ∈ ¯H∞,−1/2
+b
+( ¯X) ∈ ρC∞(∂+ ¯X)+ ¯H∞,1/2−
+b
+( ¯X).
+This proves (10.48) for (ˇgb,s1(S), ˇAb,s1(S)).
+For (ˇg∗
+b,s1(S),
+ˇA∗
+b,s1(S)), the above proof also applies. As for the regularity of (ˇg∗
+b,s1(S),
+ˇA∗
+b,s1(S)) near
+the radial point at event horizon (by the same reasoning as in the proof of statement (2) of Propo-
+sition 5.7, (ˇg∗
+b,s1(S),
+ˇA∗
+b,s1(S)) is smooth away from the radial point at even horizon), we notice that
+[Lb,γ, tb,∗](˙g∗
+b,s1(S),
+˙A∗
+b,s1(S)) is one derivative less regular than (˙g∗
+b,s1(S),
+˙A∗
+b,s1(S)) and ( �
+Lb,γ(0)∗)−1 gains
+one derivative. Therefore, (ˇg∗
+b,s1(S), ˇA∗
+b,s1(S)) has at least the same regularity as (˙g∗
+b,s1(S),
+˙A∗
+b,s1(S)), which
+is − 3
+2 − C(a, γ) in view of Proposition 5.7.
+□
+10.3. Generalized zero modes: quadratic growth. Finally, we exclude the generalized zero mode solu-
+tions with quadratic growth in tb,∗ of Lb,γ.
+Lemma 10.7. Let k ≥ 2. Then there does not exist (˙g, ˙A) = (�k
+j=1 tj
+b,∗ ˙gj, �k
+j=1 tj
+b,∗ ˙Aj), with 0 ̸= (˙gk, ˙Ak) =
+{(˙gb,s1(S), ˙Ab,s1(S)) : S ∈ S1} and (˙gj, ˙Aj) ∈ ¯H∞,−1/2−
+b
+( ¯X; S2 �
+scT ∗ ¯X ⊕ �
+scT ∗ ¯X), such that Lb,γ(˙g, ˙A) = 0.
+
+126
+LILI HE
+Proof. According to the discussion at the beginning of the previous subsection, it suffices to consider the
+case k = 2. For the simplicity of calculation, we replace tb,∗ by another smooth function t1 := t0 − 2r. Since
+tb,∗ − t1 ∈ A0−( ¯X), we assume
+(˙g, ˙A) = (t2
+1 ˙gb,s1(S) + t1 ˙g1 + ˙g0, t2
+1 ˙Ab,s1(S) + t1 ˙A1 + ˙A0)
+with S ̸= 0 and (˙gj, ˙Aj) ∈ ¯H∞,−1/2−
+b
+( ¯X) for j = 0, 1. We write Lb,γ(˙g, ˙A) as a polynomial of degree 1 in t1
+as follows
+Lb,γ(˙g, ˙A) = t1
+�
+2[Lb,γ, t1](˙gb,s1(S), ˙Ab,s1(S)) + Lb,γ(˙g1, ˙A1)
+�
++
+�
+[[Lb,γ, t1], t1](˙gb,s1(S), ˙Ab,s1(S)) + [Lb,γ, t1](˙g1, ˙A1) + Lb,γ(˙g0, ˙A0)
+�
+.
+Then Lb,γ(˙g, ˙A) = 0 is equivalent to the vanishing of the coefficients of the above polynomial. Namely,
+2[Lb,γ, t1](˙gb,s1(S), ˙Ab,s1(S)) + Lb,γ(˙g1, ˙A1) = 0,
+(10.49)
+[[Lb,γ, t1], t1](˙gb,s1(S), ˙Ab,s1(S)) + [Lb,γ, t1](˙g1, ˙A1) + Lb,γ(˙g0, ˙A0) = 0.
+(10.50)
+First, (10.49) implies Lb,γ
+�
+2t1(˙gb,s1(S), ˙Ab,s1(S)) + (˙g1, ˙A1)
+�
+= 0, and thus by Proposition 10.5
+2t1(˙gb,s1(S), ˙Ab,s1(S)) + (˙g1, ˙A1) = 2(ˆgb,s1(S), ˆAb,s1(S)) + c(˙gb,s1(S′),
+˙Ab,s1(S′))
+for some c ∈ R and S′ ∈ S1. Subtracting c(˙gb,s1(S′),
+˙Ab,s1(S′)) from (˙g, ˙A), we may assume c = 0, and this
+implies
+(˙g1, ˙A1) = 2(ˆgb,s1(S), ˆAb,s1(S)) − 2t1(˙gb,s1(S), ˙Ab,s1(S)).
+Plugging the above expression for (˙g1, ˙A1) into (10.50) yields
+Lb,γ(˙g0, ˙A0) = −[[Lb,γ, t1], t1](˙gb,s1(S), ˙Ab,s1(S))
+− [Lb,γ, t1]
+�
+2(ˆgb,s1(S), ˆAb,s1(S)) − 2t1(˙gb,s1(S), ˙Ab,s1(S))
+�
+∈ ¯H∞,3/2−
+b
+( ¯X)
+(10.51)
+Our aim is to prove that there is no solution (˙g0, ˙A0) ∈ ¯H∞,−1/2−
+b
+( ¯X) to the above equation. To this end,
+we need to verify that the pairing of the right-hand side of (10.51) with (˙g∗
+b,s1(S), ˙A∗
+b,s1(S)) is non-zero. We
+note that it suffices to check the RN case b = b0 because the general KN case with b near b0 follows directly
+by the continuity (in b). Using ˙Ab0,s1(S) = O ¯
+H∞,1/2−
+b
+( ¯
+X)(|Q0|) and the same argument as in the proof of
+Lemma 10.3, we obtain that the the pairing of the right-hand side of (10.51) with (˙g∗
+b0,s1(S), ˙A∗
+b0,s1(S)) is
+equal to
+⟨[[□gb0 ,2, t1], t1]˙gb0,s1(S), ˙g∗
+b0,s1(S)⟩ + 2⟨[□gb0 ,2, t1](ˆgb0,s1(S) − t1 ˙gb0,s1(S)), ˙g∗
+b0,s1(S)⟩ + O(|Q0| + γ)
+:= P1 + P2 + O(|Q0| + γ).
+We first calculate P1. Using ˙gb0,s1(S) = 2δ∗
+gb0 ωb0,s1(S), ˙g∗
+b0,s1(S) = 2Ggb0 δ∗
+gb0 ω∗
+b0,s1(S) and (9.53), we find
+that
+P1 = 2⟨(∇αt1∇αt1)˙gb0,s1(S), ˙g∗
+b0,s1(S)⟩ = 2⟨4(µb0 − 1)˙gb0,s1(S), ˙g∗
+b0,s1(S)⟩
+= 2⟨16(µb0 − 1)δ∗
+gb0 d
+�
+(r − m0)S
+�
+, Ggb0 δ∗
+gb0 d
+�
+(r − m0)H(r − rb0)S
+�
+⟩ + O(|Q0|2 + γ)
+= −512
+9 πm0 + O(|Q0|2 + γ).
+Next, using ˆgb0,s1(S) = 2δ∗
+gb0 ˆωb0,s1(S), (9.70) and (9.53), we compute
+P2 = 2⟨
+�
+2∇αt1∇α + □gb0 ,0t1
+�
+(ˆgb,s1(S) − t1 ˙gb,s1(S)), ˙g∗
+b0,s1(S)⟩
+= −64
+9 (7 − log 8)πm0 + O(|Q0|2 + γ).
+Therefore, the pairing is indeed non-zero if |Q0| + γ is sufficiently small.
+□
+
+LINEAR STABILITY OF KERR-NEWMAN FAMILY
+127
+11. Structure of the resolvent of the linearized gauge-fixed Einstein-Maxwell operator
+In this section, we shall prove that �
+Lb,γ(σ) is invertible for b = (m, a, Q) near b0 = (m0, 0, Q0), |Q0| ≪ m0
+and Im σ ≥ 0, σ ̸= 0. This is a generalization of the first statement in Theorem 10.1 from the RN case to
+the KN case. The proof uses the relevant conclusions of the RN case and the arguments in the proof of
+Proposition 9.10.
+Let γ > 0 be a fixed sufficiently small constant such that all the conclusions in the previous section hold.
+We recall the linearized gauge-fixed Einstein-Maxwell operator
+Lb,γ := L(gb,Ab),γ,
+LE
+b,γ := LE
+(gb,Ab),γ,
+LM
+b,γ := LM
+(gb,Ab),γ,
+and its Fourier transformed version
+�
+Lb,γ(σ) := eiσtb,∗Lb,γe−iσtb,∗
+where tb,∗ = χ0(r)(t + rb0,∗) + (1 − χ0(r))(t − r(m,0,Q),∗) is defined in (4.29). Since γ is fixed, we will drop
+the subscript γ from the notations from now on.
+Let s > 3 and −3/2 < ℓ < −1/2. We redefine
+X s,ℓ
+b
+(σ) := {(˙g, ˙A) ∈ ¯Hs,ℓ
+b ( ¯X; S2 �
+scT ∗ ¯X ⊕ �
+scT ∗ ¯X) : �
+Lb(σ)(˙g, ˙A) ∈ ¯Hs−1,ℓ+2
+b
+( ¯X; S2 �
+scT ∗ ¯X ⊕ �
+scT ∗ ¯X)}.
+As in the previous section, from now on, we will drop the notation of the bundle if it it clear from the
+context.
+We also decompose zero energy nullspace Kb and the space of zero energy dual states K∗
+b of Lb as follows
+Kb = Kb,1 ⊕ Kb,2,
+K∗
+b = K∗
+b,1 ⊕ K∗
+b,2
+(11.1)
+where
+Kb,1 := {(˙gb,s1(S1), ˙Ab,s1(S1))},
+Kb,2 := {(˙gb,s0, ˙Ab,s0)} ⊕ {(˙gb,v1(V1), ˙Ab,v1(V1))},
+(11.2)
+K∗
+b,1 := {(˙g∗
+b,s1(S1), ˙A∗
+b,s1(S1))},
+Kb,2 := {(˙g∗
+b,s0, ˙A∗
+b,s0)} ⊕ {(˙g∗
+b,v1(V1), ˙A∗
+b,v1(V1))}.
+(11.3)
+The motivation for doing such a decomposition is that �
+Lb(σ) is more singular when acting on Kb,1 because
+of the existence of the generalized solutions to Lb(˙g, ˙A) = 0 with linear growth in tb,∗ and leading order
+terms in Kb,1. Moreover, we recall the generalized zero energy states (ˆgb,s1(S1), ˆAb,s1(S1)) and dual states
+(ˆg∗
+b,s1(S1), ˆA∗
+b,s1(S1)) which is constructed in Proposition 10.5 and define
+ˇKb := {(ˇgb,s1(S), ˇAb,s1(S)) : S ∈ S1},
+ˇK∗
+b := {(ˇg∗
+b,s1(S), ˇA∗
+b,s1(S)) : S ∈ S1}
+(11.4)
+where
+(ˇgb,s1(S), ˇAb,s1(S)) = (ˆgb,s1(S), ˆAb,s1(S)) − tb,∗(˙gb,s1(S), ˙Ab,s1(S)),
+(11.5)
+(ˇg∗
+b,s1(S), ˇA∗
+b,s1(S)) = (ˆg∗
+b,s1(S), ˆA∗
+b,s1(S)) − tb,∗(˙g∗
+b,s1(S), ˙A∗
+b,s1(S)).
+(11.6)
+11.1. Construction of the reference operator ˇLb(σ). We first construct a reference operator ˇLb(σ)
+which is invertible near σ = 0 by perturbing �
+Lb(σ) by a smoothing operator.
+Lemma 11.1. Let b0 = (m0, 0, Q0) with |Q0| ≪ m0. There exist Vb ∈ Ψ−∞(X; S2 �
+scT ∗ ¯X ⊕ �
+scT ∗ ¯X) which
+is continuous in b with uniformly compactly supported Schwartz kernel, and a constant C0 > 0, such that the
+following statements hold.
+(1) The operator
+ˇLb(σ) := �
+Lb(σ) + Vb : X s,ℓ
+b (σ) → ¯Hs−1,ℓ+2
+b
+( ¯X; S2 �
+scT ∗ ¯X ⊕ �
+scT ∗ ¯X)
+is invertible for σ ∈ C, Im σ ≥ 0 with |σ| + |b − b0| ≤ C0.
+(2) The inverse operator ˇLb(σ)−1 is continuous in σ with values in Lweak( ¯Hs−1,ℓ+2
+b
+( ¯X), ¯Hs,ℓ
+b ( ¯X)) (the
+space of linear bounded operators equipped with weak operator topology), and continuous in σ with
+values in Lop( ¯Hs−1+ǫ,ℓ+2+ǫ
+b
+( ¯
+X), ¯Hs−ǫ,ℓ−ǫ
+b
+( ¯X)) (norm topology) for any ǫ > 0.
+(3) One has
+Vb(ˇgb, ˇAb) = 0
+for
+(ˇgb, ˇAb) ∈ ˇKb.
+(11.7)
+
+128
+LILI HE
+Proof. The proof closely follows [36, Lemma 11.1 and Lemma 11.7]. Concretely, we shall construct
+Vb : D′(X; S2 �
+scT ∗ ¯X ⊕ �
+scT ∗ ¯X) → C∞
+c (X; S2T ∗X ⊕ T ∗X)
+where Vb satisfy
+ˇKb ⊂ ker Vb,
+ranVb ⊂ R⊥
+b
+and
+Vb|Kb : Kb → R⊥
+b
+is invertible.
+(11.8)
+where Rb :=
+�
+ranX s,ℓ
+b
+(0)�
+Lb(0)
+�
+. We now prove that once the above requirements are satisfied, all the three
+statements in the lemma hold. As for the proof of statements (1) and (2), it suffices to prove the invertiblity
+of ˇLb(σ) = �
+Lb(σ) + Vb for the case b = b0, σ = 0 since the invertibility and continuity for (b, σ) near (b0, 0)
+follow from the same arguments as in the proof of Lemma 9.11. We split the domain X s,ℓ
+b0 (0) = K⊥
+b0 ⊕Kb0 and
+the target space ¯Hs−1,ℓ+2
+b
+( ¯X; S2 �
+scT ∗ ¯X ⊕ �
+scT ∗ ¯X) = Rb0 ⊕ R⊥
+b0. Under these splittings, ˇLb0(0) = �
+Lb0(0) + Vb
+takes the form
+ˇLb0(0) =
+�L00
+0
+V10
+V11
+�
+where L00 = �
+Lb0(0)|K⊥
+b0 : K⊥
+b0 → Rb0 is invertible. Then the invertibility of ˇLb0(0) follows from that of
+V11 : Kb0 → R⊥
+b0. We notice that the invertibility of V11 holds provided that the above conditions in (11.8)
+are satisfied. For the statement (3), it is clear that the conclusion (11.7) directly follows from the condition
+ˇKb ⊂ ker Vb.
+First, since (˙gb0,s1(S1), ˙Ab0,s1(S1)) ∈ ¯H∞,1/2−
+b
+( ¯X) (see Theorem 10.1) and (ˇgb0,s1(S1), ˇAb0,s1(S1)) have
+nonzero r−1 leading terms (see Lemma 10.6), we conclude that Kb0,1∩ ˇKb0 = 0, and thus Kb0∩ ˇKb0 = 0. By the
+continuity of Kb and ˇKb in b, this holds for b near b0 as well. This implies that fixing the basis {hb,1, · · · , hb,8}
+of Kb and {ˇhb,1, ˇhb,2, ˇhb,3} of ˇKb, both of which depend continuously on b, by the same reasoning as in the
+beginning of the seventh step in the proof of Theorem 10.1, there exists {h♭
+b,1, · · · , h♭
+b,8} ⊂ C∞
+c (X◦; S2 �
+scT ∗ ¯X⊕
+�
+scT ∗ ¯X) such that ⟨hb,i, h♭
+b,j⟩ = δij for 1 ≤ i, j ≤ 8 and ⟨ˇhb,i, h♭
+b,j⟩ = 0 for 1 ≤ i ≤ 3, 1 ≤ j ≤ 8. (One can
+verify that {h♭
+b,1, · · · , h♭
+b,8} ⊂ C∞
+c (X◦) can be chosen to be continuous in b.) Next, we pick a basis of R⊥
+b in
+the following way. We note that K∗
+b ∩ ˇK∗
+b = 0, which can be proved in a similar manner to the statement
+Kb ∩ ˇKb = 0.
+Fixing the continuous (in b) basis {h∗
+b,1, · · · , h∗
+b,8} of K∗
+b = ann(Rb) and {ˇh∗
+b,1, ˇh∗
+b,2, ˇh∗
+b,3}
+of ˇK∗
+b, by the same reasoning mentioned above, there exists {h♯
+b,1, · · · , h♯
+b,8} ⊂ C∞
+c (X; S2 �
+scT ∗ ¯X ⊕ �
+scT ∗ ¯X)
+(which can be chosen continuous in b) such that ⟨h♯
+b,i, h∗
+b,j⟩ = δij for 1 ≤ i, j ≤ 8 and ⟨h♯
+b,i, ˇh∗
+b,j⟩ = 0 for
+1 ≤ i ≤ 8, 1 ≤ j ≤ 3.
+It is easy to check that {h♯
+b,1, · · · , h♯
+b,8} are linearly independent and the space
+generated by h♯
+b,j, 1 ≤ j ≤ 8 is a complement of Rb. Therefore, we define
+Vb(h) :=
+8
+�
+i=1
+h♯
+b,i⟨h, h♭
+b,i⟩,
+which satisfies the requirement (11.8).
+□
+Using ˇLb(σ) constructed in Lemma 11.1, we define the following spaces
+�Kb,1 := ˇLb(0)Kb,1,
+�Kb,2 := ˇLb(0)Kb,2,
+�Kb := ˇLb(0)Kb = �Kb,1 ⊕ �Kb,2.
+(11.9)
+Since ˇLb(0)Kb = Vb(Kb), we have �Kb,1, �Kb,2, �Kb ⊂ C∞
+c (X; S2 �
+scT ∗ ¯X ⊕ �
+scT ∗ ¯X).
+Next, we decompose the target space ¯Hs−1,ℓ+2
+b
+( ¯X; S2 �
+scT ∗ ¯X ⊕ �
+scT ∗ ¯X) into the range of �
+Lb(0) and its
+complement in a continuous (in b) way.
+Lemma 11.2. Let b0 = (m0, 0, Q0) with |Q0| ≪ m0. There exists a linear projection map
+Π⊥
+b : ¯Hs−1,ℓ+2
+b
+( ¯X; S2 �
+scT ∗ ¯X ⊕ �
+scT ∗ ¯X) → ¯Hs−1,ℓ+2
+b
+( ¯X; S2 �
+scT ∗ ¯X ⊕ �
+scT ∗ ¯X),
+which is of rank 8 and depends continuously on b near b0 in the norm topology, such that
+⟨(I − Π⊥
+b )h, h∗⟩ = 0
+for all
+h∗ ∈ K∗
+b.
+
+LINEAR STABILITY OF KERR-NEWMAN FAMILY
+129
+Moreover, the Π⊥
+b can be chosen such that it maps ¯Hs−1,ℓ+2
+b
+( ¯X; S2 �
+scT ∗ ¯X ⊕ �
+scT ∗ ¯X) to C∞
+c (X; S2 �
+scT ∗ ¯X ⊕
+�
+scT ∗ ¯X).
+Proof. Let {h∗
+b,1, · · · , h∗
+b,8} be a basis which is continuous in b of K∗
+b.
+There exists {h♯
+b,1, · · · , h♯
+b,8} ⊂
+C∞
+c (X; S2 �
+scT ∗ ¯X ⊕ �
+scT ∗ ¯X) which is continuous in b, such that ⟨h♯
+b,i, h∗
+b,j⟩ = δij. We then define
+Π⊥
+b h =
+8
+�
+i=1
+h♯
+b,i⟨h, h∗
+b,i⟩.
+It is easy to check that Π⊥
+b has rank 8 and satisfies ⟨(I − Π⊥
+b )h, h∗⟩ = 0 for all h∗ ∈ K∗
+b.
+□
+We then define the projection onto Rb which is the range of �
+Lb(0)
+Πb = I − Π⊥
+b : ¯Hs−1,ℓ+2
+b
+( ¯X; S2 �
+scT ∗ ¯X ⊕ �
+scT ∗ ¯X) → Rb = ann K∗
+b.
+Now we are able to split the domain and target space of the operator �
+Lb(σ)ˇLb(σ)−1 : ¯Hs−1,ℓ+2
+b
+→ ¯Hs−1,ℓ+2
+b
+as follows.
+domain:
+¯Hs−1,ℓ+2
+b
+( ¯X; S2 �
+scT ∗ ¯X ⊕ �
+scT ∗ ¯X) = �K⊥
+b ⊕ �Kb,1 ⊕ �Kb,2,
+target:
+¯Hs−1,ℓ+2
+b
+( ¯X; S2 �
+scT ∗ ¯X ⊕ �
+scT ∗ ¯X) = Rb ⊕ R⊥
+b,1 ⊕ R⊥
+b,2
+(11.10)
+where R⊥
+b,1, resp. R⊥
+b,2 is a subspace of dimension 3, resp. 5 and defined as follows. Fixing the continuous
+(in b) basis {h∗(1)
+b,1 , · · · , h∗(1)
+b,3 } of K∗
+b,1 and {h∗(2)
+b,1 , · · · , h∗(2)
+b,5 } of K∗
+b,2, there exist
+{v(1)
+b,1, v(1)
+b,2, v(1)
+b,3}, {v(2)
+b,1, v(2)
+b,2, v(2)
+b,3, v(2)
+b,4, v(2)
+b,5} ⊂ C∞
+c (X; S2 �
+scT ∗ ¯X ⊕ �
+scT ∗ ¯X)
+such that ⟨v(1)
+b,i , h∗(1)
+b,j ⟩ = δij and ⟨v(2)
+b,i , h∗(2)
+b,j ⟩ = δij. We then define
+R⊥
+b,1 = {v(1)
+b,1, v(1)
+b,2, v(1)
+b,3}
+and
+R⊥
+b,2 = {v(2)
+b,1, v(2)
+b,2, v(2)
+b,3, v(2)
+b,4, v(2)
+b,5}.
+11.2. Structure of the resolvent near zero energy. Now we are at the position of establishing the mode
+stability of the operator �
+Lb(σ) for b near b0 at Im σ ≥ 0, σ ̸= 0, and we also provide a description of the
+structure of the resolvent �
+Lb(σ)−1 near σ = 0 along the way.
+Theorem 11.3. Let (gb, Ab) be the RN metric and 4-electromagnetic potential where b = (m, a, Q) is near
+b0 = (m0, 0, Q0) with |Q0| ≪ m0. Then the Fourier-transformed operator
+�
+Lb(σ) : {(˙g, ˙A) ∈ ¯Hs,ℓ
+b ( ¯X; S2 �
+scT ∗ ¯X ⊕ �
+scT ∗ ¯X) : �
+Lb(σ)(˙g, ˙A) ∈ ¯Hs,ℓ+1
+b
+( ¯X; S2 �
+scT ∗ ¯X ⊕ �
+scT ∗ ¯X)}
+→ ¯Hs,ℓ+1
+b
+( ¯X; S2 �
+scT ∗ ¯X ⊕ �
+scT ∗ ¯X)
+(11.11)
+is invertible for σ ∈ C, Im σ ≥ 0, σ ̸= 0 and s > 3, ℓ < − 1
+2, s + ℓ > − 1
+2.
+Proof. By the arguments in the last step of the proof of Theorem 8.1 and Proposition 9.10, it suffices to
+prove that X s,ℓ
+b
+(σ) → ¯Hs,ℓ+1
+b
+( ¯X; S2 �
+scT ∗ ¯X ⊕ �
+scT ∗ ¯X) is invertible for s > 3, −3/2 < ℓ < −1/2 and (b, σ) with
+Im σ ≥ 0, σ ̸= 0 near (b0, 0). To this end, we shall prove the invertibility of the operator
+�
+Lb(σ)ˇLb(σ)−1 : ¯Hs−1,ℓ+2
+b
+( ¯X; S2 �
+scT ∗ ¯X ⊕ �
+scT ∗ ¯X) → ¯Hs−1,ℓ+2
+b
+( ¯X; S2 �
+scT ∗ ¯X ⊕ �
+scT ∗ ¯X)
+for s > 3, −3/2 < ℓ < −1/2 and (b, σ) with Im σ ≥ 0, σ ̸= 0 near (b0, 0).
+Under the splitting (11.10), we express
+�
+Lb(σ)ˇLb(σ)−1 =
+
+
+L00(b, σ)
+L01(b, σ)
+L02(b, σ)
+L10(b, σ)
+L11(b, σ)
+L12(b, σ)
+L20(b, σ)
+L21(b, σ)
+L22(b, σ)
+
+ .
+(11.12)
+It is clear that Lij(b, 0) = 0 when (i, j) ̸= (0, 0) and L00(b, 0) is invertible. We will use the following resolvent
+identity (whose proof is similar to that of the resolvent identity (9.35) in the proof of Lemma 9.12) frequently
+in the subsequent analysis
+�ˇLb(σ)−1 − ˇLb(0)−1�
+(˙g, ˙A) = ˇLb(σ)−1�ˇLb(0) − ˇLb(σ)
+�ˇLb(0)−1(˙g, ˙A)
+(11.13)
+�
+(ˇLb(σ)−1)∗ − (ˇLb(0)−1)∗�
+(˙g∗, ˙A∗) = (ˇLb(σ)−1)∗�ˇLb(0)∗ − ˇLb(σ)∗�
+(ˇLb(0)−1)∗(˙g∗, ˙A∗)
+(11.14)
+
+130
+LILI HE
+for ˇLb(0)−1(˙g, ˙A) ∈ ρC∞( ¯
+X) + ¯Hs,1/2−
+b
+( ¯X) and (ˇLb(0)−1)∗(˙g∗, ˙A∗) ∈ ρC∞( ¯X) + ˙H−s+1,1/2−
+b
+( ¯X).
+• Analysis of L01 and L02. For (˜g2, ˜A2) = ˇLb(0)(˙g2, ˙A2) ∈ �Kb,2, we have
+�
+Lb(σ)ˇLb(σ)−1(˜g2, ˜A2) =
+�ˇLb(σ) − Vb
+�ˇLb(σ)−1 ˇLb(0)(˙g2, ˙A2)
+= −Vb
+�ˇLb(σ)−1 − ˇLb(0)−1�ˇLb(0)(˙g2, ˙A2)
+= Vb ˇLb(σ)−1�ˇLb(σ) − ˇLb(0)
+�ˇLb(0)−1 ˇLb(0)(˙g2, ˙A2)
+= σVb ˇLb(σ)−1∂σ�
+Lb(0)(˙g2, ˙A2) + σ2
+2 Vb ˇLb(σ)−1∂2
+σ�
+Lb(0)(˙g2, ˙A2)
+(11.15)
+where we use ˇLb(0)−1 ˇLb(0)(˙g2, ˙A2) = (˙g2, ˙A2) ∈ ρC∞( ¯X) + ¯H∞,1/2−
+b
+( ¯X) by Proposition 5.7 and
+the resolvent identity (11.13) in the third step. Since (˙g2, ˙A2) ∈ ρC∞( ¯X) + ¯H∞,1/2−
+b
+( ¯X) which is
+annihilated by normal operator of ∂σ�
+Lb(0) modulo ¯H∞,3/2−
+b
+( ¯X), it follows that
+∂σ�
+Lb(0)(˙g2, ˙A2) ∈ ¯H∞,3/2−
+b
+( ¯
+X).
+As ∂2
+σ�
+Lb(0) ∈ ρ2C∞( ¯X; End(S2 �
+scT ∗ ¯X ⊕ �
+scT ∗ ¯X)), we have
+∂2
+σ�
+Lb(0)(˙g2, ˙A2) ∈ ¯H∞,3/2−
+b
+( ¯X)
+as well. By using the continuity of ˇLb(σ)−1 in the norm operator topology
+Lop( ¯Hs−1+ǫ,ℓ+2+ǫ
+b
+( ¯X), ¯Hs−ǫ,ℓ−ǫ
+b
+( ¯X)),
+we conclude that
+ˇLb(σ)−1∂σ�
+Lb(0)(˙g2, ˙A2), ˇLb(σ)−1∂2
+σ�
+Lb(0)(˙g2, ˙A2)
+are continuous in (b, σ), and thus
+L02(b, σ) = σ�L02(b, σ)
+where �L02(b, σ) : �Kb,2 → Rb is continuous in (b, σ) in the norm operator topology.
+For (˜g1, ˜A1) = ˇLb(0)(˙g1, ˙A1) ∈ �Kb,1, the calculation in (11.15) still applies. We further calculate
+�
+Lb(σ)ˇLb(σ)−1(˜g1, ˜A1)
+= σVb ˇLb(σ)−1∂σ�
+Lb(0)(˙g1, ˙A1) + σ2
+2 Vb ˇLb(σ)−1∂2
+σ�
+Lb(0)(˙g1, ˙A1)
+= iσVb ˇLb(σ)−1�
+Lb(0)(ˇg1, ˇA1) + σ2
+2 Vb ˇLb(σ)−1∂2
+σ�
+Lb(0)(˙g1, ˙A1)
+= iσ
+�
+I − Vb ˇLb(0)−1�
+Vb(ˇg1, ˇA1) + iσVb(ˇLb(σ)−1 − ˇLb(0)−1)�
+Lb(0)(ˇg1, ˇA1)
++ σ2
+2 Vb ˇLb(σ)−1∂2
+σ�
+Lb(0)(˙g1, ˙A1)
+= iσVb(ˇLb(σ)−1 − ˇLb(0)−1)�
+Lb(0)(ˇg1, ˇA1) + σ2
+2 Vb ˇLb(σ)−1∂2
+σ�
+Lb(0)(˙g1, ˙A1)
+where we use Lemma 11.1 in the last step. Since by Lemma 11.1 and Lemma 10.6, we have
+ˇLb(0)−1�
+Lb(0)(ˇg1, ˇA1) = (ˇg1, ˇA1) ∈ ρC∞( ¯X) + ¯H∞,1/2−
+b
+( ¯X),
+and it follows that we can apply the resolvent identity (11.13) again and obtain
+�
+Lb(σ)ˇLb(σ)−1(˜g1, ˜A1)
+= iσVb ˇLb(σ)−1�ˇLb(0) − ˇLb(σ)
+�ˇLb(0)−1�
+Lb(0)(ˇg1, ˇA1) + σ2
+2 Vb ˇLb(σ)−1∂2
+σ�
+Lb(0)(˙g1, ˙A1)
+= σ2�
+− iVb ˇLb(σ)−1�
+∂σ�
+Lb(0) + σ
+2 ∂2
+σ�
+Lb(0)
+�
+(ˇg1, ˇA1) + 1
+2Vb ˇLb(σ)−1∂2
+σ�
+Lb(0)(˙g1, ˙A1)
+�
+.
+(11.16)
+By the same reasoning as in the component L02(b, σ), we find that
+L01(b, σ) = σ2�L01(b, σ)
+where �L01(b, σ) : �Kb,1 → Rb is continuous in (b, σ) in the norm operator topology.
+
+LINEAR STABILITY OF KERR-NEWMAN FAMILY
+131
+• Analysis of L10 and L20. For (˜g0, ˜A0) ∈ �K⊥
+b and (˙g∗
+2, ˙A∗
+2) ∈ K∗
+b,2, we have
+⟨�
+Lb(σ)ˇLb(σ)−1(˜g0, ˜A0), (˙g∗
+2, ˙A∗
+2)⟩
+= ⟨(˜g0, ˜A0), (ˇLb(σ)−1)∗�
+Lb(σ)∗(˙g∗
+2, ˙A∗
+2)⟩
+= ⟨(˜g0, ˜A0), (˙g∗
+2, ˙A∗
+2)⟩ − ⟨(˜g0, ˜A0), (ˇLb(σ)−1)∗V ∗
+b (˙g∗
+2, ˙A∗
+2)⟩
+= −
+�
+(˜g0, ˜A0),
+�
+(ˇLb(σ)−1)∗ − (ˇLb(0)−1)∗�
+V ∗
+b (˙g∗
+2, ˙A∗
+2)
+�
+=
+�
+(˜g0, ˜A0), (ˇLb(σ)−1)∗�ˇLb(σ)∗ − ˇLb(0)∗�
+(ˇLb(0)−1)∗V ∗
+b (˙g∗
+2, ˙A∗
+2)
+�
+= σ
+�
+(˜g0, ˜A0), (ˇLb(σ)−1)∗�
+∂σ�
+Lb(0) + σ
+2 ∂2
+σ�
+Lb(0)
+�∗(˙g∗
+2, ˙A∗
+2)
+�
+(11.17)
+where we use (ˇLb(0)−1)∗V ∗
+b (˙g∗
+2, ˙A∗
+2) = (˙g∗
+2, ˙A∗
+2) ∈ ρC∞( ¯X) + ˙H−3/2−C(a,γ),1/2−
+b
+( ¯X) where C(a, γ)
+is a small constant by Proposition 5.7 and the resolvent identity (11.14) in the fourth step. Since
+(˙g∗
+2, ˙A∗
+2) is annihilated by normal operator of (∂σ�
+Lb(0))∗ modulo ˙H−5/2−C(a,γ),3/2−
+b
+( ¯X), it follows
+that (∂σ�
+Lb(0))∗(˙g∗
+2, ˙A∗
+2) ∈ ˙H−5/2−C(a,γ),3/2−
+b
+( ¯X). As (∂2
+σ�
+Lb(0))∗ ∈ ρ2C∞( ¯X), we have
+(∂2
+σ�
+Lb(0))∗(˙g∗
+2, ˙A∗
+2) ∈ ˙H−3/2−ν−,3/2−
+b
+( ¯X).
+Since (ˇLb(σ)−1)∗ is continuous with values in Lop( ˙H−s+ǫ,−ℓ+ǫ
+b
+( ¯
+X), ˙H−s+1−ǫ,−ℓ−2−ǫ
+b
+( ¯X)), we conclude
+that
+(ˇLb(σ)−1)∗(∂σ�
+Lb(0))∗(˙g∗
+2, ˙A∗
+2), (ˇLb(σ)−1)∗(∂2
+σ�
+Lb(0))∗(˙g∗
+2, ˙A∗
+2)
+are continuous in (b, σ), and thus
+L20(b, σ) = σ�L20(b, σ)
+where �L20(b, σ) : �K⊥
+b → R⊥
+b,2 is continuous in (b, σ) in the norm operator topology.
+For (˜g0, ˜A0) ∈ �K⊥
+b and (˙g∗
+1, ˙A∗
+1) ∈ K∗
+b,1, the calculation in (11.17) still applies. We further calculate
+⟨�
+Lb(σ)ˇLb(σ)−1(˜g0, ˜A0), (˙g∗
+1, ˙A∗
+1)⟩
+= σ
+�
+(˜g0, ˜A0), (ˇLb(σ)−1)∗�
+∂σ�
+Lb(0) + σ
+2 ∂2
+σ�
+Lb(0)
+�∗(˙g∗
+1, ˙A∗
+1)
+�
+= σ
+�
+(˜g0, ˜A0), i(ˇLb(σ)−1)∗�
+Lb(0)∗(ˇg∗
+1, ˇA∗
+1)
+�
++ σ2�
+(˜g0, ˜A0), 1
+2(ˇLb(σ)−1)∗(∂2
+σ�
+Lb(0))∗(˙g∗
+1, ˙A∗
+1)
+�
+= σ
+�
+(˜g0, ˜A0), i
+�
+I − (ˇLb(0)−1)∗V ∗
+b
+�
+(ˇg∗
+1, ˇA∗
+1)
+�
++ σ
+�
+(˜g0, ˜A0), i
+�
+(ˇLb(σ)−1)∗ − (ˇLb(0)−1)∗��
+Lb(0)∗(ˇg∗
+1, ˇA∗
+1)
+�
++ σ2�
+(˜g0, ˜A0), 1
+2(ˇLb(σ)−1)∗(∂2
+σ�
+Lb(0))∗(˙g∗
+1, ˙A∗
+1)
+�
+By a normal operator argument as in the proof of Proposition 5.7, we have
+(ˇLb(0)−1)∗ : C∞
+c (X) → ρC∞( ¯X) + ˙H−3/2−C(a,γ),1/2−
+b
+( ¯X).
+Therefore, by Lemma 10.6, we conclude
+(ˇLb(0)−1)∗�
+Lb(0)∗(ˇg1, ˇA1) = (ˇg1, ˇA1) − (ˇLb(0)−1)∗V ∗
+b (ˇg1, ˇA1) ∈ ρC∞( ¯X) + ˙H−3/2−C(a,γ),1/2−
+b
+( ¯X).
+This implies that we can apply the resolvent identity (11.14) again and obtain
+⟨�
+Lb(σ)ˇLb(σ)−1(˜g0, ˜A0), (˙g∗
+1, ˙A∗
+1)⟩
+= σ
+�
+(˜g0, ˜A0), i
+�
+I − (ˇLb(0)−1)∗V ∗
+b
+�
+(ˇg∗
+2, ˇA∗
+2)
+�
+− σ2�
+(˜g0, ˜A0), i(ˇLb(σ)−1)∗(∂σ�
+Lb(0) + σ
+2 ∂2
+σ�
+Lb(0))∗�
+I − (ˇLb(0)−1)∗V ∗
+b
+�
+(ˇg∗
+2, ˇA∗
+2)
+�
++ σ2�
+(˜g0, ˜A0), 1
+2(ˇLb(σ)−1)∗(∂2
+σ�
+Lb(0))∗(˙g∗
+2, ˙A∗
+2)
+�
+.
+(11.18)
+
+132
+LILI HE
+By the same reasoning as in the component L20(b, σ), we find that
+L10(b, σ) = σ�L10(b, σ)
+where �L10(b, σ) = �L0
+10(b) + σ�Le
+10(b, σ) with �Le
+10(b, σ) : �K⊥
+b → R⊥
+b,1 continuous in (b, σ) in the norm
+operator topology.
+• Analysis of L22. Using (11.15), we find that
+⟨�
+Lb(σ)ˇLb(σ)−1(˜g2, ˜A2), (˙g∗
+2, ˙A∗
+2)⟩
+= ⟨σ∂σ�
+Lb(0)(˙g2, ˙A2) + σ2
+2 ∂2
+σ�
+Lb(0)(˙g2, ˙A2), (ˇLb(σ)−1)∗V ∗
+b (˙g∗
+2, ˙A∗
+2)⟩
+= ⟨σ∂σ�
+Lb(0)(˙g2, ˙A2) + σ2
+2 ∂2
+σ�
+Lb(0)(˙g2, ˙A2), (˙g∗
+2, ˙A∗
+2)⟩
++ ⟨σ∂σ�
+Lb(0)(˙g2, ˙A2) + σ2
+2 ∂2
+σ�
+Lb(0)(˙g2, ˙A2),
+�
+(ˇLb(σ)−1)∗ − (ˇLb(0)−1)∗�
+V ∗
+b (˙g∗
+2, ˙A∗
+2)⟩
+= ⟨σ∂σ�
+Lb(0)(˙g2, ˙A2) + σ2
+2 ∂2
+σ�
+Lb(0)(˙g2, ˙A2), (˙g∗
+2, ˙A∗
+2)⟩
++ ⟨σ∂σ�
+Lb(0)(˙g2, ˙A2) + σ2
+2 ∂2
+σ�
+Lb(0)(˙g2, ˙A2), (ˇLb(σ)−1)∗�ˇLb(0)∗ − ˇLb(σ)∗�
+(˙g∗
+2, ˙A∗
+2)⟩
+= σ⟨−i[Lb, tb,∗](˙g2, ˙A2) (˙g∗
+2, ˙A∗
+2)⟩ + σ2�
+⟨1
+2∂2
+σ�
+Lb(0)(˙g2, ˙A2), (˙g∗
+2, ˙A∗
+2)⟩
+−
+�
+∂σ�
+Lb(0)(˙g2, ˙A2) + σ
+2 ∂2
+σ�
+Lb(0)(˙g2, ˙A2), (ˇLb(σ)−1)∗(∂σ�
+Lb(0) + σ
+2 ∂2
+σ�
+Lb(0))∗(˙g∗
+2, ˙A∗
+2)
+��
+(11.19)
+where we use (ˇLb(0)−1)∗V ∗
+b (˙g∗
+2, ˙A∗
+2) = (˙g∗
+2, ˙A∗
+2) and the resolvent identity (11.14) in the third step.
+According to the proof of the non-existence of the linearly growing generalized zero modes with
+leading order term (˙g2, ˙A2) ∈ Kb,2 (see Lemma 10.3 and 10.4), the pairing on Kb,2 × K∗
+b,2
+⟨−i[Lb, tb,∗](˙g2, ˙A2), (˙g∗
+2, ˙A∗
+2)⟩
+is non-degenerate for b near b0. Therefore, we find that
+L22(b, σ) = σ�L22(b, σ)
+with
+�L22(b, σ) = �L0
+22(b) + σ�Le
+22(b, σ)
+where �L0
+22(b) is invertible and �Le
+22(b, σ) : �Kb,2 → R⊥
+b,2 is continuous in (b, σ) in the norm operator
+topology.
+
+LINEAR STABILITY OF KERR-NEWMAN FAMILY
+133
+• Analysis of L11. Using the calculation in (11.16) and (11.18), we see that
+⟨�
+Lb(σ)ˇLb(σ)−1(˜g1, ˜A1), (˙g∗
+1, ˙A∗
+1)⟩
+= σ2⟨−i
+�
+∂σ�
+Lb(0) + σ
+2 ∂2
+σ�
+Lb(0)
+�
+(ˇg1, ˇA1) + 1
+2∂2
+σ�
+Lb(0)(˙g1, ˙A1), (ˇLb(σ)−1)∗V ∗
+b (˙g∗
+1, ˙A∗
+1)⟩
+= σ2⟨−i
+�
+∂σ�
+Lb(0) + σ
+2 ∂2
+σ�
+Lb(0)
+�
+(ˇg1, ˇA1) + 1
+2∂2
+σ�
+Lb(0)(˙g1, ˙A1), (˙g∗
+1, ˙A∗
+1)⟩
++ σ2⟨−i
+�
+∂σ�
+Lb(0) + σ
+2 ∂2
+σ�
+Lb(0)
+�
+(ˇg1, ˇA1) + 1
+2∂2
+σ�
+Lb(0)(˙g1, ˙A1), (ˇLb(σ)−1)∗(ˇLb(0) − ˇLb(σ))∗(˙g∗
+1, ˙A∗
+1)⟩
+= σ2⟨−i
+�
+∂σ�
+Lb(0) + σ
+2 ∂2
+σ�
+Lb(0)
+�
+(ˇg1, ˇA1) + 1
+2∂2
+σ�
+Lb(0)(˙g1, ˙A1), (˙g∗
+1, ˙A∗
+1)⟩
+− σ3⟨−i
+�
+∂σ�
+Lb(0) + σ
+2 ∂2
+σ�
+Lb(0)
+�
+(ˇg1, ˇA1) + 1
+2∂2
+σ�
+Lb(0)(˙g1, ˙A1), i(ˇLb(σ)−1)∗�
+Lb(0)∗(ˇg∗
+1, ˇA∗
+1)⟩
+− σ4⟨−i
+�
+∂σ�
+Lb(0) + σ
+2 ∂2
+σ�
+Lb(0)
+�
+(ˇg1, ˇA1) + 1
+2∂2
+σ�
+Lb(0)(˙g1, ˙A1), (ˇLb(σ)−1)∗(1
+2∂2
+σ�
+Lb(0))∗(˙g∗
+1, ˙A∗
+1)⟩
+= σ2⟨−i∂σ�
+Lb(0)(ˇg1, ˇA1) + 1
+2∂2
+σ�
+Lb(0)(˙g1, ˙A1), (˙g∗
+2, ˙A∗
+2)⟩ + σ3⟨−i
+2 ∂2
+σ�
+Lb(0)(ˇg1, ˇA1), (˙g∗
+1, ˙A∗
+1)⟩
+− σ3⟨−i∂σ�
+Lb(0)(ˇg1, ˇA1) + 1
+2∂2
+σ�
+Lb(0)(˙g1, ˙A1), i(I − (ˇLb(0)−1)∗V ∗
+b )(ˇg∗
+1, ˇA∗
+1)⟩
+− σ4⟨−i
+2 ∂2
+σ�
+Lb(0)(ˇg1, ˇA1), i(I − (ˇLb(0)−1)∗V ∗
+b )(ˇg∗
+1, ˇA∗
+1)⟩
++ σ4�
+− i
+�
+∂σ�
+Lb(0) + σ
+2 ∂2
+σ�
+Lb(0)
+�
+(ˇg1, ˇA1) + 1
+2∂2
+σ�
+Lb(0)(˙g1, ˙A1),
+(ˇLb(σ)−1)∗(∂σ�
+Lb(0) + σ
+2 ∂2
+σ�
+Lb(0))∗(I − (ˇLb(0)−1)∗V ∗
+b )(ˇg∗
+1, ˇA∗
+1)
+�
+− σ4⟨−i
+�
+∂σ�
+Lb(0) + σ
+2 ∂2
+σ�
+Lb(0)
+�
+(ˇg1, ˇA1) + 1
+2∂2
+σ�
+Lb(0)(˙g1, ˙A1), (ˇLb(σ)−1)∗(1
+2∂2
+σ�
+Lb(0))∗(˙g∗
+1, ˙A∗
+1)⟩.
+(11.20)
+According to the proof of the non-existence of the quadratically growing generalized zero modes with
+leading order term (˙g1, ˙A1) ∈ Kb,1 (see Lemma 10.7), the pairing on Kb,1 × K∗
+b,1
+⟨−i∂σ�
+Lb(0)(ˇg1, ˇA1) + 1
+2∂2
+σ�
+Lb(0)(˙g1, ˙A1), (˙g∗
+2, ˙A∗
+2)⟩
+= −⟨[Lb, tb,∗](ˇg1, ˇA1) + 1
+2[[Lb, tb,∗], tb,∗](˙g1, ˙A1), (˙g∗
+2, ˙A∗
+2)⟩
+is non-degenerate for b near b0. Therefore, we find that
+L11(b, σ) = σ2�L11(b, σ)
+with
+�L11(b, σ) = �L0
+11(b) + σ�L1
+22(b) + σ2�Le
+11(b, σ)
+where �L0
+11(b) is invertible and �Le
+11(b, σ) : �Kb,1 → R⊥
+b,1 is continuous in (b, σ) in the norm operator
+topology.
+• Analysis of L12 and L21. For (˜g1, ˜A1) ∈ �Kb,1 and (˙g∗
+2, ˙A∗
+2) ∈ K∗
+b,2, using (11.16) and (11.19), we
+compute
+⟨�
+Lb(σ)ˇLb(σ)−1(˜g1, ˜A1), (˙g∗
+2, ˙A∗
+2)⟩
+= σ2⟨−i∂σ�
+Lb(0)(ˇg1, ˇA1) + 1
+2∂2
+σ�
+Lb(0)(˙g1, ˙A1), (˙g∗
+2, ˙A∗
+2)⟩ + σ3⟨−i
+2 ∂2
+σ�
+Lb(0)(ˇg1, ˇA1), (˙g∗
+2, ˙A∗
+2)⟩
+− σ3�
+− i
+�
+∂σ�
+Lb(0) + σ
+2 ∂2
+σ�
+Lb(0)
+�
+(ˇg1, ˇA1) + 1
+2∂2
+σ�
+Lb(0)(˙g1, ˙A1),
+(ˇLb(σ)−1)∗�
+∂σ�
+Lb(0) + σ
+2 ∂2
+σ�
+Lb(0)
+�∗(˙g∗
+2, ˙A∗
+2)
+�
+.
+(11.21)
+Therefore,
+L21(b, σ) = σ2�L21(b, σ)
+with
+�L21(b, σ) = �L0
+21(b) + σ�Le
+21(b, σ)
+where �Le
+21(b, σ) : �Kb,1 → R⊥
+b,2 is continuous in (b, σ) in the norm operator topology.
+
+134
+LILI HE
+For (˜g2, ˜A2) = ˇLb(0)(˙g2, ˙A2) ∈ �Kb,2 = ˇLb(0)Kb,2 and (˙g∗
+1, ˙A∗
+1) ∈ K∗
+b,1, using (11.18), we calculate
+⟨�
+Lb(σ)ˇLb(σ)−1(˜g2, ˜A2), (˙g∗
+1, ˙A∗
+1)⟩
+= σ
+��
+I − Vb(ˇLb(0)−1)
+�ˇLb(0)(˙g2, ˙A2), i(ˇg∗
+1, ˇA∗
+1)
+�
+− σ2�
+ˇLb(σ)−1 ˇLb(0)(˙g2, ˙A2), i(∂σ�
+Lb(0))∗�
+I − (ˇLb(0)−1)∗V ∗
+b
+�
+(ˇg∗
+1, ˇA∗
+1)
+�
++ σ2�
+ˇLb(σ)−1 ˇLb(0)(˙g2, ˙A2), 1
+2(∂2
+σ�
+Lb(0))∗(˙g∗
+1, ˙A∗
+1)
+�
+− σ3�
+ˇLb(σ)−1 ˇLb(0)(˙g2, ˙A2), i
+2(∂σ�
+Lb(0))∗�
+I − (ˇLb(0)−1)∗V ∗
+b
+�
+(ˇg∗
+1, ˇA∗
+1)
+�
+= −σ2�
+(˙g2, ˙A2), i(∂σ�
+Lb(0))∗�
+I − (ˇLb(0)−1)∗V ∗
+b
+�
+(ˇg∗
+2, ˇA∗
+2) − 1
+2(∂2
+σ�
+Lb(0))∗(˙g∗
+1, ˙A∗
+1)
+�
++ σ3�
+ˇLb(σ)−1(∂σ�
+Lb(0) + σ
+2 ∂2
+σ�
+Lb(0))(˙g2, ˙A2), i(∂σ�
+Lb(0))∗�
+I − (ˇLb(0)−1)∗V ∗
+b
+�
+(ˇg∗
+1, ˇA∗
+1)
+�
++ σ3�
+ˇLb(σ)−1(∂σ�
+Lb(0) + σ
+2 ∂2
+σ�
+Lb(0))(˙g2, ˙A2), −1
+2(∂2
+σ�
+Lb(0))∗(˙g∗
+1, ˙A∗
+1)
+�
+− σ3�
+ˇLb(σ)−1 ˇLb(0)(˙g2, ˙A2), i
+2(∂2
+σ�
+Lb(0))∗�
+I − (ˇLb(0)−1)∗V ∗
+b
+�
+(ˇg∗
+1, ˇA∗
+1)
+�
+.
+(11.22)
+Therefore,
+L12(b, σ) = σ2�L12(b, σ)
+with
+�L12(b, σ) = �L0
+12(b) + σ�Le
+12(b, σ)
+where �Le
+12(b, σ) : �Kb,2 → R⊥
+b,1 is continuous in (b, σ) in the norm operator topology.
+• Analysis of L00. Following the arguments in the first step in the proof of Lemma 9.12, we can prove
+that for (b, σ) near (b, 0), L00(b, σ) is invertible and there exists a uniform constant C > 0 such that
+∥L00(b, σ)−1∥Rb→ �
+K⊥
+b ≤ C.
+(11.23)
+Moreover, for (˜g0, ˜A0) ∈ �K⊥
+b , we write
+L00(b, σ)(˜g0, ˜A0) = �
+Lb(σ)ˇLb(σ)−1(˜g0, ˜A0) −
+�
+L10(b, σ) + L20(b, σ)
+�
+(˜g0, ˜A0)
+= (˜g0, ˜A0) − Vb ˇLb(σ)−1(˜g0, ˜A0) −
+�
+L10(b, σ) + L20(b, σ)
+�
+(˜g0, ˜A0).
+(11.24)
+Since ˇLb(σ)−1 is continuous in the norm operator topology Lop( ¯Hs−1+ǫ,ℓ+2+ǫ
+b
+( ¯X), ¯Hs−ǫ,ℓ−ǫ
+b
+( ¯X)) and
+Vb maps D′(X◦) to C∞
+c (X), it follows that L00(b, σ) : �K⊥
+b → Rb is continuous in (b, σ) in the norm
+operator topology.
+• The inverse of �
+Lb(σ)ˇLb(σ)−1. Putting all the above analysis of the components Lij(b, σ) together,
+we conclude that
+�
+Lb(σ)ˇLb(σ)−1 =
+
+
+
+L00(b, σ)
+σ2�L01(b, σ)
+σ�L02(b, σ)
+σ�L10(b, σ)
+σ2�L11(b, σ)
+σ2�L12(b, σ)
+σ�L20(b, σ)
+σ2�L21(b, σ)
+σ�L22(b, σ)
+
+
+
+=
+
+
+
+L00(b, σ)
+σ2 �L01(b, σ)
+σ�L02(b, σ)
+σ
+��L0
+10(b) + σ�Le
+10(b, σ)
+�
+σ2��L0
+11(b) + σ�L1
+11(b) + σ2�Le
+11(b, σ)
+�
+σ2��L0
+12(b) + σ�Le
+12(b, σ)
+�
+σ�L20(b, σ)
+σ2��L0
+21(b) + σ�Le
+21(b)
+�
+σ
+��L0
+22(b) + σ�Le
+22(b, σ)
+�
+
+
+
+(11.25)
+where L00(b, σ), �L0
+11(b), �L0
+22(b) are invertible, and L00(b, σ), �Lij(b, σ) with (i, j) = (0, 1), (0, 2), (2, 0),
+�Le
+ij(b, σ) with (i, j) = (1, 0), (1, 1), (1, 2), (2, 1), (2, 2) are continuous in (b, σ) in the norm operator
+topology.
+Solving the system
+�
+Lb(σ)ˇLb(σ)−1�
+(˜g0, ˜A0), (˜g1, ˜A1), (˜g2, ˜A2)
+�
+= (f0, f1, f2)
+
+LINEAR STABILITY OF KERR-NEWMAN FAMILY
+135
+yields
+R(b, σ) = (�
+Lb(σ)ˇLb(σ)−1)−1 =
+
+
+
+�R00(b, σ)
+�R01(b, σ)
+�R02(b, σ)
+σ−1 �R10(b, σ)
+σ−2 �R11(b, σ)
+σ−1 �R12(b, σ)
+�R20(b, σ)
+σ−1 �R21(b, σ)
+σ−1 �R22(b, σ)
+
+
+
+(11.26)
+where
+�R11 =
+��L♯
+11 − σ�L♯
+12(�L♭
+22)−1�L♯
+21
+�−1,
+�R10 = �R11
+�
+− �L10 + σ�L♯
+12(�L♭
+22)−1�L20
+�
+L−1
+00 ,
+�R12 = − �R11�L♯
+12(�L♭
+22)−1,
+�R21 = −(�L♭
+22)−1�L♯
+21 �R11,
+�R22 = (�L♭
+22)−1 − σ(�L♭
+22)−1�L♯
+21 �R12,
+�R20 = −(�L♭
+22)−1��L20L−1
+00 + �L♯
+21 �R10
+�
+,
+�R01 = −L−1
+00
+��L01 �R11 + �L02 �R21
+�
+,
+�R02 = −L−1
+00
+�
+σ�L01 �R12 + �L02 �R22
+�
+,
+�R00 = L−1
+00
+�
+I − σ�L01 �R10 − σ�L02 �R20
+�
+(11.27)
+with
+�L♯
+i1 = �Li1 − σ�Li0L−1
+00 �L01
+for
+i = 1, 2,
+�L♯
+12 = �L12 − �L10L−1
+00 �L02,
+�L♭
+22 = �L22 − σ�L20L−1
+00 �L02.
+(11.28)
+Since ˇLb(σ) is invertible for (b, σ) near (b0, 0), it follows that �
+Lb(σ)−1 = ˇLb(σ)−1R(b, σ), which
+implies the invertibility of �
+Lb(σ) : X s,ℓ
+b
+→ ¯Hs−1,ℓ+1
+b
+( ¯X) for (b, σ) with Im σ ≥ 0, σ ̸= 0 near (b0, 0).
+This finishes the proof of the theorem.
+□
+We now give a more detailed description of the structure of the singular part of �
+Lb(σ)−1.
+Corollary 11.4. For (b, σ) with Im σ ≥ 0 near (b0, 0), we have
+�
+Lb(σ)−1 = P(b, σ) + L−
+b (σ) : ¯Hs−1,ℓ+2
+b
+( ¯X; S2 �
+scT ∗ ¯X ⊕ �
+scT ∗ ¯X) → ¯Hs,ℓ
+b ( ¯X; S2 �
+scT ∗ ¯X ⊕ �
+scT ∗ ¯X).
+The regular part L−
+b (σ) is uniformly bounded, and is continuous in σ with values in
+Lweak( ¯Hs−1,ℓ+2
+b
+( ¯X), ¯Hs,ℓ
+b ( ¯X)) ∩ Lop( ¯Hs−1+ǫ,ℓ+2+ǫ
+b
+( ¯X), ¯Hs−ǫ,ℓ−ǫ
+b
+( ¯X))
+for ǫ > 0.
+The singular part P(b, σ) satisfies
+P(b, σ)f = (σ−2(˙g1, ˙A1) − iσ(ˇg1, ˇA1)) + iσ−1�
+(˙g2, ˙A2) + (˙g′
+1, ˙A′
+1)
+�
++ iσ−1(˙g′′
+1 (σ), ˙A′′
+1(σ))
+(11.29)
+where
+(˙g1, ˙A1), (˙g′
+1, ˙A′
+1), (˙g′′
+1(σ), ˙A′′
+1(σ)) ∈ Kb,1,
+(ˇg1, ˇA1) ∈ ˇKb
+with (˙g′′
+1 (σ), ˙A′′
+1(σ)) = o(1) in Kb,1 as σ → 0, and
+(˙g2, ˙A2) ∈ Kb,2.
+Moreover, (˙g1, ˙A1), (˙g′
+1, ˙A′
+1) and (˙g2, ˙A2) are determined by the conditions
+−
+�
+[Lb, tb,∗](ˇg1, ˇA1) + 1
+2[[Lb, tb,∗], tb,∗](˙g1, ˙A1), (˙g∗, ˙A∗)
+�
++
+�
+[Lb, tb,∗](˙g2, ˙A2), (˙g∗, ˙A∗)
+�
+= ⟨f, (˙g∗, ˙A∗)⟩
+for all
+(˙g∗, ˙A∗) ∈ K∗
+b,
+(11.30)
+�
+[Lb, tb,∗](ˇg′
+1, ˇA′
+1) + 1
+2[[Lb, tb,∗], tb,∗](˙g′
+1, ˙A′
+1), (˙g∗
+1, ˙A∗
+1)
+�
+= −⟨f, (ˇg∗
+1, ˇA∗
+1)⟩
+−
+�1
+2[[Lb, tb,∗], tb,∗](˙g1, ˙A1) + [Lb, tb,∗](˙g2 − ˇg1, ˙A2 − ˇA1), (ˇg∗
+1, ˇA∗
+1)
+�
++
+�1
+2[[Lb, tb,∗], tb,∗](ˇg1 − ˙g2, ˇA1 − ˙A2), (˙g∗
+1, ˙A∗
+1)
+�
+for all
+(˙g∗
+1, ˙A∗
+1) ∈ K∗
+b,1.
+(11.31)
+Proof. For the simplicity of notation, we define
+�K0 := �K⊥
+b ,
+�K1 := �Kb,1,
+�K2 := �Kb,2,
+R0 := Rb,
+R1 := R⊥
+b,1,
+R2 := R⊥
+b,2.
+
+136
+LILI HE
+First, we prove that if an operator A(σ) : �Ki → Rj is invertible, and continuous in σ (in norm operator
+topology) in a small neighborhood of σ = 0, then so is A(σ)−1. We write
+A(σ) = A(0) + Ae(σ) = A(0)(I + A(0)−1Ae(σ))
+with
+Ae(σ) = A(σ) − A(0) = o(1)
+in
+Lop( �Ki, Rj)
+as
+σ → 0.
+Since ∥A(0)−1Ae(σ)∥ �
+Ki→ �
+Ki < 1/2 when σ is near 0, the inverse of A(σ) is given by the Neumann series
+A(σ)−1 =
+�
+I +
+∞
+�
+k=1
+(−1)k�
+A(0)−1Ae(σ)
+�k�
+A(0)−1,
+from which we find that
+∥A(σ)−1 − A(0)−1∥Rj→ �
+Ki ≤
+� ∞
+�
+k=1
+∥A(0)−1Ae(σ)∥k
+�
+Ki→ �
+Ki
+�
+∥A(0)−1∥Rj→ �
+Ki
+≤ 2∥A(0)−1∥2
+Rj→ �
+Ki∥Ae(σ)∥ �
+Ki→Rj = o(1)
+in
+Lop(Rj, �Ki)
+as
+σ → 0.
+This proves the continuity of A(σ)−1 in the norm operator topology in a small neighborhood of σ = 0.
+Applying this result to L00 and �L♭
+22 (defined in (11.28)) yields the continuity (in the norm operator topology)
+of L−1
+00 and (�L♭
+22)−1. Then the continuity of �Rij follows from the explicit expressions in (11.27). Furthermore,
+we find that
+�R21(b, σ) = �R0
+21(b) + σ �Re
+21(b, σ)
+with
+�R0
+21(b) = −(�L0
+22(b))−1�L0
+21(b)(�L0
+11(b))−1,
+�R22(b, σ) = �R0
+22(b) + σ �Re
+22(b, σ)
+with
+�R0
+22(b) = (�L0
+22(b))−1,
+�R11(b, σ) = �R0
+11(b) + σ �Re
+11(b, σ)
+with
+�R0
+11(b) = (�L0
+11(b))−1
+(11.32)
+where �Re
+21, �Re
+22, �Re
+11 are continuous in (b, σ) for (b, σ) near (b0, 0) in the norm operator topology. Therefore,
+we rewrite
+R(b, σ) = Rsing(b, σ) + Rreg(b, σ) =
+
+
+
+0
+0
+0
+�
+R10(b,σ)
+σ
+�
+R0
+11(b)
+σ2
++
+�
+Re
+11(b,σ)
+σ
+�
+R12(b,σ)
+σ
+0
+�
+R0
+21(b)
+σ
+�
+R0
+22(b)
+σ
+
+
+ + Rreg(b, σ)
+(11.33)
+where all the entries in Rreg(b, σ) are continuous in (b, σ) in the norm operator topology.
+For (˜g, ˜A) = ˇLb(0)(˙g, ˙A) ∈ �Kb,1 with (˙g, ˙A) ∈ Kb,1, we have
+ˇLb(σ)−1(˜g, ˜A) = ˇLb(σ)−1 ˇLb(0)(˙g, ˙A) = (˙g, ˙A) − iσ ˇLb(σ)−1�
+Lb(0)(ˇg, ˇA) − σ2
+2
+ˇLb(σ)−1∂2
+σ�
+Lb(0)(˙g, ˙A)
+= (˙g, ˙A) − iσ ˇLb(σ)−1�ˇLb(σ) + �
+Lb(0) − �
+Lb(σ) − Vb
+�
+(ˇg, ˇA) − σ2
+2
+ˇLb(σ)−1∂2
+σ�
+Lb(0)(˙g, ˙A)
+= (˙g, ˙A) − iσ(ˇg, ˇA) + σ2 ˇLb(σ)−1�
+i(∂σ�
+Lb(0) + σ
+2 ∂2
+σ�
+Lb(0))(ˇg, ˇA) − 1
+2∂2
+σ�
+Lb(0)(˙g, ˙A)
+�
+(11.34)
+where the coefficient of σ2 is continuous in (b, σ) in the norm operator topology.
+For (˜g, ˜A) = ˇLb(0)(˙g, ˙A) ∈ �Kb,2 with (˙g, ˙A) ∈ Kb,2, we have
+ˇLb(σ)−1(˜g, ˜A) = ˇLb(σ)−1 ˇLb(0)(˙g, ˙A) = (˙g, ˙A) − σ ˇLb(σ)−1��
+∂σ�
+Lb(0) + σ
+2 ∂2
+σ�
+Lb(0)
+�
+(˙g, ˙A)
+�
+(11.35)
+where the coefficient of σ is continuous in (b, σ) in the norm operator topology.
+Since Rreg(b, σ) is continuous in σ in norm operator topology and ˇLb(σ)−1 is continuous in σ with
+values in Lweak( ¯Hs−1,ℓ+2
+b
+( ¯X), ¯Hs,ℓ
+b ( ¯X)) ∩ Lop( ¯Hs−1+ǫ,ℓ+2+ǫ
+b
+( ¯X), ¯Hs−ǫ,ℓ−ǫ
+b
+( ¯X)) for ǫ > 0, it follows that
+ˇLb(σ)−1Rreg(b, σ) is uniformly bounded and has the same continuity as ˇLb(σ)−1. According to (11.34) and
+(11.35), we conclude that ˇLb(σ)−1Rsing(b, σ) is equal to a singular part P(b, σ) plus a uniformly bounded
+and continuous operator with value in Lop( ¯Hs−1,ℓ+2
+b
+( ¯X), ¯Hs,ℓ
+b ( ¯X)). Moreover, P(b, σ) satisfies
+P(b, σ)f = (σ−2(˙g1, ˙A1) − iσ−1(ˇg1, ˇA1)) + iσ−1�
+(˙g2, ˙A2) + (˙g′
+1, ˙A′
+1)
+�
++ iσ−1(˙g′′
+1(σ), ˙A′′
+1(σ))
+(11.36)
+
+LINEAR STABILITY OF KERR-NEWMAN FAMILY
+137
+where (˙g1, ˙A1), (˙g′
+1, ˙A′
+1), (˙g′′
+1(σ), ˙A′′
+1(σ)) ∈ Kb,1 with (˙g′′
+1(σ), ˙A′′
+1(σ)) = o(1) in Kb,1 as σ → 0, and (˙g2, ˙A2) ∈
+Kb,2.
+It remains to prove that (˙g1, ˙A1), (˙g′
+1, ˙A′
+1), (˙g2, ˙A2) are determined by conditions (11.30) and (11.31). Let
+f ∈ ¯Hs−1,ℓ+2
+b
+( ¯X) and (˙g(σ), ˙A(σ)) := �
+Lb(σ)−1f, and then we have
+(˙g(σ), ˙A(σ)) = (σ−2(˙g1, ˙A1) − iσ−1(ˇg1, ˇA1)) + iσ−1�
+(˙g2, ˙A2) + (˙g′
+1, ˙A′
+1)
+�
++ iσ−1(˙g′′
+1(σ), ˙A′′
+1(σ))
++ (˜˙g, ˜˙A) + (˜˙g(σ), ˜˙A(σ))
+where (˜˙g, ˜˙A) ∈ ¯Hs,ℓ
+b ( ¯X) and (˜˙g(σ), ˜˙A(σ)) = o(1) in ¯Hs−ǫ,ℓ−ǫ
+b
+( ¯X) as σ → 0. Applying �
+Lb(σ) on both sides
+gives
+f = �
+Lb(σ)(˙g(σ), ˙A(σ)) = �
+Lb(0)(˙g(σ), ˙A(σ)) +
+�
+σ∂σ�
+Lb(0) + σ2
+2 ∂2
+σ�
+Lb(0)
+�
+(˙g(σ), ˙A(σ))
+=
+�1
+2∂2
+σ�
+Lb(0)(˙g1, ˙A1) − i∂σ�
+Lb(0)(ˇg1, ˇA1) − �
+Lb(0)(ˇg′
+1, ˇA′
+1) + i∂σ�
+Lb(0)(˙g2, ˙A2) + �
+Lb(0)(˜˙g, ˜˙A)
+�
++ i∂σ�
+Lb(0)(˙g′′
+1(σ), ˙A′′
+1(σ)) + �
+Lb(0)(˜˙g(σ), ˜˙A(σ))
++ iσ
+2
+�
+− ∂2
+σ�
+Lb(0)(ˇg1, ˇA1) + ∂2
+σ�
+Lb(0)(˙g′
+1, ˙A′
+1) + ∂2
+σ�
+Lb(0)(˙g2, ˙A2))
+�
++ iσ
+2 ∂2
+σ�
+Lb(0)(˙g′′
+1(σ), ˙A′′
+1(σ) +
+�
+σ∂σ�
+Lb(0) + σ2
+2 ∂2
+σ�
+Lb(0)
+�
+(˜˙g + ˜˙g(σ), ˜˙A + ˜˙A(σ)).
+(11.37)
+Letting σ → 0, we arrive at
+f = 1
+2∂2
+σ�
+Lb(0)(˙g1, ˙A1) − i∂σ�
+Lb(0)(ˇg1, ˇA1) − �
+Lb(0)(ˇg′
+1, ˇA′
+1) + i∂σ�
+Lb(0)(˙g2, ˙A2) + �
+Lb(0)(˜˙g, ˜˙A).
+(11.38)
+Pairing both sides of (11.38) with (˙g∗, ˙A∗) ∈ K∗
+b, we obtain
+⟨f, (˙g∗, ˙A∗)⟩ =
+�1
+2∂2
+σ�
+Lb(0)(˙g1, ˙A1) − i∂σ�
+Lb(0)(ˇg1, ˇA1) + i∂σ�
+Lb(0)(˙g2, ˙A2), (˙g∗, ˙A∗)
+�
+= −
+�
+[Lb, tb,∗](ˇg1, ˇA1) + 1
+2[[Lb, tb,∗], tb,∗](˙g1, ˙A1) − [Lb, tb,∗](˙g2, ˙A2), (˙g∗, ˙A∗)
+�
+(11.39)
+Since the pairing on Kb × K∗
+b on the right-hand side of (11.39) is non-degenerate, (˙g1, ˙A1) (thus (ˇg1, ˇA1))
+and (˙g2, ˙A2) are uniquely determined by (11.39). This proves the condition (11.30).
+Rewriting (11.38) gives rise to the following PDE
+�
+Lb(0)
+�
+(ˇg′
+1, ˇA′
+1) − (˜˙g, ˜˙A)
+�
+= −f + 1
+2∂2
+σ�
+Lb(0)(˙g1, ˙A1) − i∂σ�
+Lb(0)(ˇg1, ˇA1) + i∂σ�
+Lb(0)(˙g2, ˙A2)
+(11.40)
+whose right-hand side is uniquely determined by f. Due to (11.39), the pairings of the right-hand side
+of (11.40) with any (˙g∗, ˙A∗) ∈ K∗
+b is 0, which implies that (11.40) can be solved for (ˇg′
+1, ˇA′
+1) − (˜˙g, ˜˙A) and
+(ˇg′
+1, ˇA′
+1) − (˜˙g, ˜˙A) is uniquely determined by f modulo Kb. Namely,
+(ˇg′
+1, ˇA′
+1) − (˜˙g, ˜˙A) = (¯˙g, ¯˙A) + (˙g′′′, ˙A′′′)
+(11.41)
+where (¯˙g, ¯˙A) is uniquely determined by f and (˙g′′′, ˙A′′′) ∈ Kb. Pairing both sides of (11.37) with (˙g∗
+1, ˙A∗
+1) ∈
+K∗
+b,1 and using the identity (11.39), we obtain, upon multiplying σ−1 and letting σ → 0,
+0 = i
+2
+�
+− ∂2
+σ�
+Lb(0)(ˇg1, ˇA1) + ∂2
+σ�
+Lb(0)(˙g′
+1, ˙A′
+1) + ∂2
+σ�
+Lb(0)(˙g2, ˙A2) − 2i∂σ�
+Lb(0)(˜˙g, ˜˙A), (˙g∗
+1, ˙A∗
+1)
+�
+= i
+2
+�
+− ∂2
+σ�
+Lb(0)(ˇg1, ˇA1) + ∂2
+σ�
+Lb(0)(˙g2, ˙A2) + 2i∂σ�
+Lb(0)(¯˙g, ¯˙A), (˙g∗
+1, ˙A∗
+1)
+�
++ i
+2
+�
+∂2
+σ�
+Lb(0)(˙g′
+1, ˙A′
+1) − 2i∂σ�
+Lb(0)(ˇg′
+1, ˇA′
+1), (˙g∗
+1, ˙A∗
+1)
+�
++ i⟨(˙g′′′, ˙A′′′), �
+Lb(0)∗(ˇg∗
+1, ˇA∗
+1)⟩
+where we use (11.41) and −i(∂σ�
+Lb(0))∗(˙g∗
+1, ˙A∗
+1) = �
+Lb(0)∗(ˇg∗
+1, ˇA∗
+1) in the second step. Since
+⟨(˙g′′′, ˙A′′′), �
+Lb(0)∗(ˇg∗
+1, ˇA∗
+1)⟩ = ⟨�
+Lb(0)(˙g′′′, ˙A′′′), (ˇg∗
+1, ˇA∗
+1)⟩ = 0,
+
+138
+LILI HE
+we arrive at
+�
+[[Lb, tb,∗], tb,∗](˙g′
+1, ˙A′
+1) + 2[Lb, tb,∗](ˇg′
+1, ˇA′
+1), (˙g∗
+1, ˙A∗
+1)
+�
+=
+�
+[[Lb, tb,∗], tb,∗](ˇg1, ˇA1) − [[Lb, tb,∗], tb,∗](˙g2, ˙A2) + 2[Lb, tb,∗](¯˙g, ¯˙A), (˙g∗
+1, ˙A∗
+1)
+�
+.
+(11.42)
+Since the pairing on Kb,1 × K∗
+b,1 on the left-hand side of (11.42) is non-degenerate and the right-hand side is
+uniquely determined by f, so is (˙g′
+1, ˙A′
+1). Actually, we can further calculate the right-hand side
+�
+[Lb, tb,∗](¯˙g, ¯˙A), (˙g∗
+1, ˙A∗
+1)
+�
+=
+�
+(¯˙g, ¯˙A), �
+Lb(0)∗(ˇg∗
+1, ˇA∗
+1)
+�
+=
+�
+�
+Lb(0)(¯˙g, ¯˙A), (ˇg∗
+1, ˇA∗
+1)
+�
+= −⟨f + 1
+2[[Lb, tb,∗], tb,∗](˙g1, ˙A1) + [Lb, tb,∗](˙g2 − ˇg1, ˙A2 − ˇA1), (ˇg∗
+1, ˇA∗
+1)⟩
+where we use (11.40) and (11.41) in the third step. Plugging this into (11.42) proves the condition (11.31).
+□
+12. Regularity of the resolvent of the linearized gauge-fixed Einstein-Maxwell operator
+In this section, we continue using the notations from §11.
+In §12.1, we give a refined description of
+the singular part P(b, σ) and the regular part L−
+b (σ) of the operator �
+Lb(0)−1 defined in Corollary 11.4.
+Concretely, we prove that (˙g′′
+1(σ), ˙A′′
+1(σ)) defined in Corollary 11.4 is H¨older regular at σ = 0, while L−
+b (σ) is
+H¨older regular at σ = 0 only when one relaxes the target space. In §12.2, we provide estimates of the bounds
+of any derivatives (in σ) of L−
+b (σ) for σ in the region 0 ≤ Im σ ≤ C, |σ| ≥ c0 for some c0, C > 0. Finally,
+in §12.3, we discuss the conormal regularity of L−
+b (σ) at σ = 0. Specifically, after applying any number of
+derivatives σ∂σ to L−
+b (σ), it satisfies the same properties as L−
+b (σ).
+12.1. Regularity at low frequencies. In §12.1.1, we study in detail the structure of �
+Lb(σ)ˇLb(σ)−1, i.e.,
+prove the ǫ-regularity of �
+Lb(σ)ˇLb(σ)−1. In §12.1.2, we establish the ǫ-regularity of R(b, σ), which allows us
+to give a refined description of the singular part P(b, σ) of �
+Lb(σ)−1 near σ = 0. We also discuss a certain
+H¨older regularity of the regular part L−
+b (σ) at σ = 0.
+12.1.1. ǫ-regularity of �
+Lb(σ)ˇLb(σ)−1. First, we introduce several definitions.
+Definition 12.1. Let X, Y be two Banach spaces and α, β ∈ R. Suppose A(σ) : X → Y is a family of
+operators depending on parameter σ ∈ C \ {0}. Then we write
+A(σ) : |σ|αX → |σ|βY
+if and only if there exists a constant C > 0 such that ∥A(σ)∥X→Y ≤ C|σ|β−α for all σ ∈ C \ {0}.
+Let �Ki, Ri (i = 0, 1, 2) be defined as in §11.
+Definition 12.2.
+(1) An operator L(σ) : �Ki → Rj is ǫ-regular at σ = 0 if for σ ∈ C, Im σ ≥ 0 and |σ|
+small, there exists a constant C > 0 such that
+∥L(σ)∥ �
+Ki→Rj ≤ C,
+(12.1a)
+∥∂σL(σ)∥ �
+Ki→Rj ≤ C|σ|−ǫ,
+(12.1b)
+∥∂2
+σL(σ)∥ �
+Ki→Rj ≤ C|σ|−ǫ−1.
+(12.1c)
+(2) An operator L(σ) : �Ki → Rj has an ǫ-regular expansion at σ = 0 in σ up to order one if
+L(σ) = L0 + σLe(σ)
+(12.2)
+where L0 : �Ki → Rj is independent of σ and Le(σ) : �Ki → Rj is ǫ-regular at σ = 0.
+(3) An operator L(σ) : �Ki → Rj has an ǫ-regular expansion at σ = 0 in σ up to order two if
+L(σ) = L0 + σL1 + σ2Le(σ)
+(12.3)
+where L0, L1 : �Ki → Rj are independent of σ and Le(σ) : �Ki → Rj is ǫ-regular at σ = 0.
+
+LINEAR STABILITY OF KERR-NEWMAN FAMILY
+139
+We recall the analysis of the entries Lij of the operator in the proof of Theorem 11.3, and note that each en-
+try is a sum of polynomials in σ whose coefficients are either independent of σ, terms involving ˇLb(σ)−1 acting
+on an element in ¯H∞,3/2−
+b
+( ¯X), or terms involving (ˇLb(σ)−1)∗ acting on an element in ˙H−5/2−C(a,γ),3/2−
+b
+( ¯X)
+where C(a, γ0) is a sufficiently small constant depending on a, γ. Therefore, we first discuss the properties
+of the operator ˇLb(σ)−1.
+Proposition 12.3. Let − 3
+2 < ℓ < − 1
+2, 0 < ǫ < 1 with − 1
+2 < ℓ + ǫ < 1
+2, and s − ǫ > 4. For Im σ ≥ 0,
+0 < |σ| ≤ c0 with c0 small, we have
+∂σ ˇLb(σ)−1 : ¯Hs−1,ℓ+2
+b
+( ¯X) → |σ|−ǫ ¯Hs−ǫ,ℓ+ǫ−1
+b
+( ¯X),
+(12.4)
+∂2
+σ ˇLb(σ)−1 : ¯Hs−1,ℓ+2
+b
+( ¯X) → |σ|−ǫ−1 ¯Hs−1−ǫ,ℓ+ǫ−1
+b
+( ¯X).
+(12.5)
+Proof. Formally, we have
+∂σ ˇLb(σ)−1 = −ˇLb(σ)−1(∂σ ˇLb(σ))ˇLb(σ)−1
+where by Proposition 4.13
+∂σ ˇLb(σ) = ∂σ�
+Lb(σ) ∈ 2iρ(ρ∂ρ − 1) + ρ3Diff1
+b + ρ2C∞ + σρ2C∞.
+(12.6)
+Motivated by this, we first analyze ρ(ρ∂ρ − 1)ˇLb(σ)−1 and ˇLb(σ)−1ρ(ρ∂ρ − 1). Since ρ(ρ∂ρ − 1) : ¯Hs,ℓ
+b ( ¯X) →
+¯Hs−1,ℓ+1
+b
+( ¯X), it is clear that
+ρ(ρ∂ρ − 1)ˇLb(σ)−1 : ¯Hs−1,ℓ+2
+b
+( ¯X) → ¯Hs−1,ℓ+1
+b
+( ¯
+X).
+Meanwhile, in view of Proposition 4.13
+2iρ(ρ∂ρ − 1) = σ−1�ˇLb(σ) − ˇLb(0) + Q
+�
+with
+Q ∈ σρ3Diff1
+b + σρ2C∞ + σρ2C∞,
+then it follows that
+2iρ(ρ∂ρ − 1)ˇLb(σ)−1 = σ−1�
+I − (�
+Lb(0) + Vb + Q)ˇLb(σ)−1�
+: ¯Hs−1,ℓ+2
+b
+( ¯X) → |σ|−1 ¯Hs−2,ℓ+2
+b
+( ¯X).
+Then an interpolation gives
+ρ(ρ∂ρ − 1)ˇLb(σ)−1 : ¯Hs−1,ℓ+2
+b
+( ¯
+X) → |σ|−ǫ ¯Hs−1−ǫ,ℓ+1+ǫ
+b
+( ¯X),
+0 ≤ ǫ ≤ 1.
+(12.7)
+Since s − 1 − ǫ > 2 and 1/2 < ℓ + 1 + ǫ < 3/2, we have
+¯Hs−1,ℓ+2
+b
+( ¯
+X)
+2iρ(ρ∂ρ−1)ˇLb(σ)−1
+−−−−−−−−−−−−→ |σ|−ǫ ¯Hs−1−ǫ,ℓ+ǫ+1
+b
+( ¯X)
+ˇLb(σ)−1
+−−−−−→ |σ|−ǫ ¯Hs−ǫ,ℓ+ǫ−1
+b
+( ¯
+X).
+(12.8)
+We also have
+¯Hs−1,ℓ+2
+b
+( ¯X)
+ˇLb(σ)−1
+−−−−−→ ¯Hs,ℓ
+b ( ¯X)
+ρ2Diff
+1
+b+σρ2C∞
+−−−−−−−−−−→ ¯Hs−1,ℓ+2
+b
+( ¯X)
+ˇLb(σ)−1
+−−−−→ ¯Hs,ℓ
+b ( ¯X) ⊂ |σ|−ǫ ¯Hs−ǫ,ℓ+ǫ−1
+b
+( ¯X).
+(12.9)
+By (12.6), we finishes the proof of (12.4).
+For ∂2
+σ ˇLb(σ)−1, we note that
+∂2
+σ ˇLb(σ)−1 = 2ˇLb(σ)−1(∂σ ˇLb(σ))ˇLb(σ)−1(∂σ ˇLb(σ))ˇLb(σ)−1 − ˇLb(σ)−1(∂2
+σ ˇLb(σ))ˇLb(σ)−1
+:= I + II.
+Since ∂2
+σ ˇLb(σ) ∈ ρ2C∞, according to (12.9), we obtain
+II : ¯Hs−1,ℓ+2
+b
+( ¯X) → ¯Hs,ℓ
+b ( ¯X) ⊂ |σ|−ǫ−1 ¯Hs−1−ǫ,ℓ+ǫ−1
+b
+( ¯X).
+It remains to analyze I. By the assumption on s, ℓ, ǫ, we find that 1/2 < (1 + ǫ)/2 < 1, and thus using the
+reasoning in (12.8) and (12.9), we obtain
+¯Hs−1,ℓ+2
+b
+( ¯X)
+∂σ ˇLb(σ)ˇLb(σ)−1
+−−−−−−−−−−→ |σ|− 1+ǫ
+2
+¯H
+s−1− 1+ǫ
+2 ,ℓ+ 1+ǫ
+2 +1
+b
+( ¯
+X)
+∂σ ˇLb(σ)ˇLb(σ)−1
+−−−−−−−−−−→ |σ|−1−ǫ ¯Hs−2−ǫ,ℓ+ǫ+1
+b
+( ¯X)
+ˇLb(σ)−1
+−−−−−→ |σ|−1−ǫ ¯Hs−1−ǫ,ℓ+ǫ−1
+b
+( ¯X)
+where we use s − 3
+2 − ǫ
+2 > 2, 1
+2 < ℓ + 3
+2 + ǫ
+2 < 3
+2 in the second step and s − 2 − ǫ > 2, 1
+2 < ℓ + ǫ + 1 < 3
+2 in
+the last step. This finishes the proof of (12.5).
+□
+
+140
+LILI HE
+Recall the expression (11.25) of �
+Lb(σ)ˇL(σ)−1
+�
+Lb(σ)ˇLb(σ)−1 =
+
+
+
+L00(b, σ)
+σ2 �L01(b, σ)
+σ�L02(b, σ)
+σ�L10(b, σ)
+σ2 �L11(b, σ)
+σ2 �L12(b, σ)
+σ�L20(b, σ)
+σ2 �L21(b, σ)
+σ�L22(b, σ)
+
+
+
+=
+
+
+
+L00(b, σ)
+σ2�L01(b, σ)
+σ�L02(b, σ)
+σ
+��L0
+10(b) + σ�Le
+10(b, σ)
+�
+σ2��L0
+11(b) + σ�L1
+11(b) + σ2�Le
+11(b, σ)
+�
+σ2��L0
+12(b) + σ�Le
+12(b, σ)
+�
+σ�L20(b, σ)
+σ2��L0
+21(b) + σ�Le
+21(b)
+�
+σ
+��L0
+22(b) + σ�Le
+22(b, σ)
+�
+.
+
+
+
+Proposition 12.4. Let s, ℓ, ǫ be defined as in Proposition 12.3. The entries of �
+Lb(σ)ˇLb(σ)−1 satisfy the
+following properties.
+(1) L00, �L01, �L02, �L20 are ǫ-regular at σ = 0.
+(2) �L10, �L12, �L21, �L22 have an ǫ-regular expansion up to order one, that is, �Le
+10, �Le
+12, �Le
+21, �Le
+22 are ǫ-regular
+at σ = 0.
+(3) �L11 has an ǫ-regular expansion up to order two, that is, �Le
+11 is ǫ-regular at σ = 0.
+Proof. The key point in the proof was already mentioned before Proposition 12.3.
+• Analysis of �Le
+10 and �L20. Here, we only discuss �Le
+10 in detail because the proof of �L02 follows in a
+similar manner. Recall the calculation in (11.18)
+�Le
+10(˜g0, ˜A0), (˙g∗
+1, ˙A∗
+1)⟩
+= −
+�
+(˜g0, ˜A0), i(ˇLb(σ)−1)∗(∂σ�
+Lb(0) + σ
+2 ∂2
+σ�
+Lb(0))∗�
+I − (ˇLb(0)−1)∗V ∗
+b
+�
+(ˇg∗
+2, ˇA∗
+2)
+�
++
+�
+(˜g0, ˜A0), 1
+2(ˇLb(σ)−1)∗(∂2
+σ�
+Lb(0))∗(˙g∗
+2, ˙A∗
+2)
+�
+where
+(∂σ�
+Lb(0) + σ
+2 ∂2
+σ�
+Lb(0))∗�
+I − (ˇLb(0)−1)∗V ∗
+b
+�
+(ˇg∗
+2, ˇA∗
+2),
+(∂2
+σ�
+Lb(0))∗(˙g∗
+2, ˙A∗
+2) ∈ ˙H−5/2−C(a,γ),3/2−
+b
+( ¯X).
+Since (˜g0, ˜A0) ∈ ¯Hs−1,ℓ+2
+b
+( ¯X), and by Proposition (12.3)
+(ˇL−1
+b (σ))∗ ˙H−5/2−C(a,γ), 3/2−
+b
+( ¯X) ∈ ˙H−3/2−C(a,γ),−1/2−
+b
+( ¯X),
+(∂j
+σ ˇL−1
+b )∗ ˙H−5/2−C(a,γ), 3/2−
+b
+( ¯X) ∈ |σ|−ǫ−j+1 ˙H−s+1,−ℓ−2
+b
+( ¯X),
+j = 1, 2
+where we use the facts that −5/2 − C(a, γ) > −3 > −s + ǫ + 1 and −ℓ − ǫ + 1 < 3/2, the pairings
+above (and their up to second order derivatives) are well-defined. This proves the ǫ-regularity of �Le
+10.
+• Analysis of �Le
+11, �Le
+12, �Le
+22 and �Le
+22. The statements for these entries can be proved in the same way
+as in the proof of �Le
+10.
+• Analysis of �L01 and �L02. Here, we only prove the statement for �L01 as the proof of �L02 follows in an
+analogous manner. Using the calculation in (11.16), we find that
+�L01(˜g1, ˜A1)=
+�
+− iVb ˇLb(σ)−1�
+∂σ�
+Lb(0) + σ
+2 ∂2
+σ�
+Lb(0)
+�
+(ˇg1, ˇA1) + 1
+2Vb ˇLb(σ)−1∂2
+σ�
+Lb(0)(˙g1, ˙A1)
+�
+− (�L11 + �L21)(˜g1, ˜A1).
+where
+∂σ�
+Lb(0)(ˇg1, ˇA1),
+∂2
+σ�
+Lb(0)(ˇg1, ˇA1),
+∂2
+σ�
+Lb(0)(˙g1, ˙A1) ∈ ¯H∞,3/2−
+b
+( ¯X).
+Since by Proposition 12.4
+ˇL−1
+b (σ) ¯H∞,3/2−
+b
+( ¯X) ∈ ¯H∞,−1/2−
+b
+( ¯X), ∂j
+σ ˇL−1
+b (σ) ∈ |σ|−ǫ−j+1 ¯H∞,ℓ+ǫ−1
+b
+( ¯X), j = 1, 2,
+and Vb maps D′(X◦) to C∞
+c (X), it follows that the first term on the right-hand side is ǫ-regular.
+Meanwhile, it is clear that the second term is ǫ-regular. This proves the ǫ-regularity for �L01.
+
+LINEAR STABILITY OF KERR-NEWMAN FAMILY
+141
+• Analysis of L00. By the calculation in (11.24), we have
+L00(b, σ)(˜g0, ˜A0) = (˜g0, ˜A0) − Vb ˇLb(σ)−1(˜g0, ˜A0) −
+�
+σ�L0
+10(b) + σ2 �Le
+10(b, σ) + σ�L20(b, σ)
+�
+(˜g0, ˜A0).
+Again, it is clear that the third term on the right-hand side is ǫ-regular. Since by Proposition 12.4
+ˇL−1
+b (σ)(˜g0, ˜A0) ∈ ¯Hs,ℓ
+b ( ¯X), ∂j
+σ ˇL−1
+b (σ) ∈ |σ|−ǫ−j+1 ¯Hs−ǫ−j+1,ℓ+ǫ−1
+b
+( ¯X), j = 1, 2,
+and Vb maps D′(X◦) to C∞
+c (X), it follows that the second term on the right-hand side is ǫ-regular.
+This finishes the proof.
+□
+12.1.2. H¨older regularity of P(b, σ) and L−
+b (σ) at σ = 0. Now we are ready to analyze the ǫ-regularity of
+R(b, σ). To this end, we need the following lemma.
+Lemma 12.5. Let s, ℓ, ǫ be defined as in Proposition 12.3. Suppose that the operator A(σ) : �Ki → Rj with
+i, j = 0, 1, 2 has an ǫ-regular expansion up to order one, i.e.,
+A(σ) = A0 + σAe(σ)
+where A0 : �Ki → Rj is invertible and independent of σ, and Ae(σ) : �Ki → Rj is ǫ-regular. Then for |σ|
+sufficiently small, A(σ) is invertible, and its inverse B(σ) = A(σ)−1 also has an ǫ-regular expansion up to
+order one, that is,
+B(σ) = B0 + σBe(σ)
+where B0 : Rj → �Ki is independent of σ, and Be(σ) : Rj → �Ki is ǫ-regular.
+Similarly, if A(σ) is ǫ-regular and has uniformly bounded inverse B(σ), then B(σ) is ǫ-regular as well.
+Proof. Since Ae(σ) is ǫ-regular (and thus uniformly bounded), ∥σ(A0)−1Ae(σ)∥ �
+Ki→ �
+Ki < 1/2 when |σ| is
+sufficiently small, the inverse of A(σ) is given by the Neumann series
+A(σ)−1=
+�
+I +
+∞
+�
+k=1
+(−1)k�
+σ(A0)−1Ae(σ)
+�k�
+(A0)−1
+= (A0)−1 − σ
+�
+(A0)−1Ae(σ)
+∞
+�
+k=0
+(−1)k�
+σ(A0)−1Ae(σ)
+�k(A0)−1�
+:= B0 + σBe(σ).
+First, we have
+∥Be(σ)∥Rj→ �
+Ki ≤
+� ∞
+�
+k=0
+∥σ(A0)−1Ae(σ)∥k
+�
+Ki→ �
+Ki
+�
+∥(A0)−1∥2
+Rj→ �
+Ki∥Ae(σ)∥ �
+Ki→Rj
+≤ 2∥A(0)−1∥2
+Rj→ �
+Ki∥Ae(σ)∥ �
+Ki→Rj,
+which implies the uniform boundedness of Be(σ). Secondly, the first derivative of Be(σ) is a sum of the
+terms of the types
+σ−2�
+σ(A0)−1Ae(σ)
+�k(A0)−1 ≲ 2−k+2,
+k ≥ 2
+�
+σ(A0)−1Ae(σ)
+�k1 ◦
+�
+(A0)−1∂σAe(σ)
+�
+◦
+�
+σ(A0)−1Ae(σ)
+�k2(A0)−1 ≲ 2−k1−k2|σ|−ǫ,
+where we use ∥∂σAe(σ)∥ �
+Ki→Rj ≲ |σ|−ǫ. As a consequence, ∥∂σBe(σ)∥Rj→ �
+Ki ≲ |σ|−ǫ. Finally, the second
+derivative of Be(σ) is a sum of the terms of the types
+σ−3�
+σ(A0)−1Ae(σ)
+�k(A0)−1 ≲ 2−k+3,
+k ≥ 3
+σ−1�
+σ(A0)−1Ae(σ)
+�k1 ◦
+�
+(A0)−1∂σAe(σ)
+�
+◦
+�
+σ(A0)−1Ae(σ)
+�k2(A0)−1 ≲ 2−k1−k2+1|σ|−ǫ,
+k1 + k2 ≥ 1,
+σ
+�
+σ(A0)−1Ae(σ)
+�k1 ◦
+�
+(A0)−1∂σAe(σ)
+�
+◦
+�
+σ(A0)−1Ae(σ)
+�k2 ◦
+�
+(A0)−1∂σAe(σ)
+�
+◦
+�
+σ(A0)−1Ae(σ)
+�k3(A0)−1
+≲ 2−k1−k2−k3|σ|−2ǫ+1,
+�
+σ(A0)−1Ae(σ)
+�k1 ◦
+�
+(A0)−1∂2
+σAe(σ)
+�
+◦
+�
+σ(A0)−1Ae(σ)
+�k2(A0)−1 ≲ 2−k1−k2|σ|−ǫ−1,
+where we use ∂σAe(σ) ≲ |σ|−ǫ and ∂2
+σAe(σ) ≲ |σ|−ǫ−1. Therefore, ∂2
+σBe(σ) ≲ |σ|−ǫ−1. This proves that
+Be(σ) is ǫ-regular.
+
+142
+LILI HE
+The proof of the second statement is similar by using
+∂σB(σ) = −B(σ) ◦ ∂σA(σ) ◦ B(σ),
+∂2
+σB(σ) = −B(σ) ◦ ∂2
+σA(σ) ◦ B(σ) + 2B(σ) ◦ ∂σA(σ) ◦ B(σ) ◦ ∂σA(σ) ◦ B(σ).
+□
+Recall the expressions (11.26) and (11.32) of Rb,σ = (�
+Lb(σ)ˇLb(σ)−1)−1
+R(b, σ) = (�
+Lb(σ)ˇLb(σ)−1)−1 =
+
+
+
+�R00(b, σ)
+�R01(b, σ)
+�R02(b, σ)
+σ−1 �R10(b, σ)
+σ−2 �R11(b, σ)
+σ−1 �R12(b, σ)
+�R20(b, σ)
+σ−1 �R21(b, σ)
+σ−1 �R22(b, σ)
+
+
+
+=
+
+
+
+�R00(b, σ)
+�R01(b, σ)
+�R02(b, σ)
+σ−1 �R10(b, σ)
+σ−2� �R0
+11(b) + σ �Re
+11(b, σ)
+�
+σ−1 �R12(b, σ)
+�R20(b, σ)
+σ−1� �R0
+21(b) + σ �Re
+21(b, σ)
+�
+σ−1� �R0
+22(b) + σ �Re
+22(b, σ)
+�
+
+
+
+where
+�R11 =
+��L♯
+11 − σ�L♯
+12(�L♭
+22)−1�L♯
+21
+�−1,
+�R10 = �R11
+�
+− �L10 + σ�L♯
+12(�L♭
+22)−1�L20
+�
+L−1
+00 ,
+�R12 = − �R11�L♯
+12(�L♭
+22)−1,
+�R21 = −(�L♭
+22)−1�L♯
+21 �R11,
+�R22 = (�L♭
+22)−1 − σ(�L♭
+22)−1�L♯
+21 �R12,
+�R20 = −(�L♭
+22)−1��L20L−1
+00 + �L♯
+21 �R10
+�
+,
+�R01 = −L−1
+00
+��L01 �R11 + �L02 �R21
+�
+,
+�R02 = −L−1
+00
+�
+σ�L01 �R12 + �L02 �R22
+�
+,
+�R00 = L−1
+00
+�
+I − σ�L01 �R10 − σ�L02 �R20
+�
+(12.10)
+with
+�L♯
+i1 = �Li1 − σ�Li0L−1
+00 �L01
+for
+i = 1, 2,
+�L♯
+12 = �L12 − �L10L−1
+00 �L02,
+�L♭
+22 = �L22 − σ�L20L−1
+00 �L02.
+(12.11)
+Proposition 12.6. Let s, ℓ, ǫ be defined as in Proposition 12.3. The entries of R(b, σ) satisfy the following
+properties.
+(1) �R00, �R01, �R02, �R10, �R12, �R20 are ǫ-regular at σ = 0.
+(2) �R11, �R21, �R22 have an ǫ-regular expansion up to order one, that is, �Re
+11, �Re
+21, �Re
+22 are ǫ-regular at
+σ = 0.
+Proof. First, since L00 is ǫ-regular, by Lemma 12.5, so is L−1
+00 . Then by Proposition 12.4, we find that
+�L♯
+11, �L♯
+21, �L♭
+22 (and thus (�L♭
+22)−1 by Lemma 12.5) have an ǫ-regular expansion up to order one, while �L♯
+12 is
+ǫ-regular. Therefore, we see that
+�L♯
+11 − σ�L♯
+12(�L♭
+22)−1�L♯
+21
+has an ǫ-regular expansion up to order one whose coefficient of σ0 is �L0
+11(b). Then the ǫ-regular expansion
+property of �R11 follows from Lemma 12.5. Similarly, the ǫ-regular expansion property of �R21, �R22 follows
+from that of �R11, (�L♭
+22)−1, �L♯
+21 and the above expressions of �Rij. Finally, the ǫ-regularity of the remaining
+components can be obtained by using the above expressions of �Rij and Proposition 12.4.
+□
+Recall the singular part P(b, σ) of �
+Lb(σ)−1 as defined in Corollary 11.4, which satisfies
+P(b, σ)f = (σ−2(˙g1, ˙A1) − iσ−1(ˇg1, ˇA1)) + iσ−1�
+(˙g2, ˙A2) + (˙g′
+1, ˙A′
+1)
+�
++ iσ−1(˙g′′
+1(σ), ˙A′′
+1(σ))
+:= σ−2P 2(b)f + σ−1P 1(b)f + σ−1P e(b, σ)f
+(12.12)
+where (˙g′′
+1(σ), ˙A′′
+1(σ)) = P e(b, σ)f = o(1) in Kb,1 as σ → 0. With the ǫ-regularity of R(b, σ) at our dis-
+posal, now we are able to prove that P e(b, σ) is ǫ-regular as well, and thus give a refined description of
+σ−1(˙g′′
+1(σ), ˙A′′
+1(σ)).
+
+LINEAR STABILITY OF KERR-NEWMAN FAMILY
+143
+Remark 12.7. We now make a remark that there is a relationship between the Sobolev spave ˙Hk([0, c0)),
+which is the restriction to [0, c0) of elements in Hk(R) with support in [0, ∞), and the b-Sobolev space
+Hk
+b([0, c0)), which is a completion of {φ|(0,c0]} where φ ∈ C∞
+c ((0, ∞)) with respect the following norm
+∥φ∥2
+Hk
+b ([0,C0)) :=
+k
+�
+j=0
+∥(σ∂σ)jφ∥2
+L2(dσ).
+First, it is clear that σkHk
+b([0, c0)) ⊂ ˙Hk([0, c0)). Second, Hardy’s inequality tells us that for φ ∈ C∞
+c (0, ∞),
+we have
+� ∞
+0
+|φ|2
+r2 dr ≤ 4
+� ∞
+0
+|∂rf|2 dr.
+Applying Hardy’s inequality k times, we find that
+� ∞
+0
+|φ|2
+r2k dr ≤ Ck
+� ∞
+0
+|∂k
+r f|2 dr.
+for φ ∈ C∞
+c (0, ∞). This implies ˙Hk([0, c0)) ⊂ σkHk
+b([0, c0)). Therefore, ˙Hk([0, c0)) = σkHk
+b([0, c0)). Then
+an interpolation gives ˙Hα([0, c0)) = σαHα
+b ([0, c0)) for all α ≥ 0.
+As a preparation, we record the following technical lemma.
+Lemma 12.8. Suppose that u± ∈ Hs(R±) with s ≥ 0 and s ̸= 1
+2 + N0 and let u(x) = u+(x) for x > 0 and
+u(x) = u−(x) for x < 0. If ∂ju+(x) = ∂ju−(x) for all j ∈ N0, j < s − 1
+2, then u(x) ∈ Hs(R).
+Proof. See [61, Theorem 11.4 and 11.5].
+□
+Theorem 12.9. Let σ−1(˙g′′
+1(σ), ˙A′′
+1(σ)) = σ−1P e(b, σ)f ∈ Kb,1 be defined as in (12.12) and let s, ℓ, ǫ be
+defined as in Proposition 12.3. Then we have
+P e(b, σ) ∈ H
+3
+2 −ǫ−((−c0, c0); L( ¯Hs−1,ℓ+2
+b
+( ¯X), Kb,1))
+(12.13)
+where c0 > 0 is a sufficiently small constant. Moreover,
+σ−1(˙g′′
+1(σ), ˙A′′
+1(σ)) ∈
+�
+±
+A−ǫ�
+± [0, c0); Kb,1
+�
+.
+(12.14)
+Proof. Using the calculation (11.34) and (11.35) in the proof of Corollary 11.4, we find that for f =
+(f0, f1, f2) ∈ ¯Hs−1,ℓ+2
+b
+( ¯X) = R0 ⊕ R1 ⊕ R2,
+ˇLb(0)P e(b, σ)f
+=
+� �R10(b, σ) − �R10(b, 0)
+�
+f0 +
+� �Re
+11(b, σ) − �Re
+11(b, 0)
+�
+f1 +
+� �R12(b, σ) − �R12(b, 0)
+�
+f2.
+(12.15)
+According to Proposition 12.6, we have ∥∂σ �Re
+11(b, σ)∥R1→ �
+K1 ≲ |σ|−ǫ and ∥∂2
+σ �Re
+11(b, σ)∥R1→ �
+K1 ≲ |σ|−ǫ−1 for
+0 < |σ| ≤ c0 where c0 is a sufficiently small constant, which implies that
+∥∂σσ∂σ �Re
+11(b, σ)∥R1→ �
+K1 ≲ |σ|−ǫ.
+Integrating ∂σσ∂σ �Re
+11(b, σ) and ∂σ �Re
+11(b, σ) from σ ∈ (0, c0) to 0 and using σ∂σ �Re
+11(σ)|b,σ=0 = 0 yield
+σ∂σ �Re
+11(b, σ) ∈ σ1−ǫL∞([0, c0); L(R1, �K1)) ⊂ σ
+3
+2 −ǫ−H0
+b([0, c0); L(R1, �K1)),
+(12.16)
+�Re
+11(b, σ) − �Re
+11(b, 0) ∈ σ1−ǫL∞([0, c0); L(R1, �K1)) ⊂ σ
+3
+2 −ǫ−H0
+b([0, c0); L(R1, �K1)).
+(12.17)
+This implies
+( �Re
+11(σ) − �Re
+11(0)) ∈ σ
+3
+2 −ǫ−H2
+b([0, c0); L(R1, �K1)) ⊂ ˙H
+3
+2 −ǫ−([0, c0); L(R1, �K1)).
+Similarly, we also have
+( �Re
+11(σ) − �Re
+11(0)) ∈ ˙H
+3
+2 −ǫ−((−c0, 0]; L(R1, �K1)).
+Since ( �Re
+11(σ) − �Re
+11(0)) → 0 as σ → 0 from both sides, by Lemma 12.8, we obtain
+( �Re
+11(σ) − �Re
+11(0)) ∈ H
+3
+2 −ǫ−((−c0, c0); L(R1, �K1)).
+Similarly, �R10(b, σ) − �R10(b, 0) and �R12(b, σ) − �R12(b, 0) satisfy the same estimates as �Re
+11(b, σ) − �Re
+11(b, 0).
+This proves (12.13).
+
+144
+LILI HE
+Moreover, the estimate in (12.17) implies
+�Re
+11(b, σ) − �Re
+11(b, 0) ∈
+�
+±
+σ1−ǫL∞(±[0, c0); L(R1, �K1))
+(12.18)
+and this also holds for �R10(b, σ) − �R10(b, 0), �R20(b, σ) − �R20(b, 0). Using Proposition 12.16, the estimate
+(12.18) is satisfied after applying any number of derivatives σ∂σ. This proves (12.14).
+□
+Now we turn to the analysis of the regular part L−
+b (σ).
+Proposition 12.10. Let s, ℓ, ǫ be defined as in Proposition 12.3. For Im σ ≥ 0 and 0 < |σ| ≤ c0 where
+c0 > 0 is a small constant, the regular part L−
+b (σ) of the resolvent �
+Lb(σ)−1 defined in Corollary 11.4 satisfies
+∂j
+σL−
+b (σ) : ¯Hs−1,ℓ+2
+b
+( ¯X) → |σ|−ǫ−j+1 ¯Hs−ǫ−j+1,ℓ+ǫ+1
+b
+( ¯
+X),
+j = 1, 2.
+(12.19)
+Proof. According to Proposition 12.6 and the expression (12.15), we see that P(b, σ) has the form P(b, σ) =
+σ−2P 2(b) + σ−1P 1(b) + σ−1P e(b, σ) where
+P 2(b), P 1(b) : ¯Hs−1,ℓ+2
+b
+( ¯X) → ρC∞( ¯
+X) + ¯H
+∞, 1
+2 −
+b
+( ¯X)
+and P e(b, σ) :
+¯Hs−1,ℓ+2
+b
+( ¯X) → Kb,1 is ǫ-regular as proved in Theorem 12.9.
+We also write R(b, σ) =
+Rsing(b, σ) + Rreg(b, σ) where
+Rsing(b, σ) =
+
+
+0
+0
+0
+σ−1 �R10(b, σ)
+σ−2 �R0
+11(b) + σ−1 �Re
+11(b, σ)
+σ−1 �R12(b, σ)
+0
+σ−1 �R0
+21(b)
+σ−1 �R0
+22(b)
+
+
+and
+Rreg(b, σ) =
+
+
+�R00(b, σ)
+�R01(b, σ)
+�R02(b, σ)
+0
+0
+0
+�R20(b, σ)
+�Re
+21(b, σ)
+�Re
+22(b, σ)
+
+ .
+Using the calculation in (11.34), (11.35) and (11.36), we find that
+L−
+b (σ) = ˇLb(σ)−1Rreg(b, σ) − ˇLb(σ)−1��
+∂σ�
+Lb(0) + σ
+2 ∂2
+σ�
+Lb(0)
+�
+(P 1(b) + P e(b, σ)) + 1
+2∂2
+σ�
+Lb(0)P 2(b)
+�
+:= I + II.
+For the term I, we compute
+∂σI = ∂σ ˇLb(σ)−1Rreg(b, σ) + ˇLb(σ)−1∂σRreg(b, σ),
+∂2
+σI = ∂2
+σ ˇLb(σ)−1Rreg(b, σ) + 2∂σ ˇLb(σ)−1∂σRreg(b, σ) + ˇLb(σ)−1∂2
+σRreg(b, σ).
+Then by proposition 12.3 and 12.6, we conclude that
+∂j
+σI : ¯Hs−1,ℓ+2
+b
+( ¯X) → |σ|−ǫ−j+1 ¯Hs−ǫ−j+1,ℓ+ǫ+1
+b
+( ¯X),
+j = 1, 2.
+As for the term II, we first note that P 2(b), P 1(b), P e(b, σ) map to ρC( ¯X) + ¯H∞,1/2−
+b
+( ¯
+X), and thus
+∂k
+σ�
+Lb(0)P 2(b), ∂k
+σ�
+Lb(0)P 1(b), ∂k
+σ�
+Lb(0)P e(b, σ) : ¯Hs−1,ℓ+2
+b
+( ¯
+X) → ¯H
+∞, 3
+2 −
+b
+( ¯X),
+k = 1, 2.
+Next, we calculate
+∂σII = −∂σ ˇLb(σ)−1��
+∂σ�
+Lb(0) + σ
+2 ∂2
+σ�
+Lb(0)
+�
+(P 1(b) + P e(b, σ)) + 1
+2∂2
+σ�
+Lb(0)P 2(b)
+�
+− ˇLb(σ)−1��
+∂σ�
+Lb(0) + σ
+2 ∂2
+σ�
+Lb(0)
+�
+∂σP e(b, σ) + 1
+2∂2
+σ�
+Lb(0)P e(b, σ)
+�
+,
+∂2
+σII = −∂2
+σ ˇLb(σ)−1��
+∂σ�
+Lb(0) + σ
+2 ∂2
+σ�
+Lb(0)
+�
+(P 1(b) + P e(b, σ)) + 1
+2∂2
+σ�
+Lb(0)P 2(b)
+�
+− 2∂σ ˇLb(σ)−1��
+∂σ�
+Lb(0) + σ
+2 ∂2
+σ�
+Lb(0)
+�
+∂σP e(b, σ) + 1
+2∂2
+σ�
+Lb(0)P e(b, σ)
+�
+− ˇLb(σ)−1��
+∂σ�
+Lb(0) + σ
+2 ∂2
+σ�
+Lb(0)
+�
+∂2
+σP e(b, σ) + ∂2
+σ�
+Lb(0)∂σP e(b, σ)
+�
+.
+
+LINEAR STABILITY OF KERR-NEWMAN FAMILY
+145
+Therefore, using Proposition 12.3 and the fact that P e(b, σ) : ¯Hs−1,ℓ+2
+b
+→ Kb,1 is ǫ-regular, we arrive at
+∂j
+σII : ¯Hs−1,ℓ+2
+b
+( ¯X) → |σ|−ǫ−j+1 ¯Hs−ǫ−j+1,ℓ+ǫ+1
+b
+( ¯X),
+j = 1, 2.
+This completes the proof of the estimates (12.19).
+□
+Theorem 12.11. Let s, ℓ, ǫ be defined as in Proposition 12.3. Then we have
+L−
+b (σ) ∈ H
+3
+2 −ǫ−�
+(−c0, c0); L
+� ¯Hs−1,ℓ+2
+b
+( ¯X), ¯H
+s−max{ǫ, 1
+2 },ℓ+ǫ−1
+b
+( ¯X)
+��
+(12.20)
+where c0 > 0 is a sufficiently small constant.
+Proof. Using the estimates in Proposition 12.3 and following the proof at the beginning of Theorem 12.9,
+we find that
+L−
+b (σ) − L−
+b (0) ∈ σ
+3
+2 −ǫ−H2
+b
+�
+[0, c0); L
+� ¯Hs−1,ℓ+2
+b
+( ¯X), ¯Hs−1−ǫ,ℓ+ǫ−1
+b
+( ¯X)
+��
+.
+(12.21)
+Again integrating ∂σL−
+b (σ) from σ ∈ (0, c0) to 0 and using the estimate (12.19) for k = 1 yield
+L−
+b (σ) − L−
+b (0) ∈ σ
+3
+2 −ǫ−H1
+b
+�
+[0, c0); L
+� ¯Hs−1,ℓ+2
+b
+( ¯X), ¯Hs−ǫ,ℓ+ǫ−1
+b
+( ¯
+X)
+��
+.
+(12.22)
+Interpolating between (12.21) and (12.22) gives
+L−
+b (σ) − L−
+b (0) ∈ σ
+3
+2 −ǫ−H1+θ
+b
+�
+[0, c0); L
+� ¯Hs−1,ℓ+2
+b
+( ¯X), ¯Hs−θ−ǫ,ℓ+ǫ−1
+b
+( ¯X)
+��
+,
+0 ≤ θ ≤ 1.
+(12.23)
+If ǫ ∈ (0, 1/2], we take θ = 1/2 − ǫ, and thus 3/2 − ǫ = 1 + θ. If ǫ ∈ (1/2, 1), we take θ = 0. This gives
+L−
+b (σ) − L−
+b (0) ∈
+
+
+
+σ
+3
+2 −ǫ−H
+3
+2 −ǫ
+b
+�
+[0, c0); L
+� ¯Hs−1,ℓ+2
+b
+( ¯X), ¯H
+s− 1
+2 ,ℓ+ǫ−1
+b
+( ¯X)
+��
+,
+0 < ǫ ≤ 1
+2,
+σ
+3
+2 −ǫ−H1
+b
+�
+[0, c0); L
+� ¯Hs−1,ℓ+2
+b
+( ¯X), ¯Hs−ǫ,ℓ+ǫ−1
+b
+( ¯X)
+��
+,
+1
+2 < ǫ < 1,
+(12.24)
+which implies
+L−
+b (σ) − L−
+b (0) ∈
+
+
+
+˙H
+3
+2 −ǫ�
+[0, c0); L
+� ¯Hs−1,ℓ+2
+b
+( ¯X), ¯H
+s− 1
+2 ,ℓ+ǫ−1
+b
+( ¯X)
+��
+,
+0 < ǫ ≤ 1
+2,
+˙H
+3
+2 −ǫ−�
+[0, c0); L
+� ¯Hs−1,ℓ+2
+b
+( ¯X), ¯Hs−ǫ,ℓ+ǫ−1
+b
+( ¯X)
+��
+,
+1
+2 < ǫ < 1.
+(12.25)
+Similarly, we also have the above statement for the other half interval (−c0, 0]. Since L−
+b (σ) − L−
+b (0) → 0 as
+σ → 0 from both sides, by Lemma 12.8, we obtain
+L−
+b (σ) − L−
+b (0) ∈ H
+3
+2 −ǫ−�
+(−c0, c0); L
+� ¯Hs−1,ℓ+2
+b
+( ¯
+X), ¯H
+s−max{ǫ, 1
+2 },ℓ+ǫ−1
+b
+( ¯X)
+��
+,
+and thus the conclusion (12.20).
+□
+12.2. Regularity for frequencies away from 0. Now we discuss the regularity of �
+Lb(σ)−1 for |σ| ≥
+c0, 0 ≤ Im σ ≤ C for some c0, C > 0.
+Theorem 12.12. Let ℓ < − 1
+2, s > 3+m and s+ℓ > − 1
+2 +m where m ∈ N0. Then for 0 ≤ Im σ ≤ C, |σ| ≥ c0
+with c0, C > 0, the operator
+∂m
+σ �
+Lb(σ)−1 : ¯Hs,ℓ+1
+b,h
+( ¯X) → h−m ¯Hs−m,ℓ
+b,h
+( ¯X)
+(12.26)
+where h = |σ|−1, is uniformly bounded.
+Proof. The estimate (12.26) for the case m = 0 follows from high energy estimate 5.24.
+For m ≥ 1,
+∂m
+σ �
+Lb(σ)−1 is a linear combination of the operators of the type
+�
+�
+Lb(σ)−1 ◦ ∂m1
+σ �
+Lb(σ)
+�
+◦ · · · ◦
+�
+�
+Lb(σ)−1 ◦ ∂mk
+σ
+�
+Lb(σ)
+�
+◦ �
+Lb(σ)−1
+where 1 ≤ m1, · · · , mk ≤ 2 and m1 + · · · + mk = m. Since
+∂σ�
+Lb(σ) ∈ ρDiff1
+b + ρ2C∞ ⊂ h−1ρDiff1
+b,h
+and
+∂2
+σ�
+Lb(σ) ∈ ρ2C∞,
+
+146
+LILI HE
+it follows that
+¯Hs,ℓ+1
+b,h
+�
+Lb(σ)−1
+−−−−−→ ¯Hs,ℓ
+b,h
+∂
+mk
+σ
+�
+Lb(σ)
+−−−−−−→ hmk−2 ¯Hs+mk−2,ℓ+mk
+b,h
+⊂ hmk−2 ¯Hs+mk−2,ℓ+1
+b,h
+�
+Lb(σ)−1
+−−−−−→ hmk−2 ¯Hs+mk−2,ℓ
+b,h
+···
+−→ · · ·
+···
+−→ hm−m1−2(k−1) ¯Hs+m−m1−2(k−1),ℓ
+b,h
+∂m1
+σ
+�
+Lb(0)
+−−−−−−→ hm−2k ¯Hs+m−2k,ℓ+m1
+b,h
+�
+Lb(σ)−1
+−−−−−→ hm−2k ¯Hs+m−2k,ℓ
+b,h
+.
+Since 1 ≤ k ≤ m, we have hm−2k ¯Hs+m−2k,ℓ+m−k
+b,h
+⊂ h−m ¯Hs−m,ℓ
+b,h
+. This finishes the proof.
+□
+12.3. Conormal regularity. Finally, we consider the conormal regularity of L−
+b (σ) at σ = 0. Specifically,
+after applying any number of derivatives σ∂σ to L−
+b (σ), it satisfies the same properties as L−
+b (σ).
+Recall that
+L−
+b (σ) = ˇLb(σ)−1Rreg(b, σ) − ˇLb(σ)−1��
+∂σ�
+Lb(0) + σ
+2 ∂2
+σ�
+Lb(0)
+�
+(P 1(b) + P e(b, σ)) + 1
+2∂2
+σ�
+Lb(0)P 2(b)
+�
+,
+so in order to study the conormal regularity, we need to analyze that of ˇLb(σ)−1, Rreg(b, σ) and P e(b, σ).
+We first discuss the conormal regularity of ˇLb(σ)−1 and prove a analogous result of Proposition 12.3 for
+(σ∂σ)m ˇLb(σ)−1.
+Proposition 12.13. Let − 3
+2 < ℓ < − 1
+2, 0 < ǫ < 1 with − 1
+2 < ℓ + ǫ < 1
+2, and s − ǫ > 4 + m where m ∈ N.
+Define
+ˇL(m)
+b
+(σ) := (σ∂σ)m ˇLb(σ)−1.
+For 0 < |σ| ≤ c0, Im σ ≥ 0 with c0 small, we have
+ˇL(m)
+b
+(σ) : ¯Hs−1,ℓ+2
+b
+( ¯X) → ¯Hs−m,ℓ
+b
+( ¯X),
+(12.27)
+∂σ ˇL(m)
+b
+(σ) : ¯Hs−1,ℓ+2
+b
+( ¯X) → |σ|−ǫ ¯Hs−m−ǫ,ℓ+ǫ−1
+b
+( ¯X),
+(12.28)
+∂2
+σ ˇL(m)
+b
+(σ) : ¯Hs−1,ℓ+2
+b
+( ¯X) → |σ|−ǫ−1 ¯Hs−m−1−ǫ,ℓ+ǫ−1
+b
+( ¯X).
+(12.29)
+Proof. Since
+∂2
+σ ˇL(m)
+b
+(σ) = σ−1∂σ(σ∂σ − 1) ˇL(m)
+b
+(σ) = σ−1∂σ ˇL(m+1)
+b
+(σ) − σ−1∂σ ˇL(m)
+b
+(σ),
+it suffices to prove the statements (12.27) and (12.28), and then the conclusion (12.29) follows directly.
+Now we prove (12.27) and (12.28) by induction on m. For m = 1, we calculate
+ˇL(1)
+b
+= σ∂σ ˇLb(σ)−1 = −ˇLb(σ)−1(σ∂σ ˇLb(σ))ˇLb(σ)−1
+where
+σ∂σ ˇLb(σ) = ˇLb(σ) − ˇLb(0) + σ2
+2 ∂2
+σ ˇLb(0) ∈ ˇLb(σ) + ρ2Diff2
+b + σ2ρ2C∞.
+(12.30)
+As a consequence,
+ˇL(1)
+b
+= −ˇLb(σ)−1 + ˇLb(σ)−1�
+ρ2Diff2
+b + σ2ρ2C∞�ˇLb(σ)−1
+We then compute
+¯Hs−1,ℓ+2
+b
+( ¯X)
+ˇLb(σ)−1
+−−−−−→ ¯Hs,ℓ
+b ( ¯X)
+ρ2Diff
+2
+b+σ2ρ2C∞
+−−−−−−−−−−−−→ ¯Hs−2,ℓ+2
+b
+( ¯X)
+ˇLb(σ)−1
+−−−−−→ ¯Hs−1,ℓ
+b
+( ¯X).
+This proves (12.27) for m = 1. As for the first derivative ∂σ ˇL(1)
+b (σ), we calculate
+∂σ ˇL(1)
+b (σ) = −∂σ ˇLb(σ)−1 + ∂σ ˇLb(σ)−1�
+ρ2Diff2
+b + σ2ρ2C∞�ˇLb(σ)−1 + ˇLb(σ)−1�
+σρ2C∞�ˇLb(σ)−1
++ ˇLb(σ)−1�
+ρ2Diff2
+b + σ2ρ2C∞�
+∂σ ˇLb(σ)−1 := I + II + III + IV .
+
+LINEAR STABILITY OF KERR-NEWMAN FAMILY
+147
+Using Proposition 12.3, we find that
+I : ¯Hs−1,ℓ+2
+b
+( ¯
+X) → |σ|−ǫ ¯Hs−ǫ,ℓ+ǫ+1
+b
+( ¯X),
+II : ¯Hs−1,ℓ+2
+b
+( ¯
+X)
+ˇLb(σ)−1
+−−−−−→ ¯Hs,ℓ
+b ( ¯X)
+ρ2Diff
+2
+b+σ2ρ2C∞
+−−−−−−−−−−−−→ ¯Hs−2,ℓ+2
+b
+( ¯X)
+∂σ ˇLb(σ)−1
+−−−−−−−→ |σ|−ǫ ¯Hs−1−ǫ,ℓ+ǫ−1
+b
+( ¯X),
+III : ¯Hs−1,ℓ+2
+b
+( ¯
+X)
+ˇLb(σ)−1
+−−−−−→ ¯Hs,ℓ
+b ( ¯X)
+σρ2C∞
+−−−−−→ ¯Hs,ℓ+2
+b
+( ¯X)
+ˇLb(σ)−1
+−−−−−→ ¯Hs−1,ℓ
+b
+( ¯X) ⊂ |σ|−ǫ ¯Hs−1−ǫ,ℓ+ǫ−1
+b
+( ¯X),
+IV : ¯Hs−1,ℓ+2
+b
+( ¯
+X)
+∂σ ˇLb(σ)−1
+−−−−−−−→ |σ|−ǫ ¯Hs−ǫ,ℓ+ǫ−1
+b
+( ¯X)
+ρ2Diff
+2
+b+σ2ρ2C∞
+−−−−−−−−−−−−→ |σ|−ǫ ¯Hs−ǫ−2,ℓ+ǫ+1
+b
+( ¯X)
+ˇLb(σ)−1
+−−−−−→ |σ|−ǫ ¯Hs−1−ǫ,ℓ+ǫ−1
+b
+( ¯X)
+where we use 1/2 < ℓ + ǫ + 1 < 3/2 in the last step of the analysis of IV . This proves (12.28) for m = 1.
+Suppose that (12.27) and (12.28) hold for k ≤ m. Then for m + 1, by a direct calculation, we obtain that
+ˇL(m+1)
+b
+(σ) is a linear combination of the operators of the type
+L1 = ˇL(m)
+b
+(σ),
+L2 := ˇL(m1)
+b
+(σ) ◦ (ρ2Diff2
+b) ◦ ˇL(m2)
+b
+(σ)
+where m1, m2 ≥ 0 and m1 + m2 ≤ m. Then by the induction hypothesis on k ≤ m, it follows that
+L1 : ¯Hs−1,ℓ+2
+b
+( ¯X)
+ˇ
+L(m)
+b
+(σ)
+−−−−−→ ¯Hs−m,ℓ
+b
+( ¯X) ⊂ ¯Hs−m−1,ℓ
+b
+( ¯X),
+L2 : ¯Hs−1,ℓ+2
+b
+( ¯X)
+ˇ
+L(m1)
+b
+(σ)
+−−−−−−→ ¯Hs−m1,ℓ
+b
+( ¯X)
+ρ2Diff
+2
+b
+−−−−−→ ¯Hs−m1−2,ℓ+2
+b
+( ¯X)
+ˇ
+L(m1)
+b
+(σ)
+−−−−−−→ ¯Hs−m1−m2−1,ℓ
+b
+( ¯X) = ¯Hs−m−1,ℓ
+b
+( ¯X).
+This proves (12.27) for m+1. As for the first derivative ∂σ ˇL(m+1)
+b
+, it is a linear combination of the operators
+of the following types
+D1 = ∂σ ˇL(m)
+b
+(σ),
+D2 := ∂σ ˇL(m1)
+b
+(σ) ◦ (ρ2Diff2
+b) ◦ ˇL(m2)
+b
+(σ),
+D3 := ˇL(m1)
+b
+(σ) ◦ (ρ2Diff2
+b) ◦ ∂σ ˇL(m2)
+b
+(σ)
+where m1, m2 ≥ 0 and m1 + m2 ≤ m. Then we have
+D1 : ¯Hs−1,ℓ+2
+b
+( ¯X)
+∂σ ˇ
+L(m)
+b
+(σ)
+−−−−−−−→ |σ|−ǫ ¯Hs−m−ǫ,ℓ+ǫ−1
+b
+( ¯X) ⊂ |σ|−ǫ ¯Hs−m−1ǫ,ℓ+ǫ−1
+b
+( ¯X),
+D2 : ¯Hs−1,ℓ+2
+b
+( ¯X)
+ˇ
+L(m2)
+b
+(σ)
+−−−−−−→ ¯Hs−m2,ℓ
+b
+( ¯X)
+ρ2Diff
+2
+b
+−−−−−→ ¯Hs−m2−2,ℓ+2
+b
+( ¯X)
+∂σ ˇ
+L(m1)
+b
+(σ)
+−−−−−−−→ |σ|−ǫ ¯Hs−m1−1−m2−ǫ,ℓ+ǫ−1
+b
+( ¯X) ⊂ |σ|−ǫ ¯Hs−m−1−ǫ,ℓ+ǫ−1
+b
+( ¯
+X),
+D3 : ¯Hs−1,ℓ+2
+b
+( ¯X)
+∂σ ˇ
+L(m2)
+b
+(σ)
+−−−−−−−→ |σ|−ǫ ¯Hs−m2−ǫ,ℓ+ǫ−1
+b
+( ¯
+X)
+ρ2Diff
+2
+b
+−−−−−→ |σ|−ǫ ¯Hs−m2−ǫ−2,ℓ+ǫ+1
+b
+( ¯X)
+ˇ
+L(m1)
+b
+(σ)
+−−−−−−→ |σ|−ǫ ¯Hs−m1−m2−ǫ−1,ℓ+ǫ−1
+b
+( ¯X) ⊂ |σ|−ǫ ¯Hs−m−1−ǫ,ℓ+ǫ−1
+b
+( ¯X)
+where we use 1/2 < ℓ + ǫ + 1 < 3/2 in the third step of the analysis of D3. This proves (12.28) for m + 1.
+This finishes the proof of the proposition.
+□
+With the conormal regularity of ˇLb(σ)−1 with our disposal, we are ready to discuss that of the entries �Lij
+of the operator �
+Lb(σ)ˇLb(σ)−1.
+Recall the expression (11.25) of �
+Lb(σ)ˇL(σ)−1
+�
+Lb(σ)ˇLb(σ)−1 =
+
+
+
+L00(b, σ)
+σ2 �L01(b, σ)
+σ�L02(b, σ)
+σ�L10(b, σ)
+σ2 �L11(b, σ)
+σ2 �L12(b, σ)
+σ�L20(b, σ)
+σ2 �L21(b, σ)
+σ�L22(b, σ)
+
+
+
+=
+
+
+
+L00(b, σ)
+σ2�L01(b, σ)
+σ�L02(b, σ)
+σ
+��L0
+10(b) + σ�Le
+10(b, σ)
+�
+σ2��L0
+11(b) + σ�L1
+11(b) + σ2�Le
+11(b, σ)
+�
+σ2��L0
+12(b) + σ�Le
+12(b, σ)
+�
+σ�L20(b, σ)
+σ2��L0
+21(b) + σ�Le
+21(b)
+�
+σ
+��L0
+22(b) + σ�Le
+22(b, σ)
+�
+.
+
+
+
+Proposition 12.14. Let s, ℓ, ǫ, m be defined as in Proposition 12.13. Then (σ∂σ)m�Lij satisfy the following
+properties.
+
+148
+LILI HE
+(1) (σ∂σ)mL00, (σ∂σ)m�L01, (σ∂σ)m�L02, (σ∂σ)m�L20 are ǫ-regular at σ = 0.
+(2) (σ∂σ)m�L10, (σ∂σ)m�L12, (σ∂σ)m�L21, (σ∂σ)m�L22 have an ǫ-regular expansion up to order one, that is,
+(σ∂σ)m�Le
+10, (σ∂σ)m�Le
+12, (σ∂σ)m�Le
+21, (σ∂σ)m�Le
+22 are ǫ-regular at σ = 0.
+(3) (σ∂σ)m�L11 has an ǫ-regular expansion up to order two, that is, (σ∂σ)m�Le
+11 is ǫ-regular at σ = 0.
+Proof. The proof is analogous to that of Proposition 12.4, now by using Proposition 12.13 instead of Propo-
+sition 12.3. Here, we only demonstrate the proof of (σ∂σ)m�L10 in detail because the proof of the other entries
+follows in a similar manner.
+Recall the calculation in (11.18)
+(σ∂σ)m�Le
+10(˜g0, ˜A0), (˙g∗
+1, ˙A∗
+1)⟩
+= −
+�
+j+k=m
+�
+(˜g0, ˜A0), i
+� ˇL(k)
+b (σ)
+�∗�
+(σ∂σ)j�
+∂σ�
+Lb(0) + σ
+2 ∂2
+σ�
+Lb(0)
+��∗�
+I − (ˇLb(0)−1)∗V ∗
+b
+�
+(ˇg∗
+2, ˇA∗
+2)
+�
++
+�
+(˜g0, ˜A0), 1
+2( ˇL(m)
+b
+(σ))∗(∂2
+σ�
+Lb(0))∗(˙g∗
+2, ˙A∗
+2)
+�
+where
+�
+(σ∂σ)j�
+∂σ�
+Lb(0) + σ
+2 ∂2
+σ�
+Lb(0)
+��∗�
+I − (ˇLb(0)−1)∗V ∗
+b
+�
+(ˇg∗
+2, ˇA∗
+2),
+(∂2
+σ�
+Lb(0))∗(˙g∗
+2, ˙A∗
+2) ∈ ˙H−5/2−C(a,γ), 3/2−
+b
+( ¯X).
+Since (˜g0, ˜A0) ∈ ¯Hs−1,ℓ+2
+b
+( ¯X) with s − m − ǫ > 4, and by Proposition 12.13
+( ˇL(k)
+b (σ))∗ ˙H−5/2−C(a,γ), 3/2−
+b
+( ¯X) ∈ ˙H−3/2−k−C(a,γ),−1/2−
+b
+( ¯X),
+0 ≤ k ≤ m, k ∈ N0,
+(∂j
+σ ˇL(k)
+b (σ))∗ ˙H−5/2−C(a,γ), 3/2−
+b
+( ¯X) ∈ |σ|−ǫ−j+1 ˙H−s+1,−ℓ−2
+b
+( ¯X),
+j = 1, 2
+and
+0 ≤ k ≤ m, k ∈ N0
+where we use the facts that −5/2 − C(a, γ) > −3 > −s + k + ǫ + 1 for 0 ≤ k ≤ m and −ℓ − ǫ + 1 < 3/2,
+the pairings above (and their up to second order derivatives) are well-defined. This proves the ǫ-regularity
+of (σ∂σ)�Le
+10.
+□
+Next, we turn to discussing the conormal regularity of R(b, σ). To this end, we need the following lemma,
+which is an analogy of Lemma 12.5.
+Lemma 12.15. Let s, ℓ, ǫ, m be defined as in Proposition 12.13. Suppose that the operator A(σ) : �Ki → Rj
+with i, j = 0, 1, 2 has an ǫ-regular expansion up to order one, i.e.,
+A(σ) = A0 + σAe(σ)
+where A0 : �Ki → Rj is invertible and independent of σ, and (σ∂σ)kAe(σ) : �Ki → Rj is ǫ-regular for any
+0 ≤ k ≤ m, k ∈ N0. Then for |σ| sufficiently small, A(σ) is invertible with inverse B(σ) = A(σ)−1, and
+(σ∂σ)kB(σ) has an ǫ-regular expansion up to order one, that is,
+(σ∂σ)kB(σ) = B0
+k + σBe
+k(σ),
+0 ≤ k ≤ m, k ∈ N0
+where B0
+k : Rj → �Ki is independent of σ, and Be
+k(σ) : Rj → �Ki is ǫ-regular.
+Similarly, if A(σ) has uniformly bounded inverse B(σ), and (σ∂σ)kA(σ) is ǫ-regular for any 0 ≤ k ≤
+m, k ∈ N0, then (σ∂σ)kB(σ) is ǫ-regular as well.
+Proof. As established in the proof of Lemma 12.5, we have
+A(σ)−1 =
+�
+I +
+∞
+�
+k=1
+(−1)k�
+σ(A0)−1Ae(σ)
+�k�
+(A0)−1
+= (A0)−1 − σ
+�
+(A0)−1Ae(σ)
+∞
+�
+k=0
+(−1)k�
+σ(A0)−1Ae(σ)
+�k(A0)−1�
+:= B0 + σBe(σ).
+For 1 ≤ k ≤ m, we have
+(σ∂σ)kB(σ) = σ
+k
+�
+j=0
+�k
+j
+�
+(σ∂σ)jBe(σ).
+A direct calculation implies that (σ∂σ)jBe(σ) is a sum of the operators of the type σ−1L ◦ (A0)−1, where L
+is a composition of the terms of the form σ(A0)−1(σ∂σ)lAe(σ) with 0 ≤ l ≤ j. Then proceeding as in the
+
+LINEAR STABILITY OF KERR-NEWMAN FAMILY
+149
+proof of Lemma 12.5, we conclude that (σ∂σ)jBe(σ) with 0 ≤ j ≤ k is ǫ-regular, and thus (σ∂σ)kBe(σ) with
+1 ≤ k ≤ m has an ǫ-regular expansion up to order one.
+Since (σ∂σ)mB(σ) is a linear combination of the operators of the form
+(σ∂σ)m1B(σ) ◦ (σ∂σ)m2A(σ) ◦ (σ∂σ)m3B(σ)
+where 1 ≤ m2 ≤ m, 0 ≤ m1, m2 ≤ m − 1 and m1 + m2 + m3 = m, it follows that the second statement
+follows by an induction argument on m.
+□
+Recall the expressions (11.26) and (11.32) of Rb,σ = (�
+Lb(σ)ˇLb(σ)−1)−1
+R(b, σ) = (�
+Lb(σ)ˇLb(σ)−1)−1 =
+
+
+
+�R00(b, σ)
+�R01(b, σ)
+�R02(b, σ)
+σ−1 �R10(b, σ)
+σ−2 �R11(b, σ)
+σ−1 �R12(b, σ)
+�R20(b, σ)
+σ−1 �R21(b, σ)
+σ−1 �R22(b, σ)
+
+
+
+=
+
+
+
+�R00(b, σ)
+�R01(b, σ)
+�R02(b, σ)
+σ−1 �R10(b, σ)
+σ−2� �R0
+11(b) + σ �Re
+11(b, σ)
+�
+σ−1 �R12(b, σ)
+�R20(b, σ)
+σ−1� �R0
+21(b) + σ �Re
+21(b, σ)
+�
+σ−1� �R0
+22(b) + σ �Re
+22(b, σ)
+�
+
+
+
+where
+�R11 =
+��L♯
+11 − σ�L♯
+12(�L♭
+22)−1�L♯
+21
+�−1,
+�R10 = �R11
+�
+− �L10 + σ�L♯
+12(�L♭
+22)−1�L20
+�
+L−1
+00 ,
+�R12 = − �R11�L♯
+12(�L♭
+22)−1,
+�R21 = −(�L♭
+22)−1�L♯
+21 �R11,
+�R22 = (�L♭
+22)−1 − σ(�L♭
+22)−1�L♯
+21 �R12,
+�R20 = −(�L♭
+22)−1��L20L−1
+00 + �L♯
+21 �R10
+�
+,
+�R01 = −L−1
+00
+��L01 �R11 + �L02 �R21
+�
+,
+�R02 = −L−1
+00
+�
+σ�L01 �R12 + �L02 �R22
+�
+,
+�R00 = L−1
+00
+�
+I − σ�L01 �R10 − σ�L02 �R20
+�
+(12.31)
+with
+�L♯
+i1 = �Li1 − σ�Li0L−1
+00 �L01
+for
+i = 1, 2,
+�L♯
+12 = �L12 − �L10L−1
+00 �L02,
+�L♭
+22 = �L22 − σ�L20L−1
+00 �L02.
+(12.32)
+Proposition 12.16. Let s, ℓ, ǫ, m be defined as in Proposition 12.13. Then (σ∂σ)m �Rij satisfy the following
+properties.
+(1) (σ∂σ)m �R00, (σ∂σ)m �R01, (σ∂σ)m �R02, (σ∂σ)m �R10, (σ∂σ)m �R12, (σ∂σ)m �R20 are ǫ-regular at σ = 0.
+(2) (σ∂σ)m �R11, (σ∂σ)m �R21, (σ∂σ)m �R22 have an ǫ-regular expansion up to order one, that is, (σ∂σ)m �Re
+11,
+(σ∂σ)m �Re
+21, (σ∂σ)m �Re
+22 are ǫ-regular at σ = 0.
+Proof. Applying Proposition 12.14 and Lemma 12.15, we conclude that for 0 ≤ k ≤ m, (σ∂σ)kL−1
+00 is ǫ-
+regular. Then by Proposition 12.14, we find that (σ∂σ)k �L♯
+11, (σ∂σ)k �L♯
+21, (σ∂σ)k �L♭
+22 (and thus (σ∂σ)k(�L♭
+22)−1
+by Lemma 12.15) have an ǫ-regular expansion up to order one, while (σ∂σ)k �L♯
+12 is ǫ-regular. Therefore, we
+see that
+(σ∂σ)k�
+�L♯
+11 − σ�L♯
+12(�L♭
+22)−1�L♯
+21
+�
+has an ǫ-regular expansion up to order one whose coefficient of σ0 is (σ∂σ)k �L0
+11(b).
+Then the ǫ-regular
+expansion property of (σ∂σ)m �R11 follows from Lemma 12.15. Similarly, the ǫ-regular expansion property of
+(σ∂σ)m �R21, (σ∂σ)m �R22 follows from that of (σ∂σ)k �R11, (σ∂σ)k(�L♭
+22)−1, (σ∂σ)m�L♯
+21 and the above expressions
+of �Rij. Finally, the ǫ-regularity of the remaining components can be obtained by using the above expressions
+of �Rij and Proposition 12.14.
+□
+Proposition 12.17. Let L−
+b (σ) be the regular part of �
+Lb(σ)−1 defined in Corollary 11.4 and define
+L−(m)
+b
+(σ) := (σ∂σ)mL−
+b (σ).
+Let − 3
+2 < ℓ < − 1
+2, 0 < ǫ < 1, − 1
+2 < ℓ+ǫ < 1
+2 and s−ǫ > 4+m where m ∈ N0. For Im σ ≥ 0 and 0 < |σ| ≤ c0
+where c0 > 0 is a small constant, L−(m)
+b
+(σ) satisfies
+L−(m)
+b
+(σ) : ¯Hs−1,ℓ+2
+b
+( ¯X) → ¯Hs−m,ℓ+ǫ−1
+b
+( ¯X)
+(12.33)
+
+150
+LILI HE
+and
+∂j
+σL−(m)
+b
+(σ) : ¯Hs−1,ℓ+2
+b
+( ¯X) → |σ|−ǫ−j+1 ¯Hs−m−ǫ−j+1,ℓ+ǫ−1
+b
+( ¯X),
+j = 1, 2.
+(12.34)
+Proof. Recall that
+L−
+b (σ) = ˇLb(σ)−1Rreg(b, σ) − ˇLb(σ)−1��
+∂σ�
+Lb(0) + σ
+2 ∂2
+σ�
+Lb(0)
+�
+(P 1(b) + P e(b, σ)) + 1
+2∂2
+σ�
+Lb(0)P 2(b)
+�
+where P 2(b), P 1(b) : ¯Hs−1,ℓ+2
+b
+( ¯X) → ρC∞( ¯X) + ¯H
+∞, 1
+2 −
+b
+( ¯X),
+(σ∂σ)kP e(b, σ) : ¯Hs−1,ℓ+2
+b
+( ¯X) → Kb,1
+is ǫ-regular for 0 ≤ k ≤ m, and
+Rreg(b, σ) =
+
+
+�R00(b, σ)
+�R01(b, σ)
+�R02(b, σ)
+0
+0
+0
+�R20(b, σ)
+�Re
+21(b, σ)
+�Re
+22(b, σ)
+
+ .
+Then a direct calculation implies (σ∂σ)mL−
+b (σ) is a linear combination of the operators of the forms
+ˇL(m1)
+b
+(σ) ◦ (σ∂σ)m2Rreg(b, σ),
+ˇL(m1)
+b
+(σ) ◦ ∂2
+σ�
+Lb(0) ◦ P 2(b),
+ˇL(m1)
+b
+(σ) ◦ (σ∂σ)m2�
+∂σ�
+Lb(0) + σ
+2 ∂2
+σ�
+Lb(0)
+�
+◦ (σ∂σ)m3�
+P 1(b) + P e(b, σ)
+�
+.
+Applying Proposition 12.13 and 12.16, we obtain (12.33) and (12.34).
+□
+13. Decay estimates
+In this section, we continue using the notation from §12. We will study the decay of the solution (˙g, ˙A)
+to the equations
+Lb(˙g, ˙A) = f,
+f ∈ C∞
+c ((0, ∞)tb,∗; ¯Hs,ℓ+2
+b
+( ¯X)).
+(13.1)
+We define spacetime Sobolev spaces
+�Hs,ℓ,k
+b,c ,
+k ∈ N0,
+(13.2)
+which is equal to L2(t−1
+b,∗([0, ∞)); |dgb|) for s, ℓ, k = 0. Here, the index s measures the regularity with respect
+to the derivatives ∂tb,∗ and the stationary b-vector fields on ¯X (which is spanned by r∂r and rotation vector
+fields). The index ℓ is the weight in ρ = r−1, that is, �Hs,ℓ,k
+b,c
+= ρℓ �Hs,k
+b,c. The index k ∈ N0 measures the
+regularity with respect to ⟨tb,∗⟩∂tb,∗ (which is related to the conormal regularity of �
+Lb(σ)−1 on the Fourier
+transform side). Therefore, u ∈ �Hs,ℓ,k
+b,c
+if and only if (⟨tb,∗⟩∂tb,∗)ju ∈ �Hs−j,ℓ
+b,c
+:= �Hs−j,ℓ,0
+b,c
+for any 0 ≤ j ≤ k.
+Theorem 13.1. Let − 3
+2 < ℓ < − 1
+2, 0 < ǫ < 1, − 1
+2 < ℓ + ǫ < 1
+2 and s − ǫ > 4 + m where m ∈ N0. Let (˙g, ˙A)
+be the solution to the equation (13.1) with f ∈ C∞
+c ((0, ∞)tb,∗; ¯Hs,ℓ+2
+b
+( ¯X)). Then there exists a generalized
+zero mode (ˆg, ˆA) ∈ �Kb (defined in Proposition 10.5) such that
+(˙g, ˙A) = (ˆg, ˆA) + (¯g, ¯A) + (˜g, ˜A)
+(13.3)
+and (ˆg, ˆA), (¯g, ¯A) and (˜g, ˜A) satisfy the following properties.
+(1) The term (¯g, ¯A) can be written as (¯g, ¯A) = (¯g1, ¯A1)+(¯g2, ¯A2) where (¯g1, ¯A1) is a pure gauge solution.
+Moreover, for m ≥ 1 and j ≥ 0, we have the pointwise estimates
+|
+�
+⟨tb,∗⟩∂tb,∗
+�m−1V j(¯g1, ¯A1)| ≲
+�
+⟨tb,∗ + r⟩−2+ǫ + ⟨tb,∗⟩−1+ǫ⟨r⟩−2�
+∥f∥⟨tb,∗⟩− 3
+2 − �
+Hs−1,ℓ+2
+b,c
+(13.4)
+and
+|
+�
+⟨tb,∗⟩∂tb,∗
+�m−1V j(¯g2, ¯A2)| ≲ ⟨tb,∗ + r⟩−2+ǫ∥f∥⟨tb,∗⟩− 3
+2 − �
+Hs−1,ℓ+2
+b,c
+(13.5)
+where V j denotes the product of j vector fields V ∈ {∂tb,∗, r∂r, rotation fields }.
+
+LINEAR STABILITY OF KERR-NEWMAN FAMILY
+151
+(2) (˜g, ˜A) satisfies
+∥(˜g, ˜A)∥⟨tb,∗⟩− 3
+2 +ǫ �
+Hs−2,ℓ+ǫ−1,m
+b,c
+≲ ∥f∥⟨tb,∗⟩− 7
+2 +ǫ �
+Hs+m,ℓ+2,m
+b,c
+.
+(13.6)
+For m ≥ 1, we also have
+∥
+�
+⟨tb,∗⟩∂tb,∗
+�m−1(˜g, ˜A)∥⟨tb,∗⟩−2+ǫL∞(Rtb,∗ ; ¯
+Hs−2−m,ℓ+ǫ−1
+b
+( ¯
+X)) ≲ ∥f∥⟨tb,∗⟩− 7
+2 +ǫ �
+Hs+m,ℓ+2,m
+b,c
+.
+(13.7)
+For m ≥ 1 and 0 ≤ j < s − 4 − m with j ∈ N0, we have the pointwise decay estimate
+|
+�
+⟨tb,∗⟩∂tb,∗
+�m−1V j(˜g, ˜A)| ≲ ⟨tb,∗⟩−2+ǫ⟨r⟩− 1
+2 −ℓ−ǫ∥f∥⟨tb,∗⟩− 7
+2 +ǫ �
+Hs+m,ℓ+2,m
+b,c
+.
+(13.8)
+(3) Recall the definition of Kb,1, Kb,2 and K∗
+b at the beginning of §11. The leading term (ˆg, ˆA) ∈ �Kb is
+uniquely determined by f and can be expressed as follows:
+(ˆg, ˆA) = −1
+2
+�
+tb,∗(˙g1, ˙A1) + (ˇg1, ˇA1)
+�
++ 1
+2
+�
+(˙g′
+1, ˙A′
+1) + (˙g′′
+1, ˙A′′
+1) + (˙g2, ˙A2)
+�
+(13.9)
+where (˙g1, ˙A1), (˙g′
+1, ˙A′
+1), (˙g′′
+1 , ˙A′′
+1) ∈ Kb,1, and (˙g2, ˙A2) ∈ Kb,2.
+Moreover, (˙g1, ˙A1), (˙g′
+1, ˙A′
+1) and
+(˙g2, ˙A2) are determined by the conditions (11.30) and (11.31) with f there replaced by
+�
+R f(tb,∗) dtb,∗,
+and (˙g′′
+1, ˙A′′
+1) ∈ Kb,1 is uniquely determined by the condition
+−
+�
+[Lb, tb,∗](ˇg′′
+1 , ˇA′′
+1) + 1
+2[[Lb, tb,∗], tb,∗](˙g′′
+1, ˙A′′
+1), (˙g∗, ˙A∗)
+�
++
+�
+[Lb, tb,∗](˙g′′
+2, ˙A′′
+2), (˙g∗, ˙A∗)
+�
+= ⟨
+�
+R
+tb,∗f(tb,∗) dtb,∗, (˙g∗, ˙A∗)⟩
+for all
+(˙g∗, ˙A∗) ∈ K∗
+b.
+(13.10)
+for some (˙g′′
+2, ˙A′′
+2) ∈ Kb,2.
+Proof. The strategy of the proof is to take Fourier transform of the inhomogeneous equation Lb(˙g, ˙A) = f
+in tb,∗. We define the Fourier transform of (˙g, ˙A) as
+(ˆ˙g(σ), ˆ˙A(σ)) :=
+�
+R
+eitb,∗σ(˙g, ˙A)(tb,∗) dtb,∗
+(13.11)
+and likewise for f (here we drop the notation for the spatial variables.).
+First, according to the energy estimate in Proposition 5.27, we see that the solution (˙g, ˙A) satisfies
+∥(˙g, ˙A)(tb,∗, ·)∥ ¯
+H1,−1
+b
+( ¯
+X) ≲ eMtb,∗
+for some M > 0. This implies that the Fourier transform (ˆ˙g(σ), ˆ˙A(σ)) of (˙g, ˙A) is well defined for Im σ >
+M.
+Since f(tb,∗) has compact support in tb,∗, ˆf(σ) is holomorphic in the full plane σ ∈ C and decays
+superpolynomially as |Re σ| → ∞ with Im σ > 0 fixed. Fourier transforming of the equation Lb(˙g, ˙A) = f,
+we obtain
+�
+Lb(σ)(ˆ˙g(σ), ˆ˙A(σ)) = ˆf(σ),
+Im σ > M.
+(13.12)
+In view of Theorem 5.28, we see that �
+Lb(σ) is invertible in Im σ > M and
+(ˆ˙g(σ), ˆ˙A(σ)) = �
+Lb(σ)−1 ˆf(σ),
+Im σ > M.
+Therefore, we have the following integral representation
+(˙g(tb,∗),
+˙A(tb,∗)) = 1
+2π
+�
+Im σ=M+1
+e−iσtb,∗ �
+Lb(σ)−1 ˆf(σ) dσ.
+(13.13)
+Owing to Theorem 12.12, we see that the integrand in the above integral representation is holomorphic in σ
+(with values in ¯Hs,ℓ
+b ( ¯X)) in upper half plane Im σ > 0, and decay superpolynomially in |Re σ| as with Im σ
+bounded. Using Cauchy’s Theorem, we obtain
+(˙g(tb,∗),
+˙A(tb,∗)) = 1
+2π
+�
+Im σ=C
+e−iσtb,∗ �
+Lb(σ)−1 ˆf(σ) dσ.
+(13.14)
+for any C > 0. Now we let C → 0+.
+
+152
+LILI HE
+Fixing a frequency cutoff χ(s) ∈ C∞
+c (R) such that χ(s) = 1 for |s| ≤ c0/2 and χ(s) = 0 for |s| ≥ c0 where
+c0 is defined as in Theorem 12.11. We now write
+ˆf(σ) = χ(Re σ) ˆf(σ) + (1 − χ(Re σ)) ˆf(σ) := ˆf1(σ) + ˆf2(σ).
+Correspondingly, by Theorem 11.3 and Corollary 11.4, we split �
+Lb(σ)−1 ˆf(σ) into two parts
+�
+Lb(σ)−1 ˆf(σ) = (ˆ˙greg(σ), ˆ˙Areg(σ)) + (ˆ˙gsing(σ), ˆ˙Asing(σ))
+where
+(ˆ˙greg(σ), ˆ˙Areg(σ)) = L−
+b (σ) ˆf1(σ) + �
+Lb(σ)−1 ˆf2(σ),
+(ˆ˙gsing(σ), ˆ˙Asing(σ)) = P(b, σ) ˆf1(σ).
+As discussed above, we can shift the integration contour to Im σ = C for any C > 0. Letting C → 0+, we
+obtain
+(˙g(tb,∗), ˙A(tb,∗)) = (˙greg(tb,∗), ˙Areg(tb,∗)) + (˙gsing(tb,∗), ˙Asing(tb,∗))
+where
+(˙greg(tb,∗), ˙Areg(tb,∗)) = 1
+2π
+�
+R
+e−iσtb,∗(ˆ˙greg(σ), ˆ˙Areg(σ)) dσ
+and
+(˙gsing(tb,∗), ˙Asing(tb,∗)) =
+lim
+C→0+
+1
+2π
+�
+Im σ=C
+e−iσtb,∗(ˆ˙gsing(σ), ˆ˙Asing(σ)) dσ.
+• Analysis of (˙greg(tb,∗), ˙Areg(tb,∗)). Since f ∈ C∞
+c ((0, ∞)tb,∗; ¯Hs,ℓ+2
+b
+( ¯X)), we see that ˆf1(σ), ˆf2(σ) ∈
+Hs′(Rσ; ¯Hs,ℓ+2
+b
+( ¯X)) for any s′ ∈ R. As H3/2−ǫ′(R) is an algebra for any 0 < ǫ′ < 1, by Theorem
+12.11, we have
+L−
+b (σ) ˆf1(σ) ∈ H
+3
+2 −ǫ′((−c0, c0); ¯H
+s+1−max{ǫ, 1
+2 },ℓ+ǫ−1
+b
+( ¯X)),
+0 < ǫ < ǫ′ < 1.
+Using Moser estimate, we bound norm of L−
+b (σ) ˆf1(σ) as follows
+∥L−
+b (σ) ˆf1(σ)∥
+H
+3
+2 −ǫ′ (Rσ; ¯
+H
+s+1−max{ǫ, 1
+2 },ℓ+ǫ−1
+b
+( ¯
+X)) ≲ ∥ ˆf1(σ)∥H
+3
+2 −ǫ′ (Rσ; ¯
+Hs,ℓ+2
+b
+( ¯
+X))
+≲ ∥ ˆf(σ)∥H
+3
+2 −ǫ′ (Rσ; ¯
+Hs,ℓ+2
+b
+( ¯
+X)) = ∥⟨tb,∗⟩
+3
+2 −ǫ′f∥L2(Rtb,∗; ¯
+Hs,ℓ+2
+b
+( ¯
+X)).
+By exploiting the conormal regularity of L−
+b (σ) proved in Proposition 12.13, we find that for 0 ≤
+k ≤ m,
+(σ∂σ)k�
+L−
+b (σ) ˆf1(σ)
+�
+∈ H
+3
+2 −ǫ′((−c0, c0); ¯H
+s+1−k−max{ǫ, 1
+2 },ℓ+ǫ−1
+b
+( ¯X)),
+0 < ǫ < ǫ′ < 1.
+In a similar manner, we arrive at
+∥(σ∂σ)k�
+L−
+b (σ) ˆf1(σ)
+�
+∥
+H
+3
+2 −ǫ′ (Rσ; ¯
+H
+s+1−k−max{ǫ, 1
+2 },ℓ+ǫ−1
+b
+( ¯
+X))
+≲
+k
+�
+j=0
+∥(σ∂σ)j ˆf1(σ)∥H
+3
+2 −ǫ′ (Rσ; ¯
+Hs−j,ℓ+2
+b
+( ¯
+X)) ≲
+k
+�
+j=0
+∥(σ∂σ)j ˆf(σ)∥H
+3
+2 −ǫ′ (Rσ; ¯
+Hs−j,ℓ+2
+b
+( ¯
+X))
+=
+k
+�
+j=0
+∥(tb,∗∂tb,∗)j⟨tb,∗⟩
+3
+2 −ǫ′f∥L2(Rtb,∗; ¯
+Hs−j,ℓ+2
+b
+( ¯
+X)) ≲ ∥f∥⟨tb,∗⟩− 3
+2 +ǫ′ �
+Hs,ℓ+2,k
+b,c
+.
+(13.15)
+As for the second part �
+Lb(σ)−1 ˆf2(σ) in (ˆ˙greg(σ), ˆ˙Areg(σ)), by Theorem 12.12, we have
+�
+Lb(σ)−1 ∈ W j,∞�
+Rσ \ [−c0
+2 , c0
+2 ]; L
+� ¯Hs,ℓ+2
+b,h
+, h−j ¯Hs−j,ℓ
+b,h
+��
+,
+j ∈ N0
+(13.16)
+where we take h = ⟨σ⟩−1. Taking j = 2 gives
+�
+Lb(σ)−1 ˆf2(σ) ∈ H
+3
+2 −ǫ(R; ⟨σ⟩−s+2 ¯Hs−2,ℓ
+b,⟨σ⟩−1( ¯X)).
+whose norm is bounded by
+∥ ˆf2(σ)∥H
+3
+2 −ǫ(R;⟨σ⟩−s ¯
+Hs,ℓ+2
+b,⟨σ⟩−1 ( ¯
+X)) ≲ ∥f∥⟨tb,∗⟩− 3
+2 +ǫ �
+Hs,ℓ+2
+b,c
+.
+More generally, we have
+(σ∂σ)k��
+Lb(σ)−1 ˆf2(σ)
+�
+∈ H
+3
+2 −ǫ(R; ⟨σ⟩−s+2+2k ¯Hs−2−k,ℓ
+b,⟨σ⟩−1 ( ¯X)).
+
+LINEAR STABILITY OF KERR-NEWMAN FAMILY
+153
+and
+∥(σ∂σ)k��
+Lb(σ)−1 ˆf2(σ)
+�
+∥H
+3
+2 −ǫ(R;⟨σ⟩−s+2+k ¯
+Hs−2−k,ℓ
+b,⟨σ⟩−1 ( ¯
+X))
+≲
+k
+�
+j=0
+∥(σ∂σ)j ˆf2(σ)∥H
+3
+2 −ǫ(R;⟨σ⟩−s−k+2j ¯
+Hs+k−2j,ℓ+2
+b,⟨σ⟩−1
+( ¯
+X))
+≲
+k
+�
+j=0
+∥(σ∂σ)j ˆf(σ)∥H
+3
+2 −ǫ(R;⟨σ⟩−s−k+2j ¯
+Hs+k−2j,ℓ+2
+b,⟨σ⟩−1
+( ¯
+X))
+≲
+k
+�
+j=0
+∥(tb,∗∂tb,∗)jf∥⟨tb,∗⟩− 3
+2 +ǫ �
+Hs+k−2j,ℓ+2
+b,c
+≲ ∥f∥⟨tb,∗⟩− 3
+2 +ǫ �
+Hs+k,ℓ+2,k
+b,c
+.
+(13.17)
+Taking inverse Fourier transform of (ˆ˙greg(σ), ˆ˙Areg(σ)) and using the estimates (13.15) and (13.17),
+we obtain for 0 ≤ k ≤ m and 0 < ǫ < ǫ′ < 1
+∥(˙greg(tb,∗), ˙Areg(tb,∗))∥⟨tb,∗⟩− 3
+2 +ǫ′ �
+Hs−2,ℓ+ǫ−1,k
+b,c
+∼
+k
+�
+j=0
+∥(σ∂σ)j(ˆ˙greg(σ), ˆ˙Areg(σ))∥H
+3
+2 −ǫ′ (R;⟨σ⟩−s+2+j ¯
+Hs−2−j,ℓ+ǫ−1
+b,⟨σ⟩−1
+( ¯
+X))
+≲
+k
+�
+j=0
+∥(σ∂σ)j�
+L−
+b (σ) ˆf1(σ)
+�
+∥
+H
+3
+2 −ǫ′ (Rσ; ¯
+H
+s+1−j−max{ǫ, 1
+2 },ℓ+ǫ−1
+b
+( ¯
+X))
++
+k
+�
+j=0
+∥(σ∂σ)j��
+Lb(σ)−1 ˆf2(σ)
+�
+∥H
+3
+2 −ǫ′ (R;⟨σ⟩−s+2+j ¯
+Hs−2−j,ℓ
+b,⟨σ⟩−1 ( ¯
+X))
+≲ ∥f∥⟨tb,∗⟩− 3
+2 +ǫ′ �
+Hs+k,ℓ+2,k
+b,c
+.
+(13.18)
+Now we integrate ∂tb,∗(˙greg(tb,∗), ˙Areg(tb,∗)) = t−1
+b,∗(tb,∗∂tb,∗)(˙greg(tb,∗), ˙Areg(tb,∗)) and exploit the
+above estimate (13.18) for k = 1 to obtain the decay estimate of (˙greg(tb,∗), ˙Areg(tb,∗)) in tb,∗. Con-
+cretely, for tb,∗ ≥ 0 and 0 < ǫ < ǫ′ < 1, we have
+∥(˙greg(tb,∗), ˙Areg(tb,∗))∥ ¯
+Hs−3,ℓ+ǫ−1
+b
+( ¯
+X) = ∥
+� ∞
+tb,∗
+∂s
+�
+˙greg(s), ˙Areg(s)
+�
+ds∥ ¯
+Hs−3,ℓ+ǫ−1
+b
+( ¯
+X)
+≲
+� � ∞
+tb,∗
+⟨s⟩−3+2ǫ′⟨s⟩−2 ds
+� 1
+2 ∥(⟨tb,∗⟩∂tb,∗)(˙greg(tb,∗), ˙Areg(tb,∗))∥⟨tb,∗⟩− 3
+2 +ǫ′ �
+Hs−3,ℓ+ǫ−1
+b,c
+≲ ⟨tb,∗⟩−2+ǫ′∥f∥⟨tb,∗⟩− 3
+2 +ǫ′ �
+Hs+1,ℓ+2,1
+b,c
+.
+Since (˙greg(tb,∗), ˙Areg(tb,∗)) has support in tb,∗ ≥ −C for some C > 0, it follows that
+∥(˙greg(tb,∗), ˙Areg(tb,∗))∥⟨tb,∗⟩−2+ǫL∞(Rtb,∗; ¯
+Hs−3,ℓ+ǫ−1
+b
+( ¯
+X)) ≲ ∥f∥⟨tb,∗⟩− 3
+2 +ǫ′ �
+Hs+1,ℓ+2,1
+b,c
+.
+By the same reasoning, the above decay estimate can be generalized for all 0 ≤ k ≤ m,
+∥
+�
+⟨tb,∗⟩∂tb,∗
+�k−1(˙greg(tb,∗), ˙Areg(tb,∗))∥⟨tb,∗⟩−2+ǫL∞(Rtb,∗ ; ¯
+Hs−2−k,ℓ+ǫ−1
+b
+( ¯
+X)) ≲ ∥f∥⟨tb,∗⟩− 3
+2 +ǫ′ �
+Hs+k,ℓ+2,k
+b,c
+.
+(13.19)
+The estimates (13.18) and (13.19) imply that (˙greg(tb,∗), ˙Areg(tb,∗)) occurs as a part of the remainder
+(˜g, ˜A).
+• Analysis of (˙gsing(tb,∗), ˙Asing(tb,∗)). We now turn to discussing the singular part
+(˙gsing(tb,∗), ˙Asing(tb,∗)) =
+lim
+C→0+
+1
+2π
+�
+Im σ=C
+e−iσtb,∗P(b, σ) ˆf1(σ) dσ.
+Since ˆf is holomorphic in the full plane σ ∈ C, we write
+ˆf1(σ) = χ(Re σ) ˆf(σ) = χ(Re σ)
+�
+f (0) + iσf (1) + σ2f (2)(σ)) = χ(Re σ)
+�
+f (0) + iσf (1)�
++ σ2 ˆf3(σ)
+
+154
+LILI HE
+where
+f (0) = ˆf(0) =
+�
+R
+f(tb,∗) dtb,∗,
+f (1) = −i∂σ ˆf(0) =
+�
+R
+tb,∗f(tb,∗) dtb,∗,
+and f (2)(σ) is holomorphic in the full plane σ ∈ C. Moreover, by Sobolev embedding theorem, we
+have for j = 0, 1
+∥f (j)∥ ¯
+Hs,ℓ+2
+b
+( ¯
+X) ≲ ∥ ˆf(σ)∥H3/2+s′ (Rσ; ¯
+Hs,ℓ+2
+b
+( ¯
+X)) ≲ ∥f∥⟨tb,∗⟩− 3
+2 −s′ �
+Hs,ℓ
+b,c
+for any
+s′ > 0.
+As for ˆf3(σ), since ˆf3(σ) = σ−2( ˆf1(σ)−χ(Re σ)f (0)−iσχ(Re σ)f (1)), it follows that ˆf3(σ) ∈ C∞
+c (Rσ)
+and
+∥ ˆf3(σ)∥Hs′ (Rσ; ¯
+Hs,ℓ+2
+b
+( ¯
+X)) ≲ ∥ ˆf(σ)∥Hs′+2(Rσ; ¯
+Hs,ℓ+2
+b
+( ¯
+X)),
+s′ > −1
+2.
+Recall that P(b, σ) = σ−2P 2(b)+σ−1P 1(b)+σ−1P e(σ) where P 2(b), P 1(b) : ¯Hs,ℓ+2
+b
+→ ¯H∞,−1/2−
+b
+( ¯X)
+and P e(b, σ) : ¯Hs,ℓ+2
+b
+→ Kb,1 is ǫ-regular. Therefore, following the arguments in the proof of (13.15),
+we find that for 0 ≤ k ≤ m,
+(σ∂σ)k�
+P(b, σ)(σ2 ˆf3(σ))
+�
+∈ H
+3
+2 −ǫ((−c0, c0); ¯H
+∞,− 1
+2 −
+b
+( ¯X)),
+whose norm is bounded in the following manner
+∥(σ∂σ)k�
+P(b, σ)(σ2 ˆf3(σ))
+�
+∥
+H
+3
+2 −ǫ(R; ¯
+H
+∞,− 1
+2 −
+b
+( ¯
+X))
+≲
+k
+�
+j=0
+∥(σ∂σ)j ˆf3(σ)∥H
+3
+2 −ǫ(R; ¯
+Hs,ℓ+2
+b
+( ¯
+X)) ≲
+k
+�
+j=0
+∥(σ∂σ)j ˆf(σ)∥H
+7
+2 −ǫ(R; ¯
+Hs,ℓ+2
+b
+( ¯
+X))
+≲ ∥f∥⟨tb,∗⟩− 7
+2 +ǫ �
+Hs,ℓ+2,m
+b,c
+.
+(13.20)
+Similarly, we have
+(σ∂σ)k��
+P 1(b) + P e(b, σ)
+�
+(χ(σ)f (1))
+�
+∈ H
+3
+2 −ǫ((−c0, c0); ¯H
+∞,− 1
+2 −
+b
+( ¯X)),
+and
+∥(σ∂σ)k��
+P 1(b) + P e(b, σ)
+�
+(χ(σ)f (1))
+�
+∥
+H
+3
+2 −ǫ(R; ¯
+H
+∞,− 1
+2 −
+b
+( ¯
+X)) ≲ ∥f∥⟨tb,∗⟩− 3
+2 − �
+Hs,ℓ+2
+b,c
+.
+(13.21)
+As a result, it remains to analyze P(b, σ)(χ(Re σ)f (0)) and P 2(b)(iσ−1χ(Re σ)f (1)). We write
+�
+σ−2P 2(b)f (0) + σ−1P 1(b)f (0) + P 2(b)(iσ−1f (1))
+�
++ σ−1P e(b, σ)(χ(Re σ)f (0))
+=
+�
+(σ−2(˙g1, ˙A1) − iσ−1(ˇg1, ˇA1)) + iσ−1�
+(˙g2, ˙A2) + (˙g′
+1, ˙A′
+1) + (˙g′′
+1, ˙A′′
+1)
+��
++ iσ−1(˙g′′′
+1 (σ), ˙A′′′
+1 (σ)) := I + II
+where (˙g1, ˙A1), (˙g′
+1, ˙A′
+1), (˙g′′
+1 , ˙A′′
+1), (˙g′′′
+1 (σ), ˙A′′′
+1 (σ)) ∈ Kb,1 and (˙g2, ˙A2) ∈ Kb,2. Moreover, (˙g′′
+1, ˙A′′
+1) is
+uniquely determined by the condition
+−
+�
+[Lb, tb,∗](ˇg′′
+1 , ˇA′′
+1) + 1
+2[[Lb, tb,∗], tb,∗](˙g′′
+1, ˙A′′
+1), (˙g∗, ˙A∗)
+�
++
+�
+[Lb, tb,∗](˙g′′
+2, ˙A′′
+2), (˙g∗, ˙A∗)
+�
+= ⟨
+�
+R
+tb,∗f(tb,∗) dtb,∗, (˙g∗, ˙A∗)⟩
+for all
+(˙g∗, ˙A∗) ∈ K∗
+b
+(13.22)
+for some (˙g′′
+2, ˙A′′
+2) ∈ Kb,2. Since σ−j(1 − χ(σ)) ∈ Hs′(Rσ) for j = 1, 2 and any s′ ∈ R, we have
+(σ∂σ)k�
+χ(Re σ)
+�
+P(b, σ)f (0) + P 2(b)(iσ−1f (1))
+�
+− (I + II)
+�
+∈ H
+3
+2 −ǫ((−c0, c0); ¯H
+∞,− 1
+2 −
+b
+( ¯X)),
+and
+∥(σ∂σ)k�
+χ(Re σ)
+�
+P(b, σ)f (0) + P 2(b)(iσ−1f (1))
+�
+− (I + II)
+�
+∥
+H
+3
+2 −ǫ(R; ¯
+H
+∞,− 1
+2 −
+b
+( ¯
+X))
+≲ ∥f∥⟨tb,∗⟩− 3
+2 − �
+Hs,ℓ+2
+b,c
+.
+(13.23)
+As a consequence, it suffices to discuss the inverse Fourier transform of I and II.
+
+LINEAR STABILITY OF KERR-NEWMAN FAMILY
+155
+– Analysis of the leading order term (ˆg, ˆA). Since
+F−1(σ−2) =
+lim
+C→0+
+1
+2π
+�
+Im σ=C
+e−iσtb,∗σ−2 dσ = −1
+2tb,∗H(tb,∗)
+and
+F−1(iσ−1) =
+lim
+C→0+
+1
+2π
+�
+Im σ=C
+e−iσtb,∗iσ−1 dσ = 1
+2H(tb,∗),
+we obtain
+F−1(I) = lim
+C→0+
+1
+2π
+�
+Im σ=C
+e−iσtb,∗I dσ
+= −1
+2
+�
+tb,∗(˙g1, ˙A1) + (ˇg1, ˇA1)
+�
++ 1
+2
+�
+(˙g′
+1, ˙A′
+1) + (˙g′′
+1 , ˙A′′
+1) + (˙g2, ˙A2)
+�
+:= (ˆg, ˆA),
+(13.24)
+where (˙g1, ˙A1), (˙g′
+1, ˙A′
+1), (˙g′′
+1, ˙A′′
+1) ∈ Kb,1, and (˙g2, ˙A2) ∈ Kb,2. Moreover, (˙g1, ˙A1), (˙g′
+1, ˙A′
+1) and
+(˙g2, ˙A2) are determined by the conditions (11.30) and (11.31) with f there replaced by f (0) =
+�
+R f(tb,∗) dtb,∗, and (˙g′′
+1, ˙A′′
+1) ∈ Kb,1 is uniquely determined by the condition (13.22) for some
+(˙g′′
+2, ˙A′′
+2) ∈ Kb,2.
+– Analysis of the pure gauge part (¯g, ¯A). Finally, we turn to the analysis of the inverse Fourier
+transform of II = iσ−1(˙g′′′
+1 (σ), ˙A′′′
+1 (σ)) where supp (˙g′′′
+1 (σ), ˙A′′′
+1 (σ)) ⊂ (−c0, c0). By Theorem
+12.9,
+II = σ−1(˙g′′′
+1 (σ), ˙A′′′
+1 (σ)) =
+�
+±
+ψ±(σ)
+where
+ψ± ∈ A−ǫ�
+± [0, c0); Kb,1
+�
+.
+We write
+� c0
+−c0
+e−itb,∗σII dσ =
+� c0
+0
+e−itb,∗σχ(tb,∗σ)ψ+(σ) dσ +
+� c0
+0
+e−itb,∗σ(1 − χ(tb,∗σ))ψ+(σ) dσ
++
+� 0
+−c0
+e−itb,∗σχ(tb,∗σ)ψ−(σ) dσ +
+� 0
+−c0
+e−itb,∗σ(1 − χ(tb,∗σ))ψ−(σ) dσ
+:= II+
+1 + II+
+2 + II−
+1 + II−
+2 .
+First, we have
+|II+
+1 | ≤
+� c0t−1
+b,∗
+0
+|ψ+(σ)| dσ ≲
+� c0t−1
+b,∗
+0
+σ−ǫ dσ ≲ t−1+ǫ
+b,∗
+.
+Next, we compute for k ∈ N
+II+
+2 =
+� c0
+0
+e−itb,∗σ(1 − χ(tb,∗σ))ψ+(σ) dσ = (itb,∗)−k
+� c0
+0
+e−itb,∗σ∂k
+σ
+�
+(1 − χ(tb,∗σ))ψ+(σ)
+�
+dσ.
+Then we have
+|II+
+2 | ≲ t−k
+b,∗
+� c0
+c0t−1
+b,∗/2
+σ−k−ǫ dσ ≲ t−1+ǫ
+b,∗
+.
+The estimates for II+
+1 , II+
+2 also hold after applying any number of derivatives tb∗∂tb,∗. We note
+that II−
+1 , II−
+2 can be handled in a similar manner. Since
+II = σ−1(˙g′′′
+1 (σ), ˙A′′′
+1 (σ)) = (2δ∗
+gbωb,s1(σ), �LAb(ωb,s1(σ))♯ + dφb,s1(σ))
+(13.25)
+where
+(ωb,s1(σ), φb,s1(σ)) ∈
+�
+±
+A−ǫ(±[0, c0); ¯H
+∞,− 3
+2 −
+b
+( ¯X; �
+scT ∗ ¯X) ⊕ ¯H
+∞,− 1
+2 −
+b
+( ¯X; C)),
+we can write
+(¯g, ¯A) =
+� c0
+−c0
+e−itb,∗σII dσ = (¯g1, ¯A1) + (¯g2, ¯A2)
+
+156
+LILI HE
+where
+(¯g1, ¯A1) =
+�
+2δ∗
+gb
+�
+χ( ⟨r⟩
+⟨tb,∗⟩)ωb,s1(tb,∗)
+�
+, �LAb
+�
+χ( ⟨r⟩
+⟨tb,∗⟩)ωb,s1(tb,∗)
+�♯ + d
+�
+χ( ⟨r⟩
+⟨tb,∗⟩)φb,s1(tb,∗)
+��
+,
+(¯g2, ¯A2) =
+�
+2[χ( ⟨r⟩
+⟨tb,∗⟩), δ∗
+gb]ωb,s1(tb,∗), [χ( ⟨r⟩
+⟨tb,∗⟩), �LAb]
+�
+ωb,s1(tb,∗)
+�♯ + [χ( ⟨r⟩
+⟨tb,∗⟩), d]φb,s1(tb,∗)
+�
+− χ( ⟨r⟩
+⟨tb,∗⟩)
+�
+2∂tb,∗ωb,s1 ⊗s dtb,∗, ∂tb,∗
+�
+Aα(ωb,s1)♯,α + φb,s1
+�
+dtb,∗
+�
++
+�
+1 − χ( ⟨r⟩
+⟨tb,∗⟩)
+�
+(¯g, ¯A).
+where χ is a smooth nonnegative cutoff such that χ(s) = 1 for s ≤ 1/2 and χ(s) = 0 for s ≥ 1.
+Then the estimates (13.4) and (13.5) for (¯g1, ¯A1) and (¯g2, ¯A2) follow from the following estimates
+|
+�
+⟨tb,∗⟩∂tb,∗
+�m−1V jωb,s1(tb,∗)| ≲ ⟨tb,∗⟩−1+ǫ∥f∥⟨tb,∗⟩− 3
+2 − �
+Hs,ℓ+2
+b,c
+,
+|
+�
+⟨tb,∗⟩∂tb,∗
+�m−1V jφb,s1(tb,∗)| ≲ ⟨tb,∗⟩−1+ǫ⟨r⟩−1∥f∥⟨tb,∗⟩− 3
+2 − �
+Hs,ℓ+2
+b,c
+,
+|
+�
+⟨tb,∗⟩∂tb,∗
+�m−1V j(¯g, ¯A)| ≲ ⟨tb,∗⟩−1+ǫ⟨r⟩−2∥f∥⟨tb,∗⟩− 3
+2 − �
+Hs,ℓ+2
+b,c
+.
+□
+14. Proof of the main theorem
+In the initial value problem for Einstein-Maxwell system as formulated in §3, we impose the initial data
+at the Cauchy hypersurface which terminates at spatial infinity (i.e., it coincides with the hypersurface t = 0
+when r is large enough). In this section, we will discuss how to reduce this initial value problem to the
+inhomogeneous problem studied in the previous section §13.
+We recall the time function t (corresponding to gb0) defined in (4.15), which satisfies that t = t for
+r ≥ 4m0, and dt is timelike everywhere on M with respect to gb0 with b0 = (m0, 0, Q0), |Q0| ≪ m0, hence
+for gb when b = (m, a, Q) is close to b0. We define
+Σ0 := t−1(0),
+which is a spacelike (with respect to gb) Cauchy hypersurface. Identifying the region r ≥ 4m0 in Σ0 with
+a subset of R3, we can compactify Σ0 at infinity to the manifold ¯Σ0 (with two boundary components: one
+is inside the black hole and the other is the spatial infinity). We denote by �
+scT ∗¯Σ0 = scT ∗¯Σ0 ⊕ R dt the
+spacetime scattering cotangent bundle, which for large r is spanned over C∞(¯Σ0) by dt and dx1, dx2, dx3,
+with (x1, x2, x3) denoting standard coordinates on R3.
+We first study the initial value problem for linearized gauge-fixed Einstein-Maxwell system Lb(˙g, ˙A) = 0.
+Theorem 14.1. Let 0 < α < 1 and s > 8 + m, m ∈ N. Suppose
+(˙g0, ˙g1, ˙A0, ˙A1) ∈ ¯Hs,−1/2+α
+b
+(¯Σ0; S2 �
+scT ∗ ¯Σ0) ⊕ ¯Hs−1,1/2+α
+b
+(¯Σ0; S2 �
+scT ∗¯Σ0)
+⊕ ¯Hs−1,−1/2+α
+b
+(¯Σ0; �
+scT ∗ ¯Σ0) ⊕ ¯Hs−2,1/2+α
+b
+(¯Σ0; �
+scT ∗¯Σ0).
+Then the solution (˙g, ˙A) of the initial value problem
+Lb(˙g, ˙A) = 0,
+(˙g, ˙A)|¯Σ0 = (˙g0, ˙A0),
+(L∂t ˙g, L∂t ˙A)|¯Σ0 = (˙g1, ˙A1)
+satisfies the following properties. Let
+N s,α := ∥ ˙g0∥ ¯
+Hs+1,−1/2+α
+b
++ ∥g1∥ ¯
+Hs,1/2+α
+b
++ ∥A0∥ ¯
+Hs,1/2+α
+b
++ ∥A1∥ ¯
+Hs,1/2+α
+b
+.
+(1) For tb,∗ ≥ 0, the solution (˙g, ˙A) can be written as
+(˙g, ˙A) = (ˆg, ˆA) + (¯g, ¯A) + (˜g, ˜A)
+(14.1)
+where (ˆg, ˆA) ∈ �Kb is a generalized zero mode (defined in Proposition 10.5), and (¯g, ¯A), (˜g, ˜A) satisfy
+the following estimates.
+
+LINEAR STABILITY OF KERR-NEWMAN FAMILY
+157
+(i) The term (¯g, ¯A) can be written as (¯g, ¯A) = (¯g1, ¯A1) + (¯g2, ¯A2) where (¯g1, ¯A1) is a pure gauge
+solution. Moreover, for m ≥ 1 and j ≥ 0, we have the pointwise estimates
+|
+�
+⟨tb,∗⟩∂tb,∗
+�m−1V j(¯g1, ¯A1)| ≲
+�
+⟨tb,∗ + r⟩−2+ǫ + ⟨tb,∗⟩−1+ǫ⟨r⟩−2�
+N s,α
+(14.2)
+and
+|
+�
+⟨tb,∗⟩∂tb,∗
+�m−1V j(¯g2, ¯A2)| ≲ ⟨tb,∗ + r⟩−2+ǫN s,α
+(14.3)
+where V j denotes the product of j vector fields V ∈ {∂tb,∗, r∂r, rotation fields }.
+(ii) (˜g, ˜A) satisfies: for 0 < ǫ < 1 with α + ǫ > 1 and 0 ≤ m ≤ s−6
+2 , we have
+∥(˜g, ˜A)∥⟨tb,∗⟩− 3
+2 +ǫ �
+Hs−6−m,−5/2+α+ǫ−,m
+b,c
+≲ N s,α
+(14.4)
+For 1 ≤ m ≤ s−6
+2 , we also have
+∥
+�
+⟨tb,∗⟩∂tb,∗
+�m−1(˜g, ˜A)∥⟨tb,∗⟩−2+ǫL∞(Rtb,∗; ¯
+Hs−6−m,−5/2+α+ǫ−
+b
+( ¯
+X)) ≲ N s,α.
+(14.5)
+For m ≥ 1 and 0 ≤ j < s − 8 − m with j ∈ N0, we have the pointwise decay estimate
+|
+�
+⟨tb,∗⟩∂tb,∗
+�m−1V j(˜g, ˜A)| ≲ ⟨tb,∗⟩−2+ǫ⟨r⟩−α+ǫ+1+N s,α.
+(14.6)
+(2) In t ≥ 0, tb,∗ ≤ 0, we have (˙g, ˙A) = r−1Hs−3(Rtb,∗ × S2) + (˜g, ˜A) with |(˜g, ˜A)| ≲ r−1(1 + |tb,∗|)−α.
+Proof. Let T1 < T2 and let χ(tb,∗) be a nonnegative smooth cutoff such that χ = 1 for tb,∗ ≥ T2 and
+χ(tb,∗) = 0 for tb,∗ ≤ T1. Then we obtain the following equation
+Lb
+�
+χ(tb,∗)(˙g, ˙A)
+�
+= [Lb, χ(tb,∗)](˙g, ˙A),
+of which the right-hand side is supported in tb,∗ ∈ (T1, T2). In view of Theorem 13.1, it suffices to prove that
+Lb
+�
+χ(tb,∗)(˙g, ˙A)
+�
+∈ �Hs−4−k,1/2+α−,k
+b,c
+,
+k ∈ N.
+(14.7)
+Since the regularity with respect to ⟨tb,∗⟩Dtb,∗ and Dtb,∗ is equivalent in the support of Lb
+�
+χ(tb,∗)(˙g, ˙A)
+�
+,
+it suffices to prove (14.7) for k = 0. First, the global wellposedness theory of linear wave equations implies
+that (˙g, ˙A) ∈ Hs−1 for t ≥ 0, t∗ ≤ C, r ≤ C for any C; this implies that [Lb, χ] ∈ Hs−2 in such compact
+sets. Therefore, it suffices to work in an arbitrarily small neighborhood r > R0 ≫ 1 of infinity, which we
+shall do from now on.
+The proof closely follows the argument in [43, §§4–5]. Let b = (m, a, Q). Then we define b1 = (m, 0, Q)
+and introduce the incoming and outgoing null coordinates
+x0 = t + rb1,∗,
+x1 = t − rb1,∗,
+with respect to which we have
+gb1 = −µb1dx0 dx1 + r2/g
+and
+2∂0 ≡ 2∂x0 = ∂t + (1 − 2m
+r
++ Q2
+r2 )∂r,
+2∂1 ≡ 2∂x1 = ∂t − (1 − 2m
+r
++ Q2
+r2 )∂r.
+Let x2, x3 denote local coordinates on S2, and denote spherical indices by c, d = 2, 3; let ∂c ≡ ∂xc; let ¯X be
+the compactification of t−1
+b,∗(0). Since gb − gb1 ∈ ρ2C∞( ¯X; S2 �
+scT ∗ ¯X), we have
+gb(∂0, ∂0), (gb − gb1)(∂0, ∂1), gb(∂0, r−1∂c),
+gb(∂1, ∂1), gb(∂1, r−1∂c), (gb − gb1)(r−1∂b, r−1∂c) ∈ ρ2C∞.
+We let
+ρI = (rb1,∗ − t)/r,
+ρ0 = (rb1,∗ − t)−1,
+ρ = ρIρ0 = 1
+r .
+Then we have
+ρ−3Lbρ = −□ρ2gb ⊗ Id14×14 + R
+with R ∈ �
+Diff1
+b acting on sections of S2 �
+scT ∗ ¯X ⊕ �
+scT ∗ ¯X), where �
+Diffk
+b consists of up to k-fold products of the
+vector fields {ρI∂ρI, ρ0∂ρ0, rotation vector fields}. We note that switching from Lb to ρ−3Lbρ is related to
+the Friedlander rescaling for the scalar wave equation [24].
+
+158
+LILI HE
+Then one can use the multiplier W := ρ−2a0
+0
+ρ−2aI
+I
+V with V := −(1+cV )ρI∂ρI +ρ0∂ρ0 and a0 = α, aI < 0
+as introduced in [43, Lemma 4.4] to derive an energy estimate, which allows us to conclude that near
+I +, (˙g, ˙A) lies in �Hs−1,−1/2−
+b
+(see [43, Proposition 4.8] for detail. Actually, the present setting is much
+simpler than that in the reference as we do not need to get sharp decay rate for a certain component as did
+in [43, Proposition 4.8]). To further determine the leading behavior of (˙g, ˙A) near I , one rewrite the equations
+ρ−3Lbρ(ρ−1 ˙g, ρ−1 ˙A) = 0 as 2ρ−2∂0∂1(ρ−1 ˙g, ρ−1 ˙A) = ( /∆ + ˜R)(ρ−1 ˙g, ρ−1 ˙A) where ˜R ∈ ρ�
+Diff2
+b + �
+Diff1
+b. Since
+−2ρ−2∂0∂1 = ρ−1
+I (ρ0∂ρ0 − ρI∂ρI)ρI∂ρI + �
+Diff1
+b, it follows that
+(ρ0∂ρ0 − ρI∂ρI)ρI∂ρI(ρ−1 ˙g, ρ−1 ˙A) ∈ ρI �
+Diff2
+b(ρ−1 ˙g, ρ−1 ˙A) ∼ ρα
+0 ρaI+1
+I
+.
+We choose aI < 0 such that 0 < 1 + aI < α (see [43, Lemma 7.7]). Integrating first along the vector field
+ρ0∂ρ0 − ρI∂ρI from ρI ≥ ǫ, where (ρ−1 ˙g, ρ−1 ˙A) ∼ ρα
+0 , yields
+ρI∂ρI(ρ−1 ˙g, ρ−1 ˙A) ∼ ρα
+0 ρ1+aI
+I
+.
+Then integrating out the vector field ρI∂ρI gives
+(˙g, ˙A) − ρHs−3(Rtb1,∗ × S2) ∼ ρρα
+0 ρ1+aI
+I
+.
+This proves statement 2. Moreover, in view of Lemma 4.11, we see that
+[Lb, χ(tb,∗)] = 2ρχ′(t∗)(ρ∂ρ − 1) + ρ2Diff1
+sc + ρ2C∞∂t∗.
+(14.8)
+Since (ρ∂ρ − 1) annihilates the leading order term ρHs−3(Rtb1,∗ × S2) in (˙g, ˙A), it follows that Lb(χ(˙g, ˙A)) =
+[Lb, χ](˙g, ˙A) ∈ �Hs−4,1/2+α−
+b,c
+.
+Since s − 4 > 4 + m, the estimates (14.2)–(14.6) now follow from Theorem 13.1, with ℓ + 2 = 1
+2 + α−, so
+ℓ = − 3
+2 + α− and −1/2 < ℓ + ǫ < 1/2 with α + ǫ > 1, 0 < ǫ < 1.
+□
+We now state our main theorem.
+Theorem 14.2. Let 0 < α < 1 and s > 8 + m, m ∈ N. Let
+(˙h, ˙k, ˙E, ˙B) ∈ ¯Hs,−1/2+α
+b
+(Σ0; S2 scT ∗Σ0)⊕ ∈ ¯Hs−1,1/2+α
+b
+(Σ0; S2 scT ∗Σ0)
+¯Hs−1,1/2+α
+b
+(Σ0; scT ∗Σ0)⊕ ∈ ¯Hs,1/2+α
+b
+(Σ0; scT ∗Σ0)
+be the initial data for the linearized Einstein-Maxwell equations linearized around the the KN solution (gb, Ab).
+Suppose that the initial data satisfy the linearized constraint equations, which is the linearization of the
+constraint equations around the KN initial data (hb, kb, Eb, Bb) = τ(gb, dAb). Assume that the linearized
+magnetic charge vanishes, see the discussion in §3.2–3.3.
+Then there exists a solution (˙g, ˙A) of the initial value problem
+DgbRic(˙g) − 2D(gb,dAb)T (˙g, ˙A) = 0,
+Dgb,Ab(δ(·)d(·))(˙g, ˙A) = 0,
+D(gb,Ab)τ(˙g, ˙A) = (hb, kb, Eb, Bb)
+satisfying the linearized generalized wave map gauge and Lorenz gauge conditions
+Dgb �ΥE(˙g; gb) = 0,
+δgb ˙A = 0.
+Moreover, (˙g, ˙A) has the asymptotic behavior stated in Theorem 14.1.
+Proof. In view of Corollary 4.15, there exist sections
+(˙g0, ˙g1, ˙A0, ˙A1) ∈ ¯Hs,−1/2+α
+b
+(¯Σ0; S2 �
+scT ∗¯Σ0) ⊕ ¯Hs−1,1/2+α
+b
+(¯Σ0; S2 �
+scT ∗ ¯Σ0)
+⊕ ¯Hs−1,−1/2+α
+b
+(¯Σ0; �
+scT ∗¯Σ0) ⊕ ¯Hs−2,1/2+α
+b
+(¯Σ0; �
+scT ∗Σ0),
+such that they induce the data (˙h, ˙k, ˙E, ˙B) on Σ0 = {t = 0}, and satisfy the linearized gauge conditions
+Dgb �ΥE(˙g) = 0 and D(gb,Ab) �ΥM(˙g, ˙A) = −δgb ˙A = 0 at Σ0.
+Then we solve the initial value problem Lb(˙g, ˙A) = 0 with initial data (˙g0, ˙g1, ˙A0, ˙A1). Since (˙h, ˙k, ˙E, ˙B)
+satisfies the linearized constraint equations, it follows from Lemma 3.8 that
+Dgb �ΥE(˙g; gb) = 0,
+δgb ˙A = 0
+and thus (˙g, ˙A) solves the linearized Einstein-Maxwell equations.
+□
+
+LINEAR STABILITY OF KERR-NEWMAN FAMILY
+159
+Appendix A. The linearized Einstein-Maxwell operator around Reissner-Nordstr¨om
+spacetime
+A.1. Detailed calculation of the linearized Einstein-Maxwell system. We now calculate the lin-
+earized Einstein-Maxwell system
+L (˙g, ˙A) := (L1(˙g, ˙A), L2(˙g, ˙A)) := (DgRic(˙g, d ˙A) − 2D(g,dA)T (˙g, d ˙A), D(g,F ) (δgF) (˙g, ˙F)) = 0
+in more detail.
+First, following [84, §7.5], [33, §3] we see that
+DgRic(˙g) = −1
+2□g ˙g − δ∗
+gδgGg ˙g + Rg ˙g
+where
+(Rg ˙g)µν = Rα
+β
+µν ˙gαβ + 1
+2
+�
+Ric κ
+µ ˙gνκ + Ric κ
+ν ˙gµκ
+�
+Since T (g, F) = FµαF α
+ν
+− 1
+4gµνFαβF αβ, we find
+D(g,F )T (˙g, ˙F) = − ˙gαβFµαFνβ − 1
+4
+�
+˙gµνFαβF αβ − gµν ˙gκαgλβFαβFκλ − gµνgκα ˙gλβFαβFκλ
+�
++
+�
+gαβ ˙FµαFνβ + gαβFµα ˙Fνβ
+�
+− 1
+4
+�
+gµν ˙FαβF αβ + gµνFαβ ˙F αβ�
+= − ˙gαβFµαFνβ − 1
+4
+�
+˙gµνFαβF αβ−2gµν ˙gκαFαβF β
+κ
+�
++
+�
+gαβ ˙FµαFνβ + gαβFµα ˙Fνβ
+�
+− 1
+2gµν ˙FαβF αβ
+(A.1)
+and thus the Einstein part L1 of the linearized Einstein-Maxwell system is given by
+L1(˙g, d ˙A) = DgRic(˙g, d ˙A) − 2D(g,dA)T (˙g, d ˙A) = −1
+2□g ˙g − δ∗
+gδgGg ˙g + Rg ˙g − 2D(g,dA)T (˙g, d ˙A) = 0. (A.2)
+Next we analyze the Maxwell part L2 of the linearized Einstein-Maxwell system. Since
+(δgF)µ = −∇νFνµ = −gνα∇αFνµ = −gνα �
+∂αFνµ − Γκ
+ανFκµ − Γκ
+αµFνκ
+�
+,
+and
+˙Γκ
+αν(˙g) = 1
+2gκλ (∇ν ˙gαλ + ∇α ˙gνλ − ∇λ ˙gαν) ,
+we obtain
+D(g,F ) (δgF) (˙g, ˙F) = δg ˙F + ˙gνα∇αFνµ + gνα �
+˙Γκ
+αν(˙g)Fκµ + ˙Γκ
+αµ(˙g)Fνκ
+�
+= δg ˙F + ˙gνα∇αFνµ + gκλ∇α(˙gαλ − 1
+2gνα ˙gανgαλ)Fκµ + 1
+2(∇ν ˙gκ
+µ − ∇κ ˙gν
+µ)Fνκ
+= δg ˙F + ˙gνα∇αFνµ − (δgGg ˙g)κFκµ + 1
+2(∇ν ˙gκ
+µ − ∇κ ˙gν
+µ)Fνκ
+(A.3)
+where we use (∇µ ˙gνκ)Fνκ = 0 in the second equality. Therefore, the Maxwell part L2 of the linearized
+Einstein-Maxwell system is given by
+L2(˙g, d ˙A) = δgd ˙A + ˙gνα∇α(dA)νµ − (δgGg ˙g)κ(dA)κµ + 1
+2(∇ν ˙gκ
+µ − ∇κ ˙gν
+µ)(dA)νκ = 0.
+(A.4)
+We then proceed to calculate the linearized Einstein-Maxwell system L (˙g, ˙A) = 0 around a Reissner-
+Nordstr¨om spacetime g = ˚g + r2/g under the splitting (6.9) and (6.10) .
+A.2. Calculation of Rg ˙g. Here and in what follows the components we do not list are 0. Christoffel symbols
+of Reissner-Nordstr¨om metric g are,
+Γk
+ij = ˚Γk
+ij
+Γa
+ij = 0,
+Γj
+ia = 0,
+Γk
+ab = −r∂kr/gab,
+Γb
+ak = r−1∂krδb
+a,
+Γc
+ab = /Γ
+c
+ab.
+The components of Riemann curvature tensor are given by
+Rijkm = ˚
+Rijkm = 1
+2
+˚
+R(˚g) (˚gik˚gjm −˚gim˚gjk) = −1
+2µ′′
+b0 (˚gik˚gjm −˚gim˚gjk) ,
+Raijk = 0,
+Rabij = 0,
+Rabci = 0,
+Raibj = −r˚∇i˚∇jr/gab = −1
+2rµ′
+b0/gab˚gij,
+Rabcd = /Rabcd − r2µb0
+�
+/gac/gbd − /gad/gbc
+�
+= r2(1 − µb0)
+�
+/gac/gbd − /gad/gbc
+�
+.
+
+160
+LILI HE
+Correspondingly, the Ricci curvature tensor takes the form
+Ricij = ˚
+Ricij − r−1µ′
+b0˚gij = −1
+2µ′′
+b0˚gij − r−1µ′
+b0˚gij,
+Ricia = 0,
+Ricab = (1 − µb0 − rµ′
+b0)/gab.
+Now we are ready to calculate Rg ˙g in detail. Since (Rg ˙g)µν = Rα
+β
+µν ˙gαβ + 1
+2
+�
+Ric κ
+µ ˙gνκ + Ric κ
+ν ˙gµκ
+�
+, we have
+(Rg ˙g)ij = −µ′′
+b0
+�
+˙gij − 1
+2˚gkm ˙gkm˚gij
+�
+− r−1µ′
+b0 ˙gij + 1
+2r−3µ′
+b0/gab ˙gab˚gij,
+(Rg ˙g)ia = −3
+2r−1µ′
+b0 ˙gia − 1
+4µ′′
+b0 ˙gia + 1
+2r−2(1 − µb0)˙gia,
+(Rg ˙g)ab = 2r−2(1 − µb0)
+�
+˙gab − 1
+2/gcd ˙gcd/gab
+�
+− r−1µ′
+b0 ˙gab + 1
+2rµ′
+b0˚gij ˙gij/gab.
+Therefore, with respect to the splitting of S2T ∗M (6.10), we can express Rg : S2T ∗M → S2T ∗M as follows
+Rg =
+
+
+−µ′′
+b0G˚g − r−1µ′
+b0
+0
+1
+2r−3µ′
+b0˚g /tr
+0
+− 3
+2r−1µ′
+b0 − 1
+4µ′′
+b0 + 1
+2r−2(1 − µb0)
+0
+1
+2rµ′
+b0/g˚tr
+0
+2r−2(1 − µb0)G/g − r−1µ′
+b0
+
+
+(A.5)
+where (G˚gh)ij = hij − 1
+2˚gij˚trh and (G/gh)ab = hab − 1
+2/gab /trh.
+A.3. Calculation of δ∗
+gδgGg ˙g. We now calculate the first covariant derivatives on 1-forms and symmetric
+2-tensors. For η ∈ T ∗M, we compute
+∇iηj = ˚∇iηj,
+∇iηa = ∂iηa − (r−1∂ir)ηa,
+∇aηi = ∂aηi − (r−1∂ir)ηa,
+∇aηb = /∇aηb + r(∂kr)ηk/gab.
+With respect to the splitting of T ∗M (6.9), we write the symmetric gradient δ∗
+g : T ∗M → S2T ∗M with
+(δ∗
+gη)µν = 1
+2(∇µην + ∇νηµ) as follows
+δ∗
+g =
+
+
+˚δ∗
+0
+1
+2/d
+1
+2r2˚dr−2
+r/gιdr
+/δ
+∗
+
+ .
+(A.6)
+where ιdr : T ∗ ˚
+X → R is the contraction with the aspherical vector field dr♯, i.e. ιdr(·) = ˚g−1(dr, ·). We then
+calculate the first covariant derivatives on symmetric 2-tensor h ∈ S2T ∗M
+∇ihjk = ˚∇ihjk,
+∇ihja = ˚∇ihja − r−1(∂ir)hja,
+∇ihab = ∂ihab − 2r−1(∂ir)hab
+∇ahij = ∂ahij − r−1(∂ir)haj − r−1(∂jr)hia,
+∇ahib = /∇ahib + r(∂kr)hik/gab − r−1(∂ir)hab,
+∇ahbc = /∇ahbc + r(∂kr)(hkc/gab + hkb/gac).
+Therefore the negative divergence (δgh)µ = −∇νhνµ which maps symmetric 2-tensors to 1-form takes the
+following form (under the splitting (6.10) and (6.9))
+δg =
+�r−2˚δr2
+r−2/δ
+r−3(dr)/tr
+0
+r−2˚δr2
+r−2/δ
+�
+(A.7)
+where (/δh)i = − /∇
+ahai and (˚δh)a = −˚∇ihia. We can also express (Ggh)µν = hµν − 1
+2gµνtrgh as
+Gg =
+
+
+G˚g
+0
+− 1
+2r−2˚g /tr
+0
+1
+0
+− 1
+2r2/g˚tr
+0
+G/g
+
+ .
+(A.8)
+Putting (A.6), (A.7) and (A.8) together and using /δG/g = /δ + 1
+2/d/tr, ˚δG˚g = ˚δ + 1
+2 ˚d˚tr yield
+δ∗
+gδgGg =
+
+
+˚δ∗r−2˚δr2 + 1
+2˚δ∗˚d˚tr
+˚δ∗r−2/δ
+1
+2˚δ∗r−2˚
+d/tr
+1
+2r−2/d˚δr2 + 1
+4 ˚
+d/d˚tr + 1
+4r2˚dr−2/d˚tr
+1
+2r−2/d/δ + 1
+2r2˚dr−4˚δr2
+1
+4r−2˚
+d/d/tr + 1
+2r2˚dr−4/δ + 1
+4r2˚dr−4/d/tr
+r−1/gιdr˚δr2 + 1
+2r/gιdr˚d˚tr + 1
+2/δ
+∗/d˚tr
+r−1/gιdr/δ + r−2/δ
+∗˚δr2
+1
+2r−1/gιdr˚
+d/tr + r−2/δ
+∗/δ + 1
+2r−2/δ
+∗/d/tr
+
+ .
+(A.9)
+
+LINEAR STABILITY OF KERR-NEWMAN FAMILY
+161
+A.4. Calculation of □g. Next we compute the second covariant derivatives on symmetric 2-tensor h ∈
+S2T ∗M.
+Since our goal is to find the form of □g = gµν∇µ∇ν applied to h, it suffices to compute the
+following second covariant derivatives
+∇i∇jhkm = ˚∇i˚∇jhkm,
+∇a∇bhkm = /∇a /∇bhkm − r−1∂kr
+� /∇ahbm + /∇bham
+�
+− r−1∂mr
+� /∇ahkb + /∇bhka
+�
++ r∂ir˚∇ihkm/gab + 2r−2∂kr∂mrhab − /gab
+�
+∂kr∂irhim + ∂mr∂irhik
+�
+∇i∇jhkc = ˚∇i˚∇jhkc − r−1∂ir˚∇jhkc − r−1∂jr˚∇ihkc + 2r−2∂ir∂jrhkc − 1
+2r−1µ′
+b0˚gijhkc,
+∇a∇bhkc = /∇a /∇bhkc − r−1∂kr
+� /∇ahbc + /∇bhac
+�
++ r/gac∂ir /∇bhik + r/gbc∂ir /∇ahik + r/gab∂ir˚∇ihkc
+− ∂kr∂ir
+�
+/gabhic + /gachib + /gbchib
+�
+− µb0
+�
+/gabhkc + /gachkb
+�
+∇i∇jhcd = ˚∇i˚∇jhcd − 2r−1 �
+∂ir˚∇jhcd + ∂jr˚∇ihcd
+�
++ 6r−2∂ir∂jrhcd − r−1µ′
+b0˚gijhcd,
+∇a∇bhcd = /∇a /∇bhcd + r∂ir
+�
+/gbc /∇ahdi + /gbd /∇ahci + /gac /∇bhdi + /gad /∇bhci + /gab˚∇ihcd
+�
+− µb0
+�
+2/gabhcd + /gachbd + /gadhbc
+�
++ r2∂ir∂jrhij
+�
+/gac/gbd + /gad/gbc
+�
+(A.10)
+where ˚□ acting on aspherical symmetric 2-tensors is the tensor wave operator, on mixed symmetric tensors
+is the 1-form wave operator acting on the aspherical part, and on spherical symmetric tensors is the scalar
+wave operator acting on the aspherical variables; the operator /∆ is defined in a similar manner. Then we
+find
+1
+2□g =
+
+
+1
+2˚□ + 1
+2r−2 /∆
+2r−3(dr) ⊗s /δ
+r−4(dr ⊗ dr)/tr
+r−1ιdr/d
+1
+2˚□ + 1
+2r−2 /∆
+r−3(dr) ⊗ /δ
+/gιdrιdr
+2r−1ιdr/δ
+∗
+1
+2˚□ + 1
+2r−2 /∆
+
+
++
+
+
+r−1∂ir˚∇i−2r−2(dr) ⊗s ιdr
+0
+0
+0
+− 1
+2r−2(µb0 + rµ′
+b0)−2r−2(dr) ⊗ ιdr
+0
+0
+0
+−r−1∂ir˚∇i−r−1µ′
+b0
+
+ .
+(A.11)
+On the aspherical part ˚
+X, we further calculate the relevant operators on the static region where µb0 > 0
+by working in (t, r) coordinates. On the static region, the metric ˚g reads
+˚g = −µb0dt2 + µ−1
+b0 dr2.
+We further split
+T ∗ ˚
+X = ⟨˚dt⟩ ⊕ ⟨˚dr⟩,
+S2T ∗ ˚
+X = ⟨˚dt2⟩ ⊕ ⟨2˚dt˚dr⟩ ⊕ ⟨˚dr2⟩.
+(A.12)
+Since ˚Γt
+tr = ˚Γt
+rt = 1
+2µ−1
+b0 µ′
+b0,˚Γr
+tt = 1
+2µb0µ′
+b0,˚Γr
+rr = − 1
+2µ−1
+b0 µ′
+b0, with respect to the above split (A.12), we see
+that acting on scalar functions, ∂ir˚∇i = µb0∂r,
+˚d =
+�∂t
+∂r
+�
+,
+˚□ = −µ−1
+b0 ∂2
+t + µb0∂2
+r + µ′
+b0∂r = −µ−1
+b0 ∂2
+t + ∂rµb0∂r
+(A.13)
+On 1-forms, we have ιdr = (0, µb0),
+˚δ = (µ−1
+b0 ∂t, −µb0∂r − µ′
+b0) = (µ−1
+b0 ∂t, −∂rµb0),
+˚δ∗ =
+
+
+∂t
+− 1
+2µb0µ′
+b0
+1
+2µb0∂rµ−1
+b0
+1
+2∂t
+0
+µ−1/2
+b0
+∂rµ1/2
+b0
+
+ ,
+∂ir˚∇i =
+�µb0∂r − 1
+2µ′
+b0
+0
+0
+µb0∂r + 1
+2µ′
+b0
+�
+=
+�
+µ3/2
+b0 ∂rµ−1/2
+b0
+0
+0
+µ1/2
+b0 ∂rµ1/2
+b0
+�
+,
+˚∗ =
+�
+0
+−µb0
+−µ−1
+b0
+0
+�
+,
+2dr ⊗s (·) =
+
+
+0
+0
+1
+0
+0
+2
+
+ .
+(A.14)
+
+162
+LILI HE
+and
+˚□ =
+�−µ−1
+b0 ∂2
+t + µb0∂2
+r − 1
+2µ′′
+b0
+µ′
+b0∂t
+µ−2
+b0 µ′
+b0∂t
+−µ−1
+b0 ∂2
+t + µb0∂2
+r + 2µ′
+b0∂r + 1
+2µ′′
+b0
+�
+.
+(A.15)
+On symmetric 2-tensors, we find that
+˚δ =
+�
+µ−1
+b0 ∂t
+−∂rµb0
+0
+− 1
+2µ′
+b0µ−2
+b0
+µ−1
+b0 ∂t
+−µ−1/2
+b0
+∂rµ3/2
+b0
+�
+,
+ιdr =
+�0
+µb0
+0
+0
+0
+µb0
+�
+,
+∂ir˚∇i =
+
+
+µb0∂r − µ′
+b0
+0
+0
+0
+µb0∂r
+0
+0
+0
+µb0∂r + µ′
+b0
+
+ =
+
+
+µ2
+b0∂rµ−1
+b0
+0
+0
+0
+µb0∂r
+0
+0
+0
+∂rµb0
+
+ ,
+˚□ = −µ−1
+b0 ∂2
+t + µb0∂2
+r +
+
+
+−µ′
+b0∂r + 1
+2µ−1
+b0 µ′2
+b0 − µ′′
+b0
+2µ′
+b0∂t
+− 1
+2µb0µ′2
+b0
+µ−2
+b0 µ′
+b0∂t
+µ′
+b0∂r − µ−1
+b0 µ′2
+b0
+µ′
+b0∂t
+− 1
+2µ−3
+b0 µ′2
+b0
+2µ−2
+b0 µ′
+b0∂t
+3µ′
+b0∂r + 1
+2µ−1
+b0 µ′2
+b0 + µ′′
+b0
+
+ .
+(A.16)
+A.5. Calculation of D(g,dA)T (˙g, d ˙A). Recall that the Reissner-Nordstr¨om electromagnetic 4-potential is
+given by
+A = Qr−1dt∗,
+F = Qr−2dt∗ ∧ dr = Qr−2 �
+vol
+(A.17)
+Since ˚
+volij = ǫij, ˚
+volia = ˚
+volab = 0 where ǫ is the Levi-Civita symbol, i.e., ǫ12 = −ǫ21 = 1, it is clear that
+˚
+vol
+αβ = − ˚
+volαβ. A direct computation implies
+F αβFαβ = −2Q2
+r4 ,
+˙gαβFµαFνβ =
+�
+Q2
+r4
+�
+˙gµν −˚gµν˚tr ˙g
+�
+if µ, ν ∈ {i, j}
+0
+otherwise
+.
+Then according to (A.1), we find
+�
+D(g,dA)T (˙g, 0)
+�
+ij = −Q2
+r4
+�
+˙gij −˚gij˚tr˙g − 1
+2 ˙gij + 1
+2˚gij˚tr ˙g
+�
+= − Q2
+2r4
+�
+˙gij −˚gij˚tr ˙g
+�
+,
+�
+D(g,dA)T (˙g, 0)
+�
+ia = Q2
+2r4 ˙gia,
+�
+D(g,dA)T (˙g, 0)
+�
+ab = Q2
+2r4
+�
+˙gab − r2/gab˚tr ˙g
+�
+(A.18)
+which means
+D(g,dA)T (·, 0) = Q2
+2r4
+
+
+˚g˚tr−1
+0
+0
+0
+1
+0
+−r2/g˚tr
+0
+1
+
+ .
+(A.19)
+Since
+F αβ ˙Fαβ = Qr−2 ˚
+vol
+ij ˙Fij,
+gαβ ˙FiαFjβ + gαβFiα ˙Fjβ = −2Qr−2 ˙Ft∗r˚gij = 2Qr−2 ˙Ft∗r(˚∗ ˚
+vol)˚gij,
+gαβ ˙FaαFiβ = ˚gkj ˙FakFij = −Qr−2˚gkjǫij ˙Fka = Qr−2˚∗ ˙F•a
+where we use the fact ˚∗ ˚
+vol = −1 and the Hodge star operator ˚∗ only acts on the aspherical part, we have
+D(g,dA)T (0, d(·)) = −Qr−2
+
+
+−˚g˚∗˚d
+0
+˚∗/d
+−˚∗˚d
+r2/g˚∗˚d
+0
+
+ .
+(A.20)
+This finishes the calculation of L1(˙g, d ˙A).
+A.6. Calculation of D(g,dA)(δgdA)(˙g, d ˙A). It remains to consider L2(˙g, d ˙A) which is the linearization of
+δgdA. We again use the following splitting
+T ∗M = T ∗
+AS ⊕ T ∗
+S,
+∧2T ∗M = ∧2T ∗
+AS ⊕ (T ∗
+AS ∧ T ∗
+S) ⊕ ∧2T ∗
+S.
+(A.21)
+With respect to the above splitting (A.21), d : T ∗M → ∧2T ∗M and δg : ∧2T ∗M → T ∗M take the form
+d =
+
+
+˚d
+0
+−/d
+˚d
+0
+/d
+
+ ,
+δg =
+�
+r−2˚δr2
+−r−2/δ
+0
+0
+˚δ
+r−2/δ
+�
+,
+
+LINEAR STABILITY OF KERR-NEWMAN FAMILY
+163
+and thus by (A.4)
+D(g,dA)(δgdA)(0, d(·)) =
+�r−2˚δr2˚
+d + r−2/δ/d
+−r−2/δ˚d
+−˚δ/d
+˚δ˚d + r−2/δ/d
+�
+.
+(A.22)
+Finally we are left with dealing with D(g,dA)(δgdA)(˙g, 0). We first calculate the first covariant derivative on
+the 2-form F. A direct calculation implies ˚∇k ˚
+volij = 0, and then
+∇kFij = −2r−1(∂kr)Fij,
+∇iFja = 0,
+∇iFab = 0,
+∇aFij = 0,
+∇aFib = r(∂kr)/gabFik,
+∇aFbc = 0.
+Therefore we have
+˙gνα∇αFνi = −2r−1(∂jr)Fki ˙gjk + r−3/gab ˙gab(∂kr)Fki = −2Qr−3˚∗
+�
+(∂jr)˙gjk
+�
++ Qr−5(˚∗dr)i/gab ˙gab
+˙gνα∇αFνa = ˙gbi∇bFia = Q
+r3 (∂kr)(˚∗ ˙g•a)k
+We also find (δgGg ˙g)κFκa = 0 and −(δgGg ˙g)κFκi = −Qr−2˚∗ (δgGg ˙g), thus
+−(δgGg(·))κFκi = − Q
+r2
+�
+r−2˚∗˚δr2 + 1
+2˚∗˚d˚tr
+r−2˚∗/δ
+1
+2r−2˚∗˚d/tr
+0
+0
+0
+�
+.
+Since ˚gijǫikǫkl = ˚gkl −˚gkl˚tr˚g = −˚gkl and ˚∇k ˚
+volij = 0, we find
+1
+2 (∇ν ˙gκ
+i − ∇κ ˙gν
+i) Fνκ = − Q
+2r2 (˚∇k ˙glj − ˚∇l ˙gkj)˚gmnǫjmǫinǫkl
+= − Q
+2r2
+�
+˚∇k(˙gljǫjmǫkl) − ˚∇l(˙gkjǫjmǫkl)
+�
+˚gmnǫin
+= Q
+2r2
+�
+˚∇k(˙gkm −˚gkm˚tr ˙g) + ˚∇l(˙glm −˚glm˚tr ˙g)
+�
+˚gmnǫin
+= − Q
+r2˚∗
+�
+˚∇k ˙gkm − ˚
+d˚tr ˙g
+�
+.
+and using ˚∗˚∗ = (−1)k(2−k)+1I and δ = ˚∗˚d˚∗ on ∧kT ∗ ˚
+X for 1 ≤ k ≤ 2, we see that
+1
+2 (∇ν ˙gκ
+a − ∇κ ˙gν
+a) Fνκ = Q
+2r2˚gik˚gjl �
+∂k ˙gla − ∂l ˙gka − r−1(∂kr)˙gla + r−1(∂lr)˙gka
+�
+ǫij
+= − Q
+r3 (∂kr)(˚∗ ˙g•a)k + Q
+r2
+�
+2r−1(∂kr)(˚∗ ˙g•a)k +˚∗˚d˙g•a
+�
+= − Q
+r3 (∂kr)(˚∗ ˙g•a)k + Q
+r2
+�
+−r2(∂kr−2)(˚∗ ˙g•a)k + ˚δ˚∗˙g•a
+�
+= − Q
+r3 (∂kr)(˚∗ ˙g•a)k + Q˚δr−2˚∗ ˙g•a.
+Combining the above calculation and (A.4) yields
+D(g,dA)(δgdA)(˙g, 0) = Q
+r2
+� 1
+2˚∗˚d˚tr
+−r−2˚∗/δ
+− 1
+2˚∗˚dr−2 /tr
+0
+r2˚δr−2˚∗
+0
+�
+.
+(A.23)
+Lastly, we discuss the calculation of the gauge terms for the electromagnetic field. Since A = −Qr−1dt∗,
+we have
+(Lω♯A)i = ωα∂αAi + Aα∂iωα,
+(Lω♯A)a = ωα∂αAa + Aα∂aωα = ∂a(Aαωα)
+and as a consequence
+L(·)♯A =
+�˚
+dιA + ι(·)˚dA
+0
+/dιA
+0
+�
+.
+(A.24)
+
+164
+LILI HE
+Appendix B. Calculation of the subprincipal operator at trapping
+In this section, we include the discussion of the subprincipal operator of Pb0,γ, Wb0,γ and Lb0,γ at the
+trapped sets K for the sake of completeness.
+The arguments are based on [38], [39] and [40].
+In the
+subsequent discussion, we will list (without proof) the relevant results for the invariant formalism of the
+subprincipal operator with references (mostly from [38]), and then carry out all the relevant computations.
+According to [20, Theorem 1.1] and [40, Theorem 4.5] (the microlocalized version) (see also the discussion
+in [38, §2]), a sufficient condition for high energy estimates at the trapped set for a semiclassical operator
+Ph(z) acting on the covariant rank k tensor bundle Tk to hold is
+σ1,h
+� 1
+2ih(Ph(z) − Ph(z)∗)
+�
+< νmin/2
+(B.1)
+at the trapped set K, where νmin > 0 is the minimal normal expansion rate of the Hamilton flow at the
+trapping, see Proposition 5.18. Here, the adjoint is taken with respect to a positive definite inner product
+on Tk. We note that the natural inner product induced by g, with respect to which □g is symmetric, is not
+positive definite, except when k = 0, i.e. for the scalar wave equation. One can introduce a pseudodifferential
+inner product such that (B.1) can be arranged, with the adjoint taken with respect to this pseudodifferential
+inner product. We work with classical, i.e. one-step polyhomogeneous, symbols and operators, and denote
+by Sm
+hom(T ∗X \ 0) symbols which are homogeneous of degree m with respect to dilations in the fibers of
+T ∗X \ 0.
+Definition B.1 ( [38, Definition 3.1]). A pseudodifferential inner product (or Ψ-inner product) on the vector
+bundle E → X is a pseudodifferential operator B ∈ Ψ0(X; E ⊗ Ω
+1
+2 , E
+∗ ⊗ Ω
+1
+2 ) satisfying B = B∗, and such
+that moreover the principal symbol σ0(B) = b ∈ S0
+hom(T ∗X \ 0; π∗Hom(E, E
+∗)) of B satisfies
+⟨b(x, ξ)u, ι(u)⟩ > 0
+(B.2)
+for all non-zero u ∈ Ex, where x ∈ X, ξ ∈ T ∗
+xX \ 0. If the context is clear, we will also call the sesquilinear
+pairing
+C∞(X, E ⊗ Ω
+1
+2 ) × C∞(X, E ⊗ Ω
+1
+2 ) ∋ (u, v) �→
+�
+X
+⟨B(x, D)u, ι(v)⟩
+the pseudodifferential inner product associated with B.
+The use of a pseudodifferential inner product is equivalent to considering a conjugated operator QPh(z)Q−,
+where Q ∈ Ψ0
+h(X, Tk) is a carefully chosen elliptic operator with parametrix Q−. For any ǫ > 0, we can
+arrange σ1,h( 1
+2ih(QPh(z)Q− − (QPh(z)Q−)∗)) < ǫ (with the adjoint taken relative to an ordinary positive
+definite inner product on Tk), thus (B.1) holds for Ph(z) replaced by QPh(z)Q−.
+Let π : T ∗X \ 0 → X be the projection. We will work in standard pseudodifferential operator setting.
+We specialize to the case that P ∈ Ψm(X, E ⊗ Ω
+1
+2 ) has a real, scalar principal symbol, which is the case of
+interest in our application. Fix a coordinate system of X and a local trivialization of E, then the full symbol
+of P is a sum of homogeneous symbols p ∼ pm + pm−1 + . . ., with pj homogeneous of degree j and valued
+in complex N × N matrices. According to [46, Theorem 18.1.33], the subprincipal symbol
+σsub(P) = pm−1(x, ξ) − 1
+2i
+�
+j
+∂xjξjpm(x, ξ) ∈ Sm−1
+hom (T ∗X \ 0, CN×N)
+(B.3)
+is well-defined under changes of coordinates; however, it does depend on the choice of local trivialization
+of E. We would like a frame-independent notion of the subprincipal symbol since this provides us with the
+freedom to choose particularly convenient local frames in concrete computations.
+B.1. Invariant formalism for the subprincipal symbols of operators acting on bundles.
+Definition B.2 ( [38, Definition 3.8]). For P ∈ Ψm(X, E ⊗ Ω
+1
+2 ) with scalar principal symbol p, there is a
+well-defined subprincipal operator Ssub(P) ∈ Diff1(T ∗X \ 0, π∗E), homogeneous of degree m − 1 with respect
+to dilations in the fibers of T ∗X \ 0, defined as follows: if {e1(x), . . . , eN(x)} is a local frame of E, define the
+operators Pjk ∈ Ψm(X, Ω
+1
+2 ) by P(�
+k uk(x)ek(x)) = �
+jk Pjk(uk)ej(x), uk ∈ C∞(X, Ω
+1
+2 ). Then
+Ssub(P)
+��
+k
+qk(x, ξ)ek(x)
+�
+:=
+�
+jk
+(σsub(Pjk)qk)ej − i
+�
+k
+(Hpqk)ek.
+
+LINEAR STABILITY OF KERR-NEWMAN FAMILY
+165
+The symbols of commutators and imaginary parts can be expressed in a completely invariant fashion.
+Proposition B.3 ( [39, Proposition 3.11]). Let P ∈ Ψm(X, E ⊗ Ω
+1
+2 ) be a pseudodifferential operator with
+scalar principal symbol p.
+(1) Suppose Q ∈ Ψm′(X, E ⊗ Ω
+1
+2 ) is an operator acting on E-valued half-densities, with principal symbol
+q. Then
+σm+m′−1([P, Q]) = [Ssub(P), q].
+(B.4)
+If Q is elliptic with parametrix Q−, then
+Ssub(QPQ−) = qSsub(P)q−1.
+(B.5)
+(2) Suppose in addition that p is real. Let B be a Ψ-inner product (pseudodifferential inner product) on
+E with principal symbol b, then
+σm−1(Im BP) = Im bSsub(P),
+(B.6)
+where Im BP =
+1
+2i(P − P ∗B) and Im bSsub(P) =
+1
+2i
+�
+Ssub(P) − Ssub(P)∗b�
+; we take the adjoint of
+P with respect to the Ψ-inner product B and the adjoint of the differential operator Ssub(P) with
+respect to the inner product b on π∗E and the symplectic volume density on T ∗X.
+The imaginary part Im BP of an operator P with respect to a Ψ-inner product B can be interpreted in
+terms of the imaginary part of a conjugated version of P with respect to a standard inner product.
+Proposition B.4 ( [38, Proposition 3.12]). Let B be a Ψ-inner product on E.
+Then for any positive
+definite Hermitian inner product B0 ∈ C∞(X, Hom(E ⊗ Ω
+1
+2 , E
+∗ ⊗ Ω
+1
+2 )) on E, there exists an elliptic operator
+Q ∈ Ψ0(X, End(E ⊗ Ω
+1
+2 )) such that B − Q∗B0Q ∈ Ψ−∞(X, Hom(E ⊗ Ω
+1
+2 , E
+∗ ⊗ Ω
+1
+2 )).
+In particular, denoting by Q− ∈ Ψ0(X, End(E ⊗ Ω
+1
+2 )) a parametrix of Q, we have for any P ∈ Ψm(X, E ⊗
+Ω
+1
+2 ) with real and scalar principal symbol.
+Q(Im BP)Q− = Im B0(QPQ−),
+(B.7)
+and σm−1(Im BP) and σm−1(Im B0(QPQ−)) (which are self-adjoint with respect to σ0(B) and B0, respec-
+tively, hence diagonalizable) have the same eigenvalues.
+Therefore, the analysis of σm−1(Im B0(QPQ−))) is reduced to finding an inner product b such that
+(Ssub(L) − Ssub(L)∗b) has a desired form. (In our applications, the choice of b will be clear from an in-
+spection of the form of Ssub(L)).
+Let (M, g) be a smooth manifold equipped with a metric tensor g of arbitrary signature. Denote by
+TkM = �k T ∗M, k ≥ 1, the bundle of (covariant) tensors of rank k on M. The metric g induces a metric
+(which we also call g) on TkM. The subprincipal operator of ∆k = tr∇2 ∈ Diff2(M, TkM) acting on the
+bundle TkM has a nice form.
+Proposition B.5 ( [38, Proposition 4.1]). The subprincipal operator of ∆k is
+Ssub(∆k)(x, ξ) = i∇π∗TkM
+HG
+∈ Diff1(T ∗M \ 0, π∗TkM),
+(B.8)
+where ∇π∗TkM is the pullback connection, with π: T ∗M \ 0 → M being the projection, and G = |ξ|2
+g−1(x).
+B.2. Calculation of subprincipal operators. We notice that modulo terms of order 0, which are sub-
+subprincipal and thus irrelevant in the calculation of Ssub(Pb0,γ), Ssub(Wb0,γ) and Ssub(Lb0,γ) (for simplicity
+of notation, from now on, we drop the notations b0, γ, i.e., g = gb0), we have
+− 2P ≡ □g,1 − 2δgGg(˜δ∗
+g − δ∗
+g),
+−2W ≡ □g,1 − 2(˜δg − δg)Ggδ∗
+g
+(B.9)
+and
+− L(˙g, ˙A) ≡
+�
+□g ˙g − 2(˜δ∗
+g − δ∗
+g)δgGg ˙g − 2δ∗
+g(˜δg − δg)Gg ˙g + L12( ˙A), □g ˙A + L21(˙g)
+�
+,
+(B.10)
+where
+L12( ˙A) = 8tr24
+g (F ⊗s d ˙A) − 2g−1(F, d ˙A) g,
+F = dA = d(Q0
+r dt) = Q0
+r2 dt ∧ dr,
+L21(˙g) = tr12
+g (δgGg ˙g ⊗ F) − 1
+2tr24
+g tr35
+g (d∇ ˙g ⊗ F),
+(d∇ ˙g) µκ
+ν
+= ∇µ ˙g κ
+ν − ∇κ
+ν ˙g µ.
+
+166
+LILI HE
+Since ˜δ∗
+g − δ∗
+g and ˜δg − δg are compactly supported away from the trapping for Reissner-Nordstr¨om metric,
+we further have that modulo terms of order 0
+−2P ≡ −2W ≡ □g,1
+−L(˙g, ˙A) ≡
+�
+□g ˙g + L12( ˙A), □g ˙A + L21(˙g)
+�
+(B.11)
+at the trapping K.
+Let g = gb0 be the Reissner-N¨ordstrom metric with parameter b0 = (m0, 0, Q0)
+g = −α2dt2 + α−2dr2 + r2/g,
+α = √µb0 =
+�
+1 − 2m0
+r
++ Q2
+0
+r2 .
+(B.12)
+That is, we work in the coordinates (t, r, ω), ω ∈ S2 at the trapping K. Let (σ, ξ, η) be the dual variables to
+(t, r, ω). We define
+e0 := αdt,
+e1 := α−1dr
+(B.13)
+and use the following splittings of T ∗M and S2 T ∗M
+T ∗M = ⟨e0⟩ ⊕ ⟨e1⟩ ⊕ T ∗S2,
+S2 T ∗M = ⟨e0e0⟩ ⊕ ⟨2e0 ⊗s e1⟩ ⊕ (2e0 ⊗s T ∗S2) ⊕ ⟨e1e1⟩ ⊕ (2e1 ⊗s T ∗S2) ⊕ S2 T ∗S2.
+(B.14)
+In this section, we are interested in the subprincipal operators of □g,1 and L at the trapped set
+K = {r = rp, ξ = 0, σ2 = µb0r−2|η|2},
+where
+∂r(rα−1)(rp) = 0,
+|η|2 = |η|2
+/g−1.
+(B.15)
+Therefore, at the trapping K, we have
+Hα2ξ2+r−2|η|2 = 2r−3|η|2∂ξ + r−2H|η|2
+and
+σ2Hα−2 − Hα2ξ2+r−2|η|2 = −r−2H|η|2.
+Let x0 = t, and x′ = (r, ω) be coordinates on X (independent of t). We let Greek indices µ, ν, λ, . . . run
+from 0 to 3. Moreover, the canonical dual variables ξ0 =: σ and ξ′ = (ξ, η) on the fibers of T ∗M are indexed
+by decorated Greek indices �µ running from 0 to 3. For a section u of π∗T ∗M, we have
+∇π∗T ∗M
+µ
+uν = ∇M
+µ uν,
+∇π∗T ∗M
+�µ
+uν = ∂�µuν.
+B.2.1. Calculation of the subprincipal operator of □g,1. According to Proposition B.5 and using the fact
+that G = −α−2σ2 + α2ξ2 + r−2|η|2, a direct calculation implies that at trapping K = {r = rp, ξ = 0, σ2 =
+µb0r−2|η|2}
+iSsub(□g,1) =
+
+
+
+2α−2σ∂t − r−2H|η|2
+0
+0
+0
+2α−2σ∂t − r−2H|η|2
+0
+0
+0
+2α−2σ∂t − r−2∇
+π∗
+S2T ∗S2
+H|η|2
+
+
+ + S(1)
+(B.16)
+where
+S(1) =
+
+
+0
+−2r−1σ
+0
+−2r−1σ
+0
+2αr−3ιη
+0
+−2αr−1η
+0
+
+ .
+(B.17)
+For simplicity of calculation, we further decompose π∗T ∗M → T ∗M over the trapping K. For η ∈ T ∗S2\o,
+we define η⊥ := /∗η, further
+�η := ασ−1η,
+η⊥ = ασ−1η⊥,
+and write
+π∗
+S2T ∗S2 = ⟨�η⟩ ⊕ ⟨�η⊥⟩,
+π∗
+S2S2T ∗S2 = ⟨�η�η⟩ ⊕ ⟨2�η�η⊥⟩ ⊕ ⟨�η⊥�η⊥⟩,
+(B.18)
+
+LINEAR STABILITY OF KERR-NEWMAN FAMILY
+167
+inducing the decompositions
+π∗T ∗M = ⟨e0⟩ ⊕
+�
+⟨e1⟩ ⊕
+�
+⟨�η⟩ ⊕ ⟨�η⊥⟩
+��
+,
+π∗S2T ∗M =
+�
+⟨e0e0⟩ ⊕
+�
+⟨2e0e1⟩ ⊕
+�
+⟨2e0�η⟩ ⊕ ⟨2e0�η⊥⟩
+���
+⊕
+�
+⟨e1e1⟩ ⊕
+�
+⟨2e1�η⟩ ⊕ ⟨2e1�η⊥⟩
+��
+⊕
+�
+⟨�η�η⟩ ⊕ ⟨2�η�η⊥⟩ ⊕ ⟨�η⊥�η⊥⟩
+�
+,
+(B.19)
+into trivial rank 1 bundles over T ∗M near K. Then, acting on 1-forms, we have
+η =
+�α−1σ
+0
+�
+,
+r−2ιη =
+�
+α−1σ
+0
+�
+at
+K.
+Therefore, in terms of the splitting (B.19), the operator S(1) in (B.17) at trapping K is given as
+S(1) = r−1σ
+
+
+
+
+0
+−2
+0
+0
+−2
+0
+2
+0
+0
+−2
+0
+0
+0
+0
+0
+0
+
+
+
+ .
+(B.20)
+Now, for any fixed ǫ > 0, we define the change of basis matrix
+q1 =
+
+
+
+
+0
+0
+0
+1
+0
+0
+1
+ǫ
+0
+0
+− 2
+ǫ2
+0
+0
+− 4
+ǫ3
+0
+− 4
+ǫ3
+0
+
+
+
+ ,
+and then we have
+q1S(1)q−1
+1
+= r−1σ
+
+
+
+
+0
+0
+0
+0
+0
+0
+ǫ
+0
+0
+0
+0
+ǫ
+0
+0
+0
+0
+
+
+
+ .
+Since q1 is a constant coefficient operator in the region where the splitting (B.19) is well defined, in particular,
+in a neighborhood of the trapping K, it commutes with the diagonal part iSsub(□g,1) − S of Ssub(□g,1).
+Equipping the 1-form bundle over M with the Hermitian inner product B1
+0 which is given by an identity
+matrix in the splitting (B.19), q1Ssub(□g,1)q−1
+1
+has imaginary part (with respect to B0) of size O(ǫ). Put
+differently, Ssub(□g,1) has imaginary part of size (ǫ) relative to the Hermitian inner product b := B1
+0(q·, q·),
+which is the symbol of a pseudodifferential inner product on π∗T ∗M. To summarize,
+Theorem B.6. For any ǫ > 0, there exists a (positive definite) t-independent pseudodifferential inner
+product B = b(x, D) on T ∗M (thus, b is an inner product on π∗T ∗M, homogeneous of degree 0 with respect
+to dilations in the base T ∗M \ 0), such that
+sup
+K
+|σ|−1
+����
+1
+2i(Ssub(□g,1) − Ssub(□g,1)∗b)
+����
+b
+≤ ǫ,
+where K is the trapped set. Put differently, there is an elliptic pseudodifferential operator Q, invariant under
+t-translations, acting on sections of T ∗M, with parametrix Q−, such that relative to the ordinary positive
+definite inner product B1
+0 we have
+sup
+Γ
+|σ|−1
+����σ1
+� 1
+2i(Q□g,1Q− − (Q□g,1Q−)∗B1
+0)
+�����
+B1
+0
+≤ ǫ.
+B.2.2. Calculation of the subprincipal operator of L. Using Proposition B.5 again, we see that Ssub(□g,2) =
+i∇π∗S2T ∗M
+HG
+. Since π∗S2T ∗M is simply the restriction of the product connection ∇π∗T ∗M ⊗ ∇π∗T ∗M to
+π∗S2T ∗M, it follows that
+Ssub(□g,2)(w1w2) = Ssub(□g,1)w1 · w2 + w1 · Ssub(□g,1)w2
+for w1, w2 ∈ C∞(T ∗M, π∗T ∗M), under the splitting (B.14), Ssub(□g,2) has a canonical first order part,
+induced by the canonical first order part of Ssub(□g,1) which is given in (B.16). The 0-order part S(2) of
+
+168
+LILI HE
+Ssub(□g,2) is equal to the second symmetric tensor power of the 0-th order part of Ssub(□(1)
+g ) in (B.17). To
+summarize,
+iSsub(□g,2) = 2α−2σ∂t − r−2
+
+
+
+
+
+
+
+
+
+
+
+H|η|2
+0
+0
+0
+0
+0
+0
+H|η|2
+0
+0
+0
+0
+0
+0
+H
+π∗
+S2T ∗S2
+|η|2
+0
+0
+0
+0
+0
+0
+H|η|2
+0
+0
+0
+0
+0
+0
+H
+π∗
+S2T ∗S2
+|η|2
+0
+0
+0
+0
+0
+0
+H
+π∗
+S2S2 T ∗S2
+|η|2
+
+
+
+
+
+
+
+
+
+
+
++ S(2) (B.21)
+where
+S(2) =
+
+
+
+
+
+
+
+
+0
+−4r−1σ
+0
+0
+0
+0
+−2r−1σ
+0
+2αr−3iη
+−2r−1σ
+0
+0
+0
+−2αr−1η
+0
+0
+−2r−1σ
+0
+0
+−4r−1σ
+0
+0
+4αr−3iη
+0
+0
+0
+−2r−1σ
+−2αr−1η
+0
+2αr−3iη
+0
+0
+0
+0
+−4αr−1η
+0
+
+
+
+
+
+
+
+
+.
+(B.22)
+Again, we turn to the ‘microlocal splitting’ (B.18), under which we write
+η =
+
+
+α−1σ
+0
+0
+1
+2α−1σ
+0
+0
+
+ : T ∗S2 → S2T ∗S2,
+r−2iη =
+�α−1σ
+0
+0
+0
+α−1σ
+0
+�
+: S2T ∗S2 → T ∗S2
+Therefore, in terms of (B.19), we see that
+S(2) = r−1σ
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+0
+−4
+0
+0
+0
+0
+0
+0
+0
+0
+−2
+0
+2
+0
+−2
+0
+0
+0
+0
+0
+0
+−2
+0
+0
+0
+−2
+0
+0
+0
+0
+0
+0
+0
+0
+0
+0
+−2
+0
+0
+0
+0
+−4
+0
+0
+0
+4
+0
+0
+0
+0
+0
+0
+−2
+0
+−2
+0
+0
+2
+0
+0
+0
+0
+0
+−2
+0
+0
+0
+0
+2
+0
+0
+0
+0
+0
+0
+−4
+0
+0
+0
+0
+0
+0
+0
+0
+0
+0
+−2
+0
+0
+0
+0
+0
+0
+0
+0
+0
+0
+0
+0
+0
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+.
+(B.23)
+Let SL be the 0-th order part of iSsub(−L) at K, with L given in (B.10). Acting on the bundle S2T ∗M⊕
+T ∗M, we can write SL as a 2 × 2 block matrix,
+SL =
+�
+S11
+L
+S12
+L
+S21
+L
+S22
+L
+�
+where
+S11
+L = S(2), S22
+L = S(1), S12
+L = iσ1(L12), S21
+L = iσ1(L21)
+with
+L12( ˙A) = 8tr24
+g (F ⊗s d ˙A) − 2g−1(F, d ˙A) g,
+F = dA = d(Q0
+r dt) = Q0
+r2 dt ∧ dr,
+L21(˙g) = tr12
+g (δgGg ˙g ⊗ F) − 1
+2tr24
+g tr35
+g (d∇ ˙g ⊗ F),
+(d∇ ˙g) µκ
+ν
+= ∇µ ˙g κ
+ν − ∇κ
+ν ˙g µ.
+In the splitting (B.14) and
+Λ2T ∗M = ⟨e0 ∧ e1⟩ ⊕ ⟨e0 ∧ T ∗S2⟩ ⊕ ⟨e1 ∧ T ∗S2⟩ ⊕ Λ2T ∗S2,
+
+LINEAR STABILITY OF KERR-NEWMAN FAMILY
+169
+d : T ∗M → Λ2T ∗M and tr24
+g (F ⊗s (·)) : Λ2T ∗M → S2T ∗M are given by
+d =
+
+
+
+
+−α∂r − α′
+α−1∂t
+0
+−/d
+0
+α−1∂t
+0
+−/d
+α∂r
+0
+0
+/d
+
+
+
+ ,
+2tr24
+g (F ⊗s (·)) = Q0
+r2
+
+
+
+
+
+
+
+
+2
+0
+0
+0
+0
+0
+0
+0
+0
+0
+−1
+0
+−2
+0
+0
+0
+0
+−1
+0
+0
+0
+0
+0
+0
+
+
+
+
+
+
+
+
+,
+and we also have
+g−1(F, (·))g =
+
+
+
+
+
+
+
+
+2Q0
+r2
+0
+0
+0
+0
+0
+0
+0
+0
+0
+0
+0
+− 2Q0
+r2
+0
+0
+0
+0
+0
+0
+0
+− 2Q0
+r2 r2/g
+0
+0
+0
+
+
+
+
+
+
+
+
+: Λ2T ∗M → S2T ∗M.
+Therefore, we obtain
+σ1(L12) = i4Q0
+r2
+
+
+
+
+
+
+
+
+−αξ
+α−1σ
+0
+0
+0
+0
+0
+η
+−αξ
+αξ
+−α−1σ
+0
+η
+0
+−α−1σ
+−αξr2/g
+α−1σr2/g
+0
+
+
+
+
+
+
+
+
+.
+(B.24)
+Turning to the ‘microlocal splitting’ (B.18), we write /g = /g = |η|−2ηη + |η|−2η⊥η⊥ and then we have
+/g =
+
+
+r−2
+0
+r−2
+
+
+at
+K.
+Therefore, in terms of the splitting (B.18),
+iσ1(L12) = r−1σ 4Q0
+rα
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+0
+−1
+0
+0
+0
+0
+0
+0
+0
+−1
+0
+0
+0
+0
+0
+0
+0
+1
+0
+0
+−1
+0
+1
+0
+0
+0
+0
+1
+0
+−1
+0
+0
+0
+0
+0
+0
+0
+−1
+0
+0
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+at
+K.
+(B.25)
+So it remains to analyze L21(˙g). We first compute
+Gg =
+
+
+
+
+
+
+
+
+1
+2
+0
+0
+1
+2
+0
+1
+2r2 /tr
+0
+1
+0
+0
+0
+0
+0
+0
+1
+0
+0
+0
+1
+2
+0
+0
+1
+2
+0
+− 1
+2r2 /tr
+0
+0
+0
+0
+1
+0
+1
+2r2/g
+0
+0
+− 1
+2r2/g
+0
+G/g
+
+
+
+
+
+
+
+
+,
+tr12
+g ((·) ⊗ F) =
+
+
+0
+− Q0
+r2
+0
+− Q0
+r2
+0
+0
+0
+0
+0
+
+ : T ∗M → T ∗M (B.26)
+and
+σ1(δg) = i
+
+
+α−1σ
+−αξ
+−r−2ιη
+0
+0
+0
+0
+α−1σ
+0
+−αξ
+−r−2ιη
+0
+0
+0
+α−1σ
+0
+−αξ
+−r−2ιη
+
+ .
+(B.27)
+
+170
+LILI HE
+Therefore, we find that
+σ1(tr12
+g (δgGg(·) ⊗ F)) = iQ0
+r2
+
+
+1
+2αξ
+−α−1σ
+0
+1
+2αξ
+r−2ιη
+− 1
+2r2 αξ /tr
+− 1
+2α−1σ
+αξ
+r−2ιη
+− 1
+2α−1σ
+0
+− 1
+2r2 α−1σ /tr
+0
+0
+0
+0
+0
+0
+
+ .
+(B.28)
+Next, we calculate
+σ1(d∇(·) ⊗ F) = i2Q0
+r2
+
+
+αξ
+−α−1σ
+0
+0
+0
+0
+0
+αξ
+0
+−α−1σ
+0
+0
+0
+0
+αξ
+0
+−α−1σ
+0
+
+ .
+(B.29)
+In the ‘microlocal splitting’ (B.18), we have
+/tr =
+�r2
+0
+r2�
+at
+K.
+and thus we have
+iσ1(L21) = r−1σ Q0
+2rα
+
+
+
+
+0
+0
+0
+0
+0
+−2
+0
+0
+0
+0
+1
+0
+−2
+0
+−1
+0
+0
+1
+0
+1
+0
+0
+0
+0
+0
+−2
+0
+0
+0
+0
+0
+0
+0
+0
+0
+0
+−2
+0
+0
+0
+
+
+
+ .
+(B.30)
+Putting (B.20), (B.23), (B.25) and (B.30) together yields
+SL = r−1σ
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+0
+−4
+0
+0
+0
+0
+0
+0
+0
+0
+0
+−8Q′
+0
+0
+−2
+0
+2
+0
+−2
+0
+0
+0
+0
+0
+0
+0
+0
+0
+0
+−2
+0
+0
+0
+−2
+0
+0
+0
+0
+0
+−8Q′
+0
+0
+0
+0
+0
+0
+0
+0
+−2
+0
+0
+0
+0
+0
+0
+0
+0
+−4
+0
+0
+0
+4
+0
+0
+0
+0
+0
+8Q′
+0
+0
+0
+0
+−2
+0
+−2
+0
+0
+2
+0
+0
+−8Q′
+0
+8Q′
+0
+0
+0
+0
+−2
+0
+0
+0
+0
+2
+0
+0
+0
+0
+8Q′
+0
+0
+0
+0
+0
+−4
+0
+0
+0
+0
+0
+−8Q′
+0
+0
+0
+0
+0
+0
+0
+0
+−2
+0
+0
+0
+0
+0
+0
+0
+0
+0
+0
+0
+0
+0
+0
+0
+0
+0
+0
+−8Q′
+0
+0
+0
+0
+0
+0
+0
+−2Q′
+0
+0
+0
+0
+0
+−2
+0
+0
+Q′
+0
+−2Q′
+0
+−Q′
+0
+0
+Q′
+0
+Q′
+−2
+0
+2
+0
+0
+0
+0
+0
+0
+−2Q′
+0
+0
+0
+0
+0
+−2
+0
+0
+0
+0
+0
+0
+0
+0
+−2Q′
+0
+0
+0
+0
+0
+0
+0
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+(B.31)
+where Q′ = Q0
+2rα. Since SL has the following Jordan form
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+0
+0
+0
+0
+0
+0
+0
+0
+0
+0
+0
+0
+0
+0
+0
+0
+0
+0
+0
+0
+0
+0
+0
+0
+0
+0
+0
+0
+0
+0
+0
+0
+0
+0
+0
+0
+0
+0
+0
+0
+0
+0
+0
+0
+0
+0
+0
+0
+0
+0
+0
+0
+0
+0
+0
+0
+0
+0
+0
+0
+0
+0
+0
+0
+0
+0
+0
+0
+0
+0
+0
+0
+0
+0
+0
+0
+1
+0
+0
+0
+0
+0
+0
+0
+0
+0
+0
+0
+0
+0
+0
+1
+0
+0
+0
+0
+0
+0
+0
+0
+0
+0
+0
+0
+0
+0
+1
+0
+0
+0
+0
+0
+0
+0
+0
+0
+0
+0
+0
+0
+0
+1
+0
+0
+0
+0
+0
+0
+0
+0
+0
+0
+0
+0
+0
+0
+0
+0
+0
+0
+0
+0
+0
+0
+0
+0
+0
+0
+0
+0
+−4iQ′
+0
+0
+0
+0
+0
+0
+0
+0
+0
+0
+0
+0
+0
+0
+−4iQ′
+0
+0
+0
+0
+0
+0
+0
+0
+0
+0
+0
+0
+0
+0
+4iQ′
+0
+0
+0
+0
+0
+0
+0
+0
+0
+0
+0
+0
+0
+0
+4iQ′
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+,
+
+LINEAR STABILITY OF KERR-NEWMAN FAMILY
+171
+one can choose a suitable matrix qL which has constant coefficients at K such that
+qLSLq−1
+L
+=
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+0
+0
+0
+0
+0
+0
+0
+0
+0
+0
+0
+0
+0
+0
+0
+0
+0
+0
+0
+0
+0
+0
+0
+0
+0
+0
+0
+0
+0
+0
+0
+0
+0
+0
+0
+0
+0
+0
+0
+0
+0
+0
+0
+0
+0
+0
+0
+0
+0
+0
+0
+0
+0
+0
+0
+0
+0
+0
+0
+0
+0
+0
+0
+0
+0
+0
+0
+0
+0
+0
+0
+0
+0
+0
+0
+0
+ǫ
+0
+0
+0
+0
+0
+0
+0
+0
+0
+0
+0
+0
+0
+0
+ǫ
+0
+0
+0
+0
+0
+0
+0
+0
+0
+0
+0
+0
+0
+0
+ǫ
+0
+0
+0
+0
+0
+0
+0
+0
+0
+0
+0
+0
+0
+0
+ǫ
+0
+0
+0
+0
+0
+0
+0
+0
+0
+0
+0
+0
+0
+0
+0
+0
+0
+0
+0
+0
+0
+0
+0
+0
+0
+0
+0
+0
+−4iQ′
+0
+0
+0
+0
+0
+0
+0
+0
+0
+0
+0
+0
+0
+0
+−4iQ′
+0
+0
+0
+0
+0
+0
+0
+0
+0
+0
+0
+0
+0
+0
+4iQ′
+0
+0
+0
+0
+0
+0
+0
+0
+0
+0
+0
+0
+0
+0
+4iQ′
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+.
+(B.32)
+Again, equipping S2T ∗M⊕T ∗M over M with the Hermitian inner product BL
+0 which is given as an identity
+matrix in the splitting (B.14), qLSsub(−L)q−1
+L
+has imaginary part (with respect to BL
+0 ) of size O(ǫ). Put
+differently, Ssub(−L) has imaginary part of size (ǫ) relative to the Hermitian inner product b := BL
+0 (qL·, qL·),
+which is the symbol of a pseudodifferential inner product on π∗T ∗M. To summarize,
+Theorem B.7. For any ǫ > 0, there exists a (positive definite) t-independent pseudodifferential inner product
+B = b(x, D) on S2T ∗M⊕ T ∗M (thus, b is an inner product on π∗(S2T ∗M⊕ T ∗M), homogeneous of degree
+0 with respect to dilations in the base T ∗M \ 0), such that
+sup
+K
+|σ|−1
+����
+1
+2i(Ssub(−L) − Ssub(−L)∗b)
+����
+b
+≤ ǫ,
+where K is the trapped set. Put differently, there is an elliptic pseudodifferential operator Q, invariant under
+t-translations, acting on sections of S2T ∗M ⊕ T ∗M, with parametrix Q−, such that relative to the ordinary
+positive definite inner product BL
+0 we have
+sup
+Γ
+|σ|−1
+����σ1
+� 1
+2i(Q(−L)Q− − (Q(−L)Q−)∗BL
+0 )
+�����
+BL
+0
+≤ ǫ.
+Finally, we point out the relation between the operators P, W, L and their Fourier transform (semiclassi-
+cally rescaled) h2 �P(h−1z), h2 �
+W(h−1z) and h2�L(h−1z).
+Remark B.8. Given a operator P ∈ {□g,1, −L}, we define �P(σ) := eiσt∗Pe−iσt∗ and its semiclassical rescaling
+h2 �P(h−1z) with h = |σ|−1. Let p2 = σ2(P) and p2,h(z) = σ2,h(h2 �P(h−1z)). Here we also use ξ′ = (ξ, η) as
+the semiclassical dual variables of x′ = (r, ω). Then using the correspondence Re z ←→ −σ, we have
+Ssub(h2 �P(h−1z)) = Ssub(P) + iα−2σ∂t + iIm (hz)∂σp2
+and thus
+σ1,h
+� 1
+2ih(h2 �P(h−1z) − (h2 �P(h−1z))∗B)
+�
+= σ1
+� 1
+2i(P − P ∗B)
+�
++ Im (hz)∂σp2.
+Finally, we note that Im (hz)∂σp2 = Im (hz)∂zp2,h(Re z) = σ1,h
+�
+1
+2ih(h2 �
+□g,0(h−1z) − (h2 �
+□g,0(h−1z))∗)
+�
+.
+Appendix C. Calculation of the subprincipal operator at radial points at event horizon
+In this section, we calculate the threshold regularity at the radial points at event horizon. To this end, we
+discuss the subprincipal operator of Pb0,γ, Wb0,γ and Lb0,γ at radial points at event horizon. Here we again
+drop the notations b0, γ.
+We recall that modulo terms of order 0, we have
+− 2P ≡ □g,1 − 2δgGg(˜δ∗
+g − δ∗
+g),
+−2W ≡ □g,1 − 2(˜δg − δg)Ggδ∗
+g
+(C.1)
+and
+− L(˙g, ˙A) ≡
+�
+□g ˙g − 2(˜δ∗
+g − δ∗
+g)δgGg ˙g − 2δ∗
+g(˜δg − δg)Gg ˙g + L12( ˙A), □g ˙A + L21(˙g)
+�
+.
+(C.2)
+
+172
+LILI HE
+Near the radial points at the event horizon, we use the coordinates (t0, r, ω) in (4.6), in which the Reissner-
+Nordstr¨om metric g = gb0, b0 = (m0, 0, Q0) takes the form
+g = −µb0dt2
+0 + 2dt0dr + r2/g.
+Writing the covectors as
+σ dt0 + ξ dr + η,
+η ∈ T ∗S2,
+we have G = g−1(σ dt0 + ξ dr + η, σ dt0 + ξ dr + η) = 2σξ + µb0ξ2 + r−2|η| and
+HG = 2ξ∂t0 + (2σ + 2µb0ξ)∂r − µ′
+b0ξ2∂ξ + 2r−3|η|2∂ξ + r−2H|η|2.
+The radial points at event horizon is given as
+L± = {(t0, rb0, ω; 0, ξ, 0) | ±ξ > 0}.
+(C.3)
+Then at L±, we have
+HG = 2ξ∂t0 − 2κξ2∂ξ + 2µb0ξ∂r
+(C.4)
+with
+κ = 1
+2µ′
+b0(rb0).
+(C.5)
+being the surface gravity. We keep the term 2µb0ξ∂r here as it is needed in the computation of the skew
+adjoint part.
+The basis e0 = αdt, e1 = α−1dr are not defined at r = rb0. Hence, following [42, §6.3], we compute the
+transition function to a smooth trivialization of T ∗M and S2T ∗M as follows: writing a 1-form in the static
+region M as
+u = uN e0 + uT
+N e1 + uT
+T = �uN dt0 + �uT
+N dr + �uT
+T ,
+with uT
+T and �uT
+T 1-forms on S2, we have
+
+
+uN
+uT
+N
+uT
+T
+
+ = C (1)
+
+
+�uN
+�uT
+N
+�uT
+T
+
+ ,
+C (1) =
+
+
+α−1
+0
+0
+α−1
+α
+0
+0
+0
+1
+
+ .
+(C.6)
+The bundle splitting
+T ∗M = ⟨dt0⟩ ⊕ ⟨dr⟩ ⊕ T ∗S2
+(C.7)
+is smooth at r = rb0. Consider the induced bundle splitting
+S2T ∗M = ⟨dt2
+0⟩ ⊕ ⟨2dt0 dr⟩ ⊕ (2dt0 · T ∗S2) ⊕ ⟨dr2⟩ ⊕ (2dr · T ∗S2) ⊕ S2T ∗S2.
+(C.8)
+Writing a section u of S2T ∗M as
+u = uNN e0e0 + 2uNT N e0e1 + 2e0 · uNT T + uT
+NN e1e1 + 2e1 · uT
+NT + uT
+T T
+= �uNN dt2
+0 + 2�uNT N dt0 dr + 2dt0 · �uNT T + �uT
+NN dr2 + 2dr · �uT
+NT + �uT
+T T ,
+the transition matrix is defined by
+
+
+
+
+
+
+
+
+uNN
+uNT N
+uNT T
+uT
+NN
+uT
+NT
+uT
+T T
+
+
+
+
+
+
+
+
+= C (2)
+
+
+
+
+
+
+
+
+�uNN
+�uNT N
+�uNT T
+�uT
+NN
+�uT
+NT
+�uT
+T T
+
+
+
+
+
+
+
+
+,
+C (2) =
+
+
+
+
+
+
+
+
+α−2
+0
+0
+0
+0
+0
+α−2
+1
+0
+0
+0
+0
+0
+0
+α−1
+0
+0
+0
+α−2
+2
+0
+α2
+0
+0
+0
+0
+α−1
+0
+α
+0
+0
+0
+0
+0
+0
+1
+
+
+
+
+
+
+
+
+.
+(C.9)
+Then using the coordinates (t0, r, ω) and splitting (C.7), we again use Proposition B.5 to compute
+iSsub(□g,1) = −2ξ∂t0 + 2κξ2∂ξ + 2µb0ξ∂r − 2κξ
+
+
+−1
+0
+0
+0
+1
+0
+0
+0
+0
+
+
+at
+L±
+(C.10)
+
+LINEAR STABILITY OF KERR-NEWMAN FAMILY
+173
+and
+iSsub(□g,2) = −2ξ∂t0 + 2κξ2∂ξ + 2µb0ξ∂r − 2κξ
+
+
+
+
+
+
+
+
+−2
+0
+0
+0
+0
+0
+0
+0
+0
+0
+0
+0
+0
+0
+−1
+0
+0
+0
+0
+0
+0
+2
+0
+0
+0
+0
+0
+0
+1
+0
+0
+0
+0
+0
+0
+0
+
+
+
+
+
+
+
+
+at
+L±.
+(C.11)
+Next, we turn to the calculation of Ssub(L12) and Ssub(L21) at L±. This computation is easily accom-
+plished by first calculating the symbols at N ∗{r = r0} = {t, r0, ω, 0, ξ, 0}, for r0 ∈ (rb0, ∞) close to rb0, in
+the static coordinate system and bundle splittings (B.14), conjugating by C (1) or C (2) and restricting to
+µb0 = 0. Concretely, according to (B.24), in terms of the splitting (B.14), we have at N ∗{r = r0}
+iσ1(L12) = 4Q0
+r2
+
+
+
+
+
+
+
+
+αξ
+0
+0
+0
+0
+0
+0
+0
+αξ
+−αξ
+0
+0
+0
+0
+0
+αξr2/g
+0
+0
+
+
+
+
+
+
+
+
+.
+So in the smooth splittings (C.7) and (C.8) (multiplying from the left by (C (2))−1 and from the right by
+C (1) and evaluating at µ = 0),
+iσ1(L12) = 4Q0
+r2 ξ
+
+
+
+
+
+
+
+
+0
+0
+0
+−1
+0
+0
+0
+0
+0
+0
+0
+0
+0
+0
+−1
+r2/g
+0
+0
+
+
+
+
+
+
+
+
+at
+L±.
+(C.12)
+Using (B.28) and (B.29), in terms of the splitting (B.14), we have at N ∗{r = r0}
+iσ(L21) = Q0
+r2
+
+
+1
+2αξ
+0
+0
+− 1
+2αξ
+0
+1
+2r2 αξ /tr
+0
+0
+0
+0
+0
+0
+0
+0
+αξ
+0
+0
+0
+
+ .
+Multiplying this from the left by (C (1))−1 and from the right by C (2) and evaluating at µ = 0 give rise to
+iσ(L21) in the smooth splittings (C.7) and (C.8)
+iσ(L21) = Q0
+2r2 ξ
+
+
+0
+0
+0
+0
+0
+0
+0
+2
+0
+0
+0
+− 1
+r2 /tr
+0
+0
+2
+0
+0
+0
+
+
+at
+L±.
+(C.13)
+Finally, it remains to calculate the subprincipal operators of (˜δ∗
+g − δ∗
+g)δgGg ˙g and δ∗
+g(˜δg − δg)Gg ˙g. First,
+according to (B.26) and (B.27), we see that in the static splitting (B.14)
+iσ1(δgGg) =
+
+
+0
+αξ
+0
+0
+0
+0
+1
+2αξ
+0
+0
+1
+2αξ
+0
+− αξ
+2r2 /tr
+0
+0
+0
+0
+αξ
+0
+
+
+at N ∗{r = r0}, and thus
+iσ1(δgGg) = 1
+2ξ
+
+
+2
+0
+0
+0
+0
+0
+0
+0
+0
+0
+0
+− 1
+r2 /tr
+0
+0
+2
+0
+0
+0
+
+
+at
+L±.
+
+174
+LILI HE
+Also, in the smooth splitting, we have at L±
+iσ1(δ∗
+g) =
+
+
+
+
+
+
+
+
+0
+0
+0
+− ξ
+2
+0
+0
+0
+0
+0
+0
+−ξ
+0
+0
+0
+− ξ
+2
+0
+0
+0
+
+
+
+
+
+
+
+
+.
+Since ˜δ∗
+g − δ∗
+g = 2γc ⊗s (·) − 1
+2γg−1(c, ·)g and ˜δg − δg = 2γιg
+c(·) − 1
+2γctrg(·) where c = ˜χ(r)(dt0 − bdr) with
+˜χ(r) = 1 near event horizon and b > 2, it follows that in terms of the smooth splitting (C.7) and (C.8)
+˜δ∗
+g − δ∗
+g =
+
+
+
+
+
+
+
+
+2γ
+0
+0
+− 1
+2bγ
+1
+2γ
+0
+0
+0
+γ
+0
+−2bγ
+0
+0
+0
+−bγ
+1
+2bγr2/g
+− 1
+2γr2/g
+0
+
+
+
+
+
+
+
+
+,
+˜δg − δg =
+
+
+−2bγ
+γ
+0
+0
+0
+− 1
+2γr−2 /tr
+0
+−bγ
+0
+2γ
+0
+1
+2bγr−2 /tr
+0
+0
+−2bγ
+0
+2γ
+0
+
+
+at L±. Therefore, in terms of the smooth splitting, we have at L±
+iσ1((˜δ∗
+g − δ∗
+g)δgGg) = 1
+2γξ
+
+
+
+
+
+
+
+
+4
+0
+0
+0
+0
+0
+−b
+0
+0
+0
+0
+− 1
+2r2 /tr
+0
+0
+2
+0
+0
+0
+0
+0
+0
+0
+0
+2b
+r2 /tr
+0
+0
+−2b
+0
+0
+0
+br2/g
+0
+0
+0
+0
+1
+2/g /tr
+
+
+
+
+
+
+
+
+,
+(C.14)
+iσ1(δgGg(˜δ∗
+g − δ∗
+g)) = 1
+2γξ
+
+
+4
+0
+0
+−b
+1
+0
+0
+0
+2
+
+ ,
+(C.15)
+iσ1(δ∗
+g(˜δg − δg)Gg) = 1
+2γξ
+
+
+
+
+
+
+
+
+0
+0
+0
+0
+0
+0
+2b
+−2
+0
+0
+0
+1
+2r2 /tr
+0
+0
+0
+0
+0
+0
+0
+2b
+0
+−4
+0
+− b
+r2 /tr
+0
+2b
+0
+−2
+0
+0
+0
+0
+0
+0
+0
+0
+
+
+
+
+
+
+
+
+,
+(C.16)
+iσ1((˜δg − δg)Ggδ∗
+g) = 1
+2γξ
+
+
+−1
+0
+0
+b
+−4
+0
+0
+0
+−2
+
+ .
+(C.17)
+In the splitting (C.7), combining (C.10) with (C.15) and (C.17) yields
+Ssub(−2P) = −2ξ∂t0 − 2κξ2∂ξ + 2µb0ξ∂r + SP,
+Ssub(−2W) = −2ξ∂t0 − 2κξ2∂ξ + 2µb0ξ∂r + SW
+where
+SP = ξ
+
+
+2κ − 4γ
+0
+0
+bγ
+−γ − 2κ
+0
+0
+0
+−2γ
+
+ ,
+SW = ξ
+
+
+2κ + γ
+0
+0
+−bγ
+4γ − 2κ
+0
+0
+0
+2γ
+
+ .
+In the splitting (C.7), equipping T ∗M with the Hermitian inner product 1 ⊕ 1 ⊕/g−1, the first three terms of
+Ssub(−2P), Ssub(−2W) are formally self adjoint with respect to the symplectic volume form on T ∗M, and
+the last term SP resp. SW of iSsub(−2P) resp. iSsub(−2W), multiplied by ξ−1, has eigenvalues
+2(κ − 2γ), −2κ − γ, −2γ,
+resp.
+− 2(κ − 2γ), 2γ, 2κ + γ.
+(C.18)
+We perform a decomposition
+S2T ∗S2 = ⟨r2/g⟩ ⊕ /g⊥,
+so
+r2/g =
+�
+1
+0
+�
+,
+r−2 /tr =
+�2
+0�
+.
+
+LINEAR STABILITY OF KERR-NEWMAN FAMILY
+175
+and introduce the splitting
+⟨dt2
+0⟩ ⊕ ⟨2dt0 dr⟩ ⊕ (2dt0 · T ∗S2) ⊕ ⟨dr2⟩ ⊕ (2dr · T ∗S2) ⊕ ⟨r2/g⟩ ⊕ /g⊥
+⊕ ⟨dt0⟩ ⊕ ⟨dr⟩ ⊕ T ∗S2.
+(C.19)
+In the above splitting (C.19), putting (C.10),(C.11), (C.12), (C.13), (C.14) and (C.16) together yields
+iSsub(−L) = −2ξ∂t0 − 2κξ2∂ξ + 2µb0ξ∂r + SL
+where
+SL = ξ
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+4κ − 4γ
+0
+0
+0
+0
+0
+0
+0
+0
+0
+−bγ
+2γ
+0
+0
+0
+0
+0
+−4Q′′
+0
+0
+0
+0
+2κ − 2γ
+0
+0
+0
+0
+0
+0
+0
+0
+−2bγ
+0
+4γ − 4κ
+0
+−2bγ
+0
+0
+0
+0
+0
+−2bγ
+0
+2γ
+−2κ
+0
+0
+0
+0
+−4Q′′
+−bγ
+0
+0
+0
+0
+−γ
+0
+4Q′′
+0
+0
+0
+0
+0
+0
+0
+0
+0
+0
+0
+0
+0
+0
+0
+0
+0
+0
+0
+2κ
+0
+0
+0
+Q′
+0
+0
+0
+−Q′′
+0
+0
+−2κ
+0
+0
+0
+Q′′
+0
+0
+0
+0
+0
+0
+0
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+with Q′′ = Q0r−2. In the splitting (C.19), equipping S2T ∗M ⊕ T ∗M with the Hermitian inner product
+1 ⊕ 1 ⊕ /g−1 ⊕ 1 ⊕ /g−1 ⊕ 1 ⊕ 1 ⊕ 1 ⊕ 1 ⊕ /g−1, the first three terms of Ssub(−L) are formally self adjoint with
+respect to the symplectic volume form on S2T ∗M ⊕ T ∗M, and the last term SL of iSsub(−L), multiplied by
+ξ−1 has eigenvalues
+0, 0, −2κ, −2κ, 2κ, −4(κ − γ), 2(κ − γ), 4(κ − γ), −γ, 2γ.
+(C.20)
+Let P ∈ {−2P, −2W, −L}. Then the quantity ˜β for P is defined as
+|ξ|−1 1
+2i
+�
+Ssub(P) −
+�
+Ssub(P)
+�∗�
+= ∓ ˜ββ0
+at
+L±
+where β0 = 2κ. Recall that βsup = sup ˜β and βinf = inf ˜β. Then for sufficiently small γ > 0, we have
+−2P : βsup = 1 − 2γ
+κ ,
+βinf = −1 − γ
+2κ;
+−2W : βsup = 1 + γ
+2κ,
+βinf = −1 + 2γ
+κ ;
+−L : βsup = 2 − 2γ
+κ ,
+βinf = −2 + 2γ
+κ .
+(C.21)
+We again point out the relation between the operators P, W, L and their Fourier transform �P(σ), �
+W(σ) and
+�L(σ).
+Remark C.1. Given a operator P ∈ {−2P, −2W, −L}, we define �P(σ) := eiσt∗Pe−iσt∗ with σ ∈ C. The
+radial points set of �P (which we still denote by L±) is given by L± = {(rb0, ω; ξ, 0) | ±ξ > 0}. Then we have
+1
+2i
+�
+Ssub( �P) −
+�
+Ssub( �P)
+�∗�
+= 1
+2i
+�
+Ssub(P) −
+�
+Ssub(P)
+�∗�
++ σ1(
+�
+□g,0(σ) − ( �
+□g,0(σ))∗
+2i
+)
+at
+L±.
+Finally, according to the discussion around (5.59), the calculation of the threshold regularity at radial points
+at event horizon for the semiclassical operator h2 �P(h−1z) is equal to that of �P(σ).
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+Email address: lhe31@jhu.edu
+
diff --git a/k9FAT4oBgHgl3EQfbR2A/content/tmp_files/load_file.txt b/k9FAT4oBgHgl3EQfbR2A/content/tmp_files/load_file.txt
new file mode 100644
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@@ -0,0 +1,11769 @@
+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf,len=11768
+page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='08557v1  [gr-qc]  18 Jan 2023  THE LINEAR STABILITY OF WEAKLY CHARGED AND SLOWLY ROTATING  KERR-NEWMAN FAMILY OF CHARGED BLACK HOLES  LILI HE  Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' In this paper, we prove the linear stability of weakly charged and slowly rotating Kerr-Newman  black holes under coupled gravitational and electromagnetic perturbations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' We show that the solutions to  the linearized Einstein-Maxwell equations decay at an inverse polynomial rate to a linearized Kerr-Newman  solution plus a pure gauge term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' This work builds on the framework developed in [36] for the study of the  Einstein vacuum equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' We work in the generalized wave map and Lorenz gauge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' The proof involves the  analysis of the resolvent of the Fourier transformed linearized Einstein-Maxwell operator on asymptotically  flat spaces, which relies on recent advances in microlocal analysis and non-elliptic Fredholm theory developed  in [77].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' The most delicate part of the proof is the description of the resolvent at low frequencies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  Contents  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Introduction  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
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+page_content=' Invariant formalism for the subprincipal symbols of operators acting on bundles  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
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+page_content=' 165  C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Calculation of the subprincipal operator at radial points at event horizon  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
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+page_content=' 175  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Introduction  The Einstein-Maxwell system, which describes the interaction of gravity and electromagnetism, is given  by  Ric(g)µν = 2Tµν(g, F),  dF = 0,  δgF = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='1)  where g is a Lorentzian metric g with signature (−, +, +, +), Ric(g) is the Ricci curvature tensor and  Tµν(g, F) := FµαF α  ν  − 1  4gµνFαβF αβ is the energy momentum tensor associated with an electromagnetic  field represented by a 2-form F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Thus, the first equation in (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='1), Einstein’s field equation, describes the  dynamics of the spacetime metric g in the presence of an electromagnetic field F, and the last two equations,  Maxwell’s equations, describes the propagation of electromagnetic waves on the spacetime g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  Among the most interesting solutions to Einstein-Maxwell equations are the families of black hole so-  lutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' The Kerr-Newman (KN) family of solutions of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='1) describes stationary, charged, and rotating  black holes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' It was discovered by [70], following the discovery of the Kerr [51].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' The family of KN solutions  LINEAR STABILITY OF KERR-NEWMAN FAMILY  3  (gb, Fb) to the Einstein-Maxwell equations are characterized by the parameters b = (m, a, Qe, Qm) ∈ R6  where m > 0, a ∈ R3, Qe, Qm are interpreted as the mass, angular momentum, electric charge and magnetic  charge respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' The KN spacetime reduces to the Reissner-Nordstr¨om (RN) spacetime for a = 0, and  RN spacetime reduces to Schwarzschild spacetime when Qe = Qm = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Statement of the main result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' The stability problem for black holes solutions concerns the long time  behavior of solutions to the Einstein-Maxwell system with initial data close to known black hole solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  The problem of stability of black hole solutions can be roughly divided into three formulations, each of  increasing difficulty: mode stability of linearized equations, linear stability, and full nonlinear stability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  The problem of mode stability of the linearized equations involves separating the solutions into modes  ψ(t, r, θ, φ) = e−iωteimφR(r)S(θ) and aiming at proving the lack of exponentially growing (in t) modes for  all metric or curvature components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' The proof of linear stability of Einstein-Maxwell system means proving  boundedness and decay for the solutions of the linearized Einstein-Maxwell equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Finally, the nonlinear  stability problem for Einstein-Maxwell equations aims at showing that a small perturbation of a KN black  hole converges to another member of the KN family.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  In order to understand nonlinear stability, one must first understand linear stability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' In this paper, we  consider the linear stability problem of the linearized Einstein-Maxwell equations under coupled gravitational  and electromagnetic perturbations for weakly charged and slowly rotating Kerr-Newman black holes, that  is, the case |a| + |Qe| + |Qm| ≪ m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  To describe our result, we first recall the initial value problem formalism of Einstein-Maxwell equations  (see §§3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='2–3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='3 for detail).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Given 5-tuple (Σ0, h, k, E, B) where Σ0 is a smooth 3-manifold, h is a Riemannian  metric on Σ0, k is a symmetric 2-tensor on Σ0, and E, B are 1-forms on Σ0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' One seeks (M, g, F) such that  (g, F) solve the Einstein-Maxwell system and (h, k) are the metric and second fundamental form induced  by g, and (E, B) are the electric and magnetic field induced by (g, F) on Σ0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' A necessary and sufficient  condition for local solvability of Einstein-Maxwell equations (see [8, 10] and [9, §6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='10]) is that (h, k, E, B)  satisfy the constraint equations, which consist of Gauss-Codazzi equation and the pull-back of the Maxwell  equations to Σ0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Linearizing the initial value problem gives rise to the corresponding initial value problem  of linearized Einstein-Maxwell equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  We now state a first version of our linear stability result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' For a more formal statement, see Theorem 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Let b = (m, a, Qe, Qm) be the black hole parameters satisfying |a| + |Qe| + |Qm| ≪ m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Let  0 < α < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Let (˙h, ˙k, ˙E, ˙B) be an initial data set on a 3-manifold Σ0 satisfying the linearized constraint  equation, and having decay  |˙h| ≲ r−1−α,  |˙k|, | ˙E|, | ˙B| ≲ r−2−α  and similar bounds after applying a few r∂r, ∂ω derivatives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Let (˙g, ˙F) a solution to the linearized Einstein-  Maxwell equations and attain the initial data (˙h, ˙k, ˙E, ˙B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Then (˙g, ˙F) decays at an inverse polynomial rate  in time tb,∗ to a linearized Kerr-Newman solution plus a pure gauge solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' That is, there exists linearized  black hole parameters ˙b = ( ˙m, ˙a, ˙Qe, ˙Qm) ∈ R × R3 × R × R and a vector field V such that  ˙g = ˙gb(˙b) + LV gb + ˜g,  ˙F = ˙Fb(˙b) + LV Fb + ˜F  where  (˙gb(˙b), ˙Fb(˙b)) := d  ds  ���  s=0(gb+s˙b, Fb+s˙b)  is the linearized Kerr-Newman solutions and the tail (˜g, ˜F) satisfies the bound  |˜g| + | ˜F| ≲ t−α−1+ǫ  b,∗  ,  ǫ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  Remark 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Using the rotation invariance of the Maxwell equation, one can always arrange for the initial  data to have vanishing magnetic charge, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=', Qm = 0, and thus the electromagnetic 2-form can be written  as F = dA (see §3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='1 for detail).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' In this paper, we will restrict to the case where the magnetic charge is 0  and study Einstein-Maxwell equations of the following form  Ric(g)µν = 2Tµν(g, dA),  δgdA = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  We also refer the readers to Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='2 for the relation between the non-magnetically charged RN spacetime  (gb, Fb = dAb) and the magnetically charged RN spacetime (gb,m, Fb,m).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  4  LILI HE  Remark 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' The hypersurface {tb,∗ = c} terminates at null infinity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' We also note that for bounded r, the  tail (˜g, ˜F) satisfies the bound |˜g| + | ˜F| ≲ t−1−α+ǫ for all ǫ > 0 where t = t for large r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  Remark 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Upon imposing a suitable linearized generalized wave map gauge condition on ˙g, and replacing  (˙gb(˙b), ˙Fb(˙b)) by its gauge-fixed version, we can choose V such that V = V1 + V2 with V1 lying in a 6-  dimensional space (only depending on b ) of smooth vector fields (asymptotic to a linear combination of  spatial translations and boosts) on M, and V2 decaying (in tb,∗) at the rate t−α+ǫ  b,∗  , ǫ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  For charged asymptotically flat black hole spacetimes (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' the cosmology constant Λ = 0), the mode  stability of Reissner-Nordstr¨om black holes was studied in [6,7,57,67–69].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' The linear stability of full subex-  tremal Reissner-Nordstr¨om spacetime was proved by Giorgi [28, 29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' As for Kerr-Newman spacetime, the  mode stability of wave equation on the Kerr-Newman metric was established in [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' The Teukolsky and  Regge-Wheeler equations governing the linear stability of Kerr-Newman spacetime to coupled electromag-  netic and gravitational perturbations were derived in [30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' The Carter tensor and the physical-space analysis  in perturbations of Kerr-Newman spacetime were discussed in [27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' For positive cosmology case Λ > 0, the  nonlinear stability of slowly rotating Kerr-Newman-de Sitter black holes was proved by Hintz [39].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Related works.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Besides the above references for charged black holes, we also mention the closely  related results on Einstein vacuum equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Nonlinear stability problems for solutions of Einstein vacuum  equations have attracted a large amount of interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' The nonlinear stability of Minkowski spacetime was first  proved by Christodoulou-Klainerman [11] and later an alternative proof was given by Lindblad-Rodnianski  [58–60] using wave coordinates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' For the case Λ > 0, see [26] for the nonlinear stability of de Sitter spacetime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  For the Schwarzschild metric, the mode stability was studied in [4, 56, 72, 83, 88].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Dafermos-Holzegel-  Rodnianski [14] proved the linear stability of the Schwarzschild metric in a double null gauge by recon-  structing a perturbation from a certain decoupled quantity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  Later, the linear stability of Schwarzschild  metric in (generalized) harmonic gauge was studied in [47–50].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  For progress in the nonlinear stability,  Klainerman-Szeftel [55] proved the nonlinear stability of the Schwarzschild metric under axially symmetric  and polarized perturbations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Recently, Dafermos-Holzegel-Rodnianski-Taylor [15] proved the nonlinear sta-  bility of Schwarzschild spacetime under a restrictive symmetry class, which excludes rotating Kerr solutions  as final state of the evolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  For the Kerr metric, the mode stability was studied in [74, 76, 85].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Dafermos-Holzegel-Rodnianski [13]  studied the Teukolsky equation on Kerr spacetime for |a| ≪ m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Andersson-B¨ackdahl-Blue-Ma [1] proved  the linear stability of slowly rotating Kerr metric for initial data with strong decay using Newman-Penrose  formalism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' H¨afner-Hintz-Vasy [36] proved the linear stability of slowly rotating Kerr spacetime using wave  map gauge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Recently, the nonlinear stability of slowly rotating Kerr spacetime was proved by a series of work  of Klainerman-Szeftel and Giorgi-Klainerman-Szeftel [31,32,52–54].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' For large a case, Dafermos-Rodnianski-  Shlapentokh-Rothman [16] established the decay for solutions of the wave equation on full subextremal Kerr  exterior spacetimes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Shlapentokh-Rothman-Costa [75] proved the boundedness and decay for the Teukolsky  equation on Kerr in the full subextremal range |a| < m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Anderson-H¨afner-Whiting [2] gave the mode analysis  for the linearized equations on subextremal Kerr metric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  In the case of the Einstein equations with a positive cosmological constant, the global nonlinear stability of  slowly rotating Kerr-de Sitter spacetime was proved by Hintz-Vasy [43], followed by the work of Fang [22,23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Main ideas of the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' The proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='1 builds on the framework developed in [36] for  the study of the Einstein vacuum equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' The presence of electromagnetic term in the right hand side of  Einstein equations adds new difficulties to the analysis of the problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  (1) In addition to the diffeomorphism invariance, which for the Einstein vacuum equations is eliminated  by imposing a wave map gauge in [36], one needs to fix a gauge (generalized Lorenz gauge) for the  electromagnetic 4-potential A in order to express the Einstein-Maxwell system (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='1) as a quasilin-  ear system of wave equations, called the gauge-fixed Einstein-Maxwell system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Correspondingly,  the linearization of the gauge-fixed Einstein-Maxwell system gives rise to a linear system of wave  equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  (2) We need to show the (generalized) mode stability for the linearization of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='1) around a RN solution:  all generalized mode solutions are sums of linearized KN solutions and pure gauge solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' In this  paper, we give a self-contained proof in full detail of this generalized mode stability, completing and  extending work by Kodama and Ishibashi [57].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  LINEAR STABILITY OF KERR-NEWMAN FAMILY  5  (3) The fact that the KN metric is not Ricci-flat creates the major difficulty in the mode analysis of  the wave type operator δgbGgbδ∗  gb (where (δ∗  gbu)αβ = 1  2(∇αuβ + ∇βuα) is the symmetric gradient  operator and (δgbh)β = −∇αhαβ is the negative divergence operator, and Ggb = Id − 1  2gbtrgb is the  trace reversal operator) acting on 1-forms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' We note that the operator δgbGgbδ∗  gb (or its modifications)  occurs as both the gauge propagation operator and the gauge potentials operator for the Einstein  equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' To deal with this issue, we restrict to the weakly charged case, where we are able to  exploit a perturbation argument to extend the invertibility of relevant operators on Schwarzschild  metric to weakly charged Reissner-Nordstr¨om spacetime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  Concretely, we first employ the generalized wave map gauge  �ΥE(g;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' gb) := ΥE(g;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' gb) − θ(g;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' gb) = 0,  ΥE(g;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' gb) = gµνgκλ (Γ(g)µ  κλ − Γ(gb)µ  κλ)  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='2)  where θ(g;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' gb) is linear in g and θ(gb;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' gb) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' By introducing this type of gauge condition with a  suitable choice of θ, the gauge potential operator is given by ˜δgbGgbδ∗  gb where ˜δgb is a zero order  modification of δg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' In contrast to δgbGgbδ∗  gb which is the gauge potential operator corresponding  to the standard wave map gauge ΥE(g;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' gb) = 0, the operator ˜δgbGgbδ∗  gb is invertible (on suitable  function spaces) in the Schwarzschild case b = (m, 0, 0), and thus a perturbation argument enables  us to extend this invertibility to the weakly charged and slowly rotating KN metrics with sufficiently  small charge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Moreover, the generalized wave map gauge allows us to exclude a pure gauge solution  which has no geometric meaning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  Next, we implement the constraint damping, which was first discussed in [34] and played a central  role both in numerical work [71] and the stability proofs [36, 39, 42, 43].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' The reason we introduce  the constraint damping, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=', use ˜δ∗  gb in the construction of linearized gauge-fixed Einstein-Maxwell  equations, is that we need to use the invertibility of the corresponding gauge propagation operator  δgbGgb ˜δ∗  gb to exclude the generalized modes, which grows linearly in tb,∗ and whose leading term  is given by linearized (in ( ˙m, 0, ˙Q)) KN solutions, to the linearized gauge-fixed Einstein-Maxwell  equations Lb(˙g, ˙A) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' We also note that we use a perturbation argument to prove the invertibility  of the gauge propagation operator δgbGgb ˜δ∗  gb on the weakly charged and slowly rotating KN metrics  gb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Gauge fixing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' In this paper, we introduce the generalized wave map gauge and generalized Lorenz  gauge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  Concretely, one fixes a background metric g0, and requires that the identity map (M, g) → (M, g0) be a  wave map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' This is equivalent to  ΥE(g;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' g0) = gµνgκλ �  Γ(g)µ  κλ − Γ(g0)µ  κλ  �  = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  We define the generalized wave map gauge as follows  �ΥE(g;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' g0) := ΥE(g;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' g0) − θ(g;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' g0) = 0  where θ(g;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' g0) is linear in g, contains no derivatives of g and θ(g0, g0) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Then we consider the gauge-fixed  Einstein equations  P E(g, A) := Ric(g) − 2T (g, dA) − �δ∗  g �Υ(g;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' g0) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  At the linearized level, we take gb, around which we linearize the equations, as the background metric, and  thus obtain the linearized gauge-fixed Einstein equations  LE  b (˙g, ˙A) := DgbP E(˙g, ˙A) = −1  2□gb ˙g + l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='o.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='t = 0  which is a system of linear wave equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  As for the Maxwell equations, we introduce the generalized Lorenz gauge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' We fix a background metric  g0, and then a (modified) Lorenz gauge reads  ΥM(g, A;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' g0) = trgδ∗  g0A = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  We fix a background 1-form A0 and define the generalized Lorenz gauge as  �ΥM(g, A;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' g0, A0) := ΥM(g, A;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' g0) − ΥM(g, A0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' g0),  6  LILI HE  and then obtain the gauge-fixed Maxwell equations  P M(g, A) := δgdA − d�ΥM(g, A;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' g0, A0) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  Again, at the linearized level, we take (gb, Ab), around which we linearize the equations, as the background  metric and 1-form, and thus have  LM  b (˙g, ˙A) := DgbP M(˙g, ˙A) = −□gb ˙A + l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='o.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='t = 0  which is a system of linear wave equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  We see that a solution of the ungauged equations satisfying the gauge conditions gives rise to a solution  of the gauge-fixed equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' As for the other direction, suppose that (g, A) solves the gauge-fixed Einstein-  Maxwell system  �  P E(g, A), P M(g, A)  �  = 0,  and satisfies  (�ΥE(g;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' gb), �ΥM(g, A;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' gb, Ab) = 0  at Σ0  and  (L∂t �ΥE(g;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' gb), L∂t �ΥM(g, A;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' gb, Ab) = 0  at Σ0  where Σ0 = {t = 0} is the spacelike Cauchy hypersurface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Applying δg to  P M(g, A) = δgdA − d�ΥM(g, A;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' gb, Ab) = 0  gives  −δd�ΥM(g, A;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' gb, Ab) = □g �ΥM(g, A;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' gb, Ab) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  and thus according to the vanishing initial data of �ΥM(g, A;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' gb, Ab), we obtain that �ΥM(g, A;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' gb, Ab) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  Applying δgGg to P E(g, A) = Ric(g) − 2T (g, dA) − �δ∗  g �Υ(g;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' g0) = 0 and using the second Bianchi identity  and the fact δgT = 0 yields  −δgGg�δ∗  g �ΥE(g;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' gb) = 0  which is a wave-type equation on the 1-form �ΥE(g;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' gb) and thus �ΥE(g;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' gb) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' We conclude that (g, A)  satisfies the gauge conditions, and thus solves the Einstein-Maxwell system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Correspondingly, this direction  holds on the linearized level as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Fredholm framework setup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' According to the above discussion, it suffices to consider the equation  Lb(˙g, ˙A) := (2LE  b , LM  b )(˙g, ˙A) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' A simple linear theory presented in [43] reduces the initial value problem  for Lb(˙g, ˙A) = 0 to the following inhomogeneous problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  Concretely, let b = (m, a, Q) be close to b0 = (m0, 0, Q0) with |Q0| ≪ m0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Fix a function tb,∗ which  equals t + r(m0,0,Q0),∗ near event horizon and t − r(m,0,Q),∗ near null infinity I +, where rb,∗ is the tortoise  coordinate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Then we consider the following inhomogeneous equation  Lb(˙g, ˙A) = (2LE  b (˙g, ˙A), LM  b (˙g, ˙A)) = f  where f has compact support in t∗ and f decays in r at the rate r−2−α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Our strategy is to take Fourier  transform of the inhomogeneous equation Lb(˙g, ˙A) = f in tb,∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' We define the Fourier transform of (˙g, ˙A) as  (ˆ˙g(σ), ˆ˙A(σ)) :=  �  R  eitb,∗σ(˙g, ˙A)(tb,∗) dtb,∗  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='3)  and likewise for f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  According to a crude energy estimate in Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='27, we have the following integral representation  (˙g(tb,∗),  ˙A(tb,∗)) = 1  2π  �  Im σ=M+1  e−iσtb,∗�  Lb(σ)−1 ˆf(σ) dσ,  M ≫ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='4)  We expect to shift contour of the integration to Im σ = c for any c > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' To this end, we need to understand  the location of the poles of �  Lb(σ)−1 and expect invertibility of �  Lb(σ)−1 when |Re σ| ≫ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Concretely, we  need to establish the uniform Fredholm estimates (down to σ = 0) and high energy estimates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' The uniform  Fredholm estimates allow us to perform the perturbation argument (which is used a lot in this paper) while  the high energy estimates imply the invertibility of �  Lb(σ)−1 when |Re σ| ≫ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' We point out that the whole  section §5 is devoted to using microlocal tools to obtain the aforementioned uniform Fredholm estimates  and high energy estimates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' In brief, the proof of uniform Fredholm estimates uses the non-elliptic Fredholm  LINEAR STABILITY OF KERR-NEWMAN FAMILY  7  framework of [77], radial point estimates at the event horizon [77], scattering radial point estimates at infinity  ( [65,79] for non-zero σ and [81] for σ near zero), propagation of singularities estimates [18], and (for σ = 0)  elliptic b-theory [64].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' As for the proof of high energy estimates, in addition to the aforementioned estimates  at a semiclassical version, it uses estimates at normally hyperbolic trapping [20, 40, 86] and high energy  estimates at infinity [79,82].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Mode stability of the gauge-fixed Einstein-Maxwell operator for C−1 ≤ |σ| ≤ C, Im σ ≥ 0 with C > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  In view of the high energy estimates, we are left with the analysis of �  Lb(σ) for σ in a bounded region in the  closed upper half plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' For the analysis in the region C−1 ≤ |σ| ≤ C, Im σ ≥ 0 with C > 0, we prove that  �  Lb0(σ), with b0 = (m0, 0, Q0) and Q0 ≪ m0 being a Reissner-Nordstr¨om parameter, is invertible and then  use a perturbation argument (in this compact (in σ) region) to extend the invertibility to �  Lb(σ) for nearby  b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  For the proof of invertibility of �  Lb0(σ), we make use of the geometric mode stability result of Reissner-  Nordstr¨om black holes and the invertibility of the gauge progation operator and gauge potential operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  Concretely, suppose that �  Lb0(˙g, ˙A) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Then (˙g, ˙A) = e−itb0,∗σ(˙g, ˙A) is a mode solution of Lb0(˙g, ˙A) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  According to the discussion in §1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='1, we see that the solution satisfies  δgb0 d  �  D(gb0 ,Ab0) �ΥM(˙g, ˙A;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' gb0, Ab0)  �  = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  According to the mode stability of □gb0 ,0, we see that D(gb0 ,Ab0 ) �ΥM(˙g, ˙A;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' gb0, Ab0) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Therefore, (˙g, ˙A)  solves the Maxwell equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' By the discussion in §1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='1 again, we obtain that  δgb0 Ggb0 ˜δ∗  gb0  �  Dgb0 �ΥE(˙g;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' gb0)  �  = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  Using the mode stability of the gauge propagation operator (this is where we need the smallness of the  charge), we see that Dgb0 �ΥE(˙g;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' gb0) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Therefore, (˙g, ˙A) is a mode solution of the ungauged linearized  Einstein-Maxwell system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Using the geometric mode stability of Reissner-Nordstr¨om solutions, we conclude  that (˙g, ˙A) is a pure gauge mode solution, that is,  (˙g, ˙A) =  �  2δ∗  gω, Lω♯A + dφ  �  where ω is a 1-form and φ is a scalar function, both of which are modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Plugging them back into the  linearized gauge condition, and with our choice of the generalized gauge condition, we have  ˜δgb0 Ggb0 δ∗  gb0 ω = 0,  δgb0 d  �  Lω♯A + dφ  �  = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  First by the mode stability of the gauge potential operator (this is again where we need the smallness of the  charge) and next the mode stability of □gb0 ,0, we see that (ω, φ) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' This proves the injectivity of �  Lb0(σ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  Since �  Lb0(σ) has index 0, it follows that �  Lb0(σ) is invertible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' The analysis of �  Lb(σ)−1 near σ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' According to Theorem 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='1, the space Kb of zero energy modes  of Lb is 8-dimensional;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' it is the sum of a 5-dimensional space of linearized Kerr-Newman solutions corrected  by addition of pure gauge solutions to arrange the linearized gauge conditions, and a 3-dimensional space  of pure gauge solutions (2δ∗  gbωb, Lω♯  bAb + dφb) with ωb asymptotic to a spatial translation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' According to  Proposition 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='5, the space �Kb of generalized zero energy modes (with o(1) decay as r → ∞ for fixed tb,∗,  but allowing for polynomial growth in tb,∗) of Lb is the sum of Kb and a 3-dimensional space of pure gauge  solutions (2δ∗  gb ˆωb, Lˆω♯  bAb + dˆφb) with ˆωb asymptotic to a Lorentz boost and the coefficient of tb,∗ being a  stationary pure gauge solution (2δ∗  gbωb, Lω♯  bAb + dφb).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  Therefore, �  Lb(σ) is more singular when acting on the 3-dimensional space of stationary pure gauge so-  lutions because of the existence of the generalized zero energy solutions of Lb(˙g, ˙A) = 0 with linear growth  in tb,∗ and leading order terms given as a stationary pure gauge solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Motivated by this fact, we write  �  Lb(σ) (actually after multiplied by an operator which is invertible near σ = 0) as a 3 × 3 block matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  8  LILI HE  The basic mechanism underlying the analysis of �  Lb(σ)−1 near σ = 0, is the formal expansion of the  resolvent near σ = 0  �  Lb(σ)−1f = �  Lb(0)−1f + (�  Lb(σ)−1 − �  Lb(0)−1)f  = u0 + σ�  Lb(σ)−1f1,  where we use the formal resolvent identity �  Lb(σ)−1 − �  Lb(0)−1 = −�  Lb(σ)−1(�  Lb(σ) − �  Lb(0))�  Lb(0)−1 and thus  obtain  u0 = �  Lb(0)−1f,  f1 = −σ−1��  Lb(σ) − �  Lb(0)  �  u0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  The first term u0 is σ-independent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' As for the second term σ�  Lb(σ)−1f1, we gain a factor of σ (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' more  regularity in σ and thus more decay in tb,∗ after taking inverse Fourier transform).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' However, since �  Lb(0)−1  loses two orders of decay in r while σ−1(�  Lb(σ) − �  Lb(0)) = 2iρ(ρ∂ρ − 1) + ρ2C∞ with ρ = 1/r only gains  back one order decay in r, it follows that f1 has one order of decay less than f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' We expect to do the above  expansion iteratively  �  Lb(σ)−1f = u0 + σu1 + · · · + σkuk  where  uk = �  Lb(0)−1fk,  fk+1 = −σ−1��  Lb(σ) − �  Lb(0)  �  uk  as often as possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' However, this iteration requires that uk = �  Lb(0)−1fk has a certain asymptotic behavior  or decay rate as r → ∞ such that σ−1(�  Lb(σ) − �  Lb(0))uk lies in the domain of �  Lb(σ)−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' That is, we have to  stop the iteration once uk = �  Lb(0)−1fk does not satisfy this requirement and thus we obtain an error term  σkL(σ)−1fk in the expansion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  Specifically, a detailed analysis in §11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='2 implies that �  Lb(σ)−1 (up to a multiplication of an operator  invertible near σ = 0) is equal to  \uf8eb  \uf8ec  \uf8ed  L00(b, σ)  σ2 �L01(b, σ)  σ�L02(b, σ)  σ�L10(b, σ)  σ2 �L11(b, σ)  σ2 �L12(b, σ)  σ�L20(b, σ)  σ2 �L21(b, σ)  σ�L22(b, σ)  \uf8f6  \uf8f7  \uf8f8  where L00(b, σ), ˜L11(b, σ), ˜L22(b, σ) are invertible near σ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Therefore, one obtain  �  Lb(σ)−1 = σ−2P2(b) + σ−1P1(b) + R1(b, σ) + R2(b, σ),  R1(b, σ) ∼ |σ|α−1,  R2(b, σ) ∼ |σ|α  where P1(b), P2(b) are independent of σ and of finite rank with explicit range: linearized KN solutions and  pure gauge solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' The range of R1(b, σ) consists of pure gauge solutions and this allow us to rewrite  the inverse Fourier transform of R1(b, σ)f as a pure gauge solution plus a remainder term decaying faster in  time tb,∗ (in fact at the rate t−1−α  b,∗  ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Outline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' The paper is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  In §2, we introduce the b- and scattering geometric structures on manifolds with boundaries or  corners (in this paper, we work on the compactification of the spatial slice Σ of the spacetime M),  and corresponding function spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  In §3, we discuss the initial value problems formalism for Einstein-Maxwell equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  We also  introduce our choice of gauge, generalized wave map gauge and generalized Lorenz gauge, and then  derive the gauge-fixed Einstein-Maxwell system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' We include the discussion at both nonlinear and  linearized levels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  In §4, we realize the family of slowly rotating KN metric as a smooth family of stationary metrics on  a fixed manifold M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' We also introduce several vector bundles over M and discuss how to construct  the gauged Cauchy data for the gauge-fixed Einstein-Maxwell system from any given initial data  (h, k, E, B) satisfying the constraint equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  In §5, we set up the general Fredholm framework, based on tools from microlocal analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Concretely,  we prove the uniform Fredholm estimates and high energy estimates for the relevant wave type  operators (acting on both scalar functions and tensor bundles).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' We also establish a crude energy  estimate which gives an exponentially growing bound on the energy (over a spacelike hypersurface  terminating at null infinity) of solutions to the wave equations on slowly rotating Kerr-Newman  metric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  LINEAR STABILITY OF KERR-NEWMAN FAMILY  9  In §6, we introduce the spherical harmonic decompositions, which will be used in the proof of the  geometric mode stability of Reissner-Nordstr¨om spacetime in §7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  In §7, we prove the version of the geometric mode stability of Reissner-Nordstr¨om spacetime used  in the subsequent sections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  In §8, we study the modes of the scalar wave operator on RN and slowly rotating KN spacetimes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  In §9, we analyze the modes on various function spaces (which correspond to different decay rate at  spatial infinity) of the gauge propagation operator Pb,γ = δgbGgb�δ∗  gb,γ and the gauge potential wave  operator Wb,γ = �δgb,γGgbδ∗  gb, both of which are wave-type operators acting on 1-form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' From this  section on, we are restricted in the small charge case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  In §10, we prove that the linearized gauge-fixed Einstein-Maxwell system Lgb0 ,γ (linearized around  weakly charged RN metrics and 4-electromagnetic potentials (gb0, Ab0)) has no non-zero modes in  the closed upper half plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Moreover, we describe the space of zero modes and generalized zero  modes (with linearly growth in tb,∗) of Lb,γ for both RN case and slowly rotating KN case with small  charge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  In §11, we prove the mode stability of the linearized gauge-fixed Einstein-Maxwell operator Lb,γ(σ)  for b = (m, a, Q) near b0 = (m0, 0, Q0), |Q0| ≪ m0 and Im σ ≥ 0, σ ̸= 0 (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' the invertibility of the  operator �  Lb,γ(σ) on suitable function spaces for Im σ ≥ 0, σ ̸= 0), and also provide a description of  the structure of the resolvent �  Lb,γ(σ)−1 near σ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  In §12, we study the higher regularity of the resolvent ( �  Lb,γ(σ))−1 of the linearized gauge-fixed  Einstein-Maxwell operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  In §13, we put all the previous sections together to study the asymptotic behavior of the solutions  (˙g, ˙A) to the inhomogeneous linearized gauge-fixed Einstein-Maxwell equations Lb,γ(˙g, ˙A) = f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  In §14, we reduce this initial value problem for Einstein-Maxwell equations to the inhomogeneous  problem studied in the previous section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  In Appendix §A, we provide a detailed calculation of the linearized Einstein-Maxwell system around  RN black holes, which is needed in the proof of the geometric mode stability of Reissner-Nordstr¨om  spacetime in §7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' In Appendix B, we present the calculation of the subprincipal operator of the wave  operator acting on tensor bundles at trapping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' In Appendix C, we include the computation of the  subprincipal operator of the wave operator acting on tensor bundles at radial points at event horizon,  which is needed in the radial point estimates at event horizons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' List of notations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' We give a list of notation used frequently throughout the present paper, together  with a reference to the first appearance:  Aℓ  weighted L∞ conormal function space, see (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='4)  Ab0  Reissner-Nordstr¨om electromagnetic 4-potentials with parameters b0, see (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='23)  Ab  Kerr-Newman electromagnetic 4-potentials with parameters b, see (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='23)  b0  fixed Reissner-Nordstr¨om parameters, see (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='1)  b  Kerr-Newman parameters, see (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='13)  δg  negative divergence, (δgu)µ1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='µN = −∇λuλµ1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='µN  δ∗  g  symmetric gradient, (δ∗  gu)µν = 1  2(uµ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='ν + uν;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='µ)  ˜δg,γ  modification of the negative divergence, see (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='62)  ˜δ∗  g,γ  modification of the symmetric gradient, see (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='62)  gb0  Reissner-Nordstr¨om metrics with parameter b0, see (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='23)  gb  Kerr-Newman metrics with parameters b, see (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='23)  γ0  map assigning to a function on M its Cauchy data at Σ0, see (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='85)  ¯Hs,ℓ  b  weighted b-Sobolev space of extendible distributions, see (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='8)  ˙Hs,ℓ  b  weighted b-Sobolev space of supported distributions, see (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='9)  ¯Hs,ℓ  b,h  semiclassical weighted b-Sobolev space of extendible distributions, see (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='11)  ˙Hs,ℓ  b,h  semiclassical weighted b-Sobolev space of supported distributions, see (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='11)  ib  map constructing Cauchy data from geometric initial data, see Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='14  Kb  the space of zero energy modes of Lb,γ, see (10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='8)  �Kb  the space of generalized zero energy modes of Lb,γ, see Proposition 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='5  10  LILI HE  K∗  b  the space of dual zero energy modes of L∗  b,γ, see (10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='9)  �K∗  b  the space of dual generalized zero energy modes of L∗  b,γ, see Proposition 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='5  LV  Lie derivative along the vector field V  �LT V  equal to LV T , see Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='9  Lb,γ  linearized gauge-fixed Einstein-Maxwell equations (2LE  b,γ, LM  b,γ)  LE  b,γ  linearized gauge-fixed Einstein equations, see (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='63)  LM  b,γ  linearized gauge-fixed Maxwell equations, see (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='63)  M  the static region of a fixed reissner-Nordstr¨o spacetime, see (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='5)  M  extended manifold where gb is defined, see (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='8)  Pb,γ  gauge propagation operator for the generalized wave map gauge 1-form �ΥE, see (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='66)  Σ0  a spacelike Cauchy hpersurface of M, see (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='10)  t  static time coordinate, see (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='2), or Boyer-Lindquist coordinate, see (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='14)  t0  incoming Eddington-Finkelstein coordinate on a fixed Reissner-Nordstr¨om manifold, see (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='6)  tχ0  a time function interpolating t0 near the even thorizon and t near r = ∞, see (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='9)  tb,∗  a time function which is smooth across the even thorizon and transverse to null infinity near r = ∞,  see (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='29)  t∗  equal to tb0,∗, see (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='10)  t  a timelike function with respect to Kerr-Newman metric gb, which equals to t near r = ∞, see (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='15)  θ  gauge source function for the generalized wave map gauge, see (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='17)  Ub0  parameter space for slowly rotating Kerr-Newman black holes, see Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='1  Ub0,m  parameter space for slowly rotating Kerr-Newman black holes allowing for magnetic charges, see  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='24)  ΥE  wave map gauge 1-form, see (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='15)  ΥM  modified Lorenz gauge function, see (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='19)  �ΥE  generalized wave map gauge 1-form, see (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='17)  �ΥM  generalized Lorenz gauge function, see (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='20)  Wb,γ  gauge potential operator for Einstein equations, see (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='66)  X  the spatial slice of the static region M, see (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='5)  X  the spatial slice of the extended manifold M, see (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='8)  ¯X  compactification of X at r = ∞, see (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='11)  ∂− ¯X  an artificial boundary of ¯X, see (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='11)  ∂+ ¯X  the boundary at spatial infinity r = ∞ of ¯X, see (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='11)  �•(σ)  Fourier transform eitb,∗σ • e−itb,∗σ of • with respect to the time function tb,∗;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' in this paper, • =  □gb,0, Pb,γ, Wb,γ, Lb,γ  Acknowledgements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' The author would like to thank her advisor, Hans Lindblad, for his constant support  and encouragement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' The author is also grateful to Peter Hintz and Andr´as Vasy for helpful discussions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' b- and scattering geometry  In this section, we will discuss the b- and scattering geometric structures on manifolds with boundaries or  corners, and corresponding function spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Here, we follow the discussion in [36, §2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' In §2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='1, we recall the  notions of b- and scattering vector bundles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' We refer the readers to [64, §2] and [66, §2] for a more detailed  exposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' In §2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='2, we recall the notion of radial compactification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' In §2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='3, we define the relevant Sobolev,  conormal and polyhomogeneous functions defined on manifolds with boundaries or corners.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  Let ¯X be a compact n-dimensional manifold with boundary ∂ ¯X ̸= ∅, and let ρ ∈ C∞(X) denote a  boundary defining function: ∂ ¯X = ρ−1(0), dρ ̸= 0 on ∂ ¯X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' b- and scattering vector bundles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' We define the Lie algebras of b-vector fields and scattering vector  fields by  Vb( ¯X) = {V ∈ V( ¯X): V is tangent to ∂ ¯X},  Vsc( ¯X) = ρVb( ¯X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='1)  In local coordinates x ≥ 0, y ∈ Rn−1 on ¯X, with x = 0 locally defining the boundary of ¯X, elements of  Vb( ¯X) are of the form a(x, y)x∂x + �n−1  i=1 bi(x, y)∂yi, with a, bi ∈ C∞( ¯  X), while elements of Vsc( ¯X) are of  the form a(x, y)x2∂x + �n−1  i=1 bi(x, y)x∂yi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  LINEAR STABILITY OF KERR-NEWMAN FAMILY  11  Correspondingly, one can define the b-tangent bundle, resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' scattering tangent bundle  bT ¯X → ¯X,  resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  scT ¯X → ¯X,  with local frames given by {x∂x, ∂yi}, resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' {x2∂x, x∂yi}, such that Vb( ¯X) = C∞( ¯X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' bT ¯X) and Vsc( ¯X) =  C∞( ¯X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' scT ¯X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Therefore, for example, x∂x is a smooth, non-vanishing section of bT ¯X down to ∂ ¯X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Over  the interior X, these bundles are naturally isomorphic to T X, but the maps bT ¯X → T X and scT ¯X → T X  fail to be injective over ∂ ¯X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' We denote by Diffm  b ( ¯X), resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Diffm  sc(X) the space of m-th order b-, resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  scattering differential operators, consisting of linear combinations of up to m-fold products of elements of  Vb(X), resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Vsc(X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  The dual bundles bT ∗ ¯X → ¯X (b-cotangent bundle), resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' scT ∗ ¯X → ¯X (scattering cotangent bundle) have  local frames  dx  x , dyi,  resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  dx  x2 , dyi  x ,  which are smooth down to ∂ ¯X as sections of these bundles (despite their being singular as standard covectors,  i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' elements of T ∗ ¯X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' A scattering metric is then a section g ∈ C∞( ¯X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' S2 scT ∗ ¯X) which is a non-degenerate  quadratic form on each scattering tangent space scT ∗  p ¯X, p ∈ ¯X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' b-metrics are defined analogously as a section  g ∈ C∞( ¯X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' S2 bT ∗ ¯X)  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Radial compactification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' These aforementioned b- and scattering structures arise naturally on com-  pactifications of non-compact manifolds (see [66, §1]), the simplest example being the radial compactification  of Rn, defined by  Rn :=  �  Rn ⊔ ([0, 1)ρ × Sn−1)  �  / ∼  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='2)  where the relation ∼ identifies a point in Rn \\ {0}, expressed in polar coordinates as rω, r > 0, ω ∈ Sn−1,  with the point (ρ, ω) where  ρ = r−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  This has a natural smooth structure, with smoothness near ∂Rn = ρ−1(0) meaning smoothness in (ρ, ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' In  polar coordinates in r > 1, the space of b-vector fields is then locally spanned over C∞(Rn) by ρ∂ρ = −r∂r  and V(Sn−1);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' scattering vector fields are spanned by ρ2∂ρ = −∂r and ρV(Sn−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Using standard coordinates  x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' , xn on Rn, scattering vector fields on Rn are precisely those of the form  n  �  i=1  ai∂xi,  ai ∈ C∞(Rn);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  this involves the statement that {∂x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' , ∂xn}, which is a frame of T Rn, extends by continuity to a smooth  frame of scT Rn down to ∂Rn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Thus, the space of scattering vector fields on Rn is generated over C∞(Rn) by  constant coefficient (translation-invariant) vector fields on Rn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' On the other hand, Vb(Rn) is spanned over  C∞(Rn) by vector fields on Rn with coefficients which are linear functions, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' by ∂x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' , ∂xn, and xi∂xj,  1 ≤ i, j ≤ n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  On the dual side, scT ∗Rn is spanned by dxi, 1 ≤ i ≤ n, down to ∂Rn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Therefore, a scattering metric  g ∈ C∞(Rn, S2 scT ∗Rn) is a non-degenerate linear combination of dxi ⊗s dxj = 1  2(dxi ⊗ dxj + dxj ⊗ dxi)  with C∞(Rn) coefficients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' In particular, the Euclidean metric  (dx1)2 + · · · + (dxn)2 ∈ C∞(Rn;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' S2 scT ∗Rn)  is a Riemannian scattering metric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  In this manuscript, we are particularly interested in the behavior of the Hamiltonian vector field associated  to a smooth function p ∈ C∞(scT ∗ ¯X) where ¯X is a manifold with boundaries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Concretely, by extension from  T ∗X, one can define Hamilton vector fields Hp of smooth functions p ∈ C∞(scT ∗ ¯X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' In fact Hp ∈ Vsc(scT ∗ ¯X)  is a scattering vector field on scT ∗ ¯X, which is a manifold with boundary scT ∗  ∂ ¯  X ¯X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' For us, the main example  will be the Hamilton vector field HG where G(x, ξ) := |ξ|2  g(x)−1 and g ∈ C∞( ¯X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' S2 scT ∗ ¯X) is a scattering  metric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  To give a uniform discussion of the behavior of the Hamiltonian vector field Hp ∈ Vsc(scT ∗ ¯X) near the  boundary of ∂ ¯X and near the infinity in the fibers of the scattering cotangent bundle scT ∗ ¯X, we consider  the compactified scattering cotangent bundle scT ∗ ¯X as our basic microlocal space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' We note that scT ∗ ¯X is a  compact manifold with corners obtained by the radial compactification of the fibers of scT ∗ ¯X (see Figure 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  Let ρtotal = xρξ, where x, resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' ρξ is the boundary defining function of the boundary of ¯X, resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' the fiber  12  LILI HE  infinity, be the total boundary defining function of scT ∗ ¯X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Then we have Vsc(scT ∗ ¯X) = ρtotalVb(scT ∗ ¯X), and  thus the rescaled Hamiltonian vector field ρ−1  totalHp, where p ∈ C∞(scT ∗ ¯X), is a b-vector field on scT ∗ ¯X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  scT ∗  ∂ ¯  X ¯X = {x = 0}  scS∗ ¯X = {ρξ = 0}  scS∗ ¯X = {ρξ = 0}  0 section  Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' The microlocal space scT ∗ ¯X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Function spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' We next introduce Sobolev spaces corresponding to b- and scattering structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' As  an integration measure on ¯X, let us fix a scattering density, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' a positive section of scΩ1 ¯X = |Λn scT ∗ ¯X|,  which in local coordinates x ≥ 0, y ∈ Rn−1 takes the form a(x, y)| dx  x2  dy  xn−1 | with 0 < a ∈ C∞( ¯X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' (On Rn,  one can take |dx1 · · · dxn|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=') This provides us with a space L2( ¯X);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' the norm depends on the choice of density,  but all choices lead to equivalent norms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Working with a b-density on the other hand would give a different  space, namely a weighted version of L2(X);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' we therefore stress that even for b-Sobolev spaces, we work with  a scattering density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Thus, for k ∈ N, we define  Hk  (X) := {u ∈ L2(X): V1 · · · Vju ∈ L2(X) ∀ V1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' , Vj ∈ V•(X), 0 ≤ j ≤ k},  = b, sc,  called b- or scattering Sobolev space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Using a finite spanning set in V•(X), one can give this the structure of  a Hilbert space;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Hs  (X) for general s ∈ R is then defined by duality and interpolation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' If ℓ ∈ R, we denote  weighted Sobolev spaces by  Hs,ℓ  (X) = ρℓHs  (X) = {ρℓu: u ∈ Hs  (X)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  For example, Hs,ℓ  sc (R) ∼= ⟨x⟩−ℓHs(Rn) is the standard weighted Sobolev space on Rn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' The space of weighted  (L2-)conormal functions on ¯X is  H∞,ℓ  b  (X) =  �  s∈R  ¯Hs,ℓ  b (X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  Dually, we define  H−∞,ℓ  b  (X) =  �  s∈R  ¯Hs,ℓ  b (X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  Note that ¯Hs,ℓ  b (X) ⊂ C−∞( ¯  X) := ˙C∞( ¯X)∗ (where ˙C∞( ¯X) ⊂ C∞(X) is the subspace of functions vanishing  to infinite order at ∂ ¯X) consists of tempered distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' (In particular, they are extendible distributions  at ∂X in the sense of [46, Appendix B].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=') We furthermore introduce the notation  Hs,ℓ+  b  ( ¯X) :=  �  ǫ>0  Hs,ℓ+ǫ  b  ( ¯X),  Hs,ℓ−  b  ( ¯X) :=  �  ǫ>0  Hs,ℓ−ǫ  b  ( ¯X)  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='3)  for s ∈ R ∪ {±∞}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  A space closely related to H∞,ℓ  b  ( ¯X) is  Aℓ( ¯  X) := {u ∈ ρℓL∞( ¯X): Diffb( ¯X)u ⊂ ρℓL∞( ¯X)},  {ρ = 0} = ∂ ¯X  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='4)  consisting of weighted L∞-conormal functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' For ¯X = R3, we have the inclusions  H∞,ℓ  b  (R3) ⊂ Aℓ+3/2(R3),  Aℓ(R3) ⊂ H∞,ℓ−3/2−  b  (R3),  LINEAR STABILITY OF KERR-NEWMAN FAMILY  13  by Sobolev embedding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' (The shift 3  2 in the weight is due to our defining b-Sobolev spaces with respect to  scattering densities instead of b-densities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Indeed, for s > 3  2, we have  Hs,ℓ  b (R3;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' |dx1 dx2 dx3|) = Hs,ℓ+3/2  b  (R3;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' ⟨r⟩−3|dx1 dx2 dx3|) ֒→ ⟨r⟩−ℓ−3/2L∞(R3),  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='5)  with the second density here being a b-density on R3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=') We define Aℓ+( ¯X) and Aℓ−( ¯X) analogously to (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  These notions extend readily to sections of rank k vector bundles E → X: for instance, in a local trivialization  of E, an element of Hs,ℓ  (X, E) is simply a k-tuple of elements of Hs,ℓ  (X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  We next turn to the notion of E-smoothness (see [64, §5]), where E ⊂ C × N is an index set;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' the latter  means that E is a discrete subset of C × N satisfying that (z, k) ∈ E implies (z, j) ∈ E for 0 ≤ j ≤ k and  (z − i, k) ∈ E, and that any sequence (zj, kj) ∈ E with |zj| + kj → ∞ satisfies Im zj → −∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' The space  AE  phg( ¯X) ⊂ A−∞( ¯X) =  �  ℓ∈R  Aℓ( ¯X)  then consists of all u for which  �  (z,j)∈E  Im z≥−N  (ρDρ − z)u ∈ AN( ¯X)  ∀ N ∈ R,  where we define Dρ = i−1∂ρ with respect to any fixed collar neighborhood of ∂ ¯X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Equivalently, there exist  a(z,j) ∈ C∞(∂ ¯X), (z, j) ∈ E, such that  u −  �  (z,j)∈E  Im z≥−N  ρiz(log ρ)ja(z,j) ∈ AN( ¯X)  ∀ N ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='6)  We say that u ∈ A−∞(X) is polyhomogeneous if it is E-smooth for some index set E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  Suppose now ¯X′ is a compact manifold with boundary, and let ¯X ⊂ ¯X′ be a submanifold with boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  Suppose that the boundaries of ¯X decomposes into two non-empty sets  ∂ ¯X = ∂− ¯X ⊔ ∂+ ¯X,  ∂− ¯X = ∂ ¯X \\ ∂ ¯X′,  ∂+X = ∂ ¯X′;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='7)  we consider ∂+ ¯X to be a boundary ‘at infinity’, while ∂− ¯X is an interior, ‘artificial’ boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Concretely,  this means that we define (by a slight abuse of notation)  Vb( ¯  X) := {V |X : V ∈ Vb( ¯X′)},  Vsc(X) := {V |X : V ∈ Vsc( ¯  X′)};' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  these vector fields are b or scattering at infinity, but are unrestricted at ∂− ¯X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' A typical example is ¯X′ = Rn  and ¯X = {r ≥ 1} ⊂ ¯X′, in which case ∂− ¯X = {r = 1}, while ∂+ ¯X = ∂X′ is the boundary (at infinity) of  Rn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' See Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  ∂− ¯X  ∂+ ¯X = ∂ ¯X′  Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' A typical example of the setting (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='7):  ¯X (dark gray) is a submanifold of ¯X′  (the union of the dark and light gray regions) with two boundary components ∂+ ¯X = ∂ ¯X′  and ∂− ¯X ⊂ ( ¯X′)◦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' We then consider function spaces such as ¯Hs,ℓ  b ( ¯X), which measures b-  regularity of degree s at ∂+ ¯X (with decay rate ℓ), and standard regularity (regularity with  respect to incomplete vector fields) at ∂− ¯X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  Then there are now two natural classes of Sobolev spaces: those consisting of extendible distributions,  ¯Hs,ℓ  ( ¯  X) := {u| ¯  X◦ : u ∈ Hs,ℓ  ( ¯X′)},  = b, sc,  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='8)  14  LILI HE  and those consisting of supported distributions,  ˙Hs,ℓ  ( ¯X) := {u: u ∈ Hs,ℓ  ( ¯X′), supp u ⊂ ¯X}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='9)  Away from ∂− ¯X, these are the same as the standard spaces Hs,ℓ  ( ¯X);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' thus, the extendibility/support con-  siderations arise only at ∂− ¯X, not at ∂+ ¯X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  If ¯X is the ‘spatial part’ of a stationary spacetime M = Rt × ¯X with projection π ¯  X : M → ¯X, and F( ¯X)  denotes a space of distributions on ¯X such as F( ¯  X) = ¯H∞,ℓ  b  ( ¯X) or ˙H−∞,ℓ  b  ( ¯X) in the setting (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='7), we will  be interested not only in zero modes F( ¯X) ∼= π∗¯  XF( ¯X), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' t-independent distributions, but also generalized  zero modes,  Poly(t)F( ¯X) :=  �  k∈N  Polyk(t)F( ¯X),  Polyk(t)F( ¯  X) :=  \uf8f1  \uf8f2  \uf8f3  k  �  j=0  tjaj : aj ∈ F( ¯X)  \uf8fc  \uf8fd  \uf8fe .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='10)  Here, we do not require ¯X to be spacelike or dt to be timelike.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  For the high energy estimates, we will work with semiclassical b-Sobolev spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Thus, if ¯X is a manifold  with boundary, we define Hs,ℓ  b,h( ¯X) = Hs,ℓ  b ( ¯X) as a set, but with norm depending on the semiclassical  parameter h ∈ (0, 1]: if V1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' , VN ∈ Vb( ¯  X) spans Vb( ¯X) over C∞( ¯X), we let  ∥u∥2  Hk  b,h(X) :=  �  1≤i1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=',ij≤N  0≤j≤k  ∥(hVi1) · · · (hVij)u∥2  L2( ¯  X),  ∥u∥Hk,ℓ  b,h( ¯  X) := ∥ρ−ℓu∥Hk  b,h( ¯  X),  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='11)  for k ∈ N;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' for k ∈ R, we define the norm Hk  b,h( ¯X) by duality and interpolation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' (Alternatively, one can  define Hs,ℓ  b,h( ¯X) using semiclassical b-pseudodifferential operators, see [42, Appendix A].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=') On manifolds ¯X  as in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='7), one can also define semiclassical spaces ¯Hs,ℓ  b,h(X) and  ˙Hs,ℓ  b,h(X) of extendible and supported  distributions analogously to (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='8)–(2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Initial value problems formalism of Einstein-Maxwell equations  In this section, we shall discuss the initial value problems formalism for Einstein-Maxwell equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' As  a general reference, we refer the readers to [9, §6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' In §3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='1, we reduce the Einstein-Maxwell equations for  (g, F) in the form (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='1)–(3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='2) to the study of the equations for (g, A) of the form (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' In §3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='2, we formulate  the initial value problems for Einstein-Maxwell equations and then write the Einstein-Maxwell equations  with respect to a generalized wave map gauge and Lorenz gauge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' In §3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='3, we discuss the linearization around  a special solution of the Einstein-Maxwell equations, as well as the linearization of the gauge-fixed Einstein-  Maxwell equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Einstein-Maxwell equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' On a time-oriented 4-dimensional Lorentzian spacetime (M, g) with  signature (−, +, +, +), the Einstein-Maxwell system, for the metric g and the electromagnetic 2-form F,  reads  Ric(g)µν = 2T (g, F)µν,  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='1)  dF = 0,  δgF = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='2)  where T (g, F) := FµαF α  ν − 1  4gµνFαβF αβ is the energy-momentum tensor associated with the electromagnetic  2-form F and δgF = −∇αFαβ is the negative divergence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' A direct calculation implies that if F solves (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='2),  then we have δgT (g, F) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  Assuming now that (g, F) is a smooth solution of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='1)–(3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='2), let i: Σ0 ֒→ M be a spacelike hypersurface  with future timelike unit normal field N, and let h = g|Σ0 be the induced metric on Σ0 with respect ot g,  which is thus a Riemannian metric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' The electromagnetic initial data on Σ0 are the magnetic field B and  electric field E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' The 1-form B is the form induced on Σ0 by the electromagnetic field F, while E is the  1-form on Σ0 relative to the future timelike unit normal N to Σ0 in the spacetime metric g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Specifically, the  electric and magnetic fields E, B, as measured by observers with 4-velocity N, are defined by  E := −i∗ιNF = ⋆hi∗ ⋆g F,  B := i∗ιN ⋆g F = ⋆hi∗F  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='3)  where ⋆ is the Hodge star operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  LINEAR STABILITY OF KERR-NEWMAN FAMILY  15  We compute for Tg,F ≡ T (g, F)  Tg,F(N, N) = 1  2(|E|2  h + |B|2  h),  Tg,F (N, ·) = ⋆h(B ∧ E)  on  T Σ0,  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='4)  where Tg,F (N, N) is the energy density of the electromagnetic field as measured by an observer with velocity  N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  We now define the following rotation of F, E, B:  Fθ := cos(θ)F + sin(θ) ⋆g F,  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='5)  (Eθ, Bθ) := (cos(θ)E − sin(θ)B, sin(θ)E + cos(θ)B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='6)  Then Eθ and Bθ are the electric and magnetic field, respectively, corresponding to Fθ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Now we establish the  following lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' If (g, F) solves the Einstein-Maxwell equations (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='1)–(3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='2), then so does (g, Fθ) for all θ ∈ R,  with Fθ defined in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' According to the facts that ⋆g⋆g = −Id and δg = ⋆gd⋆g when acting on 2-forms, the equations (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='2)  are equivalent to dF = 0, d ⋆g F = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Then it follows that  dFθ = cos(θ)dF + sin(θ)d ⋆g F = 0,  δgFθ = ⋆g(cos(θ)d ⋆g F − sin(θ)dF) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  Since the energy momentum tensor T (g, F) can be rewritten as  T (g, F)µν = 1  2gλσ�  FµλFνσ + (⋆gF)µλ(⋆gF)νσ  �  ,  it follows that T (g, Fθ)µν = T (g, F)µν.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  This proves that (g, Fθ) solves the Einstein-Maxwell equations  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='1)–(3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  □  Let us now assume for simplicity that  M ∼= Rt × Ir × S2,  Σ0 = {t = 0} ⊂ M,  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='7)  with I ⊂ R a non-empty open interval, and assume that Σ0 is spacelike.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' If S = {t = t0, r = r0} ⊂ M (with  t0 ∈ R, r0 ∈ I) is a 2-sphere, we can define the electric and magnetic charges, associated with F solving  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='2), by  Qe(g, F) := 1  4π  �  S  ⋆gF,  Qm(F) := 1  4π  �  S  F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='8)  By Stokes’ theorem and the equations (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='2) dF = 0, d ⋆g F = 0, Qe and Qm are independent of the specific  choice of S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' In particular, we can take S ⊂ Σ0, and use (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='3) and ⋆h⋆h = Id when acting on 1-form to find  that  Qe(g, F) = Qe(h, E) := 1  4π  �  S  ⋆hE = 1  4π  �  S  ⟨E, ν⟩dσ,  Qm(F) = Qm(B) := 1  4π  �  S  ⋆hB = 1  4π  �  S  ⟨B, ν⟩dσ,  with ν the outward pointing unit normal to S (with respect to h) and dσ the volume element.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  Since Ir × S2 is not a simply connected domain, dF = 0 does not imply that F = dA for some 1-form  A globally.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' In fact, the de Rham cohomology group H2  dR(Ir × S2, R) ∼= R and F �→ Qm(F) induces an  isomorphism of the de Rham cohomology group H2  dR(Ir × S2, R).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Therefore, the vanishing of Qm(F) is  equivalent to the condition that F = dA for some 1-form A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Now we will show that the vanishing of Qm(F)  can always be arranged by the rotation transformation (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' In the setting (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='7), and given any (g, F) solving (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='2), there exists θ ∈ R such that Qm(Fθ) =  0, with Fθ defined in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='5);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' for such θ, we have  Qe(g, Fθ)2 = Qm(F)2 + Qe(g, F)2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' With Fθ = cos(θ)F + sin(θ) ⋆g F, we have Qm(Fθ) = cos(θ)Qm(F) + sin(θ)Qe(g, F).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Choosing θ ∈ R  such that (cos θ, sin θ) ⊥ (Qm(F), Qe(g, F)) finishes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  □  16  LILI HE  In view of Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='2, we may assume Qm(F) = 0, equivalently F = dA for some 1-form A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' This reduces  the study of the Einstein-Maxwell equations on M in the form (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='1)–(3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='2) to the study of the following  system in (g, A)  Ric(g) = 2T (g, dA),  δgdA = 0,  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='9)  which we continue to call Einstein-Maxwell equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' In this work, we will study the Einstein-Maxwell  equations of the above form (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Initial value problems for the nonlinear system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Initial data for the Einstein-Maxwell equations  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='9) are 5-tuples  (Σ0, h, k, E, B),  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='10)  where Σ0 is a smooth 3-manifold with Riemannian metric h, k is a symmetric 2-tensor, and E, B are 1-forms  on Σ0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' In the initial value problem, one seeks a solution (M, g, A) to (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='9) with data (Σ0, h, k, E, B) in the  sense that Σ0 ֒→ M is a spacelike embedding hypersurface with respect to g, and h, k are respectively the  induced metric and second fundamental form of Σ0 with respect to g, and F = dA induces the given fields  E, B on Σ0 via (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' We define the initial data map  τ : (g, F) �→ (h, k, E, B)  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='11)  if (g, F) induces the data (h, k, E, B) at Σ0 in the above sense.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' We note that the data (h, k, E, B) cannot be  prescribed freely and they must satisfy the following constraint equations, which are obtained by evaluating  Gg(Ric(g) − 2T (g, F)) on (N, N) where Gg = Id − 1  2gtrg and N is future timelike unit normal to Σ0, and  on (N, X) with X ∈ T Σ0, and pulling back dF = 0, d ⋆g F = 0 to Σ0, and finally adding the constraint that  the magnetic charge Qm(F) = Qm(B) = 0 vanishes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  Rh − |k|2  h + (trhk)2 = 2(|E|2  h + |B|2  h),  δhk + dtrhk = 2 ⋆h (B ∧ E);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='12)  δhE = 0,  δhB = 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='13)  Qm(B) = 1  4π  �  S  ⋆hB = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='14)  The constraints (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='12)–(3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='13) are necessary and sufficient conditions for the local solvability of the Einstein-  Maxwell equations (see [9, §6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='10]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' In the setting Σ0 ∼= Ir × S2 as in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='7), by the discussion around (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='5) and Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='2, the  constraint equations are invariant upon replacing (E, B) by (Eθ, Bθ), and for suitable θ, we indeed have  Qm(Bθ) = 0, thus we can solve the Einstein-Maxwell system for (g, A) with initial data (Eθ, Bθ), and then  F := (dA)−θ yields a solution of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='1)–(3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='2) with initial data (E, B) for F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  Under the additional assumption  �  S ⋆hB = 0, one can solve the Einstein-Maxwell equations of the form  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='9) for (g, A), with electromagnetic tensor then given by F = dA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Due to the diffeomorphism invariance  of Einstein-Maxwell equations, one has the gauge freedom and thus can fix a gauge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Here, we employ the  generalized wave map gauge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Concretely, we fix a Lorentzian background metric g0 and define the gauge  1-form  ΥE(g;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' g0) := g(g0)−1δgGgg0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='15)  In local coordinates, the above gauge 1-form can be written as  ΥE(g;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' g0)α = gακgµν(Γ(g)κ  µν − Γ(g0)κ  µν).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='16)  We note that ΥE(g) ≡ 0 if and only if the identity map Id: (M, g) → (M, g0) is a wave map, and thus  ΥE(g) ≡ 0 gives rise to the wave map gauge (see [17]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' The generalized wave map gauge is defined as  �ΥE(g;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' g0) = ΥE(g;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' g0) − θ(g;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' g0) = 0  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='17)  where θ is a 1-form linear in g, satisfying that θ(g0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' g0) = 0, and independent of the derivatives of g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' We also  point out that if the background metric g0 is Minkowski metric, one obtains the generalized wave coordinates  for θ(g;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' g0) ̸= 0 (see [25]) and wave coordinates for θ(g;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' g0) = 0 (see [9, §6]), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  The system (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='9) has an additional gauge freedom, namely, Einstein-Maxwell equations are invariant  under the gauge transformation  (g, A) �→ (φ∗g, φ∗A + da).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='18)  LINEAR STABILITY OF KERR-NEWMAN FAMILY  17  for any diffeomorphism φ : M → M and smooth function a ∈ C∞(M).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' A typical method of fixing a gauge  for A is by demanding δgA = 0, called Lorenz gauge;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' it is in fact convenient to note δg = −trgδ∗  g and to  introduce the gauge function  ΥM(g, A;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' g0) := trgδ∗  g0A,  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='19)  where g0 is a fixed background metric as above;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' thus, ΥM(g, A) equals −δgA up to terms of order 0 in A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  The generalized Lorenz gauge is defined as  �ΥM(g, A;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' g0, A0) = ΥM(g, A;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' g0) − ΥM(g, A0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' g0) = 0  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='20)  where A0 is a fixed 1-form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' We claim that any solution (g, A) to the Einstein-Maxwell equations can be transformed into  another solution (φ∗g, φ∗A + da) which satisfies the generalized wave map gauge and generalized Lorenz  gauge conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Concretely, we define new coordinates (yα) by solving  �  □gyα − θ(g;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' g0)β∇βy = 0  (yα, dyα)|Σ0 = (xα, dxα)|Σ0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  Then the pull back (φ∗g, φ∗A) of (g, A) with respect to the diffeomorphism φ(y) = x satisfies the generalized  wave map gauge condition and is also a solution to Einstein-Maxwell equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Then with (φ∗g, φ∗A), one  continues to solve the equation ΥM(φ∗g, φ∗A + da;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' g0) = ΥM(φ∗g, A0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' g0), which is a linear scalar wave  equation for a, and then (φ∗g, φ∗A + da) satisfies the generalized Lorenz gauge condition and solves the  Einstein-Maxwell equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  We now consider the gauge-fixed system  P E(g, A) := Ric(g) − ˜δ∗  g �ΥE(g;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' g0) − 2T (g, dA) = 0  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='21)  P M(g, A) := δgdA − d�ΥM(g, A;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' g0, A0) = 0  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='22)  where ˜δ∗  g is zero order modification of δ∗  g, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=', ˜δ∗  g = δ∗  g + B with B ∈ C∞(M;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Hom(S2T ∗M, T ∗M)), and  (δ∗  gu)µν = 1  2(uµ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='ν + uν;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='µ) is the symmetric gradient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Now, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='21)–(3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='22) is a quasilinear hyperbolic system  for (g, A) (which is principally scalar if one multiplies the first equation by 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  In order to relate the initial value problem for the Einstein-Maxwell system for (g, A) to that of the  quasilinear hyperbolic system (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='21)–(3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='22), we need the following lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Suppose (g, A) solves the gauge-fixed Einstein-Maxwell equation  P(g, A) = (P E(g, A), P M(g, A)) = 0  and attains the given initial data (g0, g1, A0, A1) at the initial hypersurface Σ0 = {t = 0} which satisfy  the constraint equations (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='12)–(3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='14).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' If (g, A) satisfies the generalized wave map gauge and Lorenz gauge  conditions initially at Σ0, that is ,  �ΥE(g;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' g0) = 0,  �ΥM(g, A;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' g0, A0) = 0  at  Σ0 = {t = 0},  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='23)  then (g, A) solves the Einstein-Maxwell equations with the same initial data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' We first prove that if (g, A) solves P(g, A) = 0 and satisfies the constraint equations and the gauge  conditions at Σ0, then  L∂t �ΥE(g;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' g0) = 0,  L∂t �ΥM(g, A;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' g0, A0) = 0  at  Σ0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='24)  Since the constraint equation (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='12) is equivalent to evaluating Gg(Ricg − 2T (g, dA)) on (N, N) and (N, X)  on Σ0 where N is the future timelike unit normal to Σ0 and X ∈ T Σ0, then evaluating GgP M(g, A) on  (N, X) and (N, N) and using the fact that �ΥE(g, A;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' g0) = 0 at Σ0 yield L∂t �ΥE(g, A;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' g0) = 0 at Σ0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' As for  the Maxwell part, the constraint equation implies that the evaluation of δgdA on N is zero at Σ0 and thus  we have L∂t �ΥM(g, A;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' g0, A0) = 0 at Σ0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  Applying the negative divergence operator δg on the equation P M(g, A) = 0 yields  δgd�ΥM(g, A;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' g0, A0) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  18  LILI HE  Then we obtain a linear hyperbolic system for �ΥM(g, A;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' g0, A0) with vanishing initial data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  Therefore,  �ΥM(g, A;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' g0, A0) = 0 on M, which implies that (g, A) solve the Maxwell part δgdA = 0 of the Einstein-  Maxwell equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' According to the second Bianchi identity (δgGgRic(g) = 0) and the fact that δgdA = 0  (and thus δgGgT (g, dA) = δgT (g, dA) = 0), applying first the trace reversal operator Gg = Id − 1  2gtrg and  then the negative divergence operator δg on P E(g, A) = 0 yields  δgGg˜δ∗  g �ΥE(g;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' g0) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  We again obtain a linear hyperbolic system for �ΥE(g;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' g0) with vanishing initial data, and thus �ΥE(g;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' g0) = 0  on M, which implies that (g, A) solves Einstein part Ric(g)−2T (g, dA) = 0 of the Einstein-Maxwell equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  This completes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  □  Owing to Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='5, in order to solve the initial value problem for Einstein-Maxwell equations, it suffices  to construct the gauged initial data satisfying the conditions (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='23) and solve the gauge-fixed Einstein-  Maxwell equations (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='21)–(3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='22) with the constructed gauged initial data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' We postpone the construction of  the gauged initial data to §4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Initial value problems for the linearized system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' For our stability arguments, we are interested  in understanding long time behavior of a general solution to the linearize Einstein-Maxwell equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  Thus, suppose (g, F) is a solution to the Einstein-Maxwell system (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='1)–(3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='2) on a spacetime M = R×Σ0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  Then around (g, F), the linearized Einstein-Maxwell equations are defined as  DgRic(˙g) − 2Dg,FT (˙g, ˙F) := d  ds  ���  s=0  �  (Ric − 2T )(g + s˙g, F + s ˙F)  �  = 0,  d ˙F = 0,  Dg,F (δ(·)(·))(˙g, ˙F) := d  ds(δg+s ˙g(F + s ˙F))  ���  s=0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='25)  Suppose Σ0 is spacelike for g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' We denote by (˙h, ˙k, ˙E, ˙B) the linearization of the initial data map τ around  (g, F).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' We note that (˙h, ˙k, ˙E, ˙B) are the initial data we impose in the initial value problem for the linearized  Einstein-Maxwell equations (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='25).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' As in the case of the non-linear Einstein-Maxwell equations where the  initial data should satisfy the constraint equations, the initial data (˙h, ˙k, ˙E, ˙B) for the linearized Einstein-  Maxwell equations are required to satisfy the linearized constraint equations (see [9, §7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='2]), which is defined  as the linearization of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='12)–(3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='13) around the data (h, k, E, B) induced by (g, F).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  Motivated from (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='5) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='6), we induce the following transformation  ˙Fθ := cos(θ) ˙F + sin(θ) d  ds  ���  s=0  �  ⋆g+s ˙g (F + s ˙F)  �  ,  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='26)  ( ˙Eθ, ˙Bθ) := (cos(θ) ˙E − sin(θ) ˙B, sin(θ) ˙E + cos(θ) ˙B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='27)  Then we have  (h, k, ˙Eθ, ˙Bθ) = D(g,Fθ)τ(˙g, ˙Fθ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  Now we prove the analogue of Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='1 at the linearized level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' If (˙g, ˙F) solves the linearized Einstein-Maxwell equations (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='25) linearized around (g, F), then  (˙g, ˙Fθ + cFθ+ π  2 ) solves (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='25) linearized around (g, Fθ) for any c ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Since dF = 0, d  �  d  ds  ���  s=0  �  ⋆g+s ˙g (F + s ˙F)  ��  = 0, it follows that  d( ˙Fθ + cFθ+ π  2 ) = cos(θ)d ˙F + sin(θ)d  � d  ds  ���  s=0  �  ⋆g+s ˙g (F + s ˙F)  ��  + cdFθ+ π  2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  We next calculate  d  ds  ���  s=0  �  ⋆g+s ˙g  �  Fθ + s( ˙Fθ + cFθ+ π  2 )  ��  = d  ds  ���  s=0  �  ⋆g+s ˙g  �  Fθ + s ˙Fθ  ��  + c ⋆g Fθ+ π  2  = cos(θ) d  ds  ���  s=0  �  ⋆g+s ˙g (F + s ˙F)  �  − sin(θ) ˙F + c ⋆g Fθ+ π  2  and this implies  d  ds  ���  s=0  �  δg+s ˙g  �  Fθ + s( ˙Fθ + cFθ+ π  2 )  ��  = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  LINEAR STABILITY OF KERR-NEWMAN FAMILY  19  According to the calculation for T (g, F) in the proof of Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='1, with (F + s ˙F)θ = cos(θ)(F + s ˙F) +  sin(θ) ⋆g+s ˙g (F + s ˙F) we have  T (g + s˙g, F + s ˙F) = T (g + s˙g, (F + s ˙F)θ) = T (g + s˙g, Fθ + s ˙Fθ) + O(s2)  and thus  D(g,Fθ)T (˙g, ˙Fθ + cFθ+ π  2 ) = D(g,F )T (˙g, ˙F) + c  2gλσ((Fθ)µλ(Fθ+ π  2 )νσ + (Fθ+ π  2 )µλ(Fθ)νσ)  + c  2gλσ((⋆gFθ)µλ(⋆gFθ+ π  2 )νσ + (⋆gFθ+ π  2 )µλ(⋆gFθ)νσ)  = D(g,F )T (˙g, ˙F).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  This implies that (˙g, ˙Fθ + cFθ+ π  2 ) solves the linearization of the equation Ric − 2T = 0 around (g, Fθ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  □  If M ∼= R × Σ0, we can define linearized charges as follows  ˙Qe(˙g, ˙F) := 1  4π  �  S  d  ds  ���  s=0 ⋆g+s ˙g (F + s ˙F),  ˙Qm( ˙F) := 1  4π  �  S  d  ds  ���  s=0(F + s ˙F) = 1  4π  �  S  ˙F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='28)  If (˙g, ˙F) solves the linearized Einstein-Maxwell equations, by Stokes’ theorem and the linearized equations  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='25) again, ˙Qe and ˙Qm are independent of the specific choice of S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' In particular, we can take S ⊂ Σ0, and  use the linearization of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='3) around (g, F) to find that  ˙Qe(˙g, ˙F) = ˙Qe(˙h, ˙E) := 1  4π  �  S  d  ds  ���  s=0 ⋆hs Es,  ˙Qm( ˙F) = Qm( ˙B) := 1  4π  �  S  d  ds  ���  s=0 ⋆hs Bs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  We note that the analogue of Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='2 holds for the linearized charges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' In the setting (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='7), and given any (˙g, ˙F) solving the linearized Einstein-Maxwell equations  around (g, F), there exist c, θ ∈ R such that Qm(Fθ) = 0 and ˙Qm( ˙Fθ + cFθ+ π  2 ) = 0, with Fθ+ π  2 and ˙Fθ  defined in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='5) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='26), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' First, according to Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='2, there exists θ ∈ R such that Qm(Fθ) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Since  Q2  m(Fθ+ π  2 ) = Qe(g, Fθ)2 = Qm(F)2 + Qe(g, F)2 ̸= 0,  we can choose  c = −  ˙Qm( ˙Fθ)  Qm(Fθ+ π  2 ),  with which we have ˙Qm( ˙Fθ + cFθ+ π  2 ) = ˙Qm( ˙Fθ) + cQm(Fθ+ π  2 ) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' This finishes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  □  Therefore, in addition to the linearized constraint equation, we may assume that ˙Qm( ˙F) = 0 which is  an analogue of condition (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='14) and can always be arranged by Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Concretely, without loss of  generality, suppose that F satisfies Qm(F) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Then for any (˙h, ˙k, ˙E, ˙B) satisfying the linear constraint  equations, by choosing a suitable c ∈ R such that ˙Qm( ˙Bθ) − cQm(E) = 0, the condition (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='14) is arranged.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  In the context of the present paper, given initial data satisfying the linearized constraints and having  vanishing linearized magnetic charge, we solve the following linearized Einstein-Maxwell system  L (˙g, ˙A) =  �  L1(˙g, ˙A), L2(˙g, ˙A)  �  = 0,  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='29)  where  L1(˙g, ˙A) = DgRic(˙g) − 2D(g,dA)T (˙g, ˙A),  L2(˙g, ˙A) = D(g,A)(δ(·)d(·))(˙g, ˙A),  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='30)  By choosing (g, A), around which we linearize the Einstein-Maxwell equations, as the background Lorentzian  metric and 1-form in the gauge 1-form ΥE, gauge source θ and gauge function ΥM, one can consider the  initial value problem for the corresponding linearized gauge-fixed Einstein-Maxwell equations  LE(˙g, ˙A) := Dg(Ric)(˙g) − ˜δ∗  gDg  �  ΥE − θ  �  (˙g;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' g) − 2D(g,dA)T (˙g, d ˙A) = 0,  LM(˙g, ˙A) := D(g,A)(δ(·)d(·))(˙g, ˙A) − d  �  D(g,A)ΥM(˙g, ˙A;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' g) − DgΥM(˙g, A;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' g)  �  = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='31)  20  LILI HE  We recall from [84, §7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='5] and [33, §3] that  (DgRic)(˙g) = −1  2□g ˙g − δ∗  gδgGg ˙g + Rg(˙g),  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='32)  where □g = ∇α∇α is the tensor wave operator, Rg(˙g)µν = R(g)κ  λ  µν ˙gκλ + 1  2(Ric(g)κ  µ ˙gνκ + Ric(g)κ  ν ˙gµκ), and  DgΥE(˙g;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' g) = −δgGg ˙g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='33)  Since ˜δ∗  g is a zero order modification of δ∗  g and D(g,dA)T (˙g, ˙A) is of order 0 in ˙g and order 1 in A, it follows  that the first equation in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='31) has the principal part − 1  2□g ˙g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' We define the linearized generalized wave  map gauge as  Dg �ΥE(˙g;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' g) = Dg(ΥE − θ)(˙g;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' g) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='34)  Analogously, we define the linearized generalized Lorenz gauge as  D(g,A) �ΥM(˙g, ˙A;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' g, A) = D(g,A)ΥM(˙g, ˙A;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' g) − DgΥM(˙g, A;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' g) = −δg ˙A = 0,  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='35)  and the second equation in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='31) also has principal part (δgd+dδg) ˙A, and only involves up to first derivatives  of ˙g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Thus, the linearized system (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='31) is linear hyperbolic system for (˙g, ˙A), whose initial value problem  we are able to solve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  As discussed in the non-linear case, in order to relate the initial value problem for the linearized Einstein-  Maxwell system for (˙g, ˙A) to that of the linear hyperbolic system (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='31), we need the an analogue of Lemma  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='5 at the linearized level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Suppose (˙g, ˙A) solves the linearized gauge-fixed Einstein-Maxwell equation (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='31)  L(˙g, ˙A) = (LE(˙g, ˙A), LM(˙g, ˙A)) = 0  and attains the given initial data (˙g0, ˙g1, ˙A0, ˙A1) at the initial hypersurface Σ0 = {t = 0} which satisfy the  linearized constraint equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' If (˙g, ˙A) satisfies the linearized generalized wave map gauge and Lorenz gauge  conditions initially at Σ0, that is,  Dg �ΥE(˙g;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' g) = 0,  δg ˙A = 0  at  Σ0,  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='36)  then (˙g, ˙A) solves the Einstein-Maxwell equations with the same initial data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' The proof is completely analogous to that of Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  □  Therefore, in order to solve the initial value problem for linearized Einstein-Maxwell equations, we again  need to construct the gauged initial data satisfying the conditions (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='36).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' We postpone the construction of  the gauged initial data to §4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  Finally, we discuss the pure gauge solution generated from the invariance of Einstein-Maxwell system under  the gauge transformation introduced in the non-linear setting (g, A) �→ (φ∗g, φ∗A + da).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Linearizing around  (φ, a) = (Id, 0) the gauge transformation on a solution (˙g, ˙A) of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='29) yields the following transformation  acting on (˙g, ˙A)  (˙g, ˙A) �→ (˙g + LV g, ˙A + LV A + da),  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='37)  and the linearized Einstein-Maxwell system is invariant under the above transformation (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='37).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='Equivalently,  C∞(M;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' T ∗M) × C∞(M) ∋ (ω, a) acts on (˙g, ˙A) by the same formula via the identification V = ω♯.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' We note  that Lω♯g = 2δ∗  gω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' For notational convenience, we make the following definition:  Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' For a vector field V and a tensor T , define ˜LT V := LV T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  According to Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='4 in the non-linear setting, any solution (g, A) can be transformed into another  one satisfying the generalized wave map and Lorenz gauge conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Here we state its analogue in the  linear case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Any solution (˙g, ˙A) to the Einstein-Maxwell equations can be transformed into another solution  (˙g + LV g, ˙A + LV A + da) which satisfies the linearized generalized wave map gauge and Lorenz gauge  conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Concretely, we first solve define new coordinates (yα) by solving the equation  Dg �ΥE(˙g + LV g;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' g) = 0  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='38)  LINEAR STABILITY OF KERR-NEWMAN FAMILY  21  for the vector field V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' The equation (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='38) is a wave equation since its principal part is given by −□gV ♭.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  Then we solve  δg( ˙A + LV A + da) = 0,  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='39)  whose principal part is −□ga, for the scalar function a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' If one chooses trivial Cauchy data for V and a on  Σ0, then (˙g + LV g, ˙A + LV A + da) solves (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='29), satisfies the linearized generalized wave map and Lorenz  gauge conditions, and attains the same initial data (˙g0, ˙g1, ˙A0, ˙A1) as (˙g, ˙A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Kerr-Newman black holes  In §4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='1, we introduce the Reissner-Nordstr¨om (RN) family of subextremal, non-rotating black holes, which  is parametrized by (m0, 0, Q0) ∈ R × R3 × R with m0 > 0, |Q0| < m0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' In §4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='2, we introduce the Kerr-  Newman (KN) family of black holes with parameters (m, a, Q) ∈ R× R3 × R close to (m0, 0, Q0) and realize  this family of slowly rotating KN metric as a smooth family of stationary metrics on a fixed manifold M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' In  §4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='3, we introduce several vector bundles over M and analyze the structure of differential operators, which  will be used in the subsequent sections, near infinity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' In §4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='4, we will discuss how to construct the gauged  Cauchy data for the gauge-fixed Einstein-Maxwell system from the given initial data (h, k, E, B) and also  consider its linearized version.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' The Reissner-Nordstr¨om family.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Fix a set of parameters  b0 = (m0, 0, Q0) ∈ R × R3 × R,  m0 > 0, |Q0| < m0,  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='1)  where the parameter m0 > 0 denote the mass and Q0 denotes the charge, the Reissner-Nordstr¨om (RN)  solution to the Einstein-Maxwell equations is then described by the spherically symmetric and static RN  metric  gb0 = −µb0(r) dt2 + µb0(r)−1 dr2 + r2/g,  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='2)  where /g is the round metric on S2 and  µb0(r) = 1 − 2m0  r  + Q2  0  r2 ,  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='3)  and the RN 4-potential for the electromagnetic field  ˘Ab0 = Q0  r dt,  ˘Fb0 = d ˘Ab0 = −Q0  r2 dr ∧ dt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='4)  (We reserve the name Ab0 for the 4-potential that we eventually use, see (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='23).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=') When |Q0| < m0, the  expression µb0 has two distinct positive roots m0 −  �  m2  0 − Q2  0 = rc < rb0 = m0 +  �  m2  0 − Q2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' There is a  Cauchy horizon at r = rc and an event horizon H at rb0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' The expressions (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='2) and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='4) are valid in the  static region, which sometimes is also called exterior region or domain of outer communication  M = Rt × X,  X = (rb0, ∞)r × S2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='5)  The singularity of the expression (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='2) at r = rb0 is a coordinate singularity and can be removed by change  of coordinates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' We define a new new coordinate system (t0, r, θ, ϕ) which is called incoming Eddington-  Finkelstein coordinates and make use of the function t0  t0 = t + rb0,∗,  drb0,∗  dr  =  1  µb0  ,  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='6)  where rb0,∗ is called the tortoise coordinate or Regge-Wheeler radial coordinate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' In this coordinate system,  the RN metric is given by  gb0 = −µb0(r)dt2  0 + 2 dt0dr + r2/g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='7)  Now the expression (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='7) for gb0 extends analytically beyond the event horizon H = {r = rb0}, and thus gb0  is smooth and non-degenerate metric on the extended manifold  M = Rt0 × X ⊃ M,  X = [r−, ∞)r × S2 ⊃ X,  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='8)  where r− ∈ (rc, rb0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' At the same time, we can take the RN 4-potential to be Q0r−1 dt0, which differs from  ˘Ab0 by an exact form d(  �  Q0r−1µ−1 dr).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  22  LILI HE  Since dt is timelike when r is sufficiently large, to capture this useful structure, we introduce another  coordinate  tχ0 := t +  �  χ0(r)  µb0(r) dr,  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='9)  where χ0(r) ∈ C∞(R) is smooth, is identically 1 for r ≤ 4rb0/3, and vanishes for r ≥ 3rb0/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Therefore, tχ0  is a smooth interpolation between rb0 near event horizon and t near infinity r = ∞ with a suitable choice of  the constant of integration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  In order to describe the waves near the future event horizon and future null infinity (and in between), we  introduce the following function  t∗ = tb0,∗ := t + rb0,∗χ0(r) − rb0,∗(1 − χ0(r)),  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='10)  where χ0(r) is defined as in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='9);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' it smoothly interpolates between t+rb0,∗ near the event horizon and t−rb0,∗  near null infinity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Later on, we will study forcing problems for wave equations of the type □g(m0,0,Q0)u = f  on the manifold  Rt∗ × X,  X = [r−, ∞)r × S2,  where f is supported in t∗ ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  Recall the definition of R3 from (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='2), now we compactify X in a similar manner  ¯X :=  �  X ⊔ ([0, r−1  b0 )ρ × S2)  �  / ∼  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='11)  where the relation ∼ identifies a point (r, ω) ∈ X = [r−, ∞)r × S2 with the point (ρ, ω) where ρ = r−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  Namely, we add the boundary {ρ = 0} ∼= S2 at infinity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Thus, ¯X = {r ≥ r−} ⊂ R3, and we let ∂− ¯X =  r−1(r−), ∂+ ¯X = ∂R3 = {ρ = 0} ⊂ ¯X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' We note that Rt∗ × ∂ ¯X has two components,  Σfin := Rt∗ × ∂− ¯X  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='12)  (which is a spacelike hypersurface inside the black hole) and Rt∗ ×∂+X (which is future null infinity, typically  denoted I +).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' See Figure 3 for the level sets of t, t0, tχ0, t∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  t  Σfin  t0  tχ0  t∗  i+  i−  i0  I +  I −  H−  H+  Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  Penrose diagram of the Reissner-Nordstr¨om metric (including future/past null  infinity I ±, the future/past event horizon H±, spacelike infinity i0, and future/past timelike  infinity i±), together with the level sets of the static time function t, of the null coordinate  t0 from (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='6), of the coordinate tχ0 from (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='9), and of the function t∗ from (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Also  shown are the boundaries Σfin = Rt∗ × ∂− ¯X and Rt∗ × ∂+ ¯X (future null infinity I +).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  LINEAR STABILITY OF KERR-NEWMAN FAMILY  23  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' The slowly rotating Kerr-Newman family.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Let b0 = (m0, 0, Q0) be a parameter of a RN black  hole.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Consider black hole parameters  b = (m, a, Q) ∈ R × R3 × R,  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='13)  with a ∈ R3 denoting the angular momentum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' If a = |a| ̸= 0, choose adapted polar coordinates (θ, ϕ) on  S2 such that ˆa = a/|a| is defined by θ = 0, and the vector field ∂ϕ generates counterclockwise rotation  around a/|a|, with R3 carrying the standard orientation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' For a = 0, adapted polar coordinates are simply  any polar coordinates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' A very convenient coordinate system for the Kerr-Newman metric is Boyer-Lindquist  coordinates (t, r, θ, ϕ) (see [5]), in which the Kerr-Newman metric takes the form  gBL  b  = −∆b  ρ2  b  (dt − a sin2 θ dϕ)2 + ρ2  b  �dr2  ∆b  + dθ2�  + sin2 θ  ̺2  b  �  a dt − (r2 + a2)dϕ  �2,  (gBL  b )−1 = −  1  ∆bρ2  b  �  (r2 + a2)∂t + a∂ϕ  �2 + ∆b  ρ2  b  ∂2  r + 1  ρ2  b  ∂2  θ +  1  ρ2  b sin2 θ (∂ϕ + a sin2 θ ∂t)2,  ∆b = r2 − 2mr + Q2 + a2,  ρ2  b = r2 + a2 cos2 θ,  a = |a|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='14)  In the Boyer-Lindquist coordinates, the KN 4-potential reads  ˘Ab = Qr  ρ2  b  (dt − a sin2 θ dϕ)  and the corresponding electromagnetic 2-form is  ˘Fb = − Q  ρ4  b  �  (r2 − a2 cos2 θ)dr ∧ (dt − a sin2 θ dϕ)  �  + Qr  ρ4  b  �  2a cosθ sin θ dθ ∧ (a dt − (r2 + a2) dϕ)  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  We note that (gBL  b , ˘Ab) is a solution to the Einstein-Maxwell equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  In the initial value problem for the Einstein-Maxwell equations, one places asymptotically flat initial data  on a Cauchy hypersurface  Σ0 ⊂ M,  which is spacelike and equal to t−1(0) where r is large.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' We introduce the following coordinates  t = t +  �  χ1(r)  �r2 + a2  ∆b  �  1 −  ∆b  r2 + a2  �  dr,  ϕ1 = ϕ +  �  χ1(r) a  ∆b  dr  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='15)  where χ1(r) is a nonnegative smooth function such that χ1(r) = 1 for r ≤ 3m and χ1(r) = 0 for r ≥ 4m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' In  the new coordinates (t, r, θ, ϕ1), gBL  b  and (gBL  b )−1 are smooth at event horizon, and furthermore  (gBL  b )−1(dt, dt) = −1 + χ2 − 1  ρ2  b  �(r2 + a2)2  ∆b  − r2 + a2�  =  \uf8f1  \uf8f4  \uf8f2  \uf8f4  \uf8f3  −1,  r ≤ 3m  ≤ −1,  3m ≤ r ≤ 4m  (gBL  b )−1(dt, dt),  r ≥ 4m  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  By choosing a proper integration constant we have t = t for r ≥ 4m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Then we impose asymptotically flat  initial data on the Cauchy hypersurface  Σ0 := {t = 0} ⊂ M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='16)  In the present paper, we will focus on parameters (m, a, Q) close to (m0, 0, Q0);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' in particular, we are  looking at slowly rotating (|a| ≪ m + |Q|) Kerr-Newman black holes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' As in the RN case, the singularity of  the expression (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='14) of the KN metric at  r = rb := m +  �  m2 − (|Q|2 + |a|2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='17)  is a coordinate singularity and can be removed by introducing new coordinates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' We let  tb,χ = t +  � r2 + a2  ∆b  χ(r) dr,  ϕb,χ = ϕ +  �  a  ∆b  χ(r) dr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='18)  24  LILI HE  where χ(r) ∈ C∞(R) is identically 1 near r ≤ m +  �  m2 − Q2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Then the metric gBL  b  and its inverse take  the form  gBL  b  = −∆b  ρ2  b  (dtb,χ − a sin2 θ dϕb,χ)2 + 2χ(dtb,χ − a sin2 θ dϕb,χ)dr  + ρ2  b  ∆b  (1 − χ2)dr2 + ρ2  b dθ2 + sin2 θ  ρ2  b  �  a dtb,χ − (r2 + a2)dϕb,χ  �2,  (gBL  b )−1 = ρ−2  b  �  ∆b∂2  r + χ2 − 1  ∆b  �  (r2 + a2)∂tχ + a∂ϕb,χ  �2  + ∂2  θ +  1  sin2 θ  �  ∂ϕb,χ + a sin2 θ∂tχ  �2  + 2χ  �  (r2 + a2)∂tχ + a∂ϕb,χ  �  ∂r  �  ,  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='19)  which is smooth and non-degenerate on the manifold Mb = Rtb,χ ×[r−, ∞)r ×S2  θ,ϕb,χ with r− defined in §4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  Correspondingly, we have  ˘Ab = Qr  ρ2  b  (dtb,χ − a sin2 θ dϕb,χ) − χQr  ∆b  dr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  Since the last term is an exact 1-form which makes no contribution to the electromagnetic 2-form, it follows  that one can define the 4-potential on Mb as  ABL  b  := Qr  ρ2  b  (dtb,χ − a sin2 θ dϕb,χ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='20)  Smooth coordinates on S2 near the poles θ = 0, π are x = sin θ cos ϕb,χ and y = sin θ sin ϕb,χ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Then one finds  that sin2 θ = x2 + y2 and  ∂ϕb,χ = −y∂x + x∂y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  ∂θ = cos θ cos ϕb,χ∂x + cos θ sin ϕb,χ∂y  and thus the smoothness of (gBL  b )−1 near the poles θ = 0, π (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='e, x = y = 0) follows from writing ∂2  θ +  sin−2 θ∂2  ϕb,χ as  ∂2  θ +  1  sin2 θ∂2  ϕb,χ = (1 − x2 − y2)  �  ∂2  x + ∂2  y  �  − (x∂x + y∂y)2,  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='21)  which is smooth near x = y = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Since the volume form is given by  dvolgBL  b  = (r2 + a2(1 − x2 − y2))(1 − x2 − y2)−1/2 dtb,χ dr dx dy  and thus smooth at the poles θ = 0, π, we conclude that gBL  b  indeed extends smoothly and non-degenerately  to the poles θ = 0, π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  For b = (m, a, Q) close to (m0, 0, Q0), we take χ = χ0 as defined in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='9) and choose suitable constants  of integration such that tb,χ0 ≡ t, ϕb,χ0 ≡ ϕ for r ≥ 3rb0/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Then we define the following diffeomorphism  Φb : M = Rtχ0 × X = Rtχ0 × [r−, ∞)r × S2  θ,ϕ → Mb = Rtb,χ0 × [r−, ∞)r × S2  θ,ϕb,χ0,  (tb,χ0, r, θ, ϕb,χ0)(Φb(p)) = (tχ0, r, θ, ϕ)(p)  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='22)  where tχ0 is defined in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='9) and (θ, ϕ) denote polar coordinates on S2 adapted to a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Using this family of  diffeomorphisms Φb, we define  gb = (Φb)∗(gBL  b ) ∈ C∞(M;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' S2T ∗M),  Ab = (Φb)∗(ABL  b ) ∈ C∞(M;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' T ∗M)  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='23)  which is a family of smooth and stationary metric and 1-form on M, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' For b = b0 = (m0, 0, Q0),  this pullback indeed gives rise to the RN black hole (gb0, Ab0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Following the argument in [42, Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='5],  one can prove that (gb, Ab) depends smoothly on b when b is close to b0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Let Ub0 is a sufficiently small neighborhood of b0 = (m0, 0, Q0) and let M = Rtχ0 × X with  X = {p ∈ R2 | |p| ≥ r−}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Then (gb, Ab) ∈ C∞(M;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' S2T ∗M ⊕ T ∗M) depends smoothly on b ∈ Ub0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Given (m, a, Q) with a = |a| ̸= 0, let us denote the spherical coordinate system with north pole θ = 0  at a/|a| by (θb, ϕb), so the pullbacks of the functions ∆b, ρ2  b to M are simply obtained by replacing θ by θb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  Then, if (r, θb, ϕb) are the polar coordinates of a point p ∈ X, we have  r = |p|,  a cosθb = ⟨a, p  |p|⟩,  a2 sin2 θb = |a|2 − a2 cos2 θb  LINEAR STABILITY OF KERR-NEWMAN FAMILY  25  where | · | and ⟨·, ·⟩ denote the Euclidean norm and inner product on X ⊂ R3, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Since r ̸= 0, this  shows that r, a cos θb and a2 sin2 θb, and thus the pullbacks of ∆b and ρb are smooth (in b) families of smooth  functions on M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Then the smoothness of gb and Ab in b follows from that of a sin2 θbdϕb = (ar−2∂ϕb)♭ (using  the musical isomorphism on Euclidean R3), which we will show subsequently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  We now prove the smooth dependence of the inverse metric g−1  b : in view of the expression (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='19) and the  discussion around (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='21), all we need to show is that the vector fields ∂tχ0 , ∂r, a∂ϕb and a sin θb∂θb depend  smoothly on a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  For ∂tχ0 and for the radial vector field ∂r = |p|−1p∂p, which do not depend on b, the  smoothness is clear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Further, we have  a∂ϕb = ∇a×p  at  p ∈ X,  and the expression on the right-hand side is smooth in a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Indeed, if a = a⃗e3 := (0, 0, a), both sides equal  a(x∂y − y∂x) on R3  x,y,z, and if a ∈ R3 is any given vector and R ∈ SO(R3) is a rotation with Ra = a⃗e3,  a = |a|, then (R∗(a∂ϕb))|p = (a∂ϕa⃗e3,)|R(p) which we just observed to be equal to ∇a ⃗e3×R(p) = R∗∇a×p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  In a similar manner, we have  a sin θb∂θb = |p|−1∇p×(p×a)  at  p ∈ X,  and again the expression on the right-hand side is smooth in a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' This finishes the proof of the Lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  □  Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' We consider the enlarged parameter space  Ub0,m = {(m, a, Qe, Qm)} ⊂ R × R3 × R × R;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='24)  we take Ub0,m to be a small neighborhood of the parameters  b0,m = (m0, 0, Q0,e, Q0,m)  of a RN spacetime with magnetic charge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' We define the map  β : Ub0,m ∋ (m, a, Qe, Qm) �→ (m, a,  �  Q2e + Q2m) ∈ Ub0  and then for bm = (m, a, Qe, Qm) ∈ Bb0,m, we define  gbm := gβ(bm),  Fbm := QeF(m,a,1) + Qm ⋆gbm F(m,a,1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='25)  In view of Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='1, we see that (gbm, Fbm) is a solution of the Einstein–Maxwell system (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='1)–(3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Since  the charge always appears in the second power in the KN metric, one concludes that gbm and Fbm depend  smoothly on bm ∈ Bb0,m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' The diffeomorphism used to realize the family of KN black holes as a smooth family of black  holes on a fixed manifold is not unique, and in fact another presentation is more convenient for calculations  (especially near event horizon) later on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Namely, we also consider coordinates tb,1, ϕb,1 (that is, take χ ≡ 1  in the definitions of tb,χ and ϕb,χ) and use the following diffeomorphism  Φ0  b : M = Rt0 × [r−, ∞)r × S2  θ,ϕ → Mb = Rtb,1 × [r−, ∞)r × S2  θ,ϕb,1,  (tb,1, r, θ, ϕb,χ0)(Φ0  b(p)) = (t0, r, θ, ϕ)(p)  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='26)  to define  g0  b = (Φ0  b)∗(gBL  b ) ∈ C∞(M;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' S2T ∗M),  A0  b = (Φ0  b)∗(ABL  b ) ∈ C∞(M;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' T ∗M)  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='27)  Again, (g0  b, A0  b) depends smoothly on b when b is close to b0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' for b = b0 = (m0, 0, Q0), the above pullback  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='27) produces the RN black hole (gb0, Ab0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' For b = (m, 0, Q), we have  g0  (m,0,Q) = −µ(m,0,Q)dt2  0 + 2dt0 dr + r2/g,  A0  (m,0,Q) = Q  r dt0,  µ(m,0,Q) = 1 − 2m  r  + Q  r2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='28)  For b = (m, a, Q) close to b0 = (m0, 0, Q0), we also introduce the following function  tb,∗ := t + rb0,∗χ0(r) − r(m,0,Q),∗(1 − χ0(r)),  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='29)  which generalizes t∗ = tb0,∗ defined in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='10);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' it equals t − r(m,0,Q),∗ for r ≥ 3rb0/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' In particular, in the  region r ≥ 3rb0/2 where tχ0 = t, we have for b = (m, 0, Q)  g(m,0,Q) = −µ(m,0,Q) dt2 + µ−1  (m,0,Q) dr2 + r2/g = −µ(m,0,Q) dt2  b,∗ − 2dtb,∗ dr + r2/g,  A(m,0,Q) = Q  r dt = Q  r dtb,∗ + Q  r∆b  dr,  26  LILI HE  and then dtb,∗ is null for large r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  Given the smooth families (gb, Ab), now we can define linearized KN solutions:  Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' For ˙b ∈ R × R3 × R, we define the linearized KN solutions as  ˙gb(˙b) = Dbgb(˙b) = d  ds  ���  s=0gb+s˙b,  ˙Ab(˙b) = DbAb(˙b) = d  ds  ���  s=0Ab+s˙b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='30)  Thus, (˙gb(˙b), ˙Ab(˙b)) is a solution of the linearized Einstein-Maxwell system (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='29) linearized around  (gb, Ab).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  We can also linearize the family (g0  b, A0  b) in the parameter b to obtain another version of linearized KN  solutions  ˙g0  b(˙b) = Dbg0  b(˙b) = d  ds  ���  s=0g0  b+s˙b,  ˙A0  b(˙b) = DbA0  b(˙b) = d  ds  ���  s=0A0  b+s˙b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='31)  We record the particular case  ˙g0  (m0,0,Q0)( ˙m, 0, ˙Q) =  �2 ˙m  r  − 2Q0 ˙Q  r2  �  dt2  0,  ˙A0  (m0,0,Q0)( ˙m, 0, ˙Q) =  ˙Q  r dt0  ˙g0  (m0,0,Q0)(0, ˙a, 0) = ((−4m0  r  + 2Q0  r2 )dt0 − 2dr) sin2 θ dϕ,  ˙A0  (m0,0,Q0)(0, ˙a, 0) = −Q  r sin2 θdϕ,  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='32)  where in the second line |˙a| = 1, and (θ, ϕ) are spherical coordinates adapted to ˙a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Since gb = ((Φ0  b)−1 ◦ Φb)∗g0  b, the linearized KN solution (˙gb(˙b), ˙Ab(˙b)) can be obtained from  (˙g0  b(˙b), ˙g0  b(˙b)) by subtracting off a Lie derivative of (g0  b, A0  b) along a vector field generating the flow (Φ0  b)−1◦Φb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  More precisely, in the coordinates (t∗, r, θ, ϕ) on M, we have  (Φ0  b)−1 ◦ Φb : (t∗, r, θ, ϕ) �→  �  t∗ +  � �r2 + a2  ∆b  − r2  ∆b0  �  (1 − χ0) dr, r, θ, ϕ +  �  a  ∆b  (1 − χ0) dr  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='33a)  In particular, for ˙b = ( ˙m, ˙a, ˙Q), the two versions of the linearization of the KN solution at gb0 are related by  ˙gb0(˙b) = ˙g0  b0(˙b) − LV (˙b)g0  b0,  ˙gb0(˙b) = ˙g0  b0(˙b) − LV (˙b)A0  b0,  V (˙b) = d  ds  ����  s=0  ((Φ0  b0+s˙b)−1 ◦ Φb0+s˙b)  =  �  ˙m  �� r  r0  2r3  ∆2  b0  (1 − χ0) dr  �  − ˙Q  �� r  r0  2Q0r2  ∆2  b0  (1 − χ0) dr  ��  ∂t∗ + ˙a  �� r  r0  1 − χ0  ∆b0  dr  �  ∂ϕ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='33b)  where r0 in the definition of V (˙b), can be chosen arbitrarily.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Stationarity, vector bundles, and geometric operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Now we introduce some basics of the  vector bundles and geometric operators (see [36, §3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='3]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' In the notation (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='8), denote the projection to the  spatial manifold by  πX : M → X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  this is independent of the choice of time function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Suppose that E1 → X is a vector bundle;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' then differenti-  ation along ∂t = ∂t0 = ∂t∗ is a well-defined operation on sections of the pullback bundle π∗  XE1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' The tangent  bundle of M is an important example of such a pullback bundle, as  T M ∼= π∗  X(Tt−1  0  (0)M) ∼= π∗  X(Tt−1  ∗  (0)M),  likewise for the cotangent bundle and other tensor bundles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  Let E2 → X be another vector bundle, and suppose that �L(0) ∈ Diff(X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' E1, E2) is a differential operator;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  fixing t = t∗ + F, F ∈ C∞(X), with dt ̸= 0 everywhere, we can then define its stationary extension by  assigning to u ∈ C∞(M;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' π∗  XE1) the section (Lu)(t, −) := �L(0)(u(t, −)) of π∗  XE2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' this extension does depend  on the choice of t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' The action of L on stationary functions on the other hand is independent of the choice of  t since  Lπ∗  X = π∗  X �L(0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='34)  Via stationary extension, one can consider Diffb( ¯X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' E) (for E → ¯X a smooth vector bundle down to ∂+ ¯X)  to be a subalgebra of Diff(M;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' π∗  XE);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' likewise Diffsc( ¯X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' E) ֒→ Diff(M;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' π∗  XE).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  LINEAR STABILITY OF KERR-NEWMAN FAMILY  27  Conversely, if L ∈ Diff(M;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' π∗  XE1, π∗  XE2) is stationary, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' commutes with ∂t, there exists a unique (in-  dependent of the choice of t) operator �L(0) ∈ Diff(X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' E1, E2) such that the relation (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='34) holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' More  generally, we can consider the formal conjugation of L by the Fourier transform in t,  �L(σ) := eiσtLe−iσt ∈ Diff(X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' E1, E2),  where we identify the stationary operator eiσtLe−iσt with an operator on X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Switching from t to another  time function, t + F ′, F ′ ∈ C∞(X), amounts to conjugating �L(σ) by eiσF ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  In order to describe the uniform behavior of geometric operators at ∂+X concisely, we need to define  a suitable extension of T ∗M to ‘infinity’.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' To accomplish this, note that the product decomposition (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='8)  induces a splitting  T ∗M ∼= T ∗Rt0 ⊕ T ∗X = π∗  T (T ∗Rt0) ⊕ π∗  X(T ∗X),  where πT : M0 → Rt0 is the projection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' We therefore define the extended scattering cotangent bundle of ¯X  by  �  scT ∗ ¯X := Rdt0 ⊕ scT ∗ ¯X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='35)  At this point, dt0 is merely a name for the basis of a trivial real rank 1 line bundle over ¯X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' considering the  pullback bundle π∗  X �  scT ∗ ¯X → M, we identify it with the differential of t0 ∈ C∞(M ◦), giving an isomorphism  (π∗  X �  scT ∗ ¯X)|M ∼= T ∗M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='36)  Smooth sections of �  scT ∗ ¯X → ¯X are linear combinations, with C∞( ¯X) coefficients, of dt0 and the 1-forms dxi,  where (x1, x2, x3) are standard coordinates on X ⊂ R3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' One can switch to another time function in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='35),  say, t∗, by writing dt∗ = dt0 + (dt∗ − dt0), with the second term being a smooth scattering 1-form on ¯X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  likewise for tχ0 and also for the static time t in r > rb0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  For a stationary metric g on M, there exists a unique g′ ∈ C∞(X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' S2 �  scT ∗ ¯X) such that π∗  Xg′ = g, namely  g′ is the restriction (as a section of S2 �  scT ∗ ¯X) of g to any transversal of πX, such as level sets of t0, tχ0, t∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  Identifying g with g′ and applying this to the Kerr-Newman family, we then have  gb, g0  b ∈ C∞( ¯X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' S2 �  scT ∗ ¯X);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='37)  they are non-degenerate down to ∂+X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Moreover, we have  gb − g ∈ ρC∞,  g := −dt2  χ0 + dr2 + r2/g,  g(m,a,Q) − g(m,0,Q) ∈ ρ2C∞( ¯X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' S2 �  scT ∗ ¯X),  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='38)  i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' a KN metric equals the Minkowski metric g to leading order, and is a O(ρ2) perturbation of the RN  metric of the same mass and charge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  We proceed to discuss basic geometric operators on Kerr-Newman spacetimes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' We write  (δ∗  gω)µν = 1  2(∇νωµ + ∇µων),  (δgh)α = −gµν∇µhνα,  Gg = Id − 1  2g trg,  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='39)  and furthermore denote by  □g,0, □g,1, □g,2,  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='40)  the wave operator gµν∇µ∇ν on scalar functions, 1-forms, and symmetric 2-tensors, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' When the  bundle is clear from the context, we shall simply write □g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' We use the splitting  �  scT ∗ ¯X = ⟨dt⟩ ⊕ scT ∗ ¯X,  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='41)  S2 �  scT ∗ ¯X = ⟨dt2⟩ ⊕ (2dt ⊗s scT ∗ ¯X) ⊕ S2 scT ∗ ¯X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='42)  and work in the trivialization of scT ∗ ¯X, resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' S2 scT ∗ ¯X given by dxi, 1 ≤ i ≤ 3, resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' 2dxi ⊗s dxj, 1 ≤ i ≤  j ≤ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Working in the splittings (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='41) and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='42) and the trivializations of scT ∗ ¯X and S2 scT ∗ ¯X,  we write the operators □gb,2, □gb,1, □gb,0 as  □gb,0 = g−1  b (dtχ0, dtχ0)∂2  tχ0 + �  □gb,0(0) + Q1  b,0∂tχ0 ,  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='43)  □gb,1 = □gb,0 ⊗ Id4×4 + Q1  b,1∂tχ0 + Q0  b,1,  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='44)  □gb,2 = □gb,0 ⊗ Id10×10 + Q1  b,2∂tχ0 + Q0  b,2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='45)  28  LILI HE  Then we have  �  □gb,0(0) ∈ ρ2Diff2  b( ¯X),  Q1  b,0 ∈ ρ3Diff1  b( ¯X),  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='46)  Q1  b,1 ∈ ρ2Diff0  b( ¯X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' �  scT ∗ ¯X),  Q0  b,1 ∈ ρ3Diff1  b( ¯  X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' �  scT ∗ ¯X),  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='47)  Q1  b,2 ∈ ρ2Diff0  b( ¯X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' S2 �  scT ∗ ¯X),  Q0  b,2 ∈ ρ3Diff1  b( ¯  X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' S2 �  scT ∗ ¯X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='48)  Moreover,  �  □gb,j(0) − �  □g,j(0) ∈ ρ3Diff2  b( ¯X),  g = −dt2  χ0 + dx2,  j = 0, 1, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='49)  Since the metric components are smooth away from ∂+ ¯X, it suffices to analyze □g,•, • = 0, 1, 2 near spatial  infinity ∂+ ¯X where tχ0 ≡ t is the static time coordinate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='6 is a consequence of the following  lemma (by taking h = dx2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Suppose g is a stationary Lorentzian metric on M for which  g(∂t, ∂t) ∈ −1 + ρC∞( ¯X),  g(∂t, −) ∈ ρ2C∞( ¯X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' scT ∗ ¯X),  g|scT ¯  X×scT ¯  X ∈ h + ρC∞( ¯X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' S2 �  scT ∗ ¯X),  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='50)  where h ∈ C∞( ¯X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' S2 �  scT ∗ ¯X) is Riemannian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Then the operator □g,2, □g,1, □g,0 satisfy the statements in  Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='6 with Q0  b,j ∈ ρ3Diff1  b( ¯X) replaced by Q0  b,j ∈ ρ2Diff1  b( ¯X) for j = 1, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  Moreover, �  □g,j(0) mod ρ3Diff2  b( ¯X) only depends on h for j = 0, 1, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' We introduce coordinates  x = (x0, x1, x2, x3),  where x0 = t, and x1, x2, x3 are standard coordinates on R3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' We use Latin letters i, j, k, · · · for indices from  1 to 3, and Greek letters α, β, µ, ν, · · · for indices from 0 to 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' We write Ok := ρkC∞( ¯X) and write f = Ok  in the spirit of O-notation instead of f ∈ Ok when f is a smooth function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  We first compute the Levi-Civita connection of g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Using the facts that g is stationary, ∂i ∈ ρVb(X) and  g−1(dt, dt) ∈ −1 + ρC∞( ¯X),  g−1(dt, −) ∈ ρ2C∞( ¯X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' scT ¯X),  g−1|scT ∗ ¯  X×scT ∗ ¯  X ∈ h−1 + ρC∞( ¯X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' S2 scT ¯X),  we find that  Γ0  00 = O4,  Γi  00 = O2,  Γ0  0i = O2,  Γj  0i = O3,  Γ0  ij = O3,  Γk  ij = Γk  ij(h) + O2,  Γk  ij(h) = O1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='51)  Since □g,0 = gαβ∂α∂β − gαβΓµ  αβ∂µ where gαβ is the inverse to g, we see that  □g,0 = gαβ∂α∂β + f µ∂µ,  f 0 = O3, f i = O1,  and thus we can write  □g,0 = g00∂2  t + Q∂t + �  □g,0(0)  with  Q ∈ ρ3Diff1  b( ¯X), �  □g,0(0) ∈ ρ2Diff2  b( ¯X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  Moreover, �  □g,0(0) mod ρ3Diff2  b( ¯X) only depends on h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  Next, we calculate □g,1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Since  (□g,1u)α = □g,0uα − 2gµνΓκ  µα∂νuκ + gµνΓκ  µνΓβ  καuβ + gµνΓκ  µαΓβ  κνuβ − gµν(∂µΓκ  να)uκ,  we can write  □g,1 = □g,0 ⊗ Id4×4 + Q1  g,1∂t + Q0  g,1  where Q1  g,1 ∈ ρ2Diff0  b( ¯X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' �  scT ∗ ¯X) and Q0  g,1 ∈ ρ2Diff1  b( ¯  X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' �  scT ∗ ¯X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Moreover, Q0  g,1 ∈ ρ3Diff1  b( ¯X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' �  scT ∗ ¯X) if  Γk  ij(h) vanishes, and �  □g,1(0) mod ρ3Diff2  b( ¯X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' �  scT ∗ ¯X) only depends on h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  Finally, we calculate □g,2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Since  (□g,2h)αβ = □g,0hαβ − 2gµνΓκ  µα∂νhκβ − 2gµνΓκ  µβ∂νhκα + S0  g,2  where S0  g,2 ∈ Diff0  b( ¯X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' S2 �  scT ∗ ¯X) whose elements are linear combinations of terms of the types gΓ · Γ and  g∂Γ, it follows that we can write  □g,2 = □g,0 ⊗ Id10×10 + Q1  g,2∂t + Q0  g,2  LINEAR STABILITY OF KERR-NEWMAN FAMILY  29  where Q1  g,2 ∈ ρ2Diff0  b( ¯  X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' S2 �  scT ∗ ¯X) and Q0  g,2 ∈ ρ2Diff1  b( ¯X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' S2 �  scT ∗ ¯X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Again, Q0  g,2 ∈ ρ3Diff1  b( ¯X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' S2 �  scT ∗ ¯X) if  Γk  ij(h) vanishes, and �  □g,2(0) mod ρ3Diff2  b( ¯X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' S2 �  scT ∗ ¯X) only depends on h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  □  We next discuss the curvature tensor and other geometric operators associated to gb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Let g be a metric of the form (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='50) and ˜g = −dt2 + h where h ∈ C∞( ¯  X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' S2 �  scT ∗ ¯X) is  Riemannian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Then we have  Riem(g) − Riem(˜g) ∈ ρ3C∞� ¯X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' �  scT ¯X ⊗ ( �  scT ∗ ¯X)3�  ,  Ric(g) − Ric(˜g) ∈ ρ3C∞� ¯X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' S2 �  scT ∗ ¯X  �  ,  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='52)  δg = −ιdt♯  χ0∂tχ0 + �δg(0),  �δg(0) ∈ ρDiff1  b( ¯X),  �δg(0) − �δ˜g(0) ∈ ρ2Diff1  b( ¯X),  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='53)  δ∗  g = dtχ0 ⊗s ∂tχ0 + �δ∗g(0),  �δ∗g(0) ∈ ρDiff1  b( ¯X),  �δ∗g(0) − �δ∗  ˜g(0) ∈ ρ2Diff0  b( ¯X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='54)  In particular, these hold for g = gb and ˜g = g = −dt2  χ0 + dx2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Since  Γ0  00(˜g) = Γi  00(˜g) = Γ0  0i(˜g) = Γj  0i(˜g) = Γ0  ij(˜g) = 0,  Γk  ij(˜g) = O1,  Γµ  αβ(g) = Γµ  αβ(˜g) + O2,  and g, ˜g are stationary, the proposition follows from the formulas  Riem(g)ν  µαβ = ∂αΓν  βµ(g) − ∂βΓν  αµ(g) + Γν  ακ(g)Γκ  βµ(g) − Γν  βκ(g)Γκ  αµ(g),  δgu = −gµν∂µuν + gµνΓκ  µν(g)uκ,  (δgh)α = −gµν∂µhνα + gµνΓκ  µν(g)hκα + gµνΓκ  µα(g)hνκ,  (δ∗  gu)αβ = 1  2(∂αuβ + ∂βuα) − Γκ  αβ(g)uκ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  □  In the context of (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='38), it is useful to record the following strengthening of the leading order control:  Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' If g1, g2 are two metrics of the form (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='50) and so that in addition g1−g2 ∈ ρ2C∞(X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' S2 �  scT ∗ ¯X),  then  �  □g1,j(0) − �  □g2,j(0) ∈ ρ4Diff2  b( ¯  X),  j = 0, 1, 2,  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='55)  Riem(g2) − Riem(g1) ∈ ρ4C∞� ¯X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' �  scT ¯X ⊗ ( �  scT ∗ ¯X)3�  ,  Ric(g2) − Ric(g1) ∈ ρ4C∞� ¯X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' S2 �  scT ∗ ¯X  �  ,  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='56)  �  δg1(0) − �  δg2(0) ∈ ρ3Diff1  b( ¯X),  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='57)  �  δ∗g1(0) − �  δ∗g2(0) ∈ ρ3C∞( ¯X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Hom( �  scT ∗ ¯X, S2 �  scT ∗ ¯X)),  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='58)  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' The proof follows from the facts that Γµ  αβ(g1) = O1 and Γµ  αβ(g2) − Γµ  αβ(g1) = O3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  □  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Relevant geometric operators in linearized gauge-fixed Einstein-Maxwell equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' We now study the  linearized gauge-fixed Einstein-Maxwell operator arising from the generalized wave map and Lorenz gauge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Let B ∈ C∞(M;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Hom(T ∗M, S2T ∗M)), F ∈ C∞(M;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Hom(S2T ∗M, T ∗M)) and g be a  Lorentzian metric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Then we define the modified symmetric gradient �δ∗  g,B and modified negative divergence  �δg,F as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  �δ∗  g,B = δ∗  g + Bg,  �δg,F = δg + Fg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='59)  In this paper, we will use Bg, Fg of the following form  Bg = B(g;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' c, γ) := 2γc ⊗s (•) − 1  2γg−1(c, •)g,  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='60)  Fg = F(g;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' c, γ) := 2γιg  c(•) − 1  2γctrg(•).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='61)  where γ ∈ R is a sufficiently small constant to be determined later on, c ∈ C∞  c (X, �  scT ∗ ¯X) is a stationary  1-form with compact support and ιg  c(h) = gµνcνhµα.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' With the above choices of Bg and Fg in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='60) and  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='61), we find that �δ∗  g,B and �δg,F are formally adjoint to each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' From now on, we denote  �δ∗  g,γ := �δ∗  g,B = δ∗  g + Bg,  �δg,γ := �δg,F = δg + Fg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='62)  30  LILI HE  By choosing ˜δ∗  gb = �δ∗  gb,γ and θ(g;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' gb) = F(gb;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' c, γ)[Ggbg], we obtain the linearized gauge-fixed Einstein-  Maxwell system Lb,γ(˙g, ˙A) = (2LE  b,γ(˙g, ˙A), LM  b,γ(˙g, ˙A)) = 0 around (gb, Ab)  LE  b,γ(˙g, ˙A) := Dgb(Ric)(˙g) − ˜δ∗  gb,γDgb �ΥE(˙g;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' gb) − 2D(gb,dAb)T (˙g, d ˙A),  �ΥE = ΥE − θ,  LM  b,γ(˙g, ˙A) := D(gb,Ab)(δ(·)d(·))(˙g, ˙A) − d  �  D(gb,Ab) �ΥM(˙g, ˙A;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' gb, Ab)  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='63)  According to the calculation (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='32)–(3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='35), (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='1) and (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='4), we find that  LE  b,γ(˙g, ˙A) = −1  2□gb,2 ˙g + (˜δ∗  gb,γ − δ∗  gb)δgbGgb ˙g + δ∗  gb(˜δgb,γ − δgb)Ggb ˙g  + (˜δ∗  gb,γ − δ∗  gb)(˜δgb,γ − δgb)Ggb ˙g + Rgb(˙g)µν − 2Dgb,dAbT (˙g, d ˙A),  LM  b,γ(˙g, ˙A) = (dδgb + δgbd) ˙A − (δgbGgb ˙g)κ(dAb)κµ + 1  2(∇ν ˙gκ  µ − ∇κ ˙gν  µ)(dAb)νκ + ˙gνα∇α(dAb)νµ  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='64)  where dδgb + δgbd = −□gb,1 + Ric(gb) and  (Rgb ˙g)µν = Riem(gb)α  β  µν ˙gαβ + 1  2  �  Ric(gb) κ  µ ˙gνκ + Ric(gb) κ  ν ˙gµκ  �  ,  Dgb,dAbT (˙g, d ˙A) =  �  (gb)αβ(d ˙A)µα(dAb)νβ + (gb)αβ(dAb)µα(d ˙A)νβ  �  − 1  2(gb)µν(d ˙A)αβ(dAb)αβ  − ˙gαβ(dAb)µα(dAb)νβ − 1  4  �  ˙gµν(dAb)αβ(dAb)αβ−2(gb)µν ˙gκα(dAb)αβ(dAb) β  κ  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='65)  We note that the first two terms of the second line of LE  b,γ(˙g, ˙A), the last two terms of Dgb,dAbT (˙g, d ˙A) and  the last term of LM  b,γ(˙g, ˙A) contain no derivatives of (˙g, ˙A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  For later use, we also define the gauge propagation operator Pb,γ and the gauge potential wave operator  Wb,γ for the gauge 1-form �ΥE(˙g;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' gb):  Pb,γ := δgbGgb ˜δ∗  gb,γ = −1  2□gb,1 − 1  2Ric(gb) + δgbGgb(˜δ∗  gb,γ − δ∗  gb),  Wb,γ := ˜δgb,γGgbδ∗  gb = −1  2□gb,1 − 1  2Ric(gb) + (˜δgb,γ − δgb)Ggbδ∗  gb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='66)  Then we have  Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Let Lb,γ = (2LE  b,γ, LM  b,γ), Pb,γ, Wb,γ be defined as above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Then we have  Pb,γ = −1  2□gb,1 + P 1∂tχ0 + P 0,  �  Pb,γ(0) = −1  2  �  □g,1(0) + P 0,  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='67)  Wb,γ = −1  2□gb,1 + W 1∂tχ0 + W 0,  �  Wb,γ(0) = −1  2  �  □g,1(0) + W 0,  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='68)  where  P 1, W 1 ∈ ρ∞Diff0  b( ¯X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' �  scT ∗ ¯X),  P 0, W 0 ∈ ρ4Diff1  b( ¯X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' �  scT ∗ ¯X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='69)  We also have  Lb,γ = −□gb,0 ⊗ Id14×14 + L1∂tχ0 + L0,  �  Lb,γ(0) = − �  □g,0(0) ⊗ Id14×14 + L0,  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='70)  where  L1 ∈ ρ2Diff0  b( ¯  X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' S2 �  scT ∗ ¯X ⊕ �  scT ∗ ¯X),  L0 ∈ ρ3Diff1  b( ¯X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' S2 �  scT ∗ ¯X ⊕ �  scT ∗ ¯X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='71)  Moreover, we have  − �  Lb,γ(0) − �  □g,0(0) ⊗ Id14×14 ∈ ρ3Diff2  b( ¯X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' S2 �  scT ∗ ¯X ⊕ �  scT ∗ ¯X),  g = −dt2  χ0 + dx2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='72)  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Since ˜δ∗  gb,γ − δ∗  gb and ˜δgb,γ − δgb are compactly supported and Ric(gb) = O4, then the conclusions  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='67)–(4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='69) follow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  According to (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='65) and using the facts that Riem(gb) = O3, dAb = O2, we see that Rgb ∈ ρ3Diff0  b and  D(gb,dAb)T = T 1∂tχ0 + T 0 with  T 1 ∈ ρ2Diff0  b( ¯X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' S2 �  scT ∗ ¯X ⊕ �  scT ∗ ¯X),  T 0 ∈ ρ3Diff1  b( ¯X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' S2 �  scT ∗ ¯X ⊕ �  scT ∗ ¯X),  and thus  2LE  b,γ(˙g, ˙A) = −□gb,2 ˙g + LE,1∂tχ0 + LE,0  LINEAR STABILITY OF KERR-NEWMAN FAMILY  31  with  LE,1 ∈ ρ2Diff0  b( ¯X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' S2 �  scT ∗ ¯X ⊕ �  scT ∗ ¯X),  LE,0 ∈ ρ3Diff1  b( ¯X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' S2 �  scT ∗ ¯X ⊕ �  scT ∗ ¯X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  According to the expression for LE  b,γ(˙g, ˙A) in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='64) and using Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='8, we find that  Lb,γ(˙g, ˙A) = −□gb,1 ˙A + LM,1∂tχ0 + LM,0  where  LM,1 ∈ ρ2Diff0  b( ¯X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' S2 �  scT ∗ ¯X ⊕ �  scT ∗ ¯X),  LM,0 ∈ ρ3Diff1  b( ¯X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' S2 �  scT ∗ ¯X ⊕ �  scT ∗ ¯X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  Combining the above conclusions for LE  b,γ and LM  b,γ and using the description of □gb,1, □gb,2 in Proposition  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='6, we obtain (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='70)–(4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='72).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  □  Now we let  �  Lb,γ(σ) = eitb,∗σLb,γe−itb,∗σ,  �  Pb,γ(σ) = eitb,∗σPb,γe−itb,∗σ,  �  Wb,γ(σ) = eitb,∗σWb,γe−itb,∗σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='73)  In order to study the property of �  Lb,γ(σ), �  Pb,γ(σ), �  Wb,γ(σ), we first express Lb,γ, Pb,γ, Wb,γ in the coordinates  (tb,∗, xi), i = 1, 2, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' In the coordinates (tb,∗, xi), i = 1, 2, 3, the operator □gb,0 take the form  □gb,0 = Q2∂2  tb,∗ + Q1∂tb,∗ + �  □gb,0(0)  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='74)  where Q2 ∈ ρ2C∞( ¯X), Q1 ∈ 2ρ2∂ρ − 2ρ + ρ3Diff1  b( ¯  X) and �  □gb,0(0) ∈ ρ2Diff2  b( ¯X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  Moreover, the operators Pb,γ, Wb,γ, Lb,γ have the form  Pb,γ = −1  2□gb,0 ⊗ Id4×4 + ˜P 1∂tb,∗ + ˜P 0,  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='75)  Wb,γ = −1  2□gb,0 ⊗ Id4×4 + ˜W 1∂tb,∗ + ˜W 0,  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='76)  Lb,γ = −□gb,0 ⊗ Id14×14 + ˜L1∂tb,∗ + ˜L0,  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='77)  where  ˜P 1, ˜W 1, ˜L1 ∈ ρ2Diff0  b( ¯X),  ˜P 0, ˜  W 0, ˜L0 ∈ ρ3Diff0  b( ¯X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' With parameter b = (m, a, Q), near ∂+ ¯X, we have tχ0 = t and tb,∗ = t − r(m,0,Q),∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Then changing  from (tχ0, r) coordinates to (tb,∗, r) coordinates transforms ∂tχ0 , ∂r into  ∂tb,∗,  ∂r −  1  µ(m,0,Q)  ∂tb,∗  near  ∂+ ¯X,  µ(m,0,Q) = 1 − 2m  r  + Q2  r2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  Since gb − g(m,0,Q) ∈ ρ2C∞( ¯X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' S2 �  scT ∗ ¯X), using the facts that in the coordinate (t, xi),  Γκ  αβ(g(m,0,Q)) = O2,  Γκ  αβ(gb) − Γκ  αβ(g(m,0,Q)) ∈ O3,  it follows that  □gb,0 − □g(m,0,Q),0 = ρ2C∞( ¯X)∂2  tχ0 + ρ3Diff1  b( ¯X)∂tχ0 + ρ4Diff2  b( ¯  X)  and thus  □gb,0 − □g(m,0,Q),0 = ρ2C∞( ¯X)∂2  tb,∗ + ρ3Diff1  b( ¯  X)∂tb,∗ + ρ4Diff2  b( ¯  X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  Then it remains to calculate □g(m,0,Q),0 in (tb,∗, r) coordinates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Since near ∂+ ¯X  g(m,0,Q) = −µ(m,0,Q)dt2  b,∗ − 2dtb,∗dr + r2/g,  we have for ρ = 1/r  □g(m,0,Q),0 = −2∂r∂tb,∗ + µ(m,0,Q)∂2  r + 2r − 2m  r2  ∂r − 2  r ∂tb,∗ + 1  r2 /∆  = 2ρ2∂ρ∂tb,∗ − 2ρ∂tb,∗ + ρ2Diff2  b( ¯  X)  near  ∂+ ¯X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  This completes the proof for □gb,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Then the proof for Pb,γ, Wb,γ, Lb,γ follows directly from Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='6  and Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  □  Then the following proposition is a direct result of Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  32  LILI HE  Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' The operator □gb,0 take the form  �  □gb,0(σ) = −2iσρ(ρ∂ρ − 1) + �  □gb,0(0) + σQ + σ2V  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='78)  where V ∈ ρ2C∞( ¯X), Q ∈ ρ3Diff1  b( ¯X) and �  □gb,0(0) ∈ ρ2Diff2  b( ¯X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  Moreover, the operators , Pb,γ, Wb,γ, Lb,γ have the form  Pb,γ = iσρ(ρ∂ρ − 1) ⊗ Id4×4 + �  Pb,γ(0) + σQP + σ2VP ,  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='79)  Wb,γ = iσρ(ρ∂ρ − 1) ⊗ Id4×4 + �  Wb,γ(0) + σQW + σ2VW ,  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='80)  Lb,γ = 2iσρ(ρ∂ρ − 1) ⊗ Id14×14 + �  Lb,γ(0) + σQL + σ2VL,  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='81)  where  VP , VW , VL ∈ ρ2Diff0  b( ¯X),  QP , QW , QL ∈ ρ2Diff1  sc( ¯X),  �  Pb,γ(0), �  Wb,γ(0), �  Lb,γ(0) ∈ ρ2Diff2  b( ¯X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Constructing gauged initial data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' According to Lemmas 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='5 and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='8, we now show how to construct  gauged initial data for the gauge-fixed Einstein-Maxwell system (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='22) from initial data given on Σ0 which  are close to the data (hb, kb, Eb, Bb) induced by the KN black holes with parameter b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  We use the generalized wave map gauge for the Einstein part of the system,  �ΥE(g;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' gb) = ΥE(g;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' gb) − θ(g;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' gb) = gg−1  b δgGggb − θ(g;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' gb) = 0  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='82)  i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' letting the background metric in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='15) be gb, and the generalized Lorenz gauge for the Maxwell part,  �ΥM(g, A;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' gb, Ab) = ΥM(g, A;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' gb) − ΥM(g, Ab;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' gb) = trgδ∗  gbA − trgδ∗  gbAb = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='83)  We point out that for a general treatment, we will construct the gauged initial data for any θ(g;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' gb) satisfying  that θ(gb;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' gb) = 0 and θ(g;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' gb) contains no derivatives of g, without being restricted to the specific θ(g;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' gb)  we chose in §4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  In order to capture the vanishing magnetic charge, we consider for s > 2, 0 < α < 1 the subspace of the  initial data for ungauged Einstein-Maxwell equations  Zs,α :=  �  (h, k, E, B) | d ⋆h B = 0,  �  S2 ⋆hB = 0,  (h − hb, k − kb, E − Eb, B − Bb) ∈ ¯Hs+1,−1/2+α  b  (¯Σ0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' S2  >0  scT ∗¯Σ0)  × ¯Hs,1/2+α  b  (¯Σ0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' S2scT ∗ ¯Σ0) × ¯Hs,1/2+α  b  (¯Σ0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' scT ∗¯Σ0) × ¯Hs,1/2+α  b  (¯Σ0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' scT ∗¯Σ0)  �  ,  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='84)  where S2  >0  scT ∗¯Σ0 denotes the fiber bundle of positive definite inner products on scT ¯Σ0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' We remark that the  definition of the space Zs,α incorporates (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='14) and one of the constraint equations (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='13), which is necessary  since we study the Einstein-Maxwell system using the potential A rather than the electromagnetic 2-form  F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  For the initial value problems of the gauge-fixed Einstein-Maxwell system P(g, A) = 0, which is a quasi-  linear hyperbolic system, the initial data induced by (g, A) at the spacelike hypersurface Σ0 = {t = 0} is  defined as  γ0(g, A) := (g|Σ0, L∂tg|Σ0, A|Σ0, L∂tA|Σ0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='85)  In particular, we have γ0(gb, Ab) = (gb|Σ0, 0, Ab|Σ0, 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Then we define the subspace of the initial data for  the gauge-fixed Einstein-Maxwell equations  Y s,α :=  �  (g0, g1, A0, A1) | (g0, g1, A0, A1) − γ0(gb, Ab) ∈ ¯Hs+1,−1/2+α  b  (¯Σ0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' S2 �  scT ∗¯Σ0)  × ¯Hs,1/2+α  b  (¯Σ0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' S2 �  scT ∗ ¯Σ0) × ¯Hs,−1/2+α  b  (¯Σ0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' �  scT ∗¯Σ0) × ¯Hs−1,1/2+α  b  (¯Σ0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' �  scT ∗Σ0)  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='86)  We now prove:  Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' There exist a neighborhood  (hb, kb, Eb, Bb) ∈ U ⊂ Z2,α  of KN initial data and a smooth map  ib : U ∩ Zs,α → Y s,α  LINEAR STABILITY OF KERR-NEWMAN FAMILY  33  for all s ≥ 2, 0 < α < 1, such that for (h, k, E, B) ∈ U, the sections  (g0, g1, A0, A1) = ib(h, k, E, B)  induce the data (h, k, E, B) on Σ0 = {t = 0}, and they satisfy the gauge conditions (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='82) and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='83) in the  sense that for any section (g, A) of S2T ∗M ⊕ T ∗M near Σ0 with γ0(g, A) = (g0, g1, A0, A1), it induces the  initial data (h, k, E, B) at Σ0 (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=', τ((g, dA)) = (h, k, E, B)) and the conditions (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='82) and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='83) hold at  Σ0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  Moreover, for exact KN data with parameter b, we have ib(hb, kb, Eb, Bb) = γ0(gb, Ab) = (gb|Σ0, 0, Ab|Σ0, 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' We will start with the metric components (g0, g1) of the map ib.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' We write  gb = −N 2  b (dt)2 + gb,ij(dxi + Xi  bdt)(dxj + Xj  b dt)  where Nb ∈ 1 + ρC∞(Σ0) and Xb ∈ ρ2C∞(¯Σ0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' scT ¯Σ0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Then we define the component g0 of ib(h, k, E, B) as  g0 := g|{t=0} = −N 2  b (dt)2 + hij(dxi + Xi  bdt)(dxj + Xj  b dt).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  Moreover, g0 = gb|Σ0 if h = hb and g0 satisfies the estimate  ∥g0 − gb|Σ0∥ ¯  Hs+1,−1/2+α  b  (¯Σ0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='S2 �  scT ∗ ¯Σ0) ≲ ∥h − hb∥ ¯  Hs+1,−1/2+α  b  (¯Σ0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='S2  >0  scT ∗ ¯Σ0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  We next define g1 = L∂tg|Σ0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Let ∇g0 be the Levi-Civita connection of g0 and let n resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' nb be the future  timelike unit normal to Σ0 with respect to g0 resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' gb|Σ0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Then we have  n =  −∇g0t  �  −g0(∇g0t, ∇g0t)  = (1 + N −2  b  h(Xb, Xb))−1/2( 1  Nb  , −Xi  b  Nb  ),  n − nb ∈ ¯Hs+1,1/2+α  b  (¯Σ0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' �  scT ¯Σ0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  Since we require  kij = IIg(∂i, ∂j)|Σ0 = −⟨∇g  ∂in, ∂j⟩ = −1  2Lngij = −1  2nα∂αgij − 1  2giα∂jnα − 1  2gjα∂inα,  where the Greek index α ranges over 0, 1, 2, 3 with x0 = t, we must define  (g1)ij = ∂tgij|Σ0 = −2(1 + N −2  b  h(Xb, Xb))1/2Nbkij + LXbhij  Now the initial data for the remaining components of g1, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' (g1)0α = ∂tg0α|Σ0, are fixed by the generalized  wave map gauge condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' More precisely, since θ(g;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' gb) does not contain the derivatives of g, it follows that  0 = �ΥE(g;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' gb)i = giκgµν(Γ(g)κ  µν − Γ(gb)κ  µν) − θ(g;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' gb)i  = gµν∂µgiν − 1  2gµν∂igµν − giκgµνΓ(g0)κ  µν − θ(g;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' gb)i  at  Σ0  fixes ∂tg0i|Σ0 and  0 = �ΥE(g;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' gb)0 = g0κgµν(Γ(g)κ  µν − Γ(gb)κ  µν) − θ(g;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' gb)0  = gµν∂µg0ν − 1  2gµν∂0gµν − g0κgµνΓ(gb)κ  µν − θ(g;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' gb)0  at  Σ0  fixes ∂tg00|Σ0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' This explicit construction of g1 implies that g1 = ∂tgb|Σ0 = 0 if (h, k) = (hb, kb), and  ∥g1∥ ¯  Hs,1/2+α  b  (¯Σ0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='S2 �  scT ∗ ¯Σ0) ≲ ∥h − hb∥ ¯  Hs+1,−1/2+α  b  (¯Σ0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='S2  >0  scT ∗ ¯Σ0) + ∥k − kb∥ ¯  Hs,1/2+α  b  (¯Σ0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='S2scT ∗ ¯Σ0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  So we finish the construction of the initial data (g0, g1) which induces the data (h, k) on Σ0 and satisfies the  generalized wave map gauge condition at Σ0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  We next turn to the construction of the components (A0, A1) of the map ib.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Here we closely follow the  construction in [39, §3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Motivated by the relation F = dA, we first construct a bounded linear map  A♯ :  �  u ∈ ¯Hs,ℓ  b (¯Σ0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Λ2scT ∗¯Σ0): du = 0,  �  S2 u = 0  �  → ¯Hs,ℓ−1  b  (¯Σ0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' scT ∗ ¯Σ0)  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='87)  such that d ◦ A♯ = Id with ℓ > −1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Using the structure ¯Σ0 ∼= J × S2, J = [0, 1/r−]ρ=1/r, of ¯Σ0, we define  π: ¯Σ0 → S2 to be the projection onto the second factor, and j : S2 ֒→ ¯Σ0 be the embedding ω �→ (1/r−, ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  Then the map K : ¯Hσ,ℓ  b (¯Σ0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' ΛscT ∗ ¯Σ0) → ¯Hσ,ℓ−1  b  (¯Σ0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' ΛscT ∗Σ0), σ ≥ 0, ℓ > −1/2, defined by linear extension  from  K(f(r, ω)dr ∧ π∗u)(r, ω) := (π∗u)(r, ω)  � r  r−  f(s, ω) ds,  K(f(r, ω)π∗u) := 0,  34  LILI HE  for f ∈ ¯Hσ,1/2+α  b  (¯Σ0) and u ∈ C∞(S2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' ΛT ∗S2), satisfies Id − π∗j∗ = dK + Kd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Therefore, for closed u as  in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='87), we have u = dKu + π∗(j∗u).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' According to Hodge decomposition theory and the fact that the de  Rham cohomology group H2  dR(S2) ∼= R, we see that u′ := j∗u ∈ Hs(S2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Λ2T ∗S2) can be written uniquely  as u′ = /∗u0 + /du′′ where u0 ∈ R and u′′ ∈ Hs+1(S2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' T ∗S2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Here, /d, /δ, /∗ denote the exterior differential,  codifferential and Hodge star on S2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Since K(f(r, ω)π∗u) = 0 and  �  S2 u = 0, it follows that  0 =  �  S2 π∗(j∗u) =  �  S2 u′ =  �  S2 u0 dVol/g = 4πu0,  and thus u = dA♯u with A♯u := Ku + π∗u′′, finishing the construction of the map (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='87).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  Returning to the construction of the map ib, we define  A0 = (ι∂tAb) dt + A♯  0,  A♯  0 := i∗Ab + A♯(⋆hB − ⋆hbBb) ∈ i∗Ab + ¯Hs,−1/2+α  b  (¯Σ0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' scT ∗¯Σ0),  and thus dA♯  0 = i∗(⋆hB), and A0 = Ab if (h, B) = (hb, Bb).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' We make the ansatz  A1 = a1 dt + A♯  1  with a1 ∈ ¯Hs−1,ℓ1  b  (¯Σ0) and A♯  1 ∈ ¯Hs−1,ℓ  b  (¯Σ0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' scT ∗¯Σ0) to be determined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Recall the future timelike unit  normal n to Σ0 with respect to g0 as defined above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' The requirement that −i∗ιndA = E at Σ0 then reads  E = n0(d(ι∂tAb) − A♯  1) − ιnk∂kdA♯  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Then we must define  A♯  1 := d(ι∂tAb) − (E + ιnk∂kdA♯  0)  n0  ∈ ¯Hs−1,1/2+α  b  (¯Σ0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' scT ∗ ¯Σ0)  so in particular A♯  1 = 0 if (h, k, E, B) = (hb, kb, Eb, Bb).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  Lastly, the generalized Lorenz gauge condition  0 = �ΥM(g, A;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' gb, Ab) = trg0δ∗  gb(A − Ab)  = 1  2gµν  0  �  ∂µ(Aν − Ab,ν) + ∂µ(Aν − Ab,ν) − 2Γκ  µν(g0)(Aκ − Ab,κ)  �  at  Σ0,  where gµν  0  is the inverse g0, fixes a1 = ∂tA0 ∈ ¯Hs−1,1/2+α  b  (¯Σ0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' In particular, we have a1 = 0 if the data  (h, k, E, B) are induced by (gb, Ab).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Moreover, we have  ∥A0 − Ab|Σ0∥ ¯  Hs,−1/2+α  b  (¯Σ0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' �  scT ∗ ¯Σ0) + ∥A1∥ ¯  Hs−1,1/2+α  b  (¯Σ0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' �  scT ∗ ¯Σ0)  ≲ ∥h − hb∥ ¯  Hs,−1/2+α  b  (¯Σ0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='S2  >0  scT ∗ ¯Σ0) + ∥B − Bb∥ ¯  Hs,1/2+α  b  (¯Σ0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='scT ∗ ¯Σ0) + ∥E − Eb∥ ¯  Hs,1/2+α  b  (¯Σ0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='scT ∗ ¯Σ0)  Also, according to the above explicit construction of the map ib, we see that the map ib is smooth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' This  completes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  □  We can use the map ib to construct gauged initial data for the initial value problem for the linearized  Einstein–Maxwell system (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='29), linearized around (gb, Ab) and with initial data (˙h, ˙k, ˙E, ˙B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Correspond-  ingly, we consider for s > 2, 0 < α < 1 the subspace of the initial data for ungauged linearized Einstein-  Maxwell equations around (gb, Ab)  ˙Zs,α :=  �  (˙h, ˙k, ˙E, ˙B) | D(hb,Bb)(⋆(•)(•))(˙h, ˙B) = 0,  �  S2  d  ds|s=0 ⋆hb+s˙h (Bb + s ˙B) = 0,  (˙h, ˙k, ˙E, ˙B) ∈ ¯Hs+1,−1/2+α  b  (¯Σ0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' S2  >0  scT ∗¯Σ0)  × ¯Hs,1/2+α  b  (¯Σ0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' S2scT ∗ ¯Σ0) × ¯Hs,1/2+α  b  (¯Σ0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' scT ∗¯Σ0) × ¯Hs,1/2+α  b  (¯Σ0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' scT ∗¯Σ0)  �  ,  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='88)  where S2  >0  scT ∗¯Σ0 denotes the fiber bundle of positive definite inner products on scT ¯Σ0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' We again remark  that the definition of the space Zs,α incorporates the vanishing of the linearized magnetic charge and the  linearization of one of the constraint equations (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='13).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  We also define the subspace of the initial data for the gauge-fixed linearized Einstein-Maxwell equations  ˙Y s,α :=  �  (˙g0, ˙g1, ˙A0, ˙A1) | (˙g0, ˙g1, ˙A0, ˙A1) ∈ ¯Hs+1,−1/2+α  b  (¯Σ0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' S2 �  scT ∗ ¯Σ0)  × ¯Hs,1/2+α  b  (¯Σ0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' S2 �  scT ∗ ¯Σ0) × ¯Hs,−1/2+α  b  (¯Σ0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' �  scT ∗¯Σ0) × ¯Hs−1,1/2+α  b  (¯Σ0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' �  scT ∗Σ0)  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='89)  Then we have  LINEAR STABILITY OF KERR-NEWMAN FAMILY  35  Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' For all s ≥ 2, 0 < α < 1, the map  D(hb,kb,Eb,Bb)ib : ˙Zs,α → ˙Y s,α  satisfies that for (˙h, ˙k, ˙E, ˙B) ∈ ˙Zs,α, the sections  (˙g0, ˙g1, ˙A0, ˙A1) = D(hb,kb,Eb,Bb)ib(˙h, ˙k, ˙E, ˙B)  induce the data (˙h, ˙k, ˙E, ˙B) on Σ0 = {t = 0}, and they satisfy the linearized gauge conditions Dgb �ΥE(˙g) = 0  and D(gb,Ab) �ΥM(˙g, ˙A;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' gb, Ab) = −δgb ˙A = 0 in the sense that for any section (˙g, ˙A) of S2T ∗M ⊕ T ∗M near  Σ0 with γ0(˙g, ˙A) = (˙g0, ˙g1, ˙A0, ˙A1), it induces the initial data (˙h, ˙k, ˙E, ˙B) at Σ0 and the linearized gauge  conditions hold at Σ0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' We define  h(s) = hb + s˙h, k(s) = kb + s˙k, E(s) = Eb + s ˙E, B(s) = Bb + s ˙B  and  ⋆h+s˙hB(s) = ⋆hB + sDhb,Bb(⋆(·)(·))(˙h, ˙B)  for s close to 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' thus h′(0) = ˙h etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=', and d(s) := (h(s), k(s), E(s), B(s)) ∈ U ⊂ Z2,α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Then it follows that  ib(d(s)) is well-defined and smooth in s, and  (˙g0, ˙g1, ˙A0, ˙A1) = d  dsib(d(s))  ��  s=0 = D(hb,kb,Eb,Bb)ib(˙h, ˙k, ˙E, ˙B)  gives initial data satisfying the linearized gauge conditions  Dgb �ΥE(˙g) = 0,  D(gb,Ab) �ΥM(˙g, ˙A;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' gb, Ab) = −δgb ˙A = 0,  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='90)  for ˙g and ˙A with γ0(˙g, ˙A) = (˙g0, ˙g1, ˙A0, ˙A1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  □  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Analysis of the linearized gauge-fixed Einstein-Maxwell operator  Throughout this section, we continue using the notation from §4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  In §5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='1, we will discuss the characteristic set and the global dynamics of the Hamiltonian flow of the  principal symbol p(σ) of the operator �  □gb(σ) := eitb,∗σ□gbe−itb,∗σ on full subextremal KN spacetimes (see  Figure 6 for an illustration).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Concretely, first we will show that the characteristic set of p(σ) has two parts,  one of which lies inside the ergoregion while the other is at the spatial infinity ∂+ ¯X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Then we prove that  there exist radial points, to which the nearby integral curves of the Hamiltonian vector field Hp(σ) converge  in either forward direction or backward direction, at event horizon and spatial infinity ∂+ ¯X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Finally, we  describe the global dynamics of the Hamiltonian flow of the principal symbol p(σ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  In §5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='2, we combine the global dynamics of the Hamiltonian flow of the principal symbol p(σ) established  in §5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='1 with the elliptic estimate, propagation of singularities estimate, the radial point estimate at event  horizon, the scattering radial point estimate at spatial infinity ∂+ ¯X and hyperbolic estimate to prove the  uniform Fredholm estimates for the operators �  □gb(σ), �  Pb,γ(σ), �  Wb,γ(σ), �  Lb,γ(σ) for bounded σ in the closed  upper half plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  In§5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='3, we will give a detailed description of kernel of the following wave operators: �  □gb(σ) on scalar  functions, �  Pb,γ(σ), �  Wb,γ(σ) on scattering 1-forms and the linearized gauge-fixed Einstein-Maxwell operator  �  Lb,γ(σ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  In §5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='4, we will analyze the characteristic set and the global dynamics of the Hamiltonian flow of the  semiclassical principal symbol ph,z of the semiclassical rescaled operator h2�  □gb(h−1z) on full subextremal  KN spacetimes for z ∈ R\\0 (see Figure 10 for an illustration).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Specifically, we first show that the characteristic  set of ph,z can be split into a disjoint union of two components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Then we prove that there exist radial points  at event horizon and spatial infinity ∂+ ¯X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Next, we discuss the trapped set associated to the Hamiltonian  flow of the semiclassical symbol ph,z and prove that it is normally hyperbolic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Finally, we describe the global  dynamics of the Hamiltonian flow of the semiclassical principal symbol ph,z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  In §5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='5, we combine the global dynamics of the Hamiltonian flow of the semiclassical principal symbol ph,z  established in §5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='4 with the elliptic estimate, propagation of singularities estimate, the radial point estimate  at event horizon, the scattering radial point estimate at spatial infinity ∂+ ¯X and hyperbolic estimate, all of  which are in the semiclassical version, together with the estimate at normally hyperbolic trapping to establish  36  LILI HE  the high energy estimates for the operators �  □gb(σ), �  Pb,γ(σ), �  Wb,γ(σ), �  Lb,γ(σ) for |Re σ| ≫ 1, Im σ ≤ C in the  closed upper half plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' This implies the invertibility of these operators for |Re σ| ≫ 1, Im σ ≤ C in the  closed upper half plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  In §5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='6, we shall establish the energy estimates for the solutions to various wave type equations on slowly  rotating Kerr-Newman metrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  Fix KN black hole parameters b = (m, a, Q) close to b0 = (m0, a0, Q0) and let  g = gb  Recall that with a = |a|  g−1 = ρ−2  b  �  ∆b∂2  r + χ2 − 1  ∆b  �  (r2 + a2)∂tχ + a∂ϕ  �2  + ∂2  θ +  1  sin2 θ  �  ∂ϕ + a sin2 θ∂tχ  �2  + 2χ  �  (r2 + a2)∂tχ + a∂ϕ  �  ∂r  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  We define  �  □g(σ) := eitb,∗σ□gbe−itb,∗σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  Let scT ∗ ¯X be the compact manifold with the corners of codimension two obtained by radial compactification  of the fibers scT ∗ ¯X (whose detail will be discussed later on).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Then its boundary consists of three hypersurfaces  ∂scT ∗ ¯X = scS∗ ¯X ∪ scT ∗∂− ¯  X ¯X ∪ scT ∗∂+ ¯  X ¯X  where scS∗ ¯X = (scT ∗ ¯X \\ 0)/R+ is called fiber infinity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Microlocal geometry and dynamics of Kerr-Newman spacetimes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Now, we are ready to discuss  the microlocal geometry of the KN metric g = gb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Away from ∂+ ¯X, we note that scT ∗ ¯X is isomorphic to  T ∗ ¯X and write the covectors as  ξrdr + ξθdθ + ξϕdϕ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  Then the principal symbol of �  □g(σ) is given by  p = σ2(�  □g(σ)) = −ρ−2  b  �  ∆bξ2  r + ξ2  θ +  � a2  ∆b  (χ2 − 1) +  1  sin2 θ  �  ξ2  ϕ + 2χaξϕξr  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  Near ∂+ ¯X, we write the scattering covectors as  ζρ  dρ  ρ2 + ζθ  dθ  ρ + ζϕ  dϕ  ρ ,  ρ = 1  r .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  Then the scattering principal symbol (see [65]) of �  □g(σ) at ∂+ ¯X is given by  2σζρ − ζ2  ρ − ζ2  θ −  1  sin2 θζ2  ϕ = −(ζρ − σ)2 − ζ2  θ −  1  sin2 θζ2  ϕ + σ2  To handle the poles {θ = 0} and {θ = π}, where the spherical coordinates break down, we introduce new  coordinates  x1 = sin θ cos ϕ,  x2 = sin θ sin ϕ,  and let ξ1 and ξ2 be dual variables to x1 and x2 respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Correspondingly, we have  ξθ = (x1ξ1 + x2ξ2) cot θ,  ξϕ = x1ξ2 − x2ξ1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  and note that ξϕ is a smooth function in (x1, x2, ξ1, ξ2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Since  ξ2  θ +  1  sin2 θξ2  ϕ = 1 − x2  1 − x2  2  x2  1 + x2  2  (x1ξ1 + x2ξ2)2 +  1  x2  1 + x2  2  (x1ξ2 − x2ξ1)2 = ξ2  1 + ξ2  2 − (x1ξ1 + x2ξ2)2  is a smooth function in (r, x1, x2, ξr, ξ1, ξ2) near the poles, so is p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Therefore, we can perform all symbol  calculations away from these two poles and extend the results to the poles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' From now on, we do not emphasize  the analysis near these two poles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  LINEAR STABILITY OF KERR-NEWMAN FAMILY  37  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Characteristic set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' We first study the characteristic set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Away from ∂+ ¯X, since  p = −ρ−2  b  �  ∆b(ξr + aχ  ∆b  ξϕ)2 + ξ2  θ + ∆b − a2 sin2 θ  ∆b sin2 θ  ξ2  ϕ  �  ,  it follows that the characteristic set {p = 0, ξ ̸= 0} of p satisfies  {p = 0, ξ = (ξr, ξθ, ξϕ) ̸= 0} ⊂ {∆b − a2 sin2 θ ≤ 0, ξ ̸= 0} ⊂ scT ∗ ¯X \\ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  Since χ = 1 in the region ∆b − a2 sin2 θ ≤ 0, the principal symbol can be written as  p = −ρ−2  b  �  ∆bξ2  r + 2aξrξϕ + ˜p  �  where  ˜p = ξ2  θ +  1  sin2 θξ2  ϕ,  and thus ξr ̸= 0 at the characteristic set of p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' As a consequence, in the fiber radial compactification scT ∗ ¯X  of scT ∗ ¯X, we use the following coordinates near the fiber infinity scT ∗ ¯X,  ρξ =  1  |ξr| ∈ [0, ∞),  ˆξ = (ˆξθ, ˆξϕ) = ρξ(ξθ, ξϕ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='1)  We note that ρξ is a boundary defining function of scT ∗ ¯X, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=', ρξ = 0 at ∂scT ∗ ¯X = scS∗ ¯X, and ˆξ is a  coordinate system on scS∗ ¯X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  Since p is a homogeneous polynomial of order 2 in ξ = (ξr, ξθ, ξϕ), the rescaled symbol ρ2  ξp and the  rescaled Hamiltonian vector field ρξHp extend smoothly to scT ∗ ¯X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  Therefore, we study the flow of the  rescaled Hamiltonian vector field ρξHp on the characteristic set {ρ2  ξp = 0} ⊂ scT ∗ ¯X \\ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' We now split the  characteristic set {ρ2  ξp = 0} ⊂ scT ∗ ¯X \\ 0 into two components  Σ± := {ρ2  ξp = 0} ∩ {±ξr > 0},  ∂Σ± = Σ± ∩ scS∗ ¯X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  Near ∂+ ¯X, we have χ = 0 and thus  g−1 = ρ−2  b  �  ∆b∂2  r − 1  ∆b  �  (r2 + a2)∂tχ + a∂ϕ  �2  + ∂2  θ +  1  sin2 θ  �  ∂ϕ + a sin2 θ∂tχ  �2�  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  Then using tχ = t and tb,∗ = t − r(m,a0,Q),∗ near ∂+ ¯X, we calculate  �  □g(σ) = −2iσρ2∂ρ + (ρ2∂ρ)2 + ρ2∂2  θ +  ρ2  sin2 θ ∂2  ϕ + ρDiff2  sc( ¯X),  ρ = 1  r .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  Therefore, the scattering principal symbol of �  □g(σ) at ∂+ ¯X is given by  psc(σ) = σsc(�  □g(σ)) = −(ζρ − σ)2 − ˜p + σ2,  ˜p = ζ2  θ +  1  sin2 θζ2  ϕ,  and the scattering characteristic set is the surface  Σsc(σ) = {psc(σ) = 0} ⊂ scT ∗∂+ ¯  X ¯X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  We note that Σsc(σ) is the scattering zero section o∂+ ¯  X ⊂ scT ∗∂+ ¯  X ¯X for Im σ > 0 and σ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' The radial points at event horizon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' We now analyze the behavior of the Hamiltonian flow exp(sρξHp)  on the characteristic set Σ = Σ+ ∪ Σ− ⊂ scT ∗ ¯X \\ 0, that is, we consider the portion whose projection to the  base space (r, θ, ϕ) is away from ∂+ ¯X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Near Σ = Σ+ ∪ Σ−, we see that χ = 1 and thus  p = −ρ−2  b  �  ∆bξ2  r + 2aξrξϕ + ˜p  �  where  ˜p = ξ2  θ +  1  sin2 θξ2  ϕ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  Then we calculate  Hp = −ρ−2  b  �  (2∆bξr + 2aξϕ)∂r + 2aξr∂ϕ − ∂r∆b  ∂r ξ2  r∂ξr + H˜p  �  − ρ−2  b pHρ2  b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  38  LILI HE  In the coordinates (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='1) near the fiber infinity scS∗ ¯X, we compute  ρξHp = −ρ−2  b  �  (2∆b(sgn ξr) + 2aˆξϕ)∂r + 2a(sgn ξr)∂ϕ − ∂∆b  ∂r (sgn ξr)ξr∂ξr + ρξH˜p  �  − ρ−2  b (ρξp)Hρ2  b  = −ρ−2  b  �  (2∆b(sgn ξr) + 2aˆξϕ)∂r + 2a(sgn ξr)∂ϕ + 2ˆξθ∂θ +  2  sin2 θ  ˆξϕ∂ϕ  �  − ρ−2  b  ∂∆b  ∂r (sgn ξr)  �  ρξ∂ρξ + ˆξθ∂ˆξθ + ˆξϕ∂ˆξϕ  �  − ρ−2  b  2 cos θ  sin3 θ  ˆξ2  ϕ∂ˆξθ − ρ−2  b (ρ2  ξp)ρ−1  ξ Hρ2  b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='2)  We let  Λ± = {∆b = 0, (ξθ, ξϕ) = 0, ±ξr > 0}  and  L± := ∂Λ± = Λ± ∩ scS∗ ¯X = {∆b = 0, ρξ = ± 1  ξr  = 0, ˆξ = 0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  Now we describe the property of the set L±.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' More specifically, L± are radial sources/sinks in the following  sense (see [21, Definition E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='50]):  Definition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Let κ : scT ∗ ¯X \\ 0 → scS∗ ¯X = (scT ∗ ¯X \\ 0)/R+ be the natural projection map, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=', for  each (r, θ, ϕ, ξ) ∈ scT ∗ ¯X \\ 0, the ray (r, θ, ϕ, sξ) converges to κ((r, θ, ϕ, ξ)) as s → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Then for the rescaled  Hamiltonian flow  φs := exp(sρξHp) : scT ∗ ¯X → scT ∗ ¯X,  we say that a nonempty compact φs-invariant set  L ⊂ {ρ2  ξp = 0} ∩ scS∗ ¯X  is a radial source for p, if there exists a neighborhood U ⊂ scT ∗ ¯X of L such that uniformly in (r, θ, ϕ, ξ) ∈  U ∩ scT ∗ ¯X, one have  κ(φs(r, θ, ϕ, ξ)) → L  as  s → −∞  and  |φs(r, θ, ϕ, ξ)| ≥ BeB|s||ξ|,  s ≤ 0  for some B > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Here, | · | denotes a norm on the fibers of scT ∗ ¯X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  A radial sink for p is be definition a radial source for −p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' L± are invariant under the Hamiltonian flow exp(sρξHp) on the characteristic set Σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' More-  over, L+ is a radial sink and L− is a radial source for the flow exp(sρξHp), see Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  L+  L−  Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' The flow of exp(sρξHp) near L±.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' It is projected to the (r, ξr) coordinates and  drawn in a fiber-radially compactified view.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' The horizontal coordinate is r;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' the dashed line  is the event horizon {r = rb}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' The vertical coordinate is ξr/⟨ξr⟩, so the top and bottom lines  correspond to the fiber infinity {ρξ = 0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' The midline {ξr = 0} lies outside the characteristic  set {ρ2  ξp = 0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Since {ρ2  ξ ˜p = 0} ⇔ {ˆξ = (ˆξθ, ˆξϕ) = 0}, we rewrite  L± = {∆b = 0, ρξ = ± 1  ξr  = 0, ρ2  ξ ˜p = 0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  LINEAR STABILITY OF KERR-NEWMAN FAMILY  39  Using the coordinates (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='1) and expression (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='2), we find that  ρξHpρξ = −ρ−2  b  �∂∆b  ∂r + 2rρ2  ξp  �  (sgn ξr)ρξ,  ρξHp ˆξϕ = −ρ−2  b  �∂∆b  ∂r + 2rρ2  ξp  �  (sgn ξr)ˆξϕ,  ρξHp∆b = −2ρ−2  b  ∂∆b  ∂r  �  (sgn ξr)∆b + aˆξϕ  �  ,  ρξHp(ρ2  ξ ˜p) = −ρ−2  b  �  2∂∆b  ∂r + 4rρ2  ξp  �  (sgn ξr)ρ2  ξ ˜p − 2ρ−2  b a2 sin(2θ)(ρ2  ξp)ˆξθ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='3)  This implies that ρξHp is tangent to L±, so L± is invariant under the flow exp(sρξHp).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Moreover, in a  neighborhood of L± = {∆b = 0, ρξ = ±ξ−1  r  = 0, ρ2  ξ ˜p = 0} ⊂ {ρ2  ξp = 0}, we have  ±ρξHpρ2  ξ ≤ −C0ρ2  ξ,  ±ρξHp ˆξ2  ϕ ≤ −C0 ˆξ2  ϕ,  ±ρξHp∆2  b ≤ −C0∆2  b + C1 ˆξ2  ϕ,  ±ρξHp(ρ2  ξ ˜p) ≤ −C0ρ2  ξ ˜p + C1  �  ∆2  b + ˆξ2  ϕ  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='4)  where C0, C1 > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' It follows from (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='4) that in a sufficiently small neighborhood of L±  ± ρξHpf ≤ −Cf,  f = ρ2  ξ + ˆξ2  ϕ + C2∆2  b + C3ρ2  ξ ˜p ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='5)  for some C, C2, C3 > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' This implies that for (r, θ, ϕ, ξ) in a sufficiently small neighborhood of L±, we have  f(φs((r, θ, ϕ, ξ))) ≤ e−C|s|f((r, θ, ϕ, ξ))  for  ± s > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  Therefore, we have uniformly in (r, θ, ϕ, ξ) in a neighborhood of L±  φs((r, θ, ϕ, ξ)) → L±  as  s → ±∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  Moreover, according to the first inequality in (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='4), we have  ρξ(φs((r, θ, ϕ, ξ))) ≤ e−C0|s|ρξ((r, θ, ϕ, ξ))  for  ± s ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  This proves that L+ is a radial sink and L− is a radial source.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  □  Finally, we consider the imaginary part of the operator �  □g(σ) at L±, which is needed to calculate the  threshold quantity for the order of the Sobolev spaces (here the differential order) in the radial point estimates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  Recall that near L±, the inverse metric G takes the form  g−1 = ρ−2  b  �  a2 sin2 θ∂2  tχ + 2(r2 + a2)∂tχ∂r + 2a∂tχ∂ϕ + ∆b∂2  r + 2a∂r∂ϕ + ∂2  θ +  1  sin2 θ ∂2  ϕ  �  and tχ = tb,∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' A direct calculation shows that  �  □g(σ)∗ = �  □g(¯σ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  where �  □g(σ)∗ denotes the formal adjoint of �  □g(σ) defined by ⟨�  □gb(σ)u, , v⟩ = ⟨u, �  □g(σ)∗v⟩ with respect to  L2( ¯X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  �  | det g|drdθdϕ) for u, v ∈ C∞  c (X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' We first compute  p1 := σ1  �  Im �  □g(σ)  �  = σ1  � �  □g(σ) − �  □g(σ)∗  2i  �  = ρ−2  b Im σ  �  2(r2 + a2)ξr + 2aξϕ  �  Let β0 ∈ C∞(L±) be a positive function defined as  β0 = ∓ρξHpρξ  ρξ  |L±.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  Then we define ˜β ∈ C∞(L±) as  ˜β = ∓ρξp1  β0  |L±  Let  βsup = sup ˜β,  βinf = inf ˜β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  40  LILI HE  If ˜β is a constant along L±, we may write  β = βsup = βinf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  We note that βsup and βinf are the relevant quantities in the radial point estimates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' More specifically, in our  setting where we consider the operator �  □g(σ) with Im σ ≥ 0, the threshold regularity condition in the high  regularity radial estimate is given by  s > 1  2 + βsup,  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='6)  while the low regularity radial estimate requires  s < 1  2 + βinf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='7)  Using the first equality in (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='3), we find for the operator �  □g(σ) that  ˜β = −Im σ r2  b + a2  rb − m ,  and thus  β = βsup = βinf = −Im σ r2  b + a2  rb − m ≤ 0  for  Im σ ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='8)  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' The radial points at spatial infinity ∂+ ¯X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' We now turn to the analysis near ∂+ ¯X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Recall that for the  operator �  □g(σ) with Im σ ≥ 0, the scattering characteristic set is the surface  Σsc(σ) = {(ρ = 1  r = 0, θ, ϕ, ζ)) : psc(σ) = 0} ⊂ scT ∗∂+ ¯  X ¯X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  where  psc(σ) = −(ζρ − σ)2 − ˜p + σ2,  ˜p = ζ2  θ +  1  sin2 θζ2  ϕ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  Using the identification of T ∗ ¯X and scT ∗ ¯X in ρ > 0  ζρ = ρ2ξρ = −ξr,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  (ζθ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' ζϕ) = ρ(ξθ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' ξϕ),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  we find that the scattering Hamiltonian vector field associated to p in ρ > 0 is given by  scHp = ρ2 ∂p  ∂ζρ  � ∂  ∂ρ + 2ζρ  ρ  ∂  ∂ζρ  +  � �  µ=θ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='ϕ  ζµ  ρ  ∂  ∂ζµ  ��  −  �∂p  ∂ρ + 2ζρ  ρ  ∂p  ∂ζρ  +  � �  µ=θ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='ϕ  ζµ  ρ  ∂p  ∂ζµ  ��  ρ2 ∂  ∂ζρ  +  �  µ=θ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='ϕ  ρ  � ∂p  ∂ζµ  ∂  ∂µ − ∂p  ∂µ  ∂  ∂ζµ  �  = ρ  � ∂p  ∂ζρ  �  ρ ∂  ∂ρ + (  �  µ=θ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='ϕ  ζµ  ∂  ∂ζµ  )  �  −  �  ρ∂p  ∂ρ + (  �  µ=θ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='ϕ  ζµ  ∂p  ∂ζµ  )  � ∂  ∂ζρ  +  �  µ=θ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='ϕ  � ∂p  ∂ζµ  ∂  ∂µ − ∂p  ∂µ  ∂  ∂ζµ  ��  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  By introducing the following coordinates in the fiber radial compactification scT ∗ ¯X of scT ∗ ¯X  ρζ = (1 + ζ2  r + ζ2  θ +  1  sin2 θζ2  ϕ)− 1  2 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  ˆζ = (ˆζρ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' ˆζθ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' ˆζϕ) = ρζ(ζρ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' ζθ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' ζϕ),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  the rescaled scattering Hamiltonian vector field ρζρ−1scHp can be extended to an element in Vb(scT ∗ ¯X)  taking the following form  scH2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='0  p  := ρ2−1  ζ  ρ−0−1scHp  = ρζ  � ∂p  ∂ζρ  �  ρ ∂  ∂ρ + (  �  µ=θ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='ϕ  ζµ  ∂  ∂ζµ  )  �  −  �  ρ∂p  ∂ρ + (  �  µ=θ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='ϕ  ζµ  ∂p  ∂ζµ  )  � ∂  ∂ζρ  +  �  µ=θ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='ϕ  � ∂p  ∂ζµ  ∂  ∂µ − ∂p  ∂µ  ∂  ∂ζµ  ��  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  We now discuss the integral curves of scH2,0  psc(σ) on the scattering characteristic set Σsc(σ) = {psc(σ) =  0} ⊂ scT ∗∂+ ¯  X ¯X for σ ∈ R \\ 0 (see [63]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Since Σsc(σ) has no intersection with the fiber infinity ∂scT ∗∂+ ¯  X ¯X,  we can drop the factor ρζ which is non-zero on Σsc(σ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Therefore, with psc(σ) = −(ζρ − σ)2 − ˜p + σ2, we  write  scH2,0  psc(σ) = (−2ζρ + 2σ)ρ∂ρ + 2˜p∂ζρ + (−2ζρ + 2σ)  �  µ=θ,ϕ  ζµ∂ζµ − H˜p  on  Σsc(σ)  LINEAR STABILITY OF KERR-NEWMAN FAMILY  41  where  H˜p = 2ζθ∂θ + 2ζϕ  sin2 θ∂ϕ + cos θ  sin2 θ ζ2  ϕ∂ζθ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' For 0 ̸= σ ∈ R, on the characteristic variety  Σsc(σ) = {psc(σ) = −(ζρ − σ)2 − ˜p + σ2 = 0} ⊂ scT ∗∂+ ¯  X ¯X,  ˜p = ζ2  θ +  1  sin2 θζ2  ϕ,  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='9)  there are two invariant submanifolds under the Hamiltonian flow exp(sscH2,0  psc(σ)), one of which is the zero  section R(0) = {ρ = 0, ζρ = 0, (ζθ, ζϕ) = 0} and the other is R(σ) = {ρ = 0, ζρ = 2σ, (ζθ, ζϕ) = 0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  For (θ0, ϕ0, ζ0  ρ, ζ0  θ, ζ0  ϕ) ∈ Σsc(σ) \\ (R(0) ∪ R(σ)), the integral curves exp(sscH2,0  psc(σ))(θ0, ϕ0, ζ0  ρ, ζ0  θ, ζ0  ϕ) take  the following form  ζρ(s) = σ + |σ| sin(s + s0),  ζρ(0) = ζ0  ρ,  (ζθ, ζϕ) = |σ| cos(s + s0)(˜ζθ, ˜ζϕ),  |σ| cos(s0) = ˜p(θ0, ϕ0, ζ0  θ, ζ0  ϕ)  1  2 ,  (θ, ϕ, ˜ζθ, ˜ζϕ) = exp((s + s0)H− 1  2 ˜p)(θ0, ϕ0, ˜ζ0  θ, ˜ζ0  ϕ),  (˜ζ0  θ, ˜ζ0  ϕ) = ˜p(θ0, ϕ0, ζ0  θ, ζ0  ϕ)− 1  2 · (ζ0  θ, ζ0  ϕ)  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='10)  where s0 ∈ (− π  2 , π  2 ), s ∈ (− π  2 − s0, π  2 − s0) and  ds  ds′ = 2˜p(θ, ϕ, ζθ, ζϕ)  1  2 where  d  ds′ := scH2,0  psc(σ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  This implies that for σ > 0, the zero section is a radial source and the nonzero one is a radial sink, while  for σ < 0, the zero section is a radial sink and the nonzero one is a radial source, see Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  (ζθ, ζϕ)  ζρ  R(σ)  R(0)  (ζθ, ζϕ)  ζρ  R(σ)  R(0)  Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  The integral curves of scH2,0  psc(σ) in ζρ and (ζθ, ζϕ) coordinates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' The left hand  side is the case σ > 0 and the right hand side is the case σ < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' At ∂ ¯X, the rescaled scattering Hamiltonian vector field is  scH2,0  psc(σ) = (−2ζρ + 2σ)ρ∂ρ + 2˜p∂ζρ + (−2ζρ + 2σ)  �  µ=θ,ϕ  ζµ∂ζµ − H˜p,  Introducing the polar coordinates with respect to the radial variable in (ζθ, ζϕ)  (˜ζθ, ˜ζϕ) = ˜p(θ, ϕ, ζθ, ζϕ)− 1  2 · (ζθ, ζϕ),  |(ζθ, ζϕ)| = ˜p(θ, ϕ, ζθ, ζϕ)  1  2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  Let  d  ds′ := scH2,0  psc(σ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Then we have  d  ds′ ζρ = 2˜p(θ, ϕ, ζθ, ζϕ),  d  ds′ ˜p(θ, ϕ, ζθ, ζϕ)1/2 = −2(ζρ − σ)˜p(θ, ϕ, ζθ, ζϕ)  1  2 ,  d  ds′ (˜ζθ, ˜ζϕ) = ˜p− 1  2 (∂˜p  ∂θ, ∂˜p  ∂ϕ)(θ, ϕ, ζθ, ζϕ),  d  ds′ (θ, ϕ) = −( ∂˜p  ∂ζθ  , ∂˜p  ∂ζϕ  )(θ, ϕ, ζθ, ζϕ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  This implies that  d  ds′ is tangent to R(0) and R(σ), so they are invariant under the Hamiltonian flow  exp(sscH2,0  psc(σ)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  When ˜p(θ, ϕ, ζθ, ζϕ) ̸= 0, we introduce a new parameter s satisfying  ds  ds′ = 2˜p(θ, ϕ, ζθ, ζϕ)  1  2 and obtain  d  ds(ζρ − σ) = ˜p(θ, ϕ, ζθ, ζϕ)  1  2 ,  d  ds ˜p(θ, ϕ, ζθ, ζϕ)  1  2 = −(ζρ − σ),  d  ds(˜ζθ, ˜ζϕ) = 1  2(∂˜p  ∂θ, ∂˜p  ∂ϕ)(θ, ϕ, ˜ζθ, ˜ζϕ),  d  ds(θ, ϕ) = −1  2( ∂˜p  ∂˜ζθ  , ∂˜p  ∂˜ζϕ  )(θ, ϕ, ˜ζθ, ˜ζϕ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='11)  42  LILI HE  Integrating the above system with respect to the new parameter s gives (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  □  For the propagation of singularities estimates at these radial points R(σ) and R(0), there is a threshold  quantity for the order of the Sobolev spaces (here the scattering decay order).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Let β0 be a positive function  defined at the radial points as  scH2,0  psc(σ)ρ = ±β0ρ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  Here and in what follows, we use + for radial sources and − for radial sinks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' We next define ˜β at the radial  points as  σsc(ρ−1Im �  □g(σ)) = σsc(  �  □g(σ) − �  □g(σ)∗  2iρ  ) = ±β0 ˜β  Let  βsup = sup ˜β,  βinf = inf ˜β  where the supremum and infimum are taken over the radial points set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' If ˜β is a constant along the radial  points set, we may write  β = βsup = βinf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  We note that βsup and βinf are the relevant quantities in the radial point estimates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' More specifically, in our  setting where we consider the operator �  □g(σ) with σ ∈ R \\ 0, the threshold scattering decay condition in the  high scattering decay radial estimate is given by  r > −1  2 − βinf,  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='12)  while the low scattering decay radial estimate requires  r < −1  2 − βsup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='13)  In our setting, since �  □g(σ) = �  □g(σ)∗ for σ ∈ R \\ 0, it follows that  β = βsup = βinf = 0  on  R(σ)  and  R(0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='14)  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Global dynamics of the Hamiltonian flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Now we study the global behavior of the flow of ρξHp (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=',  scH2,0  psc(σ)) away from infinity ∂+ ¯X = {ρ = 1  r = 0} (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=', at ∂+ ¯X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' The characteristic set of �  □g(σ) is a disjoint union Σ ∪ Σsc(σ) (see Figure 6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  (1) Let (r, θ, ϕ, ξ) ∈ Σ± and put φ(s) = exp(sρξHp)(r, θ, ϕ, ξ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Then φ(s) → L± as s → ±∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Moreover,  if (r, θ, ϕ, ξ) /∈ Λ± = {∆b = 0, (ξθ, ξϕ) = 0, ±ξr > 0}, then φ(s) crosses ∂− ¯X into the inward direction  of deceasing r at some s0 with ±s0 ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  (2) For σ ∈ R \\ 0, let (ρ = 0, θ, ϕ, ζ) ∈ Σsc(σ) and put φ(s) = exp(sscH2,0  psc(σ))(θ, ϕ, ζ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Then either  φ(s) → R( σ  2 + |σ|  2 ) = {ρ = 0, ζρ = σ + |σ|, ˜p = 0} as s → ∞, or φ(s) → R( σ  2 − |σ|  2 ) = {ρ = 0, ζρ =  σ − |σ|, ˜p = 0} as s → −∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Moreover, if (ρ = 0, θ, ϕ, ζ) /∈ R(0) ∪ R(σ), then φ(s) → R( σ  2 ± |σ|  2 ) as  s → ±∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' We only discuss the case Σ+ as the other case Σ− can be handled in a similar manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Let (r, θ, ϕ, ξ) ∈  Σ+ and put φ(s) = exp(sρξHp)(r, θ, ϕ, ξ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Since Σ+ is invariant under the flow exp(sρξHp), it follows that  φ(s) ∈ Σ± for all s for which it is well-defined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Put  f(s) =  �  ρ2  ξ + (∆b + 2asgn(ξr)ˆξϕ)2 + ρ2  ξ ˜p  �  (φ(s)),  ˙f(s) = ρξHpf(φ(s)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  Using the calculation in (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='3), we find that  ˙f(s) ≤ −2  �  ρ−2  b  ∂∆b  ∂r f  �  (φ(s)) ≤ −C0f(s)  on  Σ+ ⊂ {r ≥ r− | ∆b − a2 sin2 θ ≤ 0}  for some C0 > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' This gives  f(s) ≤ e−C0sf(0) = e−C0sf((r, θ, ϕ, ξ))  for  s ≥ 0  and thus  φ(s) → L+  as  s → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  LINEAR STABILITY OF KERR-NEWMAN FAMILY  43  Σsc(σ)  R(σ)  R(0)  L+  L−  Σ+  Σ−  σ > 0  Σsc(σ)  R(σ)  R(0)  L+  L−  Σ+  Σ−  σ < 0  Σsc(σ)  L+  L−  Σ+  Σ−  Im σ > 0 or σ = 0  Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' The flow of ρξHp (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  scH1,0  psc(σ)) away from infinity ∂+ ¯X = {ρ =  1  r = 0}  (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=', at ∂+ ¯X), which is projected to the coordinates (r, ξr) (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' (ρ, ζρ)) and drawn in  a fiber-radially compactified view.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' The shaded region is the characteristic set Σ = Σ+ ∪  Σ− = {ρ2  ξp = 0} away from ∂+ ¯X, while the red region is the scattering characteristic set  Σsc(σ) = {psc(σ) = 0} at ∂+ ¯X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' The horizontal coordinate is ρ and the rightmost vertical  line corresponds to ∂+ ¯X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' The dotted line is the even horizon r = rb and the vertical line  in the middle is {r = rergo} where rergo is the equatorial radius (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' greatest radius) of the  ergosphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' The vertical coordinate is ξr/⟨ξr⟩ (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' ζρ/⟨ζρ⟩), so the top and bottom lines  stand for the fiber infinity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' The midline {ξr = 0} (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' ζρ = 0) does not intersect Σ, but  intersects Σsc(σ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  Using the last expression in (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='3), we have  ρξHp(ρ2  ξ ˜p)(φ(s)) = −2  �  ρ−2  b  ∂∆b  ∂r ρ2  ξ ˜p  �  (φ(s)) ≤ −C0(ρ2  ξ ˜p)(φ(s)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  for some C0 > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Therefore  (ρ2  ξ ˜p)(φ(s)) ≥ e−C0s(ρ2  ξ ˜p)(r, θ, ϕ, ξ)  for s ≤ 0 as long as the flow remains in the region Σ+ ⊂ {r ≥ r− | ∆b − a2 sin2 θ ≤ 0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Since on Σ+  2a2 + 1  2  ˆξ2  ϕ ≥ −2aˆξϕ = ∆b + ρ2  ξ ˜p ≥ ∆b + 1  2ρ2  ξ ˜p + 1  2  ˆξ2  ϕ  where we use ∆b +2aˆξϕ +ρ2  ξ ˜p = 0 on Σ+ in the second equality and ρ2  ξ ˜p ≥ ˆξ2  ϕ in the last inequality, it follows  that  2a2 − ∆b ≥ 1  2ρ2  ξ ˜p  on  Σ+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  Since  {ρ2  ξ ˜p = 0} ∩ Σ+ = Λ+,  then if (r, θ, ϕ, ξ) ∈ Σ+ \\ Λ+, we have  (2a2 − ∆b)(φ(s)) ≥ 1  2ρ2  ξ ˜p(φ(s)) ≥ Ce−C0s,  C, C0 > 0  for s ≤ 0 as long as the flow remains in the region Σ+ ⊂ {r ≥ r− | ∆b − a2 sin2 θ ≤ 0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' As a consequence,  φ(s) cross ∂− ¯X into the inward direction of deceasing r at some s0 with s0 ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' This proves the part (i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  The proof of part (2) follows from Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  □  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Second microlocalized scattering algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' To do the analysis at the scattering zero section at ∂+ ¯X, we  introduce the second microlocalized ¯X by blowing up the zero section o∂+ ¯  X at ∂+ ¯X (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' replacing o∂+ ¯  X by  its inward pointing spherical normal bundle and thus obtaining a smooth manifold with corners), see Figure  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  44  LILI HE  scT ∗∂− ¯  X ¯X  scT ∗∂+ ¯  X ¯X  scS∗ ¯X  scS∗ ¯X  ¯X × {0}  Σsc(σ)  R(0)  R(σ)  σ > 0  scT ∗∂− ¯  X ¯X  scT ∗∂+ ¯  X ¯X  scS∗ ¯X  scS∗ ¯X  ¯X × {0}  R(σ)  bT ∗  ∂+ ¯  X ¯X  σ > 0  Figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Second microlocalized  ¯X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  The left hand side is the fiber-compactification  scT ∗ ¯X of the scattering cotangent bundle scT ∗ ¯X, and the right hand side is its blow-up  [scT ∗ ¯X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' o∂+ ¯  X] at the scattering zero section at the boundary ∂+ ¯X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Also shown is the scat-  tering characteristic set Σsc(σ) at ∂+ ¯X of �  □g(σ) for σ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' The interior of the front face of  the blow-up, shown by curved arcs, can be identified with bT ∗  ∂+ ¯  X ¯X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  On the other hand, from an analytically better behaved, but geometrically equivalent perspective, one  takes bT ∗ ¯X and blows up its corner bS∗  ∂+ ¯  X ¯X, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' the fiber infinity at ∂+ ¯X, see Figure 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' These two spaces  [scT ∗ ¯X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' o∂+ ¯  X] and [bT ∗ ¯X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' bS∗  ∂+ ¯  X ¯X] are naturally the same, see [80, Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  bT ∗∂− ¯  X ¯X  bT ∗∂+ ¯  X ¯X  bS∗ ¯X  bS∗ ¯X  ¯X × {0}  bT ∗∂− ¯  X ¯X  bS∗ ¯X  bS∗ ¯X  ¯X × {0}  bT ∗∂+ ¯  X ¯X  ff=[scT ∗∂+ ¯  X ¯X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' o∂+ ¯  X]  ff=[scT ∗∂+ ¯  X ¯X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' o∂+ ¯  X]  Figure 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' The left hand side is the fiber-compactification bT ∗ ¯X of the b-cotangent bundle  bT ∗ ¯X, and the right hand side is its blow-up [bT ∗ ¯X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' bS∗  ∂+ ¯  X ¯X] at the fiber infinity at the  boundary ∂+ ¯X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' The interior of the front face of the blow-up, shown by curved arcs, can be  identified with [scT ∗  ∂+ ¯  X ¯X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' o∂+ ¯  X].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  In order to actually do analysis, one needs to introduce the pseudodifferential operator space Ψs,r,l  sc,b( ¯X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  The three orders of Ψs,r,l  sc,b ( ¯X) are the sc-diffferential order s, the sc-decay order r and b-decay order l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' The  operators Ψs,r,l  sc,b( ¯  X) arise from the quantization of the symbols in the following class  Ss,r,l( ¯X) := ρ−l  b ρ−r  sc ρ−s  ∞ S0,0,0([bT ∗ ¯X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' bS∗  ∂+ ¯  X ¯X]) = ρ−l  b ρ−r  sc ρ−s  ∞ S0,0(bT ∗ ¯X)  where ρ∞, resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' ρsc, resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' ρb correspond to the boundary defining functions of the three boundary hypersur-  faces of [bT ∗ ¯X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' bS∗  ∂+ ¯  X ¯X] (or equivalently [scT ∗ ¯X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' o∂+ ¯  X]): the lift of b-fiber infinity (or equivalently the lift of  sc-fiber infinity), the new front face of [bT ∗ ¯X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' bS∗  ∂+ ¯  X ¯X] (or the lift of scT ∗  ∂+ ¯  X ¯X in [scT ∗ ¯X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' o∂+ ¯  X]), and the lift  of ∂+ ¯X in [bT ∗ ¯X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' bS∗  ∂+ ¯  X ¯X] (or the new front face of [scT ∗ ¯X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' o∂+ ¯  X]), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Here, the symbol spaces  S0,0(bT ∗ ¯X) (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' S0,0,0([bT ∗ ¯X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' bS∗  ∂+ ¯  X ¯X])) consist of L∞(bT ∗ ¯X) (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' L∞([bT ∗ ¯X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' bS∗  ∂+ ¯  X ¯X])) functions  which are smooth away from the boundary hypersurfaces bS∗ ¯X ∪ bT ∗∂+ ¯  X ¯X (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' bS∗ ¯X ∪ bT ∗∂+ ¯  X ¯X ∪ ff),  and they remain so under iterated applications of C∞ b-vector fields on bT ∗ ¯X (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' [bT ∗ ¯X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' bS∗ ¯X]), namely,  vector fields tangent to the aforementioned boundary hypersurfaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' In particular, we have  Ψs,s+l,l  sc,b  ( ¯X) = Ψs,l  b ( ¯X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  LINEAR STABILITY OF KERR-NEWMAN FAMILY  45  We refer the reader to [80, §5] for a more detailed description of the operators Ψs,r,l  sc,b( ¯X) and the symbols  Ss,r,l( ¯X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  Correspondingly, one can define the second microlocal Sobolev space Hs,r,l  sc,b as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Fix an elliptic  operator A ∈ Ψs,r,l  sc,b( ¯  X), let  Hs,r,l  sc,b ( ¯X) = {u ∈ C−∞( ¯  X) | Au ∈ L2}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  Therefore, we have  Hs,s+l,l  sc,b  ( ¯X) = Hs,l  b ( ¯X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Uniform Fredholm estimates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' In this section, we prove the uniform Fredholm estimates first for  operators �  □gb(σ) acting on scalar functions, then for �  Pb,γ(σ), �  Wb,γ(σ) acting on scattering 1-forms and the  linearized gauge-fixed Einstein-Maxwell operator �  Lb,γ(σ), as well as their formal adjoints for σ ∈ C, |σ| ≤ C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  To this end, we combine the global dynamics of of the Hamiltonian flow of the principal symbol pσ established  in 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='4 with the elliptic estimate, propagation of singularities estimate, the radial point estimate at event  horizon, the scattering radial point estimate at spatial infinity ∂+ ¯X and hyperbolic estimate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Uniform Fredholm estimates for scalar wave operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' We first prove uniform Fredholm estimates for  the operator �  □gb(σ) acting on scalar functions, as well as its formal adjoint with respect to volume density  L2( ¯X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  �  det |gb|drdθdϕ) for σ ∈ C, |σ| ≤ C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  B+  B−  B0  E0  Σsc(σ)  R(σ)  Bσ  R(0)  L+  L−  suppχ  B′  +  E′  +  B′  −  E′  −  L+  L−  Σsc(σ)  R(σ)  R(0)  B′  0  B′  σ  E′  σ  suppχ  supp(1 − χ)  Figure 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' A phase space picture of the proof of the estimate (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='23), on the left, and (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='24),  on the right.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' The coordinates and notations L±, R(0), R(σ) are the same as in Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  For (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='23), we use hyperbolic estimates to control u via χu;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' χ is controlled (modulo elliptic  estimates) by B±, B0, Bσ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' B0 is controlled by E0 using low b-decay radial point estimates  and E0 is again controlled by Bσ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' finally B±, Bσ are controlled using high regularity radial  points estimates at L± and high sc-decay radial point estimates at R(σ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' For (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='24), we  use hyperbolic estimates to bound (1 − χ)u;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' χ is controlled (modulo elliptic estimates) by  B′  ±, B′  0, B′  σ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' B′  ± is controlled by E′  ± using low regularity radial point estimates and E′  ± is  controlled by 1 − χ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' B′  σ is controlled by E′  σ using low sc-decay radial points estimates and  E′  σ is controlled by B′  0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' finally B′  0 is controlled using high b-decay radial points estimates  at R(0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Let b0 = (m0, a0, Q0) with |Q0| + |a0| < m0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  Then there exists ǫ > 0 such that for  b = (m, a, Q) with |b − b0| < ǫ and for s > s0 > 1  2, ℓ − 1 ≤ ℓ0 < ℓ < − 1  2 with s + ℓ > s0 + ℓ0 > − 1  2 and  ℓ ̸= − 3  2, the following holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  46  LILI HE  (1) For any fixed C1 > 0, there exist C > 0 independent of b, such that for σ ∈ C, Im σ ≥ 0, C−1  1  ≤ |σ| ≤  C1, we have  ∥u∥ ¯  Hs,ℓ  b  ( ¯  X) ≤ C  �  ∥�  □gb(σ)u∥ ¯  Hs,ℓ+1  b  ( ¯  X) + ∥u∥ ¯  Hs0,ℓ0  b  ( ¯  X)  �  ,  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='15)  ∥u∥ ˙H−s,−ℓ−1  b  ( ¯  X) ≤ C  �  ∥�  □gb(σ)∗u∥ ˙H−s,−ℓ  b  ( ¯  X) + ∥u∥ ˙H−N,−ℓ−3  b  ( ¯  X)  �  ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='16)  ∥u∥ ¯  Hs,ℓ  b  ( ¯  X) ≤ C  �  ∥�  □gb(σ)u∥ ¯  Hs−1,ℓ+2  b  ( ¯  X) + ∥u∥ ¯  Hs0,ℓ0  b  ( ¯  X)  �  ,  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='17)  ∥u∥ ˙H−s+1,−ℓ−2  b  ( ¯  X) ≤ C  �  ∥�  □gb(σ)∗u∥ ˙H−s,−ℓ  b  ( ¯  X) + ∥u∥ ˙H−N,−ℓ−3  b  ( ¯  X)  �  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='18)  where C only depends on b0, s, s0, ℓ, δ, C1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Therefore, the operators  �  □gb(σ) : {u ∈ ¯Hs,ℓ  b ( ¯X) | �  □gb(σ)u ∈ ¯Hs,ℓ+1  b  ( ¯X)} → ¯Hs,ℓ+1  b  ( ¯X)  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='19)  �  □gb(σ) : {u ∈ ¯Hs,ℓ  b ( ¯X) | �  □gb(σ)u ∈ ¯Hs−1,ℓ+2  b  ( ¯X)} → ¯Hs−1,ℓ+2  b  ( ¯X)  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='20)  are Fredholm for Im σ ≥ 0, σ ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  (2) Let ℓ ∈ (− 3  2, − 1  2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' For any fixed C1 > 0, there exist C > 0 independent of b, such that for σ ∈  C, Im σ ≥ 0, |σ| ≤ C1, we have  ∥u∥ ¯  Hs,ℓ  b  ( ¯  X) ≤ C  �  ∥�  □gb(σ)u∥ ¯  Hs−1,ℓ+2  b  ( ¯  X) + ∥u∥ ¯  Hs0,ℓ0  b  ( ¯  X)  �  ,  ∥u∥ ˙H−s+1,−ℓ−2  b  ( ¯  X) ≤ C  �  ∥�  □gb(σ)∗u∥ ˙H−s+1,−ℓ−2  b  ( ¯  X) + ∥u∥ ˙H−N,−ℓ−3  b  ( ¯  X)  �  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='21)  where C only depends on b0, s, s0, ℓ, δ, C1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Therefore, the operator  �  □gb(0) : {u ∈ ¯Hs,ℓ  b ( ¯X) | �  □gb(0)u ∈ ¯Hs−1,ℓ+2  b  ( ¯X)} → ¯Hs−1,ℓ+2  b  ( ¯X)  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='22)  is Fredholm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' We first prove the statement (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  We claim that it suffices to prove the following estimates for  s > s0 > 1  2, r > r0 > − 1  2, ℓ < − 1  2, ℓ ̸= − 3  2  ∥u∥ ¯  Hs,r,ℓ  sc,b ( ¯  X) ≤ C  �  ∥�  □gb(σ)u∥ ¯  Hs−1,r+1,ℓ+1  sc,b  ( ¯  X) + ∥u∥ ¯  Hs0,r0,ℓ−1  sc,b  ( ¯  X)  �  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='23)  and  ∥u∥ ˙H−s+1,−r−1,−ℓ−1  sc,b  ( ¯  X) ≤ C  �  ∥�  □gb(σ)∗u∥ ˙H−s,−r,−ℓ  sc,b  ( ¯  X) + ∥u∥ ˙H−N,−N,−ℓ−3  sc,b  ( ¯  X)  �  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='24)  Letting r = s + ℓ > − 1  2 and r0 = s0 + ℓ0 > − 1  2, we have  ∥u∥ ¯  Hs,s+ℓ,ℓ  sc,b  ( ¯  X) ≤ C  �  ∥�  □gb(σ)u∥ ¯  Hs−1,s+ℓ+1,ℓ+1  sc,b  ( ¯  X) + ∥u∥ ¯  Hs0,s0+ℓ0,ℓ0  sc,b  ( ¯  X)  �  and  ∥u∥ ˙H−s+1,−s−ℓ−1,−ℓ−1  sc,b  ( ¯  X) ≤ C  �  ∥�  □gb(σ)∗u∥ ˙H−s,−s−ℓ,−ℓ  sc,b  ( ¯  X) + ∥u∥ ˙H−N,−N,−ℓ−3  sc,b  ( ¯  X)  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  Then using the facts ¯Hs,s+ℓ,ℓ  sc,b  = ¯Hs,l  b , ˙Hs,s+ℓ,ℓ  sc,b  = ˙Hs,l  b ,  ¯Hs,ℓ+1  b  = ¯Hs,s+ℓ+1,ℓ+1  sc,b  ⊂ ¯Hs−1,s+ℓ+1,ℓ+1  sc,b  ,  ¯Hs−1,ℓ+2  b  = ¯Hs−1,s+ℓ+1,ℓ+2  sc,b  ⊂ ¯Hs−1,s+ℓ+1,ℓ+1  sc,b  and  ˙H−s,−ℓ−1  b  = ˙H−s,−s−ℓ−1,−ℓ−1  sc,b  ⊃ ˙H−s+1,−s−ℓ−1,−ℓ−1  sc,b  ,  ˙H−s+1,−ℓ−2  b  = ˙H−s+1,−s−ℓ−1,−ℓ−2  sc,b  ⊃ ˙H−s+1,−s−ℓ−1,−ℓ−1  sc,b  ,  we obtain (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='15), (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='16), (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='17) and (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='18).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Now we will show (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='23) and (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='24).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Here we only discuss the  case Re σ > 0 (see Figure 9 for a phase space illustration of the proof of estimates (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='23) and (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='24)), as the  case Re σ ≤ 0 can be handled in a similar manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  LINEAR STABILITY OF KERR-NEWMAN FAMILY  47  Proof of the estimate (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='23).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' For s > s0 >  1  2 ≥  1  2 − Im (σ) r2  b +a2  r2  b−m (as calculated (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='8)), using the  high regularity radial point estimate (see [77, Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='3], [78, Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='27]), there exist  B±, G±, S± ∈ Ψ0( ¯X) microlocally supported near L± with L± ⊂ ell(B±), L± ⊂ ell(S±) and satisfy-  ing that all the forward (backward) null characteristics from WF(B±) tend to L±, while remaining  in the elliptic set of G±, such that  ∥B±u∥Hs,r,ℓ  sc,b ( ¯  X) ≤ C∥G±�  □gb(σ)u∥Hs−1,r+1,ℓ+1  sc,b  ( ¯  X) + C∥S±u∥Hs0,r0,ℓ0  sc,b  ( ¯  X)  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='25)  where r0 < r, ℓ0 < ℓ and the sc-decay order r and b-decay order ℓ are actually irrelevant here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  For r > r0 > − 1  2 (as calculated in (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='14)), using the high sc-decay radial point estimates at the  non-zero scattering section R(σ) (see [79, §4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' The proof follows from a positive commutator estimate  i(�  □gb(σ)∗A − A�  □gb(σ)) = Im �  □gb(σ)A + AIm �  □gb(σ) + i[Re �  □gb(σ), A]  where  Re �  □gb(σ) =  �  □gb(σ) + �  □gb(σ)∗  2  ,  Im �  □gb(σ) =  �  □gb(σ) − �  □gb(σ)∗  2i  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  Let A ∈ Ψ2s−1,2r+1,2l+1  sc,b  ( ¯X) with principal symbol  a = χ0(ζ2  θ +  1  sin2 θζ2  ϕ)χ1((ζρ − 2Re z)2)ρ−2r+1(ζ2  θ +  1  sin2 θζ2  ϕ + ζ2  ρ)s− 1  2  where χ0, χ1 are identically 0 near 0 and have compact support sufficiently close to 0, and χ1  has relatively large support such that suppχ0 ∩ suppχ′  1 is disjoint from the characteristic set of  Re Ph(z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  Then using Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='1 and the calculation before Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='8 in [79], it follows that  (Im �  □gb(σ)A+AIm �  □gb(σ)+i[Re �  □gb(σ), A]) ∈ Ψ2s,2r,2l  sc,b  ( ¯X) whose principal symbol is positive definite  elliptic near R(σ) if r > − 1  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' This, together with a regularization argument, proves high sc-decay  radial point estimates at R(σ)), there exist Bσ, Gσ, Sσ ∈ Ψ0,0  sc ( ¯X) microlocally supported near R(σ)  with R(σ) ⊂ ellsc(Bσ), R(σ) ⊂ ellsc(Sσ) and satisfying that all the forward null characteristics from  WFsc(Bσ) tend to R(σ), while remaining in the scattering elliptic set of Gσ, such that  ∥Bσu∥Hs,r,ℓ  sc,b ( ¯  X) ≤ C∥Gσ �  □gb(σ)u∥Hs−1,r+1,ℓ+1  sc,b  ( ¯  X) + C∥Sσu∥Hs0,r0,ℓ0  sc,b  ( ¯  X)  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='26)  where s0 < s, ℓ0 < ℓ and the differential order s and b-decay order ℓ are actually irrelevant here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  For ℓ < − 1  2 (as calculated in (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='14)), using the low b-decay radial point estimates at the zero  scattering section R(0) (see [79, §4]), there exist B0, G0 ∈ Ψ0,0,0  sc,b ( ¯X) microlocally supported near  R(0) (in fact near the blown up R(0) in the second microlocalized space introduced in §5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='5) with  R(0) ⊂ ellsc,b(B0) and E0 ∈ Ψ0,0,0  sc,b ( ¯X), WFsc,b(E0)∩R(0) = ∅, and satisfying that all the forward null  characteristics from WFsc,b(B0)\\R(0) enter ellsc,b(E0), while remaining in the second microlocalized  scattering elliptic set of G0, such that for any sufficiently large N > 0  ∥B0u∥Hs,r,ℓ  sc,b ( ¯  X) ≤ Ch−1∥G0�  □gb(σ)u∥Hs−1,r+1,ℓ+1  sc,b  ( ¯  X) + C∥E0u∥Hs,r,ℓ  sc,b ( ¯  X) + C∥u∥ ¯  H−N,−N,ℓ  sc,b  ( ¯  X)  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='27)  where the differential order s is actually irrelevant here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  Let r− < r0 < rb and χ ∈ C∞  c ( ¯X) with χ = 1 near r ≥ rb and suppχ ⊂ {r ≥ r0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' By part  (1) in Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='4, propagation of singularities estimates and elliptic estimates (Concretely, if  (r, θ, ϕ, ξ)( or (ρ, θ, ϕ, ζ)) ∈ WF(χ)∩(Σ∪Σsc)c, we use elliptic estimates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' If (r, θ, ϕ, ξ)( or (ρ, θ, ϕ, ξ)) ∈  WF(χ) ∩ (Σ ∪ Σsc), then there exists s ∈ R with exp(sscH2,0  ph,z)(ρ, θ, ϕ, ξ) ∈ ellh(B±) ∪ ellh(B1) ∪  ellh(B0) ∪ ellh(Bz), and thus we use propagation of singularities estimates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' We point out that in  the set {Re psc(σ) = 0} ∩ ∂+ ¯X, the scattering symbol has a nonnegative imaginary part, so one can  propagate estimates towards R(0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Finally, by using a pseudodifferential partition of unity, χ can be  written as s sum of operators falling into the above two case), we have  ∥χu∥Hs,r,ℓ  sc,b ( ¯  X) ≤ C∥�  □gb(σ)u∥ ¯  Hs−1,r+1,ℓ+1  sc,b  ( ¯  X) + C∥B+u∥Hs,r,ℓ  sc,b ( ¯  X) + C∥B−u∥Hs,r,ℓ  sc,b ( ¯  X)  + C∥B0u∥Hs,r,ℓ  sc,b ( ¯  X) + C∥Bσu∥Hs,r,ℓ  sc,b ( ¯  X) + C∥u∥ ¯  H−N,−N,ℓ  sc,b  ( ¯  X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='28)  By the same reasoning as above in the control of χu, it follows that  ∥E0u∥Hs,r,ℓ  sc,b ( ¯  X) ≤ C∥�  □gb(σ)u∥ ¯  Hs−1,r+1,ℓ+1  sc,b  ( ¯  X) + C∥Bσu∥Hs,r,ℓ  sc,b ( ¯  X) + C∥u∥ ¯  H−N,−N,ℓ  sc,b  ( ¯  X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='29)  48  LILI HE  Putting all the above estimates together yields for s > s0 > 1  2, r > r0 > − 1  2, ℓ < − 1  2 and for any  sufficiently large N > 0  ∥χu∥Hs,r,ℓ  sc,b ( ¯  X) ≤ C∥�  □gb(σ)u∥ ¯  Hs−1,r+1,ℓ+1  sc,b  ( ¯  X) + C∥u∥ ¯  Hs0,r0,−N  sc,b  ( ¯  X) + C∥u∥ ¯  H−N,−N,ℓ  sc,b  ( ¯  X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='30)  Since  p = −ρ−2  b  �  ∆b  �  ξr + aξϕ  ∆b  �2 − a2ξ2  ϕ  ∆b  + ˜p  �  where  ˜p = ξ2  θ +  1  sin2 θ ξ2  ϕ  is hyperbolic with respect to r in the region {r− ≤ r < rb} (see [21, definition E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='55]), using the  hyperbolic estimates (see [21, Theorem E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='56]), we have for all s, r, ℓ ∈ R  ∥(1 − χ)u∥ ¯  Hs,r,ℓ  sc,b ( ¯  X) ≤ C∥�  □gb(σ)u∥ ¯  Hs−1,r+1,ℓ+1  sc,b  ( ¯  X) + ∥χu∥ ¯  Hs,r,ℓ  sc,b ( ¯  X)  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='31)  where the sc-decay order r and b-decay order ℓ are actually relevant here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Combining (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='30) with  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='31) yields for s > s0 > 1  2, r > r0 > − 1  2, ℓ < − 1  2 and for any sufficiently large N > 0  ∥u∥ ¯  Hs,r,ℓ  sc,b ( ¯  X) ≤ C∥�  □gb(σ)u∥ ¯  Hs−1,r+1,ℓ+1  sc,b  ( ¯  X) + C∥u∥ ¯  Hs0,r0,−N  sc,b  ( ¯  X) + C∥u∥ ¯  H−N,−N,ℓ  sc,b  ( ¯  X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='32)  Since near ∂+ ¯X  �  □gb(σ) − N(�  □gb(σ)) ∈ ρ2Diff2  b( ¯X),  N(�  □gb(σ)) = −2iσρ(ρ∂ρ − 1),  it follows that from [79, Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='13 and Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='16] that for ℓ < − 1  2  ∥u∥ ¯  H−N,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='−N,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='ℓ  sc,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='b  ( ¯  X) ≤ ∥u∥ ¯  H−N−ℓ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='−N,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='ℓ  sc,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='b  ( ¯  X) = ∥u∥ ¯  H−N−ℓ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='ℓ  b  ( ¯  X) ≤ C∥N(�  □gb(σ))u∥ ¯  H−N−ℓ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='ℓ+1  b  ( ¯  X)  ≤ C∥�  □gb(σ)u∥ ¯  H−N−ℓ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='ℓ+1  b  ( ¯  X) + C∥u∥ ¯  H−N−ℓ+2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='ℓ−1  b  ( ¯  X)  = C∥�  □gb(σ)u∥ ¯  H−N−ℓ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='−N+1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='ℓ+1  sc,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='b  ( ¯  X) + C∥u∥ ¯  H−N−ℓ+2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='−N+1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='ℓ−1  sc,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='b  ( ¯  X),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  and thus for s > s0 > 1  2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' r > r0 > − 1  2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' ℓ < − 1  2  ∥u∥ ¯  Hs,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='ℓ  sc,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='b ( ¯  X) ≤ C∥�  □gb(σ)u∥ ¯  Hs−1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='r+1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='ℓ+1  sc,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='b  ( ¯  X) + C∥u∥ ¯  Hs0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='r0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='ℓ−1  sc,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='b  ( ¯  X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='33)  This finishes the proof of (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='23).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  Proof of the estimate (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='24).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' For s >  1  2 ≥  1  2 − Im (σ) r2  b +a2  r2  b−m , we have 1 − s ≤  1  2 − Im (¯σ) r2  b +a2  r2  b−m .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  Then using the low regularity radial point estimates (see [77, Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='4], [78, Proposition  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='27]), there exist B′  ±, G′  ± ∈ Ψ0( ¯  X) microlocally supported near L± with L± ⊂ ell(B′  ±) and E′  ± ∈  Ψ0( ¯X), WF(E′  ±) ∩ L± = ∅, and satisfying that all the backward (forward) null characteristics from  WF(B′  ±) \\ L± reach ell(E′  ±), while remaining in the elliptic set of G′  ±, such that for any N ∈ R  ∥B′  ±u∥H1−s,−r−1,−ℓ−1  sc,b  ( ¯  X)  ≤ C∥G′�  □gb(σ)∗u∥H−s,−r,−ℓ  sc,b  ( ¯  X) + C∥E′  ±u∥H1−s,−r−1,−ℓ−1  sc,b  ( ¯  X) + C∥u∥ ˙H−N,−N,−N  sc,b  ( ¯  X)  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='34)  where the sc-decay order r and b-decay order ℓ are actually irrelevant here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  For r > − 1  2, we have −r−1 < − 1  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Then using the low sc-decay radial point estimates at the non-  zero scattering section R(σ) (see [79, §4] and the above discussion about the proof of high sc-decay  radial point estimates at R(σ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' We note that the only difference is that now the term involving χ′  0  does not have the correct sign and needs to be treated as an error term E′  σu),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' there exist B′  σ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' G′  σ ∈  Ψ0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='0  sc ( ¯X) microlocally supported near R(σ) with R(σ) ⊂ ellsc(Bσ) and E′  σ ∈ Ψ0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='0  sc ( ¯X),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' WFsc(E′  σ) ∩  R(σ) = ∅,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' and satisfying that all the backward null characteristics from WFsc(B′  σ) \\ R(σ) reach  ellsc,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='h(E′  σ),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' while remaining in the scattering elliptic set of G′  σ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' such that  ∥B′  σu∥H1−s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='−r−1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='−ℓ−1  sc,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='b  ( ¯  X)  ≤ C∥G′  σ �  □gb(σ)∗u∥H−s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='−r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='−ℓ  sc,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='b  ( ¯  X) + C∥E′  σu∥H1−s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='−r−1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='−ℓ−1  sc,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='b  ( ¯  X) + C∥u∥ ˙H−N,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='−N,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='−N  sc,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='b  ( ¯  X)  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='35)  where the differential order s and b-decay order ℓ are actually irrelevant here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  LINEAR STABILITY OF KERR-NEWMAN FAMILY  49  For ℓ < − 1  2, we have −ℓ − 1 > − 1  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Then using the high b-decay radial point estimates at the  zero scattering section R(0) (see [79, §4]), there exist B′  0, G′  0 ∈ Ψ0,0,0  sc,b ( ¯X) microlocally supported  near R(0) (in fact near the blown up R(0) in the second microlocalized space introduced in §5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='5)  with R(0) ⊂ ellsc,b(B′  0), and satisfying all the backward null characteristics from WFsc,b(B′  0) tend  to R(0), while remaining in the second microlocalized scattering elliptic set of G′  0, such that for any  N ∈ R  ∥B′  0u∥H1−s,−r−1,−ℓ−1  sc,b  ( ¯  X) ≤ C∥G′  0�  □gb(σ)∗u∥H−s,−r,−ℓ  sc,b  ( ¯  X) + C∥u∥ ˙H−N,−N,−ℓ−1  sc,b  ( ¯  X)  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='36)  where the differential order s is actually irrelevant here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  By the same reasoning as in the proof of the estimate (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='23), we have  ∥χu∥H1−s,−r−1,−ℓ−1  sc,b  ( ¯  X)  ≤ C∥�  □gb(σ)∗u∥ ˙H−s,−r,−ℓ  sc,b  ( ¯  X) + C∥B+u∥H1−s,−r−1,−ℓ−1  sc,b  ( ¯  X)  + C∥B′  −u∥H1−s,−r−1,−ℓ−1  sc,b  ( ¯  X) + C∥B′  0u∥H1−s,−r−1,−ℓ−1  sc,b  ( ¯  X)  + C∥B′  σu∥H1−s,−r−1,−ℓ−1  sc,b  ( ¯  X) + C∥u∥ ˙H−N,−N,−ℓ−1  sc,b  ( ¯  X),  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='37)  and  ∥E′  ±u∥H1−s,−r−1,−ℓ−1  sc,b,h  ( ¯  X) + ∥E′  σu∥H1−s,−r−1,−ℓ−1  sc,b,h  ( ¯  X)  ≤ C∥�  □gb(σ)∗u∥ ˙H−s,−r,−ℓ  sc,b,h  ( ¯  X) + C∥(1 − χ)u∥ ˙H1−s,−r−1,−ℓ−1  sc,b  ( ¯  X)  + C∥B′  0u∥H1−s,−r−1,−ℓ−1  sc,b  ( ¯  X) + C∥u∥ ˙H−N,−N,−ℓ−1  sc,b  ( ¯  X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='38)  Putting all the above estimates together yields for s > 1  2, r > − 1  2, ℓ < − 1  2  ∥χu∥H1−s,−r−1,−ℓ−1  sc,b  ( ¯  X) ≤ C∥�  □gb(σ)∗u∥ ˙H−s,−r,−ℓ  sc,b  ( ¯  X) + C∥(1 − χ)u∥ ˙H1−s,−r−1,−ℓ−1  sc,b  ( ¯  X)  + C∥u∥ ˙H−N,−N,−ℓ−1  sc,b  ( ¯  X),  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='39)  Again using the semiclassical hyperbolic estimates (see [21, Theorem E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='56]), we have for all s, r, ℓ ∈ R  ∥(1 − χ)u∥ ˙H1−s,−r−1,−ℓ−1  sc,b,h  ( ¯  X) ≤ C∥�  □gb(σ)∗u∥ ˙H−s,−r,−ℓ  sc,b  ( ¯  X)  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='40)  where the sc-decay order r and b-decay order ℓ are actually relevant here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Combining (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='39) with  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='40) yields for s > 1  2, r > − 1  2, ℓ < − 1  2  ∥u∥ ˙H1−s,−r−1,−ℓ−1  sc,b  ( ¯  X) ≤ C∥�  □gb(σ)∗u∥ ˙H−s,−r,−ℓ  sc,b  ( ¯  X) + C∥u∥ ˙H−N,−N,−ℓ−1  sc,b  ( ¯  X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='41)  By the same argument as in the proof of the estimate (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='23), we have for s > 1  2, r > − 1  2, ℓ < − 1  2  ∥u∥ ˙H1−s,−r−1,−ℓ−1  sc,b  ( ¯  X) ≤ C∥�  □gb(σ)∗u∥ ˙H−s,−r,−ℓ  sc,b  ( ¯  X) + C∥u∥ ˙H−N,−N,−ℓ−2  sc,b  ( ¯  X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='42)  This proves the estimate (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='24).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  Fredholm property of �  □gb(σ) for Im σ ≥ 0, σ ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' We only discuss the proof of (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='19) in detail as  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='20) can be handled in a completely analogous manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' We define the space  X(σ) := {u ∈ ¯Hs,ℓ  b ( ¯X) | �  □gb(σ)u ∈ ¯Hs,ℓ+1  b  ( ¯X)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  We note that X(σ) is a Hilbert space endowed with the norm ∥u∥X (σ) := ∥u∥ ¯  Hs,ℓ  b + ∥�  □gb(σ)u∥ ¯  Hs,ℓ+1  b  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  Let {uj} is a bounded sequence in ker �  □gb(σ) ⊂ X(σ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Since ¯Hs,ℓ  b  → ¯Hs0,ℓ0  b  is a compact embedding,  by passing to a subsequence we may assume that uj converges in ¯Hs0,ℓ0  b  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' By (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='15), we see that  uj is a Cauchy sequence in ¯Hs,ℓ  b .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' This implies that ker �  □gb(σ) is finite dimensional.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' We then show  that ranX (σ)�  □gb(σ) is closed in ¯Hs,ℓ+1  b  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Let {fj} ⊂ ranX (σ)�  □gb(σ) be a convergent sequence with  fj → f∞ ∈ ¯Hs,ℓ+1  b  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' We may write fj = �  □gb(σ)uj where uj ∈ X(σ) is in the orthogonal complement  of ker �  □gb(σ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' We claim that {uj} is bounded in X(σ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' (Otherwise, suppose that {uj} is unbounded  in X(σ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Passing to a subsequence, we may assume that ∥uj∥X (σ) → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Let ˜uj = uj/∥uj∥X (σ)  and ˜fj = �  □gb(σ)˜uj = fj/∥uj∥X (σ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Then ∥˜uj∥X (σ) = 1 and ∥ ˜fj∥ ¯  Hs,ℓ+1  b  → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' By the argument  50  LILI HE  as above, passing to a subsequence, we have that ˜uj → ˜u∞ ∈ X(σ) and thus �  □gb(σ)˜u∞ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' On  the other hand, since ˜u∞ is in the orthogonal complement of ker �  □gb(σ), we have �  □gb(σ)˜u∞ ̸= 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  this gives a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=') Then argue as above, by passing to a subsequence we may assume that  uj → u∞ ∈ X(σ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' It follows that �  □gb(σ)uj → �  □gb(σ)u∞ ∈ ¯Hs,ℓ+1  b  and thus �  □gb(σ)u∞ = f∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' This  proves that f∞ ∈ ranX (σ)�  □gb(σ) and thus ranX (σ)�  □gb(σ) is closed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Finally, it remains to show that  ranX (σ)�  □gb(σ) has finite codimension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Proceeding as in the proof of the finite dimension property of  ker �  □gb(σ) and using (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='16), we obtain that ker �  □gb(σ)∗ ⊂ ˙H−s,−ℓ−1  b  is also finite dimensional.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Let  f ∈ ¯Hs,ℓ+1  b  be such that ⟨f, v⟩ = 0 for all v ∈ ker �  □gb(σ)∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' We claim that there exists u ∈ ¯Hs,ℓ  b  such  that �  □gb(σ)u = f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' (This implies that ranX (σ)�  □gb(σ) has finite codimension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=') We define  Y(σ) := {v ∈ ˙H−s,−ℓ−1  b  ( ¯X) | �  □gb(σ)∗v ∈ ˙H−s,−ℓ  b  ( ¯X)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  Let V be a complementary space of ker �  □gb(σ)∗ in ˙H−s,−ℓ−1  b  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Then there exists a constant C2 such  that  ∥v∥ ˙H−s,−ℓ−1  b  ≤ C2∥�  □gb(σ)∗v∥ ˙H−s,−ℓ  b  ,  v ∈ Y(σ) ∩ V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  If this were not true, we could find a sequence vj ⊂ Y(σ) ∩ V such that  ∥vj∥ ˙H−s,−ℓ−1  b  = 1,  ∥vj∥ ˙H−s,−ℓ−1  b  ≥ j∥�  □gb(σ)vj∥ ˙H−s,−ℓ  b  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  By (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='16) and the compact embedding ˙H−s,−ℓ−1  b  → ˙H−N,−ℓ−3  b  , passing to a subsequence we have  vj → v∞ with ∥v∞∥ ˙H−s,−ℓ−1  b  = 1 and v∞ ∈ V ∩ ker �  □gb(σ)∗;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' this gives a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' If f ∈ ¯Hs,ℓ+1  b  satisfies ⟨f, v⟩ = 0 for all v ∈ ker �  □gb(σ)∗, we have  |⟨f, v⟩| ≤ C2∥f∥ ¯  Hs,ℓ+1  b  ∥�  □gb(σ)∗v∥ ˙H−s,−ℓ  b  ,  v ∈ Y(σ) ∩ V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  Since any v ∈ Y(σ) can be written as v = v1 + v2 with v1 ∈ ker �  □gb(σ)∗, v2 ∈ Y(σ) ∩ V , it follows  that  |⟨f, v⟩| = |⟨f, v2⟩| ≤ C2∥f∥ ¯  Hs,ℓ+1  b  ∥�  □gb(σ)∗v2∥ ˙H−s,−ℓ  b  = C2∥f∥ ¯  Hs,ℓ+1  b  ∥�  □gb(σ)∗v∥ ˙H−s,−ℓ  b  ,  v ∈ Y(σ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  By Hahn-Banach Theorem, the anti-linear form �  □gb(σ)∗v → ⟨f, v⟩ with v ∈ Y(σ) can be extended  to a continuous anti-linear form on ˙H−s,−ℓ  b  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Since ( ˙H−s,−ℓ  b  )∗ = ¯Hs,ℓ  b , there exists u ∈ ¯Hs,ℓ  b  such that  ⟨u, �  □gb(σ)∗v⟩ = ⟨f, v⟩ for v ∈ Y(σ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' In particular, for v ∈ C∞  c (X◦)  ⟨�  □gb(σ)u, v⟩ = ⟨u, �  □gb(σ)∗v⟩ = ⟨f, v⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  This proves that �  □gb(σ)u = f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  We now prove statement (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Combining the estimates near spatial infinity ∂+ ¯X (see [81, Propo-  sition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='3]) with the radial point estimates at event horizon, propagation of singularity estimates,  elliptic estimates and hyperbolic estimates as in the proof of statement (1), we obtain (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='21).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' The  Fredholm property follows as in the nonzero σ case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  □  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Uniform Fredholm estimates for tensor wave operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Now we prove an analogy to Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='5  for the operators �  Pb,γ(σ), �  Wb,γ(σ) and �  Lb,γ(σ) acting on bundles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  We note that the formal adjoint of  Pb,γ, Wb,γ, Lb,γ with respect to the natural inner product induced by gb and the volume form dvolgb is  Wb,γ,  Pb,γ,  �Ggb  0  0  4  �  Lb,γ  �Ggb  0  0  1  4  �  ,  respectively (see (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='4) and (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='5) for the derivation of the adjoint of Lb,γ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Let b0 = (m0, a0, Q0) with |Q0|+|a0| < m0 and |a0| ≪ |m0|+|Q0|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Then there exists ǫ > 0  such that for b = (m, a, Q) with |b − b0| < ǫ and for s > s0 > 2 (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' s > s0 > 3), ℓ − 1 ≤ ℓ0 < ℓ < − 1  2  with s + ℓ > s0 + ℓ0 > − 1  2 and ℓ ̸= − 3  2, the conclusions in Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='5 hold for �  Pb,γ(σ) and �  Wb,γ(σ) (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  �  Lb,γ(σ)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  LINEAR STABILITY OF KERR-NEWMAN FAMILY  51  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' The proof is analogous to that of Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='5, except for the computation of threshold regularity  in the radial point estimate at event horizon and the threshold decay rate in the radial point estimate at  spatial infinity ∂+ ¯X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  As for the calculation of threshold regularity in the radial point estimate at event horizon, the Reissner-  Nordstr¨om metric case gb = g(m0,0,Q0) was done in appendix C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' According to (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='21) and (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='8), for Im σ ≥ 0,  the threshold regularity is 3  2 − 2γ  κ for �  Pb0,γ(σ) and 3  2 + γ  2κ for �  Wb0,γ(σ) acting on 1-forms, and is 5  2 − 2γ  κ for  �  Lb,γ(σ) acting on S2 �  scT ∗ ¯X ⊕ �  scT ∗ ¯X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' This implies that the threshold regularity for nearby Kerr-Newman  metrics is close to 3  2 for �  Pb,γ(σ) and �  Wb,γ(σ), and close to 5  2 for �  Lb,γ(σ) as γ is a sufficiently small constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  For 0 ̸= σ ∈ R, the radial point estimate at R(0) and R(σ) for �  Pb,γ(σ), �  Wb,γ(σ), �  Lb,γ(σ) requires the  computation of a threshold decay rate relative to L2( ¯X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Concretely, the threshold − 1  2 from [79, Theorems 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='1  and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='3] is modified by the subprincipal symbol σsc( 1  2iρ(�•(σ) − �•(σ)∗))|R(0),R(σ) where • = Pb,γ, Wb,γ, Lb,γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  Working in the trivialization of S2 �  scT ∗ ¯X ⊕ �  scT ∗ ¯X given in terms of the differentials of standard coordinates  t, x1, x2, x3, according to Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='6 and Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='11, we see that  σsc( 1  2iρ(�•(σ) − �•(σ)∗))|R(0),R(σ) = σsc( 1  2iρ( �  □gb,0(σ) − �  □gb,0(σ)∗) ⊗ Id4×4)|R(0),R(σ) = 0,  = P, W,  σsc( 1  2iρ( �  Lb,γ(σ) − �  Lb,γ(σ)∗))|R(0),R(σ) = σsc( 1  2iρ( �  □gb,0(σ) − �  □gb,0(σ)∗) ⊗ Id14×14)|R(0),R(σ) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  Therefore, the threshold decay rate is still − 1  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  □  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Description of the kernel of the wave type operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' In this section, we give a detailed descrip-  tion of kernel of the following wave operators: �  □gb(σ) on scalar functions, �  Pb,γ(σ), �  Wb,γ(σ) on scattering  1-forms and the linearized gauge-fixed Einstein-Maxwell operator �  Lb,γ(σ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Let b = (m, a, Q) with |Q| + |a| < |m| and a = |a| ≪ |m| + |Q|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Suppose u ∈ ¯Hs,ℓ  b ( ¯X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' C)  with s > 1  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  (1) If �  □gb(0)u = 0 and u ∈ ¯Hs,ℓ  b ( ¯X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' C) with ℓ ∈ R, then u ∈ ¯H∞,ℓ  b  ( ¯X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' C) and has a polyhomogeneous  expansion with index set contained in {(z, k) | z ∈ iZ, k ∈ N}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' In particular, if − 3  2 < ℓ < − 1  2, then  u ∈ A1( ¯X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' More precisely, there exists u0 ∈ C∞(∂+ ¯X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' C) such that u − u0  r ∈ A2−( ¯X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  (2) If �  □gb(0)∗u = 0 and u ∈ ˙H−∞,ℓ  b  ( ¯  X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' C) with ℓ ∈ R, then u ∈ ˙H  1  2 −,ℓ  b  ( ¯X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Near ∂+ ¯X, u has the  same polyhomegeneous expansion as in part (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  (3) If σ ̸= 0 and u ∈ ker �  □gb(σ) ∩ ¯Hs,ℓ  b ( ¯X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' C) for some ℓ ∈ R with s + ℓ > − 1  2, then u ∈ ρC∞(∂+ ¯X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' C) +  A2−( ¯X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  The above statements also hold for �  Pb,γ(σ), �  Wb,γ acting on ω ∈ ¯Hs,ℓ  b ( ¯X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' �  scT ∗ ¯X) with s > 2 (for statement  (2), ω ∈ ˙H  − 1  2 −C(γ,a),ℓ  b  ) and �  Lb,γ(σ) acting on (˙g, ˙A) ∈ ¯Hs,ℓ  b ( ¯X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' �  scT ∗ ¯X ⊕ S2 �  scT ∗ ¯X) with s > 3 (for statement  (2), ω ∈ ˙H  − 3  2 −C(γ,a),ℓ  b  ) where C(γ, a) > 0 is a sufficiently small constant depending on γ, a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' We only discuss the proof for the operator �  □gb(σ) in detail here as the other operators �  Pb,γ(σ), �  Wb,γ,  �  Lb,γ(σ) can be handled similarly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  Proof of statement (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' According to the high regularity radial point estimate (since s > 1  2 and 1  2  is the threshold regularity at the radial points at event horizon), 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='4, propagation of singularity  estimate and elliptic estimate for b-pseudodifferential operators, we conclude that u ∈ H∞,ℓ  b  ( ¯X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  To prove the polyhomogeneous expansion of u, we exploit a typical normal operator argument  (see [64, §5]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Concretely, since  �  □gb(0) = (ρ2∂ρ)2 − 2ρ3∂ρ + ρ2(∂2  θ +  1  sin2 θ∂2  ϕ) + ρ3Diff2  b = ρ2�  ρ∂ρ(ρ∂ρ − 1) + /∆  �  + ρ3Diff2  b ∈ ρ2Diff2  b,  the normal operator N(�  □gb(0)) of �  □gb(0) is defined by factoring out an overall weight (here is ρ2)  and freezing the coefficients at the boundary ρ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Therefore, we have  N(�  □gb(0)) = ρ2�  ρ∂ρ(ρ∂ρ − 1) + /∆  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  52  LILI HE  Let χ(ρ) ∈ C∞  c (R) be identically 1 near 0 and vanishing when ρ = 1/r ≥ 1/3m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Then we have  ρ−2N(�  □gb(0))(χu) = ρ−2�  N(�  □gb(0)) − �  □gb(0)  �  (χu) + ρ−2[�  □gb(0), χ]u := f ∈ ¯H∞,ℓ+1  b  ( ¯X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='43)  For simplicity of notation, we denote ρ−2N(�  □gb(0)) by N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Using the Mellin transform in ρ, defined  by (we drop the dependence on the angular variables on ∂+ ¯X from the notation)  �  χu(ξ) :=  � ∞  0  ρ−iξ(χu)(ρ) dρ  ρ ,  the equation N(χu) = f becomes  �  N(ξ)  �  �  χu  �  (ξ) =  �  iξ(iξ − 1) + /∆  �  (�  χu)(ξ) = ˆf(ξ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  At this point, we only know that �  χu(ξ) is holomorphic and Schwartz in Re ξ (with Im ξ fixed) in  the region Im ξ > −ℓ − 3  2 and so is ˆf(ξ) in Im ξ > −ℓ − 5  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Therefore, we have the following inverse  Mellin transform  χu(ρ) = 1  2π  �  Im ξ=−ℓ− 3  2 +ǫ  ρiξ �  χu(ξ) dξ,  ǫ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  We denote by Y m  l  with l ∈ N, m ∈ Z, |m| ≤ l the spherical harmonic function of degree l and  order m which satisfies /∆Y m  l  = −l(l + 1)Y m  l .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' We denote by Sl = {Y m  l  : |m| ≤ l} the space of degree  l spherical harmonic functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Then one can conclude that {Sl : l ∈ N} form an orthogonal basis  of L2(S2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Therefore, one can expand (�  χu)(ξ) and ˆf(ξ) in terms of spherical harmonics Y m  l , i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=', we  write (�  χu)(ξ) = � um  l (ξ)Y m  l , ˆf = � f m  l (ξ)Y m  l .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Restricted to Sl, the inverse  �  N(ξ)−1 =  �  iξ(iξ − 1) + /∆  �−1  =  �  iξ(iξ − 1) − l(l + 1))−1  is meromorphic in ξ with simple poles given by ξ = −i(l+1), il.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Then in the inverse Mellin transform  χu(ρ) = 1  2π  �  Im ξ=−ℓ− 3  2 +ǫ  ρiξ( �  N(ξ))−1 ˆf(ξ) dξ,  ǫ > 0,  −ℓ − 3  2 + ǫ /∈ Z,  we can shift the contour of integration through the pole at ξ = ik to Im ξ = −ℓ − 5  2 + ǫ where k ∈ Z  and k ∈ (−ℓ − 3  2 + ǫ, −ℓ − 5  2 + ǫ), and the Residue Theorem gives  χu(ρ) = ρ−kuk + ˜uk,  ˜uk = 1  2π  �  Im ξ=−ℓ− 5  2 +ǫ  ρiξ( �  N(ξ))−1 ˆf(ξ) dξ ∈ Aℓ+ 5  2 −ǫ( ¯X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' C)  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='44)  where  uk =  ��  −k≤m≤k  −1  2k+1f m  k (ik)Y m  k ,  k ≥ 0  �  k+1≤m≤−k−1  1  2k+1f m  −k−1(−ik − i)Y m  −k−1,  k ≤ −1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  To deduce the full polyhomogeneous expansion for χu, we plug the above partial expansion (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='44)  into the equation (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='43) to obtain  N ˜uk = N(χu) = f1 + f2,  f1 ∈ ρ−k+1C∞(∂+ ¯X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' C)  and  f2 ∈ Aℓ+ 7  2 −ǫ( ¯X, C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  Now we solve the following two equations N ˜uj  k = fj, j = 1, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' First, using the Mellin transform as  before, we see that  ˜u2  k = ρ−k+1uk+1 + ˜uk  where  uk+1 ∈  �  Sk−1,  k − 1 ≥ 0  S−k,  k − 1 ≤ −1  and  ˜uk+1 ∈ Aℓ+ 7  2 −ǫ( ¯X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  As for u1  k, we can solve N ˜u1  k = f1 more directly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Since any function in C∞(∂+ ¯X) can be expressed  as a linear combination of Sl, it suffices to solve N ˜u1  k = f1 for f1 ∈ ρ−kSl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' More explicitly,  u1  k =  �  (k(k + 1) − l(l + 1))−1f1,  k(k + 1) ̸= l(l + 1)  −(2k + 1)−1f1 ln ρ,  k(k + 1) = l(l + 1) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  Therefore, we have  χu = ρ−kuk + u1  k + ρ−k+1uk+1 + ˜uk+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  LINEAR STABILITY OF KERR-NEWMAN FAMILY  53  Plugging the partial expansion into the equation (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='43) and proceeding as above iteratively give the  full polyhomogeneous expansion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  Proof of statement (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Near ∂+ ¯X, according to the elliptic estimate for b-pseudodifferential opera-  tor, we conclude that u ∈ ¯H∞,ℓ  b  there.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Away form spatial infinity ∂+ ¯X, since u is 0 in the region  r < rb, by Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='4, propagation of singularity estimate and elliptic estimate we see that u is  smooth away from the radial points at event horizon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Finally, using the low regularity radial point  estimate yields u ∈ ˙H  1  2 −,ℓ  b  ( ¯X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' The polyhomogeneous expansion of u near ∂+ ¯X can be obtained  as in the proof of statement (1) by exploiting the normal operator argument.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  Proof of statement (3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Since s + ℓ > − 1  2, according to the high sc-decay estimate at the radial  point R(σ), propagation of singularity estimate and elliptic estimate, we see that away from the  radial point R(0), u is in ¯H∞,ℓ  b  ( ¯X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  Near R(0) the low b-decay radial point estimate gives  u ∈ ¯H  ∞,min{− 1  2 −,ℓ}  b  ( ¯X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' C) at R(0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Therefore, we have u ∈ ¯H  ∞,min{− 1  2 −,ℓ}  b  ( ¯  X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Noe since σ ̸=  0, we have �  □gb(σ) = −2iσρ(ρ∂ρ − 1) + ρ2Diff2  b, and thus N(�  □gb(σ)) = −2iσρ(ρ∂ρ − 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  Then  using the typical normal operator argument as in the proof of statement (1), we obtain that u =  ρC∞(∂+ ¯X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' ¯X) + A2−( ¯  X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  Proof of �  Pb,γ(σ), �  Wb,γ and �  Lb,γ(σ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Near the spatial infinity, we work in the standard coordinate  trivialization of �  scT ∗ ¯X for �  Pb,γ(σ), �  Wb,γ (which is given in terms of the differentials dt, dx1, dx2, dx3  of the standard coordinates) and S2 �  scT ∗ ¯X ⊕ �  scT ∗ ¯X for �  Lb,γ(σ) (which is given in terms of the  differentials dt2, dtdxi, dxidxj, dt, dxi1 ≤ j ≤ i ≤ 3 of the standard coordinates).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  Then according to Propositions 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='6 and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='13, the normal operator of 2 �  Pb,γ(σ), 2�  Wb,γ (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  �  Lb,γ(σ)) is, −ρ2�  ρ∂ρ(ρ∂ρ − 1) + /∆  �  for σ = 0 and 2iσρ(ρ∂ρ − 1) for σ ̸= 0, tensored with 4 × 4  identity matrix (reps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' 14 × 14 identity matrix).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Therefore, the proof for these operators follows in a  completely analogous manner as before.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  □  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Semiclassical behavior of Kerr-Newman spacetimes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Recall that  �  □g(σ) := eitb,∗σ□gbe−itb,∗σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  where the inverse metric g−1 is given by  g−1 = ρ−2  b  �  ∆b∂2  r + χ2 − 1  ∆b  �  (r2 + a2)∂tχ + a∂ϕ  �2  + ∂2  θ +  1  sin2 θ  �  ∂ϕ + a sin2 θ∂tχ  �2  + 2χ  �  (r2 + a2)∂tχ + a∂ϕ  �  ∂r  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='45)  We introduce the semiclassically rescaled version of the operator �  □g(σ):  h2 �  □g(h−1z),  h = 1  |σ|  and  z = Re σ  |σ| + iIm σ  |σ| .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='46)  When Im σ is bounded, we write z = zR + ihzI and parametrize h2 �  □g(h−1z) by zR and zI = h−1Im z rather  than Re z and Im z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' This still gives a family of operators in Diff2  sc,h which depends smoothly on zR, zI, and its  semiclassical principal symbol does not depend on zI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Therefore, when analyzing the semiclassical principal  symbol of h2 �  □g(h−1z), it suffices to consider the case z ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  For z ∈ R, the semiclassical principal symbol of h2 �  □g(h−1z) is given by  ph,z(ξ) := σh(h2 �  □g(h−1z)) = −g−1(−zdtb,∗ + ξ, −zdtb,∗ + ξ),  ξ ∈ ∂⊥  tb,∗ = scT ∗ ¯X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='47)  As before, away from ∂+ ¯X, we write the scattering covectors as  ξ := ξrdr + ξθdθ + ξϕdϕ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  While near ∂+ ¯X, we write them as  ξ := ξρ  dρ  ρ2 + ξθ  dθ  ρ + ξϕ  dϕ  ρ ,  ρ = 1  r .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  54  LILI HE  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Characteristic set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' We first study the characteristic set  Σh := {⟨ξ⟩−2ph,z(ξ) = 0} ⊂ scT ∗ ¯X,  ⟨ξ⟩ =  �  1 + ξ2r + ξ2  θ +  1  sin2 θξ2ϕ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  We will show that for z ∈ R \\ 0, it splits into two components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' To achieve this, we make use of an auxiliary  covector dt which is timelike everywhere on ¯X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='8 (Splitting of the characteristic set).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' There exist closed z-dependent sets Σ±  h ⊂ scT ∗ ¯X such that  {⟨ξ⟩−2ph,z(ξ) = 0} = Σ+  h ⊔ Σ−  h ,  z ∈ R \\ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='48)  Moreover,  ± ⟨ξ⟩−1g−1(−zdtb,∗ + ξ, dt) > 0  on  Σ±  h  for  z ∈ R \\ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='49)  Finally,  (Σ±  h ∩ scS∗ ¯X) ∩ {∆b − a2 sin2 θ > 0} = ∅  for  z ∈ R;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='50)  Σ±  h ∩ {∆b − a2 sin2 θ > 0} = ∅  for  ∓ z > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='51)  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' We put  Σ±  h := {⟨ξ⟩−2ph,z(ξ) = 0} ∩ {±⟨ξ⟩−1g−1(−zdtb,∗ + ξ, dt) ≥ 0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  It is clear that Σ±  h are closed and their union is the characteristic set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' In order to show that Σ+  h ∩ Σ−  h = ∅  and (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='49) for z ∈ R \\ 0, we need  {⟨ξ⟩−2ph,z(ξ) = 0} ∩ {⟨ξ⟩−1g−1(−zdtb,∗ + ξ, dt) = 0} = ∅  for  z ∈ R \\ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  On the fiber infinity scS∗ ¯X, ⟨ξ⟩−1g−1(ξ, dt) = 0 implies that ⟨ξ⟩−1ξ is spacelike or 0 as dt is timelike.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  If the intersection were not empty at scS∗ ¯X, ⟨ξ⟩−1ξ must be 0, which is impossible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Also, the fact that  ⟨ξ⟩−2ph,z(ξ) = ⟨ξ⟩−2p(ξ) = 0 gives (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='50).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  In the interior scT ∗ ¯X, if the intersection were not empty, since dt is timelike, g−1(−zdtb,∗ + ξ, dt) = 0  implies that −zdtb,∗ + ξ must be spacelike or 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Then the equality g−1(−zdtb,∗ + ξ, −zdtb,∗ + ξ) = 0 forces  −zdtb,∗ + ξ to be 0, which cannot happen for ξ ∈ ∂⊥  tb,∗ and z ∈ R \\ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  On the region {∆b − a2 sin2 θ > 0}, we see that g−1(ξ, ξ) > 0 for 0 ̸= ξ ∈ scT ∗ ¯X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  For any p ∈  {∆b − a2 sin2 θ > 0}, let {dt, X1, X2, X3} be an orthogonal basis of T ∗  p ¯  M with G|p(Xi, Xi) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' We write  −zdtb,∗ + ξ = (a0 − z)dt +  3  �  i=1  aiXi,  and then g−1(−zdtb,∗+ξ, −zdtb,∗+ξ) = 0 gives (a0−z)2G(dt, dt)+�3  i=1 a2  i = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Since −zdtb,∗+ξ+zdt ∈ T ∗  p ¯X,  it follows that g−1(−zdtb,∗ + ξ + zdt, −zdtb,∗ + ξ + zdt) = a2  0G(dt, dt) + �3  i=1 a2  i > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Therefore, we have  z2 − 2a0z > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Then  ±g−1(−zdtb,∗ + ξ, dt) = ±(a0 − z)g−1(dt, dt) = ±(a0 − z  2 − z  2)g−1(dt, dt) > 0  for  ± z > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  This finishes the proof of (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='51).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  □  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' The radial points at event horizon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Now in the fiber radial compactification scT ∗ ¯X of scT ∗ ¯X, we use  the following coordinates near the fiber infinity scS∗ ¯X,  ρξ = (ξ2  r + ξ2  θ +  1  sin2 θξ2  ϕ)− 1  2 ∈ [0, ∞),  ˆξ = (ˆξr, ˆξθ, ˆξϕ) = ρξ(ξr, ξθ, ξϕ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='52)  We note that ρξ is a boundary defining function of scT ∗ ¯X, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=', ρξ = 0 at scS∗ ¯X, and  ˆξ ∈ scS∗ ¯X = {ˆξ2  r + ˆξ2  θ +  1  sin2 θ  ˆξ2  ϕ = 1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  We now analyze the behavior of the Hamiltonian flow exp(sρξHph,z) on the characteristic set Σh = Σ+  h ∪Σ−  h ,  concentrating on the behavior away from ∂+ ¯X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' With tb,∗ = tχ + c(r), we write  ph,z = −ρ−2  b  �  ∆b  �  ξr − zc′(r) + χ  �  aξϕ − (r2 + a2)z  �  ∆b  �2 − 1  ∆b  �  aξϕ − (r2 + a2)z  �2 + ˜ph,z  �  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='53)  LINEAR STABILITY OF KERR-NEWMAN FAMILY  55  where  ˜ph,z = ξ2  θ +  1  sin2 θ(ξϕ − a sin2 θz)2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='54)  Then we calculate  Hph,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='z = −ρ−2  b  ��  2∆b  �  ξr − zc′(r)  �  + 2χ  �  aξϕ − (r2 + a2)z  ��  ∂r +  �  2aχ2 − 1  ∆b  �  aξϕ − (r2 + a2)z  �  + 2χa  �  ξr − c′(r)z  ��  ∂ϕ  − ∂(−ρ2  bph,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='z)  ∂r  ∂ξr + H˜ph,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='z  �  − ρ−2  b ph,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='zHρ2  b  where  ∂(−ρ2  bph,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='z)  ∂r  = 2∆b  �  ξr − zc′(r) + χ  �  aξϕ − (r2 + a2)z  �  ∆b  �  ∂r  �  ξr − zc′(r) + χ  �  aξϕ − (r2 + a2)z  �  ∆b  �  + ∂∆b  ∂r  �  ξr − zc′(r) + χ  �  aξϕ − (r2 + a2)z  �  ∆b  �2 − ∂r  � 1  ∆b  �  aξϕ − (r2 + a2)z  �2�  In the coordinates (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='52) near the fiber infinity scS∗ ¯X, we compute  ρξHph,z = −ρ−2  b  ��  2∆b  �ˆξr − ρξzc′(r)  �  − 2ρξχ(r2 + a2)z + 2χaˆξϕ  �  ∂r  +  �  2aχ2 − 1  ∆b  �  aˆξϕ − ρξ(r2 + a2)z  �  + 2χa  �ˆξr − ρξc′(r)z  ��  ∂ϕ  − ρξ  ∂(−ρ2  bph,z)  ∂r  ∂ξr + ρξH˜ph,z  �  − ρ−2  b (ρ2  ξph,z)ρ−1  ξ Hρ2  b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='55)  We let  L± := {ρξ = 0, ˆξr = ±1, (ˆξθ, ˆξϕ) = 0, ∆b = 0} ⊂  �  Σ±  h ∩ scS∗ ¯X  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  Now we describe the property of the set L±.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' L± are invariant under the Hamiltonian flow exp(sρξHph,z) on the characteristic set Σh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  Moreover, L+ is a radial sink and L− is a radial source for the flow exp(sρξHph,z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' We rewrite  L± = {∆b = 0, ρξ = 0, ˆξr = ±1, ρ2  ξ ˜ph,z = 0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  Using the coordinates (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='52), the expression (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='55) and the facts that c(r) = 0, χ = 1 near L±, we find that  near L±  ρξHph,zρξ = −ρ−2  b  �∂∆b  ∂r  ˆξ3  r − 4rρξ ˆξ2  rz + H˜ph,zρξ  �  ρξ − ρ−2  b (ρ2  ξph,z)ρ−1  ξ Hρ2  bρξ,  ρξHph,z ˆξϕ = (Hph,zρξ)ˆξϕ,  ρξHph,z∆b = −2ρ−2  b  ∂∆b  ∂r  �  ∆b ˆξr − ρξ(r2 + a2)z + aˆξϕ  �  ,  ρξHph,z(ρ2  ξ ˜ph,z) = 2(Hph,zρξ)ρ2  ξ ˜ph,z − ρ−2  b (ρ2  ξph,z)ρξHρ2  b ˜ph,z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='56)  This implies that ρξHp is tangent to L±, so L± is invariant under the flow exp(sρξHph,z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Moreover, in a  neighborhood of L± = {∆b = 0, ρξ = 0, ˆξr = ±1, ρ2  ξ ˜ph,z = 0} ⊂ {ρ2  ξph,z = 0}, we have  ±ρξHpρ2  ξ ≤ −C0ρ2  ξ,  ±ρξHp ˆξ2  ϕ ≤ −C0ˆξ2  ϕ,  ±ρξHp∆2  b ≤ −C0∆2  b + C1(ρ2  ξ + ˆξ2  ϕ)  ±ρξHp(ρ2  ξ ˜p) ≤ −C0ρ2  ξ ˜p + C1(ρ2  ξ + ∆2  b + ˆξ2  ϕ)  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='57)  where C0, C1 > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' It follows from (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='4) that in a sufficiently small neighborhood of L±  ± ρξHpf ≤ −Cf,  f = ρ2  ξ + ˆξ2  ϕ + C2∆2  b + C3ρ2  ξ ˜p ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='58)  for some C, C2, C3 > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Proceeding as in the proof of Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='2 finishes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  □  56  LILI HE  Finally, since Hph,zρξ = Hpρξ and  ρξσh  �  h−1Im h2�  □gb(h−1z)  �  = Im (h−1z)ρ−2  b  �  2(r2 + a2)ˆξr + 2aˆξϕ)  �  = ρξσ1  �  Im �  □gb(h−1z)  �  at  L±,  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='59)  the calculation of the threshold regularity at L± in the microlocal case applies here as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' The radial points at spatial infinity ∂+ ¯X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' We now turn to the analysis near ∂+ ¯X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Recall that near  ∂+ ¯X, the semiclassical principal symbol of h−2 �  □g(h−1z) for z ∈ R is given by  ph,z(ζ) = −g−1(−zdtb,∗ + ξ, −zdtb,∗ + ξ)  = −(ξρ − z)2 + z2 − ˜p + ρ  �  0≤j,k,m≤2  0≤j+k+m≤2  ajkm(ρ, θ, ϕ)ξj  ρξk  θ ξm  ϕ ,  ˜p =  �  ξ2  θ +  1  sin2 θξ2  ϕ  �  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='60)  where ξ = ξρ  dρ  ρ2 + ξθ dθ  ρ + ξϕ  dϕ  ρ and ajkm ∈ A0( ¯X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Then the scattering Hamiltonian vector field associated  to ph,z in ρ > 0 is given by  scHph,z = ρ  �∂ph,z  ∂ξρ  �  ρ ∂  ∂ρ + (  �  µ=θ,ϕ  ξµ  ∂  ∂ξµ  )  �  −  �  ρ∂ph,z  ∂ρ  + (  �  µ=θ,ϕ  ξµ  ∂ph,z  ∂ξµ  )  � ∂  ∂ξρ  +  �  µ=θ,ϕ  �∂ph,z  ∂ξµ  ∂  ∂µ − ∂ph,z  ∂µ  ∂  ∂ξµ  ��  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  We introduce the rescaled scattering Hamiltonian vector field (here we drop the factor ⟨ρξ⟩ again)  scH2,0  ph,z := ρ−1scHph,z  = (−2ξρ + 2z)ρ∂ρ + 2˜p∂ξρ + (−2ξρ + 2z)  �  µ=θ,ϕ  ξµ∂ξµ − H˜p + ρV,  V ∈ Vb(scT ∗ ¯X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='61)  We now discuss the integral curves of scH2,0  ph,z on the semiclassical characteristic set Σh for z ∈ R \\ 0 near  ∂+ ¯X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' For z ∈ R\\0, we put φs(ρ, θ, ϕ, ξ) = exp(sscH2,0  ph,z)(ρ, θ, ϕ, ξ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Then there exists a neighborhood  V± ⊂ scT ∗ ¯X of  R(z  2 ± |z|  2 ) = {ρ = 0, ξ = (z ± |z|)dρ  ρ2 }  such that uniformly in (ρ, θ, ϕ, ξ) ∈ V± ∩ Σh, one has  φs(ρ, θ, ϕ, ξ) → R(z  2 ± |z|  2 )  as  s → ±∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  This implies that R( z  2 + |z|  2 ) is a radial sink while R( z  2 − |z|  2 ) is a radial source.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Recall that  R(z  2 ± |z|  2 ) = {ρ = 0, ˜p = 0, ξρ = z ± |z|}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  Using the expression (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='61), we find that in a neighborhood U 2ǫ  ± = {(ξρ − (z ± |z|))2 + ρ2 + ˜p < 2ǫ} of  R( z  2 ± |z|  2 ) where ǫ > 0 is a sufficiently small constant,  ±scH2,0  ph,zρ2 ≤ −C0ρ2,  ±scH2,0  ph,z ˜p ≤ −C0˜p  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='62)  for some C0 > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' According to the expression (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='60) of ph,z, there exists ǫ′ > 0 such that  {ρ2 + ˜p < ǫ′} ∩ Σh ⊂ U ǫ  + ∪ U ǫ  −  where U ǫ  ± are neighborhoods of R( z  2± |z|  2 ) satisfying U ǫ  +∩U ǫ  − = ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Then for (ρ, θ, ϕ, ξ) ∈ U ǫ  ±∩{ρ2+˜p < ǫ′}∩Σh,  since ph,z is a constant along the integral curves of scH2,0  ph,z, we have  φs(ρ, θ, ϕ, ξ) ⊂ U ǫ  ± ∩ Σh  for  ± s ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='63)  Indeed, if (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='63) failed, we could find ±s0 > 0 such that  φs0(ρ, θ, ϕ, ξ) ∈ U 2ǫ  ±  and  ± scH2,0  ph,z(ρ2 + ˜p)(φs0(ρ, θ, ϕ, ξ)) ≥ 0  LINEAR STABILITY OF KERR-NEWMAN FAMILY  57  which contradicts (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='62).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Proceeding as in the proof of Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='2, it follows that uniformly in (ρ, θ, ϕ, ξ) ∈  U ǫ  ± ∩ Σh, one has  φs(ρ, θ, ϕ, ξ) → {ρ = 0, ˜p = 0} ∩ (U ǫ  ± ∩ Σh) = R(z  2 ± |z|  2 )  as  s → ±∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  □  Finally, since ph,z at ∂+ ¯X equals to psc(z), the calculation of the threshold scattering decay order at the  radial points in the microlocal case applies here as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Structure of the trapped sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' We now study the structure of the trapped sets for KN spacetime with  |a|2 + Q2 < m2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  First, we define and investigate the incoming and outgoing tails and the trapped set  associated to the Hamiltonian flow exp(sscH2,0  ph,z) where ph,z is the semiclassical symbol given by  ph,z = −ρ−2  b  �  ∆b  �  ξr − zc′(r) + χ  �  aξϕ − (r2 + a2)z  �  ∆b  �2 − 1  ∆b  �  aξϕ − (r2 + a2)z  �2 + ˜ph,z  �  where  ˜ph,z = ξ2  θ +  1  sin2 θ(ξϕ − a sin2 θz)2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  To define them, we need an escape function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Let f(r) ∈ C∞([r−, ∞)) be defined as  f(r) = ∆b(r)  r4  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='64)  Then there exists δ0 = δ0(m, a, Q) > 0 such that for (r, θ, ϕ, ξ) ∈ scT ∗ ¯X \\ (L± ∪ ∂+ ¯X) and z ∈ R \\ 0  f(r) < δ0,  ⟨ξ⟩−2ph,z(r, θ, ϕ, ξ) = 0,  ⟨ξ⟩−1Hph,zf(r, θ, ϕ, ξ) = 0 =⇒ (⟨ξ⟩−1Hph,z)2f(r, θ, ϕ, ξ) < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='65)  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' We first show (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='65) on the fiber infinity scS∗ ¯X where ⟨ξ⟩−1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' According to (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='50) in Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='8,  we know that  {⟨ξ⟩−2ph,z = 0} ∩ scS∗ ¯X ⊂ {r ≥ r− | ∆b ≤ a2 sin2 θ} ⊂ {r− ≤ r ≤ m +  �  m2 − Q2}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  A direct calculation gives  f ′(r) = r∂r∆b(r) − 4∆b(r)  r5  = −2r2 − 6mr + 4a2 + 4Q2  r5  > 0  when  r ∈ [r−, m +  �  m2 − Q2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  Therefore, ⟨ξ⟩−1Hph,zf = 0 implies ⟨ξ⟩−1Hph,zr = 0, that is,  ⟨ξ⟩−1Hph,zr = −ρ−2  b ⟨ξ⟩−1�  2∆b  �  ξr − zc′(r)  �  − 2χ(r2 + a2)z + 2χaξϕ  �  = 0  on  scS∗ ¯X ∩ Σh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  This is equivalent to  ∆b ˆξr = −χaˆξϕ  where ˆξ is defined as in (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='52).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Plugging this back to ⟨ξ⟩−2ph,z = 0 yields  ⟨ξ⟩−2ph,z = −ρ−2  b  �  − ∆b ˆξ2  r + χ2 − 1  ∆b  a2ˆξ2  ϕ + ˆξ2  θ +  1  sin2 θ  ˆξ2  ϕ  �  = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  Since χ = 1 on [r−, rb], it follows that  {⟨ξ⟩−2ph,z = 0, ⟨ξ⟩−1Hph,zf = 0} ∩ scS∗ ¯X = L±.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  On {⟨ξ⟩−2ph,z = 0, ⟨ξ⟩−1Hph,zf = 0, r > rb} ∩ scS∗ ¯X, we calculate  (⟨ξ⟩−1Hph,z)2f = −2ρ−2  b f ′(r)∆b⟨ξ⟩−2Hph,zξr = −2f ′(r)  ρ4  b∆b  ∂∆b  ∂r a2 ˆξ2  ϕ < 0  where we use the fact that ⟨ξ⟩−1Hph,z⟨ξ⟩−1 = 0 on scS∗ ¯X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  We next prove that in the region r− ≤ r ≤ rb where f(r) ≤ 0, ⟨ξ⟩−2ph,z and ⟨ξ⟩−1Hph,zf cannot vanish  at the same time in the interior scT ∗ ¯X (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' away from the fiber infinity).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Suppose that Hph,zf = 0 and  ph,z = 0 in r− ≤ r ≤ rb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Then we have  Hph,zf = −ρ−2  b f ′(r)  �  2∆b  �  ξr − zc′(r)  �  − 2χ(r2 + a2)z + 2χaξϕ  �  = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  58  LILI HE  Since f ′(r) > 0 when r ∈ [r−, rb], Hph,zf = 0 implies  ∆b  �  ξr − zc′(r)  �  = χ(r2 + a2)z − χaξϕ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  Plugging this into ph,z = 0 and using the fact that χ = 1 in [r−, rb] yield  −ρ−2  b  �  − ∆b  �  ξr − zc′(r)  �2 + ˜ph,z  �  = 0  where  ˜ph,z = ξ2  θ +  1  sin2 θ(ξϕ − a sin2 θz)2,  and thus  ∆b(ξr − c′(r)z) = 0,  ξθ = 0,  ξϕ = a sin2 θz  in  r ∈ [r−, rb].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  Therefore, for z ∈ R \\ 0  Hph,zf = ρ−2  b f ′(r)(2r2z + 2a2 cos2 θz) ̸= 0  in  r ∈ [r−, rb],  which is a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  We now turn to the region r > rb where f > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' We return to the analysis of  f ′(r) = −2r2 − 6mr + 4a2 + 4Q2  r5  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  It is clear that f ′(r) < 0 when r > 3m and f ′(r) > 0 when rb = m+  �  m2 − a2 − Q2 ≤ r <  3m+√  9m2−8a2−8Q2  2  ,  so in the region f(r) < δ with δ = f(3m), we have f ′(r) ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Therefore, in the region f(r) < δ = f(3m),  Hph,zf = 0 is equivalent to Hph,zr = 0, which is given by  ∆b  �  ξr − c′(r)z  �  − χ  �  (r2 + a2)z − aξϕ  �  = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  Under the conditions that ph,z = 0 and Hph,zf = 0, we calculate  H2  ph,zf = −2ρ−2  b f ′(r)∆bHph,zξr = −2f ′(r)  ρ4  b∆b  (aξϕ − (r2 + a2)z)  �∂∆b  ∂r (aξϕ − (r2 + a2)z) + 4zr∆b  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  Next, we let  A = aξϕ − (r2 + a2)z,  B = ξϕ − a sin2 θz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  Then we have z = −ρ−2  b (A − aB) and  H2  ph,zf = −2f ′(r)  ρ6  b∆b  A  �∂∆b  ∂r Aρ2  b − 4r∆b(A − aB)  �  = −2f ′(r)  ρ6  b∆b  ��∂∆b  ∂r ρ2  b − 4r∆b  �  A2 + 4ar∆bAB  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  Using ph,z = 0 and Hph,zf = 0, we obtain  ph,z = −ρ−2  b  �  − 1  ∆b  A2 + ξ2  θ +  B2  sin2 θ  �  = 0  and thus  A2 ≥ ∆bB2  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='66)  where we use the fact that B = ξϕ = 0 when sin θ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' We also note that ph,z = 0 and z ∈ R \\ 0 imply  A ̸= 0 since otherwise B = 0 and z = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' We now calculate  �∂∆b  ∂r ρ2  b − 4r∆b  �  A2 + 4ar∆bAB < −2r(r2 − 3mr + (a2 + 2Q2))A2 + 4|a|r∆b|AB|  < −2r(r2 − 3mr + (a2 + 2Q2))A2 + 4mr∆1/2  b  A2  < −2r(r2 − 5mr + (a2 + 2Q2))A2  where we use |a| < m and A2 ≥ ∆bB2 in the second step and ∆b < r2 for r ≥ rb in the third step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' As a  consequence, H2  ph,zf < 0 when r > 5m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  We now assume that rb < r ≤ 5m and f(r) < δ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Then we have  ∆b(r) = r4f(r) ≤ (5m)4f(r) < (5m)4δ′  ∂r∆b = 2r − 2m > 2  �  m2 − a2 − Q2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  By choosing δ′ > 0 sufficiently small such that  2m  �  m2 − a2 − Q2 − 4(5m)4δ′ − 4m(5m)2√  δ′ ≥ 0,  LINEAR STABILITY OF KERR-NEWMAN FAMILY  59  we conclude that on {rb < r ≤ 5m} ∩ {f(r) < δ′}  r5f ′(r) = r∂r∆b − 4∆b > 2m  �  m2 − a2 − Q2 − 4(5m)4δ′ ≥ 0,  �∂∆b  ∂r ρ2  b − 4r∆b  �  A2 + 4ar∆bAB ≥ r  �∂∆b  ∂r r − 4∆b − 4m∆1/2  b  �  A2  > m  �  2m  �  m2 − a2 − Q2 − 4(5m)4δ′ − 4m(5m)2√  δ′�  A2 ≥ 0,  and thus H2  ph,zf < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Then letting δ0 = min{f(5m), δ′} finishes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  □  We now define the outgoing/incoming tails and the trapped set (see [19, Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='1], [21, Definition  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='1]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Recall that  scH2,0  ph,z = ρ−1⟨ξ⟩−1scHph,z,  ρ = r−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  Definition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Let f be the function defined as in Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Let (r, θ, ϕ, ξ) ∈ scT ∗ ¯X and φ(s) =  exp(sscH2,0  ph,z)(r, θ, ϕ, ξ) be a maximally extended integral curve with domain of definition I ⊂ R on Σh =  {⟨ξ⟩−2ph,z = 0} ⊂ scT ∗ ¯X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Let s− = inf I, s+ = sup I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' We say that φ(s) is trapped as s → ±∞, if there  exists δ > 0 and T > 0 such that f(φ(s)) > δ for all ±s ≥ T (as a consequence, s± = ±∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' We define the  incoming tail Γ− and the outgoing tail Γ+ as  Γ∓ = {(r, θ, ϕ, ξ) ∈ scT ∗ ¯X | φ(s) = exp(sscH2,0  ph,z)(r, θ, ϕ, ξ) is trapped as s → ±∞}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  Then we define the trapped set as K := Γ− ∩ Γ+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  It follows from the definition that Γ±, K are invariant under the flow exp(sscH2,0  ph,z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' We next study the  properties of Γ± and K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' We need the following lemmas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' For z ∈ R \\ 0, we have  ± ⟨ξ⟩−1Hph,zr > 0  on  (Σ±  h ∩ {r− ≤ r ≤ rb}) \\ L±.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='67)  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' We first consider the case r− ≤ r < rb and finite ξ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Since χ = 1, c′(r) = 0 on {r− ≤ r < rb}, we write  ph,z = 0 as  ph,z = −ρ−2  b  �  ∆bξ2  r − 2(r2 + a2)ξrz + 2aξrξϕ + ˜ph,z  �  = 0  where  ˜ph,z = ξ2  θ +  1  sin2 θ(ξϕ − a sin2 θz)2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  This is a quadratic equation in ξr with positive discriminant on {r− ≤ r < rb} since ∆b < 0 there.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' As  a consequence, it has two distinct roots ξ−  r  < 0 < ξ+  r and ±∂ξrph,z > 0 at ξ±  r .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' By Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='8, Σ±  h is  characterized by the conditions ±g−1(−zdtχ + ξ, dt) > 0 (because tb,∗ = tχ in the region r− ≤ r ≤ rb),  that is, with t = tχ + b(r) (according to the definition of t in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='15), we see that b′(r) ≤ 0 in the region  r− ≤ r ≤ rb), we have  ± ξr > ±b′(r)  �  (r2 + a2)z − aξϕ  �  + a  �  a sin2 θz − ξϕ  �  r2 + a2 + b′(r)∆b(r)  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='68)  If g−1(−zdtχ + ξ, dt) = 0, then −zdtχ + ξ is a nonzero spacelike vector and thus ph,z(ξ) = −g−1(−zdtχ +  ξ, −zdtχ + ξ) < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' That is, substituting the right-hand side of (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='68) into ph,z yields a negative number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  Therefore,  (r, θ, ϕ, ξ±  r , ξθ, ξϕ) ∈ Σ±  h  and  ± ⟨ξ⟩−1Hph,zr = ±⟨ξ⟩−1∂ξrph,z > 0  at  ξ±  r .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  In case of fiber infinity scS∗ ¯X ∩ {r− ≤ r < rb}, using the coordinates (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='52), we write  ρ2  ξph,z = −ρ−2  b  �  ∆b ˆξ2  r + 2aˆξr ˆξϕ + ˆξ2  θ +  1  sin2 θ  ˆξ2  ϕ  �  and  ρξg−1(−zdtχ + ξ, dt) = ρ−2  b  ��  r2 + a2 + b′(r)∆b(r)  �ˆξr +  �  a(1 + b′(r))  �ˆξϕ  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  Then the proof proceeds as in the finite ξ case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  It remains to consider the case r = rb where ∆b = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' For finite ξ, we write  ph,z = −ρ−2  b  �  − 2(r2 + a2)ξrz + 2aξrξϕ + ˜ph,z  �  = 0  where  ˜ph,z = ξ2  θ +  1  sin2 θ(ξϕ − a sin2 θz)2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  60  LILI HE  Since aξϕ − (r2 + a2)z ̸= 0 on Σh (otherwise a2 sin2 θ = r2 + a2 at rb, which is impossible), the right-hand  side of ph,z is a linear equation in ξr with nonzero leading coefficient and thus has only one root.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Again, by  Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='8, Σ±  h is characterized by the conditions ±g−1(−zdtχ + ξ, dt) > 0, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=',  ± ξr > ±−  �  a(1 + b′(r))  �  ξϕ + z  �  a2 sin2 θ + (r2 + a2)b′(r)  �  r2 + a2  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='69)  Similarly, substituting the right-hand side of (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='69) into ph,z yields a negative number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Therefore, we have  either  (r, θ, ϕ, ξr, ξθ, ξϕ) ∈ Σ+  h  and  ⟨ξ⟩−1Hph,zr = ⟨ξ⟩−1∂ξrph,z > 0  at  ξr,  or  (r, θ, ϕ, ξr, ξθ, ξϕ) ∈ Σ−  h  and  ⟨ξ⟩−1Hph,zr = ⟨ξ⟩−1∂ξrph,z < 0  at  ξr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  In case of fiber infinity scS∗ ¯X ∩ {r = rb}, using the coordinates (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='52), we write  ρ2  ξph,z = −ρ−2  b  �  2aˆξr ˆξϕ + ˆξ2  θ +  1  sin2 θ  ˆξ2  ϕ  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  Since ˆξϕ ̸= 0 on ({⟨ξ⟩−2ph,z = 0, r = rb} ∩ scS∗ ¯X) \\ L±, the right-hand side of ρξph,z s a linear equation in  ˆξr with nonzero leading coefficient and thus has only one root.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Then the proof proceeds as in the finite ξ  case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  □  Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Let z ∈ R\\0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Let δ0 > 0 and f(r) be defined as in Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Let (r, θ, ϕ, ξ) ∈ scT ∗ ¯X \\  (L± ∩ ∂+ ¯X) and φ(s) = exp(sscH2,0  ph,z)(r, θ, ϕ, ξ) be a maximally extended flow on Σh = {⟨ξ⟩−2ph,z = 0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  (1) If f(φ)(s0) < δ0 and ±∂sf(φ(s)) ≤ 0 for some ±s0 ≥ 0, then f(φ(s)) < δ0 and ±∂sf(φ(s)) < 0 for  ±s > s0 and φ(s) is not trapped as ±s → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  (2) If φ(s) is not trapped as ±s → ∞, then either f(φ(s)) < δ0 and ±∂sf(φ(s)) ≤ 0 for all sufficiently  large ±s, or φ(s) → L± (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' f(s) → 0) as ±s → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  Moreover,  (3) If φ(s) is not trapped as s → ∞, then for φ(s) ⊂ Σ−  h , φ(s) crosses ∂− ¯X into the inward direction of  decreasing r at some finite time s0 < ∞ or ρ(φ(s)) → 0 as s → ∞, while for φ(s) ⊂ Σ+  h , φ(s) → L+  or ρ(φ(s)) → 0 as s → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  (4) If φ(s) is not trapped as s → −∞, then for φ(s) ⊂ Σ−  h , φ(s) → L− or ρ(φ(s)) → 0 as s → −∞,  while for φ(s) ⊂ Σ+  h , φ(s) crosses ∂− ¯X into the inward direction of decreasing r at some finite time  s0 > −∞ or ρ(φ(s)) → 0 as s → −∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' We only prove the case ∂sf(φ(s)) ≤ 0 as the other case −∂sf(φ(s)) ≤ 0 can be handled similarly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' We  put  f(s) = f(φ(s)),  ˙f(s) = ∂sf(φ(s)) = scH2,0  ph,zf(φ(s)),  ¨f(s) = ∂2  sf(φ(s)) = (scH2,0  ph,z)2f(φ(s)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  Let s > s0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' If f(s) attains its minimum of on the interval [s0, s] at some point s1 ∈ (s0, s), then  f(s1) < δ0,  ˙f(s1) = 0,  ¨f(s1) ≥ 0,  which contradicts (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='65).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' If f(s) attains its minimum value at s0, then ˙f(s0) ≥ 0, and thus ˙f(s0) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  Again, it follows from (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='65) that s0 is a strict local maximum point, which is a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Therefore,  f(s) must attain its minimum value at s and ˙f(s) < 0 (otherwise ˙f(s) = 0 and then by (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='65), s is a strict  local maximum point, which is a contradiction), showing that  f(s) < δ0,  ˙f(s) < 0  for all  s > s0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  Suppose φ(s) is trapped as s → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Then there exists δ > 0 and T > 0 such that f(s) > δ for all s ≥ T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  Since {f(r) ≥ δ} ∩ scT ∗ ¯X is a compact set, we can take a sequence sj → ∞ such that φ(sj) converges to  some (r∞, θ∞, ϕ∞, ξ∞) ∈ {δ ≤ f(r) ≤ f(s0) < δ0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Since f(s) is non increasing and bounded for all s ≥ s0,  lims→∞ f(s) exists.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Since ¨f(s),  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  f (s) are bounded for all s ≥ max{s0, T }, by Barb˘alat’s Lemma [3], we have  scH1,0  ph,zf((r∞, θ∞, ϕ∞, ξ∞)) = (scH1,0  ph,z)2f((r∞, θ∞, ϕ∞, ξ∞)) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' This contradicts (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='65).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  We now prove the statement (2), (3) and (4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' We only prove the case φ(s) is not trapped as s → ∞ in detail  as the other case can be handled similarly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' If φ(s) ⊂ {f(r) ≤ 0} for all s ≥ 0, then φ(s) ⊂ {r− ≤ r ≤ rb}  since φ(0) /∈ ∂+ ¯X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' For the case φ(s) ⊂ Σ−  h , since φ(0) /∈ L−, it follows from Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='13 that φ(s) crosses  ∂− ¯X into the inward direction of decreasing r at finite time and thus there must exist s0 > 0 such that  LINEAR STABILITY OF KERR-NEWMAN FAMILY  61  f(s0) < δ0, ˙f(s0) < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Proceeding as in the proof of (1), we see that f(s) < δ0, ˙f(s) ≤ 0 for all s ≥ s0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' While  for the case φ(s) ⊂ Σ+  h , we claim that φ(s) → L+ as s → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Otherwise, there exists a sufficiently small  neighborhood U+ of L+ such that φ(s) ⊂ Σ+  h \\ U+ for all s > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' By Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='13, there exists δ > 0 such  that  ⟨ξ⟩−1Hph,zr ≥ δ  on  (Σ+  h ∩ {r− ≤ r ≤ rb + δ}) \\ U+  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='70)  (Otherwise, if (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='70) does not hold for any positive δ, then we can find a sequence {(rj, θj, ϕj, ξj)} such that  r− ≤ rj ≤ rb + 1  j and ⟨ξ⟩−1Hph,zr((rj, θj, ϕj, ξj)) < 1  j .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Let j → ∞, and then we obtain a limit point  (r∞, θ∞, ϕ∞, ξ∞) ∈ (Σ+  h ∩ {r− ≤ r ≤ rb}) \\ L+  such that ⟨ξ⟩−1Hph,zr((r∞, θ∞, ϕ∞, ξ∞)) ≤ 0, which contradicts Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='13).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' It follows from (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='70) that  φ(s) ⊂ {r ≥ rb + δ} for all large enough s, which contradicts the assumption that φ(s) ⊂ {r− ≤ r ≤ rb}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  On the other hand, suppose there exists s0 ≥ 0 such that f(s0) > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Since φ(s) is not trapped as s → ∞,  we can find s2 > s1 > 0 such that f(s2) < f(s1) < δ0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Then arguing as in the proof of (1), we see that f(s)  attains its minimum value on in the interval [s1, s2] at the point s2, and thus ˙f(s2) ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Proceeding again  as in the proof of (1), we see that f(s) < δ0, ˙f(s) ≤ 0 for all s ≥ s2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  Also, since φ(s) is not trapped as s → ∞, it follows that either ρ(φ(s)) → 0 as s → ∞, or φ(s) enters  the region {r ≤ rb + δ} for any δ > 0 in finite time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Using (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='70), we find that φ(s) → L+ as s → ∞ if  φ(s) ⊂ Σ+  h .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' If φ(s) ⊂ Σ−  h , suppose that φ(s) never crosses ∂− ¯X for s ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Then φ(s) is well defined for all  s ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Since (ρ, θ, ϕ, ξ) /∈ L−, then there exists a neighborhood V− of L− such that (ρ, θ, ϕ, ξ) /∈ V−.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Since  L− is a radial source, there exists a neighborhood U− of L− and T > 0 such that  φ(s)(U−) ⊂ V−  for all  s ≤ −T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  Since (ρ, θ, ϕ, ξ) /∈ V−, it follows that φ(s) /∈ U− for all s ≥ T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Proceeding as in the proof of (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='70) yields  −⟨ξ⟩−1Hph,zr ≥ δ  on  (Σ−  h ∩ {r− ≤ r ≤ rb + δ}) \\ U−  for some δ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' This forces r(φ(s)) > rb + δ for all s > T and thus ρ(φ(s)) → 0 as s → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  □  According to Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='14, it is clear that K ⊂ {f(r) ≥ δ0} ⊂ {r > rb}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Therefore, we now restrict our  analysis in the region r > rb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Recall the calculation in the proof of Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='11  ∆b > 0, Hph,zr = 0 =⇒ H2  ph,zr = 2ρ−4  b ∆b∂r  �(aξϕ − (r2 + a2)z)2  ∆b(r)  �  We define  F(r) := (aξϕ − (r2 + a2)z)2  ∆b(r)  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='71)  The key property of F(r) is given by  Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Let z ∈ R \\ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' For each r ∈ (rb, ∞), we have  ph,z = 0, ∂rF(r) = 0 =⇒ ∂2  rF(r) > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='72)  Moreover, for each fixed ξϕ, there exists a unique rξϕ ∈ (rb, ∞) such that ∂rF(rξϕ) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Since ph,z = 0, by the discussion around (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='66), we see that aξϕ − (r2 + a2)z ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Then  ∂rF(r) = −(aξϕ − (r2 + a2)z)  ∆2  b  �∂∆b  ∂r (aξϕ − (r2 + a2)z) + 4zr∆b  �  = 0  implies  ∂2  rF(r) = −(aξϕ − (r2 + a2)z)  ∆2  b  �∂2∆b  ∂r2 (aξϕ − (r2 + a2)z) + 2zr∂∆b  ∂r + 4z∆b  �  = −  8z2r  ∆b(∂r∆b)2  �  2∂2∆b  ∂r2 r∆b − r(∂r∆b)2 − 2∆b∂r∆b  �  =  32z2r  ∆b(∂r∆b)2  �  r3 − 3mr2 + 3m2r − m(a2 + Q2)  �  > 0  on  rb < r < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  Since F(r) → ∞ as r → rb and r → ∞, F(r) must attains its global minimum value at rξϕ ∈ (rb, ∞) and  ∂rF(rξϕ) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Moreover, the existence of the critical point of F is unique (otherwise, suppose that there are  62  LILI HE  two points r1, r2 ∈ (rb, ∞) with r1 < r2 such that ∂rF(r1) = ∂rF(r2) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Then let D = {r1 < r < r2 |  ∂rF(r) > 0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Since ∂rF(r1) = 0, ∂2  rF(r1) > 0, it follows that D is nonempty and we let d := sup D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Since  ∂rF(r2) = 0, ∂2  rF(r2) > 0, we see that d ∈ (r1, r2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' By continuity, we have ∂rF(d) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' By the definition of  d, we have ∂2  rF(d) ≤ 0, which is a contradiction).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  □  Let (r0, θ0, ϕ0, ξ0) ∈ Σh and (r(s), θ(s), ϕ(s), ξ(s)) = exp(sHph,z)((r0, θ0, ϕ0, ξ0)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Then we define  Φ0(r) := Φ(r;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' θ0, ϕ0, ξ0  θ, ξ0  ϕ) = F(r;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' ξ0  ϕ) − ˜ph,z((θ0, ϕ0, ξ0  θ, ξ0  ϕ)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='73)  Since ˜ph,z and ξϕ are constants along the Hamiltonian flows exp(sHph,z) on Σh, it follows that (r(s), ξr(s))  is a Hamiltonian flow trajectory of  H0(r, ξr) := ∆b  �  ξr − zc′(r) + χ(aξ0  ϕ − (r2 + a2)z)  ∆b  �2 − Φ0(r)  In particular, (r(s), ξr(s)) solves the equation H0(r, ξr) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' In the region r > rb, the outgoing tail Γ+ and incoming tail Γ− are given by  Γ± =  �  (r, θ, ϕ, ξ)|ph,z = 0, ξr − zc′(r) + χ(aξϕ − (r2 + a2)z)  ∆b  = ∓sgn(r − rξϕ)  �  Φ(r;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' θ, ϕ, ξθ, ξϕ)  ∆b  �  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='74)  where  Φ(r;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' θ, ϕ, ξθ, ξϕ) = F(r;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' ξϕ) − ˜ph,z(θ, ϕ, ξθ, ξϕ)  and rξϕ is the only solution to the equations Φ(r;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' θ, ϕ, ξθ, ξϕ) = 0 and ∂rΦ(r;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' θ, ϕ, ξθ, ξϕ) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Moreover, Γ±  are smooth codimension 1 submanifolds of Σh intersecting transversely, and their intersection is the trapped  set  K =  �  (r, θ, ϕ, ξ)|ph,z = 0, r = rξϕ, ξr − zc′(r) + χ(aξϕ − (r2 + a2)z)  ∆b  = 0  �  ,  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='75)  which is a smooth codimension 2 submanifold of Σh  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Let (r0, θ0, ϕ0, ξ0) ∈ Σh and (r(s), θ(s), ϕ(s), ξ(s)) = exp(sHph,z)((r0, θ0, ϕ0, ξ0)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' By Proposition  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='15, ∂rΦ0(rξ0ϕ) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' If Φ0(rξ0ϕ) ̸= 0, we have |Φ0(r(s))|+|∂rΦ0(r(s))| > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Suppose that (r(s), θ(s), ϕ(s), ξ(s))  is trapped as s → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Since (r(s), ξr(s)) solves the equation H0(r, ξr) = 0, arguing as in the proof of the  statement (1) of Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='14, we can find a sequence sj → ∞ such that (r(sj), ξr(sj)) converges to some  (r∞, (ξr)∞) which satisfies H0(r∞, (ξr)∞) = 0, HH0r((r∞, (ξr)∞)) = HH0ξr((r∞, (ξr)∞)) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' This contra-  dicts the fact that the Hamiltonian vector field associated to H0 is nonzero on the set H0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' This proves  that (r(s), θ(s), ϕ(s), ξ(s)) is not trapped as s → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Similarly, (r(s), θ(s), ϕ(s), ξ(s)) escapes as s → −∞ as  well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' As a result, (r0, θ0, ϕ0, ξ0) /∈ Γ±.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  On the other hand, we consider the case Φ0(rξ0ϕ) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' We note that the set of solutions to the equation  H0(r, ξr) = 0 is given by the union Γ+,0 ∪ Γ−,0 where  Γ±,0 =  �  ξr − zc′(r) + χ(aξ0  ϕ − (r2 + a2)z)  ∆b  = ∓sgn (r − rξ0ϕ)  �  Φ(r;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' θ0, ϕ0, ξ0  θ, ξ0ϕ)  ∆b  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  First, (r0, θ0, ϕ0, ξ0) ∈ Σh is equivalent to H0(r0, ξ0  r) = 0, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=', (r0, θ0, ϕ0, ξ0) ∈ Σh implies (r0, ξ0  r) ∈  Γ+,0 ∪ Γ−,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Next, if (r0, ξ0  r) ∈ Γ∓,0, we see that (r(s), θ(s), ϕ(s), ξ(s)) is trapped as s → ±∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Therefore,  we conclude that (r(s), θ(s), ϕ(s), ξ(s)) is trapped as s → ±∞ if and only if (r0, ξ0  r) ∈ Γ∓,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Then we obtain  K = Γ+ ∩ Γ− =  �  (r, θ, ϕ, ξ)|ph,z = 0, r = rξϕ, ξr − zc′(r) + χ(aξϕ − (r2 + a2)z)  ∆b  = 0  �  =  �  (r, θ, ϕ, ξ)|ph,z = 0, ∂rF = 0, Hph,zr = 0  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  We note that d(Hph,zr) ∈ {dr, dξr, dξϕ} and the coefficient of dξr is 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' In view of (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='72), d(∂rF) ∈ {dr, dξϕ}  and the coefficient of dr is positive on K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Since Hph,zr = 0 and ∂rF = 0 imply ∂ξrph,z = 0 and ∂rph,z = 0  respectively on K, we have dph,z ∈ {dθ, dξθ, dξϕ}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  Therefore, dph,z, d(∂rF) and d(Hph,zr) are linearly  independent on K and K is a smooth codimension 2 submanifold of Σh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  For any (r0, θ0, ϕ0, ξ0) ∈ K, since Φ0(r) = 1  2|∂2  rΦ0(rξ0ϕ)|(r − rξ0ϕ)2 + O((r − rξ0ϕ)3), the defining functions  of Γ±,0 are smooth functions in r, ξr and thus Γ±,0 is a smooth codimension 1 submanifold of T ∗(rb, ∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' If  LINEAR STABILITY OF KERR-NEWMAN FAMILY  63  (r, θ, ϕ, ξ) ∈ Γ±, then its projection onto (θ, ϕ, ξθ, ξφ) must lie in the set of projection of K onto (θ, ϕ, ξθ, ξφ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  Therefore, Γ± are smooth codimension 1 submanifolds of Σh intersecting transversely at K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  □  Since T K is in the kernel of d(Hph,zr) and d(∂rF), we see that the symplectic complement of T K in Σh  is (T K)⊥ = {HHph,z r, H∂rF }.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' By a direct calculation, we see that H∂rF (Hp,zr) = −2∆b∂2  rF ̸= 0 on K,  which implies T K ∩(T K)⊥ = 0 and thus K is symplectic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Now we verify that the trapped set K is normally  hyperbolic whose definition will be given below (see [21, §6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='3]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  Definition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Let K be given as above, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=', K is symplectic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Let φs = exp(sHph,z) : scT ∗ ¯X → scT ∗ ¯X  be the Hamiltonian flow and ϕ± be the defining functions of Γ±.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' We say K is normally hyperbolic if there  exist C, ν > 0 such that for all (r, θ, ϕ, ξ) ∈ K  d(ϕ± ◦ φ±s)(r, θ, ϕ, ξ)  dϕ±(r, θ, ϕ, ξ)  ≤ Ce−νs,  s ≥ 0  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='76)  Define the minimal expansion rate νmin > 0 as the supremum of all values of ν for which there exists a  constant C such that (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='76) holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Let K be defined as in (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='75).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Then K is normally hyperbolic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' It suffices to prove that the minimal expansion νmin rate defined above is positive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' By (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='74), we have  Γ± = {ϕ± = ξr − zc′(r) + χ(aξϕ − (r2 + a2)z)  ∆b  ± sgn (r − rξϕ)  �  Φ(r;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' θ, ϕ, ξθ, ξϕ)  ∆b  = 0}  where (θ, ϕ, ξθ, ξϕ) ∈ K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' That is ϕ± are defining function of Γ±.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Since ph,z = ρ−2  b ∆bϕ+ϕ−, it follows that  Hph,zϕ± = ∓ρ−2  b ∆b(Hϕ+ϕ−)ϕ± + (Hρ−2  b  ∆bϕ±)ϕ∓ϕ± := ∓ν±ϕ±  and ν+|K = ν−|K = ρ−2  b ∆b(Hϕ+ϕ−)|K = ν.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Since  ∂ξr(Hph,zϕ±) = ∂ξr(∓ν±ϕ±) = ∓  �  (∂ξrν±)ϕ± + ν±  �  ,  using Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='15 we find that  ν = −∂ξr(Hph,zϕ+)|K = ρ−2  b  �  2∆b∂2rF(rξϕ) > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  For (r, θ, ϕ, ξ) ∈ K, we calculate  ∂sd(ϕ± ◦ φ±s) = ∓d((ν±ϕ±) ◦ φ±s) = ∓(ν± ◦ φ±s)d(ϕ± ◦ φ±s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  Therefore, for all T > 0 and (r, θ, ϕ, ξ) ∈ K, we have  d(ϕ± ◦ φ±T )(r, θ, ϕ, ξ)  dϕ±(r, θ, ϕ, ξ)  = e−⟨ν⟩±  T T ,  ⟨ν⟩±  T = 1  T  � T  0  ν ◦ φ±s(r, θ, ϕ, ξ) ds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  As a result, the minimal expansion rate is given by  νmin = min{ lim inf  T →∞  inf  (r,θ,ϕ,ξ)∈K⟨ν⟩+  T ,  lim inf  T →∞  inf  (r,θ,ϕ,ξ)∈K⟨ν⟩−  T } > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='77)  □  Finally, we further determine the position of the tapped set K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' For z ∈ R \\ 0, let Σ±  h be defined as in Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Then K ⊂ Σ±  h for ±z > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Moreover,  Γ+ ∪ Γ− ⊂ Σ±  h for ±z > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' First, according to (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='51) in Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='8, we have  Σ∓  h ⊂ {r ≥ r− | ∆b ≤ a2 sin2 θ} ⊂ {r− ≤ r ≤ m +  �  m2 − Q2}  for  ± z > 0  and thus  K ∩ {r > m +  �  m2 − Q2} ⊂ Σ±  h  for  ± z > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  Since c′(r) = 0, χ = 1 on r− ≤ r ≤ m +  �  m2 − Q2, then for  (r, θ, ϕ, ξ) ∈ K ∩ {r− ≤ r ≤ m +  �  m2 − Q2} = K ∩ {rb < r ≤ m +  �  m2 − Q2},  64  LILI HE  we have  aξϕ − (r2 + a2)z = −4zr∆b  ∂r∆b  and  ξr = 4zr  ∂r∆b  ,  and thus  ±g−1(−zdtb,∗ + ξ, dt) = ±ρ−2  b  �  ξr(r2 + a2 + b′(r)∆b) + b′(r)(aξϕ − (r2 + a2)z) + (aξϕ − a2 sin2 θz)  �  = ±  4zr  ρ2  b∂r∆b  �  r2 + a2 − ∆b  �  ± z = ±  4zr  ρ2  b∂r∆b  �  2mr − Q2�  ± z > 0  for  ± z > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  This implies that  K ∩ {r− ≤ r ≤ m +  �  m2 − Q2} ⊂ Σ±  h  for  ± z > 0,  and thus proves K ⊂ Σ±  h for ±z > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  In order to prove Γ+ ∪ Γ− ⊂ Σ±  h for ±z > 0, it suffices to show that if (r, θ, ϕ, ξ) ∈ Γ±, then φ(s) =  exp(sscH2,0  ph,z)(r, θ, ϕ, ξ) → K as s → ∓∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' We only prove the case of (r, θ, ϕ, ξ) ∈ Γ− as the other case can  be handled in a similar manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Suppose that φ(s) ↛ K as s → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Then there exists a sequence sj → ∞  and a neighborhood U of K such that φ(sj) /∈ U for all j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Since φ(s) is trapped as s → ∞ (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=', there exist  δ, T > 0 such that f(φ(s)) > δ for all s > T ), passing to a subsequence we have  φ(sj) → (r∞, θ∞, ϕ∞, ξ∞)  for some  (r∞, θ∞, ϕ∞, ξ∞) /∈ K = Γ+ ∩ Γ−.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  By Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='14, we find that Γ− is closed and thus (r∞, θ∞, ϕ∞, ξ∞) ∈ Γ−.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' It follows that (r∞, θ∞, ϕ∞, ξ∞) /∈  Γ+, which means that φ(s) is not trapped as s → −∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Then there exists T1 > 0 such that f(φ∞(−T1)) < δ  where φ∞(s) = exp(sscH2,0  ph,z)(r∞, θ∞, ϕ∞, ξ∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Since φ(sj) → (r∞, θ∞, ϕ∞, ξ∞), it follows that  exp(−T1  scH2,0  ph,z)(φ(sj)) → φ∞(−T1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  Therefore, f(φ(sj − T1)) < δ for all sufficiently large j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' But this contradicts the fact that φ(s) is trapped as  s → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  □  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Global dynamics of the Hamiltonian flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Now we shall analyze the global behavior of the flow of  scH2,0  ph,z := ρ−1⟨ξ⟩−1scHph,z on the semiclassical characteristic set Σh = {⟨ξ⟩−2ph,z = 0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  We need the  following technical lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' For z ∈ R \\ 0, let (ρ, θ, ϕ, ξ) ∈ Σh and we put φ(s) = exp(sscH2,0  ph,z)(ρ, θ, ϕ, ξ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' If ρ(φ(s)) → 0  as s → ±∞, then one has  φ(s) → R(z  2 ± |z|  2 ) = {ρ = 0, ξ = (z ± |z|)dρ  ρ2 }  as  s → ±∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Recall that  ⟨ξ⟩scH2,0  ph,z = (−2ξρ + 2z)ρ∂ρ + 2˜p∂ξρ + (−2ξρ + 2z)  �  µ=θ,ϕ  ξµ∂ξµ − H˜p + ρV,  V ∈ Vb(scT ∗ ¯X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  Since ph,z = 0 along the integral curves of scH2,0  ph,z, we calculate  ⟨ξ⟩scH2,0  ph,z  �ξρ − z  ρ  �  = ⟨ξ⟩2(ξρ − z)2 + 2˜p + ρa1  ρ  = ⟨ξ⟩z2 + ρa2  ρ  ,  a1, a2 ∈ A0( ¯X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  Since ρ(φ(s)) → 0 as s → ±∞, then for any ǫ > 0, there exists T > 0 such that ρ(φ(s)) < ǫ for all  ±s > T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Picking ǫ sufficiently small, we see that scH2,0  ph,z  �  ξρ−z  ρ  �  ≥ c1 > 0 for all ±s > T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' This implies that  (ξρ − z)/ρ → ±∞ as s → ±∞;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' in particular ±(ξρ − z) ≥ 0 for all ±s > T1 where T1 > 0 is a sufficiently  large constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  According to Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='10, there exist neighborhoods V ǫ′  ± = {(ξρ − (z ± (sgn z)z)))2 + ˜p + ρ2 ≤ ǫ′} of  R( z  2 ± |z|  2 ) such that once φ(s) enters V ǫ′  ± at some time ±s0 > 0, it will converge to R( z  2 ± |z|  2 ) as ±s → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  Suppose that φ(s) /∈ V ǫ′  ± for all ±s ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Since ph,z = 0 along the integral curves of scH2,0  ph,z, it follows that  |ξρ − z| ≤ δ1  ˜p > δ2  for all  ± s > T2  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='78)  for some δ1, δ2 > 0, T2 > T1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Then for all ±s > T2, we have  scH2,0  ph,z(ξρ − z) = ⟨ξ⟩−1(2˜p + ρa2) > c2 > 0,  a2 ∈ A0( ¯X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  LINEAR STABILITY OF KERR-NEWMAN FAMILY  65  which contradicts (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='78).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Therefore, φ(s) must enter V ǫ′  ± at some time ±s0 > 0 and this finishes the proof  of the lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  □  Now we are at the position to describe the the global behavior of the flow of scH2,0  ph,z := ρ−1⟨ξ⟩−1scHph,z  on the semiclassical characteristic set Σh = {⟨ξ⟩−2ph,z = 0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  L+  L−  R(z)  R(0)  K  Γ−  Γ+  Σ−  h  Σ+  h  z > 0  L+  L−  R(z)  R(0)  K  Γ+  Γ−  Σ−  h  Σ+  h  z < 0  Figure 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' The flow of exp(sscH2,0  ph,z), which is projected to the coordinates (r, ξr) and  drawn in a fiber-radially compactified view.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' The shaded region is the semiclassical char-  acteristic set Σh = Σ+  h ⊔ Σ−  h = {⟨ξ⟩−2ph,z = 0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' The horizontal coordinate is r and the  rightmost vertical line corresponds to ∂+ ¯X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' The dotted line is the even horizon r = rb and  the vertical line in the middle is {r = rergo} where rergo is the equatorial radius (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' greatest  radius) of the ergosphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' The vertical coordinate is ξr/⟨ξr⟩, so the top and bottom lines  stand for the fiber infinity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Let ±z > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Let φ(s) = exp(sscH2,0  ph,z)(ρ, θ, ϕ, ξ) be the maximally extended integral curve  of scH2,0  ph,z on Σh with the domain of definition s ∈ I ⊂ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Let s− = inf I, s+ = sup I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' (see Figure 10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  (1) Let z > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  (i) If φ(s) ⊂ Σ−  h , then either φ(s) ⊂ L−, or φ(s) → L− as s → s− = −∞ and φ(s) crosses ∂− ¯X  into the inward direction of deceasing r at finite time s+ < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  (ii) If φ(s) ⊂ Σ+  h , then either φ(s) ⊂ L+∪R(z)∪R(0)∪K, or φ(s) → L+∪R(z)∩K as s → s+ = ∞  and φ(s) → R(0) ∪ K as s → s− = −∞ or φ(s) crosses ∂− ¯X into the inward direction of  deceasing r at finite time s− > −∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Moreover, if (ρ, θ, ϕ, ξ) /∈ K, then φ(s) cannot converge to  K in both the forward and backward direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  (2) Let z < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  (i) If φ(s) ⊂ Σ+  h , then either φ(s) ⊂ L+, or φ(s) → L+ as s → s+ = ∞ and φ(s) crosses ∂− ¯X into  the inward direction of deceasing r at finite time s− > −∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  (ii) If φ(s) ⊂ Σ−  h , then either φ(s) ⊂ L−∪R(z)∪R(0)∪K, or φ(s) → L−∪R(z)∪K as s → s− = −∞  and φ(s) → R(0)∪K as s → s+ = ∞ or φ(s) crosses ∂− ¯X into the inward direction of deceasing  r at finite time s+ < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Moreover, if (ρ, θ, ϕ, ξ) /∈ K, then φ(s) cannot converge to K in both  the forward and backward direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' We only consider the case z > 0 as the other case z < 0 can be proved similarly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' First, we note that  L+, L−, R(0), R(z), K are invariant under the flow φ(s) and if (ρ, θ, ϕ, ξ) ∈ Γ±, then φ(s) → K as ∓s → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  If (ρ, θ, ϕ, ξ) /∈ L+ ∪ L− ∪ R(0) ∪ R(z) ∪ K, then either (ρ, θ, ϕ, ξ) /∈ Γ− or (ρ, θ, ϕ, ξ) /∈ Γ+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  66  LILI HE  If (ρ, θ, ϕ, ξ) /∈ Γ−, by Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='13, for φ(s) ⊂ Σ+  h , either φ(s) → L+ or ρ(φ(s)) → 0 as s → ∞, while  for φ(s) ⊂ Σ−  h , φ(s) crosses ∂− ¯X into the inward direction of deceasing r at some finite time s0 < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' If  ρ(φ(s)) → 0 as s → ∞, according to Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='20, we have φ(s) → R(z) as s → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  If (ρ, θ, ϕ, ξ) /∈ Γ+, by Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='13, for φ(s) ⊂ Σ+  h , either ρ(φ(s)) → 0 as s → ∞ or φ(s) crosses ∂− ¯X  into the inward direction of deceasing r at some finite time s0 > −∞, while for φ(s) ⊂ Σ−  h , φ(s) → L− as  s → −∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' If ρ(φ(s)) → 0 as s → −∞, according to Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='20, we have φ(s) → R(0) as s → −∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  □  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' High energy estimates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' In this section, we prove the high energy estimates first for operators  h2�  □gb(h−1z) acting on scalar functions, then for h2 �  Pb,γ(h−1z), h2�  Wb,γ(h−1z) acting on scattering 1-forms  and the linearized gauge-fixed Einstein-Maxwell operator h2 �  Lb,γ(h−1z), as well as their formal adjoints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' To  this end, we combine the global dynamics of of the Hamiltonian flow of the semiclassical principal symbol ph,z  established in 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='21 with the elliptic estimate, propagation of singularities estimate, the radial point estimate  at event horizon, the scattering radial point estimate at spatial infinity ∂+ ¯X and hyperbolic estimate, all of  which are in the semiclassical version, together with the estimate at normally hyperbolic trapping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' High energy estimates for scalar wave operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' We first prove high energy estimates for the oper-  ator h2�  □gb(h−1z) acting on scalar functions, as well as its formal adjoint with respect to volume density  L2( ¯X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  �  det |gb|drdθdϕ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  B+  B−  B0  E0  Bz  B1  B2  B2  L+  L−  R(z)  R(0)  K  Γ−  Γ+  suppχ  B′  +  E′  +  B′  −  E′  −  B′  0  B′  z  E′  z  B′  1  B′  2  B′  2  L+  L−  R(z)  R(0)  K  Γ−  Γ+  suppχ  supp(1 − χ)  Figure 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' A phase space picture of the proof of the estimate (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='81), on the left, and (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='82),  on the right.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' The coordinates and notations L±, R(0), R(z), Γ±, K are the same as in Figure  10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' For (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='81), we use semiclassical hyperbolic estimates to control u via χu;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' χ is controlled  (modulo elliptic estimates) by B±, B1, B0, Bz;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' B0 is controlled by E0 using low b-decay  radial point estimates and E0 is again controlled by B+, B1, Bz;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' B1 is controlled by B2 by  using normally hyperbolic trapping estimate and B2 is controlled by B+, Bz;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' finally B±, Bz  are controlled using high regularity radial points estimates at L± and high sc-decay radial  point estimates at R(z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' For (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='82), we use semiclassical hyperbolic estimates to bound  (1 − χ)u;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' χ is controlled (modulo elliptic estimates) by B′  ±, B′  1, B′  0, B′  z;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' B′  ± is controlled  by E′  ± using low regularity radial point estimates and E′  ± is controlled by B′  1, B′  0, 1 − χ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  B′  z is controlled by E′  z using low sc-decay radial points estimates and E′  z is controlled by  B′  0, B′  1, 1−χ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' B′  1 is controlled by B′  2 by using normally hyperbolic trapping estimate and B′  2  is controlled by B′  +, 1−χ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' finally B′  0 is controlled using high b-decay radial points estimates  at R(0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  LINEAR STABILITY OF KERR-NEWMAN FAMILY  67  Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Let b0 = (m0, a0, Q0) with |Q0| + |a0| < m0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  Then there exists ǫ > 0 such that for  b = (m, a, Q) with |b − b0| < ǫ and for s > 1  2, ℓ < − 1  2 with s + ℓ > − 1  2, the following holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' For any fixed  C1 > 0, there exist C2 > 1, C > 0 which are independent of b, such that for σ ∈ C, Im σ ∈ [0, C1], |Re σ| > C2,  we have  ∥u∥ ¯  Hs,ℓ  b,h( ¯  X) ≤ C∥�  □gb(σ)u∥ ¯  Hs,ℓ+1  b,h  ( ¯  X),  ∥u∥ ˙H−s,−ℓ−1  b,h  ( ¯  X) ≤ C∥�  □gb(σ)∗u∥ ˙H−s,−ℓ  b,h  ( ¯  X);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='79)  ∥u∥ ¯  Hs,ℓ  b,h( ¯  X) ≤ C∥�  □gb(σ)u∥ ¯  Hs−1,ℓ+2  b,h  ( ¯  X),  ∥u∥ ˙H−s+1,−ℓ−2  b,h  ( ¯  X) ≤ C∥�  □gb(σ)∗u∥ ˙H−s,−ℓ  b,h  ( ¯  X),  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='80)  where h = |σ|−1 and C only depends on b0, s, ℓ, C1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  Therefore,  �  □gb : {u ∈ ¯Hs,ℓ  b,h( ¯X) | ¯Hs,ℓ+1  b,h  ( ¯X)} → ¯Hs,ℓ+1  b,h  ( ¯X)  and  �  □gb : {u ∈ ¯Hs,ℓ  b,h( ¯X) | ¯Hs−1,ℓ+2  b,h  ( ¯X)} → ¯Hs−1,ℓ+2  b,h  ( ¯  X)  are invertible for σ ∈ C, Im σ ∈ [0, C1], |Re σ| > C2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' For the simplicity of notations, we let Ph(z) := h2�  □gb(h−1z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' We first claim that it suffices to prove  the following estimates for s > 1  2, r > − 1  2, ℓ < − 1  2  ∥u∥ ¯  Hs,r,ℓ  sc,b,h( ¯  X) ≤ Ch−2∥Ph(z)u∥ ¯  Hs−1,r+1,ℓ+1  sc,b,h  ( ¯  X)  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='81)  and  ∥u∥ ˙H−s+1,−r−1,−ℓ−1  sc,b,h  ( ¯  X) ≤ Ch−2∥Ph(z)∗u∥ ˙H−s,−r,−ℓ  sc,b,h  ( ¯  X)  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='82)  Since ¯Hs,s+ℓ,ℓ  sc,b,h  = ¯Hs,l  b,h and ˙Hs,s+ℓ,ℓ  sc,b,h  = ˙Hs,l  b,h, letting r = s + ℓ > − 1  2 we have  ∥u∥ ¯  Hs,s+ℓ,ℓ  sc,b,h ( ¯  X) ≤ Ch−2∥Ph(z)u∥ ¯  Hs−1,s+ℓ+1,ℓ+1  sc,b,h  ( ¯  X)  and  ∥u∥ ˙H−s+1,−s−r−1,−ℓ−1  sc,b,h  ( ¯  X) ≤ Ch−2∥Ph(z)∗u∥ ˙H−s,−s−r,−ℓ  sc,b,h  ( ¯  X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  Then using the facts that  ¯Hs,ℓ+1  b,h  = ¯Hs,s+ℓ+1,ℓ+1  sc,b,h  ⊂ ¯Hs−1,s+ℓ+1,ℓ+1  sc,b,h  ,  ¯Hs−1,ℓ+2  b,h  = ¯Hs−1,s+ℓ+1,ℓ+2  sc,b,h  ⊂ ¯Hs−1,s+ℓ+1,ℓ+1  sc,b,h  ,  ˙H−s,−ℓ−1  b,h  = ˙H−s,−s−ℓ−1,−ℓ−1  sc,b,h  ⊃ ˙H−s+1,−s−ℓ−1,−ℓ−1  sc,b,h  ,  ˙H−s+1,−ℓ−2  b,h  = ˙H−s+1,−s−ℓ−1,−ℓ−2  sc,b,h  ⊃ ˙H−s+1,−s−ℓ−1,−ℓ−1  sc,b,h  ,  we obtain (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='79) and (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='80).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Now we will show (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='81) and (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='82).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Here we only discuss the case Re z > 0  (see Figure 11 for a phase space illustration of the proof of estimates (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='81) and (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='82)), as the case Re z < 0  can be handled in a similar manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  Proof of the estimate (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='81).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' For s > s0 >  1  2 ≥  1  2 − Im (h−1z) r2  b+a2  r2  b−m (as calculated in (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='59) and  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='8)), using the semiclassical high regularity radial point estimate (see [77, Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='10], [78,  Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='27]), there exist B±, G±, S± ∈ Ψ0  h( ¯X) microlocally supported near L± with L± ⊂  ellh(B±), L± ⊂ ellh(S±) and satisfying that all the forward (backward) null characteristics from  WFh(B±) tend to L±, while remaining in the elliptic set of G±, such that  ∥B±u∥Hs,r,ℓ  sc,b,h( ¯  X) ≤ Ch−1∥G±Ph(z)u∥Hs−1,r+1,ℓ+1  sc,b,h  ( ¯  X) + Chs−s0∥S±u∥Hs0,r0,ℓ0  sc,b,h  ( ¯  X)  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='83)  where r0 < r, ℓ0 < ℓ and the sc-decay order r and b-decay order ℓ are actually irrelevant here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  For r > r0 > − 1  2 (as calculated in (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='14)), using the semiclassical high sc-decay radial point esti-  mates at the non-zero scattering section R(z) (see [79].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' The proof follows from a positive commutator  estimate  i(Ph(z)∗A − APh(z)) = Im Ph(z)A + AIm Ph(z) + i[Re Ph(z), A]  where  Re Ph(z) = Ph(z) + Ph(z)∗  2  ,  Im Ph(z) = Ph(z) − Ph(z)∗  2i  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  Let A ∈ Ψ2s−1,2r+1,2l+1  sc,b,h  ( ¯X) with semiclassical principal symbol  a = χ0(ξ2  θ +  1  sin2 θξ2  ϕ)χ1((ξρ − 2Re z)2)ρ−2r+1(ξ2  θ +  1  sin2 θξ2  ϕ + ξ2  ρ)s− 1  2  68  LILI HE  where χ0, χ1 are identically 0 near 0 and have compact support sufficiently close to 0, and χ1  has relatively large support such that suppχ0 ∩ suppχ′  1 is disjoint from the characteristic set of  Re Ph(z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  Then using Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='1 and the calculation before Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='8 in [79], it follows that  h−1(Im Ph(z)A + AIm Ph(z) + i[Re Ph(z), A]) ∈ Ψ2s,2r,2l  sc,b,h ( ¯X) whose semiclassical principal symbol  is positive definite elliptic near R(z) if r > − 1  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' This,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' together with a regularization argument,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  proves semiclassical high sc-decay radial point estimates at R(z)),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' there exist Bz,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Gz,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Sz ∈ Ψ0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='0  sc,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='h( ¯X)  microlocally supported near R(z) with R(z) ⊂ ellsc,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='h(Bz),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' R(z) ⊂ ellsc,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='h(Sz) and satisfying that all  the forward null characteristics from WFsc,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='h(Bz) tend to R(z),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' while remaining in the scattering  elliptic set of Gz,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' such that  ∥Bzu∥Hs,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='ℓ  sc,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='b,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='h( ¯  X) ≤ Ch−1∥GzPh(z)u∥Hs−1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='r+1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='ℓ+1  sc,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='b,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='h  ( ¯  X) + Chs−s0∥Szu∥Hs0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='r0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='ℓ0  sc,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='b,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='h  ( ¯  X)  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='84)  where s0 < s, ℓ0 < ℓ and the differential order s and b-decay order ℓ are actually irrelevant here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  For ℓ < − 1  2 (as calculated in (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='14)), using the semiclassical low b-decay radial point estimates  at the zero scattering section R(0) (see [79, §4–5]), there exist B0, G0 ∈ Ψ0,0,0  sc,b,h( ¯X) microlocally  supported near R(0) (in fact near the blown up R(0) in the second microlocalized space introduced  in §5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='5) with R(0) ⊂ ellsc,b,h(B0) and E0 ∈ Ψ0,0,0  sc,b,h( ¯X), WFsc,b,h(E0) ∩ R(0) = ∅, and satisfying  that all the forward null characteristics from WFsc,b,h(B0)\\ R(0) enter ellsc,b,h(E0), while remaining  in the second microlocalized scattering elliptic set of G0, such that for any sufficiently large N > 0  ∥B0u∥Hs,r,ℓ  sc,b,h( ¯  X) ≤ Ch−1∥G0Ph(z)u∥Hs−1,r+1,ℓ+1  sc,b,h  ( ¯  X) + C∥E0u∥Hs,r,ℓ  sc,b,h( ¯  X) + ChN∥u∥ ¯  H−N,−N,ℓ  sc,b,h  ( ¯  X)  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='85)  where the differential order s is actually irrelevant here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='Since at the tapped set K  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='σh(h−1Im Ph(z)) = σh(Ph(z) − Ph(z)∗  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='2ih  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=') = σh(Ph(z) − Ph(¯z)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='2ih  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=')  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='= −ρ−2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='b Im σ  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='−  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='ξr − c′(r)Re z + χ(aξϕ − (r2 + a2)Re z)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='∆b  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='��  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='c′(r) + χ(r2 + a2)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='∆b  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='+ 2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='∆b  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='aξϕ − (r2 + a2)Re z  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='��  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='r2 + a2�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='− 2a  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='ξϕ − a sin2 θRe z  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='��  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='= −ρ−2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='b Im σ  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='� 2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='∆b  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='aξϕ − (r2 + a2)Re z  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='��  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='r2 + a2�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='− 2a  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='ξϕ − a sin2 θRe z  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='��  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='> 0  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='where we use ξr − c′(r)Re z + χ(aξϕ−(r2+a2)Re z)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='∆b  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='= 0 at K in the last equality and  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='aξϕ − (r2 + a2)Re z = −−4r∆bRe z  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='∂r∆b  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='< 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  (aξϕ − (r2 + a2)Re z)2 ≥ ∆b(ξϕ − a sin2 θRe z)2  at K,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  which follow from Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='15 and (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='66), in the last step, it follows that  σh(h−1Im − Ph(z)) = σh((−Ph(z)) − (−Ph(z))∗  2ih  ) < 0 < νmin  2  at  K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  Using the hyperbolic trapping estimates (see [20] and [40, Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='7]), there exist B1, G1 ∈  Ψ0  h( ¯X) microlocally supported near K with K ⊂ ellh(B1) and B2 ∈ Ψ0  h( ¯X), WFh(B2) ∩ Γ− = ∅,  and satisfying that WFh(B1) ∪ WFh(B2) is contained in the elliptic set of G1, such that for any  N, s′, r′, ℓ′ ∈ R  ∥B1u∥Hs,r,ℓ  sc,b,h( ¯  X) ≤ Ch−2∥G1Ph(z)u∥Hs−1,r+1,ℓ+1  sc,b,h  ( ¯  X) + C∥B2u∥Hs,r,ℓ  ,hsc,b( ¯  X) + ChN∥u∥ ¯  Hs′,r′,ℓ′  sc,b,h ( ¯  X)  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='86)  where the differential order s′, the sc-decay order r′ and b-decay order ℓ′ are actually irrelevant here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  Let r− < r0 < rb and χ ∈ C∞  c ( ¯X) with χ = 1 near r ≥ rb and suppχ ⊂ {r ≥ r0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  By  part (1) in Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='21, semiclassical propagation of singularities estimates and semiclassical  elliptic estimates (Concretely, when (ρ, θ, ϕ, ξ) ∈ WFh(χ) ∩ {⟨ξ⟩−2ph,z(ξ) ̸= 0}, we use semiclassical  elliptic estimates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  If (ρ, θ, ϕ, ξ) ∈ WFh(χ) ∩ {⟨ξ⟩−2ph,z(ξ) = 0}, then there exists s ∈ R with  exp(sscH2,0  ph,z)(ρ, θ, ϕ, ξ) ∈ ellh(B±) ∪ ellh(B1) ∪ ellh(B0) ∪ ellh(Bz), and thus we use semiclassical  propagation of singularities estimates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' We point out that in the set {Re ph(z) = 0} ∩ ∂+ ¯X, the  semiclassical symbol has a nonnegative imaginary part, so one can propagate estimates towards  LINEAR STABILITY OF KERR-NEWMAN FAMILY  69  R(0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Finally, by using a pseudodifferential partition of unity, χ can be written as s sum of operators  falling into the above two case), we have  ∥χu∥Hs,r,ℓ  sc,b,h( ¯  X) ≤ Ch−1∥Ph(z)u∥ ¯  Hs−1,r+1,ℓ+1  sc,b,h  ( ¯  X) + C∥B+u∥Hs,r,ℓ  sc,b,h( ¯  X) + C∥B−u∥Hs,r,ℓ  sc,b,h( ¯  X)  + C∥B1u∥Hs,r,ℓ  sc,b,h( ¯  X) + C∥B0u∥Hs,r,ℓ  sc,b,h( ¯  X) + C∥Bzu∥Hs,r,ℓ  sc,b,h( ¯  X) + ChN∥u∥ ¯  H−N,−N,ℓ  sc,b,h  ( ¯  X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='87)  By the same reasoning as above in the control of χu, it follows that  ∥E0u∥Hs,r,ℓ  sc,b,h( ¯  X) ≤ Ch−1∥Ph(z)u∥ ¯  Hs−1,r+1,ℓ+1  sc,b,h  ( ¯  X) + C∥B+u∥Hs,r,ℓ  sc,b,h( ¯  X) + C∥B1u∥Hs,r,ℓ  sc,b,h( ¯  X)  + C∥Bzu∥Hs,r,ℓ  sc,b,h( ¯  X) + ChN∥u∥ ¯  H−N,−N,ℓ  sc,b,h  ( ¯  X)  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='88)  and  ∥B2u∥Hs,r,ℓ  sc,b,h( ¯  X) ≤ Ch−1∥Ph(z)u∥ ¯  Hs−1,r+1,ℓ+1  sc,b,h  ( ¯  X) + C∥B+u∥Hs,r,ℓ  sc,b,h( ¯  X)  + C∥Bzu∥Hs,r,ℓ  sc,b,h( ¯  X) + ChN∥u∥ ¯  H−N,−N,ℓ  sc,b,h  ( ¯  X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='89)  Putting all the above estimates together yields for s > s0 > 1  2, r > r0 > − 1  2, ℓ < − 1  2  ∥χu∥Hs,r,ℓ  sc,b,h( ¯  X) ≤ Ch−2∥Ph(z)u∥ ¯  Hs−1,r+1,ℓ+1  sc,b,h  ( ¯  X) + Chs−s0∥u∥ ¯  Hs0,r0,ℓ  sc,b,h ( ¯  X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='90)  Since  ph,z = −ρ−2  b  �  ∆b  �  ξr +  �  aξϕ − (r2 + a2)z  �  ∆b  �2 − 1  ∆b  �  aξϕ − (r2 + a2)z  �2 + ˜ph,z  �  where  ˜ph,z = ξ2  θ +  1  sin2 θ(ξϕ − a sin2 θz)2  is semiclassical hyperbolic with respect to r in the region {r− ≤ r < rb} (see [21, definition E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='55]),  using the semiclassical hyperbolic estimates (see [21, Theorem E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='57]), we have for all s, r, ℓ ∈ R  ∥(1 − χ)u∥ ¯  Hs,r,ℓ  sc,b,h( ¯  X) ≤ Ch−1∥Ph(z)u∥ ¯  Hs−1,r+1,ℓ+1  sc,b,h  ( ¯  X) + ∥χu∥ ¯  Hs,r,ℓ  sc,b,h( ¯  X)  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='91)  where the sc-decay order r and b-decay order ℓ are actually relevant here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Combining (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='90) with  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='91) yields for s > s0 > 1  2, r > r0 > − 1  2, ℓ < − 1  2  ∥u∥ ¯  Hs,r,ℓ  sc,b,h( ¯  X) ≤ Ch−2∥Ph(z)u∥ ¯  Hs−1,r+1,ℓ+1  sc,b,h  ( ¯  X) + Chs−s0∥u∥ ¯  Hs0,r0,ℓ  sc,b,h ( ¯  X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='92)  Taking h small enough, the term Chs−s0∥u∥ ¯  Hs0,r0,ℓ  h,sc,b ( ¯  X) can be absorbed into the left-hand side and  this proves the estimate (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='81).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  Proof of the estimate (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='82).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' For s > 1  2 ≥ 1  2 − Im (h−1z) r2  b+a2  r2  b−m , we have 1 − s ≤ 1  2 − Im (h−1z) r2  b +a2  r2  b−m .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  Then using the semiclassical low regularity radial point estimates (see [77, Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='11], [78,  Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='27]), there exist B′  ±, G′  ± ∈ Ψ0  h( ¯X) microlocally supported near L± with L± ⊂ ellh(B′  ±)  and E′  ± ∈ Ψ0  h( ¯X), WFh(E′  ±) ∩ L± = ∅, and satisfying that all the backward (forward) null charac-  teristics from WFh(B′  ±) \\ L± reach ellh(E′  ±), while remaining in the elliptic set of G′  ±, such that for  any N ∈ R  ∥B′  ±u∥H1−s,−r−1,−ℓ−1  sc,b,h  ( ¯  X)  ≤ Ch−1∥G′Ph(z)∗u∥H−s,−r,−ℓ  sc,b,h  ( ¯  X) + C∥E′  ±u∥H1−s,−r−1,−ℓ−1  sc,b,h  ( ¯  X) + ChN∥u∥ ˙H−N,−N,−N  sc,b,h  ( ¯  X)  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='93)  where the sc-decay order r and b-decay order ℓ are actually irrelevant here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  For r > − 1  2, we have −r − 1 < − 1  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  Then using the semiclassical low sc-decay radial point  estimates at the non-zero scattering section R(z) (see [79] and the above discussion about the proof  of high sc-decay radial point estimates at R(z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  We note that the only difference is that now  the term involving χ′  0 does not have the correct sign and needs to be treated as an error term  E′  zu),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' there exist B′  z,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' G′  z ∈ Ψ0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='0  sc,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='h( ¯X) microlocally supported near R(z) with R(z) ⊂ ellsc,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='h(Bz) and  70  LILI HE  E′  z ∈ Ψ0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='0  sc,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='h( ¯X),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' WFsc,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='h(E′  z)∩R(z) = ∅,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' and satisfying that all the backward null characteristics from  WFsc,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='h(B′  z) \\ R(z) reach ellsc,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='h(E′  z),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' while remaining in the scattering elliptic set of G′  z,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' such that  ∥B′  zu∥H1−s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='−r−1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='−ℓ−1  sc,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='b,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='h  ( ¯  X)  ≤ Ch−1∥G′  zPh(z)∗u∥H−s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='−r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='−ℓ  sc,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='b,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='h  ( ¯  X) + C∥E′  zu∥H1−s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='−r−1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='−ℓ−1  sc,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='b,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='h  ( ¯  X) + ChN∥u∥ ˙H−N,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='−N,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='−N  sc,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='b,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='h  ( ¯  X)  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='94)  where the differential order s and b-decay order ℓ are actually irrelevant here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  For ℓ < − 1  2, we have −ℓ − 1 > − 1  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  Then using the semiclassical high b-decay radial point  estimates at the zero scattering section R(0) (see [79, §4–5]), there exist B′  0, G′  0 ∈ Ψ0,0,0  sc,b,h( ¯X) mi-  crolocally supported near R(0) (in fact near the blown up R(0) in the second microlocalized space  introduced in §5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='5) with R(0) ⊂ ellsc,b,h(B′  0), and satisfying all the backward null characteristics  from WFsc,b,h(B′  0) tend to R(0), while remaining in the second microlocalized scattering elliptic set  of G′  0, such that for any N ∈ R  ∥B′  0u∥H1−s,−r−1,−ℓ−1  sc,b,h  ( ¯  X) ≤ Ch−1∥G′  0Ph(z)∗u∥H−s,−r,−ℓ  sc,b,h  ( ¯  X) + ChN∥u∥ ˙H−N,−N,−ℓ−1  sc,b,h  ( ¯  X)  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='95)  where the differential order s is actually irrelevant here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  We again calculate  σh(h−1Im Ph(z)∗) = σh(Ph(¯z) − Ph(z)  2ih  ) < 0 < νmin  2  at  K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  Then using the hyperbolic trapping estimates (see [20] and [40, Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='7]), there exist B′  1, G′  1, ∈  Ψ0  h( ¯X) microlocally supported near K with K ⊂ ellh(B′  1) and B′  2 ∈ Ψ0  h( ¯X), WFh(B′  2) ∩ Γ+ = ∅,  and satisfying that WFh(B′  1) ∪ WFh(B′  2) is contained in the elliptic set of G′  1, one has for any  N, s′, r′, ℓ′ ∈ R  ∥B′  1u∥H1−s,−r−1,−ℓ−1  sc,b,h  ( ¯  X)  ≤ Ch−2∥G′  1Ph(z)∗u∥H−s,−r,−ℓ  sc,b,h  ( ¯  X) + C∥B′  2u∥H1−s,−r−1,−ℓ−1  sc,b,h  ( ¯  X) + ChN∥u∥ ˙Hs′,r′,ℓ′  sc,b,h ( ¯  X)  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='96)  where the differential order s′, the sc-decay order r′ and b-decay order ℓ′ are actually irrelevant here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  By the same reasoning as in the proof of the estimate (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='81), we have  ∥χu∥H1−s,−r−1,−ℓ−1  sc,b,h  ( ¯  X)  ≤ Ch−1∥Ph(z)∗u∥ ˙H−s,−r,−ℓ  sc,b,h  ( ¯  X) + C∥B+u∥H1−s,−r−1,−ℓ−1  sc,b,h  ( ¯  X)  + C∥B′  −u∥H1−s,−r−1,−ℓ−1  sc,b,h  ( ¯  X) + C∥B′  1u∥H1−s,−r−1,−ℓ−1  sc,b,h  ( ¯  X)  + C∥B′  0u∥H1−s,−r−1,−ℓ−1  sc,b,h  ( ¯  X) + C∥B′  zu∥H1−s,−r−1,−ℓ−1  sc,b,h  ( ¯  X) + ChN∥u∥ ˙H−N,−N,−ℓ−1  sc,b,h  ( ¯  X),  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='97)  ∥E′  ±u∥H1−s,−r−1,−ℓ−1  sc,b,h  ( ¯  X) + ∥E′  zu∥H1−s,−r−1,−ℓ−1  sc,b,h  ( ¯  X)  ≤ Ch−1∥Ph(z)∗u∥ ˙H−s,−r,−ℓ  sc,b,h  ( ¯  X) + C∥(1 − χ)u∥ ˙H1−s,−r−1,−ℓ−1  sc,b,h  ( ¯  X)  + C∥B′  1u∥H1−s,−r−1,−ℓ−1  sc,b,h  ( ¯  X) + C∥B′  0u∥H1−s,−r−1,−ℓ−1  sc,b,h  ( ¯  X) + ChN∥u∥ ˙H−N,−N,−ℓ−1  sc,b,h  ( ¯  X)  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='98)  and  ∥B′  2u∥H1−s,−r−1,−ℓ−1  sc,b,h  ( ¯  X) ≤ Ch−1∥Ph(z)∗u∥ ˙H−s,−r,−ℓ  sc,b,h  ( ¯  X) + C∥(1 − χ)u∥ ˙H1−s,−r−1,−ℓ−1  sc,b,h  ( ¯  X)  + C∥B′  0u∥H1−s,−r−1,−ℓ−1  sc,b,h  ( ¯  X) + ChN∥u∥ ˙H−N,−N,−ℓ−1  sc,b,h  ( ¯  X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='99)  Putting all the above estimates together yields for s > 1  2, r > − 1  2, ℓ < − 1  2  ∥χu∥H1−s,−r−1,−ℓ−1  sc,b,h  ( ¯  X) ≤ Ch−2∥Ph(z)∗u∥ ˙H−s,−r,−ℓ  sc,b,h  ( ¯  X) + C∥(1 − χ)u∥ ˙H1−s,−r−1,−ℓ−1  sc,b,h  ( ¯  X)  + ChN∥u∥ ˙H−N,−N,−ℓ−1  sc,b,h  ( ¯  X),  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='100)  Again using the semiclassical hyperbolic estimates (see [21, Theorem E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='57]), we have for all s, r, ℓ ∈ R  ∥(1 − χ)u∥ ˙H1−s,−r−1,−ℓ−1  sc,b,h  ( ¯  X) ≤ Ch−1∥Ph(z)∗u∥ ˙H−s,−r,−ℓ  sc,b  ( ¯  X)  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='101)  LINEAR STABILITY OF KERR-NEWMAN FAMILY  71  where the sc-decay order r and b-decay order ℓ are actually relevant here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Combining (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='100) with  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='101) yields for s > 1  2, r > − 1  2, ℓ < − 1  2  ∥u∥ ˙H1−s,−r−1,−ℓ−1  sc,b,h  ( ¯  X) ≤ Ch−2∥Ph(z)∗u∥ ˙H−s,−r,−ℓ  sc,b,h  ( ¯  X) + ChN∥u∥ ˙H−N,−N,−ℓ−1  sc,b,h  ( ¯  X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='102)  Taking h small enough, the term ChN∥u∥ ˙H−N,−N,ℓ  h,sc,b  ( ¯  X) can be absorbed into the left-hand side and  this proves the estimate (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='82).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  Invertibility of �  □gb(σ) for 0 ≤ Im σ ≤ C1, Re σ ≥ C2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' We only discuss the proof of the invertibility  of the operator �  □gb : {u ∈ ¯Hs,ℓ  b,h( ¯X) | ¯Hs,ℓ+1  b,h  ( ¯  X)} → ¯Hs,ℓ+1  b,h  ( ¯X) in detail as the operator �  □gb : {u ∈  ¯Hs,ℓ  b,h( ¯X) | ¯Hs−1,ℓ+2  b,h  ( ¯X)} → ¯Hs−1,ℓ+2  b,h  ( ¯X) can be handled in a completely analogous manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' We  define the space  X(σ) := {u ∈ ¯Hs,ℓ  b ( ¯X) | �  □gb(σ)u ∈ ¯Hs,ℓ+1  b  ( ¯X)},  which is a Hilbert space endowed with the norm ∥u∥X (σ) := ∥u∥ ¯  Hs,ℓ  b  + ∥�  □gb(σ)u∥ ¯  Hs,ℓ+1  b  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' First, the  injectivity of �  □gb(σ) : X(σ) → ¯Hs,ℓ  b  immediately follows from (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='79).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' As for the surjectivity, we need  to show that for any f ∈ ¯Hs,ℓ+1  b  , there exists u ∈ ¯Hs,ℓ  b  such that �  □gb(σ)u = f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' We define  Y(σ) := {v ∈ ˙H−s,−ℓ−1  b  ( ¯X) | �  □gb(σ)∗v ∈ ˙H−s,−ℓ  b  ( ¯X)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  According to (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='79), we have  |⟨f, v⟩| ≤ C∥f∥ ¯  Hs,ℓ+1  b  ∥�  □gb(σ)∗v∥ ˙H−s,−ℓ  b  ,  v ∈ Y(σ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  By Hahn-Banach Theorem, the anti-linear form �  □gb(σ)∗v → ⟨f, v⟩ with v ∈ Y(σ) can be extended  to a continuous anti-linear form on ˙H−s,−ℓ  b  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Since ( ˙H−s,−ℓ  b  )∗ = ¯Hs,ℓ  b , there exists u ∈ ¯Hs,ℓ  b  such that  ⟨u, �  □gb(σ)∗v⟩ = ⟨f, v⟩ for v ∈ Y(σ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' In particular, for v ∈ C∞  c (X◦)  ⟨�  □gb(σ)u, v⟩ = ⟨u, �  □gb(σ)∗v⟩ = ⟨f, v⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  This proves that �  □gb(σ)u = f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  □  Now we further determine the index of the Fredholm operators �  □gb(σ) on suitable function spaces for  σ ∈ C, Im σ ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Let s > 1  2, ℓ < − 1  2, s + ℓ > − 1  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Then the following holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  (1) For σ ∈ C, Im σ ≥ 0, σ ̸= 0, the operators  �  □gb(σ) : {u ∈ ¯Hs,ℓ  b ( ¯X) | �  □gb(σ)u ∈ ¯Hs−1,ℓ+2  b  ( ¯X)} → ¯Hs−1,ℓ+2  b  ( ¯X)  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='103)  �  □gb(σ) : {u ∈ ¯Hs,ℓ  b ( ¯X) | �  □gb(σ)u ∈ ¯Hs,ℓ+1  b  ( ¯X)} → ¯Hs,ℓ+1  b  ( ¯X)  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='104)  are Fredholm of index 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  (2) Let − 3  2 < ℓ < − 1  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Then  �  □gb(0) : {u ∈ ¯Hs,ℓ  b ( ¯X) | �  □gb(0)u ∈ ¯Hs−1,ℓ+2  b  ( ¯X)} → ¯Hs−1,ℓ+2  b  ( ¯X)  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='105)  is Fredholm of index 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' We now prove that (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='103) and (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='105) have index 0, and the index 0 property of (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='104) follows in  a similar manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  Proof of the index 0 property of (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='103).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' In view of the high energy estimates in Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='22,  �  □gb(σ) : {u ∈ ¯Hs,ℓ  b ( ¯X) | �  □gb(σ)u ∈ ¯Hs−1,ℓ+2  b  ( ¯X)} → ¯Hs−1,ℓ+2  b  ( ¯X)  is invertible and thus has index 0 for fixed Im σ = C ≥ 0 when |Re σ| ≫ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' We claim that the index  of �  □gb(σ) with Im σ ≥ 0, σ ̸= 0 is a constant on the horizontal line Im σ = C and thus is 0 for all σ  with Im σ ≥ 0, σ ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Let  X(σ) := {u ∈ ¯Hs,ℓ  b ( ¯X) : �  □gb(σ)u ∈ ¯Hs−1,ℓ+2  b  ( ¯X)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  72  LILI HE  For any fixed σ0 ̸= 0 with Im σ0 = C ≥ 0, without loss of generality, we may assume Re σ0 ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  First, there exists a C′ such that �  □gb(σ) is invertible and thus has index 0 for σ ∈ [C′ + iC, ∞ + iC).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  We next prove that for each σ′ ∈ [σ0 = Re σ0 + iC, C′ + iC], there exists ǫ > 0 such that the index  of �  □gb(σ) : X(σ) → ¯Hs−1,ℓ+2  b  ( ¯X) is a constant for σ with Im σ ≥ 0, σ ̸= 0, |σ − σ′| < ǫ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Then by  compactness of [σ0 = Re σ0 + iC, C′ + iC] we conclude that the index of �  □gb(σ0) is the same as that  of �  □gb(C′ + iC) and thus is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Suppose �  □gb(σ′) has kernel and cokernel of dimension k+ and k−,  respectively, then we establish a Grushin problem [21, Appendix C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' More specifically, we define  the following operator  L(σ) =  ��  □gb(σ)  L−  L+  0  �  : X(σ) ⊕ Ck− → ¯Hs−1,ℓ+2  b  ( ¯X) ⊕ Ck+  where L+, L− are defined as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' If ker �  □gb(σ′) is spanned by xj ∈ X(σ′), j = 1, · · · , k+, then  by Hahn-Banach Theorem we can find x∗  j ∈ ˙H−s,−ℓ  b  ( ¯X) with ∥x∗  j∥ ≤ 1 such that ⟨xi, x∗  j⟩ = δij.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' It  follows that  L+ : ¯Hs,ℓ  b ( ¯X) → Ck+,  L+(f) := (⟨f, x∗  1⟩, · · · , ⟨f, x∗  k+⟩)  restricts to an isomorphism ker �  □gb(σ′) → Ck+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' As for the construction of L−, we choose yj ∈  ¯Hs−1,ℓ+2  b  ( ¯X) with ∥yj∥ ≤ 1, j = 1, · · · , k− so that yj+ran�  □gb(σ′) form a basis of ¯Hs−1,ℓ+2  b  /ran�  □gb(σ′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  Then we define  L− : Ck− → ¯Hs−1,ℓ+2  b  ( ¯X),  L−((a1, · · · , ak−)) :=  k−  �  j=1  ajyj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  The operator L− has maximal rank and ranL− ∩ ran�  □gb(σ′) = ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' The above construction of L+, L−  implies L(σ′) has a trivial kernel and onto, and thus is invertible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Since �  □gb(σ) satisfy the uniform  Fredholm estimates (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='17) for all σ ∈ C, Im σ ∈ [0, 2C] satisfying |σ0|/2 ≤ |σ| ≤ 2(C + C′), so do  L(σ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' That is, there exists a constant ˜C (independent of b and σ) such that  ∥(u, u−)∥ ¯  Hs,ℓ  b  ( ¯  X)⊕Ck− ≤ ˜C  �  ∥L(σ)(u, u−)∥ ¯  Hs−1,ℓ+2  b  ( ¯  X)⊕Ck+ + ∥(u, u−)∥ ¯  Hs0,ℓ0  b  ( ¯  X)⊕Ck−  �  for σ ∈ C, Im σ ∈ [0, 2C] satisfying |σ0|/2 ≤ |σ| ≤ 2(C + C′), and s0 < s, ℓ0 < ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Then following  the perturbation arguments in [77, §2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='7], we prove the invertibility of L(σ) when σ is close to σ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  Concretely, suppose there exists a sequence δj → δ′ such that L(δj) is not invertible, so either  ker L(δj) on ¯Hs,ℓ  b ( ¯X) ⊕ Ck− or ker L(δj)∗ on ˙H−s+1,−ℓ−2  b  ( ¯X) ⊕ Ck+ is non-trivial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' By passing to a  subsequence, we may assume that the former possibility holds for all j, as the case of the adjoint  is completely analogous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Now, if ∥(uj, u−  j )∥ ¯  Hs,ℓ  b  ( ¯  X)⊕Ck− = 1 and L(δj)(uj, u−  j ) = 0, then the above  uniform Fredholm estimate gives ˜C−1 ≤ ∥(uj, u−  j )∥ ¯  Hs0,ℓ0  b  ( ¯  X)⊕Ck− .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Using the sequential compactness  of the unit ball in ¯Hs,ℓ  b ( ¯X) ⊕ Ck− in the weak topology, and the compactness of the inclusion  ¯Hs,ℓ  b ( ¯  X) ⊕ Ck− → ¯Hs0,ℓ0  b  ( ¯X) ⊕ Ck−, s0 < s, ℓ0 < ℓ, it follows that (uj, u−  j ) has a weakly convergent  subsequence in ¯Hs,ℓ  b ( ¯X) ⊕ Ck− to some (u0, u+  0 ) ∈ ¯Hs,ℓ  b ( ¯X) ⊕ Ck− which is norm-convergent in  ¯Hs0,ℓ0  b  ( ¯X) ⊕ Ck−.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' We also have  L(δ′)(u0, u+  0 ) = L(δ′)  �  (u0, u+  0 ) − (uj, u−  j )  �  + (L(δ′) − L(δj))(uj, u−  j ) → 0  in  ¯Hs0−2,ℓ0  b  ( ¯X) ⊕ Ck−  since L(δj) → L(δ′) as bounded operators in L( ¯Hs0,ℓ0  b  ( ¯X)⊕Ck−, ¯Hs0−2,ℓ0  b  ( ¯X)⊕Ck−) and (uj, u−  j ) →  (u0, u−  0 ) in ¯Hs0,ℓ0  b  ( ¯X) ⊕ Ck−.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Therefore, L(δ′)(u0, u−  0 ) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Since ˜C−1 ≤ ∥(u, u−)∥ ¯  Hs0,ℓ0  b  ( ¯  X)⊕Ck− ,  (u0, u−  0 ) ̸= 0 and thus ker L(σ′) is non-trivial, which leads to a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' This proves that L(σ)  is invertible for σ close to σ′, which implies �  □gb(σ) has index k+ − k− (see [21, Theorem C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='4]) for  for σ close to σ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  Proof of 0 index property of (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='105).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Suppose that − 3  2 < ℓ < − 1  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Then using the uniform Fredholm  estimates (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='21) (up to σ = 0) and the argument as above yields the index 0 property of (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='105).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  □  LINEAR STABILITY OF KERR-NEWMAN FAMILY  73  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' High energy estimates for tensor wave operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Now we prove an analogy to Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='22 for the  semicalssical operators h2 �  Pb,γ(h−1z), h2 �  Wb,γ(h−1z) and h2 �  Lb,γ(h−1z) acting on bundles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Let b0 = (m0, a0, Q0) with |Q0| + |a0| < m0 and |a0| ≪ |m0| + |Q0|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Then there exists  ǫ > 0 such that for b = (m, a, Q) with |b − b0| < ǫ and for s > s0 > 2 (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' s > s0 > 3), ℓ < − 1  2 with  s+ℓ > − 1  2, the conclusions in Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='22 hold for h2 �  Pb,γ(h−1z) and h2 �  Wb,γ(h−1z) (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' h2 �  Lb,γ(h−1z)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' The proof is analogous to that of Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='22, except for the computation of threshold regularity  in the radial point estimate at event horizon and the threshold decay rate in the radial point estimate at  spatial infinity ∂+ ¯X, and the verification of  σ1,h  � 1  2ih  �  h2�•(h−1z) − (h2�•(h−1z))∗��  < νmin  2  ,  = Pb,γ, Wb,γ, Lb,γ  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='106)  at trapping K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  The computation of threshold regularity in the radial point estimate at event horizon and the threshold  decay rate in the radial point estimate at spatial infinity ∂+ ¯X is the same as that in the fixed σ case in  Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' The verification of (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='106) at trapping K follows from the discussion in appendix B (see  Theorems B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='6, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='7 and Remark B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='8) and the relevant calculation in the proof of Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  □  Correspondingly, we have the following analogy to Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='23 for �  Pb,γ, �  Wb,γ and �  Lb,γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Let b = (m, a, Q) with |Q| + |a| < m and |a| ≪ |m| + |Q|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Let s > 2(resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' s > 3), ℓ < − 1  2  and s + ℓ > − 1  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Then the conclusions in Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='23 also hold for �  Pb,γ(σ) and �  Wb,γ(σ) (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' �  Lb,γ(σ)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' The proof is completely analogous to that of Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  □  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Energy estimates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' In this section, we shall establish the energy estimates for the solutions to various  wave type equations on slowly rotating Kerr-Newman metrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  First, we introduce another modification of the Boyer-Lindquist coordinates, which is defined by  ¯t = t + Ft(r),  ¯ϕ = ϕ + Fϕ(r).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  In the new coordinates (¯t, r, θ, ¯ϕ), the Kerr-Newman metric takes the following form  gb = −∆b  ρ2  b  �  d¯t − a sin2 θd ¯ϕ +  �  a sin2 θF ′  ϕ(r) − F ′  t(r)  �  dr  �2  + sin2 θ  ρ2  b  �  ad¯t − (r2 + a2)d ¯ϕ +  �  (r2 + a2)F ′  ϕ(r) − aF ′  t(r)  �  dr  �2  + ρ2  b(dr2  ∆b  + dθ2)  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='107)  and its inverse is  g−1  b  = −  1  ρ2  b∆b  �  (r2 + a2)∂¯t + a∂ ¯ϕ  �2  +  1  ρ2  b sin2 θ  �  ∂ ¯ϕ + a sin2 θ∂¯t  �2  + 1  ρ2  b  ∂2  θ + ∆b  ρ2  b  �  ∂r + F ′  t(r)∂¯t + F ′  ϕ(r)∂ ¯ϕ  �2  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='108)  In order for gb, g−1  b  to be smooth at event horizon, Ft(r) and Fϕ(r) are required to satisfy that F ′  t(r) =  ±(r2 + a2)/∆b, F ′  ϕ(r) = ±a/∆b (up to a smooth modification) at event horizon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Now we shall prove  Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' There exist Ft(r), Fϕ(r) such that they satisfy the following conditions  (1) F ′  ϕ(r) =  a  ∆b + ˜Fϕ(r) where ˜Fϕ(r) is smooth on [r−, ∞)r, and Fϕ(r) = 0 near r = ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  (2) F ′  t(r) = r2+a2  ∆b  + ˜Ft(r) where ˜Ft(r) is smooth on [r−, ∞), and Ft(r) = −rb,∗ + O(r−1) near r = ∞  where drb,∗  dr  =  1  ∆b .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  (3) The hypersurfaces ¯t = const are spacelike, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=', d¯t is timelike.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' More precisely,  g−1  b (d¯t, d¯t) ≤ −m2ρ−2  b  < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  With this choice of Ft(r), Fϕ(r), gb in the coordinates (¯t, r, θ, ¯ϕ) is smooth and non degenerate on the extended  manifold R¯t × [r−, ∞)r × S2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  74  LILI HE  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Let χ(r) ∈ C∞  c (R) such that χ(r) = 1 when r ≤ 3m and χ = 0 when r ≥ 4m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' We first define  Fϕ(r) = χ  �  a/∆b dr which satisfies statement (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Next, we expect  F ′  t(r) = r2 + a2  ∆b  + µ1(r)  near  r = rb;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  F ′  t(r) = −r2 + a2  ∆b  + µ2(r)  near  r = ∞  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='109)  where µ1(r) is smooth near event horizon and µ2(r) ∼ r−2 near r = ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' We let  g−1  b (d¯t, d¯t) = − 1  ρ2  b  �(r2 + a2)2  ∆b  − ∆bF ′  t(r)2 − a2 sin2 θ  �  = −m2  ρ2  b  < 0,  In order to arrange (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='109), we need  F ′  t(r) =  \uf8f1  \uf8f4  \uf8f4  \uf8f2  \uf8f4  \uf8f4  \uf8f3  ��  r2+a2  ∆b  �2  − m2+a2 sin2 θ  ∆b  = r2+a2  ∆b  �  1 − ∆b(m2+a2 sin2 θ)  (r2+a2)2  ,  r is near rb  −  ��  r2+a2  ∆b  �2  − 1+a2 sin2 θ  ∆b  = − r2+a2  ∆b  �  1 − ∆b(m2+a2 sin2 θ)  (r2+a2)2  ,  r is near ∞  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  Therefore, we define  F ′  t(r) = χ(r)  �r2 + a2  ∆b  �  1 − ∆b(m2 + a2 sin2 θ)  (r2 + a2)2  �  + (1 − χ(r))  �  − r2 + a2  ∆b  �  1 − ∆b(m2 + a2 sin2 θ)  (r2 + a2)2  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  With this choice of F ′  t(r), we have g−1  b (d¯t, d¯t) = −m2ρ−2  b  for r ≤ 3m, r ≥ 4m and g−1  b (d¯t, d¯t) ≤ −m2ρ−2  b  for 3m ≤ r ≤ 4m, and thus the requirement statement (3) is satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' By choosing a proper integration  constant we have ¯t = rb,∗ + O(r−1) near r = ∞, and thus the requirement statement (2) is satisfied  □  We now establish the energy estimates for the solutions to the equation □gbu = f on the extended Kerr-  Newman manifold M = R¯t × [r−, ∞)r × S2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' We denote by Σ¯t0 the spacelike hypersurfaces {¯t = ¯t0} ∩ M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  Here, the integral on M is taken with respect to the volume form dV = dvolgb = ρ2  b sin θd¯tdrdθdϕ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  We begin with the energy-momentum tensor  Tαβ[g] = ∂αu∂βu − 1  2gαβ∂γu∂γu  Its contraction with respect to a vector field X is denoted by  Pα[g, X] = Tαβ[g]Xβ  and its divergence is  ∇αPα[g, X] = □gu · Xu + 1  2Tαβ[g]παβ  X  and παβ  X is the deformation tensor of X, which is given in terms of the Lie derivative by  πX  αβ = ∇αXβ + ∇βXα = (LXg)αβ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  Since Σ¯t is “asymptotically null”, we shall define the (degenerate) energy at Σ¯t0 as  E[u](¯t0) :=  � ∞  r−  �  S2  1  r2 |∂¯tu(¯t0, ·)|2 + |∂ru(¯t0, ·)|2 + 1  r2  �  |∂θu(¯t0, ·)|2 +  1  sin2 θ|∂ϕu(¯t0, ·)|2�  r2 sin θ dr dθ dϕ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  Now we establish the energy estimate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Let b = (m, a, Q) with |a| ≪ |Q|+m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Let u be the solution to the following initial value  problem  □gbu = f,  u|¯t=0 = u0,  ∂¯tu|¯t=0 = u1  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='110)  where rf ∈ L1([0, ∞)¯t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' L2(Σ¯t)) and u0  r , u1  r , ∂u0 ∈ L2(Σ0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' r2 sin θ dr dθ dϕ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Then there exists a constant C(b)  which depends on the parameter b = (m, a, Q) such that  E[u](¯t) ≤ C(b)  �  ∥u0  r ∥L2(Σ0) + ∥∂u0∥L2(Σ0) + ∥u1  r ∥L2(Σ0) + ∥rf∥2  L1([0,¯t);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='L2(Σ¯t))  �  eC(b)¯t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='111)  where the integral is taken with respect the volume form r2 dr dθ dϕ and ∂ ∈ {∂r, 1  r ∂θ,  1  r sin θ∂ϕ}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  Moreover, the above statements also hold for the wave type operators Pb,γ, Wb,γ and Lb,γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  LINEAR STABILITY OF KERR-NEWMAN FAMILY  75  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' We only do the analysis for the case a = 0 here and the same argument will also work for small a  since all the calculation below near event horizon depends continuously on b = (m, a, Q) (away from the  event the horizon, we make use of the timelike property of d¯t and ∂¯t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Therefore, we let b = (m, 0, Q) and  then we have  g = gb = −∆b  r2 d¯t2 + 2∆b  r2 F ′  t(r)d¯tdr +  � r2  ∆b  − ∆b  r2 F ′  t(r)2�  dr2 + r2dθ2 + r2 sin2 θdϕ2,  g−1 = g−1  b  = −  � r2  ∆b  − ∆b  r2 F ′  t(r)2�  ∂2¯t + 2∆b  r2 F ′  t(r)∂¯t∂r + ∆b  r2 ∂2  r + 1  r2 ∂2  θ +  1  r2 sin2 θ∂2  ϕ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  By a density statement, we may assume that u is supported away from spatial infinity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  Killing multiplier: X = ∂¯t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' We compute  g−1(d¯t, P[g, ∂¯t]) =  �∆b  r2 F ′  t(r)2 − r2  ∆b  �  T (∂¯t, ∂¯t) + ∆b  r2 F ′  t(r)T (∂r, ∂¯t)  = −1  2  �� r2  ∆b  − ∆b  r2 F ′  t(r)2�  |∂¯tu|2 + ∆b  r2 |∂ru|2 + |∂θu|2 +  1  sin2 θ|∂ϕ|2  �  ∼ −  � 1  r2 |∂¯tu|2 + |∂ru|2 + 1  r2 |∂θu|2 +  1  r2 sin2 θ|∂ϕu|2�  on  [r−, ∞),  g−1(dr, P[g, ∂¯t]) = ∆b  r2 F ′  t(r)T (∂¯t, ∂¯t) + ∆b  r2 T (∂r, ∂¯t)  = ∆b(r−)  r2  −  F ′  t(r−)|∂¯tu|2 + ∆b(r−)  r2  −  ∂¯tu∂ru  at  r = r−.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='112)  We see that g−1(d¯t, P[g, ∂¯t]) gives control of | 1  r∂¯tu| and angular derivative but not ∂ru near event  horizon because ∆b(r) is nonpositive at and beyond event horizon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' We also note that the first term  of g−1(dr, P[g, ∂¯t]) is positive if r− is sufficiently close to event horizon r = rb while the second term  does not have a definite sign.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' So we need to introduce a suitable multiplier which allows us to deal  with the above issues at event horizon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  Energy estimates near event horizon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Following [62], We introduce X = b(r)∂r which is expressed in  terms of the coordinates (¯t, r, θ, ϕ) with b(r) = −χ(r) where χ(r) = 1 when r ≤ 3m and χ = 0 when  r ≥ 4m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' First we compute the boundary terms at the spacelike hypersurfaces ¯t = const  g−1(d¯t, P[g, X]) =  �∆b  r2 F ′  t(r)2 − r2  ∆b  �  T (∂¯t, b(r)∂r) + ∆b  r2 F ′  t(r)T (∂r, b(r)∂r)  = −χ(r)  �  −  �∆b  r2 F ′  t(r)2 − r2  ∆b  �  ∂¯tu∂ru + ∆b  r2 F ′  t(r)|∂ru|2  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  Since ∆b  r2 F ′  t(r) ∼ 1 near event horizon r = rb, it follows that on r ∈ [r−, ∞)  g−1(d¯t, P[g, C∂¯t + X]) ∼ −E[u](¯t)  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='113)  provided C is large enough.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' At the hypersurface r = r−, we have  g−1(dr, P[g, X]) = ∆b  r2 F ′  t(r)T (∂¯t, b(r)∂r) + ∆b  r2 T (∂r, b(r)∂r)  = −1  2χ(r)  �� r2  ∆b  − ∆b  r2 F ′  t(r)2�  |∂¯tu|2 + ∆b  r2 |∂ru|2 − 1  r2 |∂2  θu|2 −  1  r2 sin2 θ |∂ϕu|2  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  Therefore, combining the calculation of g−1(dr, P[g, X]) with (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='112) yields  g−1(dr, C∂¯t + X)  =  �  C ∆b(r−)  r2  −  F ′  t(r−) + 1  2  � r2  ∆b  − ∆b  r2 F ′  t(r)2�  − ǫ  �  |∂¯tu|2 + ǫ(∂¯tu + C∆b(r−)  2ǫr2  −  ∂ru  �2  +  �  − ∆b(r−)  2r2  −  − C2∆2  b(r−)  4ǫr4  −  �  |∂ru|2 +  1  2r2  −  |∂2  θu|2 +  1  2r2  − sin2 θ|∂ϕu|2 ≥ 0  at  r = r−.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='114)  provided C is sufficiently large, ǫ is small enough and r− is sufficiently close to rb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  76  LILI HE  Then we integrate ∇αPα[g, C∂¯t + X] = □gu · (∂¯tu + u) + 1  2Tαβ[g]παβ  X over the region  D = {0 ≤ ¯t ≤ ¯t1, r ≥ r−}  with respect to the volume form dVgb = r2 sin θd¯tdrdθdϕ and use Divergence Theorem  �  D  ∇αPα dV =  �  ∂D  ιP dV  to obtain�  D  □gu · (C∂¯tu + Xu) r2 sin θd¯tdrdθdϕ + 1  2  �  D  Tαβ[g]παβ  X r2 sin θd¯tdrdθdϕ  =  � ∞  r−  �  S2 g−1(d¯t, P[g, C∂¯t + X]) r2 sin θdrdθdϕ|  ¯t=¯t1  ¯t=0  −  � ¯t1  0  �  S2 g−1(dr, P[g, C∂¯t + X])r2 sin θd¯tdθdϕ|r=r−.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  Since g−1(dr, P[g, C∂¯t + X])|r=r− ≥ 0, and Tαβ[g]παβ  X is quadratic in ∂u, we find that  sup  0≤¯t≤¯t1  E[u](¯t) ≤ C′�  ∥∂u0∥L2(Σ0) + ∥u1  r ∥L2(Σ0) + ∥rf∥2  L1([0,¯t);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='L2(Σ¯t))  �  + C′  � ¯t1  0  E[u](¯t) d¯t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  where ∂ ∈ {∂r, 1  r∂θ,  1  r sin θ∂ϕ} and L2(Σ0) = L2(Σ0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' r2 sin θ dr dθ dϕ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  As for the L2 norm of u, we have  ∥u  r ∥L2(Σ¯t) ≲ ∥u0  r ∥L2(Σ0) +  � ¯t1  0  ∥∂¯tu  r ∥L2(Σ¯t) d¯t  where L2(Σ¯t) = L2(Σ¯t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' r2 sin θ dr dθ dϕ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Therefore, we have  sup  0≤¯t≤¯t1  E[u](¯t) ≤ C  �  ∥u0  r ∥L2(Σ0) + ∥∂u0∥L2(Σ0) + ∥u1  r ∥L2(Σ0) + ∥rf∥2  L1([0,¯t);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='L2(Σ¯t))  �  + C  � ¯t1  0  E[u](¯t) d¯t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  Finally by Gr¨onwall inequality we obtain the exponential growth of the energy  E[u](¯t) ≤ C  �  ∥u0  r ∥H1(Σ0) + ∥∂u0∥L2(Σ0) + ∥u1  r ∥L2(Σ0) + ∥rf∥2  L1([0,¯t);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='L2(Σ¯t))  �  eC¯t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  Proof for Pb,γ, Wb,γ and Lb,γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' According to Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='12, −2Pb,γ, −2Wb,γ (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  −Lb,γ) are □gb  tensored with 4 × 4 (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' 14 × 14) identity matrix plus a a matrix whose entries are first order  differential operators with coefficients decaying like r−2, and then the above arguments in the proof  for □gb still apply here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  □  We next show that if Im σ is sufficiently large, then �  □gb(σ), �  Pb,γ(σ), �  Wb,γ(σ), �  Lb,γ(σ) are invertible on  suitable function spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Let b0 = (m0, 0, Q0) with |Q0| < m0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Then there exists ǫ > 0 such that for b = (m, Q, a)  with |b − b0| < ǫ and for s > 1  2, ℓ < − 1  2 with s + ℓ > − 1  2, the operators  �  □gb(σ) : {u ∈ ¯Hs,ℓ  b ( ¯X) | �  □gb(σ)u ∈ ¯Hs,ℓ+1  b  ( ¯X)} → ¯Hs,ℓ+1  b  ( ¯X)  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='115)  �  □gb(σ) : {u ∈ ¯Hs,ℓ  b ( ¯X) | �  □gb(σ)u ∈ ¯Hs−1,ℓ+2  b  ( ¯X)} → ¯Hs−1,ℓ+2  b  ( ¯X)  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='116)  are invertible for Im σ ≥ C′(b0) where C′(b0) > 0 is a sufficiently large constant only depending on b0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  Moreover, the above statements also hold for the wave type operators Pb,γ, Wb,γ and Lb,γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  LINEAR STABILITY OF KERR-NEWMAN FAMILY  77  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' According to Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='27, there exist ǫ > 0, ˜C(b0) > 0 such that for |b − b0| < ǫ, the solution u  to the initial value problem (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='110) satisfies  E[u](¯t) ≤  �  ˜C(b0)E[u](0) + ˜C(b0)  � � ¯t  0  ∥rf∥L2(Σ¯t) d¯t  �2�  e  ˜  C(b0)¯t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  We now show that if Im σ > ˜C(b0), then (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='115) and (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='116) are invertible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' According to Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='23,  it suffices to prove that (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='115) and (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='116) are injective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Assume the contrary, by Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='7, there  exists a nontrivial solution w = ρC∞(∂+ ¯X) + A2−( ¯  X) to �  □gb(σ)w = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Then, u(tb,∗, r, θ, ϕ) = e−itb,∗σw  is a solution to □gbu = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Let χ(¯t) ∈ C∞(R) be such that suppχ ⊂ [1, ∞) and supp(1 − χ) ⊂ (−∞, 2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  Then χu is the unique solution which is supported in (1, ∞) to the inhomogeneous equation □˜u = [□gb, χ]u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  According to Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='11 and ¯t − tb,∗ = O( 1  r), we have  □˜u = [□gb, χ]u ∼  1  r3−  and thus χ(¯t)u satisfies E(χu) ≲ e ˜  C(b0)¯t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' But a direct calculation implies E(χu) ∼ eIm σtb,∗ ∼ eIm σ¯t, which  leads to a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  The proof for �  Pb,γ(σ), �  Wb,γ(σ), �  Lb,γ(σ) is completely analogous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  □  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Spherical harmonic decompositions  In this section, we shall introduce the spherical harmonic decompositions, which will be used in the proof  of the geometric mode stability of Reissner-Nordstr¨om spacetime (see §7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  Let /g be the standard metric on the unit sphere S2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' We denote the operators on S2 using a slash, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  /tr = tr/g, /δ = δ/g, /δ  ∗ = δ∗  /g, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' We denote by Y m  l  with l ∈ N, m ∈ Z, |m| ≤ l the spherical harmonic function  of degree l and order m which satisfies /∆HY m  l  = l(l+1)Y m  l  where /∆H = /d/δ+/δ/d is the Hodge Laplacian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' We  denote by Sl = {Y m  l  : |m| ≤ l} the space of degree l spherical harmonic functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Then one can conclude  that {Sl : l ∈ N} form an orthogonal basis of L2(S2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  According to Hodge decomposition Theorem and the fact that the de Rham cohomology group H1  dR(S2) =  0, we see that any 1-form ω ∈ C∞(S2, T ∗S2) can be uniquely decomposed as ω = /dφ + /δη where φ ∈  C∞(S2), η ∈ C∞(S2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Λ2T ∗S2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Since /δ = −/∗/d/∗ where /∗ is the Hodge star operator, we can rewrite ω =  /dφ + /∗/dψ where φ, ψ ∈ C∞(S2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Therefore, one can conclude that an orthogonal basis of L2(S2, T ∗S2) is  given by {/dSl, Vl = /∗/dSl : l ∈ N} where Sl, Vl are the eigen-1-forms of the Hodge Laplacian ∆H with  eigenvalue l(l + 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' We note that /δVl = 0 and /dS0 = V0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' In other words, a spectral decomposition of  the negative tensor Laplacian − /∆ = − /tr /∇a /∇b satisfying − /∆ = /∆H − 1 is given by the scalar part  /dSl ∈ ker(− /∆ − (l(l + 1) − 1))  with  l ≥ 1  (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='1)  and the vector part  Vl ∈ ker(− /∆ − (l(l + 1) − 1))  with  l ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='2)  As for the symmetric 2-tensors h ∈ C∞(S2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' S2T ∗S2), we have the transverse-traceless decomposition [87]  h = 1  2(/trh)/g + /δ  ∗  0ω + hT T where /δ  ∗  0 = /δ  ∗ + 1  2/g/δ is the traceless symmetric gradient, ω ∈ C∞(S2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' T ∗S2) and  hT T is both traceless and divergence free.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' According to [37], we know that hT T = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' We then decompose  ω further into eigen-1-forms (we also call them spherical harmonic 1-forms) of − /∆ of scalar type /dSl and  vector type Vl, and /trh into scalar spherical harmonic functions Sl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' In the spherical coordinates (θ, ϕ), since  /dY 0  1 = − sin θdθ,  /dY 1  1 = cos θ cos ϕdθ − sin θ sin ϕdϕ,  /dY −1  1  = cos θ sin ϕdθ + sin θ cos ϕdϕ,  we see that /dS1 is conformal killing and thus /δ  ∗  0/dS1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' We also have  /∗/dY 0  1 = − sin2 θdϕ,  /∗/dY 1  1 = sin θ cos θ cos ϕdϕ + sin ϕdθ,  /∗/dY −1  1  = sin θ cos θ sin ϕdϕ − cos ϕdθ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='3)  and find that V♯  1 are rotations, and thus killing, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' /δ  ∗V1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Then any symmetric 2-tensors on sphere  can be decomposed into the following spherical harmonic symmetric 2-tensors of the scalar type  Sl/g (l ≥ 0),  /δ  ∗  0/dSl (l ≥ 2)  (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='4)  and the vector type  /δ  ∗  0Vl = /δ  ∗Vl (l ≥ 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='5)  78  LILI HE  We claim that the following geometric operators on sphere S2  /g, /tr, /d, /δ, /δ  ∗, − /∆ and their compositions  preserve the the scalar and vector type of the spherical harmonics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' More precisely, these geometric operators  map the scalar type functions/1-forms/symmetric 2-tensors built out of a particular S ∈ Sl into another scalar  type tensor with the same S, likewise for the vector type 1-forms/symmetric 2-tensors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' This is obviously  true for /g acting on functions, /tr on symmetric 2-tensors, /d on functions, /δ on 1-forms, and − /∆ on functions  and 1-forms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' For the verification of /δ, /δ  ∗, − /∆, we make use of the identities /δ/δ  ∗ω = − 1  2 /∆ω + 1  2/d/δω − 1  2ω and  − /∆/δ  ∗ω = /δ  ∗(− /∆ω) − 3/δ  ∗ω − 2/g/δω to obtain  /δ(S/g) = −/dS, /δ/δ  ∗  0/dS = l(l + 1) − 2  2  /dS, /δ/δ  ∗/∗/dS = l(l + 1) − 2  2  /∗/dS;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' /δ  ∗/dS = /δ  ∗  0/dS−l(l + 1)  2  S/g;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  − /∆(S/g) = l(l + 1)S/g,  − /∆(/δ  ∗  0/dS) = (l(l + 1) − 4) (/δ  ∗  0/dS),  − /∆(δ∗/∗/dS) = (l(l + 1) − 4) (δ∗/∗/dS).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='6)  We now decompose the spacetime M into the aspherical ˚  X = Rt∗ × [r−, ∞)r and spherical part S2, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  M = ˚  X × S2,  ˚π : M → ˚  X,  /π : M → S2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='7)  We then call the functions aspherical if they are only functions of (t∗, r), and spherical if they are only  functions on S2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Alternatively, we can identify the aspherical and spherical functions as subspaces of C∞(M)  as follows:  C∞( ˚  X) ∼= ˚π∗C∞( ˚  X) ⊂ C∞(M),  C∞(S2) ∼= /π∗C∞(S2) ⊂ C∞(M).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='8)  Then functions on M, say L2(M), can be decomposed into an infinite sum of products of aspherical functions  and spherical harmonic functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  We can also split the cotangent bundle T ∗M into aspherical and spherical part as follows:  T ∗M = T ∗  AS ⊕ T ∗  S  where  T ∗  AS = ˚π∗T ∗ ˚  X  and  T ∗  S = /π∗T ∗S2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='9)  We can further write ω ∈ T ∗  S as an infinite sum of the spherical harmonic 1-forms /dSl, Vl of − /∆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Therefore,  a 1-form ω ∈ C∞(M;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' T ∗M) can be expressed as an infinite sum of products of aspherical functions/1-forms  on ˚  X and spherical harmonic functions/1-forms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  The above splitting of T ∗M induces a natural splitting of the symmetric 2-tensors,into aspherical part  S2T ∗  AS, mixed part T ∗  AS ⊗ T ∗  S and spherical part S2T ∗  S as follows:  S2T ∗M = S2T ∗  AS ⊕ (T ∗  AS ⊗ T ∗  S) ⊕ S2T ∗  S  (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='10)  where we identify a ⊗ b ∈ T ∗  AS ⊗ T ∗  S as 2a ⊗s b = a ⊗ b + b ⊗ a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Therefore, a symmetric 2-tensor h ∈  C∞(M;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' S2T ∗M) can be expressed as an infinite sum of products of aspherical functions/1-forms/symmetric  2-tensors on ˚  X and spherical harmonic functions/1-forms/symmetric 2-tensors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  The above decomposition for functions/1-form/symmetric 2-tensors on M implies that the perturbation  (˙g, ˙A) can be divided into two categories: the first consists of scalar perturbations (also called even parity  modes in [72] and closed portions in [49]):  scalar l ≥ 2 :  �  ˙g = �fS + (f ⊗s /dS) + (HLS/g + HT /δ  ∗  0/dS)  ˙A = �KS + K/dS  with  S ∈ Sl  scalar l = 1 :  �  ˙g = �fS + (f ⊗s /dS) + HLS/g  ˙A = �KS + K/dS  with  S ∈ S1  scalar l = 0 :  �  ˙g = �f + HL/g  ˙A = �K  (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='11)  where  HL, HT , K ∈ C∞( ˚  X),  f, �K ∈ C∞( ˚  X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' T ∗ ˚  X),  �f ∈ C∞( ˚  X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' S2T ∗ ˚  X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  LINEAR STABILITY OF KERR-NEWMAN FAMILY  79  The second comprises of the vector perturbations (also called odd parity modes in [72] and co-closed portions  in [49]):  vector l ≥ 2 :  �  ˙g = f ⊗s V + HT /δ  ∗V  ˙A = KV  with  V ∈ Vl  vector l = 1 :  �  ˙g = f ⊗s V  ˙A = KV  with  V ∈ V1  (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='12)  where HT , K ∈ C∞( ˚  X) and f ∈ C∞( ˚  X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' T ∗ ˚  X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  We decompose the Reissner-Nordstr¨om metric g into aspherical and spherical part as in (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='7), which takes  the form in static coordinates (t, r, ω) with ω ∈ S2 as follows  g = ˚g + r2/g  (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='13)  where ˚g is a Lorentzian metric on ˚  X = Rt∗ × [r−, ∞) and /g denote the standard metric on the unit sphere  S2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' In static coordinates (t, r), ˚g takes the form  ˚g = −µb0dt2 + µ−1  b0 dr2  with  µb0 = 1 − 2m  r  + Q2  r2 ,  (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='14)  and it can be extended beyond the event horizon by introducing (t∗, r) coordinates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' We denote the operators  associated with ˚g and /g by rings and slashes respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Here we use the Einstein summation convention  with Greek indices µ, ν, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' for coordinates on M, lower case Roman indices i, j, k, m, n for coordinates on  ˚  X and a, b, c, d, e for coordinates on S2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Indices are raised and lowered using the Reissner-Nordstr¨om metric  g, except that we write /gab = (/g−1)ab for the inverse metric to /g on S2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Mode stability of the Reissner-Nordstr¨om spacetime  In this section we shall characterize the non-decaying (generalized) modes of the linearized Einstein-  Maxwell system around the Reissner-Nordstr¨om spacetime (M, gb0, Ab0) with subextremal charge, following  the method of Kodama-Ishibashi [56,57] for the study of perturbations of black holes without (or with) charge  in high dimensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Throughout this section, we drop the subscript b0, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=', we take (g, A) = (gb0, Ab0) and  m = m0, Q = Q0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Here we adopt some notations from [36,39,57].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  Theorem 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Let σ ∈ C with Im σ ≥ 0, suppose (˙g, ˙A) is an outgoing mode solution to the linearized  Einstein-Maxwell system, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=', (˙g, ˙A) = (e−iσt∗ ˙g0, e−iσt∗ ˙A0) with (˙g0, ˙A0) ∈ ¯H∞,ℓ  b  ( ¯X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' S2 �  scT ∗ ¯X ⊕ �  scT ∗ ¯X) for  some ℓ ∈ R solves  L1(˙g, ˙A) = −1  2□g ˙g − δ∗  gδgGg ˙g + Rg ˙g − 2D(g,dA)T (˙g, d ˙A) = 0  L2(˙g, ˙A) = D(g,A) (δgdA) (˙g, ˙A) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='1)  Then there exist parameters ˙m ∈ R, ˙a ∈ R3, ˙Q ∈ R and an outgoing 1-form ω and a function φ on ¯  M, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  (ω, φ) ∈ e−iσt∗ ¯H∞,ℓ′  b  ( ¯X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' �  scT ∗ ¯X ⊕ C) for some ℓ′ ∈ R, such that  (˙g, ˙A) =  �  ˙g(m,0,Q)( ˙m, ˙a, ˙Q) + 2δ∗  gω,  ˙A(m,0,Q)( ˙m, ˙a, ˙Q) + Lω#A + dφ  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='2)  More specifically  (1) If σ ̸= 0, then  (˙g, ˙A) =  �  2δ∗  gω, Lω♯A + dφ  �  where (ω, φ) ∈ e−iσt∗ ¯H∞,ℓ′  b  ( ¯X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' �  scT ∗ ¯X ⊕ C) for some ℓ′ ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  (2) If σ = 0 and − 3  2 < ℓ < − 1  2, we decompose the perturbation (˙g, ˙A) into spherical harmonics whose  detail is given in §6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Then  (i) If (˙g, ˙A) is of scalar type with l ≥ 1 or vector type l ≥ 2, then  (˙g, ˙A) =  �  2δ∗  gω, Lω♯A + dφ  �  where (ω, φ) ∈ ¯H∞,ℓ−1  b  ( ¯X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' �  scT ∗ ¯X ⊕ C) is of the same type as (˙g, ˙A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  80  LILI HE  (ii) If (˙g, ˙A) is of scalar type with l = 0, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' spherically symmetric, then  (˙g, ˙A) =  �  ˙g(m,0,Q)( ˙m, 0, ˙Q) + 2δ∗  gω,  ˙A(m,0,Q)(0, 0, ˙Q) + Lω♯A + dφ  �  where (ω, φ) ∈ ¯H∞,ℓ−1  b  ( ¯X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' �  scT ∗ ¯X ⊕ C) is spherically symmetric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  (iii) If (˙g, ˙A) is of vector type l = 1, then  (˙g, ˙A) =  �  ˙g(m,0,Q)(0, ˙a, 0) + 2δ∗  gω,  ˙A(m,0,Q)(0, ˙a, 0) + Lω♯A + dφ  �  where (ω, φ) ∈ ¯H∞,ℓ−1  b  ( ¯X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' �  scT ∗ ¯X ⊕ C) is of vector type with l = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  The statement of stationary perturbation (˙g, ˙A) (σ = 0) of scalar type with l = 0, 1 can be extended  to the following cases:  (iv) If (˙g, ˙A) ∈ ¯H∞,ℓ  b  ( ¯X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' S2 �  scT ∗ ¯X ⊕ �  scT ∗ ¯X) with − 5  2 < ℓ < − 3  2 is a stationary perturbation of scalar  type l = 1, then  (˙g, ˙A) =  �  2δ∗  gω, Lω♯A + dφ  �  where (ω, φ) ∈ ¯H∞,ℓ−1  b  ( ¯X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' �  scT ∗ ¯X ⊕ C) is of the scalar type l = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  (v) If (˙g, ˙A) ∈ Poly(t∗)k ¯H∞,ℓ  b  ( ¯  X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' S2 �  scT ∗ ¯X ⊕ �  scT ∗ ¯X) is of scalar type l = 0, then  (˙g, ˙A) =  �  ˙g(m,0,Q)( ˙m, 0, ˙Q) + 2δ∗  gω,  ˙A(m,0,Q)(0, 0, ˙Q) + Lω♯A + dφ  �  where (ω, φ) ∈ Poly(t∗)k+1 ¯H∞,ℓ′  b  ( ¯  X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' �  scT ∗ ¯X ⊕ C) is of scalar type l = 0 for some ℓ′ < ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  Remark 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Our discussion corresponds to the case n = 2, K = 1, κ2 = 2, E0 = r−2Q, Q2 = Q2, Λ = 0 (and  thus λ = 0), in [57].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' We note that [57] discusses the scalar and vector type perturbations for non-stationary  modes (σ ̸= 0) with l ≥ 2 in detail, but the corresponding stationary perturbations (σ = 0) are only treated  in the uncharged case in [56].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' [57] also studies the scalar (Appendix D) and vector (§4) perturbations for  modes l = 1 (which is called exceptional modes there), however, they do not give a full description of the  scalar l = 0 (spherically symmetric) perturbations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Scalar type perturbations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' We shall consider the scalar type perturbations of modes l ≥ 2, l = 1  and l = 0 separately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Recall that we can write the perturbations under the splitting (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='9) and (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='10) as in  (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' We further introduce a rescaled version of the traceless part of the spherical harmonic symmetric  2-tensor /δ  ∗  0/dSl which is built from the spherical harmonic function S ∈ Sl with eigenvalues k2 = l(l + 1):  /HkS = k−2/δ  ∗  0/dS  where  S ∈ Sl with l ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Modes with l ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Suppose (˙g, ˙A) is the scalar perturbations of the following form  ˙g =  \uf8eb  \uf8ed  �fS  − r  kf ⊗ /dS  2r2(HLS/g + HT /HkS)  \uf8f6  \uf8f8 ,  ˙A =  �  �KS  − r  kK/dS  �  with  S ∈ Sl (l ≥ 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='3)  We notice that the pure gauge solutions take the form  δ ˙g = 2δ∗  gω =  \uf8eb  \uf8ed  2˚δ∗T  − r  k(− k  r T + r˚  dr−1L)/dS  2r2(r−1ιdrT + k  2rL)S/g − k  r L /HkS  \uf8f6  \uf8f8 ,  δ ˙A = Lω♯A + dφ =  �  (˚dιAT + ιT ˚dA + ˚dP)S  − r  k(− k  r ιAT − k  r P)/dS  �  (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='4)  with  ω =  �  T S  − r  kL/dS  �  ,  φ = PS  (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='5)  where T ∈ C∞( ˚  X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' T ∗ ˚  X), L, P ∈ C∞( ˚  X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' When adding (δ ˙g, δ ˙A) to (˙g, ˙A), the quantities ˜f, f, HL, HT , ˜K, K  change by  δ �f = 2˚δ∗T,  δf = −k  r T + r˚  d(r−1L),  δHL = r−1ιdrT + k  2rL,  δHT = −k  r L;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  δ �K = ˚  dιAT + ιT ˚  dA + ˚dP,  δK = −k  r (ιAT + P).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='6)  LINEAR STABILITY OF KERR-NEWMAN FAMILY  81  We define X := r  k(f + r  k ˚  dHT ), then δX = r  k(δf + r  k ˚dδHT ) = −T because δ commutes with any geometric  operators we use in this manuscript.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' As a consequence, the following quantities  �F := �f + 2˚δ∗X ∈ C∞( ˚  X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' S2T ∗ ˚  X),  J := HL + 1  2HT + r−1ιdrX ∈ C∞( ˚  X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' T ∗ ˚  X),  N := �K + ˚  d  � r  k K  �  + ιX˚dA ∈ C∞( ˚  X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' T ∗ ˚  X)  (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='7)  are gauge-invariant, that is, δ �F = 0, δJ = δN = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Conversely, if �F = 0, J = N = 0, one can verify (˙g, ˙A) is  a pure gauge solution  (˙g, ˙A) = (2δ∗ω, Lω♯A + dφ) with ω =  � −XS  r2  k2 HT /dS  �  ,  φ = − r  kK + ιAX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='8)  If (˙g, ˙A) are outgoing mode solutions with frequency σ ̸= 0, then ω and φ are modes of the same type, with  an extra factor r2 and r respectively relative to ˙g and ˙A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' If σ = 0, (ω, φ) grow at most by a factor r more  than (˙g, ˙A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  Since the linearized Einstein-Maxwell system is gauge-invariant, we can express it in terms of the gauge-  invariant quantities defined above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  Concretely, one can choose a gauge, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=', add a pure gauge solution  (δ ˙g, δ ˙A) to (˙g, ˙A) for suitable ω = ω(T, L), φ = φ(P) such that the non gauge-invariant quantities ˜f etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  take a simple form in the new gauge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' More specifically, we adds (δ ˙g, δ ˙A) built from T = X, L = r  kHT , P =  r  kK − ιAX, then X + δX = T − T = 0, HT + δHT = k  r L − k  r L = 0 which implies f + δf = 0, �f + δ �f =  ˜F, HL+δHL = J, and K +δK = K − k  r ιAX− k  r ( r  kK −ιAX) = 0 which implies �K +δ �K = N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' To summarize,  if we replace (˙g, ˙A) by (˙g + δ ˙g, ˙A + δ ˙A), in the new gauge we have ( �f, f, HL, HT , �K, K) = ( �F , 0, J, 0, N, 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  Therefore, using the detailed calculation in §6 and §A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='1 we can write the linearized Einstein-Maxwell system,  acting on the new (˙g, ˙A)  ˙g =  \uf8eb  \uf8ed  �FS  0  2r2JS/g  \uf8f6  \uf8f8 ,  ˙A =  �  NS  0  �  in terms of the gauge-invariant quantities �F, J, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  Now we express 2L1(˙g, ˙A) = 0 and L2(˙g, ˙A) = 0 in the form of (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='3) in terms of �f E, f E, HE  L , HE  T and  �KE, KE respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Then the linearized Einstein-Maxwell system reads  �f E =  �  −˚□ − 2˚δ∗˚δ − ˚δ∗˚  d˚tr  �  �F + 2r−1 �  2˚δ∗ιdr �F − (∂ir)˚∇i �F  �  − 4˚δ∗˚dJ − 8r−1(dr) ⊗s ˚dJ  + (−µ′′  b0 + k2r−2) �F + (−2Q2r−4 + µ′′  b0)˚g˚tr �F − 4Qr−2˚g˚∗˚dN = 0,  (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='9a)  − r  kf E = −˚δ �F − 2˚dJ − r˚  d(r−1˚tr �F) + 4Qr−2˚∗N = 0,  (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='9b)  2r2HE  L = −˚□(2r2J) − 2rιdr˚δ �F + 2ιdrιdr �F − rιdr˚d˚tr �F + (rµ′  b0 + k2  2 + 2Q2r−2)˚tr �F  + (2k2 − 4Q2r−2)J + 4Q˚∗˚dN = 0,  (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='9c)  2r2HE  T = −k2˚tr �F = 0,  (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='9d)  �KE = r−2˚δ(r2˚dN) + k2r−2N + 1  2Qr−2˚∗˚  d˚tr �F − 2Qr−2˚∗˚  dJ = 0,  (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='9e)  − r  kKE = −˚δN = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='9f)  By (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='9d), all terms containing ˚tr �F vanish, then plugging ˚δ �F = −2˚dJ + 4Qr−2˚∗N into (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='9a) yields the can-  cellation of the term 4˚δ∗˚  dJ and thus one obtains a wave equation of �F (coupled to J, N only via subprincipal  terms).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Moreover, by (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='9f), one adds the zero term ˚d˚δN to (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='9e) and then can obtain a wave equation for  N (coupled to J via subprincipal terms).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' In conclusion, we can rewrite equation (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='9a), (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='9c) and (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='9e) as  82  LILI HE  a principally scalar system of wave equations for ( �F, J, N)  − ˚□B − DB = 0  where  B =  \uf8eb  \uf8ed  �F  J  N  \uf8f6  \uf8f8  (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='10)  where D is a first order stationary differential operator acting on C∞( ˚  X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' S2T ∗ ˚  X ⊕ R ⊕ T ∗ ˚  X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' When (˙g, ˙A)  and thus B are smooth modes, this system of equations become a system of ODEs on the one dimensional  space t−1  ∗ (0) with a regular singular point at the event horizon r = rb0 whose solution is given by Frobenius  series, which means that the vanishing of B in the static region r > rb0 implies the vanishing of B on ˚  X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' As  a result, it suffices to prove B = 0 in the static region r > rb0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' To achieve this, we can work in the static  coordinates (t, r).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  First, since ˚δN = ˚∗˚d˚∗N = 0, we have ˚  d˚∗N = 0 and then ˚∗N = ˚dN for some N ∈ C∞( ˚  X) according to  H1  dR( ˚  X) = 0, which implies N = ˚∗˚dN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Returning to the equation (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='9e), we find  r2˚∗ �KE = ˚d(−r2˚□N) + k2˚dN − 2Q˚dJ = ˚d  �  −r2˚□N + k2N − 2QJ  �  = 0  (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='11)  and thus  −r2˚□N + k2N − 2QJ = C  for some C ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  Replacing N by N − C  k2 , we still have N = ˚∗˚  dN and meanwhile  − ˚□N + k2r−2N − 2Qr−2J = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='12)  We note that since N, and thus ˚∗N is a mode, N can also be chosen as a mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' More specifically, suppose  ˚∗N = e−it∗σ(N0dt∗ + N1dr) where N0, N1 are smooth functions of r and σ ̸= 0, we let N = e−it∗σN0 with  N0 = iσ−1N0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Then N1 = ∂rN0 = iσ−1∂rN0 will be satisfied automatically because ˚  d˚∗N = e−it∗σ(−∂rN0 −  iσN1)dt∗ ∧ dr = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  Next, according to the second Bianchi identity δgGgRic(g) = 0 (which holds for any metric g), one  concludes for all (g, F)  δgGg(Ric(g) − 2T (g, F)) = −2δgT (g, F) = −2gαβ(δgF)αFνβ − gακgβλFαβ(∇νFκλ + 2∇κFλν).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  Since dFνκλ = ∇νFκλ − ∇λFνκ + ∇κFλν and gακgβλFαβ(∇λFνκ + ∇κFλν) = 0, it follows that  δgGg(Ric(g) − 2T (g, F)) = −2δgT (g, F) = −2gαβ(δgF)αFνβ − gακgβλFαβ(dF)νκλ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='13)  Linearizing the above equation around (g, F) = (gb0, dAb0), if (˙g, ˙A) is a solution to the Maxwell part of  the linearized Einstein-Maxwell system, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=', L2(˙g, ˙A) = 0, we find by using Ric(g) − 2T (g, dA) = 0, δgdA =  0, d2A = 0 that  δgGgL1(˙g, d ˙A) = 0,  i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=',  δgGg  \uf8eb  \uf8ed  �f ES  − r  kf E ⊗ /dS  2r2(HE  L S/g + HE  T /HkS)  \uf8f6  \uf8f8 = 0,  (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='14)  which gives the following system of equations for ( �f E, f E, HE  L , HE  T )  r−2˚δ(r2 �f E) + 1  2  ˚  d˚tr �f E − k  r f E + 2r−2˚d(r2HE  L ) = 0,  1  2  ˚tr �f E −  1  kr2˚δ(r3f E) + k2 − 2  k2  HE  T = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  We define  �E = Gg �f E − 2HE  L˚g,  (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='15)  it is clear that �E = 0 implies ( �f E, HE  L ) = (h˚g, 0) for some h ∈ C∞( ˚  X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' If in addition equations (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='9b) and  (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='9d) hold (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' f E = 0 and HE  T = 0), one obtains ˚tr �f E = 0,˚δ(r2 �E) = 0, and thus ( �f E, HE  L ) = (0, 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' We  further write ˚δ(r2 �E) = 0 in (t, r) coordinates and find its dr component  − µ′  b0  2µ2  b0  �Ett + µ′  b0∂t �Etr + µ−1/2  b0  ∂r(µ3/2  b0 �Err) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  Then the vanishing of �Etr and �Err implies �Ett = 0 because µ−2  b0 µ′  b0 = µ−2  b0 (2mr−2 − 2Q2r−3) ̸= 0 in the  static region r > rb0 = m +  �  m2 − Q2 > Q2/m in the subextremal charge case m > Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' In summary,  LINEAR STABILITY OF KERR-NEWMAN FAMILY  83  ( �f E, f E, HE  L , HE  T ) = 0 is equivalent to (f E, HE  T , �Etr, �Err) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' So our goal is to prove ( �F, J, N) = 0, provided  that (f E, HE  T , �Etr, �Err, �KE, KE) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  Following [57, equation (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='27)] or [39, equation (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='37)], we write  �F − 2J˚g =  \uf8eb  \uf8ed  µb0X  −µ−1  b0 Z  −µ−1  b0 Y  \uf8f6  \uf8f8  (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='16)  in the splitting (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' From now on, we will use the fact ˚tr �F = 0 which is implied by HE  T = 0 without  specified till the end of this subsubsection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' One can recover �F, J as  �F =  \uf8eb  \uf8ed  µb0  2 (X − Y )  −µ−1  b0 Z  1  2µb0 (X − Y )  \uf8f6  \uf8f8 ,  J = X + Y  4  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='17)  Then the equation (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='9b) f E = 0 implies ˚δ( �F − 2J˚g) = 4Qr−2˚g˚dN, which we express in terms of X, Y, Z, N  as  ∂tX + ∂rZ = 4Qr−2∂tN,  − µ′  b0  2µb0  (X − Y ) − µ−2  b0 ∂tZ + ∂rY = 4Qr−2∂rN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='18)  The equation �Etr = �f E  tr = 0 reads (we use the equation (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='18) during the calculation)  − ∂t∂rX − ∂t∂rY + µ′  b0  2µb0  ∂tX + ( µ′  b0  2µb0  − 2  r )∂tY −  k2  r2µb0  Z = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='19)  Finally the equation µb0 �Err = 1  2(µ−1  b0 �f E  tt + µb0 �f E  rr) − 2HE  L = 0 becomes (we use equation (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='18) during the  calculation)  −µ−1  b0 ∂2  t X −µ−1  b0 ∂2  t Y + µ′  b0  2 ∂rX +(µ′  b0  2 + 2µb0  r  )∂rY −  4  rµb0  ∂tZ − Q2  r4 X −  �k2 − 2  r2  + 3Q2  r2  �  Y + 4k2Q  r4  N = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='20)  Therefore, it suffices to prove (X, Y, Z, N) = 0 provided that they satisfy the equations (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='12) and (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='18)–  (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='20).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  We now discuss the case that (˙g, ˙A) is a mode solution with σ ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' After taking Fourier transform with  respect to t, setting ∂t(X, Y, Z, N) = −iσ(X, Y, Z, N) and solving for (X′, Y ′, Z′) = ∂r(X, Y, Z), we obtain  from equations (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='18) and (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='19)  \uf8eb  \uf8ed  X′  Y ′  Z′  iσ  \uf8f6  \uf8f8 = T  \uf8eb  \uf8ed  X  Y  Z  iσ  \uf8f6  \uf8f8 + f  where  T =  \uf8eb  \uf8ec  \uf8ec  \uf8ed  0  µ′  b0  µb0 − 2  r  k2  r2µb0 − σ2  µ2  b0  µ′  b0  2µb0  −  µ′  b0  2µb0  σ2  µ2  b0  1  0  0  \uf8f6  \uf8f7  \uf8f7  \uf8f8 ,  f = 4Q  r2  \uf8eb  \uf8ed  −N ′  N ′  −N  \uf8f6  \uf8f8 ,  while the equation (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='20) implies the following linear constraint on X, Y, Z  γ  \uf8eb  \uf8ed  X  Y  Z  iσ  \uf8f6  \uf8f8 = h  (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='21)  where h = − 4Q  r2 (  2µb0  r N ′ + k2  r2 N) and  γ = ( σ2  µb0  − Q2  r4 + (µ′  b0)2  4µb0  + µ′  b0  r , σ2  µb0  − k2 − 2  r2  − 3Q2  r4  + (µ′  b0)2  4µb0  − 2µ′  b0  r  , − 2σ2  rµb0  + k2µ′  b0  2r2µb0  ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  Following [57] and [39], one can find a linear combination Φ of X, Y, Z  iσ which satisfies a second order  ODE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Concretely, let  Φ :=  2Z  iσ − r(X + Y )  H  with  m := k2 − 2,  x := 2m  r ,  z := Q2  r2 ,  H := m + 3x − 4z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='22)  Then the second order ODE for Φ is (see [57, equation (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='41)], [39, equation (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='49)])  (µb0∂r)2Φ + (σ2 − VΦ)Φ = FΦN  (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='23)  84  LILI HE  where VΦ and FΦ are (see [57, equations (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='43), (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='44), (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='1)], [39, equation (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='1)])  VΦ =  µb0  r2H2  �  9x3 − 9(6z − m)x2 + (72z2 − 8(4m − 3)z + 3m2)x − 32z3 + 24mz(z + 1) + m2(m + 2)  �  FΦ = 8Qµb0  r3H2  �  −3x2 + (2z + 6)x + m(m + 4)  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='24)  Conversely, X, Y, Z  iσ can be expressed in terms of Φ and N as  X = (σ2r  µb0  − PX0  2rH2 )Φ + PX1  2H Φ′ − 2QPXN  r2H2 N − 8Qµb0  rH  N ′  Y = (−σ2r  µb0  − PY0  2rH2 )Φ + PY1  2H Φ′ − 2QPY N  r2H2 N + 8Qµb0  rH  N ′  Z  iσ = PZ  2H Φ − rµb0Φ′ − 8Qµb0  rH  N ′  (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='25)  where PX0, PX1, PXN , PY0, PY1, PY N , PZ are functions of r (see [57, equations (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='46), (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='4)–(C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='8)] and [39,  equations (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='2)])  PX0 = 27x3 − 24(5z − m)x2 +  �  152z2 − 2(35m − 12)z + 3m(3m + 2)  �  x  − 64z3 + 48mz2 − 8m(m − 2)z + 2m2(m + 2),  PX1 = 9x2 − (16z − 5m + 6)x + 8z2 − 6mz − 4m,  PXN = −18x2 + 4(8z − m + 6)x − 16z2 + 4(m − 4)z + 2m(m + 6),  PY0 = 9x3 − 6(10z − m)x2 +  �  120z2 − 2(11m − 12)z + 3m(m + 2)  �  x  − 64z3 + 16(m − 4)z2 − 8m(m + 2)z,  PY1 = 3x2 − (12z + m + 6)x + 8z2 + 2(m + 8)z,  PY N = −6x2 + 4(6z − m)x − 16z2 + 4(m − 4)z − 2m(m + 2),  PZ = 3x2 + (−2z + 3m)x − (4m + 8)z − 2m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='26)  Therefore it suffices to prove (Φ, N) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Recall that N satisfies the wave equation (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='12) and since N is  also a mode, we can rewrite it as  (µb0∂r)2N + (σ2 − µb0k2  r2  )N = −2QJ  r2  = −µb0Q(X + Y )  2r2  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='27)  Using (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='25) and (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='26), we find  X + Y = −( PX0  2rH2 + PY0  2rH2 )Φ + (PX1  2H + PY1  2H )Φ′ − (2QPXN  r2H2  + 2QPY N  r2H2 )N  = −(H  r − PZ  rH )Φ − 2µb0Φ′ − 16Qµb0  r2H  N  and thus (see [57, equation (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='49)] and [39, eqaution (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='52)])  (µb0∂r)2N + (σ2 − VN )N = FN0Φ + FN1Φ′  (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='28)  with  VN = µb0k2  r2  + 8Q2µ2  b0  r4H  ,  FN0 = Qµb0  2r2 (H  r − PZ  rH ),  FN1 = Qµ2  b0  r2  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='29)  Now we have two coupled second order ODEs (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='23) and (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='28)for Φ and N and in fact we can transform  them into two decoupled second order ODEs by introducing suitable linear combinations of Φ and N (see [57,  equations (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='56)–(5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='64)] and [39, equations (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='55)–(5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='59)]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Concretely, we set  Ψ± = a±Φ + b±N  (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='30)  where a±, b± are smooth on ˚  X (which extends a little bit beyond the event horizon r = rb0)  (a+, b+) = (Qm  2˜c + Q  2r, 1),  (a−, b−) = ( ˜c  6m − 2Q2  3mr, − 4Q  3m),  ˜c = 3m +  �  9m2 + 4Q2m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='31)  LINEAR STABILITY OF KERR-NEWMAN FAMILY  85  Since  �����  Qm  2˜c + Q  2r  1  ˜c  6m − 2Q2  3mr  − 4Q  3m  ����� = − 2Q2m  3m˜c −  ˜c  6m < 0, we can recover Φ, N from Ψ±.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Next the decoupled equations  for Φ± are given as  (µb0∂r)2Ψ± + (σ2 − V±)Ψ± = 0  (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='32)  where  V± = a±FΦ + b±VN  b±  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='33)  Introducing the tortoise coordinate r∗ =  �  µ−1  b0 dr, we see that the equations for Ψ± are reduced to an  eigenvalue problem of the type AΨ = σ2Ψ on the region r∗ ∈ (−∞, ∞) where A = −∂2  r∗ + V (r) is a self-  adjoint operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' As explained in [57, §6], we expect V (r) to be non-negative on r ∈ [rb0, ∞) and then  naturally so is A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  Lemma 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Given r ∈ [rb0, ∞), we have V+ ≥ 0 for l ≥ 1, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=', k2 ≥ 2 and m ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Since rb0 = m +  �  m2 − Q2 and H = m + 2  r(3m − 2Q2  r ), in the subextremal case Q < m, we have  r ≥ rb0 > m > Q > 2Q2  3m and thus H(r) > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' We first analyze  FΦ = 8Qµb0  r3H2  �  −3x2 + (2z + 6)x + m(m + 4)  �  ,  since x = 2m  r  ∈ [0, 2), z ≥ 0, m ≥ 0, we have −3x2 + (2z + 6)x + m(m + 4) ≥ m(m + 4) ≥ 0 and thus FΦ ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  Next it is clear from (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='29) that VN ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Therefore V+ ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  □  However, the positivity of V− fails.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' To deal with this issue, we follow [57, §6] and introduce the procedure  S-deformation of V−.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' More specifically, suppose Ψ is compactly supported in r∗, we have  (Ψ, AΨ)L2(dr∗) =  �  |∂r∗Ψ|2 + V (r)|Ψ|2 dr∗ =  �  | �DΨ|2 + �V (r)|Ψ|2 dr∗  where �D = ∂r∗ + S = µb0∂r + S and �V = V + µb0∂rS − S2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' We apply the S-deformation to V− with  S = S− := µb0∂rH−  H−  = µb0∂r(log H−) where H− = k2 − 2 + ˜c  r > 0 on [rb0, ∞) for k2 ≥ 2(l ≥ 1)  and then  �V− = k2(k2 − 2)µb0  r2H−  ≥ 0 on [rb0, ∞)  for  k2 ≥ 2(l ≥ 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='34)  Namely, with H− and V− (which are also smooth near r = rb0) defined above, we write �D = ∂r∗ + S− =  e−  �  S−dr∗∂r∗e  �  S−dr∗ = H−1  − ∂r∗H− = H−µb0∂rH− and find that D∗ which is the adjoint of D with respect  to dr∗ = µ−1  b0 dr can be expressed as −H−∂r∗H−1  −  = −H−µb0∂rH−1  − .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' As a consequence, we have  �D∗ �DΨ− + �V−Ψ− = −H−µb0∂r  �  H−2  − µb0∂r(H−Ψ)  �  + �V−Ψ  = −(µb0∂r)2Ψ − ΨH−µb0∂r(H−2  − µb0∂rH−) + (V + H−1  − (µb0∂r)2H− − 2H−2  − (µb0∂rH−)2)Ψ  = −(µb0∂r)2Ψ + V Ψ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='35)  Now we are at the position to show that Ψ± = 0 on r > rb0, and thus from the previous discussion  the mode solution (˙g, ˙A) with σ ̸= 0 is a pure gauge solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' To achieve this, we first need to discuss the  behavior of Ψ± near r = rb0 and r = ∞, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=', r∗ = ∓∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Since H, a±, b± are bounded away from zero, we  only need to analyze the contributions of X, Y, Z, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Note that �F, J are smooth on ˚  X (which extends a  little bit beyond the event horizon).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Therefore, the contribution of J = (X + Y )/4 to Φ is smooth on ˚  X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' As  for the contribution from Z, we notice that ∂t, ∂r∗, which are expressed as ∂t, µb0∂r in the static coordinates  (t, r), are smooth vector fields on ˚  X, and thus Z = µb0 �Ftr = − �F(∂t, ∂r∗) is smooth on ˚  X as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' As a  result, Φ = e−iσt∗C∞([rb0, ∞)r), which is written in the static coordinates as  Φ = (t, r) = e−iσtΘ(r)  where  Θ(r) ∈ e±iσr∗C∞([rb0, ∞))  when  r∗ → ±∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  which implies the exponential decay of Φ as r∗ → −∞ when Im σ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Also, when r∗ → ∞, since ˙g = e−iσt∗ ˙g0  with ˙g0 ∈ ¯H∞,ℓ  b  ( ¯  X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' S2 �  scT ∗ ¯X), we still obtain the rapid decay of Φ when r∗ → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' As for N, from the previous  86  LILI HE  discussion, we know N is a mode, that is, N = e−iσt∗N0 with N0 ∈ C∞([rb0, ∞)r) ∩ ¯H∞,ℓ  b  ( ¯  X) as well, and  thus decays rapidly when r∗ → ±∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  Consequently, Ψ± decays rapidly as well when r∗ → ±∞ when  Im σ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  Then when Im σ > 0, by pairing (−∂2  r∗ + V+ − σ2)Ψ+ = 0 and (− �D∗ �D + �V− − σ2)Ψ− = 0 with Ψ+ and  Ψ− respectively with respect to the L2(Rr∗;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' dr∗) and then integrating by parts (whose validity is guaranteed  by the rapid decay of Ψ± when r∗ → ±∞ ), one obtains  0 = ∥∂r∗Ψ+∥L2 + ∥V 1/2  +  Ψ+∥L2 − σ2∥Ψ+∥L2,  0 = ∥ �DΨ−∥L2 + ∥�V 1/2  −  Ψ−∥L2 − σ2∥Ψ−∥L2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  When Re σ = 0 and thus σ2 < 0, we have a sum of squares and thus Ψ± = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' When Re σ ̸= 0, the imaginary  part −2iRe σIm σ∥Ψ±∥L2 = 0 and thus Ψ± = 0 on (rb0, ∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  When σ ∈ R\\{0}, we still have eiσtΦ, eiσtN ∈ e±iσr∗  �  C∞([rb0, ∞))∩ ¯H∞,ℓ  b  ( ¯X)  �  when r∗ → ±∞ for some  ℓ ∈ R, and thus so do Ψ±.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' First, by normal operator arguments as in the proof of Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='7, we find  Ψ+ = e−iσteiσr∗(a + A1−) with a ∈ C when r∗ → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Next, a standard boundary pairing argument, see the  proof of Theorem 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='1 below for the detail (starting at (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='8)), allows us to conclude limr∗→±∞ e∓iσr∗Ψ+ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  Then an indicial root argument implies Ψ+ ∈ e−iσteiσr∗A∞, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=', Ψ+ vanishes to infinite order at r∗ = ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  Lastly, the unique continuation at infinity ( [45, Theorem 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='8]) implies Ψ+ = 0 on (rb0, ∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' The proof for  Ψ− proceeds similarly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  Next we treat the stationary mode solutions, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=', σ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  We still assume l ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Now (˙g, ˙A) and thus  �F, J, N and X, Y, Z are stationary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  We first analyze the nature of N defined as ˚∗N = ˚  dN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Let ˚∗N = N0dt∗ + N1dr where N0, N1 are smooth  functions of r only, and then ˚d˚∗N = 0 gives ∂rN0 = 0 and thus N0 is a constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Therefore, we have the  unique (up to additive constants) N = N0t∗ +  � r  r− N1(s)ds, which is a generalized mode with frequency  σ = 0 and grows at most linearly in t∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Since X, Y, Z are stationary, equation (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='19) implies  Z = 0,  (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='36)  which in turn implies ∂tN = 0 by using equation (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='18).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Therefore, we conclude N0 = 0 and N is stationary  as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  Remark 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Alternatively, using N ∈ ¯Hℓ,∞  b  ( ¯X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' �  scT ∗ ¯X) ⊂ Aℓ+3/2 ⊂ A0+ (because −3/2 < ℓ < −1/2) when  σ = 0, it also follows that N0 = c = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  As for Φ, we notice that Z/(iσ) is not well-defined when σ = 0 and we need to do some remedies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Using  (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='21), Z/(iσ) can be expressed as a linear combination of X, Y, N, N ′ and thus we can rewrite Ψ± as linear  combinations of X, Y, N, N ′ whose coefficients depend on σ  Ψ±(σ) = CX±(σ)X + CY ±(σ)Y + CN±(σ)N + CN ′±(σ)N ′  (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='37)  where C•±(σ) are rational functions of r depending on σ ∈ C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' To make the dependence on σ explicit, we  write (see [39, equations (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='65), (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='3)])  CX± = a±  PX  3 �H  ,  CY ± = a±  PY  3 �H  ,  CN+ = 1 + µb0(m + 2)PN  ˜cr �H  ,  CN− = b−  �  1 + 6a−µb0(m + 2)x  r �  H  �  CN ′+ = 2µ2  b0PN  ˜c �H  ,  CN ′− = 4b−µ2  b0(˜c − 4rz)  r ˜H  (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='38)  where �H(σ) = H(k2µ′  b0 − 4rσ2) and PX, PY , PN are smooth on ˚  X  PX = (9x − 36z − 3m − 18)x + 6(4z + m + 8)z,  PY = −3(9x − 16z + 5m − 6)x − 6(4z − 3m)z + 12m,  PN = −8(˜c + mr)z,  (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='39)  Since µ′  b0 = 2r−2(m − r−1Q2) > 0 in the region r ≥ rb0 > m > m−1Q2 and thus �H(0) = Hk2µ′  b0 > 0 there  as well, we conclude that Ψ±(σ) exist down to σ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' We define Ψ± := Ψ±(0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Conversely, using the first  LINEAR STABILITY OF KERR-NEWMAN FAMILY  87  two equations in (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='25) with σ = 0, one can recover X, Y in terms of Φ, N, N ′ and thus Ψ±, N, N ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' One can  also check that Ψ± satisfy the equation (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='32) with σ = 0 (because Ψ±(σ) is holomorphic near σ = 0), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=',  (µb0∂r)2Ψ± − V±Ψ± = 0  (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='40)  with V± defined as before in (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='33), (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='31), (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='29) and (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='24).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  Since (˙g, ˙A) ∈ ¯H∞,ℓ  b  ( ¯X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' S2 �  scT ∗ ¯X ⊕ �  scT ∗ ¯X) ⊂ Aℓ+3/2, we have X, Y, N ′ ∈ Aℓ+3/2( ¯X) and N ∈ Aℓ+1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  Let ρ = 1/r, we notice that H = m + 6mρ − 4Q2ρ2 and k2µ′  b0 = 2k2mρ2 − 2k2Q2ρ3, so �H(0) ∈ 2k2mmρ2 +  ρ3C∞( ¯X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' In view of the expression in (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='39), we find PY ∈ C∞( ¯X) and PX, PN ∈ ρC∞( ¯X), and then  CY ± ∈ ρ−2C∞( ¯X), CX±, CN ′± ∈ ρ−1C∞( ¯X) and CN± ∈ C∞( ¯X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  As a consequence, we have Ψ± ∈  Aℓ−1/2 ⊂ A−2+ a priori because −3/2 < ℓ < −1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Now we are ready to describe the asymptotic behavior  of Ψ± near r = rb0 and r = ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' We only present the proof of Ψ+ because the proof of Ψ− follows analogously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  Since FΦ, VN , a±, b± are smooth across the event horizon, we have µb0(∂rΨ+)Ψ+|r=rb0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' This implies  that boundary term at r = rb0 arising in the integration by parts we will do later on vanishes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Near r = ∞,  owing to (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='24), (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='29) (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='31) and (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='33), we conclude FΦ ∈ ρ3C∞( ¯X), VN ∈  k2  r2 + ρ4C∞( ¯X) and then  V± ∈ k2  r2 + ρ3C∞( ¯X), which implies V± = k2  r2  ∗ + A3− as a function of r∗ near r∗ = ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' We set x = 1/r∗ for  the moment,  (µb0∂r)2Ψ+ − V+Ψ+ = ∂2  r∗Ψ+ − V+(r∗)Ψ = x4∂xΨ+ + 2x3∂xΨ+ − V+(1/x) = 0,  which implies x = 0(i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' r∗ = ∞) is a regular singular point and its indicial equation is λ(λ−1)+2λ−k2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  The roots to the indicial equation are λ± = (−1±  √  1 + 4k2)/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Since k2 ≥ 6, we see that λ+ ≥ 2 corresponds  to one solution Ψ+ with decay ∼ r−λ+  ∗  near r∗ = ∞, while λ− ≤ −3 implies another linearly independent  solution Ψ+ with growth ∼ r−λ−  ∗  near r∗ = ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Since we have Ψ+ ∈ A−2+ a priori, this excludes the second  solution with growth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Namely, Ψ+ ∼ r−λ+ which means that Ψ+ ∼ r−2  ∗  near r∗ = ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' With this decay rate  near r∗ = ∞, we can justify the L2(Rr∗;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' dr∗) pairing between ((µb0∂r)2 − V+)Ψ+ and Ψ+ and also find that  the boundary term at r∗ = ∞ vanishes when integrating by parts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Therefore we proceed in the same way  as in the proof of the case Im σ > 0 (use the L2(Rr∗;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' dr∗) pairing and then integrate by parts) and conclude  that Ψ± = 0 on (rb0, ∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Therefore, (˙g, ˙A) is a pure gauge solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Modes with l = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' We now let k2 = l(l + 1) = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Our goal is again to prove that an outgoing mode  solution (˙g, ˙A) is a pure gauge solution, with the gauge potential an outgoing mode as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' The scalar  perturbations with l = 1 take the form  ˙g =  \uf8eb  \uf8ed  �fS  − r  kf ⊗ /dS  2r2HLS/g  \uf8f6  \uf8f8 ,  ˙A =  �  �KS  − r  kK/dS  �  with  S ∈ Sl (l = 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='41)  where the term HT is absent because /H1S = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Now we follow the notations and expressions in the case  l ≥ 2 with HT set to be 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' First, the pure gauge solutions take the form  δ ˙g = 2δ∗  gω =  \uf8eb  \uf8ed  2˚δ∗T  − r  k(− k  r T + r˚  dr−1L)/dS  2r2(r−1ιdrT + k  2rL)S/g  \uf8f6  \uf8f8 ,  δ ˙A = Lω♯A + dφ =  �  (˚dιAT + ιT ˚dA + ˚dP)S  − r  k(− k  r ιAT − k  r P)/dS  �  (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='42)  with  ω =  �  T S  − r  kL/dS  �  ,  φ = PS  (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='43)  where T ∈ C∞( ˚  X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' T ∗ ˚  X), L, P ∈ C∞( ˚  X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' When adding (δ ˙g, δ ˙A) to (˙g, ˙A), the quantities ˜f, f, HL, ˜K, K  change by  δ �f = 2˚δ∗T,  δf = −k  r T + r˚  d(r−1L),  δHL = r−1ιdrT + k  2r L,  δ �K = ˚  dιAT + ιT ˚  dA + ˚dP,  δK = −k  r (ιAT + P).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='44)  We define X := r  kf, then δX = r  kδf = −T + r2  k ˚d(r−1L).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' We observe that for any given L, one can choose  T = X + r2  k  ˚  d(r−1L),  P = −ιAT + r  kK  (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='45)  88  LILI HE  which implies X + δX = 0, K + δK = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' If (˙g, ˙A) are outgoing mode solutions and L is a mode, then T, P  are modes of the same type which grows at most by the factor r relative to ˙g, ˙A and L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' We again define  �F := �f + 2˚δ∗X ∈ C∞( ˚  X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' S2T ∗ ˚  X),  J := HL + r−1ιdrX ∈ C∞( ˚  X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' T ∗ ˚  X),  N := �K + ˚d  � r  kK  �  + ιX˚dA ∈ C∞( ˚  X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' T ∗ ˚  X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='46)  However, �F, J, N are not gauge-invariant in this case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' In fact  δ �F = 2  k  ˚δ∗�  r2˚d(r−1L)  �  ,  δJ = k  2rL + r  k ιdr˚d(r−1L),  δN = r2  k ι˚  d(r−1L)˚dA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='47)  For any fixed L, we pick T, P as explained above and then can work in the gauge X = 0, K = 0, which  implies  ( �f, f, HL, �K, K) = ( �F, 0, J, N, 0),  and thus the linearized Einstein-Maxwell system 2L1(˙g, ˙A) = 0, L2(˙g, ˙A) = 0 is again given by (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='9a)–(7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='9f)  with the equation (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='9d) for HE  T absent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Conversely, in the gauge X = 0, K = 0, if ( �F, J, N) = 0, we have  (˙g, ˙A) = 0, which implies the original perturbations (˙g, ˙A) take the form  (˙g, ˙A) = (2δ∗  gω, Lω♯A + dφ) with ω = −  ��  X + r2  k ˚d(r−1L)  �  S  − r  kL˚dS  �  ,  φ = ιA(X + r2  k  ˚  d(r−1L)) − r  k K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='48)  If (˙g, ˙A) are outgoing mode solutions and L is a mode , then ω and φ are modes of the same type which  grow at most by a factor r more than (˙g, ˙A) and L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  Now we exploit this extra gauge freedom resulting from L to simplify the structure of the linearized  Einstein-Maxwell system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Concretely, we define HE  T := − k2  2r2 ˚tr �F and we shall arrange HE  T + δHE  T = 0 by  choosing L properly and thus the absent equation (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='9d) still holds in the case l = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' In order to achieve  this, we should choose L such that  δHE  T = − k  r2 ˚tr˚δ∗�  r2˚d(r−1L)  �  = k  r2˚δ  �  r2˚d(r−1L)  �  = −k˚□˚g(r−1L) − 2kr−1∂i(r)∂i(r−1L)  = −k˚□˚g(r−1L) + kgabΓi  ab∂i(r−1L) = −k□g(r−1L) = −HE  T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='49)  We first consider the non-stationary mode solutions with Im σ ≥ 0, σ ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' We again define N as ˚∗N =  ˚dN and X, Y, Z as in (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='16).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' As explained in the case l ≥ 2, N, X, Y, Z are modes with the same frequency  σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' We shall find a mode L satisfying the above equation (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='49).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Writing HE  T = e−iσt∗HE  T,0(r), then the  equation (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='49) for L = e−iσt∗ ˆL(r) becomes  �  □g(σ)(r−1 ˆL) = HE  T,0  k  (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='50)  which has a spherically symmetric solution since by Theorem 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='1 �  □g(σ) : ¯H∞,ℓ  b  ( ¯X) → ¯H∞,ℓ+1  b  ( ¯X) is invertible  for any ℓ < −1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Therefore, as discussed around (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='45), we add a pure gauge solution using this L and  the corresponding T, P as in (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='45) , the linearized Einstein-Maxwell system is again given by (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='9a)–(7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='9e).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  This allows us to repeat the proof of the case of non-stationary mode solutions with l ≥ 2 to conclude that  in the case l = 1, (˙g, ˙A) is a gauge solution as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' More precisely, (˙g, ˙A) take the form as in (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='48).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  Next we discuss the stationary case σ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Suppose (˙g, ˙A) ∈ ¯H∞,ℓ  b  ( ¯X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' S2 �  scT ∗ ¯X ⊕ �  scT ∗ ¯X) where we now  assume −5/2 < ℓ < −1/2, then X = rf/k ∈ ¯H∞,ℓ−1  b  ( ¯X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' �  scT ∗ ¯X) and thus ˚δ∗X ∈ ¯H∞,ℓ  b  ( ¯X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' S2 �  scT ∗ ¯X)  (because ˚δ∗ acts as ρDiff1  b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  Therefore, the right-handed side of (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='49) is in ¯H∞,ℓ+2  b  ( ¯X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  According to  Theorem 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='1,�  □g(0) : ¯H∞,ℓ  b  ( ¯  X) → ¯H∞,ℓ+2  b  ( ¯X) is surjective for −5/2 < ℓ < −1/2,and then we can solve  (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='49) for r−1L ∈ ¯H∞,ℓ  b  ( ¯X) and thus L ∈ ¯H∞,ℓ−1  b  ( ¯X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Again, according to (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='45), by adding a pure gauge  solution (2˚δ∗  gω, Lω♯A + dφ) with (ω, φ) ∈ ¯H∞,ℓ−1  b  ( ¯X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' �  scT ∗ ¯X ⊕ R), the linearized Einstein-Maxwell system is  again given by (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='9a)–(7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='9e).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  Next, we need to modify the argument in the case l ≥ 2, σ = 0 a little bit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' We again write  Ψ±(σ) = CX±(σ)X + CY ±(σ)Y + CN±(σ)N + CN ′±(σ)N ′  (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='51)  LINEAR STABILITY OF KERR-NEWMAN FAMILY  89  where C•±(σ) are rational functions of r depending on σ ∈ C (obtained by letting k2 = 2, m = 0 in (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='38))  CX± = a±  PX  3 �H  ,  CY ± = a±  PY  3 �H  ,  CN+ = 1 + 2µb0PN  ˜cr �H  ,  CN− = b−  �  1 + 12a−µb0x  r �H  �  CN ′+ = 2µ2  b0PN  ˜c �H  ,  CN ′− = 4b−µ2  b0(˜c − 4rz)  r ˜H  (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='52)  where �H(σ) = H(2µ′  b0 −4rσ2) with H(r) = 6mr−1 −4Q2r−2, (a+, b+) = ( Q  2r, 1), (a−, b−) = (1− 2Q2  3mr, − 4Q  3m),  ˜c = 6m and PX, PY , PN are smooth on ˚  X(we again let k2 = 2, m = 0 in (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='39))  PX = (9x − 36z − 18)x + 6(4z + 8)z,  PY = −3(9x − 16z − 6)x − 24z2,  PN = −8˜cz,  (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='53)  Since H(r) = 2r−1(3m − 2Q2r−1) > 0 and µ′  b0 = 2r−2(m − r−1Q2) > 0 in the region r ≥ rb0 > m >  m−1Q2 > (2/3)m−1Q2, we have �H(0) = 2Hµ′  b0 > 0 there as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  We again define Ψ± := Ψ±(0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  Conversely, using the the first two equations in (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='25) with σ = 0 and m = 0, one can recover X, Y in terms  of Φ, N, N ′ and thus Ψ±, N, N ′ as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Again, Ψ±(0) satisfy  (µb0∂r)2Ψ± − V±Ψ± = 0  (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='54)  with V± defined as before in (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='33) (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='31), (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='29) and (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='24) with m = 0, k2 = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  Since (˙g, ˙A) ∈ ¯H∞,ℓ  b  ( ¯X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' S2 �  scT ∗ ¯X ⊕ �  scT ∗ ¯X) ⊂ Aℓ+3/2( ¯X), we have X, Y, N ′ ∈ Aℓ+3/2( ¯X) and N ∈  Aℓ+1/2( ¯X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  Let ρ = 1/r, we notice that H = 6mρ − 4Q2ρ2 and k2µ′  b0 = 4mρ2 − 4Q2ρ3, so �H(0) ∈  24m2ρ3 + ρ4C∞( ¯X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' In view of the expression in (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='53), we find PX, PY ∈ ρC∞( ¯X) and PN ∈ ρ2C∞( ¯X),  and then CX+, CY +, CN ′+ ∈ ρ−1C∞( ¯X) and CN+ ∈ C∞( ¯X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' As a consequence, we have  Ψ+ ∈ Aℓ+1/2( ¯  X) ⊂ A−2+( ¯X) a priori if  − 5/2 < ℓ < −1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  Now we are ready to describe the asymptotic behavior of Ψ+ near r = rb0 and r = ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Since FΦ, VN , a±, b±  are smooth across the event horizon, we have µb0(∂rΨ+)Ψ+|r=rb0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' This implies that boundary term  at r = rb0 arising in the integration by parts we will do later on vanishes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Near r = ∞, owing to (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='24),  (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='29) (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='31) and (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='33) with m = 0, we conclude FΦ ∈ 8Q  3mρ2 + ρ3C∞( ¯X), VN ∈  2  r2 + ρ3C∞( ¯X) and then  V+ ∈  2  r2 + ρ3C∞( ¯X), which implies V+ =  2  r2  ∗ + A3− as a function of r∗ near r∗ = ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' We set x = 1/r∗ for  the moment,  (µb0∂r)2Ψ+ − V+Ψ+ = ∂2  r∗Ψ+ − V+(r∗)Ψ = x4∂xΨ+ + 2x3∂xΨ+ − V+(1/x) = 0,  which implies x = 0(i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' r∗ = ∞) is a regular singular point and its indicial equation is λ(λ−1)+2λ−2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  The roots to the indicial equation are λ+ = 1, λ− = −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Again we see that λ+ = 1 corresponds to one  solution Ψ+ with decay ∼ r−1  ∗  near r∗ = ∞, while λ− ≤ −2 implies another linearly independent solution  Ψ+ with growth ∼ r2  ∗ near r∗ = ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Since we have Ψ+ ∈ A−2+ a priori, this excludes the second solution with  growth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Namely, Ψ+ ∼ r−1  ∗  near r∗ = ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' With this decay rate near r∗ = ∞, we can justify the L2(Rr∗;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' dr∗)  pairing between ((µb0∂r)2 − V+)Ψ+ and Ψ+ and also find that the boundary term at r∗ = ∞ vanishes when  integrating by parts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Therefore we proceed in the same way as in the proof of the case Im σ > 0 (use the  L2(Rr∗;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' dr∗) pairing and then integrate by parts) and conclude that Ψ+ = 0 on (rb0, ∞) and thus on ˚  X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  As for Ψ−, recall the S-deformation we discussed around (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='35) and we can rewrite the equation ((µb0∂r)2−  V−)Ψ− = 0 as  �D∗ �DΨ− = −H−∂r  �  H−2  − µb0∂r(H−Ψ−)  �  = 0  with  H−(r) = ˜c  r = 6m  r  > 0 on r ∈ [rb0, ∞)  because �V− =  k2(k2−2)µb0  r2H−  = 0 if l = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Solving the above equation for Ψ− yields  Ψ− = c1  H−  � H2  −  µb0  dr + c2  H−  = d1 ln  �����  r − rH  r − (m −  �  m2 − Q2)  ����� + d2r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='55)  90  LILI HE  where c1, c2, d1, d2 ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Since Ψ− is smooth across the event horizon r = rb0, we conclude d1 = 0 and thus  Ψ− = d2r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Using (Ψ+, Ψ−) = (0, d2r), (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='30) and (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='31), we have (Ψ, N) = (d2r, −d2Q/2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Exploiting the  first two equations in (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='25) we obtain (X, Y ) = (−d2, −d2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' As discussed previously, the equation (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='19)  implies Z = 0 in the stationary perturbation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Finally using (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='17) and ˚∗N = ˚dN we see  �F = 0,  J = −d2  2 ,  N = 0  Since J ∈ ¯H∞,ℓ  b  ( ¯X) and 0 ̸= d2 ∈ ¯H  ∞,− 3  2 −  b  ( ¯X), we have d2 = 0 if ℓ ≥ −3/2 and thus we are done.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' If  −5/2 < ℓ < −3/2, d2 may be non-zero and then in the gauge X = 0  ˙g =  \uf8eb  \uf8ed  0  0  −2r2 d2  2 S/g  \uf8f6  \uf8f8 ,  ˙A = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  This implies in the gauge X = 0, (˙g, ˙A) is a gauge solution with T = 0, L = −d2r/k, P = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Therefore,  according to the discussion around (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='48), the original perturbations (˙g, ˙A) = (2δ∗  gω, Lω♯A+dφ) with (ω, φ) ∈  ¯H∞,ℓ−1  b  ( ¯X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' �  scT ∗ ¯X ⊕ R) for −5/2 < ℓ < −1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Modes with l = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' As stated in [36,39], the linearization of the simple proof of the Birkhoff’ theorem  [73] can be extended to the spherically symmetric Einstein-Maxwell system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Following the argument in [73],  we work in the incoming Eddington-Finkelstein coordinates (t0 = t + r∗, r) where the Reissner-Nordstr¨om  metric and the electromagnetic 4-potential take the form  g = −µb0dt2  0 + 2dt0dr + r2/g,  A = Q  r dt0  (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='56)  Remark 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' The electromagnetic 4-potential defined above differs from (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='17) by an exact 1-form f(r)dr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  We note that adding an exact 1-form to A changes neither the electromagnetic field nor the linearized  Einstein-Maxwell system because the linearized system only involves dA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  Representing the Reissner-Nordstr¨om metric and the electromagnetic 4-potential in the way (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='56), we can  linearize (gb, Ab) with respect to the parameters (m, 0, Q) in the direction ( ˙m, 0, ˙Q) to obtain the following  linearized metric and electromagnetic 4-potential  g′ = (2 ˙m  r  − 2Q ˙Q  r2  )dt2  0,  A′ =  ˙Q  r dt0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='57)  We note that (g′, A′) differs from (˙g(m,0,Q)( ˙m, 0, ˙Q), ˙A(m,0,Q)( ˙m, 0, ˙Q)) by a pure gauge term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  With this we now turn to the analysis of the spherically symmetric modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Since there is no spherical com-  ponent in 1-forms and no aspherical-spherical mixed component in symmetric 2-tensors for spherically sym-  metric perturbations, we use the following splitting of 1-forms and symmetric 2-tensors respectively:⟨dt0⟩ ⊕  ⟨dr⟩ and ⟨dt2  0⟩⊕⟨2dt0dr⟩⊕⟨dr2⟩⊕⟨/g⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Under this splitting, we write the scalar perturbations of modes l = 0  (spherically symmetric modes) in the following form  ˙g =  \uf8eb  \uf8ec  \uf8ec  \uf8ed  ˙X  ˙Y  ˙Z  2r2HL  \uf8f6  \uf8f7  \uf8f7  \uf8f8 ,  ˙A =  � ˙A0  ˙A1  �  where ˙X, ˙Y , ˙Z, HL are smooth function on Rt0 × [r−, ∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' To describe the pure gauge perturbations, we use  the following lemma  Lemma 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Let (g, A) be as defined in (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='56) , under the splitting ⟨∂t0⟩⊕⟨∂r⟩ of T M and ⟨dv2⟩⊕⟨2dt0dr⟩⊕  ⟨dr2⟩ of S2T ∗M, the map L(•)g : T M → S2T ∗M takes the form  L(•)g =  \uf8eb  \uf8ec  \uf8ec  \uf8ed  −2µb0∂t0  2∂t0 − µ′  b0  ∂t0 − µb0∂r  ∂r  2∂r  0  0  2r  \uf8f6  \uf8f7  \uf8f7  \uf8f8 ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='58)  LINEAR STABILITY OF KERR-NEWMAN FAMILY  91  the map L(•)A between sections of ⟨∂t0⟩ ⊕ ⟨∂r⟩ and ⟨dt0⟩ ⊕ ⟨dr⟩ is given by  L(•)A =  �  Qr−1∂t0  −Qr−2  Qr−1∂r  0  �  ,  (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='59)  and as a mapping between the sections of the trivial line bundle R and ⟨dt0⟩ ⊕ ⟨dr⟩ , d =  �∂t0  ∂r  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' It follows from the formula (LW g)αβ = W γ∂γgαβ +(∂αW γ)gγβ +(∂βW γ)gαγ, (LW A)α = W γ∂γAα +  (∂αW γ)Aγ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  □  Therefore, adding a pure gauge solution (LW g, LW A) to (˙g, ˙A) with W := Z1∂t0 − rHL∂r and Z1 :=  (−  � r  3m ˙Z(t0, s)ds)/2, the dr2 and spherical components of ˙g vanish.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Next, by adding another pure gauge  solution (0, dφ) with φ = − � r  3m(A1 + Qs−1∂rZ1)(t0, s)ds, the dr-component of ˙A can be eliminated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  Suppose the original perturbations (˙g, ˙A) ∈ e−iσt∗ ¯H∞,ℓ  b  ( ¯X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' S2 �  scT ∗ ¯X ⊕ �  scT ∗ ¯X) are outgoing modes, then  so are ˙X, ˙Y , ˙Z, HL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' If σ = 0, it is obvious that (ω = W ♭, φ) ∈ ¯H∞,ℓ−1  b  ( ¯  M;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' �  scT ∗ ¯X ⊕ R) are still stationary  modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' If σ ̸= 0, since t∗ = t0 + f(r) where f is smooth on [r−, ∞) and ∼ −2r∗ near infinity, we find that  (ω = W ♭, φ) ∈ e−iσt∗ ¯H∞,ℓ′  b  ( ¯X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' �  scT ∗ ¯X ⊕ R) for some ℓ′ < ℓ is also outgoing modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Further, if (˙g, ˙A) ∈  Poly(t∗)k ¯H∞,ℓ  b  ( ¯X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' S2 �  scT ∗ ¯X ⊕ �  scT ∗ ¯X) are the generalized stationary modes, the integration in the definition  of W and φ implies (ω = W ♭, φ) ∈ Poly(t∗)k ¯H∞,ℓ′  b  ( ¯X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' �  scT ∗ ¯X ⊕ R) for some ℓ′ < ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' To summarize, by  adding pure gauge solutions which are outgoing modes(or generalized stationary modes), the perturbations  are reduced to the following form  ˙g = ˙Xdt2  0 + 2 ˙Y dt0dr,  ˙A = ˙A0dt0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  Next, we analyze the linearized Einstein-Maxwell system for spherically symmetric modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Concretely, for  (ˇg, ˇA) of the general form ˇg = Xdt2  0+2Y dt0dr+r2/g and ˇA = A0dt0 where X, Y, A0 are functions of (t0, r), the  dr2-component of Ric(ˇg) − 2T (ˇg, d ˇA) = 0 implies 2∂rY/(rY ) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Linearizing it around (g, A) in (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='56), we  obtain the corresponding equation for (˙g, ˙A): ∂r ˙Y = 0, so ˙Y = ˙Y (t0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' If ˙Y is an outgoing mode with σ ̸= 0,  we must have ˙Y = ˙Y (t0) = ce−iσt0 which contradicts the definition of outgoing modes unless ˙Y = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' If ˙Y  is a stationary mode, then ˙Y = c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' However the assumption ˙Y ∈ ¯H∞,ℓ  b  ( ¯X) ⊂ A0+( ¯X) for −3/2 < ℓ < −1/2  require ˙Y = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Lastly, if ˙Y is a generalized zero mode, we then have ˙Y = ˙Y (t0) = Poly(t0)k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Letting  W1 = f∂t0 with f(t0) ∈ Poly(t0)k+1 satisfying ∂t0f(t0) = − ˙Y (t0), and adding the pure gauge solution  (LW1g, LW1A = 0) to (˙g, ˙A), dvdr-component of ˙g again vanishes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Therefore, we now have  ˙g = ˙Xdt2  0,  ˙A = ˙A0dt0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='60)  Again, for (ˇg, ˇA) the Maxwell equation becomes  δˇgd ˇA =  �  − 1  Y ∂t0∂rA0 + X  Y 2 ∂2  rA0 + 2X  rY 2 ∂rA0 − X  Y 3 ∂rY ∂rA0 + 1  Y 2 ∂rA0∂t0Y  �  dv  +  � 1  Y ∂2  rA0 + 2  rY ∂rA0 − 1  Y 2 ∂rY ∂rA0  �  dr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='61)  Linearizing around (g, A) (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' X = −µb0, Y = 1, A0 = r−1Q) and using the previously obtained fact ˙Y = 0,  the dr component tells ∂2  r ˙A0 + 2r−1∂r ˙A0 = 0, so ˙A0 = c1(t0)r−1 + c2(t0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Since c2(t0)dt0 is an exact 1-form  and thus a pure gauge solution, we may assume ˙A0 = c1(t0)r−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Then the equation for dt0-component gives  −∂t0∂r ˙A0 − µb0∂2  r ˙A0 − 2µb0r−1∂r ˙A0 = r−2∂t0c1(t0) = 0 and thus c1(t0) is a constant that we denote by ˙Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  That is, ˙A0 = r−1 ˙Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  Next, the spherical component of Ric(ˇg) − 2T (ˇg, d ˇA) = 0 is given by  1 + X  Y 2 + r  Y 2 ∂rX − rX  Y 3 ∂rY − r2  Y 2 (∂rA0)2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  Linearizing around (g, A) (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  X = −µb0, Y = 1, A0 = r−1Q) and using the previously obtained facts  ˙Y = 0, ˙A0 = r−1 ˙Q, we find ˙X + r∂r ˙X − 2r−2Q ˙Q = 0 and thus  ˙X = −2Q ˙Q  r2  + c(t0)  r  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  92  LILI HE  Lastly, the dt2  0-component of Ric(ˇg) − 2T (ˇg, d ˇA) = 0 is  − X  Y 2 ∂t0∂rY + X  2Y 2 ∂2  rX + X  Y 3 ∂t0Y ∂rY − X  2Y 3 ∂rX∂rY − 2X  rY 2 ∂t0Y + X  rY 2 ∂rX + 1  rY ∂t0X + X  Y 2 (∂rA0)2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  Again linearizing around (g, A) (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' X = −µb0, Y = 1, A0 = r−1Q) and using the previously obtained facts  ˙Y = 0, ˙A0 = r−1 ˙Q, ˙X = − 2Q ˙Q  r2  + c(v)  r , we obtain r−2∂vc(v) = 0 and thus c(v) is a constant which we denote  by 2 ˙m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Therefore, we conclude that up to pure gauge solutions  ˙g = (2 ˙m  r  − 2Q ˙Q  r2  )dt2  0 = g′,  ˙A0 =  ˙Q  r dt0 = A′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='62)  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Vector type perturbations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' We shall consider the vector type perturbations of modes l ≥ 2 and  l = 1 separately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Recall that we can write the perturbations under the splitting (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='9) and (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='10) as in (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  We denote by V ∈ Vl the spherical harmonic 1-form with eigenvalues k2 = l(l + 1) − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Modes with l ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Suppose (˙g, ˙A) is the vector perturbations of the following form  ˙g =  \uf8eb  \uf8ed  0  rf ⊗ V  − 2  kr2HT /δ  ∗V  \uf8f6  \uf8f8 ,  ˙A =  � 0  rKV  �  with  V ∈ Vl (l ≥ 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='63)  We notice that the pure gauge solutions take the form  δ ˙g = 2δ∗  gω =  \uf8eb  \uf8ed  0  r2˚d(r−1L)/dS  − 2  kr2(−kr−1L)/δ  ∗V  \uf8f6  \uf8f8 ,  δ ˙A = Lω♯A = 0  (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='64)  with  ω =  �  0  rLV  �  (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='65)  where L ∈ C∞( ˚  X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' When adding (δ ˙g, δ ˙A) to (˙g, ˙A), the quantities f, HT , K change by  δf = r˚  d(r−1L),  δHT = −kr−1L,  δK = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='66)  Then we have the following gauge-invariant quantities  J := f + r  k  ˚dHT ∈ C∞( ˚  X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' T ∗ ˚  X),  K ∈ C∞( ˚  X)  (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='67)  We also note that if J = 0, K = 0, (˙g, ˙A) is a pure gauge solution  (˙g, ˙A) = (2δ∗ω, Lω♯A) with ω =  �  0  − r2  k HT V  �  (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='68)  If (˙g, ˙A) are outgoing mode solutions, then ω is a mode of the same type, which grows by a factor r more  than (˙g, ˙A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  Again we can express the linearized Einstein-Maxwell system in terms of the gauge-invariant quantities  J, K defined above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' More specifically, we adds (δ ˙g, δ ˙A) built from L = r  kHT , then HT + δHT = 0, and thus  (f, HT , K) = (J, 0, K).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Therefore, using the detailed calculation in §6 and §A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='1 we can write the linearized  Einstein-Maxwell system, acting on the new (˙g, ˙A)  ˙g =  \uf8eb  \uf8ed  0  rJ ⊗ V  0  \uf8f6  \uf8f8 ,  ˙A =  �  0  rKV  �  in terms of the gauge-invariant quantities J, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  LINEAR STABILITY OF KERR-NEWMAN FAMILY  93  Now we express 2L1(˙g, ˙A) = 0 and L2(˙g, ˙A) = 0 in the form of (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='3) in terms of f E, HE  T and KE  respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Then the linearized Einstein-Maxwell system reads  rf E = r−2˚δ(r4˚  d(r−1J)) + k2 − 1  r  J − 4Qr−2˚∗˚d(rK) = 0,  (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='69a)  −2r2  k HE  T = −2˚δ(rJ) = 0,  (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='69b)  rKE = −˚□(rK) + k2 + 1  r  K + Q˚∗˚d(r−1J) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='69c)  By (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='69b), one adds the zero term ˚  d˚δ(rJ) to (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='69a) and then can obtain a wave equation for J (coupled  to K via subprincipal terms) as (˚δ˚  d + ˚d˚δ)J = (−˚□ − 1  2µ′′  b0)J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' In conclusion, we can rewrite equation (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='69a)  and (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='69c) as a principally scalar system of wave equations for (J, K)  − ˚□B − DB = 0  where  B =  �  J  N  �  (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='70)  where D is a first order stationary differential operator acting on C∞( ˚  X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' T ∗ ˚  X ⊕ T ∗ ˚  X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' As discussed in the  scalar perturbations, it suffices to prove B = 0 in the static region r > rb0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' To achieve this, we again work  in the static coordinates (t, r).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  First, by (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='69b) we have ˚  d˚∗(rJ) = 0 (because ˚δ = ˚∗˚d˚∗), and then ˚∗rJ = ˚d(rΦ) for some function Φ on  ˚  X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Applying ˚∗r2 on both sides of equation (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='69a) yields  ˚  d  �  r4˚δ(r−2˚  d(rΦ)) + (k2 − 1)rΦ − 4QrK  �  = 0  which implies r4˚δ(r−2˚d(rΦ)) + (k2 − 1)rΦ − 4QrK = c for some constant c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Replacing Φ by Φ −  c  (k2−1)r, we  obtain r4˚δ(r−2˚d(rΦ)) + (k2 − 1)rΦ − 4QrK = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' and thus  r˚δ(r−2˚  d(rΦ)) + (k2 − 1)  r2  Φ − 4Q  r2 K = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='71)  We can also rewrite it as  − ˚□Φ + 1  r2  �  k2 + 1 − 6m  r  + 4Q2  r2  �  Φ − 4Q  r2 K = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='72)  Next, equation (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='69c) can be rewritten as (where we use equation (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='71))  − ˚□(rK) + 1  r2  �  k2 + 1 + 4Q2  r2  �  (rK) − (k2 − 1) Q  r3 Φ = 0  (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='73)  We note that (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='72) and (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='73) recover [57, equations (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='28) and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='30)] with Ω = rΦ, A = −rK, mV =  k2 − 1, κ2 = 2, q = Q, n = 2, K = 1, ¯τ = 0, J = 0 there.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Following [57, equations(4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='34) and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='35)], we  introduce the following combinations  Ψ± = a±Φ + b±rK  (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='74)  with  (a+, b+) = (  Q(k2 − 1)  3m +  �  9m2 + 4(k2 − 1)Q2 , −1),  (a−, b−) = (1,  4Q  3m +  �  9m2 + 4(k2 − 1)Q2 ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='75)  When expressed in terms of Ψ±, equations (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='72) and (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='73) are transformed into two decoupled wave  equations (see [57, equations (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='37)–(4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='39)])  ˚□Ψ± − V±  µb0  Ψ± = 0  (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='76)  where  V± = µb0  r2  �  k2 + 1 + 4Q2  r2  − 3m  r  ±  �  9m2 + 4(k2 − 1)Q2  r  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='77)  Suppose (˙g, ˙A) ∈ e−iσt∗ ¯H∞,ℓ  b  ( ¯X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' S2 �  scT ∗ ¯X ⊕ �  scT ∗ ¯X) are modes, then (J, K) ∈ e−iσt∗ ¯H∞,ℓ−1  b  ( ¯X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' �  scT ∗ ¯X ⊕ R)  for σ ̸= 0, while (J, K) ∈ ¯H∞,ℓ  b  ( ¯X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' �  scT ∗ ¯X ⊕ R) for σ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' As discussed around (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='12) and (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='36) and using  94  LILI HE  (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='73) in the σ = 0 case, we conclude Φ, rK ∈ e−iσt∗ ¯H∞,ℓ−1  b  ( ¯X) for Im σ ≥ 0, and so are Ψ±.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Therefore in  the static coordinates (t, r), equations (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='76) take the form  (µb0∂r)2Ψ± − (V± − σ2)Ψ± = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  We observe that when r > rb0 = m +  �  m2 − Q2, V+ > 0 and  r2µ−1  b0 V− > (r2µ−1  b0 V−)(rb0) = k2 − 1 + (m −  �  9m2 + 4(k2 − 1)Q2)/rb0  = r−1  b0  �  k2m + (k2 − 1)  �  m2 − Q2 −  �  (4k2 + 5)Q2 + 9(m2 − Q2)  �  > r−1  b0  �  k2m −  �  4k2 + 5Q + (k2 − 4)  �  m2 − Q2  �  > 0  where we use k2 ≥ 5, m > Q in the last inequality, so V− > 0 in the static region as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' We also notice the  asymptotic behavior of V± is V± = (k2+1)ρ2+ρ3C∞( ¯X) where ρ = 1/r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Then we can proceed as in the scalar  perturbations l ≥ 2 case to conclude that Ψ± = 0 and thus (J, K) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' More specifically, when Im σ > 0,  we pair (µb0∂r)2Ψ± − (V± − σ2)Ψ± with Ψ± with respect to L2(Rr∗;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' dr∗) and integrate by parts, then the  positivity of V± allows us to obtain Ψ± = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' When σ ∈ R \\ {0}, we use the boundary paring argument.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  Lastly, when σ = 0, we first obtain the a priori decay rate Ψ± ∈ ¯H∞,ℓ−1  b  ( ¯X) ⊂ Aℓ+1/2( ¯X) ⊂ A−1+( ¯X)  because −3/2 < ℓ < −1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Next we use the Frobenius method (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' analyze the indicial equation) and the  decay rate of V± = (k2 +1)ρ2 +ρ3C∞( ¯  X) to conclude that Ψ± ∼ r−2 near r = ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' So the conclusion Ψ± = 0  again follows from the L2 pairing and integration by parts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Modes with l = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' We now let k2 = l(l + 1) − 1 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Suppose (˙g, ˙A) is the vector perturbations of the  following form  ˙g =  \uf8eb  \uf8ed  0  rf ⊗ V  0  \uf8f6  \uf8f8 ,  ˙A =  �  0  rKV  �  with  V ∈ Vl (l = 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='78)  We notice that the pure gauge solutions take the form  δ ˙g = 2δ∗  gω =  \uf8eb  \uf8ed  0  r2˚d(r−1L)/dS  0  \uf8f6  \uf8f8 ,  δ ˙A = Lω♯A = 0  (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='79)  with  ω =  �  0  rLV  �  (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='80)  where L ∈ C∞( ˚  X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' When adding (δ ˙g, δ ˙A) to (˙g, ˙A), the quantities f, HT , K change by  δf = r˚  d(r−1L),  δK = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='81)  Then we define the following gauge-invariant quantities  ˚d(r−1f) ∈ C∞( ˚  X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Λ2T ∗ ˚  X),  K ∈ C∞( ˚  X)  (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='82)  If ˚d(r−1f) = 0, K = 0, since ˚  X is contractible, we can find L ∈ C∞( ˚  X) such that r−1f = ˚  d(r−1L), which  implies  (˙g, ˙A) = (2δ∗ω, Lω♯A) with ω =  � 0  rLV  �  (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='83)  is a pure gauge solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' As explained around (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='12) and (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='36) again, we have L ∈ e−iσt∗ ¯H∞,ℓ  b  ( ¯X) for  σ ̸= 0, while L ∈ ¯H∞,ℓ−1  b  ( ¯X) for σ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  Again we can express the linearized Einstein-Maxwell system in terms of the gauge-invariant quantities  ˚d(r−1f), K as follows  rf E = r−2˚δ(r4˚d(r−1f)) − 4Qr−2˚∗˚d(rK) = 0,  (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='84a)  rKE = −˚□(rK) + 2  r K + Q˚∗˚d(r−1f) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='84b)  Before solving the above system, we first describe the linearized Kerr-Newman solution in which we lin-  earize the Kerr-Newman solution family (gb, Ab) around Reissner-Nordstr¨om solution (gb0, Ab0) with respect  LINEAR STABILITY OF KERR-NEWMAN FAMILY  95  to the angular momentum parameter a in the direction ˙a, which is parallel to the rotation axis of V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' We see  that in the static coordinates  g′  (m,0,Q)(0, ˙a, 0) = 2(µb0 − 1) sin2 θdtdϕ,  A′  (m,0,Q)(0, ˙a, 0) = −Q  r sin2 θdϕ  (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='85)  where |˙a| = 1 and (θ, ϕ) are the spherical coordinates adapted to ˙a, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='e, ˙a is defined by θ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' We note  that (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='85) is of the form (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='78) with f = (−2mr−2 + Q2r−3)dt, K = −Qr−2 and V = sin2 θdφ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Then the  associated gauge-invariant quantities are ˚∗˚  d(r−1f) = 6mr−4 − 4Q2r−5, K = −Qr−2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  Now we turn to solving the equations (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='84a) and (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='84b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' First (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='84a) implies ˚d  �  r4˚∗˚  d(r−1f)−4QrK  �  = 0  and thus  r4˚∗˚  d(r−1f) − 4QrK = c  (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='86)  for some constant c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Plugging this into the equation (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='84b) yields  (˚□ − V )(rK) = cQ  r4  where  V = 2  r2 + 4Q2  r4 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  Since V > 0 and the right-handed side is stationary, as discussed previously, rK = 0, c = 0 is the only mode  solution with frequency σ ̸= 0 to the above equation, and thus ˚d(r−1f) = 0 as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' As for the case σ = 0,  since V = 2ρ2 + 4Q2ρ4 where ρ = 1/r, using the Frobenius method and analyzing the indicial equation, it  follows that for each c ∈ R, there is at most one solution K ∈ ¯H∞,ℓ  b  ( ¯X) for −3/2 < ℓ < −1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' On the other  hand, one can verify  K = − cQ  6mr2 ∈ ¯H∞,ℓ  b  ( ¯  X)  is indeed a solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Returning to (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='86) we find  ˚∗˚d(r−1f) =  c  6m(6m  r4 − 4Q2  r5 ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  Therefore, replacing (˙g, ˙A) by (˙g, ˙A) −  c  6m(g′, A′), we may assume c = 0 and thus K = 0, ˚d(r−1f) = 0, which  implies (˙g, ˙A) is a pure gauge solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  This finishes the proof of Theorem 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Mode analysis of the scalar wave operator  In this section we shall discuss the modes of the scalar wave operator on slowly rotating KN spacetimes  with subextremal charge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' It not only occurs as the gauge propagation operator acting on the gauge function  D(gb,Ab) �ΥM(˙g, ˙A;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' gb, Ab) = −δgb ˙A, but an operator on a scalar function which generate the pure gauge  solutions to the Maxwell equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' We define the Fourier-transformed scalar wave operator on slowly rotating  Kerr-Newman spacetime gb with parameters b = (m, a, Q) near b0 = (m0, 0, Q0) as  �  □gb(σ) := eiσtb,∗□gbe−iσtb,∗  where tb,∗ = χ0(r)(t + rb0,∗) + (1 − χ0(r))(t − r(m,0,Q),∗) is defined as in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='29).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  For ease of notation, we drop C in the notations of the functions space ¯Hs,ℓ  b ( ¯X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' C), ¯Hs,ℓ  b ( ¯X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' C), A( ¯X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' C)  and C∞( ¯X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' C) if it is clear from the context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  Theorem 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Let b0 = (m0, 0, Q0) and g = gb0 be the RN metric with subextremal charge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' There exists  ǫ > 0 such that for b = (m, a, Q) with |b − b0| < ǫ, the following holds  (1) If Im σ ≥ 0 and σ ̸= 0,  �  □gb(σ) : {u ∈ ¯Hs,ℓ  b ( ¯X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' C) : �  □g(σ)u ∈ ¯Hs,ℓ+1  b  ( ¯X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' C)} → ¯Hs,ℓ+1  b  ( ¯X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' C)  (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='1)  is invertible for s > 1  2, ℓ < − 1  2, s + ℓ > − 1  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  (2) If s > 1  2 and − 3  2 < ℓ < − 1  2, then stationary operator  �  □gb(0) : {u ∈ ¯Hs,ℓ  b ( ¯X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' C) : �  □g(0)u ∈ ¯Hs−1,ℓ+2  b  ( ¯X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' C)} → ¯Hs−1,ℓ+2  b  ( ¯X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' C)  (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='2)  is invertible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' The proof closely follows [36, Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' We first prove (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='1) and (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='2) for the RN metric g and  then use a perturbation argument to extend them to the slowly rotating KN metric gb with b near b0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  96  LILI HE  Injectivity of �  □g(0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Suppose u ∈ ¯Hs,ℓ  b ( ¯X) with s > 1  2, ℓ ∈ (− 3  2, − 1  2) and  �  □g(0)u = □gu = 1  r2 ∂rµb0r2∂ru + 1  r2 /∆u = 0,  µb0 = 1 − 2m0  r  + Q2  0  r2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  Then by Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='7, u ∈ ρC∞( ¯X) + A2−( ¯X) where ρ = r−1, which implies |u|, |r∂ru|, | /∇u| ≲  r−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' As a consequence, the boundary term at r = ∞ in the following integration by parts  0 = −  �  S2  � ∞  rb0  □gu · u r2drd/g =  �  S2  � ∞  rb0  �  µb0(r)|r∂ru|2 + | /∇u|2�  drd/g  (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='3)  vanishes, and the boundary term at r = rb0 vanishes since u is smooth there and µb0(rb0) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  Then u is a constant in r ≥ rb0 and thus vanishes there since limr→∞ u = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Since u vanishes to  infinite order at r = rb0 and smooth in r ≤ rb0, we shall prove that u = 0 in r ≤ rb0 as well by  following the arguments in [89, Lemma 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Concretely, we first compute the following energy identity  in r− ≤ r < rb0  ∂r  �  |r − rb0|−N�  −µb0(r)|∂ru|2 + 1  r2 |/du|2  /g + |u|2��  + 2|r − rb0|−N 1  r2 /δ  �  Re (∂ru/du)  �  = N|r − rb0|−N−1�  −µb0(r)|∂ru|2 + 1  r2 |/du|2  /g + |u|2�  − |r − rb0|−N�  2Re (∂ru · �  □g(0)u) − R(u, du)  �  (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='4)  where  R(u, du) =  �  4r−1µb0(r) − µ′  b0(r)  �  |∂ru|2 + 2Re (u∂ru) − 2r−3|/du|2  /g  is a quadratic form in u and du independent of N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' As a result, R(u, du) can be controlled by the first  term on the right-hand side of the above energy identity (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='4) when N is sufficiently large.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Then for  any fixed r0 ∈ [r−, rb0) we apply Stokes’ Theorem in [r0, rb0 − ǫ] × S2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' For N large enough we have  �  S2(−µb0(r)|∂ru|2 + |/du|2 + |u|2)|r=r0 d/g  ≤ (rb0 − r0)Nǫ−N  �  S2(−µb0(r)|∂ru|2 + |/du|2 + |u|2)|r=rb0−ǫ d/g ≤ CKǫ−N+K  (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='5)  for any K as ǫ → 0+ since u is smooth and vanishes to infinite order at r = rb0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' By choosing K > N  and letting ǫ → 0+ we conclude that u = 0 at r = r0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  Surjectivity of �  □g(0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' The index 0 property of �  □g(0) established in Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='23 directly gives the  surjectivity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' But here we present an alternative proof of the surjectivity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' To prove the surjectivity  of �  □g(0), we note that it is equivalent to proving the injectivity of the adjoint �  □g(0)∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Suppose  v ∈ ˙H−s+1,−ℓ−2  b  ( ¯X) with s > 1  2, ℓ ∈ (− 3  2, − 1  2) satisfies �  □g(0)∗v = �  □g(0)v = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' By Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='7,  v is smooth in r− ≤ r < rb0 and r > rb0, and u ∈ ρC∞( ¯X) + A2−( ¯X) near r = ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' First we have  v = 0 in r < rb0 since �  □g(0)∗ is a hyperbolic operator there with r being the timelike function,  see [21, Theorem E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='56].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Next, according to [35, Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='3], v is conormal at r = rb0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Since  the normal operator of µb0 �  □g(0)∗ is (µb0∂r)2 whose boundary spectrum consists of {(0, 0), (0, 1)}, it  follows that v = H(r − rb0)(v0 + v1 log µb0 + ˜v) where v0, v1 ∈ C∞(S2) and ˜v ∈ A1−([rb0, ∞)) which  means that ˜v is conormal at r = rb0 and ˜v ∼ (r − rb0)1− as r → r+  b0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Since �  □g(0)H(r − rb0) =  r−2∂r(r2µb0(r)δ(r − rb0)) = 0 and �  □g(0)(H(r − rb0) log µb0) = µ′  b0(r)δ(r − rb0) + 2Q2  0r−4H(r − rb0),  we have  0 = �  □g(0)v = µ′  b0(r)δ(r − rb0)v1 + R  where  R = H(r − rb0)  r2  � /∆v0 + log(r − rb0) /∆v1 + 2Q2  0  r2 v1  �  + H(r − rb0)�  □g(0)˜v ∈ A0−.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  However, δ(r − rb0) /∈ A0−, and this implies v1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Since v1 = 0 implies µb0∂rv → 0 as r → r+  b0, we  find that the integration by parts (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='3) still works here and thus gives v = 0 in r > rb0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Then we  can conclude that v is supported at r = rb0, and consequently v /∈ L2 [44, Theorem 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='27] unless  v = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' However, the radial point estimates at the event horizon imply that v ∈ H1/2− near r = rb0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  Therefore, v must be 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  LINEAR STABILITY OF KERR-NEWMAN FAMILY  97  Invertibility of �  □g(σ) for σ ∈ R \\ {0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' According to Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='23, it suffices to prove the injectivity  of �  □g(σ) for non-zero real σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  We first illustrate the relationship between �  □g(σ) and �  □g(σ) =  eiσt□ge−iσt, that is, u ∈ ker �  □g(σ) gives rise to an outgoing solution  �  □g˜u = 0  where  ˜u = eiσ(t−tb0,∗)u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='6)  We next prove that ˜u = 0 and thus u = 0 in r ≥ rb0 by a boundary pairing argument (see [66, §2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='3] [41,  §3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='2] [36, §7]), and a unique continuation theorem at infinity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Concretely, let f(r) ∈ C∞([rb0, ∞))  be a nonnegative function with f(r) = r − rb0 for rb0 ≤ r < 3m0 and f(r) = r−1 for r > 4m0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Fix  a cutoff χ ∈ C∞([0, ∞)) which is 0 on [0, 1/2] and 1 on [1, ∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' For small ǫ > 0, we define  χǫ(r) := χ(f  ǫ ) ∈ C∞  c ((rb0, ∞)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='7)  We see that χǫ = 1 when r − rb0 ≥ ǫ, r ≤ ǫ−1, and r − rb0 ≥ ǫ/2, r ≤ 2ǫ−1 on suppχǫ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' We then  calculate  0 = lim  ǫ→0  � �  S2  � ∞  rb0  χǫ˜u · �  □g(σ)˜u r2drd/g −  �  S2  � ∞  rb0  χǫ �  □g(σ)˜u · ˜u r2drd/g  �  = lim  ǫ→0  �  S2  � ∞  rb0  [�  □g(σ), χǫ]˜u · ˜u r2drd/g  = lim  ǫ→0  ��  S2  � ∞  rb0  2µb0∂rχǫ · ∂r˜u · ˜u r2drd/g −  �  S2  � ∞  rb0  µb0∂rχǫ · ∂r|˜u|2 r2drd/g  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='8)  where ∂rχǫ may be nonzero only near the event horizon r = rb0 and near infinity r = ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Near r = rb0,  we have ˜u = e−iσr∗u where we write r∗ = rb0,∗ for brevity of notation, ∂rχǫ = ǫ−1χ′(  r−rb0  ǫ  ) →  δ(r − rb0) as ǫ → 0, and thus  lim  ǫ→0  ��  S2  � ∞  rb0  2µb0∂rχǫ · ∂r˜u · ˜u r2drd/g −  �  S2  � ∞  rb0  µb0∂rχǫ · ∂r|˜u|2 r2drd/g  �  =  �  S2  �  − 2iσr2|u|2 + 2µb0r2∂ru · u − r2µb0∂r|u|2�  |r=rb0 d/g  =  �  S2  �  − 2iσr2|u|2�  |r=rb0 d/g  (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='9)  where we use the smoothness of u at r = rb0 in the last step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' While near r = ∞, we have r2∂rχǫ =  −ǫ−1χ′( ρ  ǫ ) → −δ(ρ) with ρ = r−1 as ǫ → 0, ˜u = eiσr∗u where u = ρu+(ω) + e(ρ, ω) with u+(ω) ∈  C∞(S2) and e(ρ, ω) ∈ A2−( ¯X) (see Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='7), and thus  lim  ǫ→0  ��  S2  � ∞  rb0  2µb0∂rχǫ · ∂r˜u · ˜u r2drd/g −  �  S2  � ∞  rb0  µb0∂rχǫ · ∂r|˜u|2 r2drd/g  �  =  �  S2  �  − 2iσr2|u|2 − 2µb0r2∂ru · u + r2µb0∂r|u|2�  |r=∞ d/g  =  �  S2  �  − 2iσr2|u|2�  |r=∞ d/g =  �  S2−2iσ|u+|2 d/g  (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='10)  Putting (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='9) and (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='10) together yields  0 = −2iσ  ��  S2r2|u|2|r=rb0 d/g +  �  S2|u+|2 d/g  �  ,  (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='11)  and this implies u|r=rb0 = 0 and u+(ω) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' This proves that the leading order term of u at infinity  vanishes and thus u ∈ A2−( ¯X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' For the next step, we rewrite �  □g(σ)u = 0 near infinity as follows  2iσ∂r(ru) = −1  r ∂rr2µb0∂ru − 1  r /∆u = ρDiff2  bu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  Then integrating both sides from infinity to r and using ru|r=∞ = 0 yield u ∈ A3−( ¯X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Proceeding  iteratively, we find that u ∈ A∞, which means that u vanishes to infinite order at r = ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Then  using the unique continuation theorem at infinity [45, Theorem 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='8] for �  □g(σ)˜u = 0 gives ˜u = 0  98  LILI HE  and thus u = 0 in r > rb0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' By Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='7 again, u is smooth and vanishes to infinite order at  r = rb0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Therefore, we follow the arguments in [89, Lemma 1] again to prove u = 0 in r < rb0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Since  we are working in the region near the r = rb0 where tb0,∗ = t0, it would be more convenient to use  the coordinates (tb0, r, ω) with ω ∈ S2, and then we write  �  □g(σ)u = 1  r2 ∂rr2µb0∂ru + 1  r2 /∆u − 2iσ  r ∂rru.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='12)  We can again obtain the energy identity as in (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='4) with R(u, du) replaced by  R(u, du) − 2Re  �  2iσr−1∂ru · ∂r(ru)  �  which can still be controlled when N is sufficiently large.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Proceeding in the same way as in (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='5), it  follows that u = 0 in r < rb0 as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  Invertibility of �  □g(σ) for Im σ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Again according to Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='23, it suffices to prove the injectivity  of �  □g(σ) for σ ∈ C, Im σ> 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' We again use relationship between �  □g(σ) and �  □g(σ) as illustrated in  (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Now ˜u decays exponentially when rb0,∗ → ±∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' So we can apply integration by parts to  conclude  0 = −  �  S2  � ∞  rb0  �  □g(σ)˜u · ˜u r2dr/g =  �  S2  � ∞  rb0  �  µb0(r)|r∂r ˜u|2 + | /∇˜u|2 − σ2µ−1  b0 (r)|˜u|2�  drd/g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  Then by taking imaginary part for Re σ ̸= 0, while using −σ2 > 0 for Re σ = 0, it follows that ˜u = 0  and thus u = 0 in r > rb0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' The proof of u = 0 in r < rb0 is the same as that in the non-zero real σ  case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  Proof for �  □gb(σ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Finally, we consider the slowly rotating KN metric gb, with b near b0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' By Theorem  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='28 and high energy estimate 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='22, �  □gb(σ) is invertible for |σ| large, say |σ| ≥ C, when |b − b0| < ǫ1,  so we only need to consider the bounded region|σ| ≤ C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Near σ = 0, the invertibility of �  □gb(σ) for  b close to b0 follows from the perturbation argument as above and the fact that �  □g(0) is invertible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  For σ ∈ C, Im σ ≥ 0, bounded away from 0 and infinity, we prove the theorem by perturbation  arguments (starting with the invertibility of �  □g(σ)) in a compact set of Im σ ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  Concretely,  suppose there exist sequences δj → 0, bj → b0 such that ker �  □gbj (σj) is non-trivial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Then we can  find uj ∈ ¯Hs,ℓ  b ( ¯X) ∩ ker �  □gbj (σj) with ∥uj∥ ¯  Hs,ℓ  b  ( ¯  X) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' By the uniform Fredholm estimate 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='5, we  have 1 ≤ C′∥uj∥ ¯  Hs0,l0  b  ( ¯  X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Therefore, there exists a subsequence uj → u weakly in ¯Hs,ℓ  b ( ¯X) which  is norm convergent in ¯Hs0,ℓ0  b  ( ¯  X) with the limit u being non-zero and satisfying �  □g(0)u = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' But  this contradicts the invertibility of �  □g(0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' So this proves the injectivity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' The surjectivity which is  equivalent to the injectivity of the adjoint �  □gb(σ)∗ can be proved in a similar manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' This proves  that for s > 1  2, ℓ ∈ (− 3  2, 1  2) and Im σ ≥ 0, |σ| < c, |b − b0| < ǫ2 for some c, ǫ2 > 0,  �  □gb(σ) : {u ∈ ¯Hs,ℓ  b ( ¯X) : �  □gb(σ)u ∈ ¯Hs−1,ℓ+2  b  ( ¯X)} → ¯Hs−1,ℓ+2  b  ( ¯X)  is invertible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' By Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='7, this implies that �  □gb(σ) acting on the function spaces in (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='1) is  injective for Im σ ≥ 0, 0 < |σ| < c, and the invertibility follows from the fact that �  □gb(σ) has index  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' This proves the theorem for Im σ ≥ 0, 0 ≤ |σ| < c and |b − b0| < ǫ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' For Im σ ≥ 0, c ≤ |σ| ≤ C,  the fact that �  □g(σ) is invertible and a perturbation argument as above imply the invertibility of  �  □gb(σ) for (b, σ) in an open neighborhood of (b0, σ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Then the compactness of the region c ≤ |σ| ≤ C  implies that there exists an ǫ3 > 0 (depending on c, C) such that �  □gb(σ) in (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='1) is invertible for  Im σ ≥ 0, c ≤ |σ| ≤ C and |b − b0| < ǫ3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  Therefore, the theorem holds for |b − b0| < ǫ where  ǫ = min{ǫ1, ǫ2, ǫ3}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  □  8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Growing zero modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' For later use, we now discuss the explicit form of the scalar functions in  ker �  □gb(0) which are allowed to have more growth at infinity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  LINEAR STABILITY OF KERR-NEWMAN FAMILY  99  Proposition 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' For b = (m, a, Q) near b0 = (m0, 0, Q0), we have  ker �  □gb(0) ∩ ¯H∞,−3/2−  b  ( ¯  X) = ⟨ub,s0⟩,  ub,s0 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='13)  ker �  □gb(0)∗ ∩ ˙H−∞,−3/2−  b  ( ¯  X) = ⟨u∗  b,s0⟩,  u∗  b,s0 = H(r − rb).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='14)  The following spaces  ker �  □gb(0) ∩ ¯H∞,−5/2−  b  ( ¯X) = ⟨ub,s0⟩ ⊕ {ub,s1(S) : S ∈ S1},  (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='15)  ker �  □gb(0)∗ ∩ ˙H−∞,−5/2−  b  ( ¯X) = ⟨u∗  b,s0⟩ ⊕ {u∗  b,s1(S) : S ∈ S1},  (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='16)  are 4-dimensional.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Moreover, the maps b �→ ub,s1(S) and b �→ u∗  b,s1(S) can be chosen to be continuous in b  with values in the respective spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' More explicitly, we have  ub=(m,0,Q),s1(S) = (r − m)S,  ub=(m,0,Q),s1(S) = (r − m)H(r − rb)S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='17)  Remark 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' We use the notation ub,s0, ub,s1 because they are of scalar type l = 0 and l = 1 respectively  when b = (m, 0, Q) is a parameter of RN metric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  Proof of Proposition 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' We first prove (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='13).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Let u ∈ ker �  □gb(0) ∩ ¯H∞,−3/2−  b  ( ¯X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' By the normal operator  argument as in Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='7, u must have the form u = u0 + ˜u where u0 ∈ C and ˜u ∈ A1−( ¯X) ⊂  ¯H∞,−1/2−  b  ( ¯X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Then we have  0 = �  □gb(0)u = �  □gb(0)u0 + �  □gb(0)˜u = �  □gb(0)˜u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  In view of Theorem 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='1, ˜u must be 0, and thus u = u0 is a constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  Conversely, we certainly have  �  □gb(0)1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' This proves (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='13).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  The proof of (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='14) is analogous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Let u ∈ ker �  □gb(0) ∩ ˙H−∞,−3/2−  b  ( ¯X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' By the normal operator argument  again, we conclude that u = χu0 + ˜u where χ is a smooth cutoff with χ = 0 for r ≤ 3m0, χ = 1 with  r ≥ 4m0, and u0 ∈ C, ˜u ∈ ˙H∞,−1/2−  b  ( ¯X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' We calculate  −�  □gb(0)∗χu0 ∈ ˙H−∞,∞  b  ( ¯X) ⊂ ˙H−∞,3/2−  b  ( ¯X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  Owing to Theorem 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='1, there exists a unique ˜u ∈ ˙H−∞,−1/2−  b  ( ¯X) satisfying �  □gb(0)∗˜u = −�  □gb(0)∗χu0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' This  proves that the space in (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='14) is at most 1-dimensional.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Conversely, a direct calculation implies  �  □gb(0)H(r − rb) = ρ−2  b ∂r(∆bδ(r − rb)) = 0  since ∆b|r=rb = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' This proves (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='14).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  As for (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='15), since ¯H∞,−5/2−  b  ( ¯  X) ⊂ A−1−( ¯X), in the normal operator argument as in Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='7,  we shift the contour of integration through the pole i with the space of resonant states given by rS with  S ∈ S1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' That is to say, u must take the form u = χrS + ˜u where χ is the smooth cutoff as defined above  and S ∈ S1, ˜u ∈ ¯H∞,−3/2−  b  ( ¯X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  Since S1 is 3-dimensional, we find that the space in (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='15) is at most  4-dimensional.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Now we prove that it is indeed 4-dimensional.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' We compute  − �  □gb(0)χrS = −χ�  □g(0)rS + [χ, �  □g(0)]rS + (�  □g(0) − �  □gb(0))χrS = 0 + ¯H∞,∞  b  ( ¯X) + ¯H∞,1/2−  b  ( ¯X) (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='18)  where we use �  □g(0) − �  □gb(0) ∈ ρ3Diff2  b for the third summand.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Since ker �  □gb(0) ∩ ˙H−∞,−1/2−  b  ( ¯X) = 0, there  exists ˜u ∈ ¯H∞,−3/2−  b  ( ¯X), which is unique modulo a multiple of ub,s0, satisfying the above equation (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='18).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  Concretely, let v ∈ C∞  c (X◦) such that ⟨ub,s0, v⟩ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Then there exists a unique ˜u ∈ ann(v) satisfying the  equation (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='18).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Therefore, χrS + ˜u, where ˜u is uniquely determined by S and b, indeed gives rise to an  element in the space (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='15) with leading order term rS at infinity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' This proves (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='15).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' The proof of (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='16) is  completely analogous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  Now we turn to discussing the continuous dependence of ub,s1(S), u∗  b,s1(S) on b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Suppose bj → b and  ubj(S) = χrS + ˜ubj(S) ∈ ker �  □gbj ∩ ¯H∞,−5/2−  b  ( ¯X) where ˜ubj(S) is in the annihilator of v ∈ C∞  c (X◦) which  satisfies ⟨ub,s0, v⟩ ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Let u = χrS + ˜u(S) ∈ ker �  □gb(0) and ej(S) = ubj(S) − u(S) = ˜ubj(S) − ˜u(S), and we  find  �  □gb(0)ej(S) = (�  □gb(0) − �  □gbj (0))(˜ubj(S) + χrS).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  100  LILI HE  Since {˜ubj(S)} is uniformly bounded in ann(v) and �  □gb(0)− �  □gbj (0) ∈ (b−bj)ρ3Diff2  b, we see that �  □gb(0)ej(S) ∈  (b − bj) ¯H∞,1/2−  b  ( ¯X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' As �  □gb(0)|ann(v) : annv → ¯H∞,1/2−  b  ( ¯X) is invertible, it follows that ej(S) is of size  O ¯  H∞,−3/2−  b  ( ¯  X)(b − bj).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' This completes the proof of the continuity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  It remains to derive the explicit expressions in (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='17).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' We first consider the case b = b0, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' gb = g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Using  /∆S = −2S, it follow that  �  □g(0)((r − m0)S) = 1  r2  �  ∂r(r2 − 2m0r + Q2  0)S − 2(r − m0)S  �  = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  �  □g(0)∗((r − m0)H(r − rb0)S) = H(r − rb0)�  □g(0)((r − m0)S) + 2µb0δ(r − rb0)S  + r−2∂r(r2µb0δ(r − rb0))(r − m0)S = 0  (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='19)  For the general RN metric gb with parameter b = (m, 0, Q), we notice that the expression of gb is the same  as that of g with (m0, 0, Q0) replaced by (m, 0, Q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Therefore, the above calculation still follows and gives  the explicit form of u(m,0,Q),s1(S) and u∗  (m,0,Q),s1(S).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  □  9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Mode analysis of the 1-form wave-type operator  In this section we shall analyze the modes of the gauge propagation operator Pb,γ = δgbGgb�δ∗  gb,γ and the  gauge potential wave operator Wb,γ = �δgb,γGgbδ∗  gb, both of which are wave-type operators acting on 1-form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  The former one occurs as the (Einstein part) of gauge propagation operator acting on the gauge 1-form  Dgb �ΥE(˙g;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' gb) = ˜δgb,γGgb ˙g, while the latter one acts on the gauge potentials (which generate the pure gauge  solutions).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' We also note that from this section on, the smallness of the charge Q is required, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=', we are  restricted in the weakly charged RN and KN (slowly rotating) metric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Throughout this section, the bundle  we mostly deal with is scattering 1-form, so we shall drop “ �  scT ∗ ¯X” in ¯Hs,ℓ  b ( ¯X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' �  scT ∗ ¯X), ˙Hs,ℓ  b ( ¯  X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' �  scT ∗ ¯X) and  A( ¯X, �  scT ∗ ¯X) when it is clear from the context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  We first recall the linearized gauge-fixed Einstein-Maxwell system  Lb,γ = (2LE  b,γ(˙g, ˙A), LM  b,γ(˙g, ˙A)) = 0  where  LE  b,γ(˙g, ˙A) = DgbRic(˙g) − 2D(gb,dAb)T (˙g, d ˙A) + �δ∗  gb,γ�δgb,γGgb ˙g = 0,  (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='1)  LM  b,γ(˙g, ˙A) = D(gb,Ab)(δgdA)(˙g, ˙A) + dδgb ˙A = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='2)  We note that for (gb, Ab) satisfying the Einstein-Maxwell system, (2δ∗  gbω, Lω♯Ab+dφ) solves the linearized  gauge-fixed Einstein-Maxwell system (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='1) and (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='2) provided that the 1-form ω and the scalar function φ  satisfy  �δgb,γGgbδ∗  gbω = 0,  δgbdφ = −δgb(Lω♯Ab).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='3)  Therefore, zero mode solutions to the above wave-type system of equations (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='3) give rise to pure gauge zero  mode solutions to L(gb,Ab),γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  Dually, using (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='1), (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='2), (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='4) and the following calculation for (DgbRic)∗ : S2 �  scT ∗ ¯X → S2 �  scT ∗ ¯X,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  (D(gb,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='dAb)T )∗ : S2 �  scT ∗ ¯X → �  scT ∗ ¯X ⊕ S2 �  scT ∗ ¯X and (D(gb,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='Ab)(δgdA))∗ : �  scT ∗ ¯X → �  scT ∗ ¯X ⊕ S2 �  scT ∗ ¯X  (DgbRic)∗(˙g) = GgbDgbRic(Ggb ˙g)  (D(gb,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='dAb)T )∗(˙g) = (−D(gb,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='Ab)(δgdA)(˙g,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' 0),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' D(gb,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='dAb)T (˙g,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' 0) − 1  2(gb)µν ˙gαβF γ  α Fβγ + 1  2trgb ˙gFµαF α  ν ),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  = (−D(gb,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='Ab)(δgdA)(Ggb ˙g,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' 0),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' GgbD(gb,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='dAb)T (Ggb ˙g,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' 0))  (D(gb,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='Ab)(δgdA))∗( ˙A) =  �  δgbd ˙A,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' −D(gb,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='dAb)T (0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' d ˙A)  �  =  �  dδgb ˙A,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' −GgbD(gb,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='dAb)T (0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' d ˙A)  �  ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  we conclude  L∗  (gb,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='Ab),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='γ(˙g,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' ˙A) = (2L∗E  (gb,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='Ab),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='γ(˙g,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' ˙A),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' 4L∗M  (gb,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='Ab),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='γ(˙g,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' ˙A))  LINEAR STABILITY OF KERR-NEWMAN FAMILY  101  with  L∗E  (gb,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='Ab),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='γ(˙g,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' ˙A) = Ggb  �  DgbRic − 2D(gb,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='dAb)T + �δ∗  gb,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='γ�δgb,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='γGgb  �  (Ggb ˙g,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' 1  4d ˙A),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='4)  L∗M  (gb,Ab),γ(˙g, ˙A) =  �  D(gb,Ab)(δgdA) + dδgb  �  (Ggb ˙g, 1  4  ˙A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='5)  Therefore, we have L∗  (gb,Ab),γ(2Ggbδ∗  gbω∗, 4Lω∗♯Ab + dφ∗) = 0 provided that  �δgb,γGgbδ∗  gbω∗ = 0,  δgbdφ∗ = −4δgb(Lω∗♯Ab).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='6)  Such dual pure gauge zero modes (2Ggbδ∗  gbω∗, 4Lω∗♯Ab + dφ∗) give rise to zero mode solutions to  L∗  (gb,Ab),γ(˙g, ˙A) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  Remark 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' The reason we introduce the constraint damping, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=', replace δ∗  gb by δ∗  gb,γ in this manuscript, is  that we need to use the invertibility of the corresponding gauge propagation operator δgbGgb ˜δ∗  gb to exclude the  generalized modes, which grows linearly in tb,∗ and whose leading term is given by linearized (in ( ˙m, 0, ˙Q)) KN  solutions, to the linearized gauge-fixed Einstein-Maxwell equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' We also note that we use a perturbation  argument to prove the invertibility of the gauge propagation operator δgbGgb ˜δ∗  gb on the weakly charged and  slowly rotating KN metrics gb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Moreover, we use the generalized linearized gauge condition for the Einstein  part �δgb,γGgb ˙g = 0 in this paper to exclude the zero mode of no geometric significance, see [36, Remark  10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  Let ˜χ(r) ∈ C∞  c ((r−, 3m0)) such that ˜χ = 1 near r = 2m0 (and thus near the event horizon of gb with b  near (m0, 0, Q0) where |Q0| ≪ m0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' We use the following  c = ˜χ(r)(dt0 − bdr),  t0 = t + r(m0,0,Q0),∗  (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='7)  with b to be determined later, to define �δ∗  gb,γ, �δgb,γ, L(gb,Ab),γ, L∗  (gb,Ab),γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' As discussed in (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='3), we then define  gauge propagation operator  Pb,γ := δgbGgb�δ∗  gb,γ  (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='8)  and gauge potential wave operator  Wb,γ := �δgb,γGgbδ∗  gb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='9)  Remark 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' We notice that Pb,γ and Wb,γ are formally dual to each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' However, on the level of function  spaces, they are not because we need to work with the extendible b-Sobolev spaces for both when analyzing  the modes of L(gb,Ab),γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' We also note that the dual pure gauge zero mode solutions to L∗  (gb,Ab),γ(˙g, ˙A) = 0  (as discussed in (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='6)) are closely related to the zero mode solutions to Wb,γω∗ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  In order to study the modes of Pb,γ and Wb,γ for b = (m, a, Q) near b0 = (m0, 0, Q0), we define  �  Pb,γ(σ) := eiσtb,∗Pb,γe−iσtb,∗,  �  Wb,γ(σ) := eiσtb,∗Wb,γe−iσtb,∗  (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='10)  where tb,∗ = (t + rb0,∗)χ0(r) + (t − r(m,0,Q),∗)(1 − χ0(r)) is defined as in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='29).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  We first record the following result about Schwarzschild metric which we rely on in the analysis of �  Pb,γ(σ)  and �  Wb,γ(σ) on weakly charged and slowly rotating KN metrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  Proposition 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='3 ( [36, Theorem 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='1]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Let (g, M(m0,0,0)) be a Schwarzschild spacetime with mass m0 and  P0 = δgGgδ∗  g be the wave operator acting on 1-form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Define �  P0(σ) := eiσt0,∗P0e−iσt0,∗ with t0,∗ = (t +  r0,∗)χ0(r) + (t − r0,∗)(1 − χ0(r)) and r0,∗ = r + 2m0 log(r − 2m0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Then we have  (1) If Im σ ≥ 0 and σ ̸= 0, then  �  P0(σ) : {ω ∈ ¯Hs,ℓ  b ( ¯X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' �  scT ∗ ¯X) : �  P0(σ)ω ∈ ¯Hs,ℓ+1  b  ( ¯X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' �  scT ∗ ¯X)} → ¯Hs,ℓ+1  b  ( ¯X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' �  scT ∗ ¯X)  (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='11)  is invertible for s > 3  2, ℓ < − 1  2, s + ℓ > − 1  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  (2) If s > 3  2 and − 3  2 < ℓ < − 1  2, then stationary operator  �  P0(0) : {ω ∈ ¯Hs,ℓ  b ( ¯X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' �  scT ∗ ¯X) : �  P0(0)ω ∈ ¯Hs−1,ℓ+2  b  ( ¯X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' �  scT ∗ ¯X)} → ¯Hs−1,ℓ+2  b  ( ¯X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' �  scT ∗ ¯X)  (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='12)  has 1-dimensional kernel and cokernel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  102  LILI HE  More explicitly, with v = t + r0,∗ = t + r + 2m0 log(r − 2m0) we have  ker �  P0(0) ∩ ¯H∞,−1/2−  b  ( ¯X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' �  scT ∗ ¯X) = ⟨ωs0⟩,  ωs0 = r−1(dv − dr)  ker �  P0(0)∗ ∩ ˙H−∞,−1/2−  b  ( ¯X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' �  scT ∗ ¯X) = ⟨ω∗  s0⟩,  ω∗  s0 = δ(r − 2m0)dr  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='13)  Remark 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' According to (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='13), �  P0(0) restricted to 1-forms of scalar type l = 1 (and vector type l = 1  respectively) is invertible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Since �  Wb0,γ(0) also maps 1-forms of scalar type l = 1 (and vector type l = 1  respectively) to the same type and is a small perturbation of �  P0(0) for b0 = (m0, 0, Q0) with |Q0| ≪ m0  and γ ≪ 1, it follows that �  Wb0,γ(0)−1 restricted to 1-forms of scalar type l = 1 (and 1-forms of vector type  l = 1 respectively) exists with norm O(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Mode stability of the gauge propagation operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' We first record the following two propositions  about Schwarzschild metric which are needed in the course of the proof of mode stability of the gauge  propagation operator Pb,γ for the weakly charged RN metric and also slowly rotating and weakly charged  KN metric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  Proposition 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='5 ( [36, Proposition 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='3]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Let (g, M(m0,0,0)) be the Schwarzschild spacetime with mass m0  and P0,γ = δgGg˜δ∗  g,γ where ˜δ∗  g,γ is defined as in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='62) and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='60) with c = ˜χ(r)(dv − bdr), v = t + r0,∗ =  t + r + 2m0 log(r − 2m0) and b ̸= 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Then there exists γ0 > 0 such that for fixed γ with 0 < |γ| < γ0, the  following holds for �  P0,γ(σ) := eiσt0,∗P0,γe−iσt0,∗ with t0,∗ = (t + r0,∗)χ0(r) + (t − r0,∗)(1 − χ0(r)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  �  P0,γ(0) : {ω ∈ ¯Hs,ℓ  b ( ¯X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' �  scT ∗ ¯X) : �  P0,γ(0)ω ∈ ¯Hs−1,ℓ+2  b  ( ¯X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' �  scT ∗ ¯X)} → ¯Hs−1,ℓ+2  b  ( ¯X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' �  scT ∗ ¯X)  (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='14)  with s > 2, − 3  2 < ℓ < − 1  2 is invertible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  Proposition 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='6 ( [36, Proposition 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='12]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Let �  P0,γ(σ) be defined as in Proposition 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='5 with b > 1, then  we have for Im σ ≥ 0, σ ̸= 0 and s > 2, ℓ < − 1  2, s + ℓ > − 1  2  �  P0,γ(σ) : {ω ∈ ¯Hs,ℓ  b ( ¯X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' �  scT ∗ ¯X) : �  P0,γ(σ)ω ∈ ¯Hs,ℓ+1  b  ( ¯X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' �  scT ∗ ¯X)} → ¯Hs,ℓ+1  b  ( ¯X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' �  scT ∗ ¯X)  (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='15)  is invertible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  Remark 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' We note that δ∗  g,γ defined in [36, §10] is slightly different from that in this manuscript ((4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='62)  and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='60)) in the way that the coefficient of the second term in B in [36] (which is denoted by E there) is  −1 instead of −1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' However, this difference has no influence on the conclusions stated in the above two  propositions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' In particular, it does not affect the calculation in [36, equation (10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='5)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  Next we shall prove the analogues of the above two propositions for RN and slowly rotating KN metric  with small charge by a perturbation argument.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  We define c as in (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='7) with b > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' With γ0 as in Propositions 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='5 and 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='6, we fix γ ∈ (0, γ0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  Theorem 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Let b0 = (m0, 0, Q0) and b > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Fix γ ∈ (0, γ0), there exists a small constant C(γ) > 0 such  that for the weakly charged RN metric gb0 with |Q0| < C(γ), the following holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  (1) If Im σ ≥ 0 and σ ̸= 0,  �  Pb0,γ(σ) : {ω ∈ ¯Hs,ℓ  b ( ¯X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' �  scT ∗ ¯X) : �  Pb0,γ(σ)ω ∈ ¯Hs,ℓ+1  b  ( ¯X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' �  scT ∗ ¯X)} → ¯Hs,ℓ+1  b  ( ¯X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' �  scT ∗ ¯X)  (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='16)  is invertible when s > 2, ℓ < − 1  2, s + ℓ > − 1  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  (2) If s > 2 and − 3  2 < ℓ < − 1  2, then stationary operator  �  Pb0,γ(0) : {ω ∈ ¯Hs,ℓ  b ( ¯X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' �  scT ∗ ¯X) : �  Pb0,γ(0)ω ∈ ¯Hs−1,ℓ+2  b  ( ¯X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' �  scT ∗ ¯X)} → ¯Hs−1,ℓ+2  b  ( ¯X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' �  scT ∗ ¯X)  (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='17)  is invertible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  Both statements also hold for the weakly charged and slowly rotating KN metric gb with b = (m, a, Q) near  b0 = (m0, 0, Q0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' We note that a perturbation argument as in the final step in the proof of Theorem 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='1 allows us  to extend Propositions 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='5 and 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='6 to weakly charged RN metrics, which however is a smooth family of  Lorentzian metric on the fixed manifold M(m0,0,0) and is a pull back of the RN metric gb0 defined in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  LINEAR STABILITY OF KERR-NEWMAN FAMILY  103  We need to prove that Propositions 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='5 and 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='6 in fact hold for gb0 with sufficiently small charge as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  Specifically, let (g, M(m0,0,0)) be the Schwarzschild spacetime  g = −(1 − 2m0  r  )dv2 + 2dvdr + r2/g,  M(m0,0,0) = Rv × [r−, ∞) × S2  where r− ∈ (0, 2m0) is a fixed number and v = t + r0,∗ = t + r + 2m0 log(r − 2m0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' We introduce another  coordinates for (g, M(m0,0,0)): (˜tχ0, r, θ, φ) where ˜tχ0 = t +  � r  4m0 s−2(s2 − 2m0s)χ0(s) ds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Let gb0 be the RN  metric as defined in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' We then define ˜gb0 = (Ψb0)∗gb0 as the pull back of gb0 under a diffeomorphism  Ψb0  Ψb0 : M(m,0,0) → M,  (tχ0, r, θ, φ)(Ψb0(p)) = (˜tχ0, r, θ, φ)(p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  Then we have  gb0 = −µb0dt2  χ0 + 2χ0dtχ0dr + 1 − χ2  0  µb0  dr2 + r2/g,  (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='18)  ˜gb0 = −µb0d˜t2  χ0 + 2χ0d˜tχ0dr + 1 − χ2  0  µb0  dr2 + r2/g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='19)  We now define  P˜gb0 ,γ := δ˜gb0 G˜gb0 δ∗  ˜gb0 ,γ  with c = ˜χ(r)(dv − bdr) and v = t + r∗ in the definition of δ∗  ˜gb0 ,γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' In view of (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='18) and (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='19), we find that  Pb0,γ can be obtained by replacing ∂˜tχ in P˜gb0 ,γ by ∂tχ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Since  �  Pb0,γ(σ) = eiσtb0,∗Pb0,γe−iσtb0,∗,  tb0,∗ = (t + rb0,∗)χ0(r) + (t − rb0,∗)(1 − χ0(r)),  �  P˜gb0 ,γ(σ) = eiσ˜tb0,∗P˜gb0 ,γe−iσ˜tb0,∗,  ˜tb0,∗ = (t + r0,∗)χ0(r) + (t − rb0,∗)(1 − χ0(r)),  we rewrite  �  Pb0,γ(σ) = eiσ(tχ0 +F1(r))Pb0,γe−iσ(tχ0 +F1(r)),  �  P˜gb0 ,γ(σ) = eiσ(˜tχ0 +F2(r))P˜gb0 ,γe−iσ(˜tχ0 +F2(r)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  Therefore  �  Pb0,γ(σ) = eiσ(F1(r)−F2(r)) �  P˜gb0 ,γ(σ)e−iσ(F1(r)−F2(r))  where F1(r) − F2(r) ∈ C∞  c ((2m0, ∞)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' As discussed at the beginning, Propositions 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='5 and 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='6 hold for  �  P˜gb0 ,γ(σ), and thus for �  Pb0,γ(σ) as well with |Q0| < C(γ) where C(γ) is a sufficiently small constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  Finally it remains to analyze the weakly charged and slowly rotating KN case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' It follows from a pertur-  bation argument as in the final step in the proof of Theorem 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  □  9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Mode stability of the gauge potential wave operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Now we turn to the analysis of the gauge  potential wave operator Wb,γ = �δgb,γGgbδ∗  gb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' We will first prove the analogues of Propositions 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='5 and 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='6 for  W0,γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' We note that the proofs closely follow [36, Proposition 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='3, 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' However, we present the proofs  in detail for completeness here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Then proceeding as in Theorem 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='8, we can obtain its counterpart for Wb,γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  Proposition 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Let (g, M(m0,0,0)) be the Schwarzschild spacetime with mass m0 and W0,γ = ˜δg,γGgδ∗  g  where ˜δg,γ is defined as in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='62) and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='61) with c = ˜χ(r)(dv − bdr), v = t + r0,∗ = t + r + 2m0 log(r − 2m0)  and b ̸= 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  Then there exists γ0 > 0 such that for fixed γ with 0 < |γ| < γ0, the following holds for  �  W0,γ(σ) := eiσt0,∗W0,γe−iσt0,∗ with t0,∗ = (t + r0,∗)χ0(r) + (t − r0,∗)(1 − χ0(r)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  �  W0,γ(0) : {ω ∈ ¯Hs,ℓ  b ( ¯X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' �  scT ∗ ¯X) : �  W0,γ(0)ω ∈ ¯Hs−1,ℓ+2  b  ( ¯X)} → ¯Hs−1,ℓ+2  b  ( ¯X)  (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='20)  with s > 2, − 3  2 < ℓ < − 1  2 is invertible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Fix − 3  2 < ℓ < − 1  2, s > 2 and let  X s,ℓ := {u ∈ ¯Hs,ℓ  b ( ¯X) : �  W0,γ(0)u ∈ ¯Hs−1,ℓ+2  b  ( ¯X)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  In view of Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='25, we know that �  W0,γ(0) : X s,ℓ → ¯Hs−1,ℓ+2  b  ( ¯X) is Fredholm of index 0 when γ is  sufficiently small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' As �  W0,γ1(0) − �  W0,γ2(0) is a first-order differential operator with compactly supported  coefficients, the space X s,ℓ does not depend on γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  104  LILI HE  We split the domain X s,ℓ and the target space ¯Hs,ℓ  b ( ¯  X) as follows  X s,ℓ = K⊥ ⊕ K,  K = ker �  P0(0) ∩ X s,ℓ = ⟨ωs0⟩  ¯Hs,ℓ  b ( ¯X) = R ⊕ R⊥,  R = ranX s,ℓ �  P0(0) = ann ω∗  s0  where ann means annihilator, K⊥ is a complementary subspace to K inside X s,ℓ while R⊥ = ⟨η⟩ is com-  plementary to R inside ¯Hs−1,ℓ+2  b  (in other word, R⊥ is the cokernel of �  P0(0)), and ω0, ω∗  0 are defined as  in (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='13).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' We may choose η ∈ C∞  c (X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' �  scT ∗ ¯X) satisfying ⟨η, ω∗  s0⟩ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Then the operator �  W0(0) can be  rewritten as  �  W0,γ(0) =  �W00 + γW♭  00  γW♭  01  γW♭  10  γW♭  11  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='21)  We notice that W00 = �  P0(0)|K⊥ : K⊥ → R is invertible, and so is W00 + γW♭  00 provided that γ is small  enough.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Moreover, if we identify K ∼= C via cωs0 �→ c and R⊥ ∼= C via cη �→ ⟨cη, ω∗  s0⟩ = c, correspondingly,  W♭  11 can be identified as an endmorphism of C, and thus is simply a number which can be computed as  follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  W♭  11 = γ−1⟨  ��  W0,γ(0) − �  P0(0)  �  ωs0, ω∗  s0⟩  = γ−1⟨  �  FgGgδ∗  g  �  ωs0, ω∗  s0⟩ = γ−1⟨δ∗  gωs0, GgBgω∗  s0⟩  = ⟨δ∗  gωs0,  �  2c ⊗s ω∗  s0 − 1  2g−1(c, ω∗  s0)g  �  ⟩  = 4π(b − 1)  (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='22)  where we use g = −(1 − 2m0  r )dv2 + 2dvdr + r2/g, 2c ⊗s ω∗  s0 = δ(r − 2m0)˜χ(r)(2dvdr − 2bdr2), g−1(c, ω∗  s0) =  δ(r − 2m0)˜χ(r)(1 − b(1 − 2m0  r )) and δ∗  gωs0 = − 2m2  0  r4 dv2 − 2( 1  2r2 + m0  r3 )dvdr + 1  r2 dr2 − (1 − 2m0  r )/g in the last  step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  Now suppose (ω0, ω1) ∈ (K⊥ ⊕ K) ∩ ker �  W0,γ(0), then we have  ω0 = −γ(W00 + γW♭  00)−1W♭  01ω1  and thus  �  W♭  11 − γW♭  10(W00 + γW♭  00)−1W♭  01  �  ω1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  if γ is nonzero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' When b ̸= 1, W♭  11 is invertible, and thus so is W♭  11 − γW♭  10(W00 + γW♭  00)−1W♭  01 if γ is  sufficiently small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' So we must have ω1 = 0 and thus ω0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' This proves the injectivity of �  W0,γ(0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' The  surjectivity follows from the fact that �  W0,γ(0) is Fredholm of index 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  □  As explained in (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='3) and (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='6), Wb,γ acts as the gauge potential wave operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' In order to obtain the  mode stability of Lgb,γ(˙g, ˙A) = 0 for Im σ ≥ 0, σ ̸= 0, we need to prove that for the operator Wb,γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' In other  words, we need to show that for suitable choices of b and small γ, Wb,γ has no modes with σ ̸= 0 in the  closed upper half plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' We first prove this for Schwarzschild metric, and then the RN and slowly rotating  KN metric with small charge follows from a perturbation argument as in Theorem 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  Proposition 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Let �  W0,γ(σ) be defined as in Proposition 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='9 with b > 1, then there exists ˜γ0 > 0 such  that for 0 < γ < ˜γ0, Im σ ≥ 0, σ ̸= 0 and s > 2, ℓ < − 1  2, s + ℓ > − 1  2  �  W0,γ(σ) : {ω ∈ ¯Hs,ℓ  b ( ¯X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' �  scT ∗ ¯X) : �  W0,γ(σ)ω ∈ ¯Hs,ℓ+1  b  ( ¯X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' �  scT ∗ ¯X)} → ¯Hs,ℓ+1  b  ( ¯X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' �  scT ∗ ¯X)  (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='23)  is invertible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  In the course of the proof of Proposition 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='10, we closely follow the arguments in [36, §10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' As explained  there, the proof consists of two steps:  (1) We show in Lemma 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='12 that for b > 1 and for all sufficiently small γ, �  W0,γ(σ) has no modes in a  fixed neighborhood of σ = 0 in the closed upper half plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  (2) In the proof of Proposition 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='10, we proceed as in the final step in the proof of Theorem 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='1 to prove  the mode stability of �  W0,γ(σ) in the closed upper half plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  LINEAR STABILITY OF KERR-NEWMAN FAMILY  105  Fix s > 2, − 3  2 < ℓ < − 1  2, the domains  X s,ℓ(σ) := {ω ∈ ¯Hs,ℓ  b ( ¯X) : �  W0,γ(σ)ω ∈ ¯Hs−1,ℓ+2  b  ( ¯X)}  (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='24)  of the operators �  W0,γ(σ) : X s,ℓ(σ) → ¯Hs−1,ℓ+2  b  now depend on σ (but still independent of γ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' As a conse-  quence, we need the following technical lemma which allows us to pass to operators with fixed domain and  target space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  Lemma 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Let s > 2, − 3  2 < ℓ < − 1  2 and b ̸= 1 and fix γ1 satisfying 0 < |γ1| < γ0 where γ0 is as in Propo-  sition 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Then �  W0,γ1(σ) : X s,ℓ(σ) → ¯Hs−1,ℓ+2  b  ( ¯X) is invertible for σ ∈ C with Im σ ≥ 0, |σ| < c where c is  some small constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Moreover, �  W0,γ1(σ)−1 is continuous in σ with values in Lweak( ¯Hs−1,ℓ+2  b  ( ¯X), ¯Hs,ℓ  b ( ¯X))  (the space of linear bounded operators equipped with weak operator topology), while continuous in σ with  values in Lop( ¯Hs−1+ǫ,ℓ+2+ǫ  b  ( ¯X), ¯Hs−ǫ,ℓ−ǫ  b  ( ¯X)) (norm topology) for any ǫ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' By Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='6, we have the uniform Fredholm estimates for ω ∈ X s,ℓ(σ)  ∥ω∥ ¯  Hs,ℓ  b  ( ¯  X) ≤ C  �  ∥�  W0,γ1(σ)ω∥ ¯  Hs−1,ℓ+2  b  ( ¯  X) + ∥ω∥ ¯  Hs0,ℓ0  b  ( ¯  X)  �  (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='25)  for Im σ ≥ 0 with |σ| being small and s0 < s, ℓ0 < ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' We now follow the arguments in [79, Propsition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='4] to  show that the compact error term ∥ω∥ ¯  Hs0,ℓ0  b  ( ¯  X) can be dropped (while we may need to take larger constant  C) if |σ| is sufficiently small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Concretely, suppose for the sake of contradiction that there exists a sequence  δj → 0 and a sequence ωj with ∥ωj∥ ¯  Hs,ℓ  b  ( ¯  X) = 1 and �  W0,γ1(σj)ωj ∈ ¯Hs−1,ℓ+2  b  ( ¯X) such that 1 = ∥ωj∥ ¯  Hs,ℓ  b  ( ¯  X) ≥  j∥�  W0,γ1(σj)ωj∥ ¯  Hs−1,ℓ+2  b  ( ¯  X), then �  W0,γ1(σj)ωj → 0 in ¯Hs−1,ℓ+2  b  ( ¯X) and lim infj→∞ ∥ωj∥ ¯  Hs0,ℓ0  b  ( ¯  X) ≥ C−1 > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  As a consequence, there exists a subsequence ωj → ω weakly in ¯Hs,ℓ  b ( ¯X) and strongly in ¯Hs0,ℓ0  b  ( ¯X) with the  limit ω being non-zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Moreover,  �  W0,γ1(σj)ωj − �  W0,γ1(0)ω =  ��  W0,γ1(σj) − �  W0,γ1(0)  �  ωj + �  W0,γ1(0)  �  ωj − ω  �  → 0  in  ¯Hs0−2,ℓ0  b  ( ¯X)  since �  W0,γ1(σj) → �  W0,γ1(0) as bounded operators in L( ¯Hs0,ℓ0  b  ( ¯X), ¯Hs0−2,ℓ0  b  ( ¯X)) and ωj → ω in ¯Hs0,ℓ0  b  ( ¯X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  Therefore, �  W0,γ1(0)ω = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' But this contradicts Proposition 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='9, and thus proves the injectivity of �  W0,γ1(σ)  for σ with small |σ| in the closed upper half plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Since �  W0,γ1(σ) has index 0, it follows that �  W0,γ1(σ)−1  exists with a uniform bound in a small neighborhood of σ = 0 in the closed upper half plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  Next we prove the sequential continuity of �  W0,γ1(σ)−1 in σ (which is equivalent to continuity).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Suppose  δj → δ, one need to show �  W0,γ1(σj)−1 → �  W0,γ1(σ)−1 in weak operator topology of L( ¯Hs−1,ℓ+2  b  ( ¯X), ¯Hs,ℓ  b ( ¯X)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  Suppose {fj} is a sequence in ¯Hs−1,ℓ+2  b  which is norm converging to f, let ωj = �  W0,γ1(σj)−1fj and we shall  show that ωj converges weakly to �  W0,γ1(σ)−1f in ¯Hs,ℓ  b ( ¯X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' In view of the uniform bound of �  W0,γ1(σj)−1  proved in the first step above, we see that ωj is uniformly bounded in ¯Hs,ℓ  b ( ¯X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Therefore, there exists a  subsequence ωjk → ω weakly in ¯Hs,ℓ  b ( ¯X), then fjk = �  W0,γ1(σjk)ωjk → �  W0,γ1(σ)ω weakly in ¯Hs−2,ℓ  b  ( ¯X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  Meanwhile by assumption fjk → f strongly in ¯Hs−1,ℓ+2  b  ( ¯  X) and thus �  W0,γ1(σ)ω = f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' This implies that ω  is uniquely determined by f and independent of the subsequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Namely, every subsequence of ωj has a  subsequence converging weakly to the same ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' This means that the full sequence ωj → ω = �  W0,γ1(σj)−1f  weakly in ¯Hs,ℓ  b ( ¯  X) and thus proves the continuity in the weak operator topology (Otherwise, one can find  g ∈  ˙H−s,−ℓ  b  ( ¯  X)), δ > 0 and a subsequence {ωjk} ⊂ {wj} such that |g(ωjk) − g(ω)| ≥ δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  However, by  assumption {ωjk} should have a subsequence converging weakly to ω, which is impossible).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  As for the continuity in the norm topology, we shall prove it by contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Suppose there exists a  δ > 0 and a sequence σj → σ such that ∥�  W0,γ1(σj)−1 − �  W0,γ1(σ)−1∥L( ¯  Hs−1+ǫ,ℓ+2+ǫ  b  ( ¯  X), ¯  Hs−ǫ,ℓ−ǫ  b  ( ¯  X)) ≥ δ, then  one can find a bounded sequence fj ∈ ¯Hs−1+ǫ,ℓ+2+ǫ  b  such that  ∥�  W0,γ1(σj)−1fj − �  W0,γ1(σ)−1fj∥ ¯  Hs−ǫ,ℓ−ǫ  b  ( ¯  X) ≥ δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='26)  Passing to a subsequence, we may assume that fj → f weakly in ¯Hs−1+ǫ,ℓ+2+ǫ  b  ( ¯X) and strongly in ¯Hs−1,ℓ+2  b  ( ¯X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  Then it follows that �  W0,γ1(σ)−1fj → �  W0,γ1(σ)−1f in ¯Hs,ℓ  b ( ¯X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' However, by the continuity in the weak op-  erator topology, we see that �  W0,γ1(σj)−1fj → �  W0,γ1(σ)−1f weakly in ¯Hs,ℓ  b ( ¯X) and thus (passing to a  106  LILI HE  subsequence) strongly in ¯Hs−ǫ,ℓ−ǫ  b  ( ¯X), which contradicts (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='26).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' This finishes the proof of the continuity in  the norm topology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  □  With the help of the invertibility of �  W0,γ1(σ), the analysis of �  W0,γ(σ) can be converted to that of another  operator �  W0,γ(σ)�  W0,γ1(σ)−1 : ¯Hs−1,ℓ+2  b  ( ¯X) → ¯Hs−1,ℓ+2  b  ( ¯X) with fixed domain and target space independent  of σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' This will enable us to prove the invertibility of �  W0,γ(σ) for all sufficiently small γ > 0 in a fixed small  neighborhood of σ = 0 in the closed upper half plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  Lemma 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Let s > 2, − 3  2 < ℓ < − 1  2 and b > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Then there exists C0 > 0 such that for all γ > 0 and  σ ∈ C, Im σ ≥ 0 with γ + |σ| < C0, the operator �  W0,γ1(σ) : X s,ℓ(σ) → ¯Hs−1,ℓ+2  b  ( ¯X) is invertible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Let  �  W0,γ(σ)�  W0,γ1(σ)−1 : ¯Hs−1,ℓ+2  b  ( ¯X) → ¯Hs−1,ℓ+2  b  ( ¯X),  then it suffices to prove the injectivity of �  W0,γ(σ)�  W0,γ1(σ)−1 provided that b > 1 and γ > 0, Im σ ≥ 0 with  γ + |σ| < C0 where C0 is a small constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' To this end, we split the domains and the target spaces as follows  domain: ¯Hs−1,ℓ+2  b  ( ¯X) = K⊥ ⊕ K,  K = ker �  P0(0)�  W0,γ1(0)−1 ∩ ¯Hs−1,ℓ+2  b  ( ¯X) = ⟨˜ω⟩ with ˜ω = �  W0,γ1(0)ωs0  target: ¯Hs−1,ℓ+2  b  ( ¯X) = R ⊕ R⊥,  R = ranX s,ℓ(0)�  P0(0)  where K⊥ and R⊥ = ⟨η⟩ are the complementary subspaces to K and R respectively inside ¯Hs−1,ℓ+2  b  ( ¯X), and  ωs0 is defined as in (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='13).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' We notice that  ˜ω = �  W0,γ1(0)ωs0 =  ��  W0,γ1(0) − �  P0(0)  �  ωs0 = �  FgGgδgωs0 ∈ C∞  c (X◦)  and we can identify R⊥ ∼= C via cη �→ ⟨cη, ω∗  s0⟩ where ω∗  s0 is defined as in (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='13).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Under these splittings,  �  W0,γ(σ)�  W0,γ1(σ)−1 takes the form  �  W0,γ(σ)�  W0,γ1(σ)−1 =  �W00(γ, σ)  W01(γ, σ)  W10(γ, σ)  W11(γ, σ)  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='27)  We first list several basic facts: W01(0, 0), W10(0, 0), W11(0, 0) are 0, and W00(0, 0) is invertible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  Since  �  W0,γ(σ)�  W0,γ1(σ)−1 ∈ L( ¯Hs−1,ℓ+2  b  ( ¯X), ¯Hs−1,ℓ+2  b  ( ¯X)) is Fredholm of index 0 (see Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='25) for γ + |σ|  small and the three components (0, 1), (1, 0), (1, 1) are compact operators (because they all have finite rank),  we can conclude that W00(γ, σ) is Fredholm of index 0 as well for γ + |σ| small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  Suppose ω = (ω0, ω1) ∈ (K⊥ ⊕ K) ∩ ker �  W0,γ(σ)�  W0,γ1(σ)−1, we have W00(γ, σ)ω0 + W01(γ, σ)ω1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  Assume for the moment that W00(γ, σ) is invertible for γ + |σ| small, then  ω0 = −W00(γ, σ)−1W01(γ, σ)ω1  (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='28)  and thus  �  W11(γ, σ) − W10(γ, σ)W00(γ, σ)−1W01(γ, σ)  �  ω1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='29)  We want to show that ω1 = 0 and thus ω0 = 0 for γ+|σ|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Formally speaking, since W01(0, 0), W10(0, 0)  are 0, W01(γ, σ), W10(γ, σ) are of size O(γ + |σ|).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Therefore, it suffices to show that W11(γ, σ) is nonzero  modulo O(γ2 + |σ|2) for γ + |σ| small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' To make these arguments rigorous, we need to prove the injectivity of  W00(γ, σ) (and thus the invertibility in view of its 0 index property) for γ +|σ| small and the differentiability  of W01, W10 at (γ, σ) = (0, 0), and calculate the Taylor expansion of W11(γ, σ) at (0, 0) up to the first order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  Invertibility of W00.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' We shall prove that for γ + |σ| small, there exists a uniform constant C > 0  such that  ∥W00(γ, σ)−1∥R→K⊥ ≤ C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='30)  The proof is similar to that in the first step of the proof of Lemma 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Specifically, we define  Π : ¯Hs−1,ℓ+2  b  ( ¯X) → R to be the projection onto R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  Suppose (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='30) is not true, one can find  a sequence (γj, δj) → (0, 0) and a sequence {fj} ⊂ K⊥ with ∥fj∥ ¯  Hs−1,ℓ+2  b  ( ¯  X) = 1 such that for  ωj = �  W0,γ1(σj)−1fj  1 = ∥fj∥ ¯  Hs−1,ℓ+2  b  ( ¯  X) ≥ j∥Π�  W0,γj(σj)ωj∥ ¯  Hs−1,ℓ+2  b  ( ¯  X)  LINEAR STABILITY OF KERR-NEWMAN FAMILY  107  which implies Π�  W0,γj(σj)ωj → 0 in ¯Hs−1,ℓ+2  b  ( ¯X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' In view of Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='6, we have the following  uniform Fredholm estimates for γ + |σ| small  ∥ωj∥ ¯  Hs,ℓ  b  ( ¯  X) ≤ ˜C  �  ∥Π�  W0,γj(σj)ωj∥ ¯  Hs−1,ℓ+2  b  ( ¯  X) + ∥(I − Π)�  W0,γj(σj)ωj∥ ¯  Hs−1,ℓ+2  b  ( ¯  X)  + ∥ωj∥ ¯  Hs0,ℓ0  b  ( ¯  X)  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='31)  Using the identification R⊥ ∼= C introduced previously, we calculate  ∥(I − Π)�  W0,γj(σj)ωj∥ ¯  Hs−1,ℓ+2  b  ( ¯  X) ∼ |⟨�  W0,γj(σj)ωj, ω∗  s0⟩| = |⟨ωj, �  W0,γj(σj)∗ω∗  s0⟩|  = |⟨ωj,  �  �  W0,γj(σj)∗ − �  P0(0)∗�  ω∗  s0⟩| → 0  as  (γj, σj) → (0, 0)  (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='32)  where we use ω∗  s0 ∈  ˙H  − 1  2 −,∞  b  ( ¯  X) in the first equality and �  P0,γj(σj)∗ − �  P0(0)∗ → 0 as bounded  operators L( ˙H1−s,−ℓ−2  b  ( ¯X), ˙H−s,−ℓ−1  b  ( ¯X)) in the last step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  Since ωj = �  W0,γ1(σj)−1fj, we have  ∥ωj∥ ¯  Hs,ℓ  b  ( ¯  X) ∼ 1 and thus lim infj→∞ ∥ωj∥ ¯  Hs0,ℓ0  b  ( ¯  X) ≥ ˜C−1 lim infj→∞ ∥ωj∥ ¯  Hs,ℓ  b  ( ¯  X) > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' As a re-  sult, there exists a subsequence (not shown in the notation) ωj → ω ̸= 0 weakly in ¯Hs,ℓ  b ( ¯X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  Then �  W0,γj(σj)ωj → �  W0(0)ω = �  P0(0)ω weakly in ¯Hs−2,ℓ  b  ( ¯X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Since �  W0,γj(σj)ωj → 0 strongly in  ¯Hs−1,ℓ+2  b  ( ¯X), we obtain �  P0(0)ω = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  Meanwhile, since ∥fj∥ ¯  Hs−1,ℓ+2  b  ( ¯  X) = 1, passing to a subsequence, we see that fj → f weakly  in ¯Hs−1,ℓ+2  b  ( ¯X) and strongly in ¯Hs−1−ǫ,ℓ+2−ǫ  b  ( ¯X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Owing to Lemma 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='11 (the continuity in weak  operator topology), ωj = �  W0,γ1(σj)−1fj → �  W0,γ1(0)−1f weakly in ¯Hs−ǫ,ℓ−ǫ  b  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Therefore, we obtain  0 ̸= ω = �  W0,γ1(0)−1f, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=', 0 ̸= f = �  W0,γ1(0)ω ∈ K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' It is clear that one can find g ∈ ˙H1−s,ℓ−2  b  ( ¯X)  such that g(K⊥) = 0 while g( ˜w) = 1 (K = ⟨˜ω⟩), but this contradicts the fact that fj → f weakly,  and thus proves (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='30).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' From (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='30), we conclude that for γ + |σ| small, W00 is injective and then  the invertibility follows from the fact that W00 has index 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  Differentiability of W01, W10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' We define  W1(γ, σ) := W01(γ, σ) ⊕ W11(γ, σ) : K = ⟨˜ω⟩ → ¯Hs−1,ℓ+2  b  ( ¯X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='33)  We will show that W1(γ, σ) is continuous for γ + |σ| small and differentiable at (0, 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  We write  �  W0,γ(σ)�  W0,γ1(σ)−1 =  �  �  W0,γ(σ) − �  W0,γ1(σ) + �  W0,γ1(σ)  �  �  W0,γ1(σ)−1  =  �  �  W0,γ(σ) − �  W0,γ1(σ)  �  �  W0,γ1(σ)−1 + I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='34)  Then it suffices to analyze the first term on the right hand side of (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='34).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  For the continuity,  since �  W0,γ1(σ)−1˜ω ∈ ¯Hs,ℓ  b ( ¯X) depending on σ continuously for any s > 2, − 3  2 < ℓ < − 1  2 (because  ˜ω ∈ C∞  c (X◦) and �  W0,γ1(σ)−1 is continuous in σ with value in Lop( ¯Hs−1+ǫ,ℓ+2+ǫ  b  ( ¯X), ¯Hs−ǫ,ℓ−ǫ  b  ( ¯X)))  and �  W0,γ(σ)−�  W0,γ1(σ) ∈ Diff1  b with C∞  c (X) coefficients and depends on γ, σ smoothly, the continuity  of �  W0,γ(σ)�  W0,γ1(σ)−1˜ω ∈ C∞  c (X) in (γ, σ) follows  Now we turn to the analysis of the differentiability at (0, 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Since �  W0,γ(σ) − �  W0,γ1(σ) are poly-  nomials in γ, σ with coefficients being first order differential operators, it is certainly differentiable  at (0, 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' So it suffices to prove that �  W0,γ1(σ)−1˜ω is differentiable at σ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Now we claim that if  �  W0,γ1(0)−1ω ∈ ρC∞( ¯X) + ¯H∞,1/2−  b  ( ¯X), then the following resolvent identity holds  �  �  W0,γ1(σ)−1 − �  W0,γ1(0)−1�  ω = �  W0,γ1(σ)−1�  �  W0,γ1(0) − �  W0,γ1(σ)  �  �  W0,γ1(0)−1ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='35)  The proof of the above resolvent identity closely follows [81, §6] and uses a regularization argument.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  Concretely, let χǫ(ρ) = χ(ǫ−1ρ) where χ is a nonnegative smooth cutoff such that χ = 1 on [1, ∞)  108  LILI HE  and χ = 0 on (−∞, 1/2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' With the limit being strong operator limits, we have  �  W0,γ1(σ)−1 − �  W0,γ1(0)−1  = lim  ǫ→0  �  �  W0,γ1(σ)−1χǫ(ρ)�  W0,γ1(0)�  W0,γ1(0)−1 − �  W0,γ1(σ)−1�  W0,γ1(σ)χǫ(ρ)�  W0,γ1(0)−1�  = lim  ǫ→0  �  W0,γ1(σ)−1�  [χǫ(ρ), �  W0,γ1(0)] + �  W0,γ1(0)χǫ(ρ) − �  W0,γ1(σ)χǫ(ρ)  �  �  W0,γ1(0)−1  = �  W0,γ1(σ)−1 lim  ǫ→0  ��  �  W0,γ1(0) − �  W0,γ1(σ)  �  χǫ(ρ)  �  �  W0,γ1(0)−1  + �  W0,γ1(σ)−1 lim  ǫ→0  �  [χǫ(ρ), �  W0,γ1(0)]  �  �  W0,γ1(0)−1 := I + II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  As for I, since �  W0,γ1(0)−1ω ∈ ρC∞( ¯X) + ¯H∞,1/2−  b  ( ¯X) which is annihilated modulo ¯H∞,3/2−  b  ( ¯X) by  the normal operator −iσρ(ρ∂ρ − 1) of �  W0,γ1(0) − �  W0,γ1(σ), it follows that  ��  �  W0,γ1(0) − �  W0,γ1(σ)  �  χǫ(ρ)  �  �  W0,γ1(0)−1ω  = χǫ(ρ)  �  �  W0,γ1(0) − �  W0,γ1(σ)  �  �  W0,γ1(0)−1ω + ǫ−1χ′(ǫ−1ρ)  �  ρ3C∞( ¯X) + ¯H∞,5/2−  b  ( ¯X)  �  ∈ χǫ(ρ) ¯H∞,1/2−  b  ( ¯X) + ǫ−1χ′(ǫ−1ρ)  �  ρ3C∞( ¯X) + ¯H∞,5/2−  b  ( ¯X)  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  Since the ¯H∞,1/2−  b  ( ¯  X) norm of the second term ǫχ′(ǫ−1ρ)  �  ρC∞( ¯X) + ¯H∞,1/2−  b  ( ¯X)  �  goes to 0 as  ǫ → 0, we arrive at  I = �  W0,γ1(σ)−1�  �  W0,γ1(0) − �  W0,γ1(σ)  �  �  W0,γ1(0)−1ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  For II, since �  W0,γ1(0) ∈ ρ2Diff2  b, it follows that [χǫ(ρ), �  W0,γ1(0)] is uniform (in ǫ) in ρ2Diff1  b and  [χǫ(ρ), �  W0,γ1(0)] goes to 0 in the strong operator topology L( ¯Hs,ℓ  b ( ¯X), ¯Hs−1,ℓ+2  b  ( ¯X)), and thus II = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  This proves (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='35).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  Since �  W0,γ1(0)−1˜ω = ωs0 = r−1(dv − dr) ∈ ρC∞( ¯X) + ¯H∞,1/2−  b  ( ¯X), applying (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='35) yields  �  �  W0,γ1(σ)−1 − �  W0,γ1(0)−1�  ˜ω = �  W0,γ1(σ)−1�  �  W0,γ1(0) − �  W0,γ1(σ)  �  �  W0,γ1(0)−1˜ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='36)  Let  ˜ω(σ) :=  �  �  W0,γ1(0) − �  W0,γ1(σ)  �  �  W0,γ1(0)−1˜ω ∈ ¯H∞,3/2−  b  ( ¯X),  (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='37)  and write  ˜ω(σ) = −σ(∂σ �  W0,γ1(0)ωs0) + O ¯  H∞,3/2−  b  ( ¯  X)(|σ|2)  (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='38)  where the notation O ¯  H∞,3/2−  b  ( ¯  X)(|σ|2) means that the term has norm in ¯H∞,3/2−  b  ( ¯X) of size O(|σ|2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  Using the fact that �  W0,γ1(σ)−1 is continuous in σ with value in Lop( ¯Hs−1+ǫ,ℓ+2+ǫ  b  ( ¯X), ¯Hs−ǫ,ℓ−ǫ  b  ( ¯X)),  it follows that  �  �  W0,γ1(σ)−1 − �  W0,γ1(0)−1�  ˜ω = −σ�  W0,γ1(0)−1∂σ �  W0,γ1(0)ωs0 + o ¯  Hs,ℓ  b  ( ¯  X)(|σ|)  (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='39)  for any s > 2, − 3  2 < ℓ < − 1  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' This proves the differentiability of W1 and thus W01 at (γ, σ) = (0, 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  As for W10, we proceed in a similar manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' We define  W2(γ, σ) := W10(γ, σ) ⊕ W11(γ, σ) : ¯Hs−1,ℓ+2  b  ( ¯X) → R⊥ ∼= C  ω �→ ⟨�  W0,γ(σ)�  W0,γ1(σ)−1ω, ω∗  s0⟩  (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='40)  We will show that W2(γ, σ) is continuous for γ + |σ| small and differentiable at (0, 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  As ω∗  s0 = δ(r − 2m0)dr ∈ ˙H−1/2−,∞  b  ( ¯X), we can rewrite  ⟨�  W0,γ(σ)�  W0,γ1(σ)−1ω, ω∗  s0⟩ = ⟨�  W0,γ1(σ)−1ω, �  W0,γ(σ)∗ω∗  s0⟩  = ⟨�  W0,γ1(σ)−1ω,  ��  W0,γ(σ)∗ − �  W0(0)∗�  ω∗  s0⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='41)  LINEAR STABILITY OF KERR-NEWMAN FAMILY  109  Since �  W0,γ1(σ)−1 is continuous in σ with value in Lop( ¯Hs−1,ℓ+2  b  ( ¯X), ¯Hs−ǫ,ℓ−ǫ  b  ( ¯X)), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=',' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  �  W0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='γ1(σ)−1 = �  W0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='γ1(0)−1 + oL( ¯  Hs−1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='ℓ+2  b  ( ¯  X),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' ¯  Hs−ǫ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='ℓ−ǫ  b  ( ¯  X))(1),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  and �  W0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='γ(σ)∗ − �  W0(0)∗ is a polynomial in γ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' σ with coefficients in L( ˙H1−s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='−ℓ−2  b  ( ¯X),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' ˙H−s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='−ℓ−1  b  ( ¯X))  �  W0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='γ(σ)∗ − �  W0(0)∗  = σ  �  ∂σ(�  W0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='γ(σ)∗)|(0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='0)  �  + γ  �  ∂γ(�  W0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='γ(σ)∗)|(0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='0)  �  + OL( ˙H1−s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='−ℓ−2  b  ( ¯  X),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' ˙H−s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='−ℓ−1  b  ( ¯  X))((γ + |σ|)2),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  it follows that  ⟨�  W0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='γ(σ)�  W0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='γ1(σ)−1ω,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' ω∗  s0⟩  = σ⟨�  W0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='γ1(0)−1ω,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' ∂σ(�  W0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='γ(σ)∗)|(0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='0)ω∗  s0⟩ + γ⟨�  W0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='γ1(0)−1ω,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' ∂γ(�  W0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='γ(σ)∗)|(0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='0)ω∗  s0⟩  + o(γ + |σ|) · ∥ω∥ ¯  Hs−1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='ℓ+2  b  ( ¯  X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='42)  This proves the differentiability of W10 at (0, 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  Taylor expansion of W11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Since W11 : K = ⟨˜ω⟩ → R⊥ ∼= C is given by ⟨�  W0,γ(σ)�  W0,γ1(σ)−1˜ω, ω∗  s0⟩,  by (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='42) we find  W11(γ, σ) = σ⟨∂σ �  W0,γ(σ)|(0,0)ωs0, ω∗  s0⟩ + γ⟨∂γ �  W0,γ(σ)|(0,0)ωs0, ω∗  s0⟩ + o(γ + |σ|).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='43)  Since ∂γ �  W0,γ(σ)|(0,0) = γ−1��  W0,γ(0) − �  P0(0)  �  , using the calculation of W♭  11 in (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='22) gives  ⟨∂γ �  W0,γ(σ)|(0,0)ωs0, ω∗  s0⟩ = 4π(b − 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='44)  We notice that ∂σ �  W0,γ(σ)|(0,0) = ∂σ �  P0(σ)|σ=0 = −i[P0, t0,∗], so we have  ⟨∂σ �  P0,γ(σ)|(0,0)ωs0, ω∗  s0⟩ = −i⟨[P0, t0,∗]ωs0, ω∗  s0⟩ = −i⟨[P0, v]ωs0, ω∗  s0⟩ = −2πi  (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='45)  where we use  [P0, v]ωs0 = −(Ggδ∗  gωs0)αβ(∇αv) + δgGg(∇αv ⊗s ωs0)  =  �  (− 1  2r2 + m0  r3 )dv − 1  r2 dr  �  +  � 1  2r2 dv + 1  r2 dr  �  = m0  r3 dv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  Therefore, we arrive at  W11(γ, σ) = −2πiσ + 4π(b − 1)γ + o(γ + |σ|).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='46)  Final step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Owing to the previous discussion starting at (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='28), in order to prove that �  W0,γ(σ)�  W0,γ1(σ)−1  is injective, it suffices to prove the injectivity of the following operator  W11(γ, σ) − W10(γ, σ)W00(γ, σ)−1W01(γ, σ) : K → R⊥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='47)  Combining the differentiability of W01, W10 at (0, 0) and (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='46) together yields  W11(γ, σ) − W10(γ, σ)W00(γ, σ)−1W01(γ, σ) = −2πiσ + 4π(b − 1)γ + o(γ + |σ|)  which is nonzero for γ > 0, Im σ ≥ 0 with γ + |σ| small if we set b > 1, and thus is injective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Finally,  the invertibility follows from the fact that �  W0,γ(σ)�  W0,γ1(σ)−1 has index 0 for γ + |σ| small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  □  Now we are at the position to prove Proposition 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  Proof of Proposition 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' The proof proceed in a similar fashion to that in the last step in the proof of  Theorem 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Specifically, by Lemma 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='12, there exists γ′  0 > 0, C′ > 0 such that for 0 < γ < γ′  0, Im σ ≥  0, |σ| < C′ and s > 2, ℓ ∈ (− 3  2, 1  2),  �  W0,γ(σ) : {ω ∈ ¯Hs,ℓ  b ( ¯X) : �  W0,γ(σ)u ∈ ¯Hs−1,ℓ+2  b  ( ¯X)} → ¯Hs−1,ℓ+2  b  ( ¯X)  is invertible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' By Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='7, this implies for 0 < γ < γ′  0, Im σ ≥ 0, |σ| < C′ and s > 2, ℓ < − 1  2, s+ℓ > − 1  2,  �  W0,γ(σ) : {ω ∈ ¯Hs,ℓ  b ( ¯X) : �  W0,γ(σ)u ∈ ¯Hs,ℓ+1  b  ( ¯X)} → ¯Hs,ℓ+1  b  ( ¯X)  is injective as well, and the invertibility follows from the fact that �  P0,γ(σ) has index 0 (see Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='25).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  110  LILI HE  Next, according to the high energy estimates (Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='24) and energy estimate (Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='27 and  Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='28), there exists γ′′  0 > 0, C′′ > 0 such that for 0 < γ < γ′′  0 and Im σ ≥ 0, |σ| > C′′, �  W0,γ(σ)  defined as in (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='23) is invertible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  For σ with Im σ ≥ 0, C′ ≤ |σ| ≤ C′′, the fact that �  W0,0(σ) = �  P0(σ) (see Proposition 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='3) is invertible and  a perturbation argument (as in the final step in the proof of Theorem 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='1) give the invertibility of �  W0,γ(σ) for  (γ, σ) in an open neighborhood of (0, σ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Then the compactness of the region C′ ≤ |σ| ≤ C′′ implies that there  exists an γ′′′  0 > 0 (depending on C′, C′′) such that �  W0,γ(σ) in (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='23) is invertible for Im σ ≥ 0, C′ ≤ |σ| ≤ C′′  and 0 < γ < γ′′′  0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Therefore, the proposition holds for 0 < γ < ˜γ0 where ˜γ0 = min{γ′  0, γ′′  0 , γ′′′  0 }.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  □  Finally, having Propositions 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='9 and 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='10 at our disposal, we are able to establish the following theorem  for �  Wb,γ(σ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  Theorem 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Let b0 = (m0, 0, Q0) and b > 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Fix γ ∈ (0, ˜γ0), there exists a small constant C(γ) > 0  such that for the weakly charged RN metric gb0 with |Q0| < C(γ), the following holds for  (1) If Im σ ≥ 0 and σ ̸= 0,  �  Wb0,γ(σ) : {ω ∈ ¯Hs,ℓ  b ( ¯X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' �  scT ∗ ¯X) : �  Wb0,γ(σ)ω ∈ ¯Hs,ℓ+1  b  ( ¯X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' �  scT ∗ ¯X)} → ¯Hs,ℓ+1  b  ( ¯X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' �  scT ∗ ¯X)  (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='48)  is invertible when s > 2, ℓ < − 1  2, s + ℓ > − 1  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  (2) If s > 2 and − 3  2 < ℓ < − 1  2, then stationary operator  �  Wb0,γ(0) : {ω ∈ ¯Hs,ℓ  b ( ¯X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' �  scT ∗ ¯X) : �  Wb0,γ(0)ω ∈ ¯Hs−1,ℓ+2  b  ( ¯X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' �  scT ∗ ¯X)} → ¯Hs−1,ℓ+2  b  ( ¯X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' �  scT ∗ ¯X)  (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='49)  is invertible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  Both statements also hold for the weakly charged and slowly rotating Kerr Newman metric gb with b =  (m, a, Q) near b0 = (m0, 0, Q0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' The proof is completely analogous to that of Theorem 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  □  9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Growing zero modes of the gauge potential wave operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' As discussed around (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='3) and (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='6),  the zero mode solutions to Wb,γω = 0 on extendible and supported b-Sobolev spaces are related to the pure  gauge zero mode solutions of Lgb,γ(˙g, ˙A) = 0 and dual pure gauge zero mode solutions of L∗  gb,γ(˙g, ˙A) = 0,  respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' In this section we will discuss the zero mode solutions to Wb,γω = 0 which are allowed to  grow at infinity and are asymptotic to translations and rotations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' We will see in the next section that these  growing zero modes may generate (dual) pure gauge zero modes solutions to L(∗)  gb,γ(˙g, ˙A) = 0, which have the  expected decay rate at infinity (more precisely, lie in the space ¯H∞,ℓ  b  ( ¯X) or ˙H−∞,ℓ  b  ( ¯X) with − 3  2 < ℓ < − 1  2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  Throughout this subsection, we focus on the RN metric and slowly rotating metric with sufficiently small  charge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Namely, for the parameter b0 = (m0, 0, Q0) of the RN metric, we assume that  Q0 ≪ m0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  We first discuss the growing zero modes which are asymptotic to translations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  Proposition 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' For b = (m, a, Q) near b0 = (m0, 0, Q0), we obtain  ker �  Wb,γ(0) ∩ ¯H∞,−3/2−  b  ( ¯X) = ⟨ωb,s0⟩ ⊕ {ωb,s1(S) : S ∈ S1},  ωb,s0 = ∂♭  t,  (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='50)  ker �  Wb,γ(0) ∩ ˙H−∞,−3/2−  b  ( ¯X) = ⟨ω∗  b,s0⟩ ⊕ {ω∗  b,s1(S) : S ∈ S1},  (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='51)  where ♭ denotes the musical isomorphism V ♭ := gb(V, •) and ωb,s1(S) and ω∗  b,s1(S) depend on S ∈ S1 linearly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  They satisfy  δ∗  gbωb,s1 ∈ ¯H∞,1/2−  b  ( ¯X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' S2 �  scT ∗ ¯X),  Ggbδ∗  gbω∗  b,• ∈ ˙H−∞,1/2−  b  ( ¯X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' S2 �  scT ∗ ¯X)  (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='52)  Moreover, the maps b �→ ωb,s0, ωb,s1(S) and b �→ ω∗  b,s0, ω∗  b,s1(S) can be chosen to be continuous in b with values  in the respective spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  For later use, we further determine the leading term of ωb0,s1(S) and ω∗  b0,s1(S)  ωb0,s1(S) = dub0,s1(S)+O ¯  H∞,−1/2−  b  ( ¯  X)(γ+|Q0|2),  ω∗  b0,s1(S) = du∗  b0,s1(S)+O ˙H−∞,−1/2−  b  ( ¯  X)(γ+|Q0|2) (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='53)  where ub0,s1(S) = (r − m0)S and u∗  b0,s1(S) = (r − m0)H(r − rb0)S are defined as in (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='17).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  LINEAR STABILITY OF KERR-NEWMAN FAMILY  111  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' The proof exploits a normal operator argument as in Propositions 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='7 and 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' In particular, the  normal operator of −2�  Wb,γ(0) is equal to �  □g,1(0) which is the Euclidean Laplacian ∆R3 = ρ2(ρ∂ρ(ρ∂ρ−1)+ /∆)  tensored with 4 × 4 identity matrix when working in the standard coordinate trivialization of �  scT ∗ ¯X and  annihilates the following 1-forms  dt, dx1, dx2, dx3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='54)  We note that dt is of scalar type l = 0 while dxi = Sdr + rdS with S = xi/r (i = 1, 2, 3) is of scalar type  l = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  We first prove (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='50).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Let u ∈ ker �  Wb,γ(0) ∩ ˙H−∞,−3/2−  b  ( ¯X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' By the normal operator argument as in  Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='7 ( where we shift the contour of integration through the pole 0 with the space of resonant states  given by S0), ω must take the form ω = χω0+˜ω where ω0 is in the 4-dimensional space span{dt, dx1, dx2, dx3},  ˜ω ∈ ¯H∞,−1/2−  b  ( ¯X) and χ is a smooth cutoff with χ = 0 for r ≤ 3m0, χ = 1 with r ≥ 4m0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' This proves that  the space in (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='50) is at most 4-dimensional.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Now we prove that it is indeed 4-dimensional.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' We compute for  ω0 ∈ {dt, dx1, dx2, dx3}  −�  Wb,γ(0)χω0 = −(�  Wb,γ(0) − □g,1)χω0 − [□g,1, χ]ω0 ∈ ¯H∞,3/2−  b  ( ¯  X) + ¯H∞,∞  b  ( ¯X) = ¯H∞,3/2−  b  ( ¯X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  In view of Theorem 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='13, there exists a unique ˜ω ∈ ¯H∞,−1/2−  b  ( ¯X) satisfying �  Wb,γ(0)˜ω = −�  Wb,γ(0)χω0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  Therefore, χω0+˜ω gives rise to an element in the space (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='50) which we denote by ωb,s0 if ω0 = dt and ωb,s1(S)  if ω0 = d(rS) ∈ {dx1, dx2, dx3}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' The explicit expression for ωb,s0 follows from the direction calculation  �  Wb,γ∂♭  t = δgb,γGgδ∗  g∂♭  t = 0 since ∂t is killing (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' δ∗  g∂♭  t = 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' As for the symmetric gradient, we calculate  δ∗  gbωb,s1 = δgb ˜ω + (δ∗  gb − δ∗  g)χω0 + [δ∗  g, χ]ω0 ∈ ¯H∞,1/2−  b  ( ¯X) + ¯H∞,∞  b  ( ¯X) = ¯H∞,1/2−  b  ( ¯  X)  where we use δ∗  gb − δ∗  g ∈ ρ2Diff1  b, δ∗  gb ∈ ρDiff1  b and δ∗  gω0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  Now let us turn to the continuous dependence of ωb,s1(S) on b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Suppose bj → b and ωbj,s1(S) = χω0+˜ωbj(S)  where ω0 = d(rS) ∈ {dx1, dx2, dx3} and ˜ωbj(S) is uniquely determined by bj and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Let ωb,s1(S) = χω0+˜ωb(S)  and ej,s1(S) = ωbj,s1(S) − ωb,s1(S) = ˜ωbj,s1(S) − ˜ωb,s1(S), we find  �  Wb,γ(0)ej,s1(S) = (�  Wb,γ(0) − �  Wbj,γ(0))(˜ωbj,s1(S) + χω0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  Since {˜ωbj,s1(S)} is uniformly bounded in ¯H∞,−1/2−  b  ( ¯X) and �  Wb,γ(0) − �  Wbj,γ(0) ∈ (b − bj)ρ3Diff2  b, we see  that �  Wb,γ(0)ej,s1(S) ∈ (b − bj) ¯H∞,3/2−  b  , and thus ej,s1(S) ∈ ¯H∞,−1/2−  b  ( ¯X) is of size O(b − bj).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' This proves  the continuous dependence of ωb,s1(S) on b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  The proof for the dual zero modes ω∗  b,s0 and ω∗  b,s1(S )proceeds in an analogous manner (we use Theorem  9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='8 because �  Wb,γ(0) acting on supported b-Sobolev space is identical to �  Pb,γ(0)∗).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  It remains to derive the expressions in (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='53).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Since  �  Wb0,γ(0) = δgb0 Ggb0 δ∗  gb0 + Fgb0 Ggb0 δ∗  gb0 = 1  2(dδgb0 + δgb0 d) − Ric(gb0) + Fgb0 Ggb0 δ∗  gb0 ,  we have  �  Wb0,γ(0)(ωb0,s1(S) − dub,s1(S)) = Ric(g) β  α (dub0,s1(S))β − Fgb0 Ggb0 δ∗  gb0 dub0,s1(S)  = O ¯  H∞,5/2−  b  (|Q0|2) + O ¯  H∞,∞  b  ( ¯  X)(γ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='55)  Since �  Wb0,γ(0)(ωb0,s1(S) − dub,s1(S)) is of scalar type l = 1 and �  Wb0,γ(0)−1 restricted in scalar type l =  1 1-form spaces exists with norm of size O(1) (see Remark 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='4), it follows that ωb0,s1(S) − dub,s1(S) =  O ¯  H∞,−1/2−  b  ( ¯  X)(γ + |Q0|2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  □  Next we turn to the growing zero modes asymptotic to rotations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  Proposition 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' For b = (m, a, Q) near b0 = (m0, 0, Q0), there exists continuous (in b) families  b �→ ωb,v1(V) ∈ ker �  Wb,γ(0) ∩ ¯H∞,−5/2−  b  ( ¯X),  (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='56)  b �→ ω∗  b,v1(V) ∈ ker �  Wb,γ(0) ∩ ˙H−∞,−5/2−  b  ( ¯X),  (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='57)  depending linearly on V ∈ V1, such that  δ∗  gbωb,v1(V) ∈ ¯H∞,1/2−  b  ( ¯X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' S2 �  scT ∗ ¯X),  Ggbδ∗  gbω∗  b,v1(V) ∈ ˙H−∞,1/2−  b  ( ¯X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' S2 �  scT ∗ ¯X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='58)  112  LILI HE  More explicitly, we have  ωb0,v1(V) = r2V,  ω∗  b0,v1(V) = r2H(r − rb0)V + O ˙H−∞,−1/2−  b  ( ¯  X)(γ)  (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='59)  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' We first notice that, with respect to RN metric, r2V ∈ ¯H∞,−5/2−  b  ( ¯X) is dual to the rotation vector  field (see the calculation in (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='3)), and thus killing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Namely, we have δ∗  gb0 r2V = 0 and thus �  Wb0,γ(0)r2V =  δgb0,γGgb0 δ∗  gb0 r2V = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' As for KN metric gb with b = (m, a, Q) near b0 = (m0, 0, Q0), a direct calculation  implies  −�  Wb,γ(0)χr2V = −  �  �  Wb,γ(0)−�  Wb′,γ(0)  �  χr2V−[�  Wb′,γ(0), χ]r2V ∈ ¯H∞,3/2−  b  ( ¯X)+ ¯H∞,∞  b  ( ¯X) = ¯H∞,3/2−  b  ( ¯X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  where b′ = (m, 0, Q) and we use �  Wb,γ(0)−�  Wb′,γ(0) ∈ ρ4Diff2  b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' In view of Theorem 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='13, there exists a unique  ˜ωb(V) ∈ ¯H∞,−1/2−  b  ( ¯X) satisfying �  Wb,γ(0)˜ωb(V) = −�  Wb,γ(0)χr2V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' We then define ωb,v1(V) := χr2V + ˜ωb(V).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  For b = (m, a, Q), the symmetric gradient δgbωb,v1(V) is given by  δ∗  gbωb,v1(V) = δ∗  gb ˜ωb(V) + (δ∗  gb−δ∗  g(m,0,Q))χr2V + [δ∗  g(m,0,Q), χ]r2V ∈ ¯H∞,1/2−  b  ( ¯  X) + ¯H∞,∞  b  ( ¯X) = ¯H∞,1/2−  b  ( ¯X)  where we use δ∗  gb − δ∗  g(m,0,Q) ∈ ρ3Diff1  b, δ∗  gb ∈ ρDiff1  b and δ∗  g(m,0,Q)r2V = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  The proof of the continuous dependence of ωb,v1 on b proceeds as in Proposition 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Suppose bj =  (mj,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' aj,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Qj) → b = (m,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Q) and let ωbj,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='v1(V) = χr2V + ˜ωbj(V),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' ωb,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='v1(V) = χr2V + ˜ωb(V) and ej(V) =  ωbj,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='v1(V) − ωb,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='v1(V) = ˜ωbj(V) − ˜ωb(V),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' we find  �  Wb,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='γ(0)ej(V) = (�  Wb,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='γ(0) − �  Wbj,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='γ(0))(˜ωbj(V) + χr2V)  = |b − bj|ρ3Diff2  b ˜ωbj(V) +  ���  Wb,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='γ(0) − �  Wb′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='γ(0)  �  −  ��  Wbj,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='γ(0) − �  Wb′  j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='γ(0)  ��  χr2V  +  �  �  [Wb′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='γ(0) − �  Wb′  j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='γ(0),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' χ]r2V  �  = |b − bj|ρ3Diff2  b ˜ωbj(V) + |b − bj|ρ4Diff2  bχr2V + |b − bj|ρ∞Diff1  br2V  ∈ |b − bj|  �  ¯H∞,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='5/2−  b  ( ¯X) + ¯H∞,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='3/2−  b  ( ¯  X) + ¯H∞,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='∞  b  ( ¯X)  �  ⊂ |b − bj| ¯H∞,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='3/2−  b  ( ¯X)  where b′  j = (mj,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Qj) and b′ = (m,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' As a consequence, ej(V) ∈ ¯H∞,−1/2−  b  ( ¯X) is of size O(b − bj).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  This proves the continuous dependence of ωb,v1(V) on b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  The statements for dual zero modes ω∗  b,v1(V) can be proved in a similar way (we use Theorem 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='8 because  �  Wb,γ(0) acting on supported b-Sobolev space is identical to �  Pb,γ(0)∗).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Here, we present the detail of the  derivation of the expression for ω∗  b0,v1(V) in (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='59).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' First, we calculate  δgb0 Ggb0 δ∗  gb0 r2H(r − rb0)V = δgb0  �  (r2δ(r − rb0)dr) ⊗s V  �  = − 1  2r2 ∂r(r4δ(r − rb0)µb0)V = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  It follows that  �  Wb0,γ(0)(ω∗  b0,v1(V) − r2H(r − rb0)V) = −Fgb0  �  (r2δ(r − rb0)dr) ⊗s V  �  = O ¯  H−∞,∞  b  ( ¯  X)(γ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  Since �  Wb0,γ(0)(ωb0,v1(V) − r2H(r − rb0)V) is of vector type l = 1 and �  Wb0,γ(0)−1 restricted in vector type  l = 1 1-form spaces exists with norm of size O(1) (see Remark 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='4), we conclude that  �  Wb0,γ(0)(ωb0,v1(V) − r2H(r − rb0)V) = O ¯  H−∞,−1/2−  b  ( ¯  X)(γ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  □  9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Generalized zero modes of the gauge potential wave operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Apart from the growing modes,  we also need to take the generalized zero modes into consideration (the zero modes which allow for polynomial  growth in tb,∗).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Before the discussion about the generalized zero modes, we establish the following technical  lemma about another family of growing zero modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  Lemma 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' For b = (m, a, Q) near b0 = (m0, 0, Q0), there exists continuous (in b) families  b �→ ω(1)  b,s1(S) ∈ ker �  Wb,γ(0) ∩ ¯H∞,−5/2−  b  ( ¯X),  (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='60)  b �→ ω(1)∗  b,s1 (S) ∈ ker �  Wb,γ(0) ∩ ˙H−∞,−5/2−  b  ( ¯X),  (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='61)  LINEAR STABILITY OF KERR-NEWMAN FAMILY  113  depending linearly on S ∈ S1, such that  δ∗  gbω(1)  b,s1(S) = χdt ⊗s d(rS) + ¯H∞,−1/2−  b  ( ¯  X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' S2 �  scT ∗ ¯X)  δ∗  gbω(1)∗  b,s1 (S) = χdt ⊗s d(rS) + ˙H∞,−1/2−  b  ( ¯  X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' S2 �  scT ∗ ¯X)  (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='62)  Moreover, the leading terms of ω(1)  b,s1(S), ω(1)∗  b,s1 (S)are given by  ω(1)  b,s1(S) − χrSdt ∈ ¯H∞,−3/2−  b  ( ¯X),  ω(1)∗  b,s1 (S) − χrSdt ∈ ˙H−∞,−3/2−  b  ( ¯  X)  (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='63)  where χ is a smooth cutoff satisfying χ = 1 for r ≥ 4m0 and χ = 0 for r ≤ 3m0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  For later use, we further determine the leading term of ω(1)  b0,s1(S)  ω(1)  b0,s1(S) = rS(µb0dt0 − dr) + O ¯  H∞,−1/2−  b  ( ¯  X)(γ + |Q0|2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='64)  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' The proof uses the fact that the normal operator of −2�  Wb,γ(0) is equal to �  □g,1(0), which is the  Euclidean Laplacian ∆R3 = ρ2(ρ∂ρ(ρ∂ρ − 1) + /∆) tensored with 4 × 4 identity matrix when working in the  standard coordinate trivialization of �  scT ∗ ¯X and annihilates the following 1-form xidt = rSdt (i = 1, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='3)  where S = xi/r ∈ S1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Concretely, we have  −�  Wb,γ(0)χrSdt = −  �  �  Wb,γ(0) − □g,1  �  χrSdt − [□g,1, χ]rSdt ∈ ¯H∞,1/2−  b  ( ¯X) + ¯H∞,∞  b  ( ¯X) = ¯H∞,1/2−  b  ( ¯X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  Since ker �  Wb,γ(0)∗ ∩ ˙H−∞,−1/2+  b  ( ¯X) = {0} by Theorem 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='13, the above equation is solvable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' More pre-  cisely, there exists a unique ˜ω(1)  b (S) ∈ ¯H∞,−3/2−  b  ( ¯X) which is in ann{f1, · · · , f4}, where {f1, · · · , f4} ⊂  C∞  c (X◦) is a set of linearly independent functionals on �  Wb0,γ(0)∩ ¯H∞,−3/2−  b  ( ¯X), such that �  Wb,γ(0)˜ω(1)  b (S) =  −�  Wb,γ(0)χrSdt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' We then define ω(1)  b,s1(S) := χrSdt + ˜ω(1)  b (S) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' As for the symmetric gradient δgbω(1)  b,s1(S), we  find  δ∗  gbω(1)  b,s1(S) = χδ∗  grSdt + [δ∗  g, χ]rSdt + (δ∗  gb−δ∗  g)χrSdt + δ∗  gb ˜ω(1)  b (S)  = χdt ⊗s d(rS) + ¯H∞,∞  b  ( ¯  X) + ¯H∞,−1/2−  b  ( ¯X) + ¯H∞,−1/2−  b  ( ¯X)  = χdt ⊗s d(rS) + ¯H∞,−1/2−  b  ( ¯X)  where we use δ∗  gb − δ∗  g ∈ ρ2Diff1  b and δ∗  gb ∈ ρDiff1  b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  As for the continuous dependence of ω(1)  b,s1(S) on b, suppose bj → b and let ω(1)  bj,s1(S) = χrSdt + ˜ω(1)  bj (S),  ω(1)  b,s1(S) = χrSdt + ˜ω(1)  b (S) and ej(S) = ω(1)  bj,s1(S) − ω(1)  b,s1(S) = ˜ω(1)  bj (S) − ˜ω(1)  b (S), we find  �  Wb,γ(0)ej(S) = (�  Wb,γ(0) − �  Wbj,γ(0))(˜ω(1)  bj (S) + χrSdt) ∈ |b − bj| ¯H∞,1/2−  b  ( ¯X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  As �  Wb,γ(0)|ann{f1,··· ,f4} : ann{f1, · · · , f4} →  ¯H∞,1/2−  b  ( ¯  X) is invertible, it follows that ej(S) is of size  O ¯  H∞,−3/2−  b  ( ¯  X)(b − bj).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' This completes the proof of the continuity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  The proofs for dual zero modes ˆω(1)∗  b,s1 (S) are analogous where we use  ker �  Wb,γ(0)∗ ∩ ¯H∞,−1/2+  b  ( ¯X) = ker �  Pb,γ(0) ∩ ¯H∞,−1/2+  b  ( ¯X) = {0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  It remains to derive the expressions in (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='64).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Since  δgb0 Ggb0 δgb0  �  rS(µb0dt0 − dr)  �  = Q2  0  r3 S(µb0dt0 − dr) ∈ Q2  0 ¯H∞,3/2−  b  ( ¯X),  we have  �  Wb0,γ(0)  �  ω(1)  b0,s1(S) − rS(µb0dt0 − dr)  �  = O ¯  H∞,3/2−  b  (|Q0|2) + O ¯  H∞,∞  b  ( ¯  X)(γ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  Since the right-hand side of the above equation is of scalar type l = 1 and �  Wb0,γ(0)−1 restricted in scalar type  l = 1 1-form spaces exists with norm of size O(1) (see Remark 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='4), it follows that ω(1)  b0,s1(S)−rS(µb0dt0−dr) =  O ¯  H∞,−1/2−  b  ( ¯  X)(γ + |Q0|2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  □  114  LILI HE  Now we are ready to discuss (dual) generalized zero modes which have less decay rate in r at infinity  for fixed tb,∗ and grow linearly in tb,∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' In the course of the proof, we will find that they are asymptotic to  Lorentz boosts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  Proposition 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' For b = (m, a, Q) near b0 = (m0, 0, Q0), there exists continuous (in b) families  b �→ ˆωb,s1(S) ∈ ker Wb,γ ∩ Poly1(tb,∗) ¯H∞,−5/2−  b  ( ¯X),  (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='65)  b �→ ˆω∗  b,s1(S) ∈ ker Wb,γ ∩ Poly1(tb,∗) ˙H−∞,−5/2−  b  ( ¯X),  (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='66)  depending linearly on S ∈ S1, such that  δ∗  gb ˆωb,s1(S) ∈ Poly1(tb,∗) ¯H∞,−1/2−  b  ( ¯X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' S2 �  scT ∗ ¯X),  (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='67)  Ggbδ∗  gb ˆω∗  b,s1(S) ∈ Poly1(tb,∗) ˙H−∞,−1/2−  b  ( ¯X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' S2 �  scT ∗ ¯X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='68)  Moreover, the leading terms of ˆωb,s1(S) are given by  ˆω(1)  b,s1(S) − (tdxi − xidt) ∈ Poly1(tb,∗) ¯H∞,−3/2−  b  ( ¯X),  r ≫ 1  (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='69)  where S = xi/r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  For later use, we further determine the leading term of ˆωb0,s1(S)  ˆωb0,s1(S) = (t0 − r)ωb0,s1(S) − rS(µb0dt0 − dr) + m0S(dt0 + dr)  + d  �  2m0  �  2m0 + (m0 − r) log( r  m0  )  �  S  �  + O ¯  H∞,−1/2−  b  ( ¯  X)(γ + |Q0|2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='70)  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' We make the ansatz  ˆωb,s1(S) = tχ0ωb,s1(S) + ˇωb,s1(S)  (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='71)  with ˇωb,s1(S) ∈ ¯H∞,−5/2−  b  ( ¯X) to be determined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Our aim is to find  ˆωb,s1 ∈ ker Wb,γ ∩Poly1(tb,∗) ¯H∞,5/2−  b  ( ¯X)  such that δ∗  gb ˆωb,s1(S) ∈ Poly1(tb,∗) ¯H∞,−1/2−  b  ( ¯X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' To this end, since  tχ0 − tb∗ ∈ A−1( ¯X)  and  δ∗  gbωb,s1 ∈ ¯H∞,1/2−  b  ( ¯  X),  it suffices to solve the following two equations for ˇωb,s1(S) ∈ ¯H∞,−5/2−  b  ( ¯X)  [δ∗  gb, tχ0]ωb,s1(S) + δ∗  gb ˇωb,s1(S) ∈ ¯H∞,−1/2−  b  ( ¯  X),  (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='72)  �  Wb,γ(0)ˇωb,s1(S) = −[Wb,γ, tχ0]ωb,s1(S).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='73)  We first analyze (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='72).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Since  [δ∗  gb, tχ0]ωb,s1(S) = dtχ0 ⊗s ωb,s1(S),  we further make the ansatz  ˇωb,s1(S) = −ω(1)  b,s1(S) + ˜ωb,s1(S) ∈ ¯H∞,−5/2−  b  ( ¯X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='74)  with ˜ωb,s1(S) ∈ ¯H∞,−3/2−  b  ( ¯X) to be determined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Therefore, using ωb,s1(S) = χd(rS) + ¯H∞,−1/2−  b  ( ¯X) and  (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='62), it follows that  [δ∗  gb, tχ0]ωb,s1(S) + δ∗  gb ˇωb,s1 ∈ ¯H∞,−1/2−  b  ( ¯X)  and the requirement (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='72) is satisfied with the choice of ˇωb,s1(S) defined as in (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='74).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  Next we solve (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='73) to determine ˜ωb,s1(S) in (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='74).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' We rewrite (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='73) as  �  Wb,γ(0)˜ωb,s1(S) = �  Wb,γ(0)ω(1)  b,s1(S) − [Wb,γ, tχ0]ωb,s1(S)  = ˜δgb,γGgb  �  δ∗  gbω(1)  b,s1(S) − [δ∗  gb, tχ0]ωb,s1(S)  �  − [˜δgb,γ, tχ0]Ggbδ∗  gbωb,s1(S) ∈ ¯H∞,1/2−  b  ( ¯X)  (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='75)  where we use ˜δgb,γ ∈ ρDiff1  b, δ∗  gbω(1)  b,s1(S) − [δ∗  gb, t|chi0]ωb,s1(S) ∈ ¯H∞,−1/2−  b  ( ¯X) and  δ∗  gbωb,s1(S) ∈ ¯H∞,1/2−  b  ( ¯X),  [˜δgb,γ, tχ0] ∈ A0( ¯X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  LINEAR STABILITY OF KERR-NEWMAN FAMILY  115  Since ker �  Wb,γ(0)∗∩ ˙H−∞,−1/2+  b  ( ¯X) = {0}, we can solve the above equation (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='75) for ˜ωb,s1(S) ∈ ¯H∞,−3/2−  b  ( ¯X),  which is unique modulo ker �  Wb,γ(0) ∩ ¯H∞,−3/2−  b  ( ¯X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' We then define  ˆω(1)  b,s1(S) := tχ0ωb,s1(S) − ω(1)  b,s1(S) + ˜ωb,s1(S)  where ˜ωb,s1(S) is in the orthogonal complement  �  ker �  Wb,γ(0)∩ ¯H∞,−3/2−  b  ( ¯X)  �⊥ and thus uniquely determined  by S and b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' As a result, we have for r ≫ 1  ˆωb,s1(S) = tωb,s1(S) − χrSdt + ¯H∞,−3/2  b  ( ¯X) = td(rS) − rSdt + Poly1(tb,∗) ¯H∞,−3/2−  b  ( ¯X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  Finally, the continuity in b follows in the same way as in the proof of Lemma 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  The proof for ˆω∗  b,s1(S) is completely analogous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  It remains to derive the expressions in (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='70).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' For the simplicity of calculation, now we let ˜t = t0 − r and  make the ansatz  ˆωb0,s1(S) = ˜tωb0,s1(S) − ω(1)  b0,s1 + ˜ωb0,s1(S).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  Our aim is to solve the following equation for ˜ωb0,s1(S) ∈ ¯H−3/2−  b  ( ¯X)  �  Wb0,γ(0)˜ωb0,s1(S) = −[Wb0,γ, ˜t ]ωb0,s1(S) = 1  2[□gb0 , ˜t ]ωb0,s1(S) + O ¯  H∞,∞  b  ( ¯  X)(γ)  = S  �  (m0  r2 − Q2  0  r3 )dt0 + Q2  0  r3 dr  �  + m0  r (1 + m0  r  − Q2  0  r2 )/dS + O ¯  H∞,3/2−  b  ( ¯  X)(γ + |Q0|2)  (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='76)  where we use [□gb0 , ˜t ] = □gb0 ,0˜t+ 2∇α˜t∇α ∈ ρ2Diff1  b and (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='53) in the last step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' A direct calculation implies  �  Wb0,γ(0)m0(dt0 + dr) = δgb0 Ggb0 δ∗  gb0 m0(dt0 + dr) + O ¯  H∞,∞  b  ( ¯  X)(γ)  = S  �  (m0  r2 + m0Q2  0  r4  )dt0+(3m0  r2  − 2m2  0  r3 + m0Q2  0  r4  )dr  �  + m0  r (−2+ 2m0  r  − 2Q2  0  r2 )/dS +O ¯  H∞,∞  b  ( ¯  X)(γ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  Then  �  Wb0,γ(0)  �  ˜ωb0,s1(S) − m0(dt0 + dr)  �  = S(−3m0  r2  + 2m2  0  r3 )dr + m0  r (3 − m0  r )/dS + O ¯  H∞,3/2−  b  ( ¯  X)(γ + |Q0|2)  = d  �  (3m0  r  − m2  0  r2 )S  �  + O ¯  H∞,3/2−  b  ( ¯  X)(γ + |Q0|2)  Meanwhile, with u = 2m0(2m0 + (m0 − r) log(r/m0)), we have  �  Wb0,γ(0)du = 1  2dδgb0 du − Ric(g0) β  α (du)β + Fgb0 Ggb0 du  = d  �  (3m0  r  − m2  0  r2 )S  �  + O ¯  H∞,5/2−  b  ( ¯  X)(γ + |Q0|2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  Therefore, we obtain  �  Wb0,γ(0)  �  ˜ωb0,s1(S) − m0(dt0 + dr) − du  �  = O ¯  H∞,3/2−  b  ( ¯  X)(|Q0|2 + γ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  Since the right-hand side of the above equation is of scalar type l = 1 and �  Wb0,γ(0)−1 restricted in scalar type  l = 1 1-form spaces exists with norm of size O(1) (see Remark 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='4), it follows that ˜ωb0,s1(S)−m0(dt0 +dr)−  du = O ¯  H∞,−1/2−  b  ( ¯  X)(γ + |Q0|2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Therefore, (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='70) follows from the combination of this fact and (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='64).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  □  10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Modes of the linearized gauge-fixed Einstein-Maxwell system  In this section, we will discuss the modes of the linearized gauge-fixed Einstein-Maxwell system  L(gb,Ab),γ(˙g, ˙A) = (2LE  (gb,Ab),γ(˙g, ˙A), LM  (gb,Ab),γ(˙g, ˙A)) = 0  (10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='1)  where  LE  (gb,Ab),γ(˙g, ˙A) = DgbRic(˙g) − 2D(gb,dAb)T (˙g, d ˙A) + �δ∗  gb,γ�δgb,γGgb ˙g = 0,  (10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='2)  LM  (gb,Ab),γ(˙g, ˙A) = D(gb,Ab)(δgdA)(˙g, ˙A) + dδgb ˙A = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  (10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='3)  116  LILI HE  on the RN metric and 4-electromagnetic potential, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=', the case b = b0 = (m0, 0, Q0), |Q0| ≪ m0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  For the brevity of notation, we rewrite the Lie derivative of a 1-form ω with respect to the vector field X  as  �LωX := LXω  and put  Lb,γ := L(gb,Ab),γ,  LE  b,γ := LE  (gb,Ab),γ,  LM  b,γ := LM  (gb,Ab),γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  (10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='4)  For the various spaces in this section, for instance, ¯Hs,ℓ  b ( ¯X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' E), ˙Hs,ℓ  b ( ¯X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' E), A( ¯  X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' E), etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=', where E → ¯X is  a vector bundle over ¯X, we will drop the notation E of the bundle if it it clear from the context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  We shall prove that Lb0,γ has no non-zero modes in the closed upper half plane for RN metric and 4-  electromagnetic potential (gb0, Ab0) (we note that the same result for slowly rotating KN is nontrivial and will  be discussed in detail in next section).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Moreover, we will describe the space of zero modes and generalized  zero modes (with linearly growth in tb,∗) of Lb,γ for both RN case and slowly rotating KN case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Modes in the closed upper half plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Let (gb, Ab) be the KN metric and 4-electromagnetic  potential with b = (m, a, Q) near b0 = (m0, 0, Q0) and the linearized gauge-fixed Einstein-Maxwell operator  Lb,γ : ¯Hs,ℓ  b ( ¯X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' S2 �  scT ∗ ¯X ⊕ �  scT ∗ ¯X) → ¯Hs,ℓ  b ( ¯X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' S2 �  scT ∗ ¯X ⊕ �  scT ∗ ¯X) be defined as in (10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='4), (10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='1), (10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='2)  and (10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' In the definition of ˜δ∗  gb,γ, ˜δgb,γ(see (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='62), (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='60) and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='61)), c is defined as in (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='7) and γ ∈  (0, min{γ0, ˜γ0}) where γ0, ˜γ0 are as in Theorem 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='8 and 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='13 respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' We also note that we consider the  weakly charged case, namely, |Q0| ≪ m0 (more precisely, |Q0| ≤ C(γ) for some small constant C(γ)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' As in  the previous sections, we define the Fourier-transformed operator  �  Lb,γ(σ) := eiσtb,∗Lb,γe−iσtb,∗  (10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='5)  using the function tb,∗ = χ0(r)(t + rb0,∗) + (1 − χ0(r))(t − r(m,0,Q),∗) defined in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='29).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  Theorem 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Let (gb0, Ab0) be the RN metric and 4-electromagnetic potential where b0 = (m0, 0, Q0)  with |Q0| ≪ m0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Then the Fourier-transformed operator �  Lb0,γ(σ) satisfies the following statements:  (1) If Im σ ≥ 0 and σ ̸= 0,  �  Lb0,γ(σ) : {(˙g, ˙A) ∈ ¯Hs,ℓ  b ( ¯X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' S2 �  scT ∗ ¯X ⊕ �  scT ∗ ¯X) : �  Lb0,γ(σ)(˙g, ˙A) ∈ ¯Hs,ℓ+1  b  ( ¯X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' S2 �  scT ∗ ¯X ⊕ �  scT ∗ ¯X)}  → ¯Hs,ℓ+1  b  ( ¯X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' S2 �  scT ∗ ¯X ⊕ �  scT ∗ ¯X)  (10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='6)  is invertible for s > 3, ℓ < − 1  2 and s + ℓ > − 1  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  (2) If s > 3 and − 3  2 < ℓ < − 1  2, then stationary operator  �  Lb0,γ(0) : {(˙g, ˙A) ∈ ¯Hs,ℓ  b ( ¯X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' S2 �  scT ∗ ¯X ⊕ �  scT ∗ ¯X) : �  Lb0,γ(0)(˙g, ˙A) ∈ ¯Hs−1,ℓ+2  b  ( ¯X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' S2 �  scT ∗ ¯X ⊕ �  scT ∗ ¯X)}  → ¯Hs−1,ℓ+2  b  ( ¯X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' S2 �  scT ∗ ¯X ⊕ �  scT ∗ ¯X)  (10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='7)  has 8-dimensional kernel and cokernel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  The second statement also holds for the weakly charged and slowly rotating KN metric and 4-electromagnetic  potential (gb, Ab) with b = (m, a, Q) near b0 = (m0, 0, Q0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Concretely, we have  Kb := ker �  Lb,γ(0) ∩ ¯H∞,−1/2−  b  ( ¯X) = {(˙gb,s0, ˙Ab,s0)} ⊕ {(˙gb,s1(S), ˙Ab,s1(S)) : S ∈ S1}  (10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='8)  ⊕ {(˙gb,v1(V), ˙Ab,v1(V)) : V ∈ V1},  K∗  b := ker �  Lb,γ(0)∗ ∩ ˙H−∞,−1/2−  b  ( ¯X) = {(˙g∗  b,s0, ˙A∗  b,s0)} ⊕ {(˙g∗  b,s1(S), ˙A∗  b,s1(S)) : S ∈ S1}  (10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='9)  ⊕ {(˙g∗  b,v1(V), ˙A∗  b,v1(V)) : V ∈ V1},  where  (˙gb,s0, ˙Ab,s0) = (˙gb( ˙m, 0, ˙Q) + 2δgb ˜ωb,s0,  ˙Ab(0, 0, ˙Q) + �LAb ˜ω♯  b,s0 + dφb,s0),  ˙m, ˙Q ∈ R,  (10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='10)  (˙gb,s1(S), ˙Ab,s1(S)) = (2δ∗  gbωb,s1(S), �LAb(ωb,s1(S))♯ + dφb,s1(S)),  (10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='11)  (˙gb,v1(V), ˙A∗  b,v1(V)) = (˙gb(0, ˙a, 0) + 2δ∗  gb ˜ωb,v1(V),  ˙Ab(0, ˙a, 0) + �LAb(˜ωb,v1(V))♯ + dφb,v1(V)),  (10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='12)  LINEAR STABILITY OF KERR-NEWMAN FAMILY  117  and  (˙g∗  b,s0, ˙A∗  b,s0) = c1(2Ggbδ∗  gbω∗  b,s0, 4 �LAb(ω∗  b,s0)♯ + dφ∗  b,s0) + c2(0, δ(r − rb)dr),  c1, c2 ∈ R,  (10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='13)  (˙g∗  b,s1(S), ˙A∗  b,s1(S)) = (2Ggbδ∗  gbω∗  b,s1(S), 4 �LAb(ω∗  b,s1(S))♯ + dφ∗  b,s1(S)),  (10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='14)  (˙g∗  b,v1(V), ˙A∗  b,v1(V)) = (2Ggbδ∗  gbω∗  b,v1(V), 4 �LAb(ω∗  b,v1(V))♯ + dφ∗  b,v1(V))  (10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='15)  with ωb,s1(S), ω∗  b,s0 and ω∗  b,s1(S), ω∗  b,v1(V) defined as in (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='50), (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='51) and (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='57).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  Moreover, we have  (˙gb,s1(S),  ˙Ab,s1(S)),  (˙gb,v1(V),  ˙Ab,v1(V)) ∈ ¯H∞,1/2−  b  ( ¯X),  and the maps b �→ (˙gb,•(•),  ˙Ab,•(•)) and b �→ (˙g∗  b,•(•),  ˙A∗  b,•(•)) can be chosen to be continuous in b with  values in the respective spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  For later use, we further determine the leading term of (˙gb0,v1(V),  ˙Ab0,v1(V))  ˙gb0,v1(V) =  �  (−4m0  r  + 2Q2  0  r2 )dt0 +  4m0  r − (m0 −  �  m2  0 − Q2  0)  dr  �  ⊗s V + O ¯  H∞,1/2−  b  ( ¯  X)(γ),  ˙Ab0,v1(V) = O ¯  H∞,1/2−  b  ( ¯  X)(|Q0|).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  (10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='16)  Remark 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' In the course of the proof of the above theorem, we will find that the zero modes in Kb satisfy  both the linearized Einstein-Maxwell system and the linearized generalized wave map gauge and Lorenz  gauge conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  Proof of Theorem 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' We first analyze the RN case b = b0 and write (g, A) = (gb0, Ab0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  Suppose  �  Lb0,γ(σ)(˙g, ˙A) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Then by Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='7, (˙g, ˙A) ∈ ¯H∞,−1/2−  b  ( ¯X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' In view of (10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='3), applying δg to  LM  b0,γe−iσtb0,∗(˙g, ˙A) = 0 yields  δgd(δge−iσtb0,∗ ˙A) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  Since ˙A ∈ ¯H∞,−1/2−  b  ( ¯X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' �  scT ∗ ¯X), we obtain that δg(e−iσtb0,∗ ˙A) is a mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' More precisely, we have  δg(e−iσtb0,∗ ˙A) ∈  �  e−iσtb0,∗ ¯H∞,−1/2−  b  ( ¯X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' C)  if  σ ̸= 0,  ¯H∞,1/2−  b  ( ¯  X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' C)  if  σ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  By Theorem 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='1, it follows that  δg(e−iσtb0,∗ ˙A) = 0,  (10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='17)  and thus e−iσtb0,∗(˙g, ˙A) is a mode solution to the Maxwell part of the linearized Einstein-Maxwell system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  Applying δgGg to LE  b0,γe−iσtb0,∗(˙g, ˙A) = 0 and using the linearized second Bianchi identity (see the discussion  around (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='13) and (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='14)) give  δgGg˜δ∗  g,γ(˜δg,γGge−iσtb0,∗ ˙g) = Pb0,γ(˜δg,γGge−iσtb0,∗ ˙g) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  Again, since ˜δg,γGge−iσtb0,∗ ˙g is a mode in the same way as δg(e−iσtb0,∗ ˙A), it follows from Theorem 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='8 that  ˜δg,γGge−iσtb0,∗ ˙g = 0,  (10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='18)  and thus e−iσtb0,∗(˙g, ˙A) is a mode solution to the linearized Einstein-Maxwell system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  The case Im σ ≥ 0, σ ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Then we apply the mode stability result Theorem 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='1 and find that  e−iσtb0,∗(˙g, ˙A) = (2δ∗  gω, Lω♯A + dφ)  (10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='19)  where (ω, φ) = e−iσtb0,∗(ω0, φ0) ∈ e−iσtb0,∗ ¯H∞,ℓ′  b  ( ¯X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' �  scT ∗ ¯X ⊕ C) with ℓ′ < −1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Plugging (10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='19)  into the equation (10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='18), we find  �  Wb0,γ(σ)ω0 = 0,  and thus ω0 = 0 by Theorem 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' We then plug (10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='19) with ω = 0 into the equation (10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='17) and  obtain  �  □g(σ)φ0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  Therefore, we have φ0 = 0 by Theorem 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' This proves the injectivity of �  Lb0,γ(σ) for Im σ ≥ 0, σ ̸= 0,  and the surjectivity follows from the fact that �  Lb0,γ(σ) is Fredholm of index 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  118  LILI HE  Scalar type l ≥ 2 or vector type l ≥ 2 zero modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' We apply the mode stability result Theorem 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='1  and find that  (˙g, ˙A) = (2δ∗  gω, Lω♯A + dφ)  (10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='20)  where (ω, φ) ∈ ¯H∞,−3/2−  b  ( ¯X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' �  scT ∗ ¯X ⊕ C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Plugging (10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='20) into the equation (10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='18), we find  �  Wb0,γ(0)ω = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  Since ω ∈ ¯H∞,−3/2−  b  ( ¯X) is of scalar or vector type l ≥ 2, it follows that ω = 0 by Theorem 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='13 and  Proposition 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' We then plug (10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='20) with ω = 0 into the equation (10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='17) and obtain  �  □g(0)φ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  Again, since φ ∈ ¯H∞,−3/2−  b  ( ¯X) with vector or scalar type l ≥ 2, we conclude that φ = 0 by Theorem  8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='1 and Proposition 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' This proves the injectivity of �  Lb0,γ(0) when restricted on scalar or vector  type l ≥ 2 modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  Scalar type l = 1 zero modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' As in the previous case (scalar or vector l ≥ 2 case), we find that  (˙g, ˙A) = (2δ∗  gω, Lω♯A + dφ) where (ω, φ) ∈ ¯H∞,−3/2−  b  ( ¯X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' �  scT ∗ ¯X ⊕ C) and satisfies first  �  Wb0,γ(0)ω = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  Since ω ∈ ¯H∞,−3/2−  b  ( ¯  X) is of scalar type l = 1, in view of Proposition 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='14 we see that  ω = ωb0,s1(S)  for some  S ∈ S1  (10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='21)  where ωb0,s1(S) is defined as in (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='50).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' We then plug ˙A = Lω♯A + dφ and (10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='21) into the equation  (10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='17) and obtain  −�  □g(0)φ = −δg �LA(ωb0,s1(S))♯ ∈ Q0 ¯H∞,3/2−  b  ( ¯X)  where we use δg ∈ ρDiff1  b and L(·)♯A ∈ ρ2Diff1  b (see (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='24)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Owing to Theorem 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='1 and the fact  that ker �  □g(0) ∩ ¯H∞,−3/2−  b  ( ¯X) restricted to scalar type l = 1 spaces is 0, there exists a unique  φ ∈ ¯H∞,−1/2−  b  ( ¯X), which we denote by φb0,s1(S), such that the above equation holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Therefore,  the scalar type l = 1 nullspace of �  Lb0,γ(0) is 3-dimensional.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  We also note that φb0,s1(S) is of  size O ¯  H∞,−1/2−  b  ( ¯  X)(|Q0|) and thus  ˙Ab0,s1(S) = O ¯  H∞,1/2−  b  ( ¯  X)(|Q0|).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' In view of (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='52), we see that  ˙gb0,s1(S) = 2δ∗  gωb0,s1(S) ∈ ¯H∞,1/2−  b  ( ¯X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' S2 �  scT ∗ ¯X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  Scalar type l = 0 zero modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' If (˙g, ˙A) is of scalar type l = 0, by Theorem 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='1 we have  (˙g, ˙A) = (˙gb0( ˙m, 0, ˙Q) + 2δ∗  gω,  ˙Ab0(0, 0, ˙Q) + Lω♯A + dφ)  (10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='22)  where (ω, φ) ∈ ¯H∞,−3/2−  b  ( ¯X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' �  scT ∗ ¯X ⊕ C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Plugging (10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='22) into (10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='18), it follows that  2�  Wb0,γ(0)ω = −˜δg,γGg ˙gb0( ˙m, 0, ˙Q) ∈ ¯H∞,1/2−  b  ( ¯X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  In view of Theorem 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='13, ker �  Wb0,γ(0)∗ ∩ ˙H−∞,−1/2+  b  ( ¯X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' �  scT ∗ ¯X) = {0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Therefore, we can solve the  above equation for ω ∈ ¯H∞,−3/2−  b  ( ¯  X) which is of scalar type l = 0 and unique modulo ⟨∂♭  t⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' However,  since δ∗  g∂♭  t = 0 and L∂tA = 0, a multiple of ∂♭  t does not change ˙g and the equation for φ below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' This  implies ω is unique and we denote it by ˜ωb0,s0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Then we plug ˙A = ˙Ab0(0, 0, ˙Q) + Lω♯A + dφ and  ω = ˜ωb0,s0 into the equation (10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='17) and obtain  −�  □g(0)φ = −δg ˙Ab0(0, 0, ˙Q) − δgL˜ω♯  b0,s0 A ∈ ¯H∞,1/2−  b  ( ¯X)  By Theorem 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='1, we have ker �  □g(0)∗ ∩ ˙H−∞,−1/2+  b  ( ¯X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' C) = {0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Therefore, we can solve the above  equation for φ ∈ ¯H∞,−3/2−  b  ( ¯X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' C) which is of scalar type l = 0 and unique modulo ⟨1⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Again,  adding a constant to φ make no change to ˙A, which means that φ is unique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' We then denote this  unique φ by φb0,s0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' This proves that the scalar type l = 0 kernel of �  Lb0,γ(0) is 2-dimensional.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  LINEAR STABILITY OF KERR-NEWMAN FAMILY  119  Vector type l = 1 zero modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' If (˙g, ˙A) is of vector type l = 1, by Theorem 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='1 we have  (˙g, ˙A) = (˙gb0(0, ˙a, 0) + 2δ∗  gω,  ˙Ab0(0, ˙a, 0) + Lω♯A + dφ)  (10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='23)  where (ω, φ) ∈ ¯H∞,−3/2−  b  ( ¯X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' �  scT ∗ ¯X ⊕ C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Plugging (10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='23) into (10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='18), it follows that  2�  Wb0,γ(0)ω = −˜δg,γGg ˙gb0(0, ˙a, 0) ∈ ¯H∞,3/2−  b  ( ¯X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  In view of Theorem 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='13, there exists a unique ω ∈ ¯H∞,−1/2−  b  ( ¯X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' �  scT ∗ ¯X) of vector type l = 1  satisfying the above equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  We then denote this unique ω by ˜ωb0,v1(V).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  Therefore, we have  ˙gb0,v(V) ∈ ¯H∞,1/2−  b  ( ¯X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Plugging ˙A = ˙Ab0(0, ˙a, 0) + Lω♯A + dφ and ω = ˜ωb0,v1 into the equation  (10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='17), we obtain that  −�  □g(0)φ = −δg ˙Ab0(0, ˙a, 0) − δg �LA(˜ωb0,v1(V))♯ ∈ Q0 ¯H∞,3/2−  b  ( ¯X)  By Theorem 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='1, there exists a unique φ ∈ ¯H∞,−1/2−  b  of vector type l = 1 satisfying the above  equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  We then denote this unique φ by φb0,v1(V).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  Therefore, the vector l = 1 kernel of  �  Lb0,γ(0) is 3-dimensional.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' We also note that φb0,v1 is of size O ¯  H∞,−1/2−  b  ( ¯  X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='C)(|Q0|), and thus ˙Ab0,v1 =  O ¯  H∞,1/2−  b  ( ¯  X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='C)(|Q0|).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  Now we shall derive the expression for (˙gb0,v1(V), ˙Ab0,v1(V)) in (10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='16).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' We write  ˙gb0,v1(V) = ˙g0  b0(0, ˙a, 0) + 2δ∗  gω0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  When rescaling to |˙a| = 1 and letting (θ, ϕ) be the spherical coordinates adapted to ˙a, we can rewrite  ˙gb0,v1(V) = (2(µb0 − 1)dt0 − 2dr) ⊗s V + 2δ∗  gω0  with  V = sin2 θdϕ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  Plugging ˙gb0,v1(V) into (10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='18) yields  2�  Wb0,γ(0)ω0 = −˜δg,γGg  �  (2(µb0 − 1)dt0 − 2dr) ⊗s V  �  = −2r−1V + O ¯  H∞,∞  b  ( ¯  X)(γ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  We also find that  δ∗  gf(r)V =  �  1 +  2m0  r − (m0 −  �  m2  0 − Q2  0)  �  dr ⊗s V,  δgGgδ∗  gf(r)V = −r−1V  with  f(r) = (  2m0  m0 −  �  m2  0 − Q2  0  − 1)r +  2m0  (m0 −  �  m2  0 − Q2  0)2 r2 ln(1 − m0 −  �  m2  0 − Q2  0  r  ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  Then it follows that  2�  Wb0,γ(0)(ω0 − f(r)V) = O ¯  H∞,∞  b  ( ¯  X)(γ),  and thus ω0 − f(r)V = O ¯  H∞,−1/2−  b  ( ¯  X)(γ) (according to Remark 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='4, �  Wb0,γ(0)−1 restricted in vector  type l = 1 1-form spaces exists with norm of size O(1) ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' This implies  ˙gb0,v1(V)= (2(µb0 − 1)dt0 − 2dr) ⊗s V + 2δ∗  gf(r)V+O ¯  H∞,1/2−  b  ( ¯  X)(γ)  =  �  (−4m0  r  + 2Q2  0  r2 )dt0 +  4m0  r − (m0 −  �  m2  0 − Q2  0)  dr  �  ⊗s V + O ¯  H∞,1/2−  b  ( ¯  X)(γ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  (10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='24)  Therefore, ˙gb0,v1(V) ∈ ¯H∞,1/2−  b  ( ¯X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' S2 �  scT ∗ ¯X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  Dual zero modes for RN and KN case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Combining the discussion around (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='4), (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='5) and (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='6) with  the growing dual zero modes established in §9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='3, we see that  (2Ggbδ∗  gbω∗  b,•(•), 4 �LAb(ω∗  b,•(•))♯ + dφ∗  b,•(•)) ∈ ker �  Lb,γ(0)∗  provided that  − �  □gb(0)φ∗  b,•(•) = −4δgb( �LAb(ω∗  b,•(•))♯).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  (10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='25)  For ω∗  b,s•(•), in view of (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='51), δgb ∈ ρDiff1  b and L(•)♯Ab ∈ ρ2Diff1  b, the right-hand side of (10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='25) is  in ˙H−∞,3/2−  b  ( ¯X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' For the case φ∗  b,s0, by Theorem 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='1, there exists a unique φ∗  b,s0 ∈ ˙H−∞,−1/2−  b  ( ¯X)  such that (10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='25) holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  We note that adding a multiple of H(r − rb0) to φ∗  b,s0 still gives rise  120  LILI HE  to A∗  b,s0 ∈  ˙H−∞,1/2−  b  ( ¯X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  For the case φ∗  b,s1(S), S ∈ S1, by Theorem 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='1 again, there exists a  unique φ∗  b,s1(S) ∈  ˙H−∞,−1/2−  b  ( ¯X) such that (10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='25) holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  Therefore, with the above choice of  φ∗  b,•(•), we have  ˙A∗  b,s•(•) ∈  ˙H−∞,1/2−  b  ( ¯X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  As for ωb,v1(V), according to (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='57), now the right-  hand side of (10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='25) is in ˙H−∞,1/2−  b  ( ¯X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Since ker �  □gb(0) ∩ ¯H∞,−1/2+  b  ( ¯X) = {0} by Theorem 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='1,  (10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='25) is solvable for φ∗  b,v1(V).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' More precisely, there exists a unique φ∗  b,v1(V) ∈  ˙H−∞,−3/2−  b  ( ¯X),  which lies in ann(v) with v ∈ C∞  c (X) and satisfying ⟨v, H(r − rb)⟩ ̸= 0, such that (10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='25) holds,  and thus ˙A∗  b,v1 ∈ ˙H−∞,−1/2−  b  ( ¯X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' �  scT ∗ ¯X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' We also note that by (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='52) and (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='58), Ggbδ∗  gbω∗  b,•(•) ∈  ˙H−∞,1/2−  b  ( ¯X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' S2 �  scT ∗ ¯X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' As a consequence,  (2Ggbδ∗  gbω∗  b,•(•), 4 �LAb(ω∗  b,•(•))♯ + dφ∗  b,•(•))  and  (0, δ(r − rb)dr)  indeed span a 8-dimension subspace of K∗  b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  Zero modes for KN case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' First, the above construction of the dual zero modes of �  Lb,γ(0)∗ and the  index 0 property of �  Lb,γ(0) imply that Kb is at least 8-dimensional.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  Next, we shall prove by a  contradiction argument that it is at most 8-dimensional for b near b0, and thus must be 8-dimensional.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  Suppose {h1, · · · , h8} is a basis of Kb0 and choose h♭  1, · · · , h♭  8 ∈ C∞  c (X◦;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' S2 �  scT ∗ ¯X ⊕ �  scT ∗ ¯X) such that  ⟨hi, h♭  j⟩ = δij.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' (Consider the map Φ : C∞  c (X◦) → C8 defined by Φ(h♭) = (⟨h1, h♭⟩, · · · , ⟨h8, h♭⟩),  and note that Φ is surjective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  Otherwise, there would exist some a = (a1, · · · , a8) ̸= 0 ∈ C8  such that a · Φ(h♭) = ⟨�8  i=1 aihi, h♭⟩ = 0 for all h♭ ∈ C∞  c (X◦), which implies ⟨�8  i=1 aihi, h♭⟩ = 0  all h♭ ∈ ˙H−∞,−ℓ  b  ( ¯X) since C∞  c (X◦) is dense in ˙H−∞,−ℓ  b  ( ¯X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' However, this is impossible because  �8  i=1 aihi ̸= 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Assume for the sake of contradiction that there exists a sequence bj → 0 such  that for all j, the dimension of Kb is greater than 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  Then one can find uj ∈ Kbj such that  uj ∈ ann{h♭  1, · · · , h♭  8} with ∥uj∥ ¯  H∞,−1/2−  b  ( ¯  X) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' As in the proof of the first step of Lemma 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='12  (using the Fredholm estimates for �  Lb,γ(0)), there exists a subsequence uj → u ̸= 0 weakly in  ¯H∞,−1/2−  b  ( ¯X) and then u ∈ ker �  Lb0,γ(0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Therefore, one can pick h♭ ∈ span{h♭  1, · · · , h♭  8} such that  ⟨u, h♭⟩ = 1, but 0 = ⟨uj, h♭⟩ → ⟨u, h♭⟩ = 1, which leads to a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  Getting the explicit expressions for (˙gb,•(•), ˙Ab,•(•)) as in (10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='10), (10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='11) and (10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='12) requires a di-  rect argument.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' First, by (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='52), we indeed have (˙gb,s1(S), ˙Ab,s1(S)) = (2δgbωb,s1(S), �LAb(ωb,s1(S))♯ +  dφb,s1(S)) ∈ ker �  Lb,γ ∩ ¯H∞,1/2−  b  ( ¯X) for suitable φb,s1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  It remains to construct (˙gb,s0, ˙Ab,s0) and  (˙gb,v1(V), ˙Ab,v1(V)) which extend (˙gb0,s0, ˙Ab0,s0) and (˙gb0,v1(V), ˙Ab0,v1(V)), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' We make  the ansatz  (˙gb,s0, ˙Ab,s0) = (˙gb( ˙m, 0, ˙Q) + 2δ∗  gbω,  ˙Ab(0, 0, ˙Q) + Lω♯Ab + dφ)  (10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='26)  with ω, φ to be found.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Since (˙gb,s0, ˙Ab,s0) of the form in the above ansatz satisfies the linearized  Einstein-Maxwell system, we have (˙gb,s0, ˙Ab,s0) ∈ ker �  Lb,γ(0) provided that the generalized linearized  gauge conditions are satisfied, namely,  ˜δgb,γGgb ˙gb,s0 = 0  and  δgbd ˙Ab,s0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  (10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='27)  Then as in the proof of scalar type l = 0 zero modes of �  Lb0,γ(0), we can find a unique ωb,s0 ∈  ¯H∞,−3/2−  b  ( ¯X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' �  scT ∗ ¯X), which lies in ann{f1, · · · , f4} with {f1, · · · , f4} ⊂ C∞  c (X) and being a set of  linearly independent functionals on ker �  Wb0,γ(0)∩ ¯H∞,−3/2−  b  ( ¯X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' �  scT ∗ ¯X), and φb,s0 ∈ ¯H∞,−3/2−  b  ( ¯X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' C),  which lies in ann(v) with v ∈ C∞  c (X) and satisfying ⟨1, v⟩ ̸= 0, such that (10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='27) holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' The con-  struction of (˙gb,v1(V), ˙Ab,v1(V)) is analogous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' We make the ansatz  (˙gb,v1(V), ˙Ab,v1(V)) = (˙gb(0, ˙a, 0) + 2δ∗  gbω,  ˙Ab(0, ˙a, 0) + Lω♯Ab + dφ)  (10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='28)  where V ∈ V1 is dual to the rotation around the axis ˙a and ω, φ are to be found.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Then by using  the discussion above and proceeding as in the proof of vector type l = 1 zero modes of �  Lb0,γ(0), the  conclusion (10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='12) follows and (˙gb,v1(V), ˙Ab,v1(V)) ∈ ¯H∞,1/2−  b  ( ¯X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  Continuity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' The continuity (in b) of the zero modes and dual zero modes follows from the explicit  construction above (see the proof of continuity in §9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='3) and that of ωb,s1(S) and ω∗  b,•(•).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  LINEAR STABILITY OF KERR-NEWMAN FAMILY  121  □  10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Generalized zero modes: linear growth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Now we turn to discussing the generalized zero mode  solutions of Lb,γ(˙g, ˙A) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Suppose (˙g, ˙A) = (�k  j=1 tj  b,∗ ˙gj, �k  j=1 tj  b,∗ ˙Aj) and Lb,γ(˙g, ˙A) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Due to the  fact that [Lb,γ, ∂tb,∗] = 0, we find that ∂j  tb,∗(˙g, ˙A) ∈ ker Lb,γ for 0 ≤ j ≤ k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Taking j = k, we obtain that  the leading order term (˙gk, ˙Ak) of (˙g, ˙A) lies in the ker �  Lb,γ(0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Next we further determine what the leading  order term (˙gk, ˙Ak) should be.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  Lemma 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Let k ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Then there does not exist (˙g, ˙A) = (�k  j=1 tj  b,∗ ˙gj, �k  j=1 tj  b,∗ ˙Aj), with 0 ̸= (˙gk, ˙Ak) ∈  {(˙gb,v1(V), ˙Ab,v1(V)) : V ∈ V1} and (˙gj, ˙Aj) ∈ ¯H∞,−1/2−  b  ( ¯X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' S2 �  scT ∗ ¯X ⊕ �  scT ∗ ¯X), such that Lb,γ(˙g, ˙A) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' According to the discussion above, it suffices to consider the case k = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Assume  (˙g, ˙A) = (tb,∗ ˙gb,v1(V) + ˙g0, tb,∗ ˙Ab,v1(V) + ˙A0)  with V ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Then we need to show that there is no (˙g0, ˙A0) ∈ ¯H∞,−1/2−  b  ( ¯X) such that the following equation  holds  �  Lb,γ(0)(˙g0, ˙A0) = −[Lb,γ, tb,∗](˙gb,v1(V), ˙Ab,v1(V)) ∈ ¯H∞,3/2−  b  ( ¯X)  where we use [Lb,γ, tb,∗] ∈ ρDiff1  b and (˙gb,v1(V), ˙Ab,v1(V) ∈ ¯H∞,1/2−  b  ( ¯X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' To this end, we need to verify that  the pairing of the right-hand side with (˙g∗  b,v1(V), ˙A∗  b,v1(V)) is non-zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' We note that it suffices to check the  RN case b = b0 because the general KN case with b near b0 follows directly by the continuity (in b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Using  the expression (10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='16) for (˙gb0,v1(V), ˙Ab0,v1(V)), we find that  ⟨[Lb0,γ, tb0,∗](˙gb0,v1(V), ˙Ab0,v1(V)), (˙g∗  b0,v1(V′), ˙A∗  b0,v1(V′))⟩  = ⟨[LE  b0,γ, tb0,∗](˙gb0,v1(V), 0), (˙g∗  b0,v1(V′), 0)⟩ + O(|Q0|)  = ⟨[LE  b0,γ, tb0,∗](˙gb0,v1(V), 0), (2Ggδ∗  g(r2H(r − rb0)V′), 0)⟩ + O(|Q0| + γ)  (10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='29)  where we use (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='59) in the last step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Since Ggb0 δgb0 (r2H(r − rb0)V′) = r2δ(r − rb0)dr ⊗s V′ is supported at  r = rb0 where tb0,∗ = t0, we can replace tb0,∗ by t0 in the subsequent calculation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' We then compute  ⟨[LE  b0,γ, t0](˙gb0,v1(V), 0), 2r2δ(r − rb0)dr ⊗s V′⟩  = ⟨[LE  b0,γ, t0](4m0r−1(dr − dt0) ⊗s V, 0), 2r2δ(r − rb0)dr ⊗s V′⟩ + O(|Q0|2 + γ)  = −⟨[□gb0 ,2, t0](4m0r−1(dr − dt0) ⊗s V), 2r2δ(r − rb0)dr ⊗s V′⟩ + O(|Q0|2 + γ)  = ⟨8m0r−2�  (dr − (1 + m0  r )dt0)  �  ⊗s V, 2r2δ(r − rb0)dr ⊗s V′⟩ + O(|Q0|2 + γ)  = −12m0⟨V, V ′⟩L2(S2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='T ∗S2) + O(|Q0|2 + γ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  (10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='30)  where we use (10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='16) in the first step, (10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='2) and (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='2) in the second step and [□gb0 ,2, t0]h = h□gb0 t0 +  2(∇αt0)∇αh in the third step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Therefore, the pairing is indeed non-zero if V = V′ and |Q0|+ γ is sufficiently  small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  □  We now exclude the generalized zero modes whose leading order term is (˙gb,s0, ˙Ab,s0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  Lemma 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Let k ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Then there does not exist (˙g, ˙A) = (�k  j=1 tj  b,∗ ˙gj, �k  j=1 tj  b,∗ ˙Aj), with 0 ̸= (˙gk, ˙Ak) =  (˙gb,s0, ˙Ab,s0) and (˙gj, ˙Aj) ∈ ¯H∞,−1/2−  b  ( ¯  X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' S2 �  scT ∗ ¯X ⊕ �  scT ∗ ¯X), such that Lb,γ(˙g, ˙A) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' As discussed in the proof of Lemma 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='3, it suffices to prove the statement for the case k = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' We  first consider the case b = b0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Suppose  (˙g, ˙A) = (tb0,∗ ˙gb0,s0 + ˙g0, tb0,∗ ˙Ab0,s0 + ˙A0) ∈ ker Lb0,γ  with (˙g0, ˙A0) ∈ ¯H∞,−1/2−  b  ( ¯X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Applying δgb0 to LM  b0,γ(˙g, ˙A) = 0 gives δgb0 ˙A ∈ ker □gb0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' As pointed out in  Remark 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='2, we have δgb0 ˙Ab0,s0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' This implies  ker �  □gb0 (0) ∋ δgb0 ˙A = [δgb0 , tb0,∗] ˙Ab0,s0 + δgb0 ˙A0 ∈ ¯H∞,−1/2−  b  ( ¯X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  Therefore, δgb0 ˙A = 0 by Theorem 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' That is, (˙g, ˙A) satisfies the Maxwell part of the linearized Einstein-  Maxwell system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Then as in the proof of Theorem 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='1, we apply δgb0 Ggb0 to LE  b0,γ(˙g, ˙A) = 0 and use the  122  LILI HE  linearized second Bianchi identity to obtain ˜δgb0 ,γGgb0 ˙g ∈ ker Pb0,γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Since ˜δgb0 ,γGgb0 ˙gb0,s0 = 0 (see Remark  10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='2), it follows that  ker �  Pb0,γ(0) ∋ ˜δgb0 ,γGgb0 ˙g = [˜δgb0 ,γGgb0 , tb0,∗]˙gb0,s0 + ˜δgb0 ,γGgb0 ˙g0 ∈ ¯H∞,−1/2−  b  ( ¯X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' �  scT ∗ ¯X),  and thus ˜δgb0 ,γGgb0 ˙g = 0 by Theorem 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' As a consequence, (˙g, ˙A) satisfies the linearized Einstein-Maxwell  system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' By the statement (v) in mode stability Theorem 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='1, we have  (˙g, ˙A) = (˙gb0( ˙m, 0, ˙Q) + 2δ∗  gb0 ω,  ˙Ab0(0, 0, ˙Q) + Lω♯Ab0 + dφ)  where (ω, φ) ∈ Poly2(tb0,∗) ¯H∞,−3/2−  b  ( ¯X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' �  scT ∗ ¯X ⊕ C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' To make the calculation simpler, we work in (t0 = t +  rb0,∗, r, θ, ϕ) coordinates instead.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Since ˙g is linear in t0, it follows that the coefficient of t2  0 in ω and φ must be a  multiple of ∂♭  t0 and 1, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Then equating the coefficient of t0 of the equality ˙g = ˙gb0( ˙m, 0, ˙Q)+2δ∗  gb0 ω  yields  (2 ˙m  r  − 2Q0 ˙Q  r2  )dt2  0 = ˙g0  b0( ˙m, 0, ˙Q) = c(dt0) ⊗s ∂♭  t0 + δ∗  gb0 ˜ω  (10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='31)  where c ∈ R and ˜ω ∈ ¯H∞,ℓ′  b  ( ¯X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' �  scT ∗ ¯X) for some ℓ′ ∈ R is of scalar type l = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Assume ˜ω = f(r)dt0 + g(r)dr  and let S = c(dt0) ⊗s ∂♭  t0 + δ∗  gb0 ˜ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' A direct calculation implies 0 = Srr = g′(r) and thus g(r) = m for some  m ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Then 0 = St0r = c  2 + 1  2f ′(r) + m  2 µ′  b0 implies f(r) = −cr + 2mm0  r  − mQ2  0  r2  + n for some n ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Next  0 = Sθθ = r(f(r)+mµb0) = −cr2 +(m+n)r gives c = 0, m+n = 0, and thus ˜ω = −mµb0dt0 +mdr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Finally,  2 ˙m  r − 2Q0 ˙Q  r2  = St0t0 = m  2 µb0µ′  b0 − m  2 µb0µ′  b0 = 0 implies that ˙m = Q0 ˙Q = 0 and thus the leading term gb0,s0  of ˙g is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' If ˙Q ̸= 0, Q0 = 0, equating the coefficient of t0 of the equality ˙A = ˙Ab0(0, 0, ˙Q) + dφ yields  ˙Q  r dt0 = cdt0 + d˜φ  (10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='32)  where c ∈ R and ˜φ ∈ ¯H∞,ℓ′  b  ( ¯X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' C) for some ℓ′ ∈ R is of scalar type l = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' This implies that ˙Q = 0 and thus  the leading term Ab0,s0 of ˙A is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' This proves the lemma for the RN case b = b0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  This non-existence result for b = b0 in turn means that the following equation  �  Lb0,γ(0)(˙g0, ˙A0) = −[Lb0,γ, tb0,∗](˙gb0,s0, ˙Ab0,s0) ∈ ¯H∞,3/2−  b  ( ¯X)  cannot be solved for (˙g0, ˙A0) ∈ ¯H∞,−1/2−  b  ( ¯X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' S2 �  scT ∗ ¯X⊕ �  scT ∗ ¯X) (The right-hand side is in the stated Sobolev  space because by Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='7, (˙gb0,s0, ˙Ab0,s0) = ρC∞( ¯X) + ¯H∞,1/2−  b  ( ¯X) whose leading term ρC∞( ¯  X) is  annihilated by the normal operator 2ρ(ρ∂ρ − 1) of −[Lb0,γ, tb0,∗]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Suppose {(˙g1  b,s0, ˙A1  b,s0), (˙g2  b,s0, ˙A2  b,s0)} is a  basis of {(˙gb,s0, ˙Ab,s0)} and {(˙g∗1  b,s0, ˙A∗1  b,s0), (˙g∗2  b,s0, ˙A∗2  b,s0)} is a basis of {(˙g∗  b,s0, ˙A∗  b,s0)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' In terms of pairing, we  have  ⟨[Lb0,γ, tb0,∗](˙gi  b0,s0, ˙Ai  b0,s0), (˙g∗j  b0,s0, ˙A∗j  b0,s0)⟩,  1 ≤ i, j ≤ 2  is non-degenerate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Due to the continuity of (˙gb,s0, ˙Ab,s0) and (˙g∗  b,s0, ˙A∗  b,s0) in b, this pairing is still non-zero  for KN case with b near b0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' This finishes the proof of the lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  □  However, there do exist generalized zero modes (growing linearly in tb,∗) whose leading order term is given  by (˙gb,s1(S), ˙Ab,s1(S)) for S ∈ S1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  Proposition 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Let b0 = (m0, 0, Q0) with |Q0| ≪ m0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Then the following space  �Kb0 := ker Lb0,γ ∩ Poly1(tb0,∗) ¯H∞,−1/2−  b  ( ¯X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' S2 �  scT ∗ ¯X ⊕ �  scT ∗ ¯X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  (10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='33)  is 11-dimensional.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' The 3-dimensional quotient space �Kb0/Kb0 is spanned by the following continuous (in b)  families  {(ˆgb0,s1(S), ˆAb0,s1(S)) : S ∈ S1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  (10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='34)  with  (ˆgb0,s1(S), ˆAb0,s1(S)) = (2δ∗  gb0 ˆωb0,s1(S), �LAb0 (ˆωb0,s1(S))♯ + dˆφb0,s1(S))  (10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='35)  where ˆωb0,s1(S) is defined as in (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='65) and ˆφb0,s0(S) ∈ Poly1(tb0,∗) ¯H∞,−3/2−  b  ( ¯X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' C) depends linearly in S ∈ S1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  The generalized zero modes (ˆgb0,s1(S), ˆAb0,s1(S)) extend to continuous families in b  b → (ˆgb,s1(S), ˆAb,s1(S)) ∈ ker Lb,γ ∩ Poly1(tb,∗) ¯H∞,−1/2−  b  ( ¯X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' S2 �  scT ∗ ¯X ⊕ �  scT ∗ ¯X)  (10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='36)  LINEAR STABILITY OF KERR-NEWMAN FAMILY  123  with  (ˆgb,s1(S), ˆAb,s1(S)) = (2δ∗  gb ˆωb,s1(S), �LAb(ˆωb,s1(S))♯ + dˆφb,s1(S))  (10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='37)  where ˆωb,s1(S) is defined as in (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='65) and ˆφb,s0(S) ∈ Poly1(tb,∗) ¯H∞,−3/2−  b  ( ¯  X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' C) depends linearly in S ∈  S1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Moreover, (ˆgb,s1(S), ˆAb,s1(S)) satisfies both the linearized Einstein-Maxwell system and the linearized  generalized wave map gauge and generalized Lorenz gauge conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  Meanwhile, there exists continuous families (in b)  b → (ˆg∗  b,s1(S), ˆA∗  b,s1(S)) ∈ ker L∗  b,γ ∩ Poly1(tb,∗) ˙H−∞,−1/2−  b  ( ¯X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' S2 �  scT ∗ ¯X ⊕ �  scT ∗ ¯X)  (10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='38)  with  (ˆg∗  b,s1(S), ˆA∗  b,s1(S)) = (2Ggbδ∗  gb ˆω∗  b,s1(S), 4 �LAb0 (ˆω∗  b,s1(S))♯ + dˆφ∗  b,s1(S))  (10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='39)  where ˆω∗  b,s1(S) is defined as in (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='65) and ˆφ∗  b,s0(S) ∈ Poly1(tb,∗) ˙H∞,−3/2−  b  ( ¯X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Let (˙g, ˙A) = (tb,∗ ˙g1 + ˙g0, tb,∗ ˙A1 + ˙A0) ∈ ker Lb,γ with (˙gj, ˙Aj) ∈ ¯H∞,−1/2−  b  ( ¯X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  Generalized zero modes in the RN case b = b0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' In view of Lemma 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='3 and 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='4, (˙g1, ˙A1) must take  the form (2δgb0 ωb0,s1(S), �LAb0 (ωb0,s1(S))♯ + dφb0,s1(S)) for some S ∈ S1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' By the argument at the  beginning of the proof of Lemma 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='4, we see that (˙g, ˙A) satisfies the generalized linearized gauge  condition, that is,  ˜δgb0 Ggb0 ˙g = 0,  δgb0 ˙A = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  (10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='40)  Therefore, (˙g, ˙A) solves the linearized Einstein-Maxwell system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Since  ˙g = 2δ∗  gb0(tb0,∗ωb0,s1(S)) − 2[δ∗  gb0 , tb0,∗]ωb0,s1(S) + ˙g0 = 2δ∗  gb0 (tb0,∗ωb0,s1(S)) + ˙g′  0,  ˙A = �LAb0 (tb0,∗ωb0,s1(S))♯ + d  �  tb0,∗φb0,s1(S)  �  −  �  ιωb0,s1(S)♯Ab0 + φb0,s1(S)  �  dtb0,∗ + ˙A0  = �LAb0 (tb0,∗ωb0,s1(S))♯ + d  �  tb0,∗φb0,s1(S)  �  + ˙A′  0,  (10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='41)  with  ˙g′  0 = ˙g0 − 2[δ∗  gb0 , tb0,∗]ωb0,s1(S) ∈ ¯H∞,−3/2−  b  ( ¯X),  ˙A′  0 = ˙A0 −  �  ιωb0,s1 (S)♯Ab0 + φb0,s1(S)  �  dtb0,∗ ∈ ¯H∞,−1/2−  b  ( ¯X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  where we use [δ∗  gb0 , tb0,∗] ∈ A0( ¯X), ωb0,s1(S) ∈  ¯H∞,−3/2−  b  ( ¯X) and φb0,s1(S) ∈  ¯H∞,−1/2−  b  ( ¯X), it  follows that (˙g′  0, ˙A′  0) solves the linearized Einstein-Maxwell system as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' By the statement (iv) in  Theorem 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='1, we have  (˙g′  0, ˙A′  0) = (2δ∗  gb0 ω, Lω♯Ab0 + dφ)  (10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='42)  with (ω, φ) ∈ ¯H∞,−5/2−  b  ( ¯X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' �  scT ∗ ¯X ⊕ C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Plugging (10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='41) and (10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='42) into (10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='40) yields  �  Wb0,γ(0)ω = −[Wb0,γ, tb0,∗]ωb0,s1(S) ∈ ¯H∞,−1/2−  b  ( ¯  X),  (10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='43)  − �  □gb0 (0)φ = −δgb0  �  Lω♯Ab0 + (ιωb0,s1 (S)♯Ab0)dtb0,∗  �  (10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='44)  + [□gb0 , tb0,∗]φb0,s1(S) − [δgb0 , tb0,∗] �LAb0 (ωb0,s1(S))♯  We first consider (10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='43).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Since ker �  Wb0,γ(0)∗ ∩ ˙H−∞,1/2+  b  ( ¯X) = {0} by Theorem 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='13, it follows  that (10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='43) can be solved for ω ∈ ¯H∞,−5/2−  b  ( ¯X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' According to Proposition 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='17, we know that  ˆωb0,s1(S) = tb0,∗ωb0,s1(S) + ˇωb0,s1(S) ∈ ker Wb0,γ  with ˇωb0,s1(S) ∈ ¯H∞,−5/2−  b  ( ¯X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Therefore,  �  Wb0,γ(0)  �  ω − ˇωb0,s1(S)  �  = −Wb0,γ ˆωb0,s1(S) = 0  124  LILI HE  and ω − ˇωb0,s1(S) ∈ span{ω(1)  b0,s1(S) : S ∈ S1} (In fact ker �  Wb0,γ(0) ∩ ¯H∞,−5/2−  b  ( ¯X) restricted to the  scalar type l = 1 1-form is equal to span{ωb0,s1(S), ω(1)  b0,s1(S) : S ∈ S1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Since (˙g, ˙A) is in the quotient  space �Kb0/Kb0, we can exclude ωb0,s1(S)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Using (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='67), we rewrite  ˙g = 2δ∗  gb0 ˆωb0,s1(S) + 2δ∗  gb0  �  ω − ˇωb0,s1(S)  �  = Poly1(tb0,∗) ¯H∞,−1/2−  b  ( ¯X) + 2δ∗  gb0  �  ω − ˇωb0,s1(S)  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  Since we require ˙g ∈ Poly1(tb0,∗) ¯H∞,−1/2−  b  ( ¯X), it follows that δ∗  gb0  �  ω − ˇωb0,s1(S)  �  must lie in  ¯H∞,−1/2−  b  ( ¯X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Owing to (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='62), ω − ˇωb0,s1(S) must be 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' That it, ˙g = 2δgb0 ˆωb0,s1(S).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  Next we turn to (10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='44).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Since ω = ˇωb0,s1(S) ∈ ¯H∞,−5/2  b  ( ¯X) and φb0,s1 ∈ ¯H∞,−1/2−  b  ( ¯X), the right-  hand side of (10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='44) lies in ¯H∞,1/2−  b  ( ¯X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' By Theorem 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='1 (ker �  □gb0 (0)∗ ∩ ˙H−∞,−1/2+  b  ( ¯X) = {0})  and Proposition 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='2 (ker �  □gb0 (0) ∩ ¯H∞,−3/2−  b  ( ¯X) restricted to scalar type l = 1 is 0), there exists a  unique φ ∈ ¯H∞,−3/2−  b  ( ¯X) which we denote by ˇφb0,s1(S) satisfying (10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='44).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' This proves that �Kb0/Kb0  is at most 3-dimensional.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  Using ˙g = 2δ∗  gb0 ˆωb0,s1(S) ∈ Poly1(tb0,∗) ¯H∞,−1/2  b  ( ¯X) (see (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='68)) and rewriting  ˙A = �LAb0 (ˆωb0,s1(S))♯ + d  �  tb0,∗φb0,s1(S) + ˇφb0,s1(S)  �  =  �  �LAb0 (ˇωb0,s1(S))♯ + φb0,s1dtb0,∗ + ι(ωb0,s1(S))♯Ab0dtb0,∗ + dˇφb0,s1(S)  �  + tb0,∗  �  �LAb0 (ωb0,s1(S))♯ + dφb0,s1(S)  �  ∈ Poly1(tb0,∗) ¯H∞,−1/2  b  ( ¯X),  (10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='45)  we conclude that (˙g, ˙A) = (ˆgb0,s1(S), ˆAb0,s1(S)) = (2δ∗  gb0 ˆωb0,s1(S), �LAb0 (ˆωb0,s1(S))♯ + d(ˆωb0,s1(S)))  with ˆφb0,s1(S) = tb0,∗φb0,s1(S) + ˇωb0,s1(S) ∈ Poly1(tb0,∗) ¯H∞,−3/2  b  ( ¯X) indeed lies in �Kb0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  Generalized zero modes in the KN case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' For b near b0, we make the ansatz  (˙g, ˙A) = (2δ∗  gb ˆωb,s1(S), �LAb(ˆωb,s1(S))♯ + d(tb,∗φb,s1(S)) + dφ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  with φ ∈ ¯H∞,−3/2  b  ( ¯X) to be determined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' By the arguments in the proof of RN case above (the last two  paragraphs), there exists ˇφb,s1(S) ∈ ¯H∞,−3/2  b  ( ¯X) such that with ˆφb,s1(S) = tb,∗φb,s1(S) + ˇφb,s1(S) ∈  Poly1(tb,∗) ¯H∞,−3/2  b  ( ¯X),  (ˆgb,s1(S), ˆAb,s1(S))  = (2δ∗  gb ˆωb,s1(S), �LAb(ˆωb,s1(S))♯ + d(ˆφb,s1(S))) ∈ ker Lb,γ ∩ Poly1(tb,∗) ¯H∞,−1/2−  b  ( ¯X)  extends (ˆgb0,s1(S), ˆAb0,s1(S)) and is continuous in b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  Generalized dual zero modes for RN and KN cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Combining the discussion around (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='4), (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='5)  and (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='6) with the generalized dual zero mode ˆω∗  b,s1(S) = tb,∗ω∗  b,s1(S) + ˇω∗  b,s1(S) of Wb,γ established  in Proposition 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='17, we see that  (ˆg∗  b,s1(S), ˆA∗  b,s1(S)) = (2Ggbδ∗  gb ˆω∗  b,s1(S), 4 �LAb(ˆω∗  b,s1(S))♯ + d(tb,∗φ∗  b,s1(S)) + dˇφ∗  b,s1(S)) ∈ ker L∗  b,γ  provided that  −�  □gb(0)ˇφ∗  b,s1(S) = −4δgb  �  �LAb(ˇω∗  b,s1(S))♯ + (ιω∗  b,s1 (S)♯Ab)dtb,∗  �  + [□gb, tb,∗]φ∗  b,s1(S) − 4[δgb, tb,∗] �LAb(ω∗  b,s1(S))♯ ∈ ˙H−∞,1/2−  b  ( ¯X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  In view of Theorem 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='1 and Proposition 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='2, ker �  □gb(0) ∩ ¯H∞,−1/2+  b  ( ¯X) = {0}, and thus there exists  a unique ˇφ∗  b,s1(S) ∈ ˙H−∞,−3/2−  b  ( ¯X) (up to addition of a multiple of H(r − rb))) such that (10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='46)  holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' We put ˆφ∗  b,s1(S) = tb,∗φ∗  b,s1 + ˇφ∗  b,s1(S) and find that ˆAb,s1(S) ∈ poly1(tb,∗) ˙H−∞,−1/2−  b  ( ¯X)  by the argument in (10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='45).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' We also recall that Ggbδ∗  gb ˆω∗  b,s1(S) ∈ Poly1(tb,∗) ˙H−∞,−1/2−  b  ( ¯X) (see  (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='68)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' As a consequence, the continuous (in b) families (ˆg∗  b,s1(S), ˆA∗  b,s1(S)) indeed lie in ker L∗  b,γ ∩  poly1(tb,∗) ˙H−∞,−1/2−  b  ( ¯X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  LINEAR STABILITY OF KERR-NEWMAN FAMILY  125  □  Let  (ˇgb,s1(S), ˇAb,s1(S)) := (ˆgb,s1(S) − tb,∗ ˙gb,s1(S), ˆAb,s1(S) − tb,∗ ˙Ab,s1(S)) ∈ ¯H∞,−1/2−  b  ( ¯X),  (10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='46)  (ˇg∗  b,s1(S), ˇA∗  b,s1(S)) := (ˆg∗  b,s1(S) − tb,∗ ˙g∗  b,s1(S), ˆA∗  b,s1(S) − tb,∗ ˙A∗  b,s1(S)) ∈ ˙H−∞,−1/2−  b  ( ¯X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  (10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='47)  be the coefficients of t0  b,∗ of the generalized (dual) zero modes constructed above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' For later use, we determine  the leading order behavior of them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  Lemma 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' For S ∈ S1, we have  (ˇgb,s1(S),  ˇAb,s1(S)) ∈ ρC∞( ¯X) + ¯H∞,1/2−  b  ( ¯X),  (ˇg∗  b,s1(S),  ˇA∗  b,s1(S)) ∈ ρC∞( ¯X) + ˙H−3/2−C(a,γ),1/2−  b  ( ¯  X)  (10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='48)  where C(a, γ) > 0 is a sufficiently small constant depending on a, γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Arguing as in the proof of Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='7, we again exploit the normal operator argument.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' First, since  (˙gb,s1(S),  ˙Ab,s1(S)) ∈ ¯H∞,1/2−  b  ( ¯X), according to Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='7 (where we shift the contour of integration  through the pole ξ = −2i and thus the space of resonant states is S1), it follows that  (˙gb,s1(S),  ˙Ab,s1(S)) ∈ ρ2Ω1 + ¯H∞,3/2−  b  ( ¯X)  where  Ω1 ∈ span{Sdt2, Sdt ⊗s dxi, Sdxi ⊗s dxj, Sdt, Sdxi | S ∈ S1, 1 ≤ i, j ≤ 3}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  Since  − �  Lb,γ(0)(ˇgb,s1(S), ˇAb,s1(S)) = [Lb,γ, tb,∗](˙gb,s1(S),  ˙Ab,s1(S)) ∈ −2ρ3Ω1 + ¯H∞,5/2−  b  ( ¯X)  where we use the fact [Lb,γ, tb,∗] ∈ −2ρ(ρ∂ρ − 1) + ρ2Diff1  b, it follows that we can solve the above equation  for (ˇgb,s1(S), ˇAb,s1(S)) by first solving away the leading term −2ρ3Ω1 and then the error term which lies in  ¯H∞,5/2−  b  ( ¯X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  For the leading term 2ρ3Ω1, proceeding as in the proof of Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='7 (here we have k = −1, l = 1),  we need to solve the following equation  �  ρ∂ρ(ρ∂ρ − 1) + /∆  �  (˙g1, ˙A1) = −2ρΩ1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  Since now k(k + 1) = (−1) · 0 ̸= 1 · 2 = l(l + 1), we conclude that (˙g1, ˙A2) ∈ (−1/2) · (−2ρΩ1) = ρΩ1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  For the error term, we need to solve the following equation  �  ρ∂ρ(ρ∂ρ − 1) + /∆  �  (˙g2, ˙A2) = f ∈ ¯H∞,5/2−  b  ( ¯X),  (˙g2, ˙A2) ∈ ¯H∞,−1/2  b  ( ¯X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  Arguing as in the proof of Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='7, we see that (˙g2, ˙A2) ∈ ¯H∞,−1/2  b  ( ¯X) ∈ ρC∞(∂+ ¯X)+ ¯H∞,1/2−  b  ( ¯X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  This proves (10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='48) for (ˇgb,s1(S), ˇAb,s1(S)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  For (ˇg∗  b,s1(S),  ˇA∗  b,s1(S)), the above proof also applies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' As for the regularity of (ˇg∗  b,s1(S),  ˇA∗  b,s1(S)) near  the radial point at event horizon (by the same reasoning as in the proof of statement (2) of Propo-  sition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='7, (ˇg∗  b,s1(S),  ˇA∗  b,s1(S)) is smooth away from the radial point at even horizon), we notice that  [Lb,γ, tb,∗](˙g∗  b,s1(S),  ˙A∗  b,s1(S)) is one derivative less regular than (˙g∗  b,s1(S),  ˙A∗  b,s1(S)) and ( �  Lb,γ(0)∗)−1 gains  one derivative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Therefore, (ˇg∗  b,s1(S), ˇA∗  b,s1(S)) has at least the same regularity as (˙g∗  b,s1(S),  ˙A∗  b,s1(S)), which  is − 3  2 − C(a, γ) in view of Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  □  10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Generalized zero modes: quadratic growth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Finally, we exclude the generalized zero mode solu-  tions with quadratic growth in tb,∗ of Lb,γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  Lemma 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Let k ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Then there does not exist (˙g, ˙A) = (�k  j=1 tj  b,∗ ˙gj, �k  j=1 tj  b,∗ ˙Aj), with 0 ̸= (˙gk, ˙Ak) =  {(˙gb,s1(S), ˙Ab,s1(S)) : S ∈ S1} and (˙gj, ˙Aj) ∈ ¯H∞,−1/2−  b  ( ¯X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' S2 �  scT ∗ ¯X ⊕ �  scT ∗ ¯X), such that Lb,γ(˙g, ˙A) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  126  LILI HE  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' According to the discussion at the beginning of the previous subsection, it suffices to consider the  case k = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' For the simplicity of calculation, we replace tb,∗ by another smooth function t1 := t0 − 2r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Since  tb,∗ − t1 ∈ A0−( ¯X), we assume  (˙g, ˙A) = (t2  1 ˙gb,s1(S) + t1 ˙g1 + ˙g0, t2  1 ˙Ab,s1(S) + t1 ˙A1 + ˙A0)  with S ̸= 0 and (˙gj, ˙Aj) ∈ ¯H∞,−1/2−  b  ( ¯X) for j = 0, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' We write Lb,γ(˙g, ˙A) as a polynomial of degree 1 in t1  as follows  Lb,γ(˙g, ˙A) = t1  �  2[Lb,γ, t1](˙gb,s1(S), ˙Ab,s1(S)) + Lb,γ(˙g1, ˙A1)  �  +  �  [[Lb,γ, t1], t1](˙gb,s1(S), ˙Ab,s1(S)) + [Lb,γ, t1](˙g1, ˙A1) + Lb,γ(˙g0, ˙A0)  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  Then Lb,γ(˙g, ˙A) = 0 is equivalent to the vanishing of the coefficients of the above polynomial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Namely,  2[Lb,γ, t1](˙gb,s1(S), ˙Ab,s1(S)) + Lb,γ(˙g1, ˙A1) = 0,  (10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='49)  [[Lb,γ, t1], t1](˙gb,s1(S), ˙Ab,s1(S)) + [Lb,γ, t1](˙g1, ˙A1) + Lb,γ(˙g0, ˙A0) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  (10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='50)  First, (10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='49) implies Lb,γ  �  2t1(˙gb,s1(S), ˙Ab,s1(S)) + (˙g1, ˙A1)  �  = 0, and thus by Proposition 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='5  2t1(˙gb,s1(S), ˙Ab,s1(S)) + (˙g1, ˙A1) = 2(ˆgb,s1(S), ˆAb,s1(S)) + c(˙gb,s1(S′),  ˙Ab,s1(S′))  for some c ∈ R and S′ ∈ S1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Subtracting c(˙gb,s1(S′),  ˙Ab,s1(S′)) from (˙g, ˙A), we may assume c = 0, and this  implies  (˙g1, ˙A1) = 2(ˆgb,s1(S), ˆAb,s1(S)) − 2t1(˙gb,s1(S), ˙Ab,s1(S)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  Plugging the above expression for (˙g1, ˙A1) into (10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='50) yields  Lb,γ(˙g0, ˙A0) = −[[Lb,γ, t1], t1](˙gb,s1(S), ˙Ab,s1(S))  − [Lb,γ, t1]  �  2(ˆgb,s1(S), ˆAb,s1(S)) − 2t1(˙gb,s1(S), ˙Ab,s1(S))  �  ∈ ¯H∞,3/2−  b  ( ¯X)  (10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='51)  Our aim is to prove that there is no solution (˙g0, ˙A0) ∈ ¯H∞,−1/2−  b  ( ¯X) to the above equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' To this end,  we need to verify that the pairing of the right-hand side of (10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='51) with (˙g∗  b,s1(S), ˙A∗  b,s1(S)) is non-zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' We  note that it suffices to check the RN case b = b0 because the general KN case with b near b0 follows directly  by the continuity (in b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Using ˙Ab0,s1(S) = O ¯  H∞,1/2−  b  ( ¯  X)(|Q0|) and the same argument as in the proof of  Lemma 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='3, we obtain that the the pairing of the right-hand side of (10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='51) with (˙g∗  b0,s1(S), ˙A∗  b0,s1(S)) is  equal to  ⟨[[□gb0 ,2, t1], t1]˙gb0,s1(S), ˙g∗  b0,s1(S)⟩ + 2⟨[□gb0 ,2, t1](ˆgb0,s1(S) − t1 ˙gb0,s1(S)), ˙g∗  b0,s1(S)⟩ + O(|Q0| + γ)  := P1 + P2 + O(|Q0| + γ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  We first calculate P1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Using ˙gb0,s1(S) = 2δ∗  gb0 ωb0,s1(S), ˙g∗  b0,s1(S) = 2Ggb0 δ∗  gb0 ω∗  b0,s1(S) and (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='53), we find  that  P1 = 2⟨(∇αt1∇αt1)˙gb0,s1(S), ˙g∗  b0,s1(S)⟩ = 2⟨4(µb0 − 1)˙gb0,s1(S), ˙g∗  b0,s1(S)⟩  = 2⟨16(µb0 − 1)δ∗  gb0 d  �  (r − m0)S  �  , Ggb0 δ∗  gb0 d  �  (r − m0)H(r − rb0)S  �  ⟩ + O(|Q0|2 + γ)  = −512  9 πm0 + O(|Q0|2 + γ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  Next, using ˆgb0,s1(S) = 2δ∗  gb0 ˆωb0,s1(S), (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='70) and (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='53), we compute  P2 = 2⟨  �  2∇αt1∇α + □gb0 ,0t1  �  (ˆgb,s1(S) − t1 ˙gb,s1(S)), ˙g∗  b0,s1(S)⟩  = −64  9 (7 − log 8)πm0 + O(|Q0|2 + γ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  Therefore, the pairing is indeed non-zero if |Q0| + γ is sufficiently small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  □  LINEAR STABILITY OF KERR-NEWMAN FAMILY  127  11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Structure of the resolvent of the linearized gauge-fixed Einstein-Maxwell operator  In this section, we shall prove that �  Lb,γ(σ) is invertible for b = (m, a, Q) near b0 = (m0, 0, Q0), |Q0| ≪ m0  and Im σ ≥ 0, σ ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' This is a generalization of the first statement in Theorem 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='1 from the RN case to  the KN case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' The proof uses the relevant conclusions of the RN case and the arguments in the proof of  Proposition 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  Let γ > 0 be a fixed sufficiently small constant such that all the conclusions in the previous section hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  We recall the linearized gauge-fixed Einstein-Maxwell operator  Lb,γ := L(gb,Ab),γ,  LE  b,γ := LE  (gb,Ab),γ,  LM  b,γ := LM  (gb,Ab),γ,  and its Fourier transformed version  �  Lb,γ(σ) := eiσtb,∗Lb,γe−iσtb,∗  where tb,∗ = χ0(r)(t + rb0,∗) + (1 − χ0(r))(t − r(m,0,Q),∗) is defined in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='29).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Since γ is fixed, we will drop  the subscript γ from the notations from now on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  Let s > 3 and −3/2 < ℓ < −1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' We redefine  X s,ℓ  b  (σ) := {(˙g, ˙A) ∈ ¯Hs,ℓ  b ( ¯X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' S2 �  scT ∗ ¯X ⊕ �  scT ∗ ¯X) : �  Lb(σ)(˙g, ˙A) ∈ ¯Hs−1,ℓ+2  b  ( ¯X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' S2 �  scT ∗ ¯X ⊕ �  scT ∗ ¯X)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  As in the previous section, from now on, we will drop the notation of the bundle if it it clear from the  context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  We also decompose zero energy nullspace Kb and the space of zero energy dual states K∗  b of Lb as follows  Kb = Kb,1 ⊕ Kb,2,  K∗  b = K∗  b,1 ⊕ K∗  b,2  (11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='1)  where  Kb,1 := {(˙gb,s1(S1), ˙Ab,s1(S1))},  Kb,2 := {(˙gb,s0, ˙Ab,s0)} ⊕ {(˙gb,v1(V1), ˙Ab,v1(V1))},  (11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='2)  K∗  b,1 := {(˙g∗  b,s1(S1), ˙A∗  b,s1(S1))},  Kb,2 := {(˙g∗  b,s0, ˙A∗  b,s0)} ⊕ {(˙g∗  b,v1(V1), ˙A∗  b,v1(V1))}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  (11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='3)  The motivation for doing such a decomposition is that �  Lb(σ) is more singular when acting on Kb,1 because  of the existence of the generalized solutions to Lb(˙g, ˙A) = 0 with linear growth in tb,∗ and leading order  terms in Kb,1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Moreover, we recall the generalized zero energy states (ˆgb,s1(S1), ˆAb,s1(S1)) and dual states  (ˆg∗  b,s1(S1), ˆA∗  b,s1(S1)) which is constructed in Proposition 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='5 and define  ˇKb := {(ˇgb,s1(S), ˇAb,s1(S)) : S ∈ S1},  ˇK∗  b := {(ˇg∗  b,s1(S), ˇA∗  b,s1(S)) : S ∈ S1}  (11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='4)  where  (ˇgb,s1(S), ˇAb,s1(S)) = (ˆgb,s1(S), ˆAb,s1(S)) − tb,∗(˙gb,s1(S), ˙Ab,s1(S)),  (11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='5)  (ˇg∗  b,s1(S), ˇA∗  b,s1(S)) = (ˆg∗  b,s1(S), ˆA∗  b,s1(S)) − tb,∗(˙g∗  b,s1(S), ˙A∗  b,s1(S)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  (11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='6)  11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Construction of the reference operator ˇLb(σ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' We first construct a reference operator ˇLb(σ)  which is invertible near σ = 0 by perturbing �  Lb(σ) by a smoothing operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  Lemma 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Let b0 = (m0, 0, Q0) with |Q0| ≪ m0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' There exist Vb ∈ Ψ−∞(X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' S2 �  scT ∗ ¯X ⊕ �  scT ∗ ¯X) which  is continuous in b with uniformly compactly supported Schwartz kernel, and a constant C0 > 0, such that the  following statements hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  (1) The operator  ˇLb(σ) := �  Lb(σ) + Vb : X s,ℓ  b (σ) → ¯Hs−1,ℓ+2  b  ( ¯X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' S2 �  scT ∗ ¯X ⊕ �  scT ∗ ¯X)  is invertible for σ ∈ C, Im σ ≥ 0 with |σ| + |b − b0| ≤ C0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  (2) The inverse operator ˇLb(σ)−1 is continuous in σ with values in Lweak( ¯Hs−1,ℓ+2  b  ( ¯X), ¯Hs,ℓ  b ( ¯X)) (the  space of linear bounded operators equipped with weak operator topology), and continuous in σ with  values in Lop( ¯Hs−1+ǫ,ℓ+2+ǫ  b  ( ¯  X), ¯Hs−ǫ,ℓ−ǫ  b  ( ¯X)) (norm topology) for any ǫ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  (3) One has  Vb(ˇgb, ˇAb) = 0  for  (ˇgb, ˇAb) ∈ ˇKb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  (11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='7)  128  LILI HE  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' The proof closely follows [36, Lemma 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='1 and Lemma 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Concretely, we shall construct  Vb : D′(X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' S2 �  scT ∗ ¯X ⊕ �  scT ∗ ¯X) → C∞  c (X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' S2T ∗X ⊕ T ∗X)  where Vb satisfy  ˇKb ⊂ ker Vb,  ranVb ⊂ R⊥  b  and  Vb|Kb : Kb → R⊥  b  is invertible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  (11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='8)  where Rb :=  �  ranX s,ℓ  b  (0)�  Lb(0)  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' We now prove that once the above requirements are satisfied, all the three  statements in the lemma hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' As for the proof of statements (1) and (2), it suffices to prove the invertiblity  of ˇLb(σ) = �  Lb(σ) + Vb for the case b = b0, σ = 0 since the invertibility and continuity for (b, σ) near (b0, 0)  follow from the same arguments as in the proof of Lemma 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' We split the domain X s,ℓ  b0 (0) = K⊥  b0 ⊕Kb0 and  the target space ¯Hs−1,ℓ+2  b  ( ¯X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' S2 �  scT ∗ ¯X ⊕ �  scT ∗ ¯X) = Rb0 ⊕ R⊥  b0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Under these splittings, ˇLb0(0) = �  Lb0(0) + Vb  takes the form  ˇLb0(0) =  �L00  0  V10  V11  �  where L00 = �  Lb0(0)|K⊥  b0 : K⊥  b0 → Rb0 is invertible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Then the invertibility of ˇLb0(0) follows from that of  V11 : Kb0 → R⊥  b0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' We notice that the invertibility of V11 holds provided that the above conditions in (11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='8)  are satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' For the statement (3), it is clear that the conclusion (11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='7) directly follows from the condition  ˇKb ⊂ ker Vb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  First, since (˙gb0,s1(S1), ˙Ab0,s1(S1)) ∈ ¯H∞,1/2−  b  ( ¯X) (see Theorem 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='1) and (ˇgb0,s1(S1), ˇAb0,s1(S1)) have  nonzero r−1 leading terms (see Lemma 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='6), we conclude that Kb0,1∩ ˇKb0 = 0, and thus Kb0∩ ˇKb0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' By the  continuity of Kb and ˇKb in b, this holds for b near b0 as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' This implies that fixing the basis {hb,1, · · · , hb,8}  of Kb and {ˇhb,1, ˇhb,2, ˇhb,3} of ˇKb, both of which depend continuously on b, by the same reasoning as in the  beginning of the seventh step in the proof of Theorem 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='1, there exists {h♭  b,1, · · · , h♭  b,8} ⊂ C∞  c (X◦;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' S2 �  scT ∗ ¯X⊕  �  scT ∗ ¯X) such that ⟨hb,i, h♭  b,j⟩ = δij for 1 ≤ i, j ≤ 8 and ⟨ˇhb,i, h♭  b,j⟩ = 0 for 1 ≤ i ≤ 3, 1 ≤ j ≤ 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' (One can  verify that {h♭  b,1, · · · , h♭  b,8} ⊂ C∞  c (X◦) can be chosen to be continuous in b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=') Next, we pick a basis of R⊥  b in  the following way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' We note that K∗  b ∩ ˇK∗  b = 0, which can be proved in a similar manner to the statement  Kb ∩ ˇKb = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  Fixing the continuous (in b) basis {h∗  b,1, · · · , h∗  b,8} of K∗  b = ann(Rb) and {ˇh∗  b,1, ˇh∗  b,2, ˇh∗  b,3}  of ˇK∗  b, by the same reasoning mentioned above, there exists {h♯  b,1, · · · , h♯  b,8} ⊂ C∞  c (X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' S2 �  scT ∗ ¯X ⊕ �  scT ∗ ¯X)  (which can be chosen continuous in b) such that ⟨h♯  b,i, h∗  b,j⟩ = δij for 1 ≤ i, j ≤ 8 and ⟨h♯  b,i, ˇh∗  b,j⟩ = 0 for  1 ≤ i ≤ 8, 1 ≤ j ≤ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  It is easy to check that {h♯  b,1, · · · , h♯  b,8} are linearly independent and the space  generated by h♯  b,j, 1 ≤ j ≤ 8 is a complement of Rb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Therefore, we define  Vb(h) :=  8  �  i=1  h♯  b,i⟨h, h♭  b,i⟩,  which satisfies the requirement (11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  □  Using ˇLb(σ) constructed in Lemma 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='1, we define the following spaces  �Kb,1 := ˇLb(0)Kb,1,  �Kb,2 := ˇLb(0)Kb,2,  �Kb := ˇLb(0)Kb = �Kb,1 ⊕ �Kb,2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  (11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='9)  Since ˇLb(0)Kb = Vb(Kb), we have �Kb,1, �Kb,2, �Kb ⊂ C∞  c (X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' S2 �  scT ∗ ¯X ⊕ �  scT ∗ ¯X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  Next, we decompose the target space ¯Hs−1,ℓ+2  b  ( ¯X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' S2 �  scT ∗ ¯X ⊕ �  scT ∗ ¯X) into the range of �  Lb(0) and its  complement in a continuous (in b) way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  Lemma 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Let b0 = (m0, 0, Q0) with |Q0| ≪ m0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' There exists a linear projection map  Π⊥  b : ¯Hs−1,ℓ+2  b  ( ¯X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' S2 �  scT ∗ ¯X ⊕ �  scT ∗ ¯X) → ¯Hs−1,ℓ+2  b  ( ¯X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' S2 �  scT ∗ ¯X ⊕ �  scT ∗ ¯X),  which is of rank 8 and depends continuously on b near b0 in the norm topology, such that  ⟨(I − Π⊥  b )h, h∗⟩ = 0  for all  h∗ ∈ K∗  b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  LINEAR STABILITY OF KERR-NEWMAN FAMILY  129  Moreover, the Π⊥  b can be chosen such that it maps ¯Hs−1,ℓ+2  b  ( ¯X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' S2 �  scT ∗ ¯X ⊕ �  scT ∗ ¯X) to C∞  c (X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' S2 �  scT ∗ ¯X ⊕  �  scT ∗ ¯X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Let {h∗  b,1, · · · , h∗  b,8} be a basis which is continuous in b of K∗  b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  There exists {h♯  b,1, · · · , h♯  b,8} ⊂  C∞  c (X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' S2 �  scT ∗ ¯X ⊕ �  scT ∗ ¯X) which is continuous in b, such that ⟨h♯  b,i, h∗  b,j⟩ = δij.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' We then define  Π⊥  b h =  8  �  i=1  h♯  b,i⟨h, h∗  b,i⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  It is easy to check that Π⊥  b has rank 8 and satisfies ⟨(I − Π⊥  b )h, h∗⟩ = 0 for all h∗ ∈ K∗  b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  □  We then define the projection onto Rb which is the range of �  Lb(0)  Πb = I − Π⊥  b : ¯Hs−1,ℓ+2  b  ( ¯X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' S2 �  scT ∗ ¯X ⊕ �  scT ∗ ¯X) → Rb = ann K∗  b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  Now we are able to split the domain and target space of the operator �  Lb(σ)ˇLb(σ)−1 : ¯Hs−1,ℓ+2  b  → ¯Hs−1,ℓ+2  b  as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  domain:  ¯Hs−1,ℓ+2  b  ( ¯X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' S2 �  scT ∗ ¯X ⊕ �  scT ∗ ¯X) = �K⊥  b ⊕ �Kb,1 ⊕ �Kb,2,  target:  ¯Hs−1,ℓ+2  b  ( ¯X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' S2 �  scT ∗ ¯X ⊕ �  scT ∗ ¯X) = Rb ⊕ R⊥  b,1 ⊕ R⊥  b,2  (11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='10)  where R⊥  b,1, resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' R⊥  b,2 is a subspace of dimension 3, resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' 5 and defined as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Fixing the continuous  (in b) basis {h∗(1)  b,1 , · · · , h∗(1)  b,3 } of K∗  b,1 and {h∗(2)  b,1 , · · · , h∗(2)  b,5 } of K∗  b,2, there exist  {v(1)  b,1, v(1)  b,2, v(1)  b,3}, {v(2)  b,1, v(2)  b,2, v(2)  b,3, v(2)  b,4, v(2)  b,5} ⊂ C∞  c (X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' S2 �  scT ∗ ¯X ⊕ �  scT ∗ ¯X)  such that ⟨v(1)  b,i , h∗(1)  b,j ⟩ = δij and ⟨v(2)  b,i , h∗(2)  b,j ⟩ = δij.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' We then define  R⊥  b,1 = {v(1)  b,1, v(1)  b,2, v(1)  b,3}  and  R⊥  b,2 = {v(2)  b,1, v(2)  b,2, v(2)  b,3, v(2)  b,4, v(2)  b,5}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Structure of the resolvent near zero energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Now we are at the position of establishing the mode  stability of the operator �  Lb(σ) for b near b0 at Im σ ≥ 0, σ ̸= 0, and we also provide a description of the  structure of the resolvent �  Lb(σ)−1 near σ = 0 along the way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  Theorem 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Let (gb, Ab) be the RN metric and 4-electromagnetic potential where b = (m, a, Q) is near  b0 = (m0, 0, Q0) with |Q0| ≪ m0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Then the Fourier-transformed operator  �  Lb(σ) : {(˙g, ˙A) ∈ ¯Hs,ℓ  b ( ¯X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' S2 �  scT ∗ ¯X ⊕ �  scT ∗ ¯X) : �  Lb(σ)(˙g, ˙A) ∈ ¯Hs,ℓ+1  b  ( ¯X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' S2 �  scT ∗ ¯X ⊕ �  scT ∗ ¯X)}  → ¯Hs,ℓ+1  b  ( ¯X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' S2 �  scT ∗ ¯X ⊕ �  scT ∗ ¯X)  (11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='11)  is invertible for σ ∈ C, Im σ ≥ 0, σ ̸= 0 and s > 3, ℓ < − 1  2, s + ℓ > − 1  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' By the arguments in the last step of the proof of Theorem 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='1 and Proposition 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='10, it suffices to  prove that X s,ℓ  b  (σ) → ¯Hs,ℓ+1  b  ( ¯X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' S2 �  scT ∗ ¯X ⊕ �  scT ∗ ¯X) is invertible for s > 3, −3/2 < ℓ < −1/2 and (b, σ) with  Im σ ≥ 0, σ ̸= 0 near (b0, 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' To this end, we shall prove the invertibility of the operator  �  Lb(σ)ˇLb(σ)−1 : ¯Hs−1,ℓ+2  b  ( ¯X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' S2 �  scT ∗ ¯X ⊕ �  scT ∗ ¯X) → ¯Hs−1,ℓ+2  b  ( ¯X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' S2 �  scT ∗ ¯X ⊕ �  scT ∗ ¯X)  for s > 3, −3/2 < ℓ < −1/2 and (b, σ) with Im σ ≥ 0, σ ̸= 0 near (b0, 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  Under the splitting (11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='10), we express  �  Lb(σ)ˇLb(σ)−1 =  \uf8eb  \uf8ed  L00(b, σ)  L01(b, σ)  L02(b, σ)  L10(b, σ)  L11(b, σ)  L12(b, σ)  L20(b, σ)  L21(b, σ)  L22(b, σ)  \uf8f6  \uf8f8 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  (11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='12)  It is clear that Lij(b, 0) = 0 when (i, j) ̸= (0, 0) and L00(b, 0) is invertible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' We will use the following resolvent  identity (whose proof is similar to that of the resolvent identity (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='35) in the proof of Lemma 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='12) frequently  in the subsequent analysis  �ˇLb(σ)−1 − ˇLb(0)−1�  (˙g, ˙A) = ˇLb(σ)−1�ˇLb(0) − ˇLb(σ)  �ˇLb(0)−1(˙g, ˙A)  (11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='13)  �  (ˇLb(σ)−1)∗ − (ˇLb(0)−1)∗�  (˙g∗, ˙A∗) = (ˇLb(σ)−1)∗�ˇLb(0)∗ − ˇLb(σ)∗�  (ˇLb(0)−1)∗(˙g∗, ˙A∗)  (11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='14)  130  LILI HE  for ˇLb(0)−1(˙g, ˙A) ∈ ρC∞( ¯  X) + ¯Hs,1/2−  b  ( ¯X) and (ˇLb(0)−1)∗(˙g∗, ˙A∗) ∈ ρC∞( ¯X) + ˙H−s+1,1/2−  b  ( ¯X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  Analysis of L01 and L02.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' For (˜g2, ˜A2) = ˇLb(0)(˙g2, ˙A2) ∈ �Kb,2, we have  �  Lb(σ)ˇLb(σ)−1(˜g2, ˜A2) =  �ˇLb(σ) − Vb  �ˇLb(σ)−1 ˇLb(0)(˙g2, ˙A2)  = −Vb  �ˇLb(σ)−1 − ˇLb(0)−1�ˇLb(0)(˙g2, ˙A2)  = Vb ˇLb(σ)−1�ˇLb(σ) − ˇLb(0)  �ˇLb(0)−1 ˇLb(0)(˙g2, ˙A2)  = σVb ˇLb(σ)−1∂σ�  Lb(0)(˙g2, ˙A2) + σ2  2 Vb ˇLb(σ)−1∂2  σ�  Lb(0)(˙g2, ˙A2)  (11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='15)  where we use ˇLb(0)−1 ˇLb(0)(˙g2, ˙A2) = (˙g2, ˙A2) ∈ ρC∞( ¯X) + ¯H∞,1/2−  b  ( ¯X) by Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='7 and  the resolvent identity (11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='13) in the third step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Since (˙g2, ˙A2) ∈ ρC∞( ¯X) + ¯H∞,1/2−  b  ( ¯X) which is  annihilated by normal operator of ∂σ�  Lb(0) modulo ¯H∞,3/2−  b  ( ¯X), it follows that  ∂σ�  Lb(0)(˙g2, ˙A2) ∈ ¯H∞,3/2−  b  ( ¯  X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  As ∂2  σ�  Lb(0) ∈ ρ2C∞( ¯X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' End(S2 �  scT ∗ ¯X ⊕ �  scT ∗ ¯X)), we have  ∂2  σ�  Lb(0)(˙g2, ˙A2) ∈ ¯H∞,3/2−  b  ( ¯X)  as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' By using the continuity of ˇLb(σ)−1 in the norm operator topology  Lop( ¯Hs−1+ǫ,ℓ+2+ǫ  b  ( ¯X), ¯Hs−ǫ,ℓ−ǫ  b  ( ¯X)),  we conclude that  ˇLb(σ)−1∂σ�  Lb(0)(˙g2, ˙A2), ˇLb(σ)−1∂2  σ�  Lb(0)(˙g2, ˙A2)  are continuous in (b, σ), and thus  L02(b, σ) = σ�L02(b, σ)  where �L02(b, σ) : �Kb,2 → Rb is continuous in (b, σ) in the norm operator topology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  For (˜g1, ˜A1) = ˇLb(0)(˙g1, ˙A1) ∈ �Kb,1, the calculation in (11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='15) still applies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' We further calculate  �  Lb(σ)ˇLb(σ)−1(˜g1, ˜A1)  = σVb ˇLb(σ)−1∂σ�  Lb(0)(˙g1, ˙A1) + σ2  2 Vb ˇLb(σ)−1∂2  σ�  Lb(0)(˙g1, ˙A1)  = iσVb ˇLb(σ)−1�  Lb(0)(ˇg1, ˇA1) + σ2  2 Vb ˇLb(σ)−1∂2  σ�  Lb(0)(˙g1, ˙A1)  = iσ  �  I − Vb ˇLb(0)−1�  Vb(ˇg1, ˇA1) + iσVb(ˇLb(σ)−1 − ˇLb(0)−1)�  Lb(0)(ˇg1, ˇA1)  + σ2  2 Vb ˇLb(σ)−1∂2  σ�  Lb(0)(˙g1, ˙A1)  = iσVb(ˇLb(σ)−1 − ˇLb(0)−1)�  Lb(0)(ˇg1, ˇA1) + σ2  2 Vb ˇLb(σ)−1∂2  σ�  Lb(0)(˙g1, ˙A1)  where we use Lemma 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='1 in the last step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Since by Lemma 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='1 and Lemma 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='6, we have  ˇLb(0)−1�  Lb(0)(ˇg1, ˇA1) = (ˇg1, ˇA1) ∈ ρC∞( ¯X) + ¯H∞,1/2−  b  ( ¯X),  and it follows that we can apply the resolvent identity (11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='13) again and obtain  �  Lb(σ)ˇLb(σ)−1(˜g1, ˜A1)  = iσVb ˇLb(σ)−1�ˇLb(0) − ˇLb(σ)  �ˇLb(0)−1�  Lb(0)(ˇg1, ˇA1) + σ2  2 Vb ˇLb(σ)−1∂2  σ�  Lb(0)(˙g1, ˙A1)  = σ2�  − iVb ˇLb(σ)−1�  ∂σ�  Lb(0) + σ  2 ∂2  σ�  Lb(0)  �  (ˇg1, ˇA1) + 1  2Vb ˇLb(σ)−1∂2  σ�  Lb(0)(˙g1, ˙A1)  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  (11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='16)  By the same reasoning as in the component L02(b, σ), we find that  L01(b, σ) = σ2�L01(b, σ)  where �L01(b, σ) : �Kb,1 → Rb is continuous in (b, σ) in the norm operator topology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  LINEAR STABILITY OF KERR-NEWMAN FAMILY  131  Analysis of L10 and L20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' For (˜g0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' ˜A0) ∈ �K⊥  b and (˙g∗  2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' ˙A∗  2) ∈ K∗  b,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' we have  ⟨�  Lb(σ)ˇLb(σ)−1(˜g0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' ˜A0),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' (˙g∗  2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' ˙A∗  2)⟩  = ⟨(˜g0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' ˜A0),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' (ˇLb(σ)−1)∗�  Lb(σ)∗(˙g∗  2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' ˙A∗  2)⟩  = ⟨(˜g0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' ˜A0),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' (˙g∗  2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' ˙A∗  2)⟩ − ⟨(˜g0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' ˜A0),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' (ˇLb(σ)−1)∗V ∗  b (˙g∗  2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' ˙A∗  2)⟩  = −  �  (˜g0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' ˜A0),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  �  (ˇLb(σ)−1)∗ − (ˇLb(0)−1)∗�  V ∗  b (˙g∗  2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' ˙A∗  2)  �  =  �  (˜g0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' ˜A0),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' (ˇLb(σ)−1)∗�ˇLb(σ)∗ − ˇLb(0)∗�  (ˇLb(0)−1)∗V ∗  b (˙g∗  2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' ˙A∗  2)  �  = σ  �  (˜g0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' ˜A0),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' (ˇLb(σ)−1)∗�  ∂σ�  Lb(0) + σ  2 ∂2  σ�  Lb(0)  �∗(˙g∗  2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' ˙A∗  2)  �  (11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='17)  where we use (ˇLb(0)−1)∗V ∗  b (˙g∗  2, ˙A∗  2) = (˙g∗  2, ˙A∗  2) ∈ ρC∞( ¯X) + ˙H−3/2−C(a,γ),1/2−  b  ( ¯X) where C(a, γ)  is a small constant by Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='7 and the resolvent identity (11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='14) in the fourth step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Since  (˙g∗  2, ˙A∗  2) is annihilated by normal operator of (∂σ�  Lb(0))∗ modulo ˙H−5/2−C(a,γ),3/2−  b  ( ¯X), it follows  that (∂σ�  Lb(0))∗(˙g∗  2, ˙A∗  2) ∈ ˙H−5/2−C(a,γ),3/2−  b  ( ¯X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' As (∂2  σ�  Lb(0))∗ ∈ ρ2C∞( ¯X), we have  (∂2  σ�  Lb(0))∗(˙g∗  2, ˙A∗  2) ∈ ˙H−3/2−ν−,3/2−  b  ( ¯X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  Since (ˇLb(σ)−1)∗ is continuous with values in Lop( ˙H−s+ǫ,−ℓ+ǫ  b  ( ¯  X), ˙H−s+1−ǫ,−ℓ−2−ǫ  b  ( ¯X)), we conclude  that  (ˇLb(σ)−1)∗(∂σ�  Lb(0))∗(˙g∗  2, ˙A∗  2), (ˇLb(σ)−1)∗(∂2  σ�  Lb(0))∗(˙g∗  2, ˙A∗  2)  are continuous in (b, σ), and thus  L20(b, σ) = σ�L20(b, σ)  where �L20(b, σ) : �K⊥  b → R⊥  b,2 is continuous in (b, σ) in the norm operator topology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  For (˜g0, ˜A0) ∈ �K⊥  b and (˙g∗  1, ˙A∗  1) ∈ K∗  b,1, the calculation in (11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='17) still applies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' We further calculate  ⟨�  Lb(σ)ˇLb(σ)−1(˜g0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' ˜A0),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' (˙g∗  1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' ˙A∗  1)⟩  = σ  �  (˜g0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' ˜A0),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' (ˇLb(σ)−1)∗�  ∂σ�  Lb(0) + σ  2 ∂2  σ�  Lb(0)  �∗(˙g∗  1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' ˙A∗  1)  �  = σ  �  (˜g0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' ˜A0),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' i(ˇLb(σ)−1)∗�  Lb(0)∗(ˇg∗  1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' ˇA∗  1)  �  + σ2�  (˜g0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' ˜A0),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' 1  2(ˇLb(σ)−1)∗(∂2  σ�  Lb(0))∗(˙g∗  1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' ˙A∗  1)  �  = σ  �  (˜g0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' ˜A0),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' i  �  I − (ˇLb(0)−1)∗V ∗  b  �  (ˇg∗  1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' ˇA∗  1)  �  + σ  �  (˜g0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' ˜A0),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' i  �  (ˇLb(σ)−1)∗ − (ˇLb(0)−1)∗��  Lb(0)∗(ˇg∗  1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' ˇA∗  1)  �  + σ2�  (˜g0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' ˜A0),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' 1  2(ˇLb(σ)−1)∗(∂2  σ�  Lb(0))∗(˙g∗  1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' ˙A∗  1)  �  By a normal operator argument as in the proof of Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='7, we have  (ˇLb(0)−1)∗ : C∞  c (X) → ρC∞( ¯X) + ˙H−3/2−C(a,γ),1/2−  b  ( ¯X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  Therefore, by Lemma 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='6, we conclude  (ˇLb(0)−1)∗�  Lb(0)∗(ˇg1, ˇA1) = (ˇg1, ˇA1) − (ˇLb(0)−1)∗V ∗  b (ˇg1, ˇA1) ∈ ρC∞( ¯X) + ˙H−3/2−C(a,γ),1/2−  b  ( ¯X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  This implies that we can apply the resolvent identity (11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='14) again and obtain  ⟨�  Lb(σ)ˇLb(σ)−1(˜g0, ˜A0), (˙g∗  1, ˙A∗  1)⟩  = σ  �  (˜g0, ˜A0), i  �  I − (ˇLb(0)−1)∗V ∗  b  �  (ˇg∗  2, ˇA∗  2)  �  − σ2�  (˜g0, ˜A0), i(ˇLb(σ)−1)∗(∂σ�  Lb(0) + σ  2 ∂2  σ�  Lb(0))∗�  I − (ˇLb(0)−1)∗V ∗  b  �  (ˇg∗  2, ˇA∗  2)  �  + σ2�  (˜g0, ˜A0), 1  2(ˇLb(σ)−1)∗(∂2  σ�  Lb(0))∗(˙g∗  2, ˙A∗  2)  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  (11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='18)  132  LILI HE  By the same reasoning as in the component L20(b, σ), we find that  L10(b, σ) = σ�L10(b, σ)  where �L10(b, σ) = �L0  10(b) + σ�Le  10(b, σ) with �Le  10(b, σ) : �K⊥  b → R⊥  b,1 continuous in (b, σ) in the norm  operator topology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  Analysis of L22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Using (11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='15),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' we find that  ⟨�  Lb(σ)ˇLb(σ)−1(˜g2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' ˜A2),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' (˙g∗  2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' ˙A∗  2)⟩  = ⟨σ∂σ�  Lb(0)(˙g2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' ˙A2) + σ2  2 ∂2  σ�  Lb(0)(˙g2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' ˙A2),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' (ˇLb(σ)−1)∗V ∗  b (˙g∗  2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' ˙A∗  2)⟩  = ⟨σ∂σ�  Lb(0)(˙g2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' ˙A2) + σ2  2 ∂2  σ�  Lb(0)(˙g2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' ˙A2),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' (˙g∗  2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' ˙A∗  2)⟩  + ⟨σ∂σ�  Lb(0)(˙g2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' ˙A2) + σ2  2 ∂2  σ�  Lb(0)(˙g2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' ˙A2),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  �  (ˇLb(σ)−1)∗ − (ˇLb(0)−1)∗�  V ∗  b (˙g∗  2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' ˙A∗  2)⟩  = ⟨σ∂σ�  Lb(0)(˙g2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' ˙A2) + σ2  2 ∂2  σ�  Lb(0)(˙g2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' ˙A2),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' (˙g∗  2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' ˙A∗  2)⟩  + ⟨σ∂σ�  Lb(0)(˙g2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' ˙A2) + σ2  2 ∂2  σ�  Lb(0)(˙g2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' ˙A2),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' (ˇLb(σ)−1)∗�ˇLb(0)∗ − ˇLb(σ)∗�  (˙g∗  2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' ˙A∗  2)⟩  = σ⟨−i[Lb,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' tb,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='∗](˙g2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' ˙A2) (˙g∗  2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' ˙A∗  2)⟩ + σ2�  ⟨1  2∂2  σ�  Lb(0)(˙g2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' ˙A2),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' (˙g∗  2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' ˙A∗  2)⟩  −  �  ∂σ�  Lb(0)(˙g2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' ˙A2) + σ  2 ∂2  σ�  Lb(0)(˙g2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' ˙A2),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' (ˇLb(σ)−1)∗(∂σ�  Lb(0) + σ  2 ∂2  σ�  Lb(0))∗(˙g∗  2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' ˙A∗  2)  ��  (11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='19)  where we use (ˇLb(0)−1)∗V ∗  b (˙g∗  2, ˙A∗  2) = (˙g∗  2, ˙A∗  2) and the resolvent identity (11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='14) in the third step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  According to the proof of the non-existence of the linearly growing generalized zero modes with  leading order term (˙g2, ˙A2) ∈ Kb,2 (see Lemma 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='3 and 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='4), the pairing on Kb,2 × K∗  b,2  ⟨−i[Lb, tb,∗](˙g2, ˙A2), (˙g∗  2, ˙A∗  2)⟩  is non-degenerate for b near b0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Therefore, we find that  L22(b, σ) = σ�L22(b, σ)  with  �L22(b, σ) = �L0  22(b) + σ�Le  22(b, σ)  where �L0  22(b) is invertible and �Le  22(b, σ) : �Kb,2 → R⊥  b,2 is continuous in (b, σ) in the norm operator  topology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  LINEAR STABILITY OF KERR-NEWMAN FAMILY  133  Analysis of L11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Using the calculation in (11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='16) and (11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='18),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' we see that  ⟨�  Lb(σ)ˇLb(σ)−1(˜g1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' ˜A1),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' (˙g∗  1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' ˙A∗  1)⟩  = σ2⟨−i  �  ∂σ�  Lb(0) + σ  2 ∂2  σ�  Lb(0)  �  (ˇg1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' ˇA1) + 1  2∂2  σ�  Lb(0)(˙g1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' ˙A1),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' (ˇLb(σ)−1)∗V ∗  b (˙g∗  1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' ˙A∗  1)⟩  = σ2⟨−i  �  ∂σ�  Lb(0) + σ  2 ∂2  σ�  Lb(0)  �  (ˇg1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' ˇA1) + 1  2∂2  σ�  Lb(0)(˙g1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' ˙A1),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' (˙g∗  1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' ˙A∗  1)⟩  + σ2⟨−i  �  ∂σ�  Lb(0) + σ  2 ∂2  σ�  Lb(0)  �  (ˇg1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' ˇA1) + 1  2∂2  σ�  Lb(0)(˙g1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' ˙A1),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' (ˇLb(σ)−1)∗(ˇLb(0) − ˇLb(σ))∗(˙g∗  1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' ˙A∗  1)⟩  = σ2⟨−i  �  ∂σ�  Lb(0) + σ  2 ∂2  σ�  Lb(0)  �  (ˇg1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' ˇA1) + 1  2∂2  σ�  Lb(0)(˙g1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' ˙A1),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' (˙g∗  1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' ˙A∗  1)⟩  − σ3⟨−i  �  ∂σ�  Lb(0) + σ  2 ∂2  σ�  Lb(0)  �  (ˇg1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' ˇA1) + 1  2∂2  σ�  Lb(0)(˙g1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' ˙A1),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' i(ˇLb(σ)−1)∗�  Lb(0)∗(ˇg∗  1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' ˇA∗  1)⟩  − σ4⟨−i  �  ∂σ�  Lb(0) + σ  2 ∂2  σ�  Lb(0)  �  (ˇg1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' ˇA1) + 1  2∂2  σ�  Lb(0)(˙g1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' ˙A1),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' (ˇLb(σ)−1)∗(1  2∂2  σ�  Lb(0))∗(˙g∗  1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' ˙A∗  1)⟩  = σ2⟨−i∂σ�  Lb(0)(ˇg1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' ˇA1) + 1  2∂2  σ�  Lb(0)(˙g1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' ˙A1),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' (˙g∗  2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' ˙A∗  2)⟩ + σ3⟨−i  2 ∂2  σ�  Lb(0)(ˇg1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' ˇA1),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' (˙g∗  1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' ˙A∗  1)⟩  − σ3⟨−i∂σ�  Lb(0)(ˇg1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' ˇA1) + 1  2∂2  σ�  Lb(0)(˙g1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' ˙A1),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' i(I − (ˇLb(0)−1)∗V ∗  b )(ˇg∗  1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' ˇA∗  1)⟩  − σ4⟨−i  2 ∂2  σ�  Lb(0)(ˇg1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' ˇA1),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' i(I − (ˇLb(0)−1)∗V ∗  b )(ˇg∗  1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' ˇA∗  1)⟩  + σ4�  − i  �  ∂σ�  Lb(0) + σ  2 ∂2  σ�  Lb(0)  �  (ˇg1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' ˇA1) + 1  2∂2  σ�  Lb(0)(˙g1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' ˙A1),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  (ˇLb(σ)−1)∗(∂σ�  Lb(0) + σ  2 ∂2  σ�  Lb(0))∗(I − (ˇLb(0)−1)∗V ∗  b )(ˇg∗  1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' ˇA∗  1)  �  − σ4⟨−i  �  ∂σ�  Lb(0) + σ  2 ∂2  σ�  Lb(0)  �  (ˇg1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' ˇA1) + 1  2∂2  σ�  Lb(0)(˙g1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' ˙A1),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' (ˇLb(σ)−1)∗(1  2∂2  σ�  Lb(0))∗(˙g∗  1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' ˙A∗  1)⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  (11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='20)  According to the proof of the non-existence of the quadratically growing generalized zero modes with  leading order term (˙g1, ˙A1) ∈ Kb,1 (see Lemma 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='7), the pairing on Kb,1 × K∗  b,1  ⟨−i∂σ�  Lb(0)(ˇg1, ˇA1) + 1  2∂2  σ�  Lb(0)(˙g1, ˙A1), (˙g∗  2, ˙A∗  2)⟩  = −⟨[Lb, tb,∗](ˇg1, ˇA1) + 1  2[[Lb, tb,∗], tb,∗](˙g1, ˙A1), (˙g∗  2, ˙A∗  2)⟩  is non-degenerate for b near b0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Therefore, we find that  L11(b, σ) = σ2�L11(b, σ)  with  �L11(b, σ) = �L0  11(b) + σ�L1  22(b) + σ2�Le  11(b, σ)  where �L0  11(b) is invertible and �Le  11(b, σ) : �Kb,1 → R⊥  b,1 is continuous in (b, σ) in the norm operator  topology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  Analysis of L12 and L21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' For (˜g1, ˜A1) ∈ �Kb,1 and (˙g∗  2, ˙A∗  2) ∈ K∗  b,2, using (11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='16) and (11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='19), we  compute  ⟨�  Lb(σ)ˇLb(σ)−1(˜g1, ˜A1), (˙g∗  2, ˙A∗  2)⟩  = σ2⟨−i∂σ�  Lb(0)(ˇg1, ˇA1) + 1  2∂2  σ�  Lb(0)(˙g1, ˙A1), (˙g∗  2, ˙A∗  2)⟩ + σ3⟨−i  2 ∂2  σ�  Lb(0)(ˇg1, ˇA1), (˙g∗  2, ˙A∗  2)⟩  − σ3�  − i  �  ∂σ�  Lb(0) + σ  2 ∂2  σ�  Lb(0)  �  (ˇg1, ˇA1) + 1  2∂2  σ�  Lb(0)(˙g1, ˙A1),  (ˇLb(σ)−1)∗�  ∂σ�  Lb(0) + σ  2 ∂2  σ�  Lb(0)  �∗(˙g∗  2, ˙A∗  2)  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  (11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='21)  Therefore,  L21(b, σ) = σ2�L21(b, σ)  with  �L21(b, σ) = �L0  21(b) + σ�Le  21(b, σ)  where �Le  21(b, σ) : �Kb,1 → R⊥  b,2 is continuous in (b, σ) in the norm operator topology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  134  LILI HE  For (˜g2, ˜A2) = ˇLb(0)(˙g2, ˙A2) ∈ �Kb,2 = ˇLb(0)Kb,2 and (˙g∗  1, ˙A∗  1) ∈ K∗  b,1, using (11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='18),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' we calculate  ⟨�  Lb(σ)ˇLb(σ)−1(˜g2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' ˜A2),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' (˙g∗  1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' ˙A∗  1)⟩  = σ  ��  I − Vb(ˇLb(0)−1)  �ˇLb(0)(˙g2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' ˙A2),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' i(ˇg∗  1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' ˇA∗  1)  �  − σ2�  ˇLb(σ)−1 ˇLb(0)(˙g2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' ˙A2),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' i(∂σ�  Lb(0))∗�  I − (ˇLb(0)−1)∗V ∗  b  �  (ˇg∗  1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' ˇA∗  1)  �  + σ2�  ˇLb(σ)−1 ˇLb(0)(˙g2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' ˙A2),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' 1  2(∂2  σ�  Lb(0))∗(˙g∗  1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' ˙A∗  1)  �  − σ3�  ˇLb(σ)−1 ˇLb(0)(˙g2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' ˙A2),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' i  2(∂σ�  Lb(0))∗�  I − (ˇLb(0)−1)∗V ∗  b  �  (ˇg∗  1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' ˇA∗  1)  �  = −σ2�  (˙g2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' ˙A2),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' i(∂σ�  Lb(0))∗�  I − (ˇLb(0)−1)∗V ∗  b  �  (ˇg∗  2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' ˇA∗  2) − 1  2(∂2  σ�  Lb(0))∗(˙g∗  1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' ˙A∗  1)  �  + σ3�  ˇLb(σ)−1(∂σ�  Lb(0) + σ  2 ∂2  σ�  Lb(0))(˙g2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' ˙A2),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' i(∂σ�  Lb(0))∗�  I − (ˇLb(0)−1)∗V ∗  b  �  (ˇg∗  1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' ˇA∗  1)  �  + σ3�  ˇLb(σ)−1(∂σ�  Lb(0) + σ  2 ∂2  σ�  Lb(0))(˙g2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' ˙A2),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' −1  2(∂2  σ�  Lb(0))∗(˙g∗  1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' ˙A∗  1)  �  − σ3�  ˇLb(σ)−1 ˇLb(0)(˙g2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' ˙A2),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' i  2(∂2  σ�  Lb(0))∗�  I − (ˇLb(0)−1)∗V ∗  b  �  (ˇg∗  1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' ˇA∗  1)  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  (11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='22)  Therefore,  L12(b, σ) = σ2�L12(b, σ)  with  �L12(b, σ) = �L0  12(b) + σ�Le  12(b, σ)  where �Le  12(b, σ) : �Kb,2 → R⊥  b,1 is continuous in (b, σ) in the norm operator topology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  Analysis of L00.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Following the arguments in the first step in the proof of Lemma 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='12, we can prove  that for (b, σ) near (b, 0), L00(b, σ) is invertible and there exists a uniform constant C > 0 such that  ∥L00(b, σ)−1∥Rb→ �  K⊥  b ≤ C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  (11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='23)  Moreover, for (˜g0, ˜A0) ∈ �K⊥  b , we write  L00(b, σ)(˜g0, ˜A0) = �  Lb(σ)ˇLb(σ)−1(˜g0, ˜A0) −  �  L10(b, σ) + L20(b, σ)  �  (˜g0, ˜A0)  = (˜g0, ˜A0) − Vb ˇLb(σ)−1(˜g0, ˜A0) −  �  L10(b, σ) + L20(b, σ)  �  (˜g0, ˜A0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  (11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='24)  Since ˇLb(σ)−1 is continuous in the norm operator topology Lop( ¯Hs−1+ǫ,ℓ+2+ǫ  b  ( ¯X), ¯Hs−ǫ,ℓ−ǫ  b  ( ¯X)) and  Vb maps D′(X◦) to C∞  c (X), it follows that L00(b, σ) : �K⊥  b → Rb is continuous in (b, σ) in the norm  operator topology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  The inverse of �  Lb(σ)ˇLb(σ)−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Putting all the above analysis of the components Lij(b,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' σ) together,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  we conclude that  �  Lb(σ)ˇLb(σ)−1 =  \uf8eb  \uf8ec  \uf8ed  L00(b,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' σ)  σ2�L01(b,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' σ)  σ�L02(b,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' σ)  σ�L10(b,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' σ)  σ2�L11(b,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' σ)  σ2�L12(b,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' σ)  σ�L20(b,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' σ)  σ2�L21(b,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' σ)  σ�L22(b,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' σ)  \uf8f6  \uf8f7  \uf8f8  =  \uf8eb  \uf8ec  \uf8ed  L00(b,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' σ)  σ2 �L01(b,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' σ)  σ�L02(b,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' σ)  σ  ��L0  10(b) + σ�Le  10(b,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' σ)  �  σ2��L0  11(b) + σ�L1  11(b) + σ2�Le  11(b,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' σ)  �  σ2��L0  12(b) + σ�Le  12(b,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' σ)  �  σ�L20(b,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' σ)  σ2��L0  21(b) + σ�Le  21(b)  �  σ  ��L0  22(b) + σ�Le  22(b,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' σ)  �  \uf8f6  \uf8f7  \uf8f8  (11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='25)  where L00(b, σ), �L0  11(b), �L0  22(b) are invertible, and L00(b, σ), �Lij(b, σ) with (i, j) = (0, 1), (0, 2), (2, 0),  �Le  ij(b, σ) with (i, j) = (1, 0), (1, 1), (1, 2), (2, 1), (2, 2) are continuous in (b, σ) in the norm operator  topology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  Solving the system  �  Lb(σ)ˇLb(σ)−1�  (˜g0, ˜A0), (˜g1, ˜A1), (˜g2, ˜A2)  �  = (f0, f1, f2)  LINEAR STABILITY OF KERR-NEWMAN FAMILY  135  yields  R(b, σ) = (�  Lb(σ)ˇLb(σ)−1)−1 =  \uf8eb  \uf8ec  \uf8ed  �R00(b, σ)  �R01(b, σ)  �R02(b, σ)  σ−1 �R10(b, σ)  σ−2 �R11(b, σ)  σ−1 �R12(b, σ)  �R20(b, σ)  σ−1 �R21(b, σ)  σ−1 �R22(b, σ)  \uf8f6  \uf8f7  \uf8f8  (11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='26)  where  �R11 =  ��L♯  11 − σ�L♯  12(�L♭  22)−1�L♯  21  �−1,  �R10 = �R11  �  − �L10 + σ�L♯  12(�L♭  22)−1�L20  �  L−1  00 ,  �R12 = − �R11�L♯  12(�L♭  22)−1,  �R21 = −(�L♭  22)−1�L♯  21 �R11,  �R22 = (�L♭  22)−1 − σ(�L♭  22)−1�L♯  21 �R12,  �R20 = −(�L♭  22)−1��L20L−1  00 + �L♯  21 �R10  �  ,  �R01 = −L−1  00  ��L01 �R11 + �L02 �R21  �  ,  �R02 = −L−1  00  �  σ�L01 �R12 + �L02 �R22  �  ,  �R00 = L−1  00  �  I − σ�L01 �R10 − σ�L02 �R20  �  (11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='27)  with  �L♯  i1 = �Li1 − σ�Li0L−1  00 �L01  for  i = 1, 2,  �L♯  12 = �L12 − �L10L−1  00 �L02,  �L♭  22 = �L22 − σ�L20L−1  00 �L02.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  (11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='28)  Since ˇLb(σ) is invertible for (b, σ) near (b0, 0), it follows that �  Lb(σ)−1 = ˇLb(σ)−1R(b, σ), which  implies the invertibility of �  Lb(σ) : X s,ℓ  b  → ¯Hs−1,ℓ+1  b  ( ¯X) for (b, σ) with Im σ ≥ 0, σ ̸= 0 near (b0, 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  This finishes the proof of the theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  □  We now give a more detailed description of the structure of the singular part of �  Lb(σ)−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  Corollary 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' For (b, σ) with Im σ ≥ 0 near (b0, 0), we have  �  Lb(σ)−1 = P(b, σ) + L−  b (σ) : ¯Hs−1,ℓ+2  b  ( ¯X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' S2 �  scT ∗ ¯X ⊕ �  scT ∗ ¯X) → ¯Hs,ℓ  b ( ¯X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' S2 �  scT ∗ ¯X ⊕ �  scT ∗ ¯X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  The regular part L−  b (σ) is uniformly bounded, and is continuous in σ with values in  Lweak( ¯Hs−1,ℓ+2  b  ( ¯X), ¯Hs,ℓ  b ( ¯X)) ∩ Lop( ¯Hs−1+ǫ,ℓ+2+ǫ  b  ( ¯X), ¯Hs−ǫ,ℓ−ǫ  b  ( ¯X))  for ǫ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  The singular part P(b, σ) satisfies  P(b, σ)f = (σ−2(˙g1, ˙A1) − iσ(ˇg1, ˇA1)) + iσ−1�  (˙g2, ˙A2) + (˙g′  1, ˙A′  1)  �  + iσ−1(˙g′′  1 (σ), ˙A′′  1(σ))  (11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='29)  where  (˙g1, ˙A1), (˙g′  1, ˙A′  1), (˙g′′  1(σ), ˙A′′  1(σ)) ∈ Kb,1,  (ˇg1, ˇA1) ∈ ˇKb  with (˙g′′  1 (σ), ˙A′′  1(σ)) = o(1) in Kb,1 as σ → 0, and  (˙g2, ˙A2) ∈ Kb,2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  Moreover, (˙g1, ˙A1), (˙g′  1, ˙A′  1) and (˙g2, ˙A2) are determined by the conditions  −  �  [Lb, tb,∗](ˇg1, ˇA1) + 1  2[[Lb, tb,∗], tb,∗](˙g1, ˙A1), (˙g∗, ˙A∗)  �  +  �  [Lb, tb,∗](˙g2, ˙A2), (˙g∗, ˙A∗)  �  = ⟨f, (˙g∗, ˙A∗)⟩  for all  (˙g∗, ˙A∗) ∈ K∗  b,  (11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='30)  �  [Lb, tb,∗](ˇg′  1, ˇA′  1) + 1  2[[Lb, tb,∗], tb,∗](˙g′  1, ˙A′  1), (˙g∗  1, ˙A∗  1)  �  = −⟨f, (ˇg∗  1, ˇA∗  1)⟩  −  �1  2[[Lb, tb,∗], tb,∗](˙g1, ˙A1) + [Lb, tb,∗](˙g2 − ˇg1, ˙A2 − ˇA1), (ˇg∗  1, ˇA∗  1)  �  +  �1  2[[Lb, tb,∗], tb,∗](ˇg1 − ˙g2, ˇA1 − ˙A2), (˙g∗  1, ˙A∗  1)  �  for all  (˙g∗  1, ˙A∗  1) ∈ K∗  b,1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  (11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='31)  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' For the simplicity of notation, we define  �K0 := �K⊥  b ,  �K1 := �Kb,1,  �K2 := �Kb,2,  R0 := Rb,  R1 := R⊥  b,1,  R2 := R⊥  b,2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  136  LILI HE  First, we prove that if an operator A(σ) : �Ki → Rj is invertible, and continuous in σ (in norm operator  topology) in a small neighborhood of σ = 0, then so is A(σ)−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' We write  A(σ) = A(0) + Ae(σ) = A(0)(I + A(0)−1Ae(σ))  with  Ae(σ) = A(σ) − A(0) = o(1)  in  Lop( �Ki, Rj)  as  σ → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  Since ∥A(0)−1Ae(σ)∥ �  Ki→ �  Ki < 1/2 when σ is near 0, the inverse of A(σ) is given by the Neumann series  A(σ)−1 =  �  I +  ∞  �  k=1  (−1)k�  A(0)−1Ae(σ)  �k�  A(0)−1,  from which we find that  ∥A(σ)−1 − A(0)−1∥Rj→ �  Ki ≤  � ∞  �  k=1  ∥A(0)−1Ae(σ)∥k  �  Ki→ �  Ki  �  ∥A(0)−1∥Rj→ �  Ki  ≤ 2∥A(0)−1∥2  Rj→ �  Ki∥Ae(σ)∥ �  Ki→Rj = o(1)  in  Lop(Rj, �Ki)  as  σ → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  This proves the continuity of A(σ)−1 in the norm operator topology in a small neighborhood of σ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  Applying this result to L00 and �L♭  22 (defined in (11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='28)) yields the continuity (in the norm operator topology)  of L−1  00 and (�L♭  22)−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Then the continuity of �Rij follows from the explicit expressions in (11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='27).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Furthermore,  we find that  �R21(b, σ) = �R0  21(b) + σ �Re  21(b, σ)  with  �R0  21(b) = −(�L0  22(b))−1�L0  21(b)(�L0  11(b))−1,  �R22(b, σ) = �R0  22(b) + σ �Re  22(b, σ)  with  �R0  22(b) = (�L0  22(b))−1,  �R11(b, σ) = �R0  11(b) + σ �Re  11(b, σ)  with  �R0  11(b) = (�L0  11(b))−1  (11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='32)  where �Re  21, �Re  22, �Re  11 are continuous in (b, σ) for (b, σ) near (b0, 0) in the norm operator topology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Therefore,  we rewrite  R(b, σ) = Rsing(b, σ) + Rreg(b, σ) =  \uf8eb  \uf8ec  \uf8ed  0  0  0  �  R10(b,σ)  σ  �  R0  11(b)  σ2  +  �  Re  11(b,σ)  σ  �  R12(b,σ)  σ  0  �  R0  21(b)  σ  �  R0  22(b)  σ  \uf8f6  \uf8f7  \uf8f8 + Rreg(b, σ)  (11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='33)  where all the entries in Rreg(b, σ) are continuous in (b, σ) in the norm operator topology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  For (˜g, ˜A) = ˇLb(0)(˙g, ˙A) ∈ �Kb,1 with (˙g, ˙A) ∈ Kb,1, we have  ˇLb(σ)−1(˜g, ˜A) = ˇLb(σ)−1 ˇLb(0)(˙g, ˙A) = (˙g, ˙A) − iσ ˇLb(σ)−1�  Lb(0)(ˇg, ˇA) − σ2  2  ˇLb(σ)−1∂2  σ�  Lb(0)(˙g, ˙A)  = (˙g, ˙A) − iσ ˇLb(σ)−1�ˇLb(σ) + �  Lb(0) − �  Lb(σ) − Vb  �  (ˇg, ˇA) − σ2  2  ˇLb(σ)−1∂2  σ�  Lb(0)(˙g, ˙A)  = (˙g, ˙A) − iσ(ˇg, ˇA) + σ2 ˇLb(σ)−1�  i(∂σ�  Lb(0) + σ  2 ∂2  σ�  Lb(0))(ˇg, ˇA) − 1  2∂2  σ�  Lb(0)(˙g, ˙A)  �  (11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='34)  where the coefficient of σ2 is continuous in (b, σ) in the norm operator topology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  For (˜g, ˜A) = ˇLb(0)(˙g, ˙A) ∈ �Kb,2 with (˙g, ˙A) ∈ Kb,2, we have  ˇLb(σ)−1(˜g, ˜A) = ˇLb(σ)−1 ˇLb(0)(˙g, ˙A) = (˙g, ˙A) − σ ˇLb(σ)−1��  ∂σ�  Lb(0) + σ  2 ∂2  σ�  Lb(0)  �  (˙g, ˙A)  �  (11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='35)  where the coefficient of σ is continuous in (b, σ) in the norm operator topology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  Since Rreg(b, σ) is continuous in σ in norm operator topology and ˇLb(σ)−1 is continuous in σ with  values in Lweak( ¯Hs−1,ℓ+2  b  ( ¯X), ¯Hs,ℓ  b ( ¯X)) ∩ Lop( ¯Hs−1+ǫ,ℓ+2+ǫ  b  ( ¯X), ¯Hs−ǫ,ℓ−ǫ  b  ( ¯X)) for ǫ > 0, it follows that  ˇLb(σ)−1Rreg(b, σ) is uniformly bounded and has the same continuity as ˇLb(σ)−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' According to (11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='34) and  (11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='35), we conclude that ˇLb(σ)−1Rsing(b, σ) is equal to a singular part P(b, σ) plus a uniformly bounded  and continuous operator with value in Lop( ¯Hs−1,ℓ+2  b  ( ¯X), ¯Hs,ℓ  b ( ¯X)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Moreover, P(b, σ) satisfies  P(b, σ)f = (σ−2(˙g1, ˙A1) − iσ−1(ˇg1, ˇA1)) + iσ−1�  (˙g2, ˙A2) + (˙g′  1, ˙A′  1)  �  + iσ−1(˙g′′  1(σ), ˙A′′  1(σ))  (11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='36)  LINEAR STABILITY OF KERR-NEWMAN FAMILY  137  where (˙g1, ˙A1), (˙g′  1, ˙A′  1), (˙g′′  1(σ), ˙A′′  1(σ)) ∈ Kb,1 with (˙g′′  1(σ), ˙A′′  1(σ)) = o(1) in Kb,1 as σ → 0, and (˙g2, ˙A2) ∈  Kb,2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  It remains to prove that (˙g1, ˙A1), (˙g′  1, ˙A′  1), (˙g2, ˙A2) are determined by conditions (11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='30) and (11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='31).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Let  f ∈ ¯Hs−1,ℓ+2  b  ( ¯X) and (˙g(σ), ˙A(σ)) := �  Lb(σ)−1f, and then we have  (˙g(σ), ˙A(σ)) = (σ−2(˙g1, ˙A1) − iσ−1(ˇg1, ˇA1)) + iσ−1�  (˙g2, ˙A2) + (˙g′  1, ˙A′  1)  �  + iσ−1(˙g′′  1(σ), ˙A′′  1(σ))  + (˜˙g, ˜˙A) + (˜˙g(σ), ˜˙A(σ))  where (˜˙g, ˜˙A) ∈ ¯Hs,ℓ  b ( ¯X) and (˜˙g(σ), ˜˙A(σ)) = o(1) in ¯Hs−ǫ,ℓ−ǫ  b  ( ¯X) as σ → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Applying �  Lb(σ) on both sides  gives  f = �  Lb(σ)(˙g(σ),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' ˙A(σ)) = �  Lb(0)(˙g(σ),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' ˙A(σ)) +  �  σ∂σ�  Lb(0) + σ2  2 ∂2  σ�  Lb(0)  �  (˙g(σ),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' ˙A(σ))  =  �1  2∂2  σ�  Lb(0)(˙g1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' ˙A1) − i∂σ�  Lb(0)(ˇg1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' ˇA1) − �  Lb(0)(ˇg′  1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' ˇA′  1) + i∂σ�  Lb(0)(˙g2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' ˙A2) + �  Lb(0)(˜˙g,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' ˜˙A)  �  + i∂σ�  Lb(0)(˙g′′  1(σ),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' ˙A′′  1(σ)) + �  Lb(0)(˜˙g(σ),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' ˜˙A(σ))  + iσ  2  �  − ∂2  σ�  Lb(0)(ˇg1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' ˇA1) + ∂2  σ�  Lb(0)(˙g′  1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' ˙A′  1) + ∂2  σ�  Lb(0)(˙g2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' ˙A2))  �  + iσ  2 ∂2  σ�  Lb(0)(˙g′′  1(σ),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' ˙A′′  1(σ) +  �  σ∂σ�  Lb(0) + σ2  2 ∂2  σ�  Lb(0)  �  (˜˙g + ˜˙g(σ),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' ˜˙A + ˜˙A(σ)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  (11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='37)  Letting σ → 0, we arrive at  f = 1  2∂2  σ�  Lb(0)(˙g1, ˙A1) − i∂σ�  Lb(0)(ˇg1, ˇA1) − �  Lb(0)(ˇg′  1, ˇA′  1) + i∂σ�  Lb(0)(˙g2, ˙A2) + �  Lb(0)(˜˙g, ˜˙A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  (11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='38)  Pairing both sides of (11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='38) with (˙g∗, ˙A∗) ∈ K∗  b, we obtain  ⟨f, (˙g∗, ˙A∗)⟩ =  �1  2∂2  σ�  Lb(0)(˙g1, ˙A1) − i∂σ�  Lb(0)(ˇg1, ˇA1) + i∂σ�  Lb(0)(˙g2, ˙A2), (˙g∗, ˙A∗)  �  = −  �  [Lb, tb,∗](ˇg1, ˇA1) + 1  2[[Lb, tb,∗], tb,∗](˙g1, ˙A1) − [Lb, tb,∗](˙g2, ˙A2), (˙g∗, ˙A∗)  �  (11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='39)  Since the pairing on Kb × K∗  b on the right-hand side of (11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='39) is non-degenerate, (˙g1, ˙A1) (thus (ˇg1, ˇA1))  and (˙g2, ˙A2) are uniquely determined by (11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='39).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' This proves the condition (11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='30).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  Rewriting (11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='38) gives rise to the following PDE  �  Lb(0)  �  (ˇg′  1, ˇA′  1) − (˜˙g, ˜˙A)  �  = −f + 1  2∂2  σ�  Lb(0)(˙g1, ˙A1) − i∂σ�  Lb(0)(ˇg1, ˇA1) + i∂σ�  Lb(0)(˙g2, ˙A2)  (11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='40)  whose right-hand side is uniquely determined by f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Due to (11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='39), the pairings of the right-hand side  of (11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='40) with any (˙g∗, ˙A∗) ∈ K∗  b is 0, which implies that (11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='40) can be solved for (ˇg′  1, ˇA′  1) − (˜˙g, ˜˙A) and  (ˇg′  1, ˇA′  1) − (˜˙g, ˜˙A) is uniquely determined by f modulo Kb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Namely,  (ˇg′  1, ˇA′  1) − (˜˙g, ˜˙A) = (¯˙g, ¯˙A) + (˙g′′′, ˙A′′′)  (11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='41)  where (¯˙g, ¯˙A) is uniquely determined by f and (˙g′′′, ˙A′′′) ∈ Kb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Pairing both sides of (11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='37) with (˙g∗  1, ˙A∗  1) ∈  K∗  b,1 and using the identity (11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='39), we obtain, upon multiplying σ−1 and letting σ → 0,  0 = i  2  �  − ∂2  σ�  Lb(0)(ˇg1, ˇA1) + ∂2  σ�  Lb(0)(˙g′  1, ˙A′  1) + ∂2  σ�  Lb(0)(˙g2, ˙A2) − 2i∂σ�  Lb(0)(˜˙g, ˜˙A), (˙g∗  1, ˙A∗  1)  �  = i  2  �  − ∂2  σ�  Lb(0)(ˇg1, ˇA1) + ∂2  σ�  Lb(0)(˙g2, ˙A2) + 2i∂σ�  Lb(0)(¯˙g, ¯˙A), (˙g∗  1, ˙A∗  1)  �  + i  2  �  ∂2  σ�  Lb(0)(˙g′  1, ˙A′  1) − 2i∂σ�  Lb(0)(ˇg′  1, ˇA′  1), (˙g∗  1, ˙A∗  1)  �  + i⟨(˙g′′′, ˙A′′′), �  Lb(0)∗(ˇg∗  1, ˇA∗  1)⟩  where we use (11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='41) and −i(∂σ�  Lb(0))∗(˙g∗  1, ˙A∗  1) = �  Lb(0)∗(ˇg∗  1, ˇA∗  1) in the second step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Since  ⟨(˙g′′′, ˙A′′′), �  Lb(0)∗(ˇg∗  1, ˇA∗  1)⟩ = ⟨�  Lb(0)(˙g′′′, ˙A′′′), (ˇg∗  1, ˇA∗  1)⟩ = 0,  138  LILI HE  we arrive at  �  [[Lb, tb,∗], tb,∗](˙g′  1, ˙A′  1) + 2[Lb, tb,∗](ˇg′  1, ˇA′  1), (˙g∗  1, ˙A∗  1)  �  =  �  [[Lb, tb,∗], tb,∗](ˇg1, ˇA1) − [[Lb, tb,∗], tb,∗](˙g2, ˙A2) + 2[Lb, tb,∗](¯˙g, ¯˙A), (˙g∗  1, ˙A∗  1)  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  (11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='42)  Since the pairing on Kb,1 × K∗  b,1 on the left-hand side of (11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='42) is non-degenerate and the right-hand side is  uniquely determined by f, so is (˙g′  1, ˙A′  1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Actually, we can further calculate the right-hand side  �  [Lb, tb,∗](¯˙g, ¯˙A), (˙g∗  1, ˙A∗  1)  �  =  �  (¯˙g, ¯˙A), �  Lb(0)∗(ˇg∗  1, ˇA∗  1)  �  =  �  �  Lb(0)(¯˙g, ¯˙A), (ˇg∗  1, ˇA∗  1)  �  = −⟨f + 1  2[[Lb, tb,∗], tb,∗](˙g1, ˙A1) + [Lb, tb,∗](˙g2 − ˇg1, ˙A2 − ˇA1), (ˇg∗  1, ˇA∗  1)⟩  where we use (11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='40) and (11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='41) in the third step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Plugging this into (11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='42) proves the condition (11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='31).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  □  12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Regularity of the resolvent of the linearized gauge-fixed Einstein-Maxwell operator  In this section, we continue using the notations from §11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  In §12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='1, we give a refined description of  the singular part P(b, σ) and the regular part L−  b (σ) of the operator �  Lb(0)−1 defined in Corollary 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  Concretely, we prove that (˙g′′  1(σ), ˙A′′  1(σ)) defined in Corollary 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='4 is H¨older regular at σ = 0, while L−  b (σ) is  H¨older regular at σ = 0 only when one relaxes the target space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' In §12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='2, we provide estimates of the bounds  of any derivatives (in σ) of L−  b (σ) for σ in the region 0 ≤ Im σ ≤ C, |σ| ≥ c0 for some c0, C > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Finally,  in §12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='3, we discuss the conormal regularity of L−  b (σ) at σ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Specifically, after applying any number of  derivatives σ∂σ to L−  b (σ), it satisfies the same properties as L−  b (σ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Regularity at low frequencies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' In §12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='1, we study in detail the structure of �  Lb(σ)ˇLb(σ)−1, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=',  prove the ǫ-regularity of �  Lb(σ)ˇLb(σ)−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' In §12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='2, we establish the ǫ-regularity of R(b, σ), which allows us  to give a refined description of the singular part P(b, σ) of �  Lb(σ)−1 near σ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' We also discuss a certain  H¨older regularity of the regular part L−  b (σ) at σ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' ǫ-regularity of �  Lb(σ)ˇLb(σ)−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' First, we introduce several definitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  Definition 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Let X, Y be two Banach spaces and α, β ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Suppose A(σ) : X → Y is a family of  operators depending on parameter σ ∈ C \\ {0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Then we write  A(σ) : |σ|αX → |σ|βY  if and only if there exists a constant C > 0 such that ∥A(σ)∥X→Y ≤ C|σ|β−α for all σ ∈ C \\ {0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  Let �Ki, Ri (i = 0, 1, 2) be defined as in §11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  Definition 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  (1) An operator L(σ) : �Ki → Rj is ǫ-regular at σ = 0 if for σ ∈ C, Im σ ≥ 0 and |σ|  small, there exists a constant C > 0 such that  ∥L(σ)∥ �  Ki→Rj ≤ C,  (12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='1a)  ∥∂σL(σ)∥ �  Ki→Rj ≤ C|σ|−ǫ,  (12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='1b)  ∥∂2  σL(σ)∥ �  Ki→Rj ≤ C|σ|−ǫ−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  (12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='1c)  (2) An operator L(σ) : �Ki → Rj has an ǫ-regular expansion at σ = 0 in σ up to order one if  L(σ) = L0 + σLe(σ)  (12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='2)  where L0 : �Ki → Rj is independent of σ and Le(σ) : �Ki → Rj is ǫ-regular at σ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  (3) An operator L(σ) : �Ki → Rj has an ǫ-regular expansion at σ = 0 in σ up to order two if  L(σ) = L0 + σL1 + σ2Le(σ)  (12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='3)  where L0, L1 : �Ki → Rj are independent of σ and Le(σ) : �Ki → Rj is ǫ-regular at σ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  LINEAR STABILITY OF KERR-NEWMAN FAMILY  139  We recall the analysis of the entries Lij of the operator in the proof of Theorem 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='3, and note that each en-  try is a sum of polynomials in σ whose coefficients are either independent of σ, terms involving ˇLb(σ)−1 acting  on an element in ¯H∞,3/2−  b  ( ¯X), or terms involving (ˇLb(σ)−1)∗ acting on an element in ˙H−5/2−C(a,γ),3/2−  b  ( ¯X)  where C(a, γ0) is a sufficiently small constant depending on a, γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Therefore, we first discuss the properties  of the operator ˇLb(σ)−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  Proposition 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Let − 3  2 < ℓ < − 1  2, 0 < ǫ < 1 with − 1  2 < ℓ + ǫ < 1  2, and s − ǫ > 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' For Im σ ≥ 0,  0 < |σ| ≤ c0 with c0 small, we have  ∂σ ˇLb(σ)−1 : ¯Hs−1,ℓ+2  b  ( ¯X) → |σ|−ǫ ¯Hs−ǫ,ℓ+ǫ−1  b  ( ¯X),  (12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='4)  ∂2  σ ˇLb(σ)−1 : ¯Hs−1,ℓ+2  b  ( ¯X) → |σ|−ǫ−1 ¯Hs−1−ǫ,ℓ+ǫ−1  b  ( ¯X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  (12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='5)  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Formally, we have  ∂σ ˇLb(σ)−1 = −ˇLb(σ)−1(∂σ ˇLb(σ))ˇLb(σ)−1  where by Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='13  ∂σ ˇLb(σ) = ∂σ�  Lb(σ) ∈ 2iρ(ρ∂ρ − 1) + ρ3Diff1  b + ρ2C∞ + σρ2C∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  (12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='6)  Motivated by this, we first analyze ρ(ρ∂ρ − 1)ˇLb(σ)−1 and ˇLb(σ)−1ρ(ρ∂ρ − 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Since ρ(ρ∂ρ − 1) : ¯Hs,ℓ  b ( ¯X) →  ¯Hs−1,ℓ+1  b  ( ¯X), it is clear that  ρ(ρ∂ρ − 1)ˇLb(σ)−1 : ¯Hs−1,ℓ+2  b  ( ¯X) → ¯Hs−1,ℓ+1  b  ( ¯  X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  Meanwhile, in view of Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='13  2iρ(ρ∂ρ − 1) = σ−1�ˇLb(σ) − ˇLb(0) + Q  �  with  Q ∈ σρ3Diff1  b + σρ2C∞ + σρ2C∞,  then it follows that  2iρ(ρ∂ρ − 1)ˇLb(σ)−1 = σ−1�  I − (�  Lb(0) + Vb + Q)ˇLb(σ)−1�  : ¯Hs−1,ℓ+2  b  ( ¯X) → |σ|−1 ¯Hs−2,ℓ+2  b  ( ¯X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  Then an interpolation gives  ρ(ρ∂ρ − 1)ˇLb(σ)−1 : ¯Hs−1,ℓ+2  b  ( ¯  X) → |σ|−ǫ ¯Hs−1−ǫ,ℓ+1+ǫ  b  ( ¯X),  0 ≤ ǫ ≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  (12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='7)  Since s − 1 − ǫ > 2 and 1/2 < ℓ + 1 + ǫ < 3/2, we have  ¯Hs−1,ℓ+2  b  ( ¯  X)  2iρ(ρ∂ρ−1)ˇLb(σ)−1  −−−−−−−−−−−−→ |σ|−ǫ ¯Hs−1−ǫ,ℓ+ǫ+1  b  ( ¯X)  ˇLb(σ)−1  −−−−−→ |σ|−ǫ ¯Hs−ǫ,ℓ+ǫ−1  b  ( ¯  X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  (12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='8)  We also have  ¯Hs−1,ℓ+2  b  ( ¯X)  ˇLb(σ)−1  −−−−−→ ¯Hs,ℓ  b ( ¯X)  ρ2Diff  1  b+σρ2C∞  −−−−−−−−−−→ ¯Hs−1,ℓ+2  b  ( ¯X)  ˇLb(σ)−1  −−−−→ ¯Hs,ℓ  b ( ¯X) ⊂ |σ|−ǫ ¯Hs−ǫ,ℓ+ǫ−1  b  ( ¯X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  (12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='9)  By (12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='6), we finishes the proof of (12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  For ∂2  σ ˇLb(σ)−1, we note that  ∂2  σ ˇLb(σ)−1 = 2ˇLb(σ)−1(∂σ ˇLb(σ))ˇLb(σ)−1(∂σ ˇLb(σ))ˇLb(σ)−1 − ˇLb(σ)−1(∂2  σ ˇLb(σ))ˇLb(σ)−1  := I + II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  Since ∂2  σ ˇLb(σ) ∈ ρ2C∞, according to (12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='9), we obtain  II : ¯Hs−1,ℓ+2  b  ( ¯X) → ¯Hs,ℓ  b ( ¯X) ⊂ |σ|−ǫ−1 ¯Hs−1−ǫ,ℓ+ǫ−1  b  ( ¯X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  It remains to analyze I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' By the assumption on s, ℓ, ǫ, we find that 1/2 < (1 + ǫ)/2 < 1, and thus using the  reasoning in (12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='8) and (12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='9), we obtain  ¯Hs−1,ℓ+2  b  ( ¯X)  ∂σ ˇLb(σ)ˇLb(σ)−1  −−−−−−−−−−→ |σ|− 1+ǫ  2  ¯H  s−1− 1+ǫ  2 ,ℓ+ 1+ǫ  2 +1  b  ( ¯  X)  ∂σ ˇLb(σ)ˇLb(σ)−1  −−−−−−−−−−→ |σ|−1−ǫ ¯Hs−2−ǫ,ℓ+ǫ+1  b  ( ¯X)  ˇLb(σ)−1  −−−−−→ |σ|−1−ǫ ¯Hs−1−ǫ,ℓ+ǫ−1  b  ( ¯X)  where we use s − 3  2 − ǫ  2 > 2, 1  2 < ℓ + 3  2 + ǫ  2 < 3  2 in the second step and s − 2 − ǫ > 2, 1  2 < ℓ + ǫ + 1 < 3  2 in  the last step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' This finishes the proof of (12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  □  140  LILI HE  Recall the expression (11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='25) of �  Lb(σ)ˇL(σ)−1  �  Lb(σ)ˇLb(σ)−1 =  \uf8eb  \uf8ec  \uf8ed  L00(b, σ)  σ2 �L01(b, σ)  σ�L02(b, σ)  σ�L10(b, σ)  σ2 �L11(b, σ)  σ2 �L12(b, σ)  σ�L20(b, σ)  σ2 �L21(b, σ)  σ�L22(b, σ)  \uf8f6  \uf8f7  \uf8f8  =  \uf8eb  \uf8ec  \uf8ed  L00(b, σ)  σ2�L01(b, σ)  σ�L02(b, σ)  σ  ��L0  10(b) + σ�Le  10(b, σ)  �  σ2��L0  11(b) + σ�L1  11(b) + σ2�Le  11(b, σ)  �  σ2��L0  12(b) + σ�Le  12(b, σ)  �  σ�L20(b, σ)  σ2��L0  21(b) + σ�Le  21(b)  �  σ  ��L0  22(b) + σ�Le  22(b, σ)  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  \uf8f6  \uf8f7  \uf8f8  Proposition 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Let s, ℓ, ǫ be defined as in Proposition 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' The entries of �  Lb(σ)ˇLb(σ)−1 satisfy the  following properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  (1) L00, �L01, �L02, �L20 are ǫ-regular at σ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  (2) �L10, �L12, �L21, �L22 have an ǫ-regular expansion up to order one, that is, �Le  10, �Le  12, �Le  21, �Le  22 are ǫ-regular  at σ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  (3) �L11 has an ǫ-regular expansion up to order two, that is, �Le  11 is ǫ-regular at σ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' The key point in the proof was already mentioned before Proposition 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  Analysis of �Le  10 and �L20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Here, we only discuss �Le  10 in detail because the proof of �L02 follows in a  similar manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Recall the calculation in (11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='18)  �Le  10(˜g0, ˜A0), (˙g∗  1, ˙A∗  1)⟩  = −  �  (˜g0, ˜A0), i(ˇLb(σ)−1)∗(∂σ�  Lb(0) + σ  2 ∂2  σ�  Lb(0))∗�  I − (ˇLb(0)−1)∗V ∗  b  �  (ˇg∗  2, ˇA∗  2)  �  +  �  (˜g0, ˜A0), 1  2(ˇLb(σ)−1)∗(∂2  σ�  Lb(0))∗(˙g∗  2, ˙A∗  2)  �  where  (∂σ�  Lb(0) + σ  2 ∂2  σ�  Lb(0))∗�  I − (ˇLb(0)−1)∗V ∗  b  �  (ˇg∗  2, ˇA∗  2),  (∂2  σ�  Lb(0))∗(˙g∗  2, ˙A∗  2) ∈ ˙H−5/2−C(a,γ),3/2−  b  ( ¯X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  Since (˜g0, ˜A0) ∈ ¯Hs−1,ℓ+2  b  ( ¯X), and by Proposition (12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='3)  (ˇL−1  b (σ))∗ ˙H−5/2−C(a,γ), 3/2−  b  ( ¯X) ∈ ˙H−3/2−C(a,γ),−1/2−  b  ( ¯X),  (∂j  σ ˇL−1  b )∗ ˙H−5/2−C(a,γ), 3/2−  b  ( ¯X) ∈ |σ|−ǫ−j+1 ˙H−s+1,−ℓ−2  b  ( ¯X),  j = 1, 2  where we use the facts that −5/2 − C(a, γ) > −3 > −s + ǫ + 1 and −ℓ − ǫ + 1 < 3/2, the pairings  above (and their up to second order derivatives) are well-defined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' This proves the ǫ-regularity of �Le  10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  Analysis of �Le  11, �Le  12, �Le  22 and �Le  22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' The statements for these entries can be proved in the same way  as in the proof of �Le  10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  Analysis of �L01 and �L02.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Here, we only prove the statement for �L01 as the proof of �L02 follows in an  analogous manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Using the calculation in (11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='16), we find that  �L01(˜g1, ˜A1)=  �  − iVb ˇLb(σ)−1�  ∂σ�  Lb(0) + σ  2 ∂2  σ�  Lb(0)  �  (ˇg1, ˇA1) + 1  2Vb ˇLb(σ)−1∂2  σ�  Lb(0)(˙g1, ˙A1)  �  − (�L11 + �L21)(˜g1, ˜A1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  where  ∂σ�  Lb(0)(ˇg1, ˇA1),  ∂2  σ�  Lb(0)(ˇg1, ˇA1),  ∂2  σ�  Lb(0)(˙g1, ˙A1) ∈ ¯H∞,3/2−  b  ( ¯X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  Since by Proposition 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='4  ˇL−1  b (σ) ¯H∞,3/2−  b  ( ¯X) ∈ ¯H∞,−1/2−  b  ( ¯X), ∂j  σ ˇL−1  b (σ) ∈ |σ|−ǫ−j+1 ¯H∞,ℓ+ǫ−1  b  ( ¯X), j = 1, 2,  and Vb maps D′(X◦) to C∞  c (X), it follows that the first term on the right-hand side is ǫ-regular.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  Meanwhile, it is clear that the second term is ǫ-regular.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' This proves the ǫ-regularity for �L01.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  LINEAR STABILITY OF KERR-NEWMAN FAMILY  141  Analysis of L00.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' By the calculation in (11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='24), we have  L00(b, σ)(˜g0, ˜A0) = (˜g0, ˜A0) − Vb ˇLb(σ)−1(˜g0, ˜A0) −  �  σ�L0  10(b) + σ2 �Le  10(b, σ) + σ�L20(b, σ)  �  (˜g0, ˜A0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  Again, it is clear that the third term on the right-hand side is ǫ-regular.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Since by Proposition 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='4  ˇL−1  b (σ)(˜g0, ˜A0) ∈ ¯Hs,ℓ  b ( ¯X), ∂j  σ ˇL−1  b (σ) ∈ |σ|−ǫ−j+1 ¯Hs−ǫ−j+1,ℓ+ǫ−1  b  ( ¯X), j = 1, 2,  and Vb maps D′(X◦) to C∞  c (X), it follows that the second term on the right-hand side is ǫ-regular.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  This finishes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  □  12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' H¨older regularity of P(b, σ) and L−  b (σ) at σ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Now we are ready to analyze the ǫ-regularity of  R(b, σ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' To this end, we need the following lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  Lemma 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Let s, ℓ, ǫ be defined as in Proposition 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Suppose that the operator A(σ) : �Ki → Rj with  i, j = 0, 1, 2 has an ǫ-regular expansion up to order one, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=',  A(σ) = A0 + σAe(σ)  where A0 : �Ki → Rj is invertible and independent of σ, and Ae(σ) : �Ki → Rj is ǫ-regular.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Then for |σ|  sufficiently small, A(σ) is invertible, and its inverse B(σ) = A(σ)−1 also has an ǫ-regular expansion up to  order one, that is,  B(σ) = B0 + σBe(σ)  where B0 : Rj → �Ki is independent of σ, and Be(σ) : Rj → �Ki is ǫ-regular.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  Similarly, if A(σ) is ǫ-regular and has uniformly bounded inverse B(σ), then B(σ) is ǫ-regular as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Since Ae(σ) is ǫ-regular (and thus uniformly bounded), ∥σ(A0)−1Ae(σ)∥ �  Ki→ �  Ki < 1/2 when |σ| is  sufficiently small, the inverse of A(σ) is given by the Neumann series  A(σ)−1=  �  I +  ∞  �  k=1  (−1)k�  σ(A0)−1Ae(σ)  �k�  (A0)−1  = (A0)−1 − σ  �  (A0)−1Ae(σ)  ∞  �  k=0  (−1)k�  σ(A0)−1Ae(σ)  �k(A0)−1�  := B0 + σBe(σ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  First, we have  ∥Be(σ)∥Rj→ �  Ki ≤  � ∞  �  k=0  ∥σ(A0)−1Ae(σ)∥k  �  Ki→ �  Ki  �  ∥(A0)−1∥2  Rj→ �  Ki∥Ae(σ)∥ �  Ki→Rj  ≤ 2∥A(0)−1∥2  Rj→ �  Ki∥Ae(σ)∥ �  Ki→Rj,  which implies the uniform boundedness of Be(σ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Secondly, the first derivative of Be(σ) is a sum of the  terms of the types  σ−2�  σ(A0)−1Ae(σ)  �k(A0)−1 ≲ 2−k+2,  k ≥ 2  �  σ(A0)−1Ae(σ)  �k1 ◦  �  (A0)−1∂σAe(σ)  � �  σ(A0)−1Ae(σ)  �k2(A0)−1 ≲ 2−k1−k2|σ|−ǫ,  where we use ∥∂σAe(σ)∥ �  Ki→Rj ≲ |σ|−ǫ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' As a consequence, ∥∂σBe(σ)∥Rj→ �  Ki ≲ |σ|−ǫ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Finally,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' the second  derivative of Be(σ) is a sum of the terms of the types  σ−3�  σ(A0)−1Ae(σ)  �k(A0)−1 ≲ 2−k+3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  k ≥ 3  σ−1�  σ(A0)−1Ae(σ)  �k1 ◦  �  (A0)−1∂σAe(σ)  � �  σ(A0)−1Ae(σ)  �k2(A0)−1 ≲ 2−k1−k2+1|σ|−ǫ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  k1 + k2 ≥ 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  σ  �  σ(A0)−1Ae(σ)  �k1 ◦  �  (A0)−1∂σAe(σ)  � �  σ(A0)−1Ae(σ)  �k2 ◦  �  (A0)−1∂σAe(σ)  � �  σ(A0)−1Ae(σ)  �k3(A0)−1  ≲ 2−k1−k2−k3|σ|−2ǫ+1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  �  σ(A0)−1Ae(σ)  �k1 ◦  �  (A0)−1∂2  σAe(σ)  � �  σ(A0)−1Ae(σ)  �k2(A0)−1 ≲ 2−k1−k2|σ|−ǫ−1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  where we use ∂σAe(σ) ≲ |σ|−ǫ and ∂2  σAe(σ) ≲ |σ|−ǫ−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Therefore, ∂2  σBe(σ) ≲ |σ|−ǫ−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' This proves that  Be(σ) is ǫ-regular.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  142  LILI HE  The proof of the second statement is similar by using  ∂σB(σ) = −B(σ) ◦ ∂σA(σ) ◦ B(σ),  ∂2  σB(σ) = −B(σ) ◦ ∂2  σA(σ) ◦ B(σ) + 2B(σ) ◦ ∂σA(σ) ◦ B(σ) ◦ ∂σA(σ) ◦ B(σ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  □  Recall the expressions (11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='26) and (11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='32) of Rb,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='σ = (�  Lb(σ)ˇLb(σ)−1)−1  R(b,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' σ) = (�  Lb(σ)ˇLb(σ)−1)−1 =  \uf8eb  \uf8ec  \uf8ed  �R00(b,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' σ)  �R01(b,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' σ)  �R02(b,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' σ)  σ−1 �R10(b,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' σ)  σ−2 �R11(b,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' σ)  σ−1 �R12(b,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' σ)  �R20(b,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' σ)  σ−1 �R21(b,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' σ)  σ−1 �R22(b,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' σ)  \uf8f6  \uf8f7  \uf8f8  =  \uf8eb  \uf8ec  \uf8ed  �R00(b,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' σ)  �R01(b,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' σ)  �R02(b,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' σ)  σ−1 �R10(b,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' σ)  σ−2� �R0  11(b) + σ �Re  11(b,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' σ)  �  σ−1 �R12(b,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' σ)  �R20(b,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' σ)  σ−1� �R0  21(b) + σ �Re  21(b,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' σ)  �  σ−1� �R0  22(b) + σ �Re  22(b,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' σ)  �  \uf8f6  \uf8f7  \uf8f8  where  �R11 =  ��L♯  11 − σ�L♯  12(�L♭  22)−1�L♯  21  �−1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  �R10 = �R11  �  − �L10 + σ�L♯  12(�L♭  22)−1�L20  �  L−1  00 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  �R12 = − �R11�L♯  12(�L♭  22)−1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  �R21 = −(�L♭  22)−1�L♯  21 �R11,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  �R22 = (�L♭  22)−1 − σ(�L♭  22)−1�L♯  21 �R12,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  �R20 = −(�L♭  22)−1��L20L−1  00 + �L♯  21 �R10  �  ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  �R01 = −L−1  00  ��L01 �R11 + �L02 �R21  �  ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  �R02 = −L−1  00  �  σ�L01 �R12 + �L02 �R22  �  ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  �R00 = L−1  00  �  I − σ�L01 �R10 − σ�L02 �R20  �  (12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='10)  with  �L♯  i1 = �Li1 − σ�Li0L−1  00 �L01  for  i = 1, 2,  �L♯  12 = �L12 − �L10L−1  00 �L02,  �L♭  22 = �L22 − σ�L20L−1  00 �L02.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  (12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='11)  Proposition 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Let s, ℓ, ǫ be defined as in Proposition 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' The entries of R(b, σ) satisfy the following  properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  (1) �R00, �R01, �R02, �R10, �R12, �R20 are ǫ-regular at σ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  (2) �R11, �R21, �R22 have an ǫ-regular expansion up to order one, that is, �Re  11, �Re  21, �Re  22 are ǫ-regular at  σ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' First, since L00 is ǫ-regular, by Lemma 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='5, so is L−1  00 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Then by Proposition 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='4, we find that  �L♯  11, �L♯  21, �L♭  22 (and thus (�L♭  22)−1 by Lemma 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='5) have an ǫ-regular expansion up to order one, while �L♯  12 is  ǫ-regular.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Therefore, we see that  �L♯  11 − σ�L♯  12(�L♭  22)−1�L♯  21  has an ǫ-regular expansion up to order one whose coefficient of σ0 is �L0  11(b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Then the ǫ-regular expansion  property of �R11 follows from Lemma 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Similarly, the ǫ-regular expansion property of �R21, �R22 follows  from that of �R11, (�L♭  22)−1, �L♯  21 and the above expressions of �Rij.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Finally, the ǫ-regularity of the remaining  components can be obtained by using the above expressions of �Rij and Proposition 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  □  Recall the singular part P(b, σ) of �  Lb(σ)−1 as defined in Corollary 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='4, which satisfies  P(b, σ)f = (σ−2(˙g1, ˙A1) − iσ−1(ˇg1, ˇA1)) + iσ−1�  (˙g2, ˙A2) + (˙g′  1, ˙A′  1)  �  + iσ−1(˙g′′  1(σ), ˙A′′  1(σ))  := σ−2P 2(b)f + σ−1P 1(b)f + σ−1P e(b, σ)f  (12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='12)  where (˙g′′  1(σ), ˙A′′  1(σ)) = P e(b, σ)f = o(1) in Kb,1 as σ → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' With the ǫ-regularity of R(b, σ) at our dis-  posal, now we are able to prove that P e(b, σ) is ǫ-regular as well, and thus give a refined description of  σ−1(˙g′′  1(σ), ˙A′′  1(σ)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  LINEAR STABILITY OF KERR-NEWMAN FAMILY  143  Remark 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' We now make a remark that there is a relationship between the Sobolev spave ˙Hk([0, c0)),  which is the restriction to [0, c0) of elements in Hk(R) with support in [0, ∞), and the b-Sobolev space  Hk  b([0, c0)), which is a completion of {φ|(0,c0]} where φ ∈ C∞  c ((0, ∞)) with respect the following norm  ∥φ∥2  Hk  b ([0,C0)) :=  k  �  j=0  ∥(σ∂σ)jφ∥2  L2(dσ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  First, it is clear that σkHk  b([0, c0)) ⊂ ˙Hk([0, c0)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Second, Hardy’s inequality tells us that for φ ∈ C∞  c (0, ∞),  we have  � ∞  0  |φ|2  r2 dr ≤ 4  � ∞  0  |∂rf|2 dr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  Applying Hardy’s inequality k times, we find that  � ∞  0  |φ|2  r2k dr ≤ Ck  � ∞  0  |∂k  r f|2 dr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  for φ ∈ C∞  c (0, ∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' This implies ˙Hk([0, c0)) ⊂ σkHk  b([0, c0)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Therefore, ˙Hk([0, c0)) = σkHk  b([0, c0)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Then  an interpolation gives ˙Hα([0, c0)) = σαHα  b ([0, c0)) for all α ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  As a preparation, we record the following technical lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  Lemma 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Suppose that u± ∈ Hs(R±) with s ≥ 0 and s ̸= 1  2 + N0 and let u(x) = u+(x) for x > 0 and  u(x) = u−(x) for x < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' If ∂ju+(x) = ∂ju−(x) for all j ∈ N0, j < s − 1  2, then u(x) ∈ Hs(R).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' See [61, Theorem 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='4 and 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  □  Theorem 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Let σ−1(˙g′′  1(σ), ˙A′′  1(σ)) = σ−1P e(b, σ)f ∈ Kb,1 be defined as in (12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='12) and let s, ℓ, ǫ be  defined as in Proposition 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Then we have  P e(b, σ) ∈ H  3  2 −ǫ−((−c0, c0);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' L( ¯Hs−1,ℓ+2  b  ( ¯X), Kb,1))  (12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='13)  where c0 > 0 is a sufficiently small constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Moreover,  σ−1(˙g′′  1(σ), ˙A′′  1(σ)) ∈  �  ±  A−ǫ�  ± [0, c0);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Kb,1  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  (12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='14)  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Using the calculation (11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='34) and (11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='35) in the proof of Corollary 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='4, we find that for f =  (f0, f1, f2) ∈ ¯Hs−1,ℓ+2  b  ( ¯X) = R0 ⊕ R1 ⊕ R2,  ˇLb(0)P e(b, σ)f  =  � �R10(b, σ) − �R10(b, 0)  �  f0 +  � �Re  11(b, σ) − �Re  11(b, 0)  �  f1 +  � �R12(b, σ) − �R12(b, 0)  �  f2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  (12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='15)  According to Proposition 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='6, we have ∥∂σ �Re  11(b, σ)∥R1→ �  K1 ≲ |σ|−ǫ and ∥∂2  σ �Re  11(b, σ)∥R1→ �  K1 ≲ |σ|−ǫ−1 for  0 < |σ| ≤ c0 where c0 is a sufficiently small constant, which implies that  ∥∂σσ∂σ �Re  11(b, σ)∥R1→ �  K1 ≲ |σ|−ǫ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  Integrating ∂σσ∂σ �Re  11(b, σ) and ∂σ �Re  11(b, σ) from σ ∈ (0, c0) to 0 and using σ∂σ �Re  11(σ)|b,σ=0 = 0 yield  σ∂σ �Re  11(b, σ) ∈ σ1−ǫL∞([0, c0);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' L(R1, �K1)) ⊂ σ  3  2 −ǫ−H0  b([0, c0);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' L(R1, �K1)),  (12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='16)  �Re  11(b, σ) − �Re  11(b, 0) ∈ σ1−ǫL∞([0, c0);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' L(R1, �K1)) ⊂ σ  3  2 −ǫ−H0  b([0, c0);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' L(R1, �K1)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  (12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='17)  This implies  ( �Re  11(σ) − �Re  11(0)) ∈ σ  3  2 −ǫ−H2  b([0, c0);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' L(R1, �K1)) ⊂ ˙H  3  2 −ǫ−([0, c0);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' L(R1, �K1)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  Similarly, we also have  ( �Re  11(σ) − �Re  11(0)) ∈ ˙H  3  2 −ǫ−((−c0, 0];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' L(R1, �K1)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  Since ( �Re  11(σ) − �Re  11(0)) → 0 as σ → 0 from both sides, by Lemma 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='8, we obtain  ( �Re  11(σ) − �Re  11(0)) ∈ H  3  2 −ǫ−((−c0, c0);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' L(R1, �K1)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  Similarly, �R10(b, σ) − �R10(b, 0) and �R12(b, σ) − �R12(b, 0) satisfy the same estimates as �Re  11(b, σ) − �Re  11(b, 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  This proves (12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='13).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  144  LILI HE  Moreover, the estimate in (12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='17) implies  �Re  11(b, σ) − �Re  11(b, 0) ∈  �  ±  σ1−ǫL∞(±[0, c0);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' L(R1, �K1))  (12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='18)  and this also holds for �R10(b, σ) − �R10(b, 0), �R20(b, σ) − �R20(b, 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Using Proposition 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='16, the estimate  (12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='18) is satisfied after applying any number of derivatives σ∂σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' This proves (12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='14).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  □  Now we turn to the analysis of the regular part L−  b (σ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  Proposition 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Let s, ℓ, ǫ be defined as in Proposition 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' For Im σ ≥ 0 and 0 < |σ| ≤ c0 where  c0 > 0 is a small constant, the regular part L−  b (σ) of the resolvent �  Lb(σ)−1 defined in Corollary 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='4 satisfies  ∂j  σL−  b (σ) : ¯Hs−1,ℓ+2  b  ( ¯X) → |σ|−ǫ−j+1 ¯Hs−ǫ−j+1,ℓ+ǫ+1  b  ( ¯  X),  j = 1, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  (12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='19)  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' According to Proposition 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='6 and the expression (12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='15), we see that P(b, σ) has the form P(b, σ) =  σ−2P 2(b) + σ−1P 1(b) + σ−1P e(b, σ) where  P 2(b), P 1(b) : ¯Hs−1,ℓ+2  b  ( ¯X) → ρC∞( ¯  X) + ¯H  ∞, 1  2 −  b  ( ¯X)  and P e(b, σ) :  ¯Hs−1,ℓ+2  b  ( ¯X) → Kb,1 is ǫ-regular as proved in Theorem 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  We also write R(b, σ) =  Rsing(b, σ) + Rreg(b, σ) where  Rsing(b, σ) =  \uf8eb  \uf8ed  0  0  0  σ−1 �R10(b, σ)  σ−2 �R0  11(b) + σ−1 �Re  11(b, σ)  σ−1 �R12(b, σ)  0  σ−1 �R0  21(b)  σ−1 �R0  22(b)  \uf8f6  \uf8f8  and  Rreg(b, σ) =  \uf8eb  \uf8ed  �R00(b, σ)  �R01(b, σ)  �R02(b, σ)  0  0  0  �R20(b, σ)  �Re  21(b, σ)  �Re  22(b, σ)  \uf8f6  \uf8f8 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  Using the calculation in (11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='34), (11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='35) and (11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='36), we find that  L−  b (σ) = ˇLb(σ)−1Rreg(b, σ) − ˇLb(σ)−1��  ∂σ�  Lb(0) + σ  2 ∂2  σ�  Lb(0)  �  (P 1(b) + P e(b, σ)) + 1  2∂2  σ�  Lb(0)P 2(b)  �  := I + II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  For the term I, we compute  ∂σI = ∂σ ˇLb(σ)−1Rreg(b, σ) + ˇLb(σ)−1∂σRreg(b, σ),  ∂2  σI = ∂2  σ ˇLb(σ)−1Rreg(b, σ) + 2∂σ ˇLb(σ)−1∂σRreg(b, σ) + ˇLb(σ)−1∂2  σRreg(b, σ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  Then by proposition 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='3 and 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='6, we conclude that  ∂j  σI : ¯Hs−1,ℓ+2  b  ( ¯X) → |σ|−ǫ−j+1 ¯Hs−ǫ−j+1,ℓ+ǫ+1  b  ( ¯X),  j = 1, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  As for the term II, we first note that P 2(b), P 1(b), P e(b, σ) map to ρC( ¯X) + ¯H∞,1/2−  b  ( ¯  X), and thus  ∂k  σ�  Lb(0)P 2(b), ∂k  σ�  Lb(0)P 1(b), ∂k  σ�  Lb(0)P e(b, σ) : ¯Hs−1,ℓ+2  b  ( ¯  X) → ¯H  ∞, 3  2 −  b  ( ¯X),  k = 1, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  Next,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' we calculate  ∂σII = −∂σ ˇLb(σ)−1��  ∂σ�  Lb(0) + σ  2 ∂2  σ�  Lb(0)  �  (P 1(b) + P e(b,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' σ)) + 1  2∂2  σ�  Lb(0)P 2(b)  �  − ˇLb(σ)−1��  ∂σ�  Lb(0) + σ  2 ∂2  σ�  Lb(0)  �  ∂σP e(b,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' σ) + 1  2∂2  σ�  Lb(0)P e(b,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' σ)  �  ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  ∂2  σII = −∂2  σ ˇLb(σ)−1��  ∂σ�  Lb(0) + σ  2 ∂2  σ�  Lb(0)  �  (P 1(b) + P e(b,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' σ)) + 1  2∂2  σ�  Lb(0)P 2(b)  �  − 2∂σ ˇLb(σ)−1��  ∂σ�  Lb(0) + σ  2 ∂2  σ�  Lb(0)  �  ∂σP e(b,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' σ) + 1  2∂2  σ�  Lb(0)P e(b,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' σ)  �  − ˇLb(σ)−1��  ∂σ�  Lb(0) + σ  2 ∂2  σ�  Lb(0)  �  ∂2  σP e(b,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' σ) + ∂2  σ�  Lb(0)∂σP e(b,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' σ)  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  LINEAR STABILITY OF KERR-NEWMAN FAMILY  145  Therefore, using Proposition 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='3 and the fact that P e(b, σ) : ¯Hs−1,ℓ+2  b  → Kb,1 is ǫ-regular, we arrive at  ∂j  σII : ¯Hs−1,ℓ+2  b  ( ¯X) → |σ|−ǫ−j+1 ¯Hs−ǫ−j+1,ℓ+ǫ+1  b  ( ¯X),  j = 1, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  This completes the proof of the estimates (12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='19).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  □  Theorem 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Let s, ℓ, ǫ be defined as in Proposition 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Then we have  L−  b (σ) ∈ H  3  2 −ǫ−�  (−c0, c0);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' L  � ¯Hs−1,ℓ+2  b  ( ¯X), ¯H  s−max{ǫ, 1  2 },ℓ+ǫ−1  b  ( ¯X)  ��  (12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='20)  where c0 > 0 is a sufficiently small constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Using the estimates in Proposition 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='3 and following the proof at the beginning of Theorem 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='9,  we find that  L−  b (σ) − L−  b (0) ∈ σ  3  2 −ǫ−H2  b  �  [0, c0);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' L  � ¯Hs−1,ℓ+2  b  ( ¯X), ¯Hs−1−ǫ,ℓ+ǫ−1  b  ( ¯X)  ��  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  (12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='21)  Again integrating ∂σL−  b (σ) from σ ∈ (0, c0) to 0 and using the estimate (12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='19) for k = 1 yield  L−  b (σ) − L−  b (0) ∈ σ  3  2 −ǫ−H1  b  �  [0, c0);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' L  � ¯Hs−1,ℓ+2  b  ( ¯X), ¯Hs−ǫ,ℓ+ǫ−1  b  ( ¯  X)  ��  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  (12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='22)  Interpolating between (12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='21) and (12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='22) gives  L−  b (σ) − L−  b (0) ∈ σ  3  2 −ǫ−H1+θ  b  �  [0, c0);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' L  � ¯Hs−1,ℓ+2  b  ( ¯X), ¯Hs−θ−ǫ,ℓ+ǫ−1  b  ( ¯X)  ��  ,  0 ≤ θ ≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  (12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='23)  If ǫ ∈ (0, 1/2], we take θ = 1/2 − ǫ, and thus 3/2 − ǫ = 1 + θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' If ǫ ∈ (1/2, 1), we take θ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' This gives  L−  b (σ) − L−  b (0) ∈  \uf8f1  \uf8f2  \uf8f3  σ  3  2 −ǫ−H  3  2 −ǫ  b  �  [0, c0);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' L  � ¯Hs−1,ℓ+2  b  ( ¯X), ¯H  s− 1  2 ,ℓ+ǫ−1  b  ( ¯X)  ��  ,  0 < ǫ ≤ 1  2,  σ  3  2 −ǫ−H1  b  �  [0, c0);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' L  � ¯Hs−1,ℓ+2  b  ( ¯X), ¯Hs−ǫ,ℓ+ǫ−1  b  ( ¯X)  ��  ,  1  2 < ǫ < 1,  (12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='24)  which implies  L−  b (σ) − L−  b (0) ∈  \uf8f1  \uf8f2  \uf8f3  ˙H  3  2 −ǫ�  [0, c0);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' L  � ¯Hs−1,ℓ+2  b  ( ¯X), ¯H  s− 1  2 ,ℓ+ǫ−1  b  ( ¯X)  ��  ,  0 < ǫ ≤ 1  2,  ˙H  3  2 −ǫ−�  [0, c0);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' L  � ¯Hs−1,ℓ+2  b  ( ¯X), ¯Hs−ǫ,ℓ+ǫ−1  b  ( ¯X)  ��  ,  1  2 < ǫ < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  (12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='25)  Similarly, we also have the above statement for the other half interval (−c0, 0].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Since L−  b (σ) − L−  b (0) → 0 as  σ → 0 from both sides, by Lemma 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='8, we obtain  L−  b (σ) − L−  b (0) ∈ H  3  2 −ǫ−�  (−c0, c0);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' L  � ¯Hs−1,ℓ+2  b  ( ¯  X), ¯H  s−max{ǫ, 1  2 },ℓ+ǫ−1  b  ( ¯X)  ��  ,  and thus the conclusion (12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='20).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  □  12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Regularity for frequencies away from 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Now we discuss the regularity of �  Lb(σ)−1 for |σ| ≥  c0, 0 ≤ Im σ ≤ C for some c0, C > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  Theorem 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Let ℓ < − 1  2, s > 3+m and s+ℓ > − 1  2 +m where m ∈ N0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Then for 0 ≤ Im σ ≤ C, |σ| ≥ c0  with c0, C > 0, the operator  ∂m  σ �  Lb(σ)−1 : ¯Hs,ℓ+1  b,h  ( ¯X) → h−m ¯Hs−m,ℓ  b,h  ( ¯X)  (12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='26)  where h = |σ|−1, is uniformly bounded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' The estimate (12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='26) for the case m = 0 follows from high energy estimate 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  For m ≥ 1,  ∂m  σ �  Lb(σ)−1 is a linear combination of the operators of the type  �  �  Lb(σ)−1 ◦ ∂m1  σ �  Lb(σ)  �  · · · ◦  �  �  Lb(σ)−1 ◦ ∂mk  σ  �  Lb(σ)  �  �  Lb(σ)−1  where 1 ≤ m1, · · · , mk ≤ 2 and m1 + · · · + mk = m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Since  ∂σ�  Lb(σ) ∈ ρDiff1  b + ρ2C∞ ⊂ h−1ρDiff1  b,h  and  ∂2  σ�  Lb(σ) ∈ ρ2C∞,  146  LILI HE  it follows that  ¯Hs,ℓ+1  b,h  �  Lb(σ)−1  −−−−−→ ¯Hs,ℓ  b,h  ∂  mk  σ  �  Lb(σ)  −−−−−−→ hmk−2 ¯Hs+mk−2,ℓ+mk  b,h  ⊂ hmk−2 ¯Hs+mk−2,ℓ+1  b,h  �  Lb(σ)−1  −−−−−→ hmk−2 ¯Hs+mk−2,ℓ  b,h  ···  −→ · · ·  ···  −→ hm−m1−2(k−1) ¯Hs+m−m1−2(k−1),ℓ  b,h  ∂m1  σ  �  Lb(0)  −−−−−−→ hm−2k ¯Hs+m−2k,ℓ+m1  b,h  �  Lb(σ)−1  −−−−−→ hm−2k ¯Hs+m−2k,ℓ  b,h  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  Since 1 ≤ k ≤ m, we have hm−2k ¯Hs+m−2k,ℓ+m−k  b,h  ⊂ h−m ¯Hs−m,ℓ  b,h  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' This finishes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  □  12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Conormal regularity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Finally, we consider the conormal regularity of L−  b (σ) at σ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Specifically,  after applying any number of derivatives σ∂σ to L−  b (σ), it satisfies the same properties as L−  b (σ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  Recall that  L−  b (σ) = ˇLb(σ)−1Rreg(b, σ) − ˇLb(σ)−1��  ∂σ�  Lb(0) + σ  2 ∂2  σ�  Lb(0)  �  (P 1(b) + P e(b, σ)) + 1  2∂2  σ�  Lb(0)P 2(b)  �  ,  so in order to study the conormal regularity, we need to analyze that of ˇLb(σ)−1, Rreg(b, σ) and P e(b, σ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  We first discuss the conormal regularity of ˇLb(σ)−1 and prove a analogous result of Proposition 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='3 for  (σ∂σ)m ˇLb(σ)−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  Proposition 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Let − 3  2 < ℓ < − 1  2, 0 < ǫ < 1 with − 1  2 < ℓ + ǫ < 1  2, and s − ǫ > 4 + m where m ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  Define  ˇL(m)  b  (σ) := (σ∂σ)m ˇLb(σ)−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  For 0 < |σ| ≤ c0, Im σ ≥ 0 with c0 small, we have  ˇL(m)  b  (σ) : ¯Hs−1,ℓ+2  b  ( ¯X) → ¯Hs−m,ℓ  b  ( ¯X),  (12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='27)  ∂σ ˇL(m)  b  (σ) : ¯Hs−1,ℓ+2  b  ( ¯X) → |σ|−ǫ ¯Hs−m−ǫ,ℓ+ǫ−1  b  ( ¯X),  (12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='28)  ∂2  σ ˇL(m)  b  (σ) : ¯Hs−1,ℓ+2  b  ( ¯X) → |σ|−ǫ−1 ¯Hs−m−1−ǫ,ℓ+ǫ−1  b  ( ¯X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  (12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='29)  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Since  ∂2  σ ˇL(m)  b  (σ) = σ−1∂σ(σ∂σ − 1) ˇL(m)  b  (σ) = σ−1∂σ ˇL(m+1)  b  (σ) − σ−1∂σ ˇL(m)  b  (σ),  it suffices to prove the statements (12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='27) and (12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='28), and then the conclusion (12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='29) follows directly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  Now we prove (12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='27) and (12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='28) by induction on m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' For m = 1, we calculate  ˇL(1)  b  = σ∂σ ˇLb(σ)−1 = −ˇLb(σ)−1(σ∂σ ˇLb(σ))ˇLb(σ)−1  where  σ∂σ ˇLb(σ) = ˇLb(σ) − ˇLb(0) + σ2  2 ∂2  σ ˇLb(0) ∈ ˇLb(σ) + ρ2Diff2  b + σ2ρ2C∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  (12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='30)  As a consequence,  ˇL(1)  b  = −ˇLb(σ)−1 + ˇLb(σ)−1�  ρ2Diff2  b + σ2ρ2C∞�ˇLb(σ)−1  We then compute  ¯Hs−1,ℓ+2  b  ( ¯X)  ˇLb(σ)−1  −−−−−→ ¯Hs,ℓ  b ( ¯X)  ρ2Diff  2  b+σ2ρ2C∞  −−−−−−−−−−−−→ ¯Hs−2,ℓ+2  b  ( ¯X)  ˇLb(σ)−1  −−−−−→ ¯Hs−1,ℓ  b  ( ¯X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  This proves (12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='27) for m = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' As for the first derivative ∂σ ˇL(1)  b (σ), we calculate  ∂σ ˇL(1)  b (σ) = −∂σ ˇLb(σ)−1 + ∂σ ˇLb(σ)−1�  ρ2Diff2  b + σ2ρ2C∞�ˇLb(σ)−1 + ˇLb(σ)−1�  σρ2C∞�ˇLb(σ)−1  + ˇLb(σ)−1�  ρ2Diff2  b + σ2ρ2C∞�  ∂σ ˇLb(σ)−1 := I + II + III + IV .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  LINEAR STABILITY OF KERR-NEWMAN FAMILY  147  Using Proposition 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' we find that  I : ¯Hs−1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='ℓ+2  b  ( ¯  X) → |σ|−ǫ ¯Hs−ǫ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='ℓ+ǫ+1  b  ( ¯X),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  II : ¯Hs−1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='ℓ+2  b  ( ¯  X)  ˇLb(σ)−1  −−−−−→ ¯Hs,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='ℓ  b ( ¯X)  ρ2Diff  2  b+σ2ρ2C∞  −−−−−−−−−−−−→ ¯Hs−2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='ℓ+2  b  ( ¯X)  ∂σ ˇLb(σ)−1  −−−−−−−→ |σ|−ǫ ¯Hs−1−ǫ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='ℓ+ǫ−1  b  ( ¯X),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  III : ¯Hs−1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='ℓ+2  b  ( ¯  X)  ˇLb(σ)−1  −−−−−→ ¯Hs,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='ℓ  b ( ¯X)  σρ2C∞  −−−−−→ ¯Hs,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='ℓ+2  b  ( ¯X)  ˇLb(σ)−1  −−−−−→ ¯Hs−1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='ℓ  b  ( ¯X) ⊂ |σ|−ǫ ¯Hs−1−ǫ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='ℓ+ǫ−1  b  ( ¯X),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  IV : ¯Hs−1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='ℓ+2  b  ( ¯  X)  ∂σ ˇLb(σ)−1  −−−−−−−→ |σ|−ǫ ¯Hs−ǫ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='ℓ+ǫ−1  b  ( ¯X)  ρ2Diff  2  b+σ2ρ2C∞  −−−−−−−−−−−−→ |σ|−ǫ ¯Hs−ǫ−2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='ℓ+ǫ+1  b  ( ¯X)  ˇLb(σ)−1  −−−−−→ |σ|−ǫ ¯Hs−1−ǫ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='ℓ+ǫ−1  b  ( ¯X)  where we use 1/2 < ℓ + ǫ + 1 < 3/2 in the last step of the analysis of IV .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' This proves (12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='28) for m = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  Suppose that (12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='27) and (12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='28) hold for k ≤ m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Then for m + 1, by a direct calculation, we obtain that  ˇL(m+1)  b  (σ) is a linear combination of the operators of the type  L1 = ˇL(m)  b  (σ),  L2 := ˇL(m1)  b  (σ) ◦ (ρ2Diff2  b) ◦ ˇL(m2)  b  (σ)  where m1, m2 ≥ 0 and m1 + m2 ≤ m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Then by the induction hypothesis on k ≤ m, it follows that  L1 : ¯Hs−1,ℓ+2  b  ( ¯X)  ˇ  L(m)  b  (σ)  −−−−−→ ¯Hs−m,ℓ  b  ( ¯X) ⊂ ¯Hs−m−1,ℓ  b  ( ¯X),  L2 : ¯Hs−1,ℓ+2  b  ( ¯X)  ˇ  L(m1)  b  (σ)  −−−−−−→ ¯Hs−m1,ℓ  b  ( ¯X)  ρ2Diff  2  b  −−−−−→ ¯Hs−m1−2,ℓ+2  b  ( ¯X)  ˇ  L(m1)  b  (σ)  −−−−−−→ ¯Hs−m1−m2−1,ℓ  b  ( ¯X) = ¯Hs−m−1,ℓ  b  ( ¯X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  This proves (12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='27) for m+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' As for the first derivative ∂σ ˇL(m+1)  b  , it is a linear combination of the operators  of the following types  D1 = ∂σ ˇL(m)  b  (σ),  D2 := ∂σ ˇL(m1)  b  (σ) ◦ (ρ2Diff2  b) ◦ ˇL(m2)  b  (σ),  D3 := ˇL(m1)  b  (σ) ◦ (ρ2Diff2  b) ◦ ∂σ ˇL(m2)  b  (σ)  where m1, m2 ≥ 0 and m1 + m2 ≤ m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Then we have  D1 : ¯Hs−1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='ℓ+2  b  ( ¯X)  ∂σ ˇ  L(m)  b  (σ)  −−−−−−−→ |σ|−ǫ ¯Hs−m−ǫ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='ℓ+ǫ−1  b  ( ¯X) ⊂ |σ|−ǫ ¯Hs−m−1ǫ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='ℓ+ǫ−1  b  ( ¯X),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  D2 : ¯Hs−1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='ℓ+2  b  ( ¯X)  ˇ  L(m2)  b  (σ)  −−−−−−→ ¯Hs−m2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='ℓ  b  ( ¯X)  ρ2Diff  2  b  −−−−−→ ¯Hs−m2−2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='ℓ+2  b  ( ¯X)  ∂σ ˇ  L(m1)  b  (σ)  −−−−−−−→ |σ|−ǫ ¯Hs−m1−1−m2−ǫ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='ℓ+ǫ−1  b  ( ¯X) ⊂ |σ|−ǫ ¯Hs−m−1−ǫ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='ℓ+ǫ−1  b  ( ¯  X),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  D3 : ¯Hs−1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='ℓ+2  b  ( ¯X)  ∂σ ˇ  L(m2)  b  (σ)  −−−−−−−→ |σ|−ǫ ¯Hs−m2−ǫ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='ℓ+ǫ−1  b  ( ¯  X)  ρ2Diff  2  b  −−−−−→ |σ|−ǫ ¯Hs−m2−ǫ−2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='ℓ+ǫ+1  b  ( ¯X)  ˇ  L(m1)  b  (σ)  −−−−−−→ |σ|−ǫ ¯Hs−m1−m2−ǫ−1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='ℓ+ǫ−1  b  ( ¯X) ⊂ |σ|−ǫ ¯Hs−m−1−ǫ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='ℓ+ǫ−1  b  ( ¯X)  where we use 1/2 < ℓ + ǫ + 1 < 3/2 in the third step of the analysis of D3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' This proves (12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='28) for m + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  This finishes the proof of the proposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  □  With the conormal regularity of ˇLb(σ)−1 with our disposal, we are ready to discuss that of the entries �Lij  of the operator �  Lb(σ)ˇLb(σ)−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  Recall the expression (11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='25) of �  Lb(σ)ˇL(σ)−1  �  Lb(σ)ˇLb(σ)−1 =  \uf8eb  \uf8ec  \uf8ed  L00(b, σ)  σ2 �L01(b, σ)  σ�L02(b, σ)  σ�L10(b, σ)  σ2 �L11(b, σ)  σ2 �L12(b, σ)  σ�L20(b, σ)  σ2 �L21(b, σ)  σ�L22(b, σ)  \uf8f6  \uf8f7  \uf8f8  =  \uf8eb  \uf8ec  \uf8ed  L00(b, σ)  σ2�L01(b, σ)  σ�L02(b, σ)  σ  ��L0  10(b) + σ�Le  10(b, σ)  �  σ2��L0  11(b) + σ�L1  11(b) + σ2�Le  11(b, σ)  �  σ2��L0  12(b) + σ�Le  12(b, σ)  �  σ�L20(b, σ)  σ2��L0  21(b) + σ�Le  21(b)  �  σ  ��L0  22(b) + σ�Le  22(b, σ)  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  \uf8f6  \uf8f7  \uf8f8  Proposition 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Let s, ℓ, ǫ, m be defined as in Proposition 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Then (σ∂σ)m�Lij satisfy the following  properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  148  LILI HE  (1) (σ∂σ)mL00, (σ∂σ)m�L01, (σ∂σ)m�L02, (σ∂σ)m�L20 are ǫ-regular at σ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  (2) (σ∂σ)m�L10, (σ∂σ)m�L12, (σ∂σ)m�L21, (σ∂σ)m�L22 have an ǫ-regular expansion up to order one, that is,  (σ∂σ)m�Le  10, (σ∂σ)m�Le  12, (σ∂σ)m�Le  21, (σ∂σ)m�Le  22 are ǫ-regular at σ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  (3) (σ∂σ)m�L11 has an ǫ-regular expansion up to order two, that is, (σ∂σ)m�Le  11 is ǫ-regular at σ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' The proof is analogous to that of Proposition 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='4, now by using Proposition 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='13 instead of Propo-  sition 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Here, we only demonstrate the proof of (σ∂σ)m�L10 in detail because the proof of the other entries  follows in a similar manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  Recall the calculation in (11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='18)  (σ∂σ)m�Le  10(˜g0, ˜A0), (˙g∗  1, ˙A∗  1)⟩  = −  �  j+k=m  �  (˜g0, ˜A0), i  � ˇL(k)  b (σ)  �∗�  (σ∂σ)j�  ∂σ�  Lb(0) + σ  2 ∂2  σ�  Lb(0)  ��∗�  I − (ˇLb(0)−1)∗V ∗  b  �  (ˇg∗  2, ˇA∗  2)  �  +  �  (˜g0, ˜A0), 1  2( ˇL(m)  b  (σ))∗(∂2  σ�  Lb(0))∗(˙g∗  2, ˙A∗  2)  �  where  �  (σ∂σ)j�  ∂σ�  Lb(0) + σ  2 ∂2  σ�  Lb(0)  ��∗�  I − (ˇLb(0)−1)∗V ∗  b  �  (ˇg∗  2, ˇA∗  2),  (∂2  σ�  Lb(0))∗(˙g∗  2, ˙A∗  2) ∈ ˙H−5/2−C(a,γ), 3/2−  b  ( ¯X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  Since (˜g0, ˜A0) ∈ ¯Hs−1,ℓ+2  b  ( ¯X) with s − m − ǫ > 4, and by Proposition 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='13  ( ˇL(k)  b (σ))∗ ˙H−5/2−C(a,γ), 3/2−  b  ( ¯X) ∈ ˙H−3/2−k−C(a,γ),−1/2−  b  ( ¯X),  0 ≤ k ≤ m, k ∈ N0,  (∂j  σ ˇL(k)  b (σ))∗ ˙H−5/2−C(a,γ), 3/2−  b  ( ¯X) ∈ |σ|−ǫ−j+1 ˙H−s+1,−ℓ−2  b  ( ¯X),  j = 1, 2  and  0 ≤ k ≤ m, k ∈ N0  where we use the facts that −5/2 − C(a, γ) > −3 > −s + k + ǫ + 1 for 0 ≤ k ≤ m and −ℓ − ǫ + 1 < 3/2,  the pairings above (and their up to second order derivatives) are well-defined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' This proves the ǫ-regularity  of (σ∂σ)�Le  10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  □  Next, we turn to discussing the conormal regularity of R(b, σ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' To this end, we need the following lemma,  which is an analogy of Lemma 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  Lemma 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Let s, ℓ, ǫ, m be defined as in Proposition 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Suppose that the operator A(σ) : �Ki → Rj  with i, j = 0, 1, 2 has an ǫ-regular expansion up to order one, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=',  A(σ) = A0 + σAe(σ)  where A0 : �Ki → Rj is invertible and independent of σ, and (σ∂σ)kAe(σ) : �Ki → Rj is ǫ-regular for any  0 ≤ k ≤ m, k ∈ N0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Then for |σ| sufficiently small, A(σ) is invertible with inverse B(σ) = A(σ)−1, and  (σ∂σ)kB(σ) has an ǫ-regular expansion up to order one, that is,  (σ∂σ)kB(σ) = B0  k + σBe  k(σ),  0 ≤ k ≤ m, k ∈ N0  where B0  k : Rj → �Ki is independent of σ, and Be  k(σ) : Rj → �Ki is ǫ-regular.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  Similarly, if A(σ) has uniformly bounded inverse B(σ), and (σ∂σ)kA(σ) is ǫ-regular for any 0 ≤ k ≤  m, k ∈ N0, then (σ∂σ)kB(σ) is ǫ-regular as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' As established in the proof of Lemma 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='5, we have  A(σ)−1 =  �  I +  ∞  �  k=1  (−1)k�  σ(A0)−1Ae(σ)  �k�  (A0)−1  = (A0)−1 − σ  �  (A0)−1Ae(σ)  ∞  �  k=0  (−1)k�  σ(A0)−1Ae(σ)  �k(A0)−1�  := B0 + σBe(σ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  For 1 ≤ k ≤ m, we have  (σ∂σ)kB(σ) = σ  k  �  j=0  �k  j  �  (σ∂σ)jBe(σ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  A direct calculation implies that (σ∂σ)jBe(σ) is a sum of the operators of the type σ−1L ◦ (A0)−1, where L  is a composition of the terms of the form σ(A0)−1(σ∂σ)lAe(σ) with 0 ≤ l ≤ j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Then proceeding as in the  LINEAR STABILITY OF KERR-NEWMAN FAMILY  149  proof of Lemma 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='5, we conclude that (σ∂σ)jBe(σ) with 0 ≤ j ≤ k is ǫ-regular, and thus (σ∂σ)kBe(σ) with  1 ≤ k ≤ m has an ǫ-regular expansion up to order one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  Since (σ∂σ)mB(σ) is a linear combination of the operators of the form  (σ∂σ)m1B(σ) ◦ (σ∂σ)m2A(σ) ◦ (σ∂σ)m3B(σ)  where 1 ≤ m2 ≤ m, 0 ≤ m1, m2 ≤ m − 1 and m1 + m2 + m3 = m, it follows that the second statement  follows by an induction argument on m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  □  Recall the expressions (11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='26) and (11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='32) of Rb,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='σ = (�  Lb(σ)ˇLb(σ)−1)−1  R(b,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' σ) = (�  Lb(σ)ˇLb(σ)−1)−1 =  \uf8eb  \uf8ec  \uf8ed  �R00(b,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' σ)  �R01(b,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' σ)  �R02(b,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' σ)  σ−1 �R10(b,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' σ)  σ−2 �R11(b,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' σ)  σ−1 �R12(b,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' σ)  �R20(b,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' σ)  σ−1 �R21(b,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' σ)  σ−1 �R22(b,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' σ)  \uf8f6  \uf8f7  \uf8f8  =  \uf8eb  \uf8ec  \uf8ed  �R00(b,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' σ)  �R01(b,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' σ)  �R02(b,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' σ)  σ−1 �R10(b,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' σ)  σ−2� �R0  11(b) + σ �Re  11(b,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' σ)  �  σ−1 �R12(b,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' σ)  �R20(b,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' σ)  σ−1� �R0  21(b) + σ �Re  21(b,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' σ)  �  σ−1� �R0  22(b) + σ �Re  22(b,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' σ)  �  \uf8f6  \uf8f7  \uf8f8  where  �R11 =  ��L♯  11 − σ�L♯  12(�L♭  22)−1�L♯  21  �−1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  �R10 = �R11  �  − �L10 + σ�L♯  12(�L♭  22)−1�L20  �  L−1  00 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  �R12 = − �R11�L♯  12(�L♭  22)−1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  �R21 = −(�L♭  22)−1�L♯  21 �R11,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  �R22 = (�L♭  22)−1 − σ(�L♭  22)−1�L♯  21 �R12,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  �R20 = −(�L♭  22)−1��L20L−1  00 + �L♯  21 �R10  �  ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  �R01 = −L−1  00  ��L01 �R11 + �L02 �R21  �  ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  �R02 = −L−1  00  �  σ�L01 �R12 + �L02 �R22  �  ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  �R00 = L−1  00  �  I − σ�L01 �R10 − σ�L02 �R20  �  (12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='31)  with  �L♯  i1 = �Li1 − σ�Li0L−1  00 �L01  for  i = 1, 2,  �L♯  12 = �L12 − �L10L−1  00 �L02,  �L♭  22 = �L22 − σ�L20L−1  00 �L02.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  (12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='32)  Proposition 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Let s, ℓ, ǫ, m be defined as in Proposition 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Then (σ∂σ)m �Rij satisfy the following  properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  (1) (σ∂σ)m �R00, (σ∂σ)m �R01, (σ∂σ)m �R02, (σ∂σ)m �R10, (σ∂σ)m �R12, (σ∂σ)m �R20 are ǫ-regular at σ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  (2) (σ∂σ)m �R11, (σ∂σ)m �R21, (σ∂σ)m �R22 have an ǫ-regular expansion up to order one, that is, (σ∂σ)m �Re  11,  (σ∂σ)m �Re  21, (σ∂σ)m �Re  22 are ǫ-regular at σ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Applying Proposition 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='14 and Lemma 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='15, we conclude that for 0 ≤ k ≤ m, (σ∂σ)kL−1  00 is ǫ-  regular.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Then by Proposition 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='14, we find that (σ∂σ)k �L♯  11, (σ∂σ)k �L♯  21, (σ∂σ)k �L♭  22 (and thus (σ∂σ)k(�L♭  22)−1  by Lemma 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='15) have an ǫ-regular expansion up to order one, while (σ∂σ)k �L♯  12 is ǫ-regular.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Therefore, we  see that  (σ∂σ)k�  �L♯  11 − σ�L♯  12(�L♭  22)−1�L♯  21  �  has an ǫ-regular expansion up to order one whose coefficient of σ0 is (σ∂σ)k �L0  11(b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  Then the ǫ-regular  expansion property of (σ∂σ)m �R11 follows from Lemma 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Similarly, the ǫ-regular expansion property of  (σ∂σ)m �R21, (σ∂σ)m �R22 follows from that of (σ∂σ)k �R11, (σ∂σ)k(�L♭  22)−1, (σ∂σ)m�L♯  21 and the above expressions  of �Rij.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Finally, the ǫ-regularity of the remaining components can be obtained by using the above expressions  of �Rij and Proposition 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  □  Proposition 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Let L−  b (σ) be the regular part of �  Lb(σ)−1 defined in Corollary 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='4 and define  L−(m)  b  (σ) := (σ∂σ)mL−  b (σ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  Let − 3  2 < ℓ < − 1  2, 0 < ǫ < 1, − 1  2 < ℓ+ǫ < 1  2 and s−ǫ > 4+m where m ∈ N0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' For Im σ ≥ 0 and 0 < |σ| ≤ c0  where c0 > 0 is a small constant, L−(m)  b  (σ) satisfies  L−(m)  b  (σ) : ¯Hs−1,ℓ+2  b  ( ¯X) → ¯Hs−m,ℓ+ǫ−1  b  ( ¯X)  (12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='33)  150  LILI HE  and  ∂j  σL−(m)  b  (σ) : ¯Hs−1,ℓ+2  b  ( ¯X) → |σ|−ǫ−j+1 ¯Hs−m−ǫ−j+1,ℓ+ǫ−1  b  ( ¯X),  j = 1, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  (12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='34)  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Recall that  L−  b (σ) = ˇLb(σ)−1Rreg(b, σ) − ˇLb(σ)−1��  ∂σ�  Lb(0) + σ  2 ∂2  σ�  Lb(0)  �  (P 1(b) + P e(b, σ)) + 1  2∂2  σ�  Lb(0)P 2(b)  �  where P 2(b), P 1(b) : ¯Hs−1,ℓ+2  b  ( ¯X) → ρC∞( ¯X) + ¯H  ∞, 1  2 −  b  ( ¯X),  (σ∂σ)kP e(b, σ) : ¯Hs−1,ℓ+2  b  ( ¯X) → Kb,1  is ǫ-regular for 0 ≤ k ≤ m, and  Rreg(b, σ) =  \uf8eb  \uf8ed  �R00(b, σ)  �R01(b, σ)  �R02(b, σ)  0  0  0  �R20(b, σ)  �Re  21(b, σ)  �Re  22(b, σ)  \uf8f6  \uf8f8 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  Then a direct calculation implies (σ∂σ)mL−  b (σ) is a linear combination of the operators of the forms  ˇL(m1)  b  (σ) ◦ (σ∂σ)m2Rreg(b, σ),  ˇL(m1)  b  (σ) ◦ ∂2  σ�  Lb(0) ◦ P 2(b),  ˇL(m1)  b  (σ) ◦ (σ∂σ)m2�  ∂σ�  Lb(0) + σ  2 ∂2  σ�  Lb(0)  �  (σ∂σ)m3�  P 1(b) + P e(b, σ)  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  Applying Proposition 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='13 and 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='16, we obtain (12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='33) and (12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='34).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  □  13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Decay estimates  In this section, we continue using the notation from §12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' We will study the decay of the solution (˙g, ˙A)  to the equations  Lb(˙g, ˙A) = f,  f ∈ C∞  c ((0, ∞)tb,∗;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' ¯Hs,ℓ+2  b  ( ¯X)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  (13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='1)  We define spacetime Sobolev spaces  �Hs,ℓ,k  b,c ,  k ∈ N0,  (13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='2)  which is equal to L2(t−1  b,∗([0, ∞));' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' |dgb|) for s, ℓ, k = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Here, the index s measures the regularity with respect  to the derivatives ∂tb,∗ and the stationary b-vector fields on ¯X (which is spanned by r∂r and rotation vector  fields).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' The index ℓ is the weight in ρ = r−1, that is, �Hs,ℓ,k  b,c  = ρℓ �Hs,k  b,c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' The index k ∈ N0 measures the  regularity with respect to ⟨tb,∗⟩∂tb,∗ (which is related to the conormal regularity of �  Lb(σ)−1 on the Fourier  transform side).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Therefore, u ∈ �Hs,ℓ,k  b,c  if and only if (⟨tb,∗⟩∂tb,∗)ju ∈ �Hs−j,ℓ  b,c  := �Hs−j,ℓ,0  b,c  for any 0 ≤ j ≤ k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  Theorem 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Let − 3  2 < ℓ < − 1  2, 0 < ǫ < 1, − 1  2 < ℓ + ǫ < 1  2 and s − ǫ > 4 + m where m ∈ N0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Let (˙g, ˙A)  be the solution to the equation (13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='1) with f ∈ C∞  c ((0, ∞)tb,∗;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' ¯Hs,ℓ+2  b  ( ¯X)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Then there exists a generalized  zero mode (ˆg, ˆA) ∈ �Kb (defined in Proposition 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='5) such that  (˙g, ˙A) = (ˆg, ˆA) + (¯g, ¯A) + (˜g, ˜A)  (13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='3)  and (ˆg, ˆA), (¯g, ¯A) and (˜g, ˜A) satisfy the following properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  (1) The term (¯g, ¯A) can be written as (¯g, ¯A) = (¯g1, ¯A1)+(¯g2, ¯A2) where (¯g1, ¯A1) is a pure gauge solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  Moreover, for m ≥ 1 and j ≥ 0, we have the pointwise estimates  |  �  ⟨tb,∗⟩∂tb,∗  �m−1V j(¯g1, ¯A1)| ≲  �  ⟨tb,∗ + r⟩−2+ǫ + ⟨tb,∗⟩−1+ǫ⟨r⟩−2�  ∥f∥⟨tb,∗⟩− 3  2 − �  Hs−1,ℓ+2  b,c  (13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='4)  and  |  �  ⟨tb,∗⟩∂tb,∗  �m−1V j(¯g2, ¯A2)| ≲ ⟨tb,∗ + r⟩−2+ǫ∥f∥⟨tb,∗⟩− 3  2 − �  Hs−1,ℓ+2  b,c  (13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='5)  where V j denotes the product of j vector fields V ∈ {∂tb,∗, r∂r, rotation fields }.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  LINEAR STABILITY OF KERR-NEWMAN FAMILY  151  (2) (˜g, ˜A) satisfies  ∥(˜g, ˜A)∥⟨tb,∗⟩− 3  2 +ǫ �  Hs−2,ℓ+ǫ−1,m  b,c  ≲ ∥f∥⟨tb,∗⟩− 7  2 +ǫ �  Hs+m,ℓ+2,m  b,c  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  (13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='6)  For m ≥ 1, we also have  ∥  �  ⟨tb,∗⟩∂tb,∗  �m−1(˜g, ˜A)∥⟨tb,∗⟩−2+ǫL∞(Rtb,∗ ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' ¯  Hs−2−m,ℓ+ǫ−1  b  ( ¯  X)) ≲ ∥f∥⟨tb,∗⟩− 7  2 +ǫ �  Hs+m,ℓ+2,m  b,c  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  (13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='7)  For m ≥ 1 and 0 ≤ j < s − 4 − m with j ∈ N0, we have the pointwise decay estimate  |  �  ⟨tb,∗⟩∂tb,∗  �m−1V j(˜g, ˜A)| ≲ ⟨tb,∗⟩−2+ǫ⟨r⟩− 1  2 −ℓ−ǫ∥f∥⟨tb,∗⟩− 7  2 +ǫ �  Hs+m,ℓ+2,m  b,c  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  (13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='8)  (3) Recall the definition of Kb,1, Kb,2 and K∗  b at the beginning of §11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' The leading term (ˆg, ˆA) ∈ �Kb is  uniquely determined by f and can be expressed as follows:  (ˆg, ˆA) = −1  2  �  tb,∗(˙g1, ˙A1) + (ˇg1, ˇA1)  �  + 1  2  �  (˙g′  1, ˙A′  1) + (˙g′′  1, ˙A′′  1) + (˙g2, ˙A2)  �  (13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='9)  where (˙g1, ˙A1), (˙g′  1, ˙A′  1), (˙g′′  1 , ˙A′′  1) ∈ Kb,1, and (˙g2, ˙A2) ∈ Kb,2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  Moreover, (˙g1, ˙A1), (˙g′  1, ˙A′  1) and  (˙g2, ˙A2) are determined by the conditions (11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='30) and (11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='31) with f there replaced by  �  R f(tb,∗) dtb,∗,  and (˙g′′  1, ˙A′′  1) ∈ Kb,1 is uniquely determined by the condition  −  �  [Lb, tb,∗](ˇg′′  1 , ˇA′′  1) + 1  2[[Lb, tb,∗], tb,∗](˙g′′  1, ˙A′′  1), (˙g∗, ˙A∗)  �  +  �  [Lb, tb,∗](˙g′′  2, ˙A′′  2), (˙g∗, ˙A∗)  �  = ⟨  �  R  tb,∗f(tb,∗) dtb,∗, (˙g∗, ˙A∗)⟩  for all  (˙g∗, ˙A∗) ∈ K∗  b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  (13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='10)  for some (˙g′′  2, ˙A′′  2) ∈ Kb,2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' The strategy of the proof is to take Fourier transform of the inhomogeneous equation Lb(˙g, ˙A) = f  in tb,∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' We define the Fourier transform of (˙g, ˙A) as  (ˆ˙g(σ), ˆ˙A(σ)) :=  �  R  eitb,∗σ(˙g, ˙A)(tb,∗) dtb,∗  (13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='11)  and likewise for f (here we drop the notation for the spatial variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  First, according to the energy estimate in Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='27, we see that the solution (˙g, ˙A) satisfies  ∥(˙g, ˙A)(tb,∗, ·)∥ ¯  H1,−1  b  ( ¯  X) ≲ eMtb,∗  for some M > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' This implies that the Fourier transform (ˆ˙g(σ), ˆ˙A(σ)) of (˙g, ˙A) is well defined for Im σ >  M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  Since f(tb,∗) has compact support in tb,∗, ˆf(σ) is holomorphic in the full plane σ ∈ C and decays  superpolynomially as |Re σ| → ∞ with Im σ > 0 fixed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Fourier transforming of the equation Lb(˙g, ˙A) = f,  we obtain  �  Lb(σ)(ˆ˙g(σ), ˆ˙A(σ)) = ˆf(σ),  Im σ > M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  (13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='12)  In view of Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='28, we see that �  Lb(σ) is invertible in Im σ > M and  (ˆ˙g(σ), ˆ˙A(σ)) = �  Lb(σ)−1 ˆf(σ),  Im σ > M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  Therefore, we have the following integral representation  (˙g(tb,∗),  ˙A(tb,∗)) = 1  2π  �  Im σ=M+1  e−iσtb,∗ �  Lb(σ)−1 ˆf(σ) dσ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  (13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='13)  Owing to Theorem 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='12, we see that the integrand in the above integral representation is holomorphic in σ  (with values in ¯Hs,ℓ  b ( ¯X)) in upper half plane Im σ > 0, and decay superpolynomially in |Re σ| as with Im σ  bounded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Using Cauchy’s Theorem, we obtain  (˙g(tb,∗),  ˙A(tb,∗)) = 1  2π  �  Im σ=C  e−iσtb,∗ �  Lb(σ)−1 ˆf(σ) dσ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  (13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='14)  for any C > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Now we let C → 0+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  152  LILI HE  Fixing a frequency cutoff χ(s) ∈ C∞  c (R) such that χ(s) = 1 for |s| ≤ c0/2 and χ(s) = 0 for |s| ≥ c0 where  c0 is defined as in Theorem 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' We now write  ˆf(σ) = χ(Re σ) ˆf(σ) + (1 − χ(Re σ)) ˆf(σ) := ˆf1(σ) + ˆf2(σ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  Correspondingly, by Theorem 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='3 and Corollary 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='4, we split �  Lb(σ)−1 ˆf(σ) into two parts  �  Lb(σ)−1 ˆf(σ) = (ˆ˙greg(σ), ˆ˙Areg(σ)) + (ˆ˙gsing(σ), ˆ˙Asing(σ))  where  (ˆ˙greg(σ), ˆ˙Areg(σ)) = L−  b (σ) ˆf1(σ) + �  Lb(σ)−1 ˆf2(σ),  (ˆ˙gsing(σ), ˆ˙Asing(σ)) = P(b, σ) ˆf1(σ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  As discussed above, we can shift the integration contour to Im σ = C for any C > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Letting C → 0+, we  obtain  (˙g(tb,∗), ˙A(tb,∗)) = (˙greg(tb,∗), ˙Areg(tb,∗)) + (˙gsing(tb,∗), ˙Asing(tb,∗))  where  (˙greg(tb,∗), ˙Areg(tb,∗)) = 1  2π  �  R  e−iσtb,∗(ˆ˙greg(σ), ˆ˙Areg(σ)) dσ  and  (˙gsing(tb,∗), ˙Asing(tb,∗)) =  lim  C→0+  1  2π  �  Im σ=C  e−iσtb,∗(ˆ˙gsing(σ), ˆ˙Asing(σ)) dσ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  Analysis of (˙greg(tb,∗), ˙Areg(tb,∗)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Since f ∈ C∞  c ((0, ∞)tb,∗;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' ¯Hs,ℓ+2  b  ( ¯X)), we see that ˆf1(σ), ˆf2(σ) ∈  Hs′(Rσ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' ¯Hs,ℓ+2  b  ( ¯X)) for any s′ ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' As H3/2−ǫ′(R) is an algebra for any 0 < ǫ′ < 1, by Theorem  12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='11, we have  L−  b (σ) ˆf1(σ) ∈ H  3  2 −ǫ′((−c0, c0);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' ¯H  s+1−max{ǫ, 1  2 },ℓ+ǫ−1  b  ( ¯X)),  0 < ǫ < ǫ′ < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  Using Moser estimate, we bound norm of L−  b (σ) ˆf1(σ) as follows  ∥L−  b (σ) ˆf1(σ)∥  H  3  2 −ǫ′ (Rσ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' ¯  H  s+1−max{ǫ, 1  2 },ℓ+ǫ−1  b  ( ¯  X)) ≲ ∥ ˆf1(σ)∥H  3  2 −ǫ′ (Rσ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' ¯  Hs,ℓ+2  b  ( ¯  X))  ≲ ∥ ˆf(σ)∥H  3  2 −ǫ′ (Rσ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' ¯  Hs,ℓ+2  b  ( ¯  X)) = ∥⟨tb,∗⟩  3  2 −ǫ′f∥L2(Rtb,∗;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' ¯  Hs,ℓ+2  b  ( ¯  X)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  By exploiting the conormal regularity of L−  b (σ) proved in Proposition 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='13, we find that for 0 ≤  k ≤ m,  (σ∂σ)k�  L−  b (σ) ˆf1(σ)  �  ∈ H  3  2 −ǫ′((−c0, c0);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' ¯H  s+1−k−max{ǫ, 1  2 },ℓ+ǫ−1  b  ( ¯X)),  0 < ǫ < ǫ′ < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  In a similar manner, we arrive at  ∥(σ∂σ)k�  L−  b (σ) ˆf1(σ)  �  ∥  H  3  2 −ǫ′ (Rσ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' ¯  H  s+1−k−max{ǫ, 1  2 },ℓ+ǫ−1  b  ( ¯  X))  ≲  k  �  j=0  ∥(σ∂σ)j ˆf1(σ)∥H  3  2 −ǫ′ (Rσ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' ¯  Hs−j,ℓ+2  b  ( ¯  X)) ≲  k  �  j=0  ∥(σ∂σ)j ˆf(σ)∥H  3  2 −ǫ′ (Rσ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' ¯  Hs−j,ℓ+2  b  ( ¯  X))  =  k  �  j=0  ∥(tb,∗∂tb,∗)j⟨tb,∗⟩  3  2 −ǫ′f∥L2(Rtb,∗;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' ¯  Hs−j,ℓ+2  b  ( ¯  X)) ≲ ∥f∥⟨tb,∗⟩− 3  2 +ǫ′ �  Hs,ℓ+2,k  b,c  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  (13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='15)  As for the second part �  Lb(σ)−1 ˆf2(σ) in (ˆ˙greg(σ), ˆ˙Areg(σ)), by Theorem 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='12, we have  �  Lb(σ)−1 ∈ W j,∞�  Rσ \\ [−c0  2 , c0  2 ];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' L  � ¯Hs,ℓ+2  b,h  , h−j ¯Hs−j,ℓ  b,h  ��  ,  j ∈ N0  (13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='16)  where we take h = ⟨σ⟩−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Taking j = 2 gives  �  Lb(σ)−1 ˆf2(σ) ∈ H  3  2 −ǫ(R;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' ⟨σ⟩−s+2 ¯Hs−2,ℓ  b,⟨σ⟩−1( ¯X)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  whose norm is bounded by  ∥ ˆf2(σ)∥H  3  2 −ǫ(R;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='⟨σ⟩−s ¯  Hs,ℓ+2  b,⟨σ⟩−1 ( ¯  X)) ≲ ∥f∥⟨tb,∗⟩− 3  2 +ǫ �  Hs,ℓ+2  b,c  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  More generally, we have  (σ∂σ)k��  Lb(σ)−1 ˆf2(σ)  �  ∈ H  3  2 −ǫ(R;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' ⟨σ⟩−s+2+2k ¯Hs−2−k,ℓ  b,⟨σ⟩−1 ( ¯X)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  LINEAR STABILITY OF KERR-NEWMAN FAMILY  153  and  ∥(σ∂σ)k��  Lb(σ)−1 ˆf2(σ)  �  ∥H  3  2 −ǫ(R;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='⟨σ⟩−s+2+k ¯  Hs−2−k,ℓ  b,⟨σ⟩−1 ( ¯  X))  ≲  k  �  j=0  ∥(σ∂σ)j ˆf2(σ)∥H  3  2 −ǫ(R;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='⟨σ⟩−s−k+2j ¯  Hs+k−2j,ℓ+2  b,⟨σ⟩−1  ( ¯  X))  ≲  k  �  j=0  ∥(σ∂σ)j ˆf(σ)∥H  3  2 −ǫ(R;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='⟨σ⟩−s−k+2j ¯  Hs+k−2j,ℓ+2  b,⟨σ⟩−1  ( ¯  X))  ≲  k  �  j=0  ∥(tb,∗∂tb,∗)jf∥⟨tb,∗⟩− 3  2 +ǫ �  Hs+k−2j,ℓ+2  b,c  ≲ ∥f∥⟨tb,∗⟩− 3  2 +ǫ �  Hs+k,ℓ+2,k  b,c  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  (13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='17)  Taking inverse Fourier transform of (ˆ˙greg(σ), ˆ˙Areg(σ)) and using the estimates (13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='15) and (13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='17),  we obtain for 0 ≤ k ≤ m and 0 < ǫ < ǫ′ < 1  ∥(˙greg(tb,∗), ˙Areg(tb,∗))∥⟨tb,∗⟩− 3  2 +ǫ′ �  Hs−2,ℓ+ǫ−1,k  b,c  ∼  k  �  j=0  ∥(σ∂σ)j(ˆ˙greg(σ), ˆ˙Areg(σ))∥H  3  2 −ǫ′ (R;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='⟨σ⟩−s+2+j ¯  Hs−2−j,ℓ+ǫ−1  b,⟨σ⟩−1  ( ¯  X))  ≲  k  �  j=0  ∥(σ∂σ)j�  L−  b (σ) ˆf1(σ)  �  ∥  H  3  2 −ǫ′ (Rσ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' ¯  H  s+1−j−max{ǫ, 1  2 },ℓ+ǫ−1  b  ( ¯  X))  +  k  �  j=0  ∥(σ∂σ)j��  Lb(σ)−1 ˆf2(σ)  �  ∥H  3  2 −ǫ′ (R;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='⟨σ⟩−s+2+j ¯  Hs−2−j,ℓ  b,⟨σ⟩−1 ( ¯  X))  ≲ ∥f∥⟨tb,∗⟩− 3  2 +ǫ′ �  Hs+k,ℓ+2,k  b,c  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  (13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='18)  Now we integrate ∂tb,∗(˙greg(tb,∗), ˙Areg(tb,∗)) = t−1  b,∗(tb,∗∂tb,∗)(˙greg(tb,∗), ˙Areg(tb,∗)) and exploit the  above estimate (13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='18) for k = 1 to obtain the decay estimate of (˙greg(tb,∗), ˙Areg(tb,∗)) in tb,∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Con-  cretely, for tb,∗ ≥ 0 and 0 < ǫ < ǫ′ < 1, we have  ∥(˙greg(tb,∗), ˙Areg(tb,∗))∥ ¯  Hs−3,ℓ+ǫ−1  b  ( ¯  X) = ∥  � ∞  tb,∗  ∂s  �  ˙greg(s), ˙Areg(s)  �  ds∥ ¯  Hs−3,ℓ+ǫ−1  b  ( ¯  X)  ≲  � � ∞  tb,∗  ⟨s⟩−3+2ǫ′⟨s⟩−2 ds  � 1  2 ∥(⟨tb,∗⟩∂tb,∗)(˙greg(tb,∗), ˙Areg(tb,∗))∥⟨tb,∗⟩− 3  2 +ǫ′ �  Hs−3,ℓ+ǫ−1  b,c  ≲ ⟨tb,∗⟩−2+ǫ′∥f∥⟨tb,∗⟩− 3  2 +ǫ′ �  Hs+1,ℓ+2,1  b,c  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  Since (˙greg(tb,∗), ˙Areg(tb,∗)) has support in tb,∗ ≥ −C for some C > 0, it follows that  ∥(˙greg(tb,∗), ˙Areg(tb,∗))∥⟨tb,∗⟩−2+ǫL∞(Rtb,∗;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' ¯  Hs−3,ℓ+ǫ−1  b  ( ¯  X)) ≲ ∥f∥⟨tb,∗⟩− 3  2 +ǫ′ �  Hs+1,ℓ+2,1  b,c  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  By the same reasoning, the above decay estimate can be generalized for all 0 ≤ k ≤ m,  ∥  �  ⟨tb,∗⟩∂tb,∗  �k−1(˙greg(tb,∗), ˙Areg(tb,∗))∥⟨tb,∗⟩−2+ǫL∞(Rtb,∗ ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' ¯  Hs−2−k,ℓ+ǫ−1  b  ( ¯  X)) ≲ ∥f∥⟨tb,∗⟩− 3  2 +ǫ′ �  Hs+k,ℓ+2,k  b,c  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  (13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='19)  The estimates (13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='18) and (13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='19) imply that (˙greg(tb,∗), ˙Areg(tb,∗)) occurs as a part of the remainder  (˜g, ˜A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  Analysis of (˙gsing(tb,∗), ˙Asing(tb,∗)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' We now turn to discussing the singular part  (˙gsing(tb,∗), ˙Asing(tb,∗)) =  lim  C→0+  1  2π  �  Im σ=C  e−iσtb,∗P(b, σ) ˆf1(σ) dσ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  Since ˆf is holomorphic in the full plane σ ∈ C, we write  ˆf1(σ) = χ(Re σ) ˆf(σ) = χ(Re σ)  �  f (0) + iσf (1) + σ2f (2)(σ)) = χ(Re σ)  �  f (0) + iσf (1)�  + σ2 ˆf3(σ)  154  LILI HE  where  f (0) = ˆf(0) =  �  R  f(tb,∗) dtb,∗,  f (1) = −i∂σ ˆf(0) =  �  R  tb,∗f(tb,∗) dtb,∗,  and f (2)(σ) is holomorphic in the full plane σ ∈ C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Moreover, by Sobolev embedding theorem, we  have for j = 0, 1  ∥f (j)∥ ¯  Hs,ℓ+2  b  ( ¯  X) ≲ ∥ ˆf(σ)∥H3/2+s′ (Rσ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' ¯  Hs,ℓ+2  b  ( ¯  X)) ≲ ∥f∥⟨tb,∗⟩− 3  2 −s′ �  Hs,ℓ  b,c  for any  s′ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  As for ˆf3(σ), since ˆf3(σ) = σ−2( ˆf1(σ)−χ(Re σ)f (0)−iσχ(Re σ)f (1)), it follows that ˆf3(σ) ∈ C∞  c (Rσ)  and  ∥ ˆf3(σ)∥Hs′ (Rσ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' ¯  Hs,ℓ+2  b  ( ¯  X)) ≲ ∥ ˆf(σ)∥Hs′+2(Rσ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' ¯  Hs,ℓ+2  b  ( ¯  X)),  s′ > −1  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  Recall that P(b, σ) = σ−2P 2(b)+σ−1P 1(b)+σ−1P e(σ) where P 2(b), P 1(b) : ¯Hs,ℓ+2  b  → ¯H∞,−1/2−  b  ( ¯X)  and P e(b, σ) : ¯Hs,ℓ+2  b  → Kb,1 is ǫ-regular.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Therefore, following the arguments in the proof of (13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='15),  we find that for 0 ≤ k ≤ m,  (σ∂σ)k�  P(b, σ)(σ2 ˆf3(σ))  �  ∈ H  3  2 −ǫ((−c0, c0);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' ¯H  ∞,− 1  2 −  b  ( ¯X)),  whose norm is bounded in the following manner  ∥(σ∂σ)k�  P(b, σ)(σ2 ˆf3(σ))  �  ∥  H  3  2 −ǫ(R;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' ¯  H  ∞,− 1  2 −  b  ( ¯  X))  ≲  k  �  j=0  ∥(σ∂σ)j ˆf3(σ)∥H  3  2 −ǫ(R;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' ¯  Hs,ℓ+2  b  ( ¯  X)) ≲  k  �  j=0  ∥(σ∂σ)j ˆf(σ)∥H  7  2 −ǫ(R;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' ¯  Hs,ℓ+2  b  ( ¯  X))  ≲ ∥f∥⟨tb,∗⟩− 7  2 +ǫ �  Hs,ℓ+2,m  b,c  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  (13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='20)  Similarly, we have  (σ∂σ)k��  P 1(b) + P e(b, σ)  �  (χ(σ)f (1))  �  ∈ H  3  2 −ǫ((−c0, c0);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' ¯H  ∞,− 1  2 −  b  ( ¯X)),  and  ∥(σ∂σ)k��  P 1(b) + P e(b, σ)  �  (χ(σ)f (1))  �  ∥  H  3  2 −ǫ(R;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' ¯  H  ∞,− 1  2 −  b  ( ¯  X)) ≲ ∥f∥⟨tb,∗⟩− 3  2 − �  Hs,ℓ+2  b,c  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  (13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='21)  As a result, it remains to analyze P(b, σ)(χ(Re σ)f (0)) and P 2(b)(iσ−1χ(Re σ)f (1)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' We write  �  σ−2P 2(b)f (0) + σ−1P 1(b)f (0) + P 2(b)(iσ−1f (1))  �  + σ−1P e(b, σ)(χ(Re σ)f (0))  =  �  (σ−2(˙g1, ˙A1) − iσ−1(ˇg1, ˇA1)) + iσ−1�  (˙g2, ˙A2) + (˙g′  1, ˙A′  1) + (˙g′′  1, ˙A′′  1)  ��  + iσ−1(˙g′′′  1 (σ), ˙A′′′  1 (σ)) := I + II  where (˙g1, ˙A1), (˙g′  1, ˙A′  1), (˙g′′  1 , ˙A′′  1), (˙g′′′  1 (σ), ˙A′′′  1 (σ)) ∈ Kb,1 and (˙g2, ˙A2) ∈ Kb,2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Moreover, (˙g′′  1, ˙A′′  1) is  uniquely determined by the condition  −  �  [Lb, tb,∗](ˇg′′  1 , ˇA′′  1) + 1  2[[Lb, tb,∗], tb,∗](˙g′′  1, ˙A′′  1), (˙g∗, ˙A∗)  �  +  �  [Lb, tb,∗](˙g′′  2, ˙A′′  2), (˙g∗, ˙A∗)  �  = ⟨  �  R  tb,∗f(tb,∗) dtb,∗, (˙g∗, ˙A∗)⟩  for all  (˙g∗, ˙A∗) ∈ K∗  b  (13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='22)  for some (˙g′′  2, ˙A′′  2) ∈ Kb,2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Since σ−j(1 − χ(σ)) ∈ Hs′(Rσ) for j = 1, 2 and any s′ ∈ R, we have  (σ∂σ)k�  χ(Re σ)  �  P(b, σ)f (0) + P 2(b)(iσ−1f (1))  �  − (I + II)  �  ∈ H  3  2 −ǫ((−c0, c0);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' ¯H  ∞,− 1  2 −  b  ( ¯X)),  and  ∥(σ∂σ)k�  χ(Re σ)  �  P(b, σ)f (0) + P 2(b)(iσ−1f (1))  �  − (I + II)  �  ∥  H  3  2 −ǫ(R;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' ¯  H  ∞,− 1  2 −  b  ( ¯  X))  ≲ ∥f∥⟨tb,∗⟩− 3  2 − �  Hs,ℓ+2  b,c  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  (13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='23)  As a consequence, it suffices to discuss the inverse Fourier transform of I and II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  LINEAR STABILITY OF KERR-NEWMAN FAMILY  155  – Analysis of the leading order term (ˆg, ˆA).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Since  F−1(σ−2) =  lim  C→0+  1  2π  �  Im σ=C  e−iσtb,∗σ−2 dσ = −1  2tb,∗H(tb,∗)  and  F−1(iσ−1) =  lim  C→0+  1  2π  �  Im σ=C  e−iσtb,∗iσ−1 dσ = 1  2H(tb,∗),  we obtain  F−1(I) = lim  C→0+  1  2π  �  Im σ=C  e−iσtb,∗I dσ  = −1  2  �  tb,∗(˙g1, ˙A1) + (ˇg1, ˇA1)  �  + 1  2  �  (˙g′  1, ˙A′  1) + (˙g′′  1 , ˙A′′  1) + (˙g2, ˙A2)  �  := (ˆg, ˆA),  (13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='24)  where (˙g1, ˙A1), (˙g′  1, ˙A′  1), (˙g′′  1, ˙A′′  1) ∈ Kb,1, and (˙g2, ˙A2) ∈ Kb,2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Moreover, (˙g1, ˙A1), (˙g′  1, ˙A′  1) and  (˙g2, ˙A2) are determined by the conditions (11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='30) and (11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='31) with f there replaced by f (0) =  �  R f(tb,∗) dtb,∗, and (˙g′′  1, ˙A′′  1) ∈ Kb,1 is uniquely determined by the condition (13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='22) for some  (˙g′′  2, ˙A′′  2) ∈ Kb,2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  – Analysis of the pure gauge part (¯g, ¯A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Finally, we turn to the analysis of the inverse Fourier  transform of II = iσ−1(˙g′′′  1 (σ), ˙A′′′  1 (σ)) where supp (˙g′′′  1 (σ), ˙A′′′  1 (σ)) ⊂ (−c0, c0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' By Theorem  12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='9,  II = σ−1(˙g′′′  1 (σ), ˙A′′′  1 (σ)) =  �  ±  ψ±(σ)  where  ψ± ∈ A−ǫ�  ± [0, c0);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Kb,1  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  We write  � c0  −c0  e−itb,∗σII dσ =  � c0  0  e−itb,∗σχ(tb,∗σ)ψ+(σ) dσ +  � c0  0  e−itb,∗σ(1 − χ(tb,∗σ))ψ+(σ) dσ  +  � 0  −c0  e−itb,∗σχ(tb,∗σ)ψ−(σ) dσ +  � 0  −c0  e−itb,∗σ(1 − χ(tb,∗σ))ψ−(σ) dσ  := II+  1 + II+  2 + II−  1 + II−  2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  First, we have  |II+  1 | ≤  � c0t−1  b,∗  0  |ψ+(σ)| dσ ≲  � c0t−1  b,∗  0  σ−ǫ dσ ≲ t−1+ǫ  b,∗  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  Next, we compute for k ∈ N  II+  2 =  � c0  0  e−itb,∗σ(1 − χ(tb,∗σ))ψ+(σ) dσ = (itb,∗)−k  � c0  0  e−itb,∗σ∂k  σ  �  (1 − χ(tb,∗σ))ψ+(σ)  �  dσ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  Then we have  |II+  2 | ≲ t−k  b,∗  � c0  c0t−1  b,∗/2  σ−k−ǫ dσ ≲ t−1+ǫ  b,∗  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  The estimates for II+  1 , II+  2 also hold after applying any number of derivatives tb∗∂tb,∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' We note  that II−  1 , II−  2 can be handled in a similar manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Since  II = σ−1(˙g′′′  1 (σ), ˙A′′′  1 (σ)) = (2δ∗  gbωb,s1(σ), �LAb(ωb,s1(σ))♯ + dφb,s1(σ))  (13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='25)  where  (ωb,s1(σ), φb,s1(σ)) ∈  �  ±  A−ǫ(±[0, c0);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' ¯H  ∞,− 3  2 −  b  ( ¯X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' �  scT ∗ ¯X) ⊕ ¯H  ∞,− 1  2 −  b  ( ¯X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' C)),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  we can write  (¯g,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' ¯A) =  � c0  −c0  e−itb,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='∗σII dσ = (¯g1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' ¯A1) + (¯g2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' ¯A2)  156  LILI HE  where  (¯g1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' ¯A1) =  �  2δ∗  gb  �  χ( ⟨r⟩  ⟨tb,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='∗⟩)ωb,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='s1(tb,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='∗)  �  ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' �LAb  �  χ( ⟨r⟩  ⟨tb,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='∗⟩)ωb,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='s1(tb,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='∗)  �♯ + d  �  χ( ⟨r⟩  ⟨tb,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='∗⟩)φb,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='s1(tb,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='∗)  ��  ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  (¯g2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' ¯A2) =  �  2[χ( ⟨r⟩  ⟨tb,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='∗⟩),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' δ∗  gb]ωb,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='s1(tb,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='∗),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' [χ( ⟨r⟩  ⟨tb,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='∗⟩),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' �LAb]  �  ωb,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='s1(tb,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='∗)  �♯ + [χ( ⟨r⟩  ⟨tb,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='∗⟩),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' d]φb,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='s1(tb,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='∗)  �  − χ( ⟨r⟩  ⟨tb,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='∗⟩)  �  2∂tb,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='∗ωb,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='s1 ⊗s dtb,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='∗,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' ∂tb,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='∗  �  Aα(ωb,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='s1)♯,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='α + φb,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='s1  �  dtb,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='∗  �  +  �  1 − χ( ⟨r⟩  ⟨tb,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='∗⟩)  �  (¯g,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' ¯A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  where χ is a smooth nonnegative cutoff such that χ(s) = 1 for s ≤ 1/2 and χ(s) = 0 for s ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  Then the estimates (13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='4) and (13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='5) for (¯g1, ¯A1) and (¯g2, ¯A2) follow from the following estimates  |  �  ⟨tb,∗⟩∂tb,∗  �m−1V jωb,s1(tb,∗)| ≲ ⟨tb,∗⟩−1+ǫ∥f∥⟨tb,∗⟩− 3  2 − �  Hs,ℓ+2  b,c  ,  |  �  ⟨tb,∗⟩∂tb,∗  �m−1V jφb,s1(tb,∗)| ≲ ⟨tb,∗⟩−1+ǫ⟨r⟩−1∥f∥⟨tb,∗⟩− 3  2 − �  Hs,ℓ+2  b,c  ,  |  �  ⟨tb,∗⟩∂tb,∗  �m−1V j(¯g, ¯A)| ≲ ⟨tb,∗⟩−1+ǫ⟨r⟩−2∥f∥⟨tb,∗⟩− 3  2 − �  Hs,ℓ+2  b,c  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  □  14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Proof of the main theorem  In the initial value problem for Einstein-Maxwell system as formulated in §3, we impose the initial data  at the Cauchy hypersurface which terminates at spatial infinity (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=', it coincides with the hypersurface t = 0  when r is large enough).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' In this section, we will discuss how to reduce this initial value problem to the  inhomogeneous problem studied in the previous section §13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  We recall the time function t (corresponding to gb0) defined in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='15), which satisfies that t = t for  r ≥ 4m0, and dt is timelike everywhere on M with respect to gb0 with b0 = (m0, 0, Q0), |Q0| ≪ m0, hence  for gb when b = (m, a, Q) is close to b0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' We define  Σ0 := t−1(0),  which is a spacelike (with respect to gb) Cauchy hypersurface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Identifying the region r ≥ 4m0 in Σ0 with  a subset of R3, we can compactify Σ0 at infinity to the manifold ¯Σ0 (with two boundary components: one  is inside the black hole and the other is the spatial infinity).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' We denote by �  scT ∗¯Σ0 = scT ∗¯Σ0 ⊕ R dt the  spacetime scattering cotangent bundle, which for large r is spanned over C∞(¯Σ0) by dt and dx1, dx2, dx3,  with (x1, x2, x3) denoting standard coordinates on R3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  We first study the initial value problem for linearized gauge-fixed Einstein-Maxwell system Lb(˙g, ˙A) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  Theorem 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Let 0 < α < 1 and s > 8 + m, m ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Suppose  (˙g0, ˙g1, ˙A0, ˙A1) ∈ ¯Hs,−1/2+α  b  (¯Σ0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' S2 �  scT ∗ ¯Σ0) ⊕ ¯Hs−1,1/2+α  b  (¯Σ0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' S2 �  scT ∗¯Σ0)  ⊕ ¯Hs−1,−1/2+α  b  (¯Σ0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' �  scT ∗ ¯Σ0) ⊕ ¯Hs−2,1/2+α  b  (¯Σ0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' �  scT ∗¯Σ0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  Then the solution (˙g, ˙A) of the initial value problem  Lb(˙g, ˙A) = 0,  (˙g, ˙A)|¯Σ0 = (˙g0, ˙A0),  (L∂t ˙g, L∂t ˙A)|¯Σ0 = (˙g1, ˙A1)  satisfies the following properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Let  N s,α := ∥ ˙g0∥ ¯  Hs+1,−1/2+α  b  + ∥g1∥ ¯  Hs,1/2+α  b  + ∥A0∥ ¯  Hs,1/2+α  b  + ∥A1∥ ¯  Hs,1/2+α  b  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  (1) For tb,∗ ≥ 0, the solution (˙g, ˙A) can be written as  (˙g, ˙A) = (ˆg, ˆA) + (¯g, ¯A) + (˜g, ˜A)  (14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='1)  where (ˆg, ˆA) ∈ �Kb is a generalized zero mode (defined in Proposition 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='5), and (¯g, ¯A), (˜g, ˜A) satisfy  the following estimates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  LINEAR STABILITY OF KERR-NEWMAN FAMILY  157  (i) The term (¯g, ¯A) can be written as (¯g, ¯A) = (¯g1, ¯A1) + (¯g2, ¯A2) where (¯g1, ¯A1) is a pure gauge  solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Moreover, for m ≥ 1 and j ≥ 0, we have the pointwise estimates  |  �  ⟨tb,∗⟩∂tb,∗  �m−1V j(¯g1, ¯A1)| ≲  �  ⟨tb,∗ + r⟩−2+ǫ + ⟨tb,∗⟩−1+ǫ⟨r⟩−2�  N s,α  (14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='2)  and  |  �  ⟨tb,∗⟩∂tb,∗  �m−1V j(¯g2, ¯A2)| ≲ ⟨tb,∗ + r⟩−2+ǫN s,α  (14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='3)  where V j denotes the product of j vector fields V ∈ {∂tb,∗, r∂r, rotation fields }.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  (ii) (˜g, ˜A) satisfies: for 0 < ǫ < 1 with α + ǫ > 1 and 0 ≤ m ≤ s−6  2 , we have  ∥(˜g, ˜A)∥⟨tb,∗⟩− 3  2 +ǫ �  Hs−6−m,−5/2+α+ǫ−,m  b,c  ≲ N s,α  (14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='4)  For 1 ≤ m ≤ s−6  2 , we also have  ∥  �  ⟨tb,∗⟩∂tb,∗  �m−1(˜g, ˜A)∥⟨tb,∗⟩−2+ǫL∞(Rtb,∗;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' ¯  Hs−6−m,−5/2+α+ǫ−  b  ( ¯  X)) ≲ N s,α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  (14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='5)  For m ≥ 1 and 0 ≤ j < s − 8 − m with j ∈ N0, we have the pointwise decay estimate  |  �  ⟨tb,∗⟩∂tb,∗  �m−1V j(˜g, ˜A)| ≲ ⟨tb,∗⟩−2+ǫ⟨r⟩−α+ǫ+1+N s,α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  (14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='6)  (2) In t ≥ 0, tb,∗ ≤ 0, we have (˙g, ˙A) = r−1Hs−3(Rtb,∗ × S2) + (˜g, ˜A) with |(˜g, ˜A)| ≲ r−1(1 + |tb,∗|)−α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Let T1 < T2 and let χ(tb,∗) be a nonnegative smooth cutoff such that χ = 1 for tb,∗ ≥ T2 and  χ(tb,∗) = 0 for tb,∗ ≤ T1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Then we obtain the following equation  Lb  �  χ(tb,∗)(˙g, ˙A)  �  = [Lb, χ(tb,∗)](˙g, ˙A),  of which the right-hand side is supported in tb,∗ ∈ (T1, T2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' In view of Theorem 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='1, it suffices to prove that  Lb  �  χ(tb,∗)(˙g, ˙A)  �  ∈ �Hs−4−k,1/2+α−,k  b,c  ,  k ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  (14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='7)  Since the regularity with respect to ⟨tb,∗⟩Dtb,∗ and Dtb,∗ is equivalent in the support of Lb  �  χ(tb,∗)(˙g, ˙A)  �  ,  it suffices to prove (14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='7) for k = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' First, the global wellposedness theory of linear wave equations implies  that (˙g, ˙A) ∈ Hs−1 for t ≥ 0, t∗ ≤ C, r ≤ C for any C;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' this implies that [Lb, χ] ∈ Hs−2 in such compact  sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Therefore, it suffices to work in an arbitrarily small neighborhood r > R0 ≫ 1 of infinity, which we  shall do from now on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  The proof closely follows the argument in [43, §§4–5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Let b = (m, a, Q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Then we define b1 = (m, 0, Q)  and introduce the incoming and outgoing null coordinates  x0 = t + rb1,∗,  x1 = t − rb1,∗,  with respect to which we have  gb1 = −µb1dx0 dx1 + r2/g  and  2∂0 ≡ 2∂x0 = ∂t + (1 − 2m  r  + Q2  r2 )∂r,  2∂1 ≡ 2∂x1 = ∂t − (1 − 2m  r  + Q2  r2 )∂r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  Let x2, x3 denote local coordinates on S2, and denote spherical indices by c, d = 2, 3;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' let ∂c ≡ ∂xc;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' let ¯X be  the compactification of t−1  b,∗(0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Since gb − gb1 ∈ ρ2C∞( ¯X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' S2 �  scT ∗ ¯X), we have  gb(∂0, ∂0), (gb − gb1)(∂0, ∂1), gb(∂0, r−1∂c),  gb(∂1, ∂1), gb(∂1, r−1∂c), (gb − gb1)(r−1∂b, r−1∂c) ∈ ρ2C∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  We let  ρI = (rb1,∗ − t)/r,  ρ0 = (rb1,∗ − t)−1,  ρ = ρIρ0 = 1  r .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  Then we have  ρ−3Lbρ = −□ρ2gb ⊗ Id14×14 + R  with R ∈ �  Diff1  b acting on sections of S2 �  scT ∗ ¯X ⊕ �  scT ∗ ¯X), where �  Diffk  b consists of up to k-fold products of the  vector fields {ρI∂ρI, ρ0∂ρ0, rotation vector fields}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' We note that switching from Lb to ρ−3Lbρ is related to  the Friedlander rescaling for the scalar wave equation [24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  158  LILI HE  Then one can use the multiplier W := ρ−2a0  0  ρ−2aI  I  V with V := −(1+cV )ρI∂ρI +ρ0∂ρ0 and a0 = α, aI < 0  as introduced in [43, Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='4] to derive an energy estimate, which allows us to conclude that near  I +, (˙g, ˙A) lies in �Hs−1,−1/2−  b  (see [43, Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='8] for detail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Actually, the present setting is much  simpler than that in the reference as we do not need to get sharp decay rate for a certain component as did  in [43, Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='8]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' To further determine the leading behavior of (˙g, ˙A) near I , one rewrite the equations  ρ−3Lbρ(ρ−1 ˙g, ρ−1 ˙A) = 0 as 2ρ−2∂0∂1(ρ−1 ˙g, ρ−1 ˙A) = ( /∆ + ˜R)(ρ−1 ˙g, ρ−1 ˙A) where ˜R ∈ ρ�  Diff2  b + �  Diff1  b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Since  −2ρ−2∂0∂1 = ρ−1  I (ρ0∂ρ0 − ρI∂ρI)ρI∂ρI + �  Diff1  b, it follows that  (ρ0∂ρ0 − ρI∂ρI)ρI∂ρI(ρ−1 ˙g, ρ−1 ˙A) ∈ ρI �  Diff2  b(ρ−1 ˙g, ρ−1 ˙A) ∼ ρα  0 ρaI+1  I  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  We choose aI < 0 such that 0 < 1 + aI < α (see [43, Lemma 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='7]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Integrating first along the vector field  ρ0∂ρ0 − ρI∂ρI from ρI ≥ ǫ, where (ρ−1 ˙g, ρ−1 ˙A) ∼ ρα  0 , yields  ρI∂ρI(ρ−1 ˙g, ρ−1 ˙A) ∼ ρα  0 ρ1+aI  I  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  Then integrating out the vector field ρI∂ρI gives  (˙g, ˙A) − ρHs−3(Rtb1,∗ × S2) ∼ ρρα  0 ρ1+aI  I  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  This proves statement 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Moreover, in view of Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='11, we see that  [Lb, χ(tb,∗)] = 2ρχ′(t∗)(ρ∂ρ − 1) + ρ2Diff1  sc + ρ2C∞∂t∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  (14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='8)  Since (ρ∂ρ − 1) annihilates the leading order term ρHs−3(Rtb1,∗ × S2) in (˙g, ˙A), it follows that Lb(χ(˙g, ˙A)) =  [Lb, χ](˙g, ˙A) ∈ �Hs−4,1/2+α−  b,c  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  Since s − 4 > 4 + m, the estimates (14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='2)–(14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='6) now follow from Theorem 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='1, with ℓ + 2 = 1  2 + α−, so  ℓ = − 3  2 + α− and −1/2 < ℓ + ǫ < 1/2 with α + ǫ > 1, 0 < ǫ < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  □  We now state our main theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  Theorem 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Let 0 < α < 1 and s > 8 + m, m ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Let  (˙h, ˙k, ˙E, ˙B) ∈ ¯Hs,−1/2+α  b  (Σ0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' S2 scT ∗Σ0)⊕ ∈ ¯Hs−1,1/2+α  b  (Σ0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' S2 scT ∗Σ0)  ¯Hs−1,1/2+α  b  (Σ0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' scT ∗Σ0)⊕ ∈ ¯Hs,1/2+α  b  (Σ0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' scT ∗Σ0)  be the initial data for the linearized Einstein-Maxwell equations linearized around the the KN solution (gb, Ab).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  Suppose that the initial data satisfy the linearized constraint equations, which is the linearization of the  constraint equations around the KN initial data (hb, kb, Eb, Bb) = τ(gb, dAb).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Assume that the linearized  magnetic charge vanishes, see the discussion in §3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='2–3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  Then there exists a solution (˙g, ˙A) of the initial value problem  DgbRic(˙g) − 2D(gb,dAb)T (˙g, ˙A) = 0,  Dgb,Ab(δ(·)d(·))(˙g, ˙A) = 0,  D(gb,Ab)τ(˙g, ˙A) = (hb, kb, Eb, Bb)  satisfying the linearized generalized wave map gauge and Lorenz gauge conditions  Dgb �ΥE(˙g;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' gb) = 0,  δgb ˙A = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  Moreover, (˙g, ˙A) has the asymptotic behavior stated in Theorem 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' In view of Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='15, there exist sections  (˙g0, ˙g1, ˙A0, ˙A1) ∈ ¯Hs,−1/2+α  b  (¯Σ0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' S2 �  scT ∗¯Σ0) ⊕ ¯Hs−1,1/2+α  b  (¯Σ0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' S2 �  scT ∗ ¯Σ0)  ⊕ ¯Hs−1,−1/2+α  b  (¯Σ0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' �  scT ∗¯Σ0) ⊕ ¯Hs−2,1/2+α  b  (¯Σ0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' �  scT ∗Σ0),  such that they induce the data (˙h, ˙k, ˙E, ˙B) on Σ0 = {t = 0}, and satisfy the linearized gauge conditions  Dgb �ΥE(˙g) = 0 and D(gb,Ab) �ΥM(˙g, ˙A) = −δgb ˙A = 0 at Σ0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  Then we solve the initial value problem Lb(˙g, ˙A) = 0 with initial data (˙g0, ˙g1, ˙A0, ˙A1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Since (˙h, ˙k, ˙E, ˙B)  satisfies the linearized constraint equations, it follows from Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='8 that  Dgb �ΥE(˙g;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' gb) = 0,  δgb ˙A = 0  and thus (˙g, ˙A) solves the linearized Einstein-Maxwell equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  □  LINEAR STABILITY OF KERR-NEWMAN FAMILY  159  Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' The linearized Einstein-Maxwell operator around Reissner-Nordstr¨om  spacetime  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Detailed calculation of the linearized Einstein-Maxwell system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' We now calculate the lin-  earized Einstein-Maxwell system  L (˙g, ˙A) := (L1(˙g, ˙A), L2(˙g, ˙A)) := (DgRic(˙g, d ˙A) − 2D(g,dA)T (˙g, d ˙A), D(g,F ) (δgF) (˙g, ˙F)) = 0  in more detail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  First, following [84, §7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='5],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' [33,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' §3] we see that  DgRic(˙g) = −1  2□g ˙g − δ∗  gδgGg ˙g + Rg ˙g  where  (Rg ˙g)µν = Rα  β  µν ˙gαβ + 1  2  �  Ric κ  µ ˙gνκ + Ric κ  ν ˙gµκ  �  Since T (g,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' F) = FµαF α  ν  − 1  4gµνFαβF αβ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' we find  D(g,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='F )T (˙g,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' ˙F) = − ˙gαβFµαFνβ − 1  4  �  ˙gµνFαβF αβ − gµν ˙gκαgλβFαβFκλ − gµνgκα ˙gλβFαβFκλ  �  +  �  gαβ ˙FµαFνβ + gαβFµα ˙Fνβ  �  − 1  4  �  gµν ˙FαβF αβ + gµνFαβ ˙F αβ�  = − ˙gαβFµαFνβ − 1  4  �  ˙gµνFαβF αβ−2gµν ˙gκαFαβF β  κ  �  +  �  gαβ ˙FµαFνβ + gαβFµα ˙Fνβ  �  − 1  2gµν ˙FαβF αβ  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='1)  and thus the Einstein part L1 of the linearized Einstein-Maxwell system is given by  L1(˙g, d ˙A) = DgRic(˙g, d ˙A) − 2D(g,dA)T (˙g, d ˙A) = −1  2□g ˙g − δ∗  gδgGg ˙g + Rg ˙g − 2D(g,dA)T (˙g, d ˙A) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='2)  Next we analyze the Maxwell part L2 of the linearized Einstein-Maxwell system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Since  (δgF)µ = −∇νFνµ = −gνα∇αFνµ = −gνα �  ∂αFνµ − Γκ  ανFκµ − Γκ  αµFνκ  �  ,  and  ˙Γκ  αν(˙g) = 1  2gκλ (∇ν ˙gαλ + ∇α ˙gνλ − ∇λ ˙gαν) ,  we obtain  D(g,F ) (δgF) (˙g, ˙F) = δg ˙F + ˙gνα∇αFνµ + gνα �  ˙Γκ  αν(˙g)Fκµ + ˙Γκ  αµ(˙g)Fνκ  �  = δg ˙F + ˙gνα∇αFνµ + gκλ∇α(˙gαλ − 1  2gνα ˙gανgαλ)Fκµ + 1  2(∇ν ˙gκ  µ − ∇κ ˙gν  µ)Fνκ  = δg ˙F + ˙gνα∇αFνµ − (δgGg ˙g)κFκµ + 1  2(∇ν ˙gκ  µ − ∇κ ˙gν  µ)Fνκ  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='3)  where we use (∇µ ˙gνκ)Fνκ = 0 in the second equality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Therefore, the Maxwell part L2 of the linearized  Einstein-Maxwell system is given by  L2(˙g, d ˙A) = δgd ˙A + ˙gνα∇α(dA)νµ − (δgGg ˙g)κ(dA)κµ + 1  2(∇ν ˙gκ  µ − ∇κ ˙gν  µ)(dA)νκ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='4)  We then proceed to calculate the linearized Einstein-Maxwell system L (˙g, ˙A) = 0 around a Reissner-  Nordstr¨om spacetime g = ˚g + r2/g under the splitting (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='9) and (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='10) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Calculation of Rg ˙g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Here and in what follows the components we do not list are 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Christoffel symbols  of Reissner-Nordstr¨om metric g are,  Γk  ij = ˚Γk  ij  Γa  ij = 0,  Γj  ia = 0,  Γk  ab = −r∂kr/gab,  Γb  ak = r−1∂krδb  a,  Γc  ab = /Γ  c  ab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  The components of Riemann curvature tensor are given by  Rijkm = ˚  Rijkm = 1  2  ˚  R(˚g) (˚gik˚gjm −˚gim˚gjk) = −1  2µ′′  b0 (˚gik˚gjm −˚gim˚gjk) ,  Raijk = 0,  Rabij = 0,  Rabci = 0,  Raibj = −r˚∇i˚∇jr/gab = −1  2rµ′  b0/gab˚gij,  Rabcd = /Rabcd − r2µb0  �  /gac/gbd − /gad/gbc  �  = r2(1 − µb0)  �  /gac/gbd − /gad/gbc  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  160  LILI HE  Correspondingly, the Ricci curvature tensor takes the form  Ricij = ˚  Ricij − r−1µ′  b0˚gij = −1  2µ′′  b0˚gij − r−1µ′  b0˚gij,  Ricia = 0,  Ricab = (1 − µb0 − rµ′  b0)/gab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  Now we are ready to calculate Rg ˙g in detail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Since (Rg ˙g)µν = Rα  β  µν ˙gαβ + 1  2  �  Ric κ  µ ˙gνκ + Ric κ  ν ˙gµκ  �  , we have  (Rg ˙g)ij = −µ′′  b0  �  ˙gij − 1  2˚gkm ˙gkm˚gij  �  − r−1µ′  b0 ˙gij + 1  2r−3µ′  b0/gab ˙gab˚gij,  (Rg ˙g)ia = −3  2r−1µ′  b0 ˙gia − 1  4µ′′  b0 ˙gia + 1  2r−2(1 − µb0)˙gia,  (Rg ˙g)ab = 2r−2(1 − µb0)  �  ˙gab − 1  2/gcd ˙gcd/gab  �  − r−1µ′  b0 ˙gab + 1  2rµ′  b0˚gij ˙gij/gab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  Therefore, with respect to the splitting of S2T ∗M (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='10), we can express Rg : S2T ∗M → S2T ∗M as follows  Rg =  \uf8eb  \uf8ed  −µ′′  b0G˚g − r−1µ′  b0  0  1  2r−3µ′  b0˚g /tr  0  − 3  2r−1µ′  b0 − 1  4µ′′  b0 + 1  2r−2(1 − µb0)  0  1  2rµ′  b0/g˚tr  0  2r−2(1 − µb0)G/g − r−1µ′  b0  \uf8f6  \uf8f8  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='5)  where (G˚gh)ij = hij − 1  2˚gij˚trh and (G/gh)ab = hab − 1  2/gab /trh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Calculation of δ∗  gδgGg ˙g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' We now calculate the first covariant derivatives on 1-forms and symmetric  2-tensors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' For η ∈ T ∗M, we compute  ∇iηj = ˚∇iηj,  ∇iηa = ∂iηa − (r−1∂ir)ηa,  ∇aηi = ∂aηi − (r−1∂ir)ηa,  ∇aηb = /∇aηb + r(∂kr)ηk/gab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  With respect to the splitting of T ∗M (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='9), we write the symmetric gradient δ∗  g : T ∗M → S2T ∗M with  (δ∗  gη)µν = 1  2(∇µην + ∇νηµ) as follows  δ∗  g =  \uf8eb  \uf8ed  ˚δ∗  0  1  2/d  1  2r2˚dr−2  r/gιdr  /δ  ∗  \uf8f6  \uf8f8 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='6)  where ιdr : T ∗ ˚  X → R is the contraction with the aspherical vector field dr♯, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' ιdr(·) = ˚g−1(dr, ·).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' We then  calculate the first covariant derivatives on symmetric 2-tensor h ∈ S2T ∗M  ∇ihjk = ˚∇ihjk,  ∇ihja = ˚∇ihja − r−1(∂ir)hja,  ∇ihab = ∂ihab − 2r−1(∂ir)hab  ∇ahij = ∂ahij − r−1(∂ir)haj − r−1(∂jr)hia,  ∇ahib = /∇ahib + r(∂kr)hik/gab − r−1(∂ir)hab,  ∇ahbc = /∇ahbc + r(∂kr)(hkc/gab + hkb/gac).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  Therefore the negative divergence (δgh)µ = −∇νhνµ which maps symmetric 2-tensors to 1-form takes the  following form (under the splitting (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='10) and (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='9))  δg =  �r−2˚δr2  r−2/δ  r−3(dr)/tr  0  r−2˚δr2  r−2/δ  �  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='7)  where (/δh)i = − /∇  ahai and (˚δh)a = −˚∇ihia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' We can also express (Ggh)µν = hµν − 1  2gµνtrgh as  Gg =  \uf8eb  \uf8ed  G˚g  0  − 1  2r−2˚g /tr  0  1  0  − 1  2r2/g˚tr  0  G/g  \uf8f6  \uf8f8 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='8)  Putting (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='6), (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='7) and (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='8) together and using /δG/g = /δ + 1  2/d/tr, ˚δG˚g = ˚δ + 1  2 ˚d˚tr yield  δ∗  gδgGg =  \uf8eb  \uf8ed  ˚δ∗r−2˚δr2 + 1  2˚δ∗˚d˚tr  ˚δ∗r−2/δ  1  2˚δ∗r−2˚  d/tr  1  2r−2/d˚δr2 + 1  4 ˚  d/d˚tr + 1  4r2˚dr−2/d˚tr  1  2r−2/d/δ + 1  2r2˚dr−4˚δr2  1  4r−2˚  d/d/tr + 1  2r2˚dr−4/δ + 1  4r2˚dr−4/d/tr  r−1/gιdr˚δr2 + 1  2r/gιdr˚d˚tr + 1  2/δ  ∗/d˚tr  r−1/gιdr/δ + r−2/δ  ∗˚δr2  1  2r−1/gιdr˚  d/tr + r−2/δ  ∗/δ + 1  2r−2/δ  ∗/d/tr  \uf8f6  \uf8f8 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='9)  LINEAR STABILITY OF KERR-NEWMAN FAMILY  161  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Calculation of □g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Next we compute the second covariant derivatives on symmetric 2-tensor h ∈  S2T ∗M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  Since our goal is to find the form of □g = gµν∇µ∇ν applied to h,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' it suffices to compute the  following second covariant derivatives  ∇i∇jhkm = ˚∇i˚∇jhkm,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  ∇a∇bhkm = /∇a /∇bhkm − r−1∂kr  � /∇ahbm + /∇bham  �  − r−1∂mr  � /∇ahkb + /∇bhka  �  + r∂ir˚∇ihkm/gab + 2r−2∂kr∂mrhab − /gab  �  ∂kr∂irhim + ∂mr∂irhik  �  ∇i∇jhkc = ˚∇i˚∇jhkc − r−1∂ir˚∇jhkc − r−1∂jr˚∇ihkc + 2r−2∂ir∂jrhkc − 1  2r−1µ′  b0˚gijhkc,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  ∇a∇bhkc = /∇a /∇bhkc − r−1∂kr  � /∇ahbc + /∇bhac  �  + r/gac∂ir /∇bhik + r/gbc∂ir /∇ahik + r/gab∂ir˚∇ihkc  − ∂kr∂ir  �  /gabhic + /gachib + /gbchib  �  − µb0  �  /gabhkc + /gachkb  �  ∇i∇jhcd = ˚∇i˚∇jhcd − 2r−1 �  ∂ir˚∇jhcd + ∂jr˚∇ihcd  �  + 6r−2∂ir∂jrhcd − r−1µ′  b0˚gijhcd,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  ∇a∇bhcd = /∇a /∇bhcd + r∂ir  �  /gbc /∇ahdi + /gbd /∇ahci + /gac /∇bhdi + /gad /∇bhci + /gab˚∇ihcd  �  − µb0  �  2/gabhcd + /gachbd + /gadhbc  �  + r2∂ir∂jrhij  �  /gac/gbd + /gad/gbc  �  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='10)  where ˚□ acting on aspherical symmetric 2-tensors is the tensor wave operator, on mixed symmetric tensors  is the 1-form wave operator acting on the aspherical part, and on spherical symmetric tensors is the scalar  wave operator acting on the aspherical variables;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' the operator /∆ is defined in a similar manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Then we  find  1  2□g =  \uf8eb  \uf8ed  1  2˚□ + 1  2r−2 /∆  2r−3(dr) ⊗s /δ  r−4(dr ⊗ dr)/tr  r−1ιdr/d  1  2˚□ + 1  2r−2 /∆  r−3(dr) ⊗ /δ  /gιdrιdr  2r−1ιdr/δ  ∗  1  2˚□ + 1  2r−2 /∆  \uf8f6  \uf8f8  +  \uf8eb  \uf8ed  r−1∂ir˚∇i−2r−2(dr) ⊗s ιdr  0  0  0  − 1  2r−2(µb0 + rµ′  b0)−2r−2(dr) ⊗ ιdr  0  0  0  −r−1∂ir˚∇i−r−1µ′  b0  \uf8f6  \uf8f8 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='11)  On the aspherical part ˚  X, we further calculate the relevant operators on the static region where µb0 > 0  by working in (t, r) coordinates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' On the static region, the metric ˚g reads  ˚g = −µb0dt2 + µ−1  b0 dr2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  We further split  T ∗ ˚  X = ⟨˚dt⟩ ⊕ ⟨˚dr⟩,  S2T ∗ ˚  X = ⟨˚dt2⟩ ⊕ ⟨2˚dt˚dr⟩ ⊕ ⟨˚dr2⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='12)  Since ˚Γt  tr = ˚Γt  rt = 1  2µ−1  b0 µ′  b0,˚Γr  tt = 1  2µb0µ′  b0,˚Γr  rr = − 1  2µ−1  b0 µ′  b0, with respect to the above split (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='12), we see  that acting on scalar functions, ∂ir˚∇i = µb0∂r,  ˚d =  �∂t  ∂r  �  ,  ˚□ = −µ−1  b0 ∂2  t + µb0∂2  r + µ′  b0∂r = −µ−1  b0 ∂2  t + ∂rµb0∂r  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='13)  On 1-forms, we have ιdr = (0, µb0),  ˚δ = (µ−1  b0 ∂t, −µb0∂r − µ′  b0) = (µ−1  b0 ∂t, −∂rµb0),  ˚δ∗ =  \uf8eb  \uf8ed  ∂t  − 1  2µb0µ′  b0  1  2µb0∂rµ−1  b0  1  2∂t  0  µ−1/2  b0  ∂rµ1/2  b0  \uf8f6  \uf8f8 ,  ∂ir˚∇i =  �µb0∂r − 1  2µ′  b0  0  0  µb0∂r + 1  2µ′  b0  �  =  �  µ3/2  b0 ∂rµ−1/2  b0  0  0  µ1/2  b0 ∂rµ1/2  b0  �  ,  ˚∗ =  �  0  −µb0  −µ−1  b0  0  �  ,  2dr ⊗s (·) =  \uf8eb  \uf8ed  0  0  1  0  0  2  \uf8f6  \uf8f8 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='14)  162  LILI HE  and  ˚□ =  �−µ−1  b0 ∂2  t + µb0∂2  r − 1  2µ′′  b0  µ′  b0∂t  µ−2  b0 µ′  b0∂t  −µ−1  b0 ∂2  t + µb0∂2  r + 2µ′  b0∂r + 1  2µ′′  b0  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='15)  On symmetric 2-tensors,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' we find that  ˚δ =  �  µ−1  b0 ∂t  −∂rµb0  0  − 1  2µ′  b0µ−2  b0  µ−1  b0 ∂t  −µ−1/2  b0  ∂rµ3/2  b0  �  ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  ιdr =  �0  µb0  0  0  0  µb0  �  ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  ∂ir˚∇i =  \uf8eb  \uf8ed  µb0∂r − µ′  b0  0  0  0  µb0∂r  0  0  0  µb0∂r + µ′  b0  \uf8f6  \uf8f8 =  \uf8eb  \uf8ed  µ2  b0∂rµ−1  b0  0  0  0  µb0∂r  0  0  0  ∂rµb0  \uf8f6  \uf8f8 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  ˚□ = −µ−1  b0 ∂2  t + µb0∂2  r +  \uf8eb  \uf8ed  −µ′  b0∂r + 1  2µ−1  b0 µ′2  b0 − µ′′  b0  2µ′  b0∂t  − 1  2µb0µ′2  b0  µ−2  b0 µ′  b0∂t  µ′  b0∂r − µ−1  b0 µ′2  b0  µ′  b0∂t  − 1  2µ−3  b0 µ′2  b0  2µ−2  b0 µ′  b0∂t  3µ′  b0∂r + 1  2µ−1  b0 µ′2  b0 + µ′′  b0  \uf8f6  \uf8f8 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='16)  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Calculation of D(g,dA)T (˙g, d ˙A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Recall that the Reissner-Nordstr¨om electromagnetic 4-potential is  given by  A = Qr−1dt∗,  F = Qr−2dt∗ ∧ dr = Qr−2 �  vol  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='17)  Since ˚  volij = ǫij, ˚  volia = ˚  volab = 0 where ǫ is the Levi-Civita symbol, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=', ǫ12 = −ǫ21 = 1, it is clear that  ˚  vol  αβ = − ˚  volαβ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' A direct computation implies  F αβFαβ = −2Q2  r4 ,  ˙gαβFµαFνβ =  �  Q2  r4  �  ˙gµν −˚gµν˚tr ˙g  �  if µ, ν ∈ {i, j}  0  otherwise  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  Then according to (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='1), we find  �  D(g,dA)T (˙g, 0)  �  ij = −Q2  r4  �  ˙gij −˚gij˚tr˙g − 1  2 ˙gij + 1  2˚gij˚tr ˙g  �  = − Q2  2r4  �  ˙gij −˚gij˚tr ˙g  �  ,  �  D(g,dA)T (˙g, 0)  �  ia = Q2  2r4 ˙gia,  �  D(g,dA)T (˙g, 0)  �  ab = Q2  2r4  �  ˙gab − r2/gab˚tr ˙g  �  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='18)  which means  D(g,dA)T (·, 0) = Q2  2r4  \uf8eb  \uf8ed  ˚g˚tr−1  0  0  0  1  0  −r2/g˚tr  0  1  \uf8f6  \uf8f8 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='19)  Since  F αβ ˙Fαβ = Qr−2 ˚  vol  ij ˙Fij,  gαβ ˙FiαFjβ + gαβFiα ˙Fjβ = −2Qr−2 ˙Ft∗r˚gij = 2Qr−2 ˙Ft∗r(˚∗ ˚  vol)˚gij,  gαβ ˙FaαFiβ = ˚gkj ˙FakFij = −Qr−2˚gkjǫij ˙Fka = Qr−2˚∗ ˙F•a  where we use the fact ˚∗ ˚  vol = −1 and the Hodge star operator ˚∗ only acts on the aspherical part, we have  D(g,dA)T (0, d(·)) = −Qr−2  \uf8eb  \uf8ed  −˚g˚∗˚d  0  ˚∗/d  −˚∗˚d  r2/g˚∗˚d  0  \uf8f6  \uf8f8 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='20)  This finishes the calculation of L1(˙g, d ˙A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Calculation of D(g,dA)(δgdA)(˙g, d ˙A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' It remains to consider L2(˙g, d ˙A) which is the linearization of  δgdA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' We again use the following splitting  T ∗M = T ∗  AS ⊕ T ∗  S,  ∧2T ∗M = ∧2T ∗  AS ⊕ (T ∗  AS ∧ T ∗  S) ⊕ ∧2T ∗  S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='21)  With respect to the above splitting (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='21), d : T ∗M → ∧2T ∗M and δg : ∧2T ∗M → T ∗M take the form  d =  \uf8eb  \uf8ed  ˚d  0  −/d  ˚d  0  /d  \uf8f6  \uf8f8 ,  δg =  �  r−2˚δr2  −r−2/δ  0  0  ˚δ  r−2/δ  �  ,  LINEAR STABILITY OF KERR-NEWMAN FAMILY  163  and thus by (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='4)  D(g,dA)(δgdA)(0, d(·)) =  �r−2˚δr2˚  d + r−2/δ/d  −r−2/δ˚d  −˚δ/d  ˚δ˚d + r−2/δ/d  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='22)  Finally we are left with dealing with D(g,dA)(δgdA)(˙g, 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' We first calculate the first covariant derivative on  the 2-form F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' A direct calculation implies ˚∇k ˚  volij = 0, and then  ∇kFij = −2r−1(∂kr)Fij,  ∇iFja = 0,  ∇iFab = 0,  ∇aFij = 0,  ∇aFib = r(∂kr)/gabFik,  ∇aFbc = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  Therefore we have  ˙gνα∇αFνi = −2r−1(∂jr)Fki ˙gjk + r−3/gab ˙gab(∂kr)Fki = −2Qr−3˚∗  �  (∂jr)˙gjk  �  + Qr−5(˚∗dr)i/gab ˙gab  ˙gνα∇αFνa = ˙gbi∇bFia = Q  r3 (∂kr)(˚∗ ˙g•a)k  We also find (δgGg ˙g)κFκa = 0 and −(δgGg ˙g)κFκi = −Qr−2˚∗ (δgGg ˙g), thus  −(δgGg(·))κFκi = − Q  r2  �  r−2˚∗˚δr2 + 1  2˚∗˚d˚tr  r−2˚∗/δ  1  2r−2˚∗˚d/tr  0  0  0  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  Since ˚gijǫikǫkl = ˚gkl −˚gkl˚tr˚g = −˚gkl and ˚∇k ˚  volij = 0, we find  1  2 (∇ν ˙gκ  i − ∇κ ˙gν  i) Fνκ = − Q  2r2 (˚∇k ˙glj − ˚∇l ˙gkj)˚gmnǫjmǫinǫkl  = − Q  2r2  �  ˚∇k(˙gljǫjmǫkl) − ˚∇l(˙gkjǫjmǫkl)  �  ˚gmnǫin  = Q  2r2  �  ˚∇k(˙gkm −˚gkm˚tr ˙g) + ˚∇l(˙glm −˚glm˚tr ˙g)  �  ˚gmnǫin  = − Q  r2˚∗  �  ˚∇k ˙gkm − ˚  d˚tr ˙g  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  and using ˚∗˚∗ = (−1)k(2−k)+1I and δ = ˚∗˚d˚∗ on ∧kT ∗ ˚  X for 1 ≤ k ≤ 2, we see that  1  2 (∇ν ˙gκ  a − ∇κ ˙gν  a) Fνκ = Q  2r2˚gik˚gjl �  ∂k ˙gla − ∂l ˙gka − r−1(∂kr)˙gla + r−1(∂lr)˙gka  �  ǫij  = − Q  r3 (∂kr)(˚∗ ˙g•a)k + Q  r2  �  2r−1(∂kr)(˚∗ ˙g•a)k +˚∗˚d˙g•a  �  = − Q  r3 (∂kr)(˚∗ ˙g•a)k + Q  r2  �  −r2(∂kr−2)(˚∗ ˙g•a)k + ˚δ˚∗˙g•a  �  = − Q  r3 (∂kr)(˚∗ ˙g•a)k + Q˚δr−2˚∗ ˙g•a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  Combining the above calculation and (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='4) yields  D(g,dA)(δgdA)(˙g, 0) = Q  r2  � 1  2˚∗˚d˚tr  −r−2˚∗/δ  − 1  2˚∗˚dr−2 /tr  0  r2˚δr−2˚∗  0  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='23)  Lastly, we discuss the calculation of the gauge terms for the electromagnetic field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Since A = −Qr−1dt∗,  we have  (Lω♯A)i = ωα∂αAi + Aα∂iωα,  (Lω♯A)a = ωα∂αAa + Aα∂aωα = ∂a(Aαωα)  and as a consequence  L(·)♯A =  �˚  dιA + ι(·)˚dA  0  /dιA  0  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='24)  164  LILI HE  Appendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Calculation of the subprincipal operator at trapping  In this section, we include the discussion of the subprincipal operator of Pb0,γ, Wb0,γ and Lb0,γ at the  trapped sets K for the sake of completeness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  The arguments are based on [38], [39] and [40].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  In the  subsequent discussion, we will list (without proof) the relevant results for the invariant formalism of the  subprincipal operator with references (mostly from [38]), and then carry out all the relevant computations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  According to [20, Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='1] and [40, Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='5] (the microlocalized version) (see also the discussion  in [38, §2]), a sufficient condition for high energy estimates at the trapped set for a semiclassical operator  Ph(z) acting on the covariant rank k tensor bundle Tk to hold is  σ1,h  � 1  2ih(Ph(z) − Ph(z)∗)  �  < νmin/2  (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='1)  at the trapped set K, where νmin > 0 is the minimal normal expansion rate of the Hamilton flow at the  trapping, see Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Here, the adjoint is taken with respect to a positive definite inner product  on Tk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' We note that the natural inner product induced by g, with respect to which □g is symmetric, is not  positive definite, except when k = 0, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' for the scalar wave equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' One can introduce a pseudodifferential  inner product such that (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='1) can be arranged, with the adjoint taken with respect to this pseudodifferential  inner product.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' We work with classical, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' one-step polyhomogeneous, symbols and operators, and denote  by Sm  hom(T ∗X \\ 0) symbols which are homogeneous of degree m with respect to dilations in the fibers of  T ∗X \\ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  Definition B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='1 ( [38, Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='1]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' A pseudodifferential inner product (or Ψ-inner product) on the vector  bundle E → X is a pseudodifferential operator B ∈ Ψ0(X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' E ⊗ Ω  1  2 , E  ∗ ⊗ Ω  1  2 ) satisfying B = B∗, and such  that moreover the principal symbol σ0(B) = b ∈ S0  hom(T ∗X \\ 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' π∗Hom(E, E  ∗)) of B satisfies  ⟨b(x, ξ)u, ι(u)⟩ > 0  (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='2)  for all non-zero u ∈ Ex, where x ∈ X, ξ ∈ T ∗  xX \\ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' If the context is clear, we will also call the sesquilinear  pairing  C∞(X, E ⊗ Ω  1  2 ) × C∞(X, E ⊗ Ω  1  2 ) ∋ (u, v) �→  �  X  ⟨B(x, D)u, ι(v)⟩  the pseudodifferential inner product associated with B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  The use of a pseudodifferential inner product is equivalent to considering a conjugated operator QPh(z)Q−,  where Q ∈ Ψ0  h(X, Tk) is a carefully chosen elliptic operator with parametrix Q−.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' For any ǫ > 0, we can  arrange σ1,h( 1  2ih(QPh(z)Q− − (QPh(z)Q−)∗)) < ǫ (with the adjoint taken relative to an ordinary positive  definite inner product on Tk), thus (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='1) holds for Ph(z) replaced by QPh(z)Q−.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  Let π : T ∗X \\ 0 → X be the projection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' We will work in standard pseudodifferential operator setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  We specialize to the case that P ∈ Ψm(X, E ⊗ Ω  1  2 ) has a real, scalar principal symbol, which is the case of  interest in our application.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Fix a coordinate system of X and a local trivialization of E, then the full symbol  of P is a sum of homogeneous symbols p ∼ pm + pm−1 + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=', with pj homogeneous of degree j and valued  in complex N × N matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' According to [46, Theorem 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='33], the subprincipal symbol  σsub(P) = pm−1(x, ξ) − 1  2i  �  j  ∂xjξjpm(x, ξ) ∈ Sm−1  hom (T ∗X \\ 0, CN×N)  (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='3)  is well-defined under changes of coordinates;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' however, it does depend on the choice of local trivialization  of E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' We would like a frame-independent notion of the subprincipal symbol since this provides us with the  freedom to choose particularly convenient local frames in concrete computations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Invariant formalism for the subprincipal symbols of operators acting on bundles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  Definition B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='2 ( [38, Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='8]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' For P ∈ Ψm(X, E ⊗ Ω  1  2 ) with scalar principal symbol p, there is a  well-defined subprincipal operator Ssub(P) ∈ Diff1(T ∗X \\ 0, π∗E), homogeneous of degree m − 1 with respect  to dilations in the fibers of T ∗X \\ 0, defined as follows: if {e1(x), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' , eN(x)} is a local frame of E, define the  operators Pjk ∈ Ψm(X, Ω  1  2 ) by P(�  k uk(x)ek(x)) = �  jk Pjk(uk)ej(x), uk ∈ C∞(X, Ω  1  2 ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Then  Ssub(P)  ��  k  qk(x, ξ)ek(x)  �  :=  �  jk  (σsub(Pjk)qk)ej − i  �  k  (Hpqk)ek.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  LINEAR STABILITY OF KERR-NEWMAN FAMILY  165  The symbols of commutators and imaginary parts can be expressed in a completely invariant fashion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  Proposition B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='3 ( [39, Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='11]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Let P ∈ Ψm(X, E ⊗ Ω  1  2 ) be a pseudodifferential operator with  scalar principal symbol p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  (1) Suppose Q ∈ Ψm′(X, E ⊗ Ω  1  2 ) is an operator acting on E-valued half-densities, with principal symbol  q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Then  σm+m′−1([P, Q]) = [Ssub(P), q].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='4)  If Q is elliptic with parametrix Q−, then  Ssub(QPQ−) = qSsub(P)q−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='5)  (2) Suppose in addition that p is real.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Let B be a Ψ-inner product (pseudodifferential inner product) on  E with principal symbol b, then  σm−1(Im BP) = Im bSsub(P),  (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='6)  where Im BP =  1  2i(P − P ∗B) and Im bSsub(P) =  1  2i  �  Ssub(P) − Ssub(P)∗b�  ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' we take the adjoint of  P with respect to the Ψ-inner product B and the adjoint of the differential operator Ssub(P) with  respect to the inner product b on π∗E and the symplectic volume density on T ∗X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  The imaginary part Im BP of an operator P with respect to a Ψ-inner product B can be interpreted in  terms of the imaginary part of a conjugated version of P with respect to a standard inner product.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  Proposition B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='4 ( [38, Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='12]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Let B be a Ψ-inner product on E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  Then for any positive  definite Hermitian inner product B0 ∈ C∞(X, Hom(E ⊗ Ω  1  2 , E  ∗ ⊗ Ω  1  2 )) on E, there exists an elliptic operator  Q ∈ Ψ0(X, End(E ⊗ Ω  1  2 )) such that B − Q∗B0Q ∈ Ψ−∞(X, Hom(E ⊗ Ω  1  2 , E  ∗ ⊗ Ω  1  2 )).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  In particular, denoting by Q− ∈ Ψ0(X, End(E ⊗ Ω  1  2 )) a parametrix of Q, we have for any P ∈ Ψm(X, E ⊗  Ω  1  2 ) with real and scalar principal symbol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  Q(Im BP)Q− = Im B0(QPQ−),  (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='7)  and σm−1(Im BP) and σm−1(Im B0(QPQ−)) (which are self-adjoint with respect to σ0(B) and B0, respec-  tively, hence diagonalizable) have the same eigenvalues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  Therefore, the analysis of σm−1(Im B0(QPQ−))) is reduced to finding an inner product b such that  (Ssub(L) − Ssub(L)∗b) has a desired form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' (In our applications, the choice of b will be clear from an in-  spection of the form of Ssub(L)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  Let (M, g) be a smooth manifold equipped with a metric tensor g of arbitrary signature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Denote by  TkM = �k T ∗M, k ≥ 1, the bundle of (covariant) tensors of rank k on M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' The metric g induces a metric  (which we also call g) on TkM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' The subprincipal operator of ∆k = tr∇2 ∈ Diff2(M, TkM) acting on the  bundle TkM has a nice form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  Proposition B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='5 ( [38, Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='1]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' The subprincipal operator of ∆k is  Ssub(∆k)(x, ξ) = i∇π∗TkM  HG  ∈ Diff1(T ∗M \\ 0, π∗TkM),  (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='8)  where ∇π∗TkM is the pullback connection, with π: T ∗M \\ 0 → M being the projection, and G = |ξ|2  g−1(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Calculation of subprincipal operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' We notice that modulo terms of order 0, which are sub-  subprincipal and thus irrelevant in the calculation of Ssub(Pb0,γ), Ssub(Wb0,γ) and Ssub(Lb0,γ) (for simplicity  of notation, from now on, we drop the notations b0, γ, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=', g = gb0), we have  − 2P ≡ □g,1 − 2δgGg(˜δ∗  g − δ∗  g),  −2W ≡ □g,1 − 2(˜δg − δg)Ggδ∗  g  (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='9)  and  − L(˙g, ˙A) ≡  �  □g ˙g − 2(˜δ∗  g − δ∗  g)δgGg ˙g − 2δ∗  g(˜δg − δg)Gg ˙g + L12( ˙A), □g ˙A + L21(˙g)  �  ,  (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='10)  where  L12( ˙A) = 8tr24  g (F ⊗s d ˙A) − 2g−1(F, d ˙A) g,  F = dA = d(Q0  r dt) = Q0  r2 dt ∧ dr,  L21(˙g) = tr12  g (δgGg ˙g ⊗ F) − 1  2tr24  g tr35  g (d∇ ˙g ⊗ F),  (d∇ ˙g) µκ  ν  = ∇µ ˙g κ  ν − ∇κ  ν ˙g µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  166  LILI HE  Since ˜δ∗  g − δ∗  g and ˜δg − δg are compactly supported away from the trapping for Reissner-Nordstr¨om metric,  we further have that modulo terms of order 0  −2P ≡ −2W ≡ □g,1  −L(˙g, ˙A) ≡  �  □g ˙g + L12( ˙A), □g ˙A + L21(˙g)  �  (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='11)  at the trapping K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  Let g = gb0 be the Reissner-N¨ordstrom metric with parameter b0 = (m0, 0, Q0)  g = −α2dt2 + α−2dr2 + r2/g,  α = √µb0 =  �  1 − 2m0  r  + Q2  0  r2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='12)  That is, we work in the coordinates (t, r, ω), ω ∈ S2 at the trapping K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Let (σ, ξ, η) be the dual variables to  (t, r, ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' We define  e0 := αdt,  e1 := α−1dr  (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='13)  and use the following splittings of T ∗M and S2 T ∗M  T ∗M = ⟨e0⟩ ⊕ ⟨e1⟩ ⊕ T ∗S2,  S2 T ∗M = ⟨e0e0⟩ ⊕ ⟨2e0 ⊗s e1⟩ ⊕ (2e0 ⊗s T ∗S2) ⊕ ⟨e1e1⟩ ⊕ (2e1 ⊗s T ∗S2) ⊕ S2 T ∗S2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='14)  In this section, we are interested in the subprincipal operators of □g,1 and L at the trapped set  K = {r = rp, ξ = 0, σ2 = µb0r−2|η|2},  where  ∂r(rα−1)(rp) = 0,  |η|2 = |η|2  /g−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='15)  Therefore, at the trapping K, we have  Hα2ξ2+r−2|η|2 = 2r−3|η|2∂ξ + r−2H|η|2  and  σ2Hα−2 − Hα2ξ2+r−2|η|2 = −r−2H|η|2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  Let x0 = t, and x′ = (r, ω) be coordinates on X (independent of t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' We let Greek indices µ, ν, λ, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' run  from 0 to 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Moreover, the canonical dual variables ξ0 =: σ and ξ′ = (ξ, η) on the fibers of T ∗M are indexed  by decorated Greek indices �µ running from 0 to 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' For a section u of π∗T ∗M, we have  ∇π∗T ∗M  µ  uν = ∇M  µ uν,  ∇π∗T ∗M  �µ  uν = ∂�µuν.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Calculation of the subprincipal operator of □g,1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' According to Proposition B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='5 and using the fact  that G = −α−2σ2 + α2ξ2 + r−2|η|2, a direct calculation implies that at trapping K = {r = rp, ξ = 0, σ2 =  µb0r−2|η|2}  iSsub(□g,1) =  \uf8eb  \uf8ec  \uf8ed  2α−2σ∂t − r−2H|η|2  0  0  0  2α−2σ∂t − r−2H|η|2  0  0  0  2α−2σ∂t − r−2∇  π∗  S2T ∗S2  H|η|2  \uf8f6  \uf8f7  \uf8f8 + S(1)  (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='16)  where  S(1) =  \uf8eb  \uf8ed  0  −2r−1σ  0  −2r−1σ  0  2αr−3ιη  0  −2αr−1η  0  \uf8f6  \uf8f8 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='17)  For simplicity of calculation, we further decompose π∗T ∗M → T ∗M over the trapping K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' For η ∈ T ∗S2\\o,  we define η⊥ := /∗η, further  �η := ασ−1η,  η⊥ = ασ−1η⊥,  and write  π∗  S2T ∗S2 = ⟨�η⟩ ⊕ ⟨�η⊥⟩,  π∗  S2S2T ∗S2 = ⟨�η�η⟩ ⊕ ⟨2�η�η⊥⟩ ⊕ ⟨�η⊥�η⊥⟩,  (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='18)  LINEAR STABILITY OF KERR-NEWMAN FAMILY  167  inducing the decompositions  π∗T ∗M = ⟨e0⟩ ⊕  �  ⟨e1⟩ ⊕  �  ⟨�η⟩ ⊕ ⟨�η⊥⟩  ��  ,  π∗S2T ∗M =  �  ⟨e0e0⟩ ⊕  �  ⟨2e0e1⟩ ⊕  �  ⟨2e0�η⟩ ⊕ ⟨2e0�η⊥⟩  ���  ⊕  �  ⟨e1e1⟩ ⊕  �  ⟨2e1�η⟩ ⊕ ⟨2e1�η⊥⟩  ��  ⊕  �  ⟨�η�η⟩ ⊕ ⟨2�η�η⊥⟩ ⊕ ⟨�η⊥�η⊥⟩  �  ,  (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='19)  into trivial rank 1 bundles over T ∗M near K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Then, acting on 1-forms, we have  η =  �α−1σ  0  �  ,  r−2ιη =  �  α−1σ  0  �  at  K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  Therefore, in terms of the splitting (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='19), the operator S(1) in (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='17) at trapping K is given as  S(1) = r−1σ  \uf8eb  \uf8ec  \uf8ec  \uf8ed  0  −2  0  0  −2  0  2  0  0  −2  0  0  0  0  0  0  \uf8f6  \uf8f7  \uf8f7  \uf8f8 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='20)  Now, for any fixed ǫ > 0, we define the change of basis matrix  q1 =  \uf8eb  \uf8ec  \uf8ec  \uf8ed  0  0  0  1  0  0  1  ǫ  0  0  − 2  ǫ2  0  0  − 4  ǫ3  0  − 4  ǫ3  0  \uf8f6  \uf8f7  \uf8f7  \uf8f8 ,  and then we have  q1S(1)q−1  1  = r−1σ  \uf8eb  \uf8ec  \uf8ec  \uf8ed  0  0  0  0  0  0  ǫ  0  0  0  0  ǫ  0  0  0  0  \uf8f6  \uf8f7  \uf8f7  \uf8f8 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  Since q1 is a constant coefficient operator in the region where the splitting (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='19) is well defined, in particular,  in a neighborhood of the trapping K, it commutes with the diagonal part iSsub(□g,1) − S of Ssub(□g,1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  Equipping the 1-form bundle over M with the Hermitian inner product B1  0 which is given by an identity  matrix in the splitting (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='19), q1Ssub(□g,1)q−1  1  has imaginary part (with respect to B0) of size O(ǫ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Put  differently, Ssub(□g,1) has imaginary part of size (ǫ) relative to the Hermitian inner product b := B1  0(q·, q·),  which is the symbol of a pseudodifferential inner product on π∗T ∗M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' To summarize,  Theorem B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' For any ǫ > 0, there exists a (positive definite) t-independent pseudodifferential inner  product B = b(x, D) on T ∗M (thus, b is an inner product on π∗T ∗M, homogeneous of degree 0 with respect  to dilations in the base T ∗M \\ 0), such that  sup  K  |σ|−1  ����  1  2i(Ssub(□g,1) − Ssub(□g,1)∗b)  ����  b  ≤ ǫ,  where K is the trapped set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Put differently, there is an elliptic pseudodifferential operator Q, invariant under  t-translations, acting on sections of T ∗M, with parametrix Q−, such that relative to the ordinary positive  definite inner product B1  0 we have  sup  Γ  |σ|−1  ����σ1  � 1  2i(Q□g,1Q− − (Q□g,1Q−)∗B1  0)  �����  B1  0  ≤ ǫ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Calculation of the subprincipal operator of L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Using Proposition B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='5 again, we see that Ssub(□g,2) =  i∇π∗S2T ∗M  HG  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Since π∗S2T ∗M is simply the restriction of the product connection ∇π∗T ∗M ⊗ ∇π∗T ∗M to  π∗S2T ∗M, it follows that  Ssub(□g,2)(w1w2) = Ssub(□g,1)w1 · w2 + w1 · Ssub(□g,1)w2  for w1, w2 ∈ C∞(T ∗M, π∗T ∗M), under the splitting (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='14), Ssub(□g,2) has a canonical first order part,  induced by the canonical first order part of Ssub(□g,1) which is given in (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='16).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' The 0-order part S(2) of  168  LILI HE  Ssub(□g,2) is equal to the second symmetric tensor power of the 0-th order part of Ssub(□(1)  g ) in (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='17).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' To  summarize,  iSsub(□g,2) = 2α−2σ∂t − r−2  \uf8eb  \uf8ec  \uf8ec  \uf8ec  \uf8ec  \uf8ec  \uf8ec  \uf8ec  \uf8ec  \uf8ec  \uf8ed  H|η|2  0  0  0  0  0  0  H|η|2  0  0  0  0  0  0  H  π∗  S2T ∗S2  |η|2  0  0  0  0  0  0  H|η|2  0  0  0  0  0  0  H  π∗  S2T ∗S2  |η|2  0  0  0  0  0  0  H  π∗  S2S2 T ∗S2  |η|2  \uf8f6  \uf8f7  \uf8f7  \uf8f7  \uf8f7  \uf8f7  \uf8f7  \uf8f7  \uf8f7  \uf8f7  \uf8f8  + S(2) (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='21)  where  S(2) =  \uf8eb  \uf8ec  \uf8ec  \uf8ec  \uf8ec  \uf8ec  \uf8ec  \uf8ed  0  −4r−1σ  0  0  0  0  −2r−1σ  0  2αr−3iη  −2r−1σ  0  0  0  −2αr−1η  0  0  −2r−1σ  0  0  −4r−1σ  0  0  4αr−3iη  0  0  0  −2r−1σ  −2αr−1η  0  2αr−3iη  0  0  0  0  −4αr−1η  0  \uf8f6  \uf8f7  \uf8f7  \uf8f7  \uf8f7  \uf8f7  \uf8f7  \uf8f8  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='22)  Again, we turn to the ‘microlocal splitting’ (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='18), under which we write  η =  \uf8eb  \uf8ed  α−1σ  0  0  1  2α−1σ  0  0  \uf8f6  \uf8f8 : T ∗S2 → S2T ∗S2,  r−2iη =  �α−1σ  0  0  0  α−1σ  0  �  : S2T ∗S2 → T ∗S2  Therefore, in terms of (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='19), we see that  S(2) = r−1σ  \uf8eb  \uf8ec  \uf8ec  \uf8ec  \uf8ec  \uf8ec  \uf8ec  \uf8ec  \uf8ec  \uf8ec  \uf8ec  \uf8ec  \uf8ec  \uf8ec  \uf8ec  \uf8ed  0  −4  0  0  0  0  0  0  0  0  −2  0  2  0  −2  0  0  0  0  0  0  −2  0  0  0  −2  0  0  0  0  0  0  0  0  0  0  −2  0  0  0  0  −4  0  0  0  4  0  0  0  0  0  0  −2  0  −2  0  0  2  0  0  0  0  0  −2  0  0  0  0  2  0  0  0  0  0  0  −4  0  0  0  0  0  0  0  0  0  0  −2  0  0  0  0  0  0  0  0  0  0  0  0  0  \uf8f6  \uf8f7  \uf8f7  \uf8f7  \uf8f7  \uf8f7  \uf8f7  \uf8f7  \uf8f7  \uf8f7  \uf8f7  \uf8f7  \uf8f7  \uf8f7  \uf8f7  \uf8f8  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='23)  Let SL be the 0-th order part of iSsub(−L) at K, with L given in (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Acting on the bundle S2T ∗M⊕  T ∗M, we can write SL as a 2 × 2 block matrix,  SL =  �  S11  L  S12  L  S21  L  S22  L  �  where  S11  L = S(2), S22  L = S(1), S12  L = iσ1(L12), S21  L = iσ1(L21)  with  L12( ˙A) = 8tr24  g (F ⊗s d ˙A) − 2g−1(F, d ˙A) g,  F = dA = d(Q0  r dt) = Q0  r2 dt ∧ dr,  L21(˙g) = tr12  g (δgGg ˙g ⊗ F) − 1  2tr24  g tr35  g (d∇ ˙g ⊗ F),  (d∇ ˙g) µκ  ν  = ∇µ ˙g κ  ν − ∇κ  ν ˙g µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  In the splitting (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='14) and  Λ2T ∗M = ⟨e0 ∧ e1⟩ ⊕ ⟨e0 ∧ T ∗S2⟩ ⊕ ⟨e1 ∧ T ∗S2⟩ ⊕ Λ2T ∗S2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  LINEAR STABILITY OF KERR-NEWMAN FAMILY  169  d : T ∗M → Λ2T ∗M and tr24  g (F ⊗s (·)) : Λ2T ∗M → S2T ∗M are given by  d =  \uf8eb  \uf8ec  \uf8ec  \uf8ed  −α∂r − α′  α−1∂t  0  −/d  0  α−1∂t  0  −/d  α∂r  0  0  /d  \uf8f6  \uf8f7  \uf8f7  \uf8f8 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  2tr24  g (F ⊗s (·)) = Q0  r2  \uf8eb  \uf8ec  \uf8ec  \uf8ec  \uf8ec  \uf8ec  \uf8ec  \uf8ed  2  0  0  0  0  0  0  0  0  0  −1  0  −2  0  0  0  0  −1  0  0  0  0  0  0  \uf8f6  \uf8f7  \uf8f7  \uf8f7  \uf8f7  \uf8f7  \uf8f7  \uf8f8  ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  and we also have  g−1(F,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' (·))g =  \uf8eb  \uf8ec  \uf8ec  \uf8ec  \uf8ec  \uf8ec  \uf8ec  \uf8ed  2Q0  r2  0  0  0  0  0  0  0  0  0  0  0  − 2Q0  r2  0  0  0  0  0  0  0  − 2Q0  r2 r2/g  0  0  0  \uf8f6  \uf8f7  \uf8f7  \uf8f7  \uf8f7  \uf8f7  \uf8f7  \uf8f8  : Λ2T ∗M → S2T ∗M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  Therefore, we obtain  σ1(L12) = i4Q0  r2  \uf8eb  \uf8ec  \uf8ec  \uf8ec  \uf8ec  \uf8ec  \uf8ec  \uf8ed  −αξ  α−1σ  0  0  0  0  0  η  −αξ  αξ  −α−1σ  0  η  0  −α−1σ  −αξr2/g  α−1σr2/g  0  \uf8f6  \uf8f7  \uf8f7  \uf8f7  \uf8f7  \uf8f7  \uf8f7  \uf8f8  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='24)  Turning to the ‘microlocal splitting’ (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='18), we write /g = /g = |η|−2ηη + |η|−2η⊥η⊥ and then we have  /g =  \uf8eb  \uf8ed  r−2  0  r−2  \uf8f6  \uf8f8  at  K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  Therefore, in terms of the splitting (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='18),  iσ1(L12) = r−1σ 4Q0  rα  \uf8eb  \uf8ec  \uf8ec  \uf8ec  \uf8ec  \uf8ec  \uf8ec  \uf8ec  \uf8ec  \uf8ec  \uf8ec  \uf8ec  \uf8ec  \uf8ec  \uf8ec  \uf8ed  0  −1  0  0  0  0  0  0  0  −1  0  0  0  0  0  0  0  1  0  0  −1  0  1  0  0  0  0  1  0  −1  0  0  0  0  0  0  0  −1  0  0  \uf8f6  \uf8f7  \uf8f7  \uf8f7  \uf8f7  \uf8f7  \uf8f7  \uf8f7  \uf8f7  \uf8f7  \uf8f7  \uf8f7  \uf8f7  \uf8f7  \uf8f7  \uf8f8  at  K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='25)  So it remains to analyze L21(˙g).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' We first compute  Gg =  \uf8eb  \uf8ec  \uf8ec  \uf8ec  \uf8ec  \uf8ec  \uf8ec  \uf8ed  1  2  0  0  1  2  0  1  2r2 /tr  0  1  0  0  0  0  0  0  1  0  0  0  1  2  0  0  1  2  0  − 1  2r2 /tr  0  0  0  0  1  0  1  2r2/g  0  0  − 1  2r2/g  0  G/g  \uf8f6  \uf8f7  \uf8f7  \uf8f7  \uf8f7  \uf8f7  \uf8f7  \uf8f8  ,  tr12  g ((·) ⊗ F) =  \uf8eb  \uf8ed  0  − Q0  r2  0  − Q0  r2  0  0  0  0  0  \uf8f6  \uf8f8 : T ∗M → T ∗M (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='26)  and  σ1(δg) = i  \uf8eb  \uf8ed  α−1σ  −αξ  −r−2ιη  0  0  0  0  α−1σ  0  −αξ  −r−2ιη  0  0  0  α−1σ  0  −αξ  −r−2ιη  \uf8f6  \uf8f8 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='27)  170  LILI HE  Therefore, we find that  σ1(tr12  g (δgGg(·) ⊗ F)) = iQ0  r2  \uf8eb  \uf8ed  1  2αξ  −α−1σ  0  1  2αξ  r−2ιη  − 1  2r2 αξ /tr  − 1  2α−1σ  αξ  r−2ιη  − 1  2α−1σ  0  − 1  2r2 α−1σ /tr  0  0  0  0  0  0  \uf8f6  \uf8f8 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='28)  Next, we calculate  σ1(d∇(·) ⊗ F) = i2Q0  r2  \uf8eb  \uf8ed  αξ  −α−1σ  0  0  0  0  0  αξ  0  −α−1σ  0  0  0  0  αξ  0  −α−1σ  0  \uf8f6  \uf8f8 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='29)  In the ‘microlocal splitting’ (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='18), we have  /tr =  �r2  0  r2�  at  K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  and thus we have  iσ1(L21) = r−1σ Q0  2rα  \uf8eb  \uf8ec  \uf8ec  \uf8ed  0  0  0  0  0  −2  0  0  0  0  1  0  −2  0  −1  0  0  1  0  1  0  0  0  0  0  −2  0  0  0  0  0  0  0  0  0  0  −2  0  0  0  \uf8f6  \uf8f7  \uf8f7  \uf8f8 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='30)  Putting (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='20), (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='23), (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='25) and (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='30) together yields  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='SL = r−1σ  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='\uf8eb  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
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+page_content='\uf8f7  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='\uf8f7  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='\uf8f7  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='\uf8f7  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='\uf8f7  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='\uf8f7  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='\uf8f7  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='\uf8f8  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='32)  Again, equipping S2T ∗M⊕T ∗M over M with the Hermitian inner product BL  0 which is given as an identity  matrix in the splitting (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='14), qLSsub(−L)q−1  L  has imaginary part (with respect to BL  0 ) of size O(ǫ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Put  differently, Ssub(−L) has imaginary part of size (ǫ) relative to the Hermitian inner product b := BL  0 (qL·, qL·),  which is the symbol of a pseudodifferential inner product on π∗T ∗M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' To summarize,  Theorem B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' For any ǫ > 0, there exists a (positive definite) t-independent pseudodifferential inner product  B = b(x, D) on S2T ∗M⊕ T ∗M (thus, b is an inner product on π∗(S2T ∗M⊕ T ∗M), homogeneous of degree  0 with respect to dilations in the base T ∗M \\ 0), such that  sup  K  |σ|−1  ����  1  2i(Ssub(−L) − Ssub(−L)∗b)  ����  b  ≤ ǫ,  where K is the trapped set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Put differently, there is an elliptic pseudodifferential operator Q, invariant under  t-translations, acting on sections of S2T ∗M ⊕ T ∗M, with parametrix Q−, such that relative to the ordinary  positive definite inner product BL  0 we have  sup  Γ  |σ|−1  ����σ1  � 1  2i(Q(−L)Q− − (Q(−L)Q−)∗BL  0 )  �����  BL  0  ≤ ǫ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  Finally, we point out the relation between the operators P, W, L and their Fourier transform (semiclassi-  cally rescaled) h2 �P(h−1z), h2 �  W(h−1z) and h2�L(h−1z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  Remark B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Given a operator P ∈ {□g,1, −L}, we define �P(σ) := eiσt∗Pe−iσt∗ and its semiclassical rescaling  h2 �P(h−1z) with h = |σ|−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Let p2 = σ2(P) and p2,h(z) = σ2,h(h2 �P(h−1z)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Here we also use ξ′ = (ξ, η) as  the semiclassical dual variables of x′ = (r, ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Then using the correspondence Re z ←→ −σ, we have  Ssub(h2 �P(h−1z)) = Ssub(P) + iα−2σ∂t + iIm (hz)∂σp2  and thus  σ1,h  � 1  2ih(h2 �P(h−1z) − (h2 �P(h−1z))∗B)  �  = σ1  � 1  2i(P − P ∗B)  �  + Im (hz)∂σp2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  Finally, we note that Im (hz)∂σp2 = Im (hz)∂zp2,h(Re z) = σ1,h  �  1  2ih(h2 �  □g,0(h−1z) − (h2 �  □g,0(h−1z))∗)  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  Appendix C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Calculation of the subprincipal operator at radial points at event horizon  In this section, we calculate the threshold regularity at the radial points at event horizon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' To this end, we  discuss the subprincipal operator of Pb0,γ, Wb0,γ and Lb0,γ at radial points at event horizon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Here we again  drop the notations b0, γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  We recall that modulo terms of order 0, we have  − 2P ≡ □g,1 − 2δgGg(˜δ∗  g − δ∗  g),  −2W ≡ □g,1 − 2(˜δg − δg)Ggδ∗  g  (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='1)  and  − L(˙g, ˙A) ≡  �  □g ˙g − 2(˜δ∗  g − δ∗  g)δgGg ˙g − 2δ∗  g(˜δg − δg)Gg ˙g + L12( ˙A), □g ˙A + L21(˙g)  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='2)  172  LILI HE  Near the radial points at the event horizon, we use the coordinates (t0, r, ω) in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='6), in which the Reissner-  Nordstr¨om metric g = gb0, b0 = (m0, 0, Q0) takes the form  g = −µb0dt2  0 + 2dt0dr + r2/g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  Writing the covectors as  σ dt0 + ξ dr + η,  η ∈ T ∗S2,  we have G = g−1(σ dt0 + ξ dr + η, σ dt0 + ξ dr + η) = 2σξ + µb0ξ2 + r−2|η| and  HG = 2ξ∂t0 + (2σ + 2µb0ξ)∂r − µ′  b0ξ2∂ξ + 2r−3|η|2∂ξ + r−2H|η|2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  The radial points at event horizon is given as  L± = {(t0, rb0, ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' 0, ξ, 0) | ±ξ > 0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='3)  Then at L±, we have  HG = 2ξ∂t0 − 2κξ2∂ξ + 2µb0ξ∂r  (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='4)  with  κ = 1  2µ′  b0(rb0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='5)  being the surface gravity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' We keep the term 2µb0ξ∂r here as it is needed in the computation of the skew  adjoint part.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  The basis e0 = αdt, e1 = α−1dr are not defined at r = rb0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Hence, following [42, §6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='3], we compute the  transition function to a smooth trivialization of T ∗M and S2T ∗M as follows: writing a 1-form in the static  region M as  u = uN e0 + uT  N e1 + uT  T = �uN dt0 + �uT  N dr + �uT  T ,  with uT  T and �uT  T 1-forms on S2, we have  \uf8eb  \uf8ed  uN  uT  N  uT  T  \uf8f6  \uf8f8 = C (1)  \uf8eb  \uf8ed  �uN  �uT  N  �uT  T  \uf8f6  \uf8f8 ,  C (1) =  \uf8eb  \uf8ed  α−1  0  0  α−1  α  0  0  0  1  \uf8f6  \uf8f8 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='6)  The bundle splitting  T ∗M = ⟨dt0⟩ ⊕ ⟨dr⟩ ⊕ T ∗S2  (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='7)  is smooth at r = rb0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Consider the induced bundle splitting  S2T ∗M = ⟨dt2  0⟩ ⊕ ⟨2dt0 dr⟩ ⊕ (2dt0 · T ∗S2) ⊕ ⟨dr2⟩ ⊕ (2dr · T ∗S2) ⊕ S2T ∗S2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='8)  Writing a section u of S2T ∗M as  u = uNN e0e0 + 2uNT N e0e1 + 2e0 · uNT T + uT  NN e1e1 + 2e1 · uT  NT + uT  T T  = �uNN dt2  0 + 2�uNT N dt0 dr + 2dt0 · �uNT T + �uT  NN dr2 + 2dr · �uT  NT + �uT  T T ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  the transition matrix is defined by  \uf8eb  \uf8ec  \uf8ec  \uf8ec  \uf8ec  \uf8ec  \uf8ec  \uf8ed  uNN  uNT N  uNT T  uT  NN  uT  NT  uT  T T  \uf8f6  \uf8f7  \uf8f7  \uf8f7  \uf8f7  \uf8f7  \uf8f7  \uf8f8  = C (2)  \uf8eb  \uf8ec  \uf8ec  \uf8ec  \uf8ec  \uf8ec  \uf8ec  \uf8ed  �uNN  �uNT N  �uNT T  �uT  NN  �uT  NT  �uT  T T  \uf8f6  \uf8f7  \uf8f7  \uf8f7  \uf8f7  \uf8f7  \uf8f7  \uf8f8  ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  C (2) =  \uf8eb  \uf8ec  \uf8ec  \uf8ec  \uf8ec  \uf8ec  \uf8ec  \uf8ed  α−2  0  0  0  0  0  α−2  1  0  0  0  0  0  0  α−1  0  0  0  α−2  2  0  α2  0  0  0  0  α−1  0  α  0  0  0  0  0  0  1  \uf8f6  \uf8f7  \uf8f7  \uf8f7  \uf8f7  \uf8f7  \uf8f7  \uf8f8  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='9)  Then using the coordinates (t0, r, ω) and splitting (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='7), we again use Proposition B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='5 to compute  iSsub(□g,1) = −2ξ∂t0 + 2κξ2∂ξ + 2µb0ξ∂r − 2κξ  \uf8eb  \uf8ed  −1  0  0  0  1  0  0  0  0  \uf8f6  \uf8f8  at  L±  (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='10)  LINEAR STABILITY OF KERR-NEWMAN FAMILY  173  and  iSsub(□g,2) = −2ξ∂t0 + 2κξ2∂ξ + 2µb0ξ∂r − 2κξ  \uf8eb  \uf8ec  \uf8ec  \uf8ec  \uf8ec  \uf8ec  \uf8ec  \uf8ed  −2  0  0  0  0  0  0  0  0  0  0  0  0  0  −1  0  0  0  0  0  0  2  0  0  0  0  0  0  1  0  0  0  0  0  0  0  \uf8f6  \uf8f7  \uf8f7  \uf8f7  \uf8f7  \uf8f7  \uf8f7  \uf8f8  at  L±.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='11)  Next, we turn to the calculation of Ssub(L12) and Ssub(L21) at L±.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' This computation is easily accom-  plished by first calculating the symbols at N ∗{r = r0} = {t, r0, ω, 0, ξ, 0}, for r0 ∈ (rb0, ∞) close to rb0, in  the static coordinate system and bundle splittings (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='14), conjugating by C (1) or C (2) and restricting to  µb0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Concretely, according to (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='24), in terms of the splitting (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='14), we have at N ∗{r = r0}  iσ1(L12) = 4Q0  r2  \uf8eb  \uf8ec  \uf8ec  \uf8ec  \uf8ec  \uf8ec  \uf8ec  \uf8ed  αξ  0  0  0  0  0  0  0  αξ  −αξ  0  0  0  0  0  αξr2/g  0  0  \uf8f6  \uf8f7  \uf8f7  \uf8f7  \uf8f7  \uf8f7  \uf8f7  \uf8f8  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  So in the smooth splittings (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='7) and (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='8) (multiplying from the left by (C (2))−1 and from the right by  C (1) and evaluating at µ = 0),  iσ1(L12) = 4Q0  r2 ξ  \uf8eb  \uf8ec  \uf8ec  \uf8ec  \uf8ec  \uf8ec  \uf8ec  \uf8ed  0  0  0  −1  0  0  0  0  0  0  0  0  0  0  −1  r2/g  0  0  \uf8f6  \uf8f7  \uf8f7  \uf8f7  \uf8f7  \uf8f7  \uf8f7  \uf8f8  at  L±.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='12)  Using (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='28) and (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='29), in terms of the splitting (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='14), we have at N ∗{r = r0}  iσ(L21) = Q0  r2  \uf8eb  \uf8ed  1  2αξ  0  0  − 1  2αξ  0  1  2r2 αξ /tr  0  0  0  0  0  0  0  0  αξ  0  0  0  \uf8f6  \uf8f8 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  Multiplying this from the left by (C (1))−1 and from the right by C (2) and evaluating at µ = 0 give rise to  iσ(L21) in the smooth splittings (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='7) and (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='8)  iσ(L21) = Q0  2r2 ξ  \uf8eb  \uf8ed  0  0  0  0  0  0  0  2  0  0  0  − 1  r2 /tr  0  0  2  0  0  0  \uf8f6  \uf8f8  at  L±.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='13)  Finally, it remains to calculate the subprincipal operators of (˜δ∗  g − δ∗  g)δgGg ˙g and δ∗  g(˜δg − δg)Gg ˙g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' First,  according to (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='26) and (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='27), we see that in the static splitting (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='14)  iσ1(δgGg) =  \uf8eb  \uf8ed  0  αξ  0  0  0  0  1  2αξ  0  0  1  2αξ  0  − αξ  2r2 /tr  0  0  0  0  αξ  0  \uf8f6  \uf8f8  at N ∗{r = r0}, and thus  iσ1(δgGg) = 1  2ξ  \uf8eb  \uf8ed  2  0  0  0  0  0  0  0  0  0  0  − 1  r2 /tr  0  0  2  0  0  0  \uf8f6  \uf8f8  at  L±.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  174  LILI HE  Also, in the smooth splitting, we have at L±  iσ1(δ∗  g) =  \uf8eb  \uf8ec  \uf8ec  \uf8ec  \uf8ec  \uf8ec  \uf8ec  \uf8ed  0  0  0  − ξ  2  0  0  0  0  0  0  −ξ  0  0  0  − ξ  2  0  0  0  \uf8f6  \uf8f7  \uf8f7  \uf8f7  \uf8f7  \uf8f7  \uf8f7  \uf8f8  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  Since ˜δ∗  g − δ∗  g = 2γc ⊗s (·) − 1  2γg−1(c, ·)g and ˜δg − δg = 2γιg  c(·) − 1  2γctrg(·) where c = ˜χ(r)(dt0 − bdr) with  ˜χ(r) = 1 near event horizon and b > 2, it follows that in terms of the smooth splitting (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='7) and (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='8)  ˜δ∗  g − δ∗  g =  \uf8eb  \uf8ec  \uf8ec  \uf8ec  \uf8ec  \uf8ec  \uf8ec  \uf8ed  2γ  0  0  − 1  2bγ  1  2γ  0  0  0  γ  0  −2bγ  0  0  0  −bγ  1  2bγr2/g  − 1  2γr2/g  0  \uf8f6  \uf8f7  \uf8f7  \uf8f7  \uf8f7  \uf8f7  \uf8f7  \uf8f8  ,  ˜δg − δg =  \uf8eb  \uf8ed  −2bγ  γ  0  0  0  − 1  2γr−2 /tr  0  −bγ  0  2γ  0  1  2bγr−2 /tr  0  0  −2bγ  0  2γ  0  \uf8f6  \uf8f8  at L±.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Therefore, in terms of the smooth splitting, we have at L±  iσ1((˜δ∗  g − δ∗  g)δgGg) = 1  2γξ  \uf8eb  \uf8ec  \uf8ec  \uf8ec  \uf8ec  \uf8ec  \uf8ec  \uf8ed  4  0  0  0  0  0  −b  0  0  0  0  − 1  2r2 /tr  0  0  2  0  0  0  0  0  0  0  0  2b  r2 /tr  0  0  −2b  0  0  0  br2/g  0  0  0  0  1  2/g /tr  \uf8f6  \uf8f7  \uf8f7  \uf8f7  \uf8f7  \uf8f7  \uf8f7  \uf8f8  ,  (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='14)  iσ1(δgGg(˜δ∗  g − δ∗  g)) = 1  2γξ  \uf8eb  \uf8ed  4  0  0  −b  1  0  0  0  2  \uf8f6  \uf8f8 ,  (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='15)  iσ1(δ∗  g(˜δg − δg)Gg) = 1  2γξ  \uf8eb  \uf8ec  \uf8ec  \uf8ec  \uf8ec  \uf8ec  \uf8ec  \uf8ed  0  0  0  0  0  0  2b  −2  0  0  0  1  2r2 /tr  0  0  0  0  0  0  0  2b  0  −4  0  − b  r2 /tr  0  2b  0  −2  0  0  0  0  0  0  0  0  \uf8f6  \uf8f7  \uf8f7  \uf8f7  \uf8f7  \uf8f7  \uf8f7  \uf8f8  ,  (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='16)  iσ1((˜δg − δg)Ggδ∗  g) = 1  2γξ  \uf8eb  \uf8ed  −1  0  0  b  −4  0  0  0  −2  \uf8f6  \uf8f8 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='17)  In the splitting (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='7), combining (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='10) with (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='15) and (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='17) yields  Ssub(−2P) = −2ξ∂t0 − 2κξ2∂ξ + 2µb0ξ∂r + SP,  Ssub(−2W) = −2ξ∂t0 − 2κξ2∂ξ + 2µb0ξ∂r + SW  where  SP = ξ  \uf8eb  \uf8ed  2κ − 4γ  0  0  bγ  −γ − 2κ  0  0  0  −2γ  \uf8f6  \uf8f8 ,  SW = ξ  \uf8eb  \uf8ed  2κ + γ  0  0  −bγ  4γ − 2κ  0  0  0  2γ  \uf8f6  \uf8f8 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  In the splitting (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='7), equipping T ∗M with the Hermitian inner product 1 ⊕ 1 ⊕/g−1, the first three terms of  Ssub(−2P), Ssub(−2W) are formally self adjoint with respect to the symplectic volume form on T ∗M, and  the last term SP resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' SW of iSsub(−2P) resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' iSsub(−2W), multiplied by ξ−1, has eigenvalues  2(κ − 2γ), −2κ − γ, −2γ,  resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  − 2(κ − 2γ), 2γ, 2κ + γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='18)  We perform a decomposition  S2T ∗S2 = ⟨r2/g⟩ ⊕ /g⊥,  so  r2/g =  �  1  0  �  ,  r−2 /tr =  �2  0�  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  LINEAR STABILITY OF KERR-NEWMAN FAMILY  175  and introduce the splitting  ⟨dt2  0⟩ ⊕ ⟨2dt0 dr⟩ ⊕ (2dt0 · T ∗S2) ⊕ ⟨dr2⟩ ⊕ (2dr · T ∗S2) ⊕ ⟨r2/g⟩ ⊕ /g⊥  ⊕ ⟨dt0⟩ ⊕ ⟨dr⟩ ⊕ T ∗S2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='19)  In the above splitting (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='19), putting (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='10),(C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='11), (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='12), (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='13), (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='14) and (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='16) together yields  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='iSsub(−L) = −2ξ∂t0 − 2κξ2∂ξ + 2µb0ξ∂r + SL  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='where  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='SL = ξ  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='\uf8eb  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='\uf8ec  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='\uf8ec  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='\uf8ec  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='\uf8ec  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='\uf8ec  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='\uf8ec  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='\uf8ec  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='\uf8ec  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='\uf8ec  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='\uf8ec  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='\uf8ec  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='\uf8ec  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='\uf8ec  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='\uf8ec  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='\uf8ed  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='4κ − 4γ  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='0  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='0  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='0  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='0  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='0  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='0  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='0  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='0  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='0  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='−bγ  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='2γ  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='0  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='0  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='0  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='0  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='0  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='−4Q′′  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='0  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='0  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='0  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='0  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='2κ − 2γ  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='0  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='0  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='0  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='0  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='0  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='0  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='0  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='0  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='−2bγ  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='0  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
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+page_content='\uf8f7  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='\uf8f8  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='with Q′′ = Q0r−2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' In the splitting (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='19), equipping S2T ∗M ⊕ T ∗M with the Hermitian inner product  1 ⊕ 1 ⊕ /g−1 ⊕ 1 ⊕ /g−1 ⊕ 1 ⊕ 1 ⊕ 1 ⊕ 1 ⊕ /g−1, the first three terms of Ssub(−L) are formally self adjoint with  respect to the symplectic volume form on S2T ∗M ⊕ T ∗M, and the last term SL of iSsub(−L), multiplied by  ξ−1 has eigenvalues  0, 0, −2κ, −2κ, 2κ, −4(κ − γ), 2(κ − γ), 4(κ − γ), −γ, 2γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='20)  Let P ∈ {−2P, −2W, −L}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Then the quantity ˜β for P is defined as  |ξ|−1 1  2i  �  Ssub(P) −  �  Ssub(P)  �∗�  = ∓ ˜ββ0  at  L±  where β0 = 2κ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Recall that βsup = sup ˜β and βinf = inf ˜β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Then for sufficiently small γ > 0, we have  −2P : βsup = 1 − 2γ  κ ,  βinf = −1 − γ  2κ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  −2W : βsup = 1 + γ  2κ,  βinf = −1 + 2γ  κ ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  −L : βsup = 2 − 2γ  κ ,  βinf = −2 + 2γ  κ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='21)  We again point out the relation between the operators P, W, L and their Fourier transform �P(σ), �  W(σ) and  �L(σ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  Remark C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Given a operator P ∈ {−2P, −2W, −L}, we define �P(σ) := eiσt∗Pe−iσt∗ with σ ∈ C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' The  radial points set of �P (which we still denote by L±) is given by L± = {(rb0, ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' ξ, 0) | ±ξ > 0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Then we have  1  2i  �  Ssub( �P) −  �  Ssub( �P)  �∗�  = 1  2i  �  Ssub(P) −  �  Ssub(P)  �∗�  + σ1(  �  □g,0(σ) − ( �  □g,0(σ))∗  2i  )  at  L±.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  Finally, according to the discussion around (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='59), the calculation of the threshold regularity at radial points  at event horizon for the semiclassical operator h2 �P(h−1z) is equal to that of �P(σ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  References  [1] Lars Andersson, Thomas B¨ackdahl, Pieter Blue, and Siyuan Ma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Stability for linearized gravity on the Kerr spacetime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  arXiv:1903.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='03859.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  [2] Lars Andersson, Dietrich H¨afner, and Bernard F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Whiting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Mode analysis for the linearized Einstein equations on the Kerr  metric: the large a case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' arXiv:2207.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='12952.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  [3] I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Barb˘alat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
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+page_content=' ), 21:319–332, 1974.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  [88] Frank J Zerilli.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Effective potential for even-parity Regge-Wheeler gravitational perturbation equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Physical Review  Letters, 24(13):737, 1970.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  [89] Maciej Zworski.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Resonances for asymptotically hyperbolic manifolds: Vasy’s method revisited.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Spectr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content=' Theory, 6(4):1087–  1114, 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='  Department of Mathematics, Johns Hopkins University, Baltimore, MD 21218, USA  Email address: lhe31@jhu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
+page_content='edu' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FAT4oBgHgl3EQfbR2A/content/2301.08557v1.pdf'}
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+NeRF in the Palm of Your Hand:
+Corrective Augmentation for Robotics via Novel-View Synthesis
+Allan Zhou*
+Stanford
+Moo Jin Kim∗
+Stanford
+Lirui Wang
+MIT CSAIL
+Pete Florence
+Google
+Chelsea Finn
+Stanford
+Abstract
+Expert demonstrations are a rich source of supervision
+for training visual robotic manipulation policies, but imita-
+tion learning methods often require either a large number
+of demonstrations or expensive online expert supervision to
+learn reactive closed-loop behaviors. In this work, we in-
+troduce SPARTN (Synthetic Perturbations for Augmenting
+Robot Trajectories via NeRF): a fully-offline data augmen-
+tation scheme for improving robot policies that use eye-in-
+hand cameras. Our approach leverages neural radiance
+fields (NeRFs) to synthetically inject corrective noise into
+visual demonstrations, using NeRFs to generate perturbed
+viewpoints while simultaneously calculating the corrective
+actions. This requires no additional expert supervision or
+environment interaction, and distills the geometric informa-
+tion in NeRFs into a real-time reactive RGB-only policy.
+In a simulated 6-DoF visual grasping benchmark, SPARTN
+improves success rates by 2.8× over imitation learning
+without the corrective augmentations and even outperforms
+some methods that use online supervision. It additionally
+closes the gap between RGB-only and RGB-D success rates,
+eliminating the previous need for depth sensors. In real-
+world 6-DoF robotic grasping experiments from limited hu-
+man demonstrations, our method improves absolute success
+rates by 22.5% on average, including objects that are tra-
+ditionally challenging for depth-based methods. See video
+results at https://bland.website/spartn.
+1. Introduction
+Object grasping is a central problem in vision-based con-
+trol and is fundamental to many robotic manipulation prob-
+lems.
+While there has been significant progress in top-
+down bin picking settings [21, 34], 6-DoF grasping of ar-
+bitrary objects amidst clutter remains an open problem, and
+is especially challenging for shiny or reflective objects that
+are not visible to depth cameras. For example, the task of
+grasping a wine glass from the stem shown in Figure 1 re-
+*Equal contribution. Correspondence to ayz@stanford.edu.
+Figure 1. SPARTN is an offline data augmentation method for be-
+havior cloning eye-in-hand visual policies. It simulates recovery
+in a demonstration by using NeRFs to render high-fidelity obser-
+vations (right) from noisy states, then generates corrective action
+labels.
+quires precise 6-DoF control (using full 3D translation and
+3D rotation of the gripper) and closed-loop perception of
+a transparent object. Traditional 6-DoF grasping pipelines
+[8, 57] synthesize only one grasp pose and use a motion
+planner to generate a collision-free trajectory to reach the
+grasp [38,40,54,60]. However, the use of open-loop trajec-
+tory execution prevents the system from using perceptual
+feedback for reactive, precise grasping behavior. In this pa-
+per, we study how to learn closed-loop policies for 6-DoF
+object grasping from RGB images, which can be trained
+with imitation or reinforcement learning methods [58].
+Imitation learning from expert demonstrations is a sim-
+ple and promising approach to this problem, but is known
+to suffer from compounding errors [46]. As a result, com-
+plex vision-based tasks can require online expert supervi-
+sion [19, 46] or environment interaction [13, 44], both of
+which are expensive and time-consuming to collect. On the
+other hand, offline “feedback augmentation” methods [14,
+22] can be effective at combating compounding errors, but
+are severely limited in scope and thus far have not been ap-
+1
+arXiv:2301.08556v1  [cs.LG]  18 Jan 2023
+
+NeRFrenderings
+--Demonstration
+Corrective actionplied to visual observations. Other recent works have found
+that using eye-in-hand cameras mounted on a robot’s wrist
+can significantly improve the performance of visuomotor
+policies trained with imitation learning [17,20,35], but still
+do not address the underlying issue of compounding errors.
+We develop an approach that helps address compounding
+errors to improve vision-based policies, while building on
+the success of eye-in-hand cameras.
+To improve imitation learning for quasi-static tasks like
+grasping, we propose a simple yet effective offline data aug-
+mentation technique. For an eye-in-hand camera, the im-
+ages in each demonstration trajectory form a collection of
+views of the demonstration scene, which we use to train
+neural radiance fields (NeRFs) [37] of each scene. Then, we
+can augment the demonstration data with corrective feed-
+back by injecting noise into the camera poses along the
+demonstration and using the demonstration’s NeRF to ren-
+der observations from the new camera pose. Because the
+camera to end-effector transform is known, we can compute
+corrective action labels for the newly rendered observations
+by considering the action that would return the gripper to
+the expert trajectory. The augmented data can be combined
+with the original demonstrations to train a reactive, real-
+time policy. Since the NeRFs are trained on the original
+demonstrations, this method effectively “distills” the 3D in-
+formation from each NeRF into the policy.
+The main contribution of this work is a NeRF-based data
+augmentation technique, called SPARTN (Synthetic Pertur-
+bations for Augmenting Robot Trajectories via NeRF), that
+improves behavior cloning for eye-in-hand visual grasping
+policies. By leveraging view-synthesis methods like NeRF,
+SPARTN extends the idea of corrective feedback augmen-
+tation to the visual domain. The resulting approach can pro-
+duce (i) reactive, (ii) real-time, and (iii) RGB-only policies
+for 6-DoF grasping. The data augmentation is fully offline
+and does not require additional effort from expert demon-
+strators nor online environment interactions. We evaluate
+SPARTN on 6-DoF robotic grasping tasks both in simula-
+tion and in the real world. On a previously-proposed sim-
+ulated 6-DoF grasping benchmark [58], the augmentation
+from SPARTN improves grasp success rates by 2.8× com-
+pared to training without SPARTN, and even outperforms
+some methods that use expensive online supervision. On
+eight challenging real-world grasping tasks with a Franka
+Emika Panda robot, SPARTN improves the absolute aver-
+age success rate by 22.5%.
+2. Related Work
+Robotic Grasping.
+Grasping is a long-studied topic in
+robotics [50]; see the multitude of survey articles for a
+complete review [2, 4, 26]. Most data-driven grasping sys-
+tems focus on learning how to predict some parameterized
+“grasp” (whether a full 6-DoF pose, 2-DoF table-top po-
+Method
+No Depth
+Full 6-DoF
+Closed-Loop
+[34,45]
+[53]
+✓
+✓
+[18,24,40,54]
+✓
+[21,32]
+✓
+✓
+[43]
+✓
+[39]
+✓
+[56,58]
+✓
+✓
+SPARTN (ours)
+✓
+✓
+✓
+Table 1. Comparison of our approach with related grasping work.
+SPARTN is the only approach to learn closed-loop 6-DoF grasping
+policies from only RGB inputs.
+sition, etc.), and leave intermediate motion generation to
+be open-loop, handled either through motion planners or
+simple heuristics, e.g. [12, 34, 42, 43, 52, 57]. Other works
+have trained closed-loop grasping policies [21, 39, 53, 58],
+and bring all the benefits of closed-loop policies: includ-
+ing for example, the ability to avoid obstacles, to perform
+precise grasping without precise calibration, and to react to
+dynamic objects. Additionally, grasping policies are often
+designed for top-down (2- or 3-DoF) grasping [21, 31, 43],
+while 6-DoF grasping typically requires depth or 3D infor-
+mation [39, 42, 53, 58]. In contrast, our method trains a re-
+active 6-DoF grasping policy with only RGB data. See Ta-
+ble 1 for a summary of the assumptions of the most related
+grasping works.
+Imitation learning and data augmentation.
+Behavior
+cloning is known to struggle with covariate shift: small er-
+rors cause imitation policies to fall slightly off of the data
+distribution and it is then difficult to correct the mistake
+back onto the data manifold.
+DAgger [46] and its vari-
+ants [16,23,36] mitigate this issue by obtaining expert cor-
+rections throughout training. Alternatively, DART [29] in-
+jects noise during expert demonstration collection, which
+is especially effective with algorithmic experts but inter-
+feres with human demonstration collection. Previous works
+[14, 22] have injected noise into the low-dimensional sys-
+tem state after data collection (in a fully offline manner),
+but the visual observations are left out, limiting the help-
+fulness of noise injection. Our method can be seen as a vi-
+sual, fully-offline version of noise injection that does not re-
+quire perturbing the expert during demonstration collection,
+using NeRF to synthetically render perturbed states post-
+hoc. Unlike standard image augmentations for policy learn-
+ing [30, 61], our method uses NeRF to learn a 3D model
+of the demonstration scene, which enables us to generate
+high-fidelity novel views for data augmentation. In addi-
+tion, while standard image augmentation approaches do not
+modify the action labels, we leverage hand-eye coordination
+to calculate corrective actions for augmented observations.
+NeRF for Robotics. A number of recent works have in-
+2
+
+vestigated applications of NeRF and related methods in
+robotics, including localization [63], navigation [1], dynam-
+ics modeling [7, 9, 33], reinforcement learning [10], and
+data generation for other learning-based methods [18, 62].
+NeRF-Supervision [62], for example, generates pixel-level
+correspondence to learn dense object descriptors, which are
+useful for manipulation tasks. For grasping, various meth-
+ods have leveraged NeRF [3, 18, 24] for open-loop grasp
+synthesis.
+In contrast, our method uses NeRF offline to
+augment data for grasping and distills a reactive, real-time,
+RGB-only, closed-loop policy.
+3. Methodology
+We now introduce SPARTN, which augments an eye-in-
+hand robot demonstration dataset using NeRF. We first re-
+view preliminaries, then describe a method overview, fol-
+lowed by details of training NeRFs and augmenting correc-
+tive behavior.
+3.1. Preliminaries
+Imitation learning. In imitation learning, we assume ac-
+cess to a dataset of N expert trajectories D = {τ}N
+i=1,
+where each trajectory consists of a sequence of state-action
+pairs, τ = {(sk, ak)}K
+k=0. In purely offline imitation learn-
+ing, there exists no other primary data-collection assump-
+tions other than this dataset D, i.e. no reward labels or on-
+line interactions. A standard method for this offline setting
+is behavior cloning (BC), which trains a policy πθ to mimic
+the expert via supervised learning, by minimizing the ob-
+jective L(θ) = E(s,a)∼D[ℓ
+�
+πθ(s), a
+�
+], where ℓ is some loss
+function in the action space.
+NeRF. Our method uses novel-view synthesis as a build-
+ing block, for which we use Neural Radiance Fields [37].
+For each scene, NeRF performs novel-view synthesis by
+training a scene-specific neural radiance field FΘ from
+a “training set” of posed images {(Ik, Tk)}, where each
+Ik ∈ Rw×h×3, Tk ∈ SE(3). After FΘ is trained, through
+volume rendering NeRF can render new views of a scene
+from any requested pose, which can be summarized as
+I = NeRF-Render(T; FΘ) – in particular, this works best
+“near” the training set of poses. Since we train many NeRFs
+(one per demonstration), we use an accelerated implemen-
+tation of NeRF (Instant-NeRF [41]).
+Corrective Noise Augmentation. A simple method which
+has been shown to improve the robustness of behavior-
+cloned policies is to perform corrective noise augmenta-
+tion [14, 22].
+The idea is to take a nominal trajectory
+τ = {(sk, ak)}K
+k=0 and create a noise-distributed corrective
+trajectory ˜τ = {(˜sk, ˜ak)}K
+k=0, where each sampled state ˜sk
+is a noisy version of the measured state, i.e. ˜sk ∼ sk + ϵ
+and ϵ is sampled noise. To perform corrective feedback aug-
+mentation, ˜ak is chosen such that it will return the state to
+the nominal trajectory: given the true inverse dynamics f −1
+Algorithm 1 SPARTN (Simplified overview)
+Input: D = {τ}N
+i=1 - expert demonstrations
+Output: Augmented transition dataset ˜D
+1: ˜D ← {}
+2: for τ = {(Ik, Tk, ak)}K
+k=1 ∈ D do
+3:
+F τ
+Θ ← NeRF({(Ik, Tk)}K
+k=1) # train NeRF
+4:
+for k ∈ (0, 1, . . . , K) do
+5:
+for i = 1 : Naug do
+6:
+ε ← NoiseDist(SE(3))
+7:
+˜Tk ← Perturb(Tk, ε) # perturb pose
+8:
+˜ak ← CorrectAction(Tk, ˜Tk, ak)
+9:
+˜Ik ← NeRF-Render( ˜Tk, F τ
+Θ)
+10:
+˜D ← ˜D ∪ {(˜Ik, ˜Tk, ˜ak)}
+of the environment, then ˜ak = f −1(˜sk, sk+1), or compute
+the commanded control that can be tracked with a stabiliz-
+ing controller to sk+1. An easy way to parameterize this
+is by choosing the action space of the learned policy to be
+the input to a stabilizing controller, as in [14,22]. Note that
+it is also common in the imitation and reinforcement learn-
+ing literature to apply noise to inputs, which is typically
+interpreted as a way to regularize the policy [27,30]. Mean-
+while, in addition to potential regularization effects, the cor-
+rective noise augmentation has been interpreted to specifi-
+cally reduce compounding errors [14,22], but of course has
+limits if the scale of perturbations is too large or in highly
+non-smooth dynamics regimes. A critical limitation of prior
+works using corrective noise augmentation is that they have
+not been applied to visual observations.
+3.2. Overview: Visual Corrective Augmentation
+Consider the case where an agent receives partial visual
+observations I instead of the full state of the world, and also
+receives direct proprioceptive state measurements, srobot, as
+is commonly the case in robotics. In general, the correc-
+tive augmentation of Sec. 3.1 requires obtaining the visual
+observation ˜Ik for each noisy state ˜sk, which could be ex-
+pensive or impossible in the actual environment.
+An insight of this work is that for eye-in-hand robot
+policies (where the visual observation comes from a cam-
+era mounted on the wrist) in static scenes, we can readily
+generate novel visual observations using novel-view synthe-
+sis (i.e., NeRF) without further interactions in the environ-
+ment. In this setting, a primary subset of the observations
+are posed images (I, T), together with some actions a.
+The key intuition of our method can be grasped by con-
+sidering how to perform visually corrective augmentation
+via NeRF, as illustrated in Figure 2. Noisy states and cor-
+rective actions ( ˜Tk, ˜ak) can be generated for a trajectory
+of posed observations and actions τ = {(Ik, Tk, ak)}K
+k=0
+(Sec. 3.1).
+The key pre-processing step is to train a
+trajectory-specific NeRF F τ
+Θ for each demonstration τ
+(Sec. 3.3). These trajectory-specific NeRFs enable us to
+3
+
+Figure 2. An illustration of how SPARTN creates augmentations from an original demonstration (in reality, this process is repeated for
+every available demonstration). (i): The eye-in-hand demonstration contains posed images {(Ik, Tk)}K
+k=1. (ii): We train a neural radiance
+field (NeRF) of the demonstration scene on the posed images. (iii): We sample perturbations around each pose to simulate noise in
+the demonstration, and calculate the corrective action (in magenta) that would stabilize the trajectory. (iv): We use the NeRF to render
+observations for the perturbed poses. The end result is augmented image-action pairs for improving behavior cloning.
+Figure 3. An overview of the SPARTN training process. A NeRF
+is trained for each of the original demonstrations in D. We use
+these NeRFs to generate visual corrective augmentations for each
+demonstration and collect them in ˜D. The policy πθ can be trained
+on D and ˜D using standard behavior cloning methods.
+render observations ˜Ik for noisy states ˜Tk, completing the
+augmentation process and resulting in visually corrective
+transitions (˜Ik, ˜Tk, ˜ak) (Sec. 3.4). Algorithm 1 and Figure 3
+overview this process.
+3.3. Training NeRFs from Robot Demonstrations
+SPARTN uses novel-view synthesis, in particular NeRF,
+to generate observations ˜Ik for noisy states without environ-
+ment interaction. We train a NeRF F τ
+Θ for each demonstra-
+tion trajectory using the image observations (I1, · · · , IK) ∈
+τ as the training set of views. After training F τ
+Θ, we create
+observations ˜Ik for perturbed robot states ˜Tk by rendering
+the view from the perturbed camera pose using F τ
+Θ.
+An important detail is that the end-effector reference
+frame used for control in demonstrations may differ from
+the reference frame for the camera itself, but through stan-
+dard eye-in-hand calibration we can transform all visual ob-
+servations used for training and augmenting the NeRFs into
+the cameras frame. Given a transform toT from
+k
+which trans-
+forms between two frames, we simply transform all NeRF-
+poses to the world frame: W T C
+k = W T E
+k
+ET C, where W is
+the world frame, E is the end-effector frame which changes
+at each step k, and C is the camera (NeRF) frame stati-
+cally linked to the end-effector frame, and ET C is acquired
+through hand-eye calibration.
+COLMAP camera poses.
+Real-world calibration error
+means that our camera-to-world transforms {W T C
+k }K
+k=1 are
+noisy and we obtain higher-quality NeRFs by using cam-
+era poses estimated by COLMAP [48,49]. Up to noise and
+a scale factor β, the only difference from the world frame
+camera transforms is that COLMAP uses an arbitrary ref-
+erence frame V ̸= W. We denote the COLMAP outputs
+{V HC
+k }K
+k=1, using H instead of T because of the difference
+in scale. We now introduce notation to separate the rotation
+and translation components of a transform:
+aT b :=
+�
+Rot
+�aT b�
+, Trans
+�aT b��
+(1)
+Since we train the NeRFs on COLMAP’s camera poses, we
+must convert perturbed camera poses W T ˜
+C
+k to COLMAP’s
+frame in order to render the observations. In other words,
+we must call NeRF-Render(V H ˜
+C
+k , F τ
+Θ), where:
+V H
+˜
+C
+k =
+�
+Rot
+�
+V T
+˜
+C
+k
+�
+, β Trans
+�
+V T
+˜
+C
+k
+��
+(2)
+V T
+˜
+C
+k = V T W W T
+˜
+C
+k .
+(3)
+Both β and
+V T W
+can be estimated from the pairs
+{(W T C
+k , V HC
+k )}K
+k=1, as described in Appendix D.2.
+Static Scene Assumption. An additional consideration for
+robotic manipulation is that the standard NeRF formulation
+4
+
+Trajectory
+F
+of Posed Images
+T
+Ik
+Eye-in-hand
+cameraDemonstrations D
+NeRF training
+Corrective augmentation D
+(1)
+T1
+F(2)
+T2
+0
+7(3)
+T3
+0
+Train(π)Figure 4. An illustration of how the gripper is inserted into the
+result of the NeRF rendering process. Gray regions indicate pixels
+being masked out by the binary gripper mask M ∈ {0, 1}w×h,
+which denotes the pixels where the gripper is located in all frames.
+assumes a static scene, while manipulation tasks such as
+grasping will usually move objects in the scene. To address
+this, we apply the NeRF training and augmentation process
+to only the subsets of each demonstration trajectory where
+no robot-object interaction occurs, instead of all timesteps.
+For grasping, a simple and effective heuristic is to only ap-
+ply SPARTN to the portions of each demonstration before
+the expert closes the gripper.
+Masking Out the Robot Gripper.
+The robot’s gripper
+is often within the view of an eye-in-hand camera, which
+breaks the static scene assumption.
+To address this, we
+leverage that the gripper is in the same location in each im-
+age assuming the camera is rigidly mounted and the gripper
+is open. Further, since NeRF is trained per-ray, we can sim-
+ply mask pixels from training images. We construct a single
+binary mask M ∈ {0, 1}w×h, where a 1 indicates gripper
+pixels to mask out. Figure 4 shows how we use the same
+mask to splice the gripper back into each NeRF rendering
+output, ˜Ik ← ¬M ⊙ ˜Ik +M ⊙Ik, where ⊙ is element-wise
+multiplication broadcasted across the color channels.
+NeRF Quality.
+Even when the training views from a
+demonstration are suboptimal for NeRF training, SPARTN
+benefits from the fact that our augmentation is local and
+the perturbations ε are typically small, so we only need the
+NeRF to generalize in a small region around the demon-
+stration trajectory. As we will verify in the experiments,
+SPARTN can be effective even with a limited number of
+training views compared to other NeRF applications.
+3.4. NeRFing Corrective Noise Augmentation
+Given a visual augmentation model (Sec. 3.3), we can
+adapt methods for corrective augmentation (Sec. 3.1) into
+the visual domain. Our goal is to create noise-distributed
+corrective transitions {(˜Ik, ˜Tk, ˜ak)}. First, we describe this
+simply in the global frame. In order to sample from the
+noise-distributed corrective trajectory, one can first apply
+noise to the measured end-effector pose, ˜Tk := Tkε, where
+ε ∼ NoiseDist(SE(3)) is a randomly sampled rotation and
+translation.
+The high-fidelity perturbed image ˜Ik corre-
+sponding to ˜Tk can then be rendered using the trajectory-
+specific NeRF F τ
+Θ, without requiring access to the actual
+environment. For the actions, the simplest case is when they
+are inputs to a global-frame stabilizing Cartesian controller
+controller [25], in which case ˜ak = ak = ˆTk will provide
+stabilization to the nominal trajectory, where ˆTk is the de-
+sired pose sent to the lower-level controller.
+Corrective Relative Actions.
+As is common in prior
+works, we observe better performance by parameteriz-
+ing the learned policy as a relative rather than global ac-
+tion. Consider the as-discussed global-frame version, with
+(1) observations as a global-frame measured SE(3) end-
+effector pose W T E
+k , where W refers to world-frame, and
+E to the end-effector frame at timestep k, and (2) action as
+a global-frame desired SE(3) end-effector pose W T ˆ
+E
+k . To
+switch to a relative action space, we adjust the action to
+ak = ET
+ˆ
+E
+k = (W T E
+k )−1 W T
+ˆ
+E
+k = ET W
+k
+W T
+ˆ
+E
+k .
+(4)
+To additionally formulate the corrective noise augmentation
+in the relative frame, we consider the SE(3)-noise ε as trans-
+forming from the noisy end-effector frame to the measured
+end-effector frame, i.e.
+ET ˜
+E := ε. This accordingly ad-
+justs the observation as W T ˜
+E
+k = W T E
+k
+ET ˜
+E
+k = W T E
+k ε and
+the relative action as:
+˜ak =
+˜
+ET
+ˆ
+E
+k = (W T E
+k
+ET
+˜
+E
+k )−1 W T
+ˆ
+E
+k = ε−1 ak.
+(5)
+Concisely, this amounts to post-pending ε to the measured
+pose, and pre-prending ε−1 to the un-noised relative action.
+Non-SE(3) actions. Thus far we have assumed that the ac-
+tions a ∈ SE(3) only command desired pose transforms.
+In practice, the action space may contain additional dimen-
+sions, for example to open and close the gripper itself. The
+SPARTN augmentation does not concern these aspects of
+the action, so the corrective actions simply copy the origi-
+nal action for non-SE(3) dimensions.
+Summary. Figure 2 summarizes our augmentation proce-
+dure: given a demonstration dataset D, we first train all
+the neural radiance fields {F τ
+Θ}τ∈D and save the weights
+to disk. We then augment each transition in the original
+dataset with Naug noisy corrective transitions produced us-
+ing the process we described above, and save these transi-
+tions into an augmented dataset ˜D. Appendix Algorithm 2
+describes the precise augmentation procedure in detail.
+After the augmented dataset has been created, it can sim-
+ply be combined with the original dataset to augment BC
+training. Various sampling strategies are possible, but in our
+experiments we simply construct mini-batches for BC train-
+ing by sampling from the original and augmented datasets
+with equal probability.
+4. Simulation Experiments
+We evaluate SPARTN and related approaches in the sim-
+ulated 6-DoF grasping benchmark first introduced in [58],
+which features a simulated Franka Emika Panda arm with
+5
+
+M o Ik
+-M O NeRF-Render(Tk, F)
++Figure 5. Example tasks in the simulated 6-DoF grasping bench-
+mark, first introduced in [58]. There are ∼ 1,500 ShapeNet ob-
+jects in training demonstrations, and held out YCB objects for
+evaluation. A camera mounted to the Franka Panda robot’s arm
+provides observations, and the policy controls the 6-DoF end-
+effector pose.
+a parallel-jaw gripper. Objects are placed on a table and
+must be lifted above a height threshold in a successful grasp.
+Policies receive either RGB, RGBD, or point cloud observa-
+tions from a wrist camera, and control the gripper by com-
+manding relative pose changes in 6-DoF end-effector space.
+4.1. Data Collection and Evaluation Protocol
+We follow the training and evaluation procedure
+from [58]. The training dataset includes 2,500 demonstra-
+tions of grasping 1,500 ShapeNet [6] objects. Demonstra-
+tions are up to 20 timesteps and are generated by trajectory
+optimization to precomputed grasps from the ACRONYM
+dataset [11]. Policies are evaluated on grasping held-out ob-
+jects from the YCB [5] or ShapeNet datasets. Though it is
+more “out of distribution” relative to the training objects,
+the YCB evaluation is more realistic for tabletop grasping
+scenarios (ShapeNet includes a wide variety of objects, e.g.
+airplanes and bicycles). Each held-out object is evaluated
+ten times, each with a different initial robot configuration
+and object’s initial pose.
+4.2. Comparisons
+We compare SPARTN against other approaches ranging
+from simple behavior cloning to online-supervised imita-
+tion and reinforcement learning:
+Behavior Cloning (BC): Trains policies via supervised
+learning on the demonstration dataset.
+DART [29]: Introduces a modified demonstration setup
+where a continuous Gaussian noise is injected into the
+robot’s state during the expert demonstration, then trains
+policies on the modified demonstrations with BC.
+Homography Augmentation (HA): A simplification of
+SPARTN where perturbations can be 3D rotations, but not
+1Note that the “BC” results in [58] actually use DART.
+Supervision
+Method
+Input
+YCB SR(%)
+SN SR(%)
+Offline
+BC
+RGB
+28.9 ± 2.4
+57.4 ± 0.2
+DART†
+RGB
+51.2 ± 3.2
+57.8 ± 2.2
+RGBD
+46.3
+45.3
+Point cloud
+65.6
+73.6
+HA
+RGB
+29.7 ± 2.4
+57.5 ± 0.8
+SPARTN (ours)
+RGB
+74.7 ± 2.4
+66.9 ± 1.6
+Online*
+DAgger
+RGB
+52.3
+52.1
+RGBD
+67.1
+60.4
+Point cloud
+77.2
+75.8
+GA-DDPG
+Point cloud
+88.2
+91.3
+Table 2. Grasping success rates (SR) on held-out objects from
+YCB [5] or ShapeNet (SN) [6] in a simulated 6-DoF grasping
+benchmark [58].
+We bold the best offline RGB-only results,
+though we include online and non-RGB methods for comparison.
+Online* requires additional environment interactions, while Of-
+fline only uses demonstration data. DART† is offline but requires
+a special demonstration collection setup. SPARTN outperforms
+other offline RGB-only methods, while RL (GA-DDPG) performs
+best overall while requiring millions of interactions. We calcu-
+late average success rates and standard error over 4 random seeds.
+Italicized success rates were reported in prior work1 [58].
+Method
+Image aug.
+YCB SR(%)
+BC
+Without
+25.3 ± 1.6
+With
+28.9 ± 2.4
+DART
+Without
+51.2 ± 3.2
+With
+48.6 ± 2.8
+HA
+Without
+23.9 ± 4.0
+With
+29.7 ± 2.4
+SPARTN
+Without
+68.3 ± 3.6
+With
+74.7 ± 2.4
+Table 3. Ablating the effect of standard image augmentation on
+each method for RGB policies. Average success rates and standard
+errors are calculated over four seeds.
+translations, of the camera. For pure rotation, we can cal-
+culate the homography transform for rendering the rotated
+view without NeRF. Augmented actions are computed sim-
+ilarly to SPARTN.
+DAgger [46]: An online-supervised method where a policy
+is first trained using BC on offline demos, and then the ex-
+pert provides action labels for states from the policy’s roll-
+outs throughout further policy training.
+GA-DDPG [58]: Jointly trains policies via BC and fine-
+tunes them with reinforcement learning (DDPG [51]).
+Of the methods considered, DAgger and GA-DDPG re-
+quire online environment interaction and supervision from
+an expert or reward function, respectively. The other meth-
+ods, including SPARTN, train only on pre-collected demon-
+stration datasets.
+Since the DART data collection setup
+noisily perturbs the robot as the expert demonstrates, DART
+can interfere with human expert demonstration, though this
+challenge does not arise here with our simulated expert.
+6
+
+4.3. Training Details
+We follow the RGB architectural and training hyper-
+parameters of [58], with policies consisting of a ResNet-
+18 [15] image encoder followed by an MLP. We apply ran-
+dom crop and color jitter to the training images. We re-use
+the same training BC hyperparameters for SPARTN. Ap-
+pendix C.1 describes the complete training details.
+For SPARTN, we create Naug = 100 augmented transi-
+tions from each original transition and save the augmented
+dataset to disk before BC training. To sample the pertur-
+bations ε ∼ NoiseDist(SE(3)), we parameterize the rota-
+tions in terms of Euler angles (φ, θ, ϕ) and uniformly sam-
+ple both rotation and translation parameters:
+(φ, θ, ϕ), (tx, ty, tz) ∼ U(−α, α), U(−β, β)
+(6)
+ε := (R(φ, θ, ϕ), (tx, ty, tz))
+(7)
+In simulation, we set α = 0.2 radians and β = 3 mm.
+Following the DART hyperparameters in [58], we only aug-
+ment timesteps 5 − 13 of each demonstration. Appendix B
+contains samples of SPARTN’s perturbed observations ren-
+dered via NeRF.
+4.4. Results
+Table 2 shows grasping success rates on held-out objects
+from either the YCB or ShapeNet (SN) datasets. SPARTN
+significantly outperforms the other two offline-supervised
+approaches for RGB inputs. Since DART injects noise dur-
+ing expert collection, the amount of augmentation is lim-
+ited by the number of demonstrations the expert can col-
+lect. Meanwhile, SPARTN can cheaply generate an arbi-
+trary number of augmented examples on top of the exist-
+ing demonstration dataset, leading to more data diversity
+without any additional effort from the expert. See the sup-
+plementary material for videos of rollouts for SPARTN and
+BC policies, with success and failure cases.
+Naive BC performs relatively well on ShapeNet evalua-
+tion, likely because the evaluation set is “in-distribution”.
+Correspondingly, corrective methods (DART, SPARTN)
+achieve smaller improvements over BC on ShapeNet and
+larger improvements on YCB.
+On YCB, SPARTN’s performance is closer to RL (GA-
+DDPG) than to other offline methods. It actually outper-
+forms DAgger for RGB policies, perhaps because DAgger’s
+data distribution changes online presenting a more challeng-
+ing learning problem. Notably, on YCB SPARTN outper-
+forms DART with point clouds and is comparable to DAg-
+ger with point clouds. Since SPARTN itself only requires
+RGB images during both NeRF training and policy train-
+ing, it significantly closes the gap between the performance
+of RGB-only and the best depth-based methods. Since con-
+sumer depth cameras can struggle with common thin and
+reflective items [62], improving RGB-only grasping can en-
+able more robust grapsing of such objects.
+Figure 6. Real-world grasping environments. In each environ-
+ment, the task is to grasp the labeled target object in a particular
+way. The target objects have various geometric shapes and exhibit
+a diverse range of characteristics, including reflectiveness, trans-
+parency, and radial symmetry.
+Target Object
+# Demos
+BC SR(%)
+SPARTN SR(%)
+Banana
+14
+55
+75
+Thin box
+20
+35
+65
+Steel water bottle
+15
+20
+40
+Wine glass
+25
+70
+90
+Lubriderm bottle
+17
+25
+60
+White tape roll
+15
+40
+45
+Tupperware
+20
+40
+75
+Fork
+20
+25
+40
+Average
+--
+38.75
+61.25
+Table 4. Success rates (SR) of behavior cloning (BC) and SPARTN
+6-DoF grasping policies on a suite of eight real-world target ob-
+jects. SPARTN outperforms BC in every environment, achieving
+an average absolute performance boost of 22.5%. Each success
+rate is computed over 20 trials.
+Effect of Image Augmentations. We try ablating the effect
+of standard image augmentations (random crop and color
+jitter) for the BC, DART, HA, and SPARTN methods. Ta-
+ble 3 shows that the image augmentations have a small ef-
+fect on the performance of most methods, relative to the
+larger effect of SPARTN’s corrective augmentations.
+5. Real-World Experiments
+The simulation benchmark results show that SPARTN
+can improve grasping generalization in imitation learning
+without online supervision. Here, we verify that SPARTN
+can enable real-world robotic grasping of challenging ob-
+jects from limited human demonstration. See the website in
+the supplement for video results.
+5.1. Experimental details
+Robot setup. The robotic manipulator is a Franka Emika
+Panda robot arm with a wrist-mounted consumer grade we-
+bcam. Policies take images of size 256 × 256 as input and
+output a 6-DoF action representing the desired change in
+end-effector position and orientation, as well as a binary
+7
+
+Banana
+Thin box
+Steelwaterbottle
+Wine glass
+Lubriderm bottle
+White tape roll
+Tupperware
+Forkopen/close gripper action. We use a Cartesian impedance
+controller to command the pose changes at a frequency of 4
+Hz. We task the robot to grasp a target object in eight dif-
+ferent environments depicted in Figure 6. The target objects
+include natural shapes that are common in the real world
+and exhibit a diverse range of attributes, such as reflective-
+ness, transparency, and radial symmetry.
+Comparisons and Evaluation. In each environment, we
+collect a small number of expert grasping demonstrations
+with a virtual reality controller. Because DART is difficult
+to use with human demonstration collection, we compare
+SPARTN to a vanilla BC policy trained on the same set of
+demonstrations. Policies are evaluated with the same ob-
+jects seen in the demonstrations. Initial object and robot
+configurations are randomized during both data collection
+and evaluation.
+Training. To improve NeRF quality for SPARTN, we pro-
+gram the robot to collect a few images of the scene from
+a fixed set of poses before the start of each demonstration.
+This automatic process is only used to improve COLMAP’s
+pose estimation and the subsequent NeRF training.
+For
+SPARTN, we generate Naug
+=
+50 augmented transi-
+tions from each original transition in the demonstrations.
+We sample perturbations ϵ according to Eq. 6 with α =
+0.05 radians and β = 0.4 mm. Appendix D.2 describes
+COLMAP and NeRF training in more detail, and Ap-
+pendix B shows sample SPARTN observations rendered by
+NeRF. Aside from using the augmented dataset, SPARTN
+policies are trained using the same BC architecture and
+training hyperparameters described in Appendix D.1.
+5.2. Results
+Table 4 shows grasping success rates in the eight real-
+world environments. Quantitatively, SPARTN policies out-
+perform the baseline BC policies across the board, on aver-
+age achieving an absolute 22.5% increase in success rate.
+Figure 7 shows qualitative differences in performance
+between the BC and SPARTN policies. SPARTN generally
+exhibits more reactive behaviors than the baseline policy: it
+navigates towards the target object better while occasionally
+avoiding obstacles, executes actions with greater precision,
+and even reattempts the grasp more successfully after an
+initial miss. In some cases, the differences are stark: for in-
+stance, SPARTN may successfully move toward and grasp
+the target object while the baseline fails to even reach it. We
+present further analysis of policy rollouts in Appendix D.3,
+showing how SPARTN qualitatively performs better than
+BC even in cases where both methods fail to grasp the tar-
+get object. The supplementary videos of real-world pol-
+icy rollouts illustrate all of these differences, revealing that
+SPARTN induces important reactive closed-loop behaviors
+that enable the manipulator to successfully execute grasps
+in the real world.
+Figure 7.
+Sample real-world policy rollouts illustrating how
+SPARTN succeeds (green) in cases where BC fails (red). Top
+left: While BC fails to reach the steel bottle, SPARTN success-
+fully reaches and grasps it. Top right: While BC collides into the
+orange bottle (a distractor object), SPARTN takes a more rounded
+path to avoid it before grasping the white Lubriderm bottle. Bot-
+tom left: BC fails to recover after a missed grasp, while SPARTN
+successfully reattempts the grasp after failing the first time. Bot-
+tom right: SPARTN operates with higher precision than BC and
+successfully completes a difficult fork grasping task.
+6. Conclusion
+We introduce SPARTN, which augments eye-in-hand
+demonstrations with perturbed visual observations and cor-
+rective actions.
+Our augmentation leverages novel-view
+synthesis, in particular NeRF, to produce these augmenta-
+tions during training-time. SPARTN can improve behavior
+cloning training of robust, real-time, and closed-loop 6-DoF
+visual control policies. We show that SPARTN-trained poli-
+cies outperform other offline-supervised methods in a sim-
+ulated 6-DoF grasping generalization benchmark. Our poli-
+cies can also perform on par with imitation methods that re-
+quire depth information and online supervision. We verify
+that SPARTN can train policies to grasp a variety of objects
+in the real world from limited human demonstrations.
+Despite its strong performance, SPARTN also has some
+limitations. First, SPARTN is limited to tasks with static
+scenes like grasping: extending to a broader set of manip-
+ulation tasks would require effective view synthesis for dy-
+namic scenes. Second, training a neural radiance field for
+every demonstration before training is computationally ex-
+pensive, a limitation that may be mitigated through amor-
+tized NeRF models [55, 59, 64]. Finally, the noise distri-
+bution must be tuned for a given platform, e.g. it is tuned
+separately for the simulated and real experiments. An in-
+teresting direction for future work is to use policy actions
+to generate the noise, which would result in a fully offline
+variant of DAgger [46] that uses NeRF as a simulator.
+8
+
+Time
+Time
+BC
+SPARTN
+Time
+Time
+BC
+SPARTN7. Acknowledgements
+We thank Jimmy Wu, Kaylee Burns, Bohan Wu, and
+Suraj Nair for technical advice on various aspects of the real
+robot setup, and Archit Sharma for helpful conceptual dis-
+cussions. This project was funded by ONR grant N00014-
+21-1-2685. AZ acknowledges the support of the NSF Grad-
+uate Research Fellowship. CF is a fellow of the CIFAR
+Learning in Machines and Brains Program.
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+11
+
+A. Full Algorithm
+Algorithm 2 SPARTN: Corrective Augmentation via NeRF
+Input: D = {τ}N
+i=1 - expert demonstrations
+Input: ET C - Transform between camera and end-effector
+Input: M ∈ {0, 1}w×h - mask for robot gripper
+Output: Augmented transition dataset ˜D
+1: ˜D ← {}
+2: for τ = {(Ik, W T E
+k ), (ET ˆ
+E
+k )}K
+k=1 ∈ D do
+3:
+# EE pose → camera pose
+4:
+W T C
+k ← W T E
+k
+ET C, for k = 0, · · · , K
+5:
+{V T C
+k }K
+k=1 ← COLMAP(I0, · · · , IK)
+6:
+β ← SOLVE(Eq. 8) # Estimate β
+7:
+V T W
+k
+= V T C
+k (W T C
+k )−1, for k = 0, · · · , K
+8:
+# train NeRF
+9:
+F τ
+Θ ← NeRF({(Ik, V T C
+k )}K
+k=1, M)
+10:
+for k ∈ (0, 1, . . . , K) do
+11:
+for i = 1 : Naug do
+12:
+ε ← NoiseDist(SE(3))
+13:
+W T ˜
+E
+k ←
+W T E
+k ε # perturbed end-effector pose
+14:
+˜
+ET ˆ
+E
+k ← ε−1 ET ˆ
+E
+k # corrective relative action
+15:
+W T ˜
+C
+k ← W T ˜
+E
+k
+ET C # perturbed camera pose
+16:
+V T ˜
+C
+k ← V T W
+k
+W T ˜
+C
+k
+17:
+˜Ik ← NeRF-Render(V T ˜
+C
+k ; F τ
+Θ)
+18:
+# splice gripper into rendered image
+19:
+˜Ik ← ¬M ⊙ ˜Ik + M ⊙ Ik
+20:
+˜D ← ˜D ∪ {(˜Ik, W T ˜
+E
+k ), ( ˜
+ET ˆ
+E
+k )}
+B. Sample Perturbed Observations
+As described in the main text, image observations ˜I for
+perturbed camera poses are rendered via novel-view synthe-
+sis techniques, in particular NeRF. The section “Example
+NeRF renders from augmented poses” in the supplemen-
+tary website shows videos of the rendering output for both
+simulation and real-world augmentation.
+C. Simulation Details
+C.1. Training for Simulation Experiments
+We train all of our RGB policies (SPARTN, DART, and
+HA) closely following the architectural and optimization
+hyperparameters of [58]. The image encoder is a ResNet-
+18 [15] pretrained on ImageNet [47], followed by a 3-layer
+MLP with 512 hidden units each and ReLU activations. The
+MLP outputs the predicted action as a 6-D vector of pre-
+dicted translations and Euler angles. The final layer has
+a tanh activation to scale the output between [−1, 1], then
+each dimension is scaled to match the environment’s action
+bounds.
+For behavior cloning, we use the same 3D point-
+matching loss as [58].
+We optimize the objective using
+the Adam optimizer with batch size 100 for 100, 000 steps.
+We train only the MLP (and not the pre-trained ResNet) for
+the first 20, 000 steps, since “freeze-then-train” fine-tuning
+techniques have been shown to be more robust [28]. After
+20, 000 steps, the ResNet is unfrozen and the entire network
+is trained as usual.
+C.2. NeRF Training
+We use Instant-NGP [41] to train a NeRF for each of the
+2,500 demonstration scenes. We train each NeRF for 3,500
+steps, which takes 30 seconds on an NVIDIA GeFORCE
+RTX 2080 Ti. To reduce total training time, we train the
+NeRFs in parallel: with 4 GPUs, we can train all 2,500
+NeRFs in ∼ 7 hours. Because camera calibration is exact in
+simulation, we use the world-frame camera poses given by
+calibration instead of COLMAP to train NeRF.
+D. Real-World Details
+D.1. Policy Training
+For both BC and SPARTN, the image encoder for each
+policy is a four-layer convolutional network followed by
+two feedforward layers, and uses batch normalization and
+ReLU activation functions. The image embedding is fed
+into a three-layer ReLU MLP policy head, which outputs
+the predicted action. We train the policy from scratch with
+mean squared error, using Adam with learning rate 5×10−4
+and a batch size of 64. No image augmentations like ran-
+dom crop are applied.
+D.2. COLMAP and NeRF Training
+In the real world, we train our NeRF using cam-
+era poses {V HC
+k }K
+k=1 estimated by COLMAP [48, 49].
+From the camera calibration, we also have pose transforms
+{W T C
+k }K
+k=1 which are too noisy to train a good NeRF from,
+but will allow us to convert between world and COLMAP
+frames. Conversion is necessary because we want to ren-
+der from perturbed camera poses W T ˜
+C, but we must call
+NeRF-Render using the transform V H ˜
+C (COLMAP’s ref-
+erence frame) instead. From Eq. 2, we can do this conver-
+sion if we have the transform V T W and the scale factor β
+which distinguishes T and H.
+We can estimate both quantities using the (W T C
+k , V HC
+k )
+pairs in each demonstration.
+Recalling that
+aT b
+=
+(bT a)−1, we have for any j ̸= k:
+β Trans
+�CT W
+j
+W T C
+k
+�
+= Trans
+�CHV
+j
+V HC
+k
+�
+(8)
+This leads to an overconstrained linear system in one un-
+known β, which can be solved with linear regression. After
+solving for β, we can rescale from H → T:
+V T C
+k :=
+�
+Rot
+�CHV
+k
+�
+, (1/β) Trans
+�CHV
+k
+��
+(9)
+12
+
+Figure 8.
+Sample real-world policy rollouts illustrating cases
+where both BC and SPARTN policies fail. Top left: BC fails
+to reach the thin box, while SPARTN successfully reaches it and
+nearly completes the grasp. Top right: While BC completely
+misses the white tape roll, SPARTN makes contact but fails to
+grasp it. Bottom left: BC fails to reach the steel water bottle,
+while SPARTN contacts it but fails to grasp it. Bottom right:
+BC reaches the Lubriderm bottle over but knocks it over before
+grasping it, while SPARTN nearly completes the grasp as the bot-
+tle barely slips away from the robot’s fingers.
+Now solving for V T W is simple:
+V T W
+k
+= V T C
+k (W T C
+k )−1.
+(10)
+In theory, V T W is constant for all k, but in practice there
+is noise from both COLMAP and the real-world pose esti-
+mates, so we simply compute V T W
+k
+separately for each k.
+D.3. Additional Qualitative Analysis
+Figure 8 shows sample evaluation trials where both
+SPARTN and BC policies fail, given the same initial ob-
+ject and end-effector configurations.
+Even in these fail-
+ure cases, SPARTN qualitatively performs better than BC,
+nearly grasping the object in some cases, and dropping the
+object soon after grasping in other cases. See the videos on
+the supplemental website for additional qualitative results.
+D.4. Real-World Environments
+The real-world environments shown in Figure 6 each in-
+clude a single target object to grasp among other distractor
+objects. The initial positions of all objects and the robotic
+end-effector are randomized while collecting expert demon-
+strations and during test time. The initial orientations of the
+objects are also randomized such that all policies must per-
+form end-effector orientation control to complete the task.
+To make a fair comparison, we evaluate different policies
+given the same set of initial configurations (as shown in
+Figure 7, Figure 8, and the videos on the supplemental web-
+site); we do this by resetting the robot and environment to
+the same conditions and alternating the rollouts of differ-
+ent policies before moving on to the next test configuration.
+Rollouts are considered successful if the robot grasps the
+target object and lifts it steadily into the air. We describe
+the details of each environment below:
+• Banana: The environment includes an artificial ba-
+nana resting on a white ceramic plate, a red cup, an ar-
+tificial avocado, an artificial tomato, and a plastic fork
+and knife resting on a blue cloth.
+• Thin box: The environment includes a thin brown box
+resting on a thicker white toy box.
+• Steel water bottle: The environment includes a steel
+water bottle, a red water bottle, and an aluminum can.
+The steel water bottle is highly reflective.
+• Wine glass: The environment includes a wine glass,
+red water bottle, and blue water bottle all placed
+upside-down and leaning against a brown box. The
+wine glass is transparent.
+• Lubriderm bottle: The environment includes a white
+Lubriderm bottle, an orange Neutrogena bottle, and a
+small white container of moisturizer.
+• White tape roll: The environment includes a roll of
+white tape resting on a roll of blue tape, as well as a
+pair of black scissors. The rolls of tape are radially
+symmetric.
+• Tupperware: The environment includes a plastic tup-
+perware container with a red lid, as well as three artifi-
+cial food items: a donut, a cookie, and a slice of pizza.
+• Fork: The environment includes a fork resting on top
+of a plastic tupperware container. The fork’s location
+and orientation relative to the surface of the tupper-
+ware container is fixed, but the position of the entire
+assembly is randomized.
+13
+
+Time
+Time
+BC
+SPARTN
+Time
+Time
+BC
+SPARTN
\ No newline at end of file
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new file mode 100644
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@@ -0,0 +1,700 @@
+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFAT4oBgHgl3EQfbR0j/content/2301.08556v1.pdf,len=699
+page_content='NeRF in the Palm of Your Hand:  Corrective Augmentation for Robotics via Novel-View Synthesis  Allan Zhou*  Stanford  Moo Jin Kim∗  Stanford  Lirui Wang  MIT CSAIL  Pete Florence  Google  Chelsea Finn  Stanford  Abstract  Expert demonstrations are a rich source of supervision  for training visual robotic manipulation policies, but imita-  tion learning methods often require either a large number  of demonstrations or expensive online expert supervision to  learn reactive closed-loop behaviors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFAT4oBgHgl3EQfbR0j/content/2301.08556v1.pdf'}
+page_content=' In this work, we in-  troduce SPARTN (Synthetic Perturbations for Augmenting  Robot Trajectories via NeRF): a fully-offline data augmen-  tation scheme for improving robot policies that use eye-in-  hand cameras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFAT4oBgHgl3EQfbR0j/content/2301.08556v1.pdf'}
+page_content=' Our approach leverages neural radiance  fields (NeRFs) to synthetically inject corrective noise into  visual demonstrations, using NeRFs to generate perturbed  viewpoints while simultaneously calculating the corrective  actions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFAT4oBgHgl3EQfbR0j/content/2301.08556v1.pdf'}
+page_content=' This requires no additional expert supervision or  environment interaction, and distills the geometric informa-  tion in NeRFs into a real-time reactive RGB-only policy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFAT4oBgHgl3EQfbR0j/content/2301.08556v1.pdf'}
+page_content='  In a simulated 6-DoF visual grasping benchmark, SPARTN  improves success rates by 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFAT4oBgHgl3EQfbR0j/content/2301.08556v1.pdf'}
+page_content='8× over imitation learning  without the corrective augmentations and even outperforms  some methods that use online supervision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFAT4oBgHgl3EQfbR0j/content/2301.08556v1.pdf'}
+page_content=' It additionally  closes the gap between RGB-only and RGB-D success rates,  eliminating the previous need for depth sensors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFAT4oBgHgl3EQfbR0j/content/2301.08556v1.pdf'}
+page_content=' In real-  world 6-DoF robotic grasping experiments from limited hu-  man demonstrations, our method improves absolute success  rates by 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFAT4oBgHgl3EQfbR0j/content/2301.08556v1.pdf'}
+page_content='5% on average, including objects that are tra-  ditionally challenging for depth-based methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFAT4oBgHgl3EQfbR0j/content/2301.08556v1.pdf'}
+page_content=' See video  results at https://bland.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFAT4oBgHgl3EQfbR0j/content/2301.08556v1.pdf'}
+page_content='website/spartn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFAT4oBgHgl3EQfbR0j/content/2301.08556v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFAT4oBgHgl3EQfbR0j/content/2301.08556v1.pdf'}
+page_content=' Introduction  Object grasping is a central problem in vision-based con-  trol and is fundamental to many robotic manipulation prob-  lems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFAT4oBgHgl3EQfbR0j/content/2301.08556v1.pdf'}
+page_content='  While there has been significant progress in top-  down bin picking settings [21, 34], 6-DoF grasping of ar-  bitrary objects amidst clutter remains an open problem, and  is especially challenging for shiny or reflective objects that  are not visible to depth cameras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFAT4oBgHgl3EQfbR0j/content/2301.08556v1.pdf'}
+page_content=' For example, the task of  grasping a wine glass from the stem shown in Figure 1 re-  Equal contribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFAT4oBgHgl3EQfbR0j/content/2301.08556v1.pdf'}
+page_content=' Correspondence to ayz@stanford.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFAT4oBgHgl3EQfbR0j/content/2301.08556v1.pdf'}
+page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFAT4oBgHgl3EQfbR0j/content/2301.08556v1.pdf'}
+page_content='  Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFAT4oBgHgl3EQfbR0j/content/2301.08556v1.pdf'}
+page_content=' SPARTN is an offline data augmentation method for be-  havior cloning eye-in-hand visual policies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFAT4oBgHgl3EQfbR0j/content/2301.08556v1.pdf'}
+page_content=' It simulates recovery  in a demonstration by using NeRFs to render high-fidelity obser-  vations (right) from noisy states, then generates corrective action  labels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFAT4oBgHgl3EQfbR0j/content/2301.08556v1.pdf'}
+page_content='  quires precise 6-DoF control (using full 3D translation and  3D rotation of the gripper) and closed-loop perception of  a transparent object.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFAT4oBgHgl3EQfbR0j/content/2301.08556v1.pdf'}
+page_content=' Traditional 6-DoF grasping pipelines  [8, 57] synthesize only one grasp pose and use a motion  planner to generate a collision-free trajectory to reach the  grasp [38,40,54,60].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFAT4oBgHgl3EQfbR0j/content/2301.08556v1.pdf'}
+page_content=' However, the use of open-loop trajec-  tory execution prevents the system from using perceptual  feedback for reactive, precise grasping behavior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFAT4oBgHgl3EQfbR0j/content/2301.08556v1.pdf'}
+page_content=' In this pa-  per, we study how to learn closed-loop policies for 6-DoF  object grasping from RGB images, which can be trained  with imitation or reinforcement learning methods [58].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFAT4oBgHgl3EQfbR0j/content/2301.08556v1.pdf'}
+page_content='  Imitation learning from expert demonstrations is a sim-  ple and promising approach to this problem, but is known  to suffer from compounding errors [46].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFAT4oBgHgl3EQfbR0j/content/2301.08556v1.pdf'}
+page_content=' As a result, com-  plex vision-based tasks can require online expert supervi-  sion [19, 46] or environment interaction [13, 44], both of  which are expensive and time-consuming to collect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFAT4oBgHgl3EQfbR0j/content/2301.08556v1.pdf'}
+page_content=' On the  other hand, offline “feedback augmentation” methods [14,  22] can be effective at combating compounding errors, but  are severely limited in scope and thus far have not been ap-  1  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFAT4oBgHgl3EQfbR0j/content/2301.08556v1.pdf'}
+page_content='08556v1  [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFAT4oBgHgl3EQfbR0j/content/2301.08556v1.pdf'}
+page_content='LG]  18 Jan 2023  NeRFrenderings  --Demonstration  Corrective actionplied to visual observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFAT4oBgHgl3EQfbR0j/content/2301.08556v1.pdf'}
+page_content=' Other recent works have found  that using eye-in-hand cameras mounted on a robot’s wrist  can significantly improve the performance of visuomotor  policies trained with imitation learning [17,20,35], but still  do not address the underlying issue of compounding errors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFAT4oBgHgl3EQfbR0j/content/2301.08556v1.pdf'}
+page_content='  We develop an approach that helps address compounding  errors to improve vision-based policies, while building on  the success of eye-in-hand cameras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFAT4oBgHgl3EQfbR0j/content/2301.08556v1.pdf'}
+page_content='  To improve imitation learning for quasi-static tasks like  grasping, we propose a simple yet effective offline data aug-  mentation technique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFAT4oBgHgl3EQfbR0j/content/2301.08556v1.pdf'}
+page_content=' For an eye-in-hand camera, the im-  ages in each demonstration trajectory form a collection of  views of the demonstration scene, which we use to train  neural radiance fields (NeRFs) [37] of each scene.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFAT4oBgHgl3EQfbR0j/content/2301.08556v1.pdf'}
+page_content=' Then, we  can augment the demonstration data with corrective feed-  back by injecting noise into the camera poses along the  demonstration and using the demonstration’s NeRF to ren-  der observations from the new camera pose.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFAT4oBgHgl3EQfbR0j/content/2301.08556v1.pdf'}
+page_content=' Because the  camera to end-effector transform is known, we can compute  corrective action labels for the newly rendered observations  by considering the action that would return the gripper to  the expert trajectory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFAT4oBgHgl3EQfbR0j/content/2301.08556v1.pdf'}
+page_content=' The augmented data can be combined  with the original demonstrations to train a reactive, real-  time policy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFAT4oBgHgl3EQfbR0j/content/2301.08556v1.pdf'}
+page_content=' Since the NeRFs are trained on the original  demonstrations, this method effectively “distills” the 3D in-  formation from each NeRF into the policy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFAT4oBgHgl3EQfbR0j/content/2301.08556v1.pdf'}
+page_content='  The main contribution of this work is a NeRF-based data  augmentation technique, called SPARTN (Synthetic Pertur-  bations for Augmenting Robot Trajectories via NeRF), that  improves behavior cloning for eye-in-hand visual grasping  policies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFAT4oBgHgl3EQfbR0j/content/2301.08556v1.pdf'}
+page_content=' By leveraging view-synthesis methods like NeRF,  SPARTN extends the idea of corrective feedback augmen-  tation to the visual domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFAT4oBgHgl3EQfbR0j/content/2301.08556v1.pdf'}
+page_content=' The resulting approach can pro-  duce (i) reactive, (ii) real-time, and (iii) RGB-only policies  for 6-DoF grasping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFAT4oBgHgl3EQfbR0j/content/2301.08556v1.pdf'}
+page_content=' The data augmentation is fully offline  and does not require additional effort from expert demon-  strators nor online environment interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFAT4oBgHgl3EQfbR0j/content/2301.08556v1.pdf'}
+page_content=' We evaluate  SPARTN on 6-DoF robotic grasping tasks both in simula-  tion and in the real world.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFAT4oBgHgl3EQfbR0j/content/2301.08556v1.pdf'}
+page_content=' On a previously-proposed sim-  ulated 6-DoF grasping benchmark [58], the augmentation  from SPARTN improves grasp success rates by 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFAT4oBgHgl3EQfbR0j/content/2301.08556v1.pdf'}
+page_content='8× com-  pared to training without SPARTN, and even outperforms  some methods that use expensive online supervision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFAT4oBgHgl3EQfbR0j/content/2301.08556v1.pdf'}
+page_content=' On  eight challenging real-world grasping tasks with a Franka  Emika Panda robot, SPARTN improves the absolute aver-  age success rate by 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFAT4oBgHgl3EQfbR0j/content/2301.08556v1.pdf'}
+page_content='5%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFAT4oBgHgl3EQfbR0j/content/2301.08556v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFAT4oBgHgl3EQfbR0j/content/2301.08556v1.pdf'}
+page_content=' Related Work  Robotic Grasping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFAT4oBgHgl3EQfbR0j/content/2301.08556v1.pdf'}
+page_content='  Grasping is a long-studied topic in  robotics [50];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFAT4oBgHgl3EQfbR0j/content/2301.08556v1.pdf'}
+page_content=' see the multitude of survey articles for a  complete review [2, 4, 26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFAT4oBgHgl3EQfbR0j/content/2301.08556v1.pdf'}
+page_content=' Most data-driven grasping sys-  tems focus on learning how to predict some parameterized  “grasp” (whether a full 6-DoF pose, 2-DoF table-top po-  Method  No Depth  Full 6-DoF  Closed-Loop  [34,45]  [53]  ✓  ✓  [18,24,40,54]  ✓  [21,32]  ✓  ✓  [43]  ✓  [39]  ✓  [56,58]  ✓  ✓  SPARTN (ours)  ✓  ✓  ✓  Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFAT4oBgHgl3EQfbR0j/content/2301.08556v1.pdf'}
+page_content=' Comparison of our approach with related grasping work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFAT4oBgHgl3EQfbR0j/content/2301.08556v1.pdf'}
+page_content='  SPARTN is the only approach to learn closed-loop 6-DoF grasping  policies from only RGB inputs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFAT4oBgHgl3EQfbR0j/content/2301.08556v1.pdf'}
+page_content='  sition, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFAT4oBgHgl3EQfbR0j/content/2301.08556v1.pdf'}
+page_content=' ), and leave intermediate motion generation to  be open-loop, handled either through motion planners or  simple heuristics, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFAT4oBgHgl3EQfbR0j/content/2301.08556v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFAT4oBgHgl3EQfbR0j/content/2301.08556v1.pdf'}
+page_content=' [12, 34, 42, 43, 52, 57].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFAT4oBgHgl3EQfbR0j/content/2301.08556v1.pdf'}
+page_content=' Other works  have trained closed-loop grasping policies [21, 39, 53, 58],  and bring all the benefits of closed-loop policies: includ-  ing for example, the ability to avoid obstacles, to perform  precise grasping without precise calibration, and to react to  dynamic objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFAT4oBgHgl3EQfbR0j/content/2301.08556v1.pdf'}
+page_content=' Additionally, grasping policies are often  designed for top-down (2- or 3-DoF) grasping [21, 31, 43],  while 6-DoF grasping typically requires depth or 3D infor-  mation [39, 42, 53, 58].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFAT4oBgHgl3EQfbR0j/content/2301.08556v1.pdf'}
+page_content=' In contrast, our method trains a re-  active 6-DoF grasping policy with only RGB data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFAT4oBgHgl3EQfbR0j/content/2301.08556v1.pdf'}
+page_content=' See Ta-  ble 1 for a summary of the assumptions of the most related  grasping works.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFAT4oBgHgl3EQfbR0j/content/2301.08556v1.pdf'}
+page_content='  Imitation learning and data augmentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFAT4oBgHgl3EQfbR0j/content/2301.08556v1.pdf'}
+page_content='  Behavior  cloning is known to struggle with covariate shift: small er-  rors cause imitation policies to fall slightly off of the data  distribution and it is then difficult to correct the mistake  back onto the data manifold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFAT4oBgHgl3EQfbR0j/content/2301.08556v1.pdf'}
+page_content='  DAgger [46] and its vari-  ants [16,23,36] mitigate this issue by obtaining expert cor-  rections throughout training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFAT4oBgHgl3EQfbR0j/content/2301.08556v1.pdf'}
+page_content=' Alternatively, DART [29] in-  jects noise during expert demonstration collection, which  is especially effective with algorithmic experts but inter-  feres with human demonstration collection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFAT4oBgHgl3EQfbR0j/content/2301.08556v1.pdf'}
+page_content=' Previous works  [14, 22] have injected noise into the low-dimensional sys-  tem state after data collection (in a fully offline manner),  but the visual observations are left out, limiting the help-  fulness of noise injection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFAT4oBgHgl3EQfbR0j/content/2301.08556v1.pdf'}
+page_content=' Our method can be seen as a vi-  sual, fully-offline version of noise injection that does not re-  quire perturbing the expert during demonstration collection,  using NeRF to synthetically render perturbed states post-  hoc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFAT4oBgHgl3EQfbR0j/content/2301.08556v1.pdf'}
+page_content=' Unlike standard image augmentations for policy learn-  ing [30, 61], our method uses NeRF to learn a 3D model  of the demonstration scene, which enables us to generate  high-fidelity novel views for data augmentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFAT4oBgHgl3EQfbR0j/content/2301.08556v1.pdf'}
+page_content=' In addi-  tion, while standard image augmentation approaches do not  modify the action labels, we leverage hand-eye coordination  to calculate corrective actions for augmented observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFAT4oBgHgl3EQfbR0j/content/2301.08556v1.pdf'}
+page_content='  NeRF for Robotics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFAT4oBgHgl3EQfbR0j/content/2301.08556v1.pdf'}
+page_content=' A number of recent works have in-  2  vestigated applications of NeRF and related methods in  robotics, including localization [63], navigation [1], dynam-  ics modeling [7, 9, 33], reinforcement learning [10], and  data generation for other learning-based methods [18, 62].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFAT4oBgHgl3EQfbR0j/content/2301.08556v1.pdf'}
+page_content='  NeRF-Supervision [62], for example, generates pixel-level  correspondence to learn dense object descriptors, which are  useful for manipulation tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFAT4oBgHgl3EQfbR0j/content/2301.08556v1.pdf'}
+page_content=' For grasping, various meth-  ods have leveraged NeRF [3, 18, 24] for open-loop grasp  synthesis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFAT4oBgHgl3EQfbR0j/content/2301.08556v1.pdf'}
+page_content='  In contrast, our method uses NeRF offline to  augment data for grasping and distills a reactive, real-time,  RGB-only, closed-loop policy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFAT4oBgHgl3EQfbR0j/content/2301.08556v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFAT4oBgHgl3EQfbR0j/content/2301.08556v1.pdf'}
+page_content=' Methodology  We now introduce SPARTN, which augments an eye-in-  hand robot demonstration dataset using NeRF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFAT4oBgHgl3EQfbR0j/content/2301.08556v1.pdf'}
+page_content=' We first re-  view preliminaries, then describe a method overview, fol-  lowed by details of training NeRFs and augmenting correc-  tive behavior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFAT4oBgHgl3EQfbR0j/content/2301.08556v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFAT4oBgHgl3EQfbR0j/content/2301.08556v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFAT4oBgHgl3EQfbR0j/content/2301.08556v1.pdf'}
+page_content=' Preliminaries  Imitation learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFAT4oBgHgl3EQfbR0j/content/2301.08556v1.pdf'}
+page_content=' In imitation learning, we assume ac-  cess to a dataset of N expert trajectories D = {τ}N  i=1,  where each trajectory consists of a sequence of state-action  pairs, τ = {(sk, ak)}K  k=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFAT4oBgHgl3EQfbR0j/content/2301.08556v1.pdf'}
+page_content=' In purely offline imitation learn-  ing, there exists no other primary data-collection assump-  tions other than this dataset D, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFAT4oBgHgl3EQfbR0j/content/2301.08556v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFAT4oBgHgl3EQfbR0j/content/2301.08556v1.pdf'}
+page_content=' no reward labels or on-  line interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFAT4oBgHgl3EQfbR0j/content/2301.08556v1.pdf'}
+page_content=' A standard method for this offline setting  is behavior cloning (BC), which trains a policy πθ to mimic  the expert via supervised learning, by minimizing the ob-  jective L(θ) = E(s,a)∼D[ℓ  �  πθ(s), a  �  ], where ℓ is some loss  function in the action space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFAT4oBgHgl3EQfbR0j/content/2301.08556v1.pdf'}
+page_content='  NeRF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFAT4oBgHgl3EQfbR0j/content/2301.08556v1.pdf'}
+page_content=' Our method uses novel-view synthesis as a build-  ing block, for which we use Neural Radiance Fields [37].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFAT4oBgHgl3EQfbR0j/content/2301.08556v1.pdf'}
+page_content='  For each scene, NeRF performs novel-view synthesis by  training a scene-specific neural radiance field FΘ from  a “training set” of posed images {(Ik, Tk)}, where each  Ik ∈ Rw×h×3, Tk ∈ SE(3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFAT4oBgHgl3EQfbR0j/content/2301.08556v1.pdf'}
+page_content=' After FΘ is trained, through  volume rendering NeRF can render new views of a scene  from any requested pose, which can be summarized as  I = NeRF-Render(T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFAT4oBgHgl3EQfbR0j/content/2301.08556v1.pdf'}
+page_content=' FΘ) – in particular, this works best  “near” the training set of poses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFAT4oBgHgl3EQfbR0j/content/2301.08556v1.pdf'}
+page_content=' Since we train many NeRFs  (one per demonstration), we use an accelerated implemen-  tation of NeRF (Instant-NeRF [41]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFAT4oBgHgl3EQfbR0j/content/2301.08556v1.pdf'}
+page_content='  Corrective Noise Augmentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFAT4oBgHgl3EQfbR0j/content/2301.08556v1.pdf'}
+page_content=' A simple method which  has been shown to improve the robustness of behavior-  cloned policies is to perform corrective noise augmenta-  tion [14, 22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFAT4oBgHgl3EQfbR0j/content/2301.08556v1.pdf'}
+page_content='  The idea is to take a nominal trajectory  τ = {(sk, ak)}K  k=0 and create a noise-distributed corrective  trajectory ˜τ = {(˜sk, ˜ak)}K  k=0, where each sampled state ˜sk  is a noisy version of the measured state, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFAT4oBgHgl3EQfbR0j/content/2301.08556v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFAT4oBgHgl3EQfbR0j/content/2301.08556v1.pdf'}
+page_content=' ˜sk ∼ sk + ϵ  and ϵ is sampled noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFAT4oBgHgl3EQfbR0j/content/2301.08556v1.pdf'}
+page_content=' To perform corrective feedback aug-  mentation, ˜ak is chosen such that it will return the state to  the nominal trajectory: given the true inverse dynamics f −1  Algorithm 1 SPARTN (Simplified overview)  Input: D = {τ}N  i=1 - expert demonstrations  Output: Augmented transition dataset ˜D  1: ˜D ← {}  2: for τ = {(Ik, Tk, ak)}K  k=1 ∈ D do  3:  F τ  Θ ← NeRF({(Ik, Tk)}K  k=1) # train NeRF  4:  for k ∈ (0, 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFAT4oBgHgl3EQfbR0j/content/2301.08556v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFAT4oBgHgl3EQfbR0j/content/2301.08556v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFAT4oBgHgl3EQfbR0j/content/2301.08556v1.pdf'}
+page_content=' , K) do  5:  for i = 1 : Naug do  6:  ε ← NoiseDist(SE(3))  7:  ˜Tk ← Perturb(Tk, ε) # perturb pose  8:  ˜ak ← CorrectAction(Tk, ˜Tk, ak)  9:  ˜Ik ← NeRF-Render( ˜Tk, F τ  Θ)  10:  ˜D ← ˜D ∪ {(˜Ik, ˜Tk, ˜ak)}  of the environment, then ˜ak = f −1(˜sk, sk+1), or compute  the commanded control that can be tracked with a stabiliz-  ing controller to sk+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFAT4oBgHgl3EQfbR0j/content/2301.08556v1.pdf'}
+page_content=' An easy way to parameterize this  is by choosing the action space of the learned policy to be  the input to a stabilizing controller, as in [14,22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFAT4oBgHgl3EQfbR0j/content/2301.08556v1.pdf'}
+page_content=' Note that  it is also common in the imitation and reinforcement learn-  ing literature to apply noise to inputs, which is typically  interpreted as a way to regularize the policy [27,30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFAT4oBgHgl3EQfbR0j/content/2301.08556v1.pdf'}
+page_content=' Mean-  while, in addition to potential regularization effects, the cor-  rective noise augmentation has been interpreted to specifi-  cally reduce compounding errors [14,22], but of course has  limits if the scale of perturbations is too large or in highly  non-smooth dynamics regimes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFAT4oBgHgl3EQfbR0j/content/2301.08556v1.pdf'}
+page_content=' A critical limitation of prior  works using corrective noise augmentation is that they have  not been applied to visual observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFAT4oBgHgl3EQfbR0j/content/2301.08556v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFAT4oBgHgl3EQfbR0j/content/2301.08556v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFAT4oBgHgl3EQfbR0j/content/2301.08556v1.pdf'}
+page_content=' Overview: Visual Corrective Augmentation  Consider the case where an agent receives partial visual  observations I instead of the full state of the world, and also  receives direct proprioceptive state measurements, srobot, as  is commonly the case in robotics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFAT4oBgHgl3EQfbR0j/content/2301.08556v1.pdf'}
+page_content=' In general, the correc-  tive augmentation of Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFAT4oBgHgl3EQfbR0j/content/2301.08556v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFAT4oBgHgl3EQfbR0j/content/2301.08556v1.pdf'}
+page_content='1 requires obtaining the visual  observation ˜Ik for each noisy state ˜sk, which could be ex-  pensive or impossible in the actual environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFAT4oBgHgl3EQfbR0j/content/2301.08556v1.pdf'}
+page_content='  An insight of this work is that for eye-in-hand robot  policies (where the visual observation comes from a cam-  era mounted on the wrist) in static scenes, we can readily  generate novel visual observations using novel-view synthe-  sis (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFAT4oBgHgl3EQfbR0j/content/2301.08556v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFAT4oBgHgl3EQfbR0j/content/2301.08556v1.pdf'}
+page_content=', NeRF) without further interactions in the environ-  ment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFAT4oBgHgl3EQfbR0j/content/2301.08556v1.pdf'}
+page_content=' In this setting, a primary subset of the observations  are posed images (I, T), together with some actions a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFAT4oBgHgl3EQfbR0j/content/2301.08556v1.pdf'}
+page_content='  The key intuition of our method can be grasped by con-  sidering how to perform visually corrective augmentation  via NeRF, as illustrated in Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFAT4oBgHgl3EQfbR0j/content/2301.08556v1.pdf'}
+page_content=' Noisy states and cor-  rective actions ( ˜Tk, ˜ak) can be generated for a trajectory  of posed observations and actions τ = {(Ik, Tk, ak)}K  k=0  (Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFAT4oBgHgl3EQfbR0j/content/2301.08556v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFAT4oBgHgl3EQfbR0j/content/2301.08556v1.pdf'}
+page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFAT4oBgHgl3EQfbR0j/content/2301.08556v1.pdf'}
+page_content='  The key pre-processing step is to train a  trajectory-specific NeRF F τ  Θ for each demonstration τ  (Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFAT4oBgHgl3EQfbR0j/content/2301.08556v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFAT4oBgHgl3EQfbR0j/content/2301.08556v1.pdf'}
+page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFAT4oBgHgl3EQfbR0j/content/2301.08556v1.pdf'}
+page_content=' These trajectory-specific NeRFs enable us to  3  Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFAT4oBgHgl3EQfbR0j/content/2301.08556v1.pdf'}
+page_content=' An illustration of how SPARTN creates augmentations from an original demonstration (in reality, this process is repeated for  every available demonstration).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFAT4oBgHgl3EQfbR0j/content/2301.08556v1.pdf'}
+page_content=' (i): The eye-in-hand demonstration contains posed images {(Ik, Tk)}K  k=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFAT4oBgHgl3EQfbR0j/content/2301.08556v1.pdf'}
+page_content=' (ii): We train a neural radiance  field (NeRF) of the demonstration scene on the posed images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFAT4oBgHgl3EQfbR0j/content/2301.08556v1.pdf'}
+page_content=' (iii): We sample perturbations around each pose to simulate noise in  the demonstration, and calculate the corrective action (in magenta) that would stabilize the trajectory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFAT4oBgHgl3EQfbR0j/content/2301.08556v1.pdf'}
+page_content=' (iv): We use the NeRF to render  observations for the perturbed poses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFAT4oBgHgl3EQfbR0j/content/2301.08556v1.pdf'}
+page_content=' The end result is augmented image-action pairs for improving behavior cloning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFAT4oBgHgl3EQfbR0j/content/2301.08556v1.pdf'}
+page_content='  Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFAT4oBgHgl3EQfbR0j/content/2301.08556v1.pdf'}
+page_content=' An overview of the SPARTN training process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFAT4oBgHgl3EQfbR0j/content/2301.08556v1.pdf'}
+page_content=' A NeRF  is trained for each of the original demonstrations in D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFAT4oBgHgl3EQfbR0j/content/2301.08556v1.pdf'}
+page_content=' We use  these NeRFs to generate visual corrective augmentations for each  demonstration and collect them in ˜D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFAT4oBgHgl3EQfbR0j/content/2301.08556v1.pdf'}
+page_content=' The policy πθ can be trained  on D and ˜D using standard behavior cloning methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFAT4oBgHgl3EQfbR0j/content/2301.08556v1.pdf'}
+page_content='  render observations ˜Ik for noisy states ˜Tk, completing the  augmentation process and resulting in visually corrective  transitions (˜Ik, ˜Tk, ˜ak) (Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFAT4oBgHgl3EQfbR0j/content/2301.08556v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFAT4oBgHgl3EQfbR0j/content/2301.08556v1.pdf'}
+page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFAT4oBgHgl3EQfbR0j/content/2301.08556v1.pdf'}
+page_content=' Algorithm 1 and Figure 3  overview this process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFAT4oBgHgl3EQfbR0j/content/2301.08556v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFAT4oBgHgl3EQfbR0j/content/2301.08556v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFAT4oBgHgl3EQfbR0j/content/2301.08556v1.pdf'}
+page_content=' Training NeRFs from Robot Demonstrations  SPARTN uses novel-view synthesis, in particular NeRF,  to generate observations ˜Ik for noisy states without environ-  ment interaction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFAT4oBgHgl3EQfbR0j/content/2301.08556v1.pdf'}
+page_content=' We train a NeRF F τ  Θ for each demonstra-  tion trajectory using the image observations (I1, · · · , IK) ∈  τ as the training set of views.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFAT4oBgHgl3EQfbR0j/content/2301.08556v1.pdf'}
+page_content=' After training F τ  Θ, we create  observations ˜Ik for perturbed robot states ˜Tk by rendering  the view from the perturbed camera pose using F τ  Θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFAT4oBgHgl3EQfbR0j/content/2301.08556v1.pdf'}
+page_content='  An important detail is that the end-effector reference  frame used for control in demonstrations may differ from  the reference frame for the camera itself, but through stan-  dard eye-in-hand calibration we can transform all visual ob-  servations used for training and augmenting the NeRFs into  the cameras frame.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFAT4oBgHgl3EQfbR0j/content/2301.08556v1.pdf'}
+page_content=' Given a transform toT from  k  which trans-  forms between two frames, we simply transform all NeRF-  poses to the world frame: W T C  k = W T E  k  ET C, where W is  the world frame, E is the end-effector frame which changes  at each step k, and C is the camera (NeRF) frame stati-  cally linked to the end-effector frame, and ET C is acquired  through hand-eye calibration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFAT4oBgHgl3EQfbR0j/content/2301.08556v1.pdf'}
+page_content='  COLMAP camera poses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFAT4oBgHgl3EQfbR0j/content/2301.08556v1.pdf'}
+page_content='  Real-world calibration error  means that our camera-to-world transforms {W T C  k }K  k=1 are  noisy and we obtain higher-quality NeRFs by using cam-  era poses estimated by COLMAP [48,49].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFAT4oBgHgl3EQfbR0j/content/2301.08556v1.pdf'}
+page_content=' Up to noise and  a scale factor β, the only difference from the world frame  camera transforms is that COLMAP uses an arbitrary ref-  erence frame V ̸= W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFAT4oBgHgl3EQfbR0j/content/2301.08556v1.pdf'}
+page_content=' We denote the COLMAP outputs  {V HC  k }K  k=1, using H instead of T because of the difference  in scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFAT4oBgHgl3EQfbR0j/content/2301.08556v1.pdf'}
+page_content=' We now introduce notation to separate the rotation  and translation components of a transform:  aT b :=  �  Rot  �aT b�  , Trans  �aT b��  (1)  Since we train the NeRFs on COLMAP’s camera poses, we  must convert perturbed camera poses W T ˜  C  k to COLMAP’s  frame in order to render the observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFAT4oBgHgl3EQfbR0j/content/2301.08556v1.pdf'}
+page_content=' In other words,  we must call NeRF-Render(V H ˜  C  k , F τ  Θ), where:  V H  ˜  C  k =  �  Rot  �  V T  ˜  C  k  �  , β Trans  �  V T  ˜  C  k  ��  (2)  V T  ˜  C  k = V T W W T  ˜  C  k .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFAT4oBgHgl3EQfbR0j/content/2301.08556v1.pdf'}
+page_content='  (3)  Both β and  V T W  can be estimated from the pairs  {(W T C  k , V HC  k )}K  k=1, as described in Appendix D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFAT4oBgHgl3EQfbR0j/content/2301.08556v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFAT4oBgHgl3EQfbR0j/content/2301.08556v1.pdf'}
+page_content='  Static Scene Assumption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFAT4oBgHgl3EQfbR0j/content/2301.08556v1.pdf'}
+page_content=' An additional consideration for  robotic manipulation is that the standard NeRF formulation  4  Trajectory  F  of Posed Images  T  Ik  Eye-in-hand  cameraDemonstrations D  NeRF training  Corrective augmentation D  (1)  T1  F(2)  T2  0  7(3)  T3  0  Train(π)Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFAT4oBgHgl3EQfbR0j/content/2301.08556v1.pdf'}
+page_content=' An illustration of how the gripper is inserted into the  result of the NeRF rendering process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFAT4oBgHgl3EQfbR0j/content/2301.08556v1.pdf'}
+page_content=' Gray regions indicate pixels  being masked out by the binary gripper mask M ∈ {0, 1}w×h,  which denotes the pixels where the gripper is located in all frames.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFAT4oBgHgl3EQfbR0j/content/2301.08556v1.pdf'}
+page_content='  assumes a static scene, while manipulation tasks such as  grasping will usually move objects in the scene.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFAT4oBgHgl3EQfbR0j/content/2301.08556v1.pdf'}
+page_content=' To address  this, we apply the NeRF training and augmentation process  to only the subsets of each demonstration trajectory where  no robot-object interaction occurs, instead of all timesteps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFAT4oBgHgl3EQfbR0j/content/2301.08556v1.pdf'}
+page_content='  For grasping, a simple and effective heuristic is to only ap-  ply SPARTN to the portions of each demonstration before  the expert closes the gripper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFAT4oBgHgl3EQfbR0j/content/2301.08556v1.pdf'}
+page_content='  Masking Out the Robot Gripper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFAT4oBgHgl3EQfbR0j/content/2301.08556v1.pdf'}
+page_content='  The robot’s gripper  is often within the view of an eye-in-hand camera, which  breaks the static scene assumption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFAT4oBgHgl3EQfbR0j/content/2301.08556v1.pdf'}
+page_content='  To address this, we  leverage that the gripper is in the same location in each im-  age assuming the camera is rigidly mounted and the gripper  is open.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFAT4oBgHgl3EQfbR0j/content/2301.08556v1.pdf'}
+page_content=' Further, since NeRF is trained per-ray, we can sim-  ply mask pixels from training images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFAT4oBgHgl3EQfbR0j/content/2301.08556v1.pdf'}
+page_content=' We construct a single  binary mask M ∈ {0, 1}w×h, where a 1 indicates gripper  pixels to mask out.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFAT4oBgHgl3EQfbR0j/content/2301.08556v1.pdf'}
+page_content=' Figure 4 shows how we use the same  mask to splice the gripper back into each NeRF rendering  output, ˜Ik ← ¬M ⊙ ˜Ik +M ⊙Ik, where ⊙ is element-wise  multiplication broadcasted across the color channels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFAT4oBgHgl3EQfbR0j/content/2301.08556v1.pdf'}
+page_content='  NeRF Quality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFAT4oBgHgl3EQfbR0j/content/2301.08556v1.pdf'}
+page_content='  Even when the training views from a  demonstration are suboptimal for NeRF training, SPARTN  benefits from the fact that our augmentation is local and  the perturbations ε are typically small, so we only need the  NeRF to generalize in a small region around the demon-  stration trajectory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFAT4oBgHgl3EQfbR0j/content/2301.08556v1.pdf'}
+page_content=' As we will verify in the experiments,  SPARTN can be effective even with a limited number of  training views compared to other NeRF applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFAT4oBgHgl3EQfbR0j/content/2301.08556v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFAT4oBgHgl3EQfbR0j/content/2301.08556v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFAT4oBgHgl3EQfbR0j/content/2301.08556v1.pdf'}
+page_content=' NeRFing Corrective Noise Augmentation  Given a visual augmentation model (Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFAT4oBgHgl3EQfbR0j/content/2301.08556v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFAT4oBgHgl3EQfbR0j/content/2301.08556v1.pdf'}
+page_content='3), we can  adapt methods for corrective augmentation (Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFAT4oBgHgl3EQfbR0j/content/2301.08556v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFAT4oBgHgl3EQfbR0j/content/2301.08556v1.pdf'}
+page_content='1) into  the visual domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFAT4oBgHgl3EQfbR0j/content/2301.08556v1.pdf'}
+page_content=' Our goal is to create noise-distributed  corrective transitions {(˜Ik, ˜Tk, ˜ak)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFAT4oBgHgl3EQfbR0j/content/2301.08556v1.pdf'}
+page_content=' First, we describe this  simply in the global frame.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFAT4oBgHgl3EQfbR0j/content/2301.08556v1.pdf'}
+page_content=' In order to sample from the  noise-distributed corrective trajectory, one can first apply  noise to the measured end-effector pose, ˜Tk := Tkε, where  ε ∼ NoiseDist(SE(3)) is a randomly sampled rotation and  translation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFAT4oBgHgl3EQfbR0j/content/2301.08556v1.pdf'}
+page_content='  The high-fidelity perturbed image ˜Ik corre-  sponding to ˜Tk can then be rendered using the trajectory-  specific NeRF F τ  Θ, without requiring access to the actual  environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFAT4oBgHgl3EQfbR0j/content/2301.08556v1.pdf'}
+page_content=' For the actions, the simplest case is when they  are inputs to a global-frame stabilizing Cartesian controller  controller [25], in which case ˜ak = ak = ˆTk will provide  stabilization to the nominal trajectory, where ˆTk is the de-  sired pose sent to the lower-level controller.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFAT4oBgHgl3EQfbR0j/content/2301.08556v1.pdf'}
+page_content='  Corrective Relative Actions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFAT4oBgHgl3EQfbR0j/content/2301.08556v1.pdf'}
+page_content='  As is common in prior  works, we observe better performance by parameteriz-  ing the learned policy as a relative rather than global ac-  tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFAT4oBgHgl3EQfbR0j/content/2301.08556v1.pdf'}
+page_content=' Consider the as-discussed global-frame version, with  (1) observations as a global-frame measured SE(3) end-  effector pose W T E  k , where W refers to world-frame, and  E to the end-effector frame at timestep k, and (2) action as  a global-frame desired SE(3) end-effector pose W T ˆ  E  k .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFAT4oBgHgl3EQfbR0j/content/2301.08556v1.pdf'}
+page_content=' To  switch to a relative action space, we adjust the action to  ak = ET  ˆ  E  k = (W T E  k )−1 W T  ˆ  E  k = ET W  k  W T  ˆ  E  k .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFAT4oBgHgl3EQfbR0j/content/2301.08556v1.pdf'}
+page_content='  (4)  To additionally formulate the corrective noise augmentation  in the relative frame, we consider the SE(3)-noise ε as trans-  forming from the noisy end-effector frame to the measured  end-effector frame, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFAT4oBgHgl3EQfbR0j/content/2301.08556v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFAT4oBgHgl3EQfbR0j/content/2301.08556v1.pdf'}
+page_content='  ET ˜  E := ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFAT4oBgHgl3EQfbR0j/content/2301.08556v1.pdf'}
+page_content=' This accordingly ad-  justs the observation as W T ˜  E  k = W T E  k  ET ˜  E  k = W T E  k ε and  the relative action as:  ˜ak =  ˜  ET  ˆ  E  k = (W T E  k  ET  ˜  E  k )−1 W T  ˆ  E  k = ε−1 ak.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFAT4oBgHgl3EQfbR0j/content/2301.08556v1.pdf'}
+page_content='  (5)  Concisely, this amounts to post-pending ε to the measured  pose, and pre-prending ε−1 to the un-noised relative action.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFAT4oBgHgl3EQfbR0j/content/2301.08556v1.pdf'}
+page_content='  Non-SE(3) actions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFAT4oBgHgl3EQfbR0j/content/2301.08556v1.pdf'}
+page_content=' Thus far we have assumed that the ac-  tions a ∈ SE(3) only command desired pose transforms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFAT4oBgHgl3EQfbR0j/content/2301.08556v1.pdf'}
+page_content='  In practice, the action space may contain additional dimen-  sions, for example to open and close the gripper itself.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFAT4oBgHgl3EQfbR0j/content/2301.08556v1.pdf'}
+page_content=' The  SPARTN augmentation does not concern these aspects of  the action, so the corrective actions simply copy the origi-  nal action for non-SE(3) dimensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFAT4oBgHgl3EQfbR0j/content/2301.08556v1.pdf'}
+page_content='  Summary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFAT4oBgHgl3EQfbR0j/content/2301.08556v1.pdf'}
+page_content=' Figure 2 summarizes our augmentation proce-  dure: given a demonstration dataset D, we first train all  the neural radiance fields {F τ  Θ}τ∈D and save the weights  to disk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFAT4oBgHgl3EQfbR0j/content/2301.08556v1.pdf'}
+page_content=' We then augment each transition in the original  dataset with Naug noisy corrective transitions produced us-  ing the process we described above, and save these transi-  tions into an augmented dataset ˜D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFAT4oBgHgl3EQfbR0j/content/2301.08556v1.pdf'}
+page_content=' Appendix Algorithm 2  describes the precise augmentation procedure in detail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFAT4oBgHgl3EQfbR0j/content/2301.08556v1.pdf'}
+page_content='  After the augmented dataset has been created, it can sim-  ply be combined with the original dataset to augment BC  training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFAT4oBgHgl3EQfbR0j/content/2301.08556v1.pdf'}
+page_content=' Various sampling strategies are possible, but in our  experiments we simply construct mini-batches for BC train-  ing by sampling from the original and augmented datasets  with equal probability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFAT4oBgHgl3EQfbR0j/content/2301.08556v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFAT4oBgHgl3EQfbR0j/content/2301.08556v1.pdf'}
+page_content=' Simulation Experiments  We evaluate SPARTN and related approaches in the sim-  ulated 6-DoF grasping benchmark first introduced in [58],  which features a simulated Franka Emika Panda arm with  5  M o Ik  M O NeRF-Render(Tk, F)  +Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFAT4oBgHgl3EQfbR0j/content/2301.08556v1.pdf'}
+page_content=' Example tasks in the simulated 6-DoF grasping bench-  mark, first introduced in [58].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFAT4oBgHgl3EQfbR0j/content/2301.08556v1.pdf'}
+page_content=' There are ∼ 1,500 ShapeNet ob-  jects in training demonstrations, and held out YCB objects for  evaluation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFAT4oBgHgl3EQfbR0j/content/2301.08556v1.pdf'}
+page_content=' A camera mounted to the Franka Panda robot’s arm  provides observations, and the policy controls the 6-DoF end-  effector pose.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFAT4oBgHgl3EQfbR0j/content/2301.08556v1.pdf'}
+page_content='  a parallel-jaw gripper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFAT4oBgHgl3EQfbR0j/content/2301.08556v1.pdf'}
+page_content=' Objects are placed on a table and  must be lifted above a height threshold in a successful grasp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFAT4oBgHgl3EQfbR0j/content/2301.08556v1.pdf'}
+page_content='  Policies receive either RGB, RGBD, or point cloud observa-  tions from a wrist camera, and control the gripper by com-  manding relative pose changes in 6-DoF end-effector space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFAT4oBgHgl3EQfbR0j/content/2301.08556v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFAT4oBgHgl3EQfbR0j/content/2301.08556v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFAT4oBgHgl3EQfbR0j/content/2301.08556v1.pdf'}
+page_content=' Data Collection and Evaluation Protocol  We follow the training and evaluation procedure  from [58].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFAT4oBgHgl3EQfbR0j/content/2301.08556v1.pdf'}
+page_content=' The training dataset includes 2,500 demonstra-  tions of grasping 1,500 ShapeNet [6] objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFAT4oBgHgl3EQfbR0j/content/2301.08556v1.pdf'}
+page_content=' Demonstra-  tions are up to 20 timesteps and are generated by trajectory  optimization to precomputed grasps from the ACRONYM  dataset [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFAT4oBgHgl3EQfbR0j/content/2301.08556v1.pdf'}
+page_content=' Policies are evaluated on grasping held-out ob-  jects from the YCB [5] or ShapeNet datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFAT4oBgHgl3EQfbR0j/content/2301.08556v1.pdf'}
+page_content=' Though it is  more “out of distribution” relative to the training objects,  the YCB evaluation is more realistic for tabletop grasping  scenarios (ShapeNet includes a wide variety of objects, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFAT4oBgHgl3EQfbR0j/content/2301.08556v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFAT4oBgHgl3EQfbR0j/content/2301.08556v1.pdf'}
+page_content='  airplanes and bicycles).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFAT4oBgHgl3EQfbR0j/content/2301.08556v1.pdf'}
+page_content=' Each held-out object is evaluated  ten times, each with a different initial robot configuration  and object’s initial pose.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFAT4oBgHgl3EQfbR0j/content/2301.08556v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFAT4oBgHgl3EQfbR0j/content/2301.08556v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFAT4oBgHgl3EQfbR0j/content/2301.08556v1.pdf'}
+page_content=' Comparisons  We compare SPARTN against other approaches ranging  from simple behavior cloning to online-supervised imita-  tion and reinforcement learning:  Behavior Cloning (BC): Trains policies via supervised  learning on the demonstration dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFAT4oBgHgl3EQfbR0j/content/2301.08556v1.pdf'}
+page_content='  DART [29]: Introduces a modified demonstration setup  where a continuous Gaussian noise is injected into the  robot’s state during the expert demonstration, then trains  policies on the modified demonstrations with BC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFAT4oBgHgl3EQfbR0j/content/2301.08556v1.pdf'}
+page_content='  Homography Augmentation (HA): A simplification of  SPARTN where perturbations can be 3D rotations, but not  1Note that the “BC” results in [58] actually use DART.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFAT4oBgHgl3EQfbR0j/content/2301.08556v1.pdf'}
+page_content='  Supervision  Method  Input  YCB SR(%)  SN SR(%)  Offline  BC  RGB  28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFAT4oBgHgl3EQfbR0j/content/2301.08556v1.pdf'}
+page_content='9 ± 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFAT4oBgHgl3EQfbR0j/content/2301.08556v1.pdf'}
+page_content='4  57.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFAT4oBgHgl3EQfbR0j/content/2301.08556v1.pdf'}
+page_content='4 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFAT4oBgHgl3EQfbR0j/content/2301.08556v1.pdf'}
+page_content='2  DART†  RGB  51.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFAT4oBgHgl3EQfbR0j/content/2301.08556v1.pdf'}
+page_content='2 ± 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFAT4oBgHgl3EQfbR0j/content/2301.08556v1.pdf'}
+page_content='2  57.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFAT4oBgHgl3EQfbR0j/content/2301.08556v1.pdf'}
+page_content='8 ± 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFAT4oBgHgl3EQfbR0j/content/2301.08556v1.pdf'}
+page_content='2  RGBD  46.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFAT4oBgHgl3EQfbR0j/content/2301.08556v1.pdf'}
+page_content='3  45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFAT4oBgHgl3EQfbR0j/content/2301.08556v1.pdf'}
+page_content='3  Point cloud  65.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFAT4oBgHgl3EQfbR0j/content/2301.08556v1.pdf'}
+page_content='6  73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFAT4oBgHgl3EQfbR0j/content/2301.08556v1.pdf'}
+page_content='6  HA  RGB  29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFAT4oBgHgl3EQfbR0j/content/2301.08556v1.pdf'}
+page_content='7 ± 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFAT4oBgHgl3EQfbR0j/content/2301.08556v1.pdf'}
+page_content='4  57.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFAT4oBgHgl3EQfbR0j/content/2301.08556v1.pdf'}
+page_content='5 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFAT4oBgHgl3EQfbR0j/content/2301.08556v1.pdf'}
+page_content='8  SPARTN (ours)  RGB  74.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFAT4oBgHgl3EQfbR0j/content/2301.08556v1.pdf'}
+page_content='7 ± 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFAT4oBgHgl3EQfbR0j/content/2301.08556v1.pdf'}
+page_content='4  66.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFAT4oBgHgl3EQfbR0j/content/2301.08556v1.pdf'}
+page_content='9 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFAT4oBgHgl3EQfbR0j/content/2301.08556v1.pdf'}
+page_content='6  Online*  DAgger  RGB  52.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFAT4oBgHgl3EQfbR0j/content/2301.08556v1.pdf'}
+page_content='3  52.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFAT4oBgHgl3EQfbR0j/content/2301.08556v1.pdf'}
+page_content='1  RGBD  67.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFAT4oBgHgl3EQfbR0j/content/2301.08556v1.pdf'}
+page_content='1  60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFAT4oBgHgl3EQfbR0j/content/2301.08556v1.pdf'}
+page_content='4  Point cloud  77.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFAT4oBgHgl3EQfbR0j/content/2301.08556v1.pdf'}
+page_content='2  75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFAT4oBgHgl3EQfbR0j/content/2301.08556v1.pdf'}
+page_content='8  GA-DDPG  Point cloud  88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFAT4oBgHgl3EQfbR0j/content/2301.08556v1.pdf'}
+page_content='2  91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFAT4oBgHgl3EQfbR0j/content/2301.08556v1.pdf'}
+page_content='3  Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFAT4oBgHgl3EQfbR0j/content/2301.08556v1.pdf'}
+page_content=' Grasping success rates (SR) on held-out objects from  YCB [5] or ShapeNet (SN) [6] in a simulated 6-DoF grasping  benchmark [58].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFAT4oBgHgl3EQfbR0j/content/2301.08556v1.pdf'}
+page_content='  We bold the best offline RGB-only results,  though we include online and non-RGB methods for comparison.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFAT4oBgHgl3EQfbR0j/content/2301.08556v1.pdf'}
+page_content='  Online* requires additional environment interactions, while Of-  fline only uses demonstration data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFAT4oBgHgl3EQfbR0j/content/2301.08556v1.pdf'}
+page_content=' DART† is offline but requires  a special demonstration collection setup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFAT4oBgHgl3EQfbR0j/content/2301.08556v1.pdf'}
+page_content=' SPARTN outperforms  other offline RGB-only methods, while RL (GA-DDPG) performs  best overall while requiring millions of interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFAT4oBgHgl3EQfbR0j/content/2301.08556v1.pdf'}
+page_content=' We calcu-  late average success rates and standard error over 4 random seeds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFAT4oBgHgl3EQfbR0j/content/2301.08556v1.pdf'}
+page_content='  Italicized success rates were reported in prior work1 [58].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFAT4oBgHgl3EQfbR0j/content/2301.08556v1.pdf'}
+page_content='  Method  Image aug.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFAT4oBgHgl3EQfbR0j/content/2301.08556v1.pdf'}
+page_content='  YCB SR(%)  BC  Without  25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFAT4oBgHgl3EQfbR0j/content/2301.08556v1.pdf'}
+page_content='3 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFAT4oBgHgl3EQfbR0j/content/2301.08556v1.pdf'}
+page_content='6  With  28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFAT4oBgHgl3EQfbR0j/content/2301.08556v1.pdf'}
+page_content='9 ± 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFAT4oBgHgl3EQfbR0j/content/2301.08556v1.pdf'}
+page_content='4  DART  Without  51.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFAT4oBgHgl3EQfbR0j/content/2301.08556v1.pdf'}
+page_content='2 ± 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFAT4oBgHgl3EQfbR0j/content/2301.08556v1.pdf'}
+page_content='2  With  48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFAT4oBgHgl3EQfbR0j/content/2301.08556v1.pdf'}
+page_content='6 ± 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFAT4oBgHgl3EQfbR0j/content/2301.08556v1.pdf'}
+page_content='8  HA  Without  23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFAT4oBgHgl3EQfbR0j/content/2301.08556v1.pdf'}
+page_content='9 ± 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFAT4oBgHgl3EQfbR0j/content/2301.08556v1.pdf'}
+page_content='0  With  29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFAT4oBgHgl3EQfbR0j/content/2301.08556v1.pdf'}
+page_content='7 ± 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFAT4oBgHgl3EQfbR0j/content/2301.08556v1.pdf'}
+page_content='4  SPARTN  Without  68.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFAT4oBgHgl3EQfbR0j/content/2301.08556v1.pdf'}
+page_content='3 ± 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFAT4oBgHgl3EQfbR0j/content/2301.08556v1.pdf'}
+page_content='6  With  74.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFAT4oBgHgl3EQfbR0j/content/2301.08556v1.pdf'}
+page_content='7 ± 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFAT4oBgHgl3EQfbR0j/content/2301.08556v1.pdf'}
+page_content='4  Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFAT4oBgHgl3EQfbR0j/content/2301.08556v1.pdf'}
+page_content=' Ablating the effect of standard image augmentation on  each method for RGB policies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFAT4oBgHgl3EQfbR0j/content/2301.08556v1.pdf'}
+page_content=' Average success rates and standard  errors are calculated over four seeds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFAT4oBgHgl3EQfbR0j/content/2301.08556v1.pdf'}
+page_content='  translations, of the camera.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFAT4oBgHgl3EQfbR0j/content/2301.08556v1.pdf'}
+page_content=' For pure rotation, we can cal-  culate the homography transform for rendering the rotated  view without NeRF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFAT4oBgHgl3EQfbR0j/content/2301.08556v1.pdf'}
+page_content=' Augmented actions are computed sim-  ilarly to SPARTN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFAT4oBgHgl3EQfbR0j/content/2301.08556v1.pdf'}
+page_content='  DAgger [46]: An online-supervised method where a policy  is first trained using BC on offline demos, and then the ex-  pert provides action labels for states from the policy’s roll-  outs throughout further policy training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFAT4oBgHgl3EQfbR0j/content/2301.08556v1.pdf'}
+page_content='  GA-DDPG [58]: Jointly trains policies via BC and fine-  tunes them with reinforcement learning (DDPG [51]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFAT4oBgHgl3EQfbR0j/content/2301.08556v1.pdf'}
+page_content='  Of the methods considered, DAgger and GA-DDPG re-  quire online environment interaction and supervision from  an expert or reward function, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFAT4oBgHgl3EQfbR0j/content/2301.08556v1.pdf'}
+page_content=' The other meth-  ods, including SPARTN, train only on pre-collected demon-  stration datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFAT4oBgHgl3EQfbR0j/content/2301.08556v1.pdf'}
+page_content='  Since the DART data collection setup  noisily perturbs the robot as the expert demonstrates, DART  can interfere with human expert demonstration, though this  challenge does not arise here with our simulated expert.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFAT4oBgHgl3EQfbR0j/content/2301.08556v1.pdf'}
+page_content='  6  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFAT4oBgHgl3EQfbR0j/content/2301.08556v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFAT4oBgHgl3EQfbR0j/content/2301.08556v1.pdf'}
+page_content=' Training Details  We follow the RGB architectural and training hyper-  parameters of [58], with policies consisting of a ResNet-  18 [15] image encoder followed by an MLP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFAT4oBgHgl3EQfbR0j/content/2301.08556v1.pdf'}
+page_content=' We apply ran-  dom crop and color jitter to the training images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFAT4oBgHgl3EQfbR0j/content/2301.08556v1.pdf'}
+page_content=' We re-use  the same training BC hyperparameters for SPARTN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFAT4oBgHgl3EQfbR0j/content/2301.08556v1.pdf'}
+page_content=' Ap-  pendix C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFAT4oBgHgl3EQfbR0j/content/2301.08556v1.pdf'}
+page_content='1 describes the complete training details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFAT4oBgHgl3EQfbR0j/content/2301.08556v1.pdf'}
+page_content='  For SPARTN, we create Naug = 100 augmented transi-  tions from each original transition and save the augmented  dataset to disk before BC training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFAT4oBgHgl3EQfbR0j/content/2301.08556v1.pdf'}
+page_content=' To sample the pertur-  bations ε ∼ NoiseDist(SE(3)), we parameterize the rota-  tions in terms of Euler angles (φ, θ, ϕ) and uniformly sam-  ple both rotation and translation parameters:  (φ, θ, ϕ), (tx, ty, tz) ∼ U(−α, α), U(−β, β)  (6)  ε := (R(φ, θ, ϕ), (tx, ty, tz))  (7)  In simulation, we set α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFAT4oBgHgl3EQfbR0j/content/2301.08556v1.pdf'}
+page_content='2 radians and β = 3 mm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFAT4oBgHgl3EQfbR0j/content/2301.08556v1.pdf'}
+page_content='  Following the DART hyperparameters in [58], we only aug-  ment timesteps 5 − 13 of each demonstration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFAT4oBgHgl3EQfbR0j/content/2301.08556v1.pdf'}
+page_content=' Appendix B  contains samples of SPARTN’s perturbed observations ren-  dered via NeRF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFAT4oBgHgl3EQfbR0j/content/2301.08556v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFAT4oBgHgl3EQfbR0j/content/2301.08556v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFAT4oBgHgl3EQfbR0j/content/2301.08556v1.pdf'}
+page_content=' Results  Table 2 shows grasping success rates on held-out objects  from either the YCB or ShapeNet (SN) datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFAT4oBgHgl3EQfbR0j/content/2301.08556v1.pdf'}
+page_content=' SPARTN  significantly outperforms the other two offline-supervised  approaches for RGB inputs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFAT4oBgHgl3EQfbR0j/content/2301.08556v1.pdf'}
+page_content=' Since DART injects noise dur-  ing expert collection, the amount of augmentation is lim-  ited by the number of demonstrations the expert can col-  lect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFAT4oBgHgl3EQfbR0j/content/2301.08556v1.pdf'}
+page_content=' Meanwhile, SPARTN can cheaply generate an arbi-  trary number of augmented examples on top of the exist-  ing demonstration dataset, leading to more data diversity  without any additional effort from the expert.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFAT4oBgHgl3EQfbR0j/content/2301.08556v1.pdf'}
+page_content=' See the sup-  plementary material for videos of rollouts for SPARTN and  BC policies, with success and failure cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFAT4oBgHgl3EQfbR0j/content/2301.08556v1.pdf'}
+page_content='  Naive BC performs relatively well on ShapeNet evalua-  tion, likely because the evaluation set is “in-distribution”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFAT4oBgHgl3EQfbR0j/content/2301.08556v1.pdf'}
+page_content='  Correspondingly, corrective methods (DART, SPARTN)  achieve smaller improvements over BC on ShapeNet and  larger improvements on YCB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFAT4oBgHgl3EQfbR0j/content/2301.08556v1.pdf'}
+page_content='  On YCB, SPARTN’s performance is closer to RL (GA-  DDPG) than to other offline methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFAT4oBgHgl3EQfbR0j/content/2301.08556v1.pdf'}
+page_content=' It actually outper-  forms DAgger for RGB policies, perhaps because DAgger’s  data distribution changes online presenting a more challeng-  ing learning problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFAT4oBgHgl3EQfbR0j/content/2301.08556v1.pdf'}
+page_content=' Notably, on YCB SPARTN outper-  forms DART with point clouds and is comparable to DAg-  ger with point clouds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFAT4oBgHgl3EQfbR0j/content/2301.08556v1.pdf'}
+page_content=' Since SPARTN itself only requires  RGB images during both NeRF training and policy train-  ing, it significantly closes the gap between the performance  of RGB-only and the best depth-based methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFAT4oBgHgl3EQfbR0j/content/2301.08556v1.pdf'}
+page_content=' Since con-  sumer depth cameras can struggle with common thin and  reflective items [62], improving RGB-only grasping can en-  able more robust grapsing of such objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFAT4oBgHgl3EQfbR0j/content/2301.08556v1.pdf'}
+page_content='  Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFAT4oBgHgl3EQfbR0j/content/2301.08556v1.pdf'}
+page_content=' Real-world grasping environments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFAT4oBgHgl3EQfbR0j/content/2301.08556v1.pdf'}
+page_content=' In each environ-  ment, the task is to grasp the labeled target object in a particular  way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFAT4oBgHgl3EQfbR0j/content/2301.08556v1.pdf'}
+page_content=' The target objects have various geometric shapes and exhibit  a diverse range of characteristics, including reflectiveness, trans-  parency, and radial symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFAT4oBgHgl3EQfbR0j/content/2301.08556v1.pdf'}
+page_content='  Target Object  # Demos  BC SR(%)  SPARTN SR(%)  Banana  14  55  75  Thin box  20  35  65  Steel water bottle  15  20  40  Wine glass  25  70  90  Lubriderm bottle  17  25  60  White tape roll  15  40  45  Tupperware  20  40  75  Fork  20  25  40  Average  --  38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFAT4oBgHgl3EQfbR0j/content/2301.08556v1.pdf'}
+page_content='75  61.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFAT4oBgHgl3EQfbR0j/content/2301.08556v1.pdf'}
+page_content='25  Table 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFAT4oBgHgl3EQfbR0j/content/2301.08556v1.pdf'}
+page_content=' Success rates (SR) of behavior cloning (BC) and SPARTN  6-DoF grasping policies on a suite of eight real-world target ob-  jects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFAT4oBgHgl3EQfbR0j/content/2301.08556v1.pdf'}
+page_content=' SPARTN outperforms BC in every environment, achieving  an average absolute performance boost of 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFAT4oBgHgl3EQfbR0j/content/2301.08556v1.pdf'}
+page_content='5%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFAT4oBgHgl3EQfbR0j/content/2301.08556v1.pdf'}
+page_content=' Each success  rate is computed over 20 trials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFAT4oBgHgl3EQfbR0j/content/2301.08556v1.pdf'}
+page_content='  Effect of Image Augmentations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFAT4oBgHgl3EQfbR0j/content/2301.08556v1.pdf'}
+page_content=' We try ablating the effect  of standard image augmentations (random crop and color  jitter) for the BC, DART, HA, and SPARTN methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFAT4oBgHgl3EQfbR0j/content/2301.08556v1.pdf'}
+page_content=' Ta-  ble 3 shows that the image augmentations have a small ef-  fect on the performance of most methods, relative to the  larger effect of SPARTN’s corrective augmentations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFAT4oBgHgl3EQfbR0j/content/2301.08556v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFAT4oBgHgl3EQfbR0j/content/2301.08556v1.pdf'}
+page_content=' Real-World Experiments  The simulation benchmark results show that SPARTN  can improve grasping generalization in imitation learning  without online supervision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFAT4oBgHgl3EQfbR0j/content/2301.08556v1.pdf'}
+page_content=' Here, we verify that SPARTN  can enable real-world robotic grasping of challenging ob-  jects from limited human demonstration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFAT4oBgHgl3EQfbR0j/content/2301.08556v1.pdf'}
+page_content=' See the website in  the supplement for video results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFAT4oBgHgl3EQfbR0j/content/2301.08556v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFAT4oBgHgl3EQfbR0j/content/2301.08556v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFAT4oBgHgl3EQfbR0j/content/2301.08556v1.pdf'}
+page_content=' Experimental details  Robot setup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFAT4oBgHgl3EQfbR0j/content/2301.08556v1.pdf'}
+page_content=' The robotic manipulator is a Franka Emika  Panda robot arm with a wrist-mounted consumer grade we-  bcam.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFAT4oBgHgl3EQfbR0j/content/2301.08556v1.pdf'}
+page_content=' Policies take images of size 256 × 256 as input and  output a 6-DoF action representing the desired change in  end-effector position and orientation, as well as a binary  7  Banana  Thin box  Steelwaterbottle  Wine glass  Lubriderm bottle  White tape roll  Tupperware  Forkopen/close gripper action.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFAT4oBgHgl3EQfbR0j/content/2301.08556v1.pdf'}
+page_content=' We use a Cartesian impedance  controller to command the pose changes at a frequency of 4  Hz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFAT4oBgHgl3EQfbR0j/content/2301.08556v1.pdf'}
+page_content=' We task the robot to grasp a target object in eight dif-  ferent environments depicted in Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFAT4oBgHgl3EQfbR0j/content/2301.08556v1.pdf'}
+page_content=' The target objects  include natural shapes that are common in the real world  and exhibit a diverse range of attributes, such as reflective-  ness, transparency, and radial symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFAT4oBgHgl3EQfbR0j/content/2301.08556v1.pdf'}
+page_content='  Comparisons and Evaluation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFAT4oBgHgl3EQfbR0j/content/2301.08556v1.pdf'}
+page_content=' In each environment, we  collect a small number of expert grasping demonstrations  with a virtual reality controller.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFAT4oBgHgl3EQfbR0j/content/2301.08556v1.pdf'}
+page_content=' Because DART is difficult  to use with human demonstration collection, we compare  SPARTN to a vanilla BC policy trained on the same set of  demonstrations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFAT4oBgHgl3EQfbR0j/content/2301.08556v1.pdf'}
+page_content=' Policies are evaluated with the same ob-  jects seen in the demonstrations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFAT4oBgHgl3EQfbR0j/content/2301.08556v1.pdf'}
+page_content=' Initial object and robot  configurations are randomized during both data collection  and evaluation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFAT4oBgHgl3EQfbR0j/content/2301.08556v1.pdf'}
+page_content='  Training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFAT4oBgHgl3EQfbR0j/content/2301.08556v1.pdf'}
+page_content=' To improve NeRF quality for SPARTN, we pro-  gram the robot to collect a few images of the scene from  a fixed set of poses before the start of each demonstration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFAT4oBgHgl3EQfbR0j/content/2301.08556v1.pdf'}
+page_content='  This automatic process is only used to improve COLMAP’s  pose estimation and the subsequent NeRF training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFAT4oBgHgl3EQfbR0j/content/2301.08556v1.pdf'}
+page_content='  For  SPARTN, we generate Naug  =  50 augmented transi-  tions from each original transition in the demonstrations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFAT4oBgHgl3EQfbR0j/content/2301.08556v1.pdf'}
+page_content='  We sample perturbations ϵ according to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFAT4oBgHgl3EQfbR0j/content/2301.08556v1.pdf'}
+page_content=' 6 with α =  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFAT4oBgHgl3EQfbR0j/content/2301.08556v1.pdf'}
+page_content='05 radians and β = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFAT4oBgHgl3EQfbR0j/content/2301.08556v1.pdf'}
+page_content='4 mm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFAT4oBgHgl3EQfbR0j/content/2301.08556v1.pdf'}
+page_content=' Appendix D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFAT4oBgHgl3EQfbR0j/content/2301.08556v1.pdf'}
+page_content='2 describes  COLMAP and NeRF training in more detail, and Ap-  pendix B shows sample SPARTN observations rendered by  NeRF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFAT4oBgHgl3EQfbR0j/content/2301.08556v1.pdf'}
+page_content=' Aside from using the augmented dataset, SPARTN  policies are trained using the same BC architecture and  training hyperparameters described in Appendix D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFAT4oBgHgl3EQfbR0j/content/2301.08556v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFAT4oBgHgl3EQfbR0j/content/2301.08556v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFAT4oBgHgl3EQfbR0j/content/2301.08556v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFAT4oBgHgl3EQfbR0j/content/2301.08556v1.pdf'}
+page_content=' Results  Table 4 shows grasping success rates in the eight real-  world environments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFAT4oBgHgl3EQfbR0j/content/2301.08556v1.pdf'}
+page_content=' Quantitatively, SPARTN policies out-  perform the baseline BC policies across the board, on aver-  age achieving an absolute 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFAT4oBgHgl3EQfbR0j/content/2301.08556v1.pdf'}
+page_content='5% increase in success rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFAT4oBgHgl3EQfbR0j/content/2301.08556v1.pdf'}
+page_content='  Figure 7 shows qualitative differences in performance  between the BC and SPARTN policies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFAT4oBgHgl3EQfbR0j/content/2301.08556v1.pdf'}
+page_content=' SPARTN generally  exhibits more reactive behaviors than the baseline policy: it  navigates towards the target object better while occasionally  avoiding obstacles, executes actions with greater precision,  and even reattempts the grasp more successfully after an  initial miss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFAT4oBgHgl3EQfbR0j/content/2301.08556v1.pdf'}
+page_content=' In some cases, the differences are stark: for in-  stance, SPARTN may successfully move toward and grasp  the target object while the baseline fails to even reach it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFAT4oBgHgl3EQfbR0j/content/2301.08556v1.pdf'}
+page_content=' We  present further analysis of policy rollouts in Appendix D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFAT4oBgHgl3EQfbR0j/content/2301.08556v1.pdf'}
+page_content='3,  showing how SPARTN qualitatively performs better than  BC even in cases where both methods fail to grasp the tar-  get object.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFAT4oBgHgl3EQfbR0j/content/2301.08556v1.pdf'}
+page_content=' The supplementary videos of real-world pol-  icy rollouts illustrate all of these differences, revealing that  SPARTN induces important reactive closed-loop behaviors  that enable the manipulator to successfully execute grasps  in the real world.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFAT4oBgHgl3EQfbR0j/content/2301.08556v1.pdf'}
+page_content='  Figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFAT4oBgHgl3EQfbR0j/content/2301.08556v1.pdf'}
+page_content='  Sample real-world policy rollouts illustrating how  SPARTN succeeds (green) in cases where BC fails (red).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFAT4oBgHgl3EQfbR0j/content/2301.08556v1.pdf'}
+page_content=' Top  left: While BC fails to reach the steel bottle, SPARTN success-  fully reaches and grasps it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFAT4oBgHgl3EQfbR0j/content/2301.08556v1.pdf'}
+page_content=' Top right: While BC collides into the  orange bottle (a distractor object), SPARTN takes a more rounded  path to avoid it before grasping the white Lubriderm bottle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFAT4oBgHgl3EQfbR0j/content/2301.08556v1.pdf'}
+page_content=' Bot-  tom left: BC fails to recover after a missed grasp, while SPARTN  successfully reattempts the grasp after failing the first time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFAT4oBgHgl3EQfbR0j/content/2301.08556v1.pdf'}
+page_content=' Bot-  tom right: SPARTN operates with higher precision than BC and  successfully completes a difficult fork grasping task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFAT4oBgHgl3EQfbR0j/content/2301.08556v1.pdf'}
+page_content='  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFAT4oBgHgl3EQfbR0j/content/2301.08556v1.pdf'}
+page_content=' Conclusion  We introduce SPARTN, which augments eye-in-hand  demonstrations with perturbed visual observations and cor-  rective actions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFAT4oBgHgl3EQfbR0j/content/2301.08556v1.pdf'}
+page_content='  Our augmentation leverages novel-view  synthesis, in particular NeRF, to produce these augmenta-  tions during training-time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFAT4oBgHgl3EQfbR0j/content/2301.08556v1.pdf'}
+page_content=' SPARTN can improve behavior  cloning training of robust, real-time, and closed-loop 6-DoF  visual control policies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFAT4oBgHgl3EQfbR0j/content/2301.08556v1.pdf'}
+page_content=' We show that SPARTN-trained poli-  cies outperform other offline-supervised methods in a sim-  ulated 6-DoF grasping generalization benchmark.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFAT4oBgHgl3EQfbR0j/content/2301.08556v1.pdf'}
+page_content=' Our poli-  cies can also perform on par with imitation methods that re-  quire depth information and online supervision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFAT4oBgHgl3EQfbR0j/content/2301.08556v1.pdf'}
+page_content=' We verify  that SPARTN can train policies to grasp a variety of objects  in the real world from limited human demonstrations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFAT4oBgHgl3EQfbR0j/content/2301.08556v1.pdf'}
+page_content='  Despite its strong performance, SPARTN also has some  limitations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFAT4oBgHgl3EQfbR0j/content/2301.08556v1.pdf'}
+page_content=' First, SPARTN is limited to tasks with static  scenes like grasping: extending to a broader set of manip-  ulation tasks would require effective view synthesis for dy-  namic scenes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFAT4oBgHgl3EQfbR0j/content/2301.08556v1.pdf'}
+page_content=' Second, training a neural radiance field for  every demonstration before training is computationally ex-  pensive, a limitation that may be mitigated through amor-  tized NeRF models [55, 59, 64].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFAT4oBgHgl3EQfbR0j/content/2301.08556v1.pdf'}
+page_content=' Finally, the noise distri-  bution must be tuned for a given platform, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFAT4oBgHgl3EQfbR0j/content/2301.08556v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFAT4oBgHgl3EQfbR0j/content/2301.08556v1.pdf'}
+page_content=' it is tuned  separately for the simulated and real experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFAT4oBgHgl3EQfbR0j/content/2301.08556v1.pdf'}
+page_content=' An in-  teresting direction for future work is to use policy actions  to generate the noise, which would result in a fully offline  variant of DAgger [46] that uses NeRF as a simulator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFAT4oBgHgl3EQfbR0j/content/2301.08556v1.pdf'}
+page_content='  8  Time  Time  BC  SPARTN  Time  Time  BC  SPARTN7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFAT4oBgHgl3EQfbR0j/content/2301.08556v1.pdf'}
+page_content=' Acknowledgements  We thank Jimmy Wu, Kaylee Burns, Bohan Wu, and  Suraj Nair for technical advice on various aspects of the real  robot setup, and Archit Sharma for helpful conceptual dis-  cussions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFAT4oBgHgl3EQfbR0j/content/2301.08556v1.pdf'}
+page_content=' This project was funded by ONR grant N00014-  21-1-2685.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFAT4oBgHgl3EQfbR0j/content/2301.08556v1.pdf'}
+page_content=' AZ acknowledges the support of the NSF Grad-  uate Research Fellowship.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFAT4oBgHgl3EQfbR0j/content/2301.08556v1.pdf'}
+page_content=' CF is a fellow of the CIFAR  Learning in Machines and Brains Program.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFAT4oBgHgl3EQfbR0j/content/2301.08556v1.pdf'}
+page_content='  References  [1] Michal Adamkiewicz, Timothy Chen, Adam Caccavale,  Rachel Gardner, Preston Culbertson, Jeannette Bohg, and  Mac Schwager.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFAT4oBgHgl3EQfbR0j/content/2301.08556v1.pdf'}
+page_content='  Vision-only robot navigation in a neural  radiance world.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFAT4oBgHgl3EQfbR0j/content/2301.08556v1.pdf'}
+page_content='  IEEE Robotics and Automation Letters,  7(2):4606–4613, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFAT4oBgHgl3EQfbR0j/content/2301.08556v1.pdf'}
+page_content=' 3  [2] Antonio Bicchi and Vijay Kumar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFAT4oBgHgl3EQfbR0j/content/2301.08556v1.pdf'}
+page_content='  Robotic grasping and  contact: A review.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFAT4oBgHgl3EQfbR0j/content/2301.08556v1.pdf'}
+page_content=' In Proceedings 2000 ICRA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFAT4oBgHgl3EQfbR0j/content/2301.08556v1.pdf'}
+page_content=' Millennium  conference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFAT4oBgHgl3EQfbR0j/content/2301.08556v1.pdf'}
+page_content=' IEEE international conference on robotics and  automation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFAT4oBgHgl3EQfbR0j/content/2301.08556v1.pdf'}
+page_content=' Symposia proceedings (Cat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFAT4oBgHgl3EQfbR0j/content/2301.08556v1.pdf'}
+page_content=' No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFAT4oBgHgl3EQfbR0j/content/2301.08556v1.pdf'}
+page_content=' 00CH37065),  volume 1, pages 348–353.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFAT4oBgHgl3EQfbR0j/content/2301.08556v1.pdf'}
+page_content=' IEEE, 2000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFAT4oBgHgl3EQfbR0j/content/2301.08556v1.pdf'}
+page_content=' 2  [3] Valts Blukis, Taeyeop Lee, Jonathan Tremblay, Bowen Wen,  In So Kweon, Kuk-Jin Yoon, Dieter Fox, and Stan Birchfield.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFAT4oBgHgl3EQfbR0j/content/2301.08556v1.pdf'}
+page_content='  Neural fields for robotic object manipulation from a single  image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFAT4oBgHgl3EQfbR0j/content/2301.08556v1.pdf'}
+page_content=' arXiv preprint arXiv:2210.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFAT4oBgHgl3EQfbR0j/content/2301.08556v1.pdf'}
+page_content='12126, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFAT4oBgHgl3EQfbR0j/content/2301.08556v1.pdf'}
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+page_content=' Image aug-  mentation is all you need: Regularizing deep reinforcement  learning from pixels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFAT4oBgHgl3EQfbR0j/content/2301.08556v1.pdf'}
+page_content=' In International Conference on Learn-  ing Representations, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFAT4oBgHgl3EQfbR0j/content/2301.08556v1.pdf'}
+page_content=' 2  [62] Lin Yen-Chen, Pete Florence, Jonathan T Barron, Tsung-Yi  Lin, Alberto Rodriguez, and Phillip Isola.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFAT4oBgHgl3EQfbR0j/content/2301.08556v1.pdf'}
+page_content=' Nerf-supervision:  Learning dense object descriptors from neural radiance  fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFAT4oBgHgl3EQfbR0j/content/2301.08556v1.pdf'}
+page_content=' arXiv preprint arXiv:2203.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFAT4oBgHgl3EQfbR0j/content/2301.08556v1.pdf'}
+page_content='01913, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFAT4oBgHgl3EQfbR0j/content/2301.08556v1.pdf'}
+page_content=' 3, 7  [63] Lin Yen-Chen, Pete Florence, Jonathan T Barron, Alberto  Rodriguez, Phillip Isola, and Tsung-Yi Lin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFAT4oBgHgl3EQfbR0j/content/2301.08556v1.pdf'}
+page_content=' inerf: Inverting  neural radiance fields for pose estimation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFAT4oBgHgl3EQfbR0j/content/2301.08556v1.pdf'}
+page_content=' In 2021 IEEE/RSJ  International Conference on Intelligent Robots and Systems  (IROS), pages 1323–1330.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFAT4oBgHgl3EQfbR0j/content/2301.08556v1.pdf'}
+page_content=' IEEE, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFAT4oBgHgl3EQfbR0j/content/2301.08556v1.pdf'}
+page_content=' 3  [64] Alex Yu, Vickie Ye, Matthew Tancik, and Angjoo Kanazawa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFAT4oBgHgl3EQfbR0j/content/2301.08556v1.pdf'}
+page_content='  pixelNeRF: Neural radiance fields from one or few images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFAT4oBgHgl3EQfbR0j/content/2301.08556v1.pdf'}
+page_content='  In CVPR, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFAT4oBgHgl3EQfbR0j/content/2301.08556v1.pdf'}
+page_content=' 8  11  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFAT4oBgHgl3EQfbR0j/content/2301.08556v1.pdf'}
+page_content=' Full Algorithm  Algorithm 2 SPARTN: Corrective Augmentation via NeRF  Input: D = {τ}N  i=1 - expert demonstrations  Input: ET C - Transform between camera and end-effector  Input: M ∈ {0, 1}w×h - mask for robot gripper  Output: Augmented transition dataset ˜D  1: ˜D ← {}  2: for τ = {(Ik, W T E  k ), (ET ˆ  E  k )}K  k=1 ∈ D do  3:  # EE pose → camera pose  4:  W T C  k ← W T E  k  ET C, for k = 0, · · · , K  5:  {V T C  k }K  k=1 ← COLMAP(I0, · · · , IK)  6:  β ← SOLVE(Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFAT4oBgHgl3EQfbR0j/content/2301.08556v1.pdf'}
+page_content=' 8) # Estimate β  7:  V T W  k  = V T C  k (W T C  k )−1, for k = 0, · · · , K  8:  # train NeRF  9:  F τ  Θ ← NeRF({(Ik, V T C  k )}K  k=1, M)  10:  for k ∈ (0, 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFAT4oBgHgl3EQfbR0j/content/2301.08556v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFAT4oBgHgl3EQfbR0j/content/2301.08556v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFAT4oBgHgl3EQfbR0j/content/2301.08556v1.pdf'}
+page_content=' , K) do  11:  for i = 1 : Naug do  12:  ε ← NoiseDist(SE(3))  13:  W T ˜  E  k ←  W T E  k ε # perturbed end-effector pose  14:  ˜  ET ˆ  E  k ← ε−1 ET ˆ  E  k # corrective relative action  15:  W T ˜  C  k ← W T ˜  E  k  ET C # perturbed camera pose  16:  V T ˜  C  k ← V T W  k  W T ˜  C  k  17:  ˜Ik ← NeRF-Render(V T ˜  C  k ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFAT4oBgHgl3EQfbR0j/content/2301.08556v1.pdf'}
+page_content=' F τ  Θ)  18:  # splice gripper into rendered image  19:  ˜Ik ← ¬M ⊙ ˜Ik + M ⊙ Ik  20:  ˜D ← ˜D ∪ {(˜Ik, W T ˜  E  k ), ( ˜  ET ˆ  E  k )}  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFAT4oBgHgl3EQfbR0j/content/2301.08556v1.pdf'}
+page_content=' Sample Perturbed Observations  As described in the main text, image observations ˜I for  perturbed camera poses are rendered via novel-view synthe-  sis techniques, in particular NeRF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFAT4oBgHgl3EQfbR0j/content/2301.08556v1.pdf'}
+page_content=' The section “Example  NeRF renders from augmented poses” in the supplemen-  tary website shows videos of the rendering output for both  simulation and real-world augmentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFAT4oBgHgl3EQfbR0j/content/2301.08556v1.pdf'}
+page_content='  C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFAT4oBgHgl3EQfbR0j/content/2301.08556v1.pdf'}
+page_content=' Simulation Details  C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFAT4oBgHgl3EQfbR0j/content/2301.08556v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFAT4oBgHgl3EQfbR0j/content/2301.08556v1.pdf'}
+page_content=' Training for Simulation Experiments  We train all of our RGB policies (SPARTN, DART, and  HA) closely following the architectural and optimization  hyperparameters of [58].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFAT4oBgHgl3EQfbR0j/content/2301.08556v1.pdf'}
+page_content=' The image encoder is a ResNet-  18 [15] pretrained on ImageNet [47], followed by a 3-layer  MLP with 512 hidden units each and ReLU activations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFAT4oBgHgl3EQfbR0j/content/2301.08556v1.pdf'}
+page_content=' The  MLP outputs the predicted action as a 6-D vector of pre-  dicted translations and Euler angles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFAT4oBgHgl3EQfbR0j/content/2301.08556v1.pdf'}
+page_content=' The final layer has  a tanh activation to scale the output between [−1, 1], then  each dimension is scaled to match the environment’s action  bounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFAT4oBgHgl3EQfbR0j/content/2301.08556v1.pdf'}
+page_content='  For behavior cloning, we use the same 3D point-  matching loss as [58].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFAT4oBgHgl3EQfbR0j/content/2301.08556v1.pdf'}
+page_content='  We optimize the objective using  the Adam optimizer with batch size 100 for 100, 000 steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFAT4oBgHgl3EQfbR0j/content/2301.08556v1.pdf'}
+page_content='  We train only the MLP (and not the pre-trained ResNet) for  the first 20, 000 steps, since “freeze-then-train” fine-tuning  techniques have been shown to be more robust [28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFAT4oBgHgl3EQfbR0j/content/2301.08556v1.pdf'}
+page_content=' After  20, 000 steps, the ResNet is unfrozen and the entire network  is trained as usual.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFAT4oBgHgl3EQfbR0j/content/2301.08556v1.pdf'}
+page_content='  C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFAT4oBgHgl3EQfbR0j/content/2301.08556v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFAT4oBgHgl3EQfbR0j/content/2301.08556v1.pdf'}
+page_content=' NeRF Training  We use Instant-NGP [41] to train a NeRF for each of the  2,500 demonstration scenes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFAT4oBgHgl3EQfbR0j/content/2301.08556v1.pdf'}
+page_content=' We train each NeRF for 3,500  steps, which takes 30 seconds on an NVIDIA GeFORCE  RTX 2080 Ti.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFAT4oBgHgl3EQfbR0j/content/2301.08556v1.pdf'}
+page_content=' To reduce total training time, we train the  NeRFs in parallel: with 4 GPUs, we can train all 2,500  NeRFs in ∼ 7 hours.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFAT4oBgHgl3EQfbR0j/content/2301.08556v1.pdf'}
+page_content=' Because camera calibration is exact in  simulation, we use the world-frame camera poses given by  calibration instead of COLMAP to train NeRF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFAT4oBgHgl3EQfbR0j/content/2301.08556v1.pdf'}
+page_content='  D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFAT4oBgHgl3EQfbR0j/content/2301.08556v1.pdf'}
+page_content=' Real-World Details  D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFAT4oBgHgl3EQfbR0j/content/2301.08556v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFAT4oBgHgl3EQfbR0j/content/2301.08556v1.pdf'}
+page_content=' Policy Training  For both BC and SPARTN, the image encoder for each  policy is a four-layer convolutional network followed by  two feedforward layers, and uses batch normalization and  ReLU activation functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFAT4oBgHgl3EQfbR0j/content/2301.08556v1.pdf'}
+page_content=' The image embedding is fed  into a three-layer ReLU MLP policy head, which outputs  the predicted action.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFAT4oBgHgl3EQfbR0j/content/2301.08556v1.pdf'}
+page_content=' We train the policy from scratch with  mean squared error, using Adam with learning rate 5×10−4  and a batch size of 64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFAT4oBgHgl3EQfbR0j/content/2301.08556v1.pdf'}
+page_content=' No image augmentations like ran-  dom crop are applied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFAT4oBgHgl3EQfbR0j/content/2301.08556v1.pdf'}
+page_content='  D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFAT4oBgHgl3EQfbR0j/content/2301.08556v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFAT4oBgHgl3EQfbR0j/content/2301.08556v1.pdf'}
+page_content=' COLMAP and NeRF Training  In the real world, we train our NeRF using cam-  era poses {V HC  k }K  k=1 estimated by COLMAP [48, 49].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFAT4oBgHgl3EQfbR0j/content/2301.08556v1.pdf'}
+page_content='  From the camera calibration, we also have pose transforms  {W T C  k }K  k=1 which are too noisy to train a good NeRF from,  but will allow us to convert between world and COLMAP  frames.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFAT4oBgHgl3EQfbR0j/content/2301.08556v1.pdf'}
+page_content=' Conversion is necessary because we want to ren-  der from perturbed camera poses W T ˜  C, but we must call  NeRF-Render using the transform V H ˜  C (COLMAP’s ref-  erence frame) instead.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFAT4oBgHgl3EQfbR0j/content/2301.08556v1.pdf'}
+page_content=' From Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFAT4oBgHgl3EQfbR0j/content/2301.08556v1.pdf'}
+page_content=' 2, we can do this conver-  sion if we have the transform V T W and the scale factor β  which distinguishes T and H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFAT4oBgHgl3EQfbR0j/content/2301.08556v1.pdf'}
+page_content='  We can estimate both quantities using the (W T C  k , V HC  k )  pairs in each demonstration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFAT4oBgHgl3EQfbR0j/content/2301.08556v1.pdf'}
+page_content='  Recalling that  aT b  =  (bT a)−1, we have for any j ̸= k:  β Trans  �CT W  j  W T C  k  �  = Trans  �CHV  j  V HC  k  �  (8)  This leads to an overconstrained linear system in one un-  known β, which can be solved with linear regression.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFAT4oBgHgl3EQfbR0j/content/2301.08556v1.pdf'}
+page_content=' After  solving for β, we can rescale from H → T:  V T C  k :=  �  Rot  �CHV  k  �  , (1/β) Trans  �CHV  k  ��  (9)  12  Figure 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFAT4oBgHgl3EQfbR0j/content/2301.08556v1.pdf'}
+page_content='  Sample real-world policy rollouts illustrating cases  where both BC and SPARTN policies fail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFAT4oBgHgl3EQfbR0j/content/2301.08556v1.pdf'}
+page_content=' Top left: BC fails  to reach the thin box, while SPARTN successfully reaches it and  nearly completes the grasp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFAT4oBgHgl3EQfbR0j/content/2301.08556v1.pdf'}
+page_content=' Top right: While BC completely  misses the white tape roll, SPARTN makes contact but fails to  grasp it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFAT4oBgHgl3EQfbR0j/content/2301.08556v1.pdf'}
+page_content=' Bottom left: BC fails to reach the steel water bottle,  while SPARTN contacts it but fails to grasp it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFAT4oBgHgl3EQfbR0j/content/2301.08556v1.pdf'}
+page_content=' Bottom right:  BC reaches the Lubriderm bottle over but knocks it over before  grasping it, while SPARTN nearly completes the grasp as the bot-  tle barely slips away from the robot’s fingers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFAT4oBgHgl3EQfbR0j/content/2301.08556v1.pdf'}
+page_content='  Now solving for V T W is simple:  V T W  k  = V T C  k (W T C  k )−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFAT4oBgHgl3EQfbR0j/content/2301.08556v1.pdf'}
+page_content='  (10)  In theory, V T W is constant for all k, but in practice there  is noise from both COLMAP and the real-world pose esti-  mates, so we simply compute V T W  k  separately for each k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFAT4oBgHgl3EQfbR0j/content/2301.08556v1.pdf'}
+page_content='  D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFAT4oBgHgl3EQfbR0j/content/2301.08556v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFAT4oBgHgl3EQfbR0j/content/2301.08556v1.pdf'}
+page_content=' Additional Qualitative Analysis  Figure 8 shows sample evaluation trials where both  SPARTN and BC policies fail, given the same initial ob-  ject and end-effector configurations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFAT4oBgHgl3EQfbR0j/content/2301.08556v1.pdf'}
+page_content='  Even in these fail-  ure cases, SPARTN qualitatively performs better than BC,  nearly grasping the object in some cases, and dropping the  object soon after grasping in other cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFAT4oBgHgl3EQfbR0j/content/2301.08556v1.pdf'}
+page_content=' See the videos on  the supplemental website for additional qualitative results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFAT4oBgHgl3EQfbR0j/content/2301.08556v1.pdf'}
+page_content='  D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFAT4oBgHgl3EQfbR0j/content/2301.08556v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFAT4oBgHgl3EQfbR0j/content/2301.08556v1.pdf'}
+page_content=' Real-World Environments  The real-world environments shown in Figure 6 each in-  clude a single target object to grasp among other distractor  objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFAT4oBgHgl3EQfbR0j/content/2301.08556v1.pdf'}
+page_content=' The initial positions of all objects and the robotic  end-effector are randomized while collecting expert demon-  strations and during test time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFAT4oBgHgl3EQfbR0j/content/2301.08556v1.pdf'}
+page_content=' The initial orientations of the  objects are also randomized such that all policies must per-  form end-effector orientation control to complete the task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFAT4oBgHgl3EQfbR0j/content/2301.08556v1.pdf'}
+page_content='  To make a fair comparison, we evaluate different policies  given the same set of initial configurations (as shown in  Figure 7, Figure 8, and the videos on the supplemental web-  site);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFAT4oBgHgl3EQfbR0j/content/2301.08556v1.pdf'}
+page_content=' we do this by resetting the robot and environment to  the same conditions and alternating the rollouts of differ-  ent policies before moving on to the next test configuration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFAT4oBgHgl3EQfbR0j/content/2301.08556v1.pdf'}
+page_content='  Rollouts are considered successful if the robot grasps the  target object and lifts it steadily into the air.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFAT4oBgHgl3EQfbR0j/content/2301.08556v1.pdf'}
+page_content=' We describe  the details of each environment below:  Banana: The environment includes an artificial ba-  nana resting on a white ceramic plate, a red cup, an ar-  tificial avocado, an artificial tomato, and a plastic fork  and knife resting on a blue cloth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFAT4oBgHgl3EQfbR0j/content/2301.08556v1.pdf'}
+page_content='  Thin box: The environment includes a thin brown box  resting on a thicker white toy box.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFAT4oBgHgl3EQfbR0j/content/2301.08556v1.pdf'}
+page_content='  Steel water bottle: The environment includes a steel  water bottle, a red water bottle, and an aluminum can.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFAT4oBgHgl3EQfbR0j/content/2301.08556v1.pdf'}
+page_content='  The steel water bottle is highly reflective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFAT4oBgHgl3EQfbR0j/content/2301.08556v1.pdf'}
+page_content='  Wine glass: The environment includes a wine glass,  red water bottle, and blue water bottle all placed  upside-down and leaning against a brown box.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFAT4oBgHgl3EQfbR0j/content/2301.08556v1.pdf'}
+page_content=' The  wine glass is transparent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFAT4oBgHgl3EQfbR0j/content/2301.08556v1.pdf'}
+page_content='  Lubriderm bottle: The environment includes a white  Lubriderm bottle, an orange Neutrogena bottle, and a  small white container of moisturizer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFAT4oBgHgl3EQfbR0j/content/2301.08556v1.pdf'}
+page_content='  White tape roll: The environment includes a roll of  white tape resting on a roll of blue tape, as well as a  pair of black scissors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFAT4oBgHgl3EQfbR0j/content/2301.08556v1.pdf'}
+page_content=' The rolls of tape are radially  symmetric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFAT4oBgHgl3EQfbR0j/content/2301.08556v1.pdf'}
+page_content='  Tupperware: The environment includes a plastic tup-  perware container with a red lid, as well as three artifi-  cial food items: a donut, a cookie, and a slice of pizza.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFAT4oBgHgl3EQfbR0j/content/2301.08556v1.pdf'}
+page_content='  Fork: The environment includes a fork resting on top  of a plastic tupperware container.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFAT4oBgHgl3EQfbR0j/content/2301.08556v1.pdf'}
+page_content=' The fork’s location  and orientation relative to the surface of the tupper-  ware container is fixed, but the position of the entire  assembly is randomized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFAT4oBgHgl3EQfbR0j/content/2301.08556v1.pdf'}
+page_content='  13  Time  Time  BC  SPARTN  Time  Time  BC  SPARTN' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFAT4oBgHgl3EQfbR0j/content/2301.08556v1.pdf'}
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+arXiv:2301.05648v1  [eess.SP]  13 Jan 2023
+Reconfigurable Intelligent Surface Empowered
+Rate-Splitting Multiple Access for Simultaneous
+Wireless Information and Power Transfer
+Chengzhong Tian∗, Yijie Mao∗, Kangchun Zhao∗, Yuanming Shi∗, Bruno Clerckx¶†
+∗School of Information Science and Technology, ShanghaiTech University, Shanghai, China
+¶Department of Electrical and Electronic Engineering, Imperial College London, United Kingdom
+†Silicon Austria Labs (SAL), Graz A-8010, Austria
+Email: {tianchzh, maoyj, zhaokch, shiym}@shanghaitech.edu.cn, b.clerckx@imperial.ac.uk
+Abstract—Rate-splitting multiple access (RSMA) and recon-
+figurable intelligent surface (RIS) have been both recognized
+as promising techniques for 6G. The benefits of combining the
+two techniques to enhance the spectral and energy efficiency
+have been recently exploited in communication-only networks.
+Inspired by the recent advances on RIS empowered RSMA, in
+this work we investigate the use of RIS empowered RSMA for
+simultaneous wireless information and power transfer (SWIPT)
+with one transmitter concurrently sending information to mul-
+tiple information receivers (IRs) and transferring energy to
+multiple energy receivers (ERs). Specifically, we jointly optimize
+the transmit beamformers and the RIS reflection coefficients
+to maximize the weighted sum-rate (WSR) of IRs under the
+harvested energy constraint of ERs and the transmit power
+constraint. An alternating optimization and successive convex
+approximation (SCA)-based optimization framework is then
+proposed to solve the problem. Numerical results show that by
+marrying the benefits of RSMA and RIS, the proposed RIS
+empowered RSMA achieves a better tradeoff between the WSR
+of IRs and energy harvested at ERs. Therefore, we conclude that
+RIS empowered RSMA is a promising strategy for SWIPT.
+Index Terms—Simultaneous wireless information and power
+transfer (SWIPT), reconfigurable intelligent surface (RIS), rate
+splitting multiple access (RSMA), weighted sum-rate (WSR).
+I. INTRODUCTION
+In recent years, with the rise of the concept of the internet
+of things (IoT), low-power communications specially catered
+for IoT nodes with non-replaceable batteries have become
+indispensable for 6G and beyond. As radio frequency signals
+carry both information and energy, simultaneous wireless
+information and power transfer (SWIPT), which concurrently
+transmits information to the information receivers (IRs) and
+provides energy for wireless charging for energy receivers
+(ERs), has become a feasible solution to achieve ubiqui-
+tous and self-sustainable wireless communications for future
+ultra-dense networks with short communication distances [1].
+Meanwhile, as a promising interference management and
+powerful multiple access technique for 6G, rate-splitting mul-
+tiple access (RSMA) has significant advantages in improving
+the system performance in terms of spectral efficiency (SE),
+This work has been supported in part by the National Nature Science
+Foundation of China under Grant 62201347; and in part by Shanghai Sailing
+Program under Grant 22YF1428400.
+energy efficiency (EE), quality of service (QoS), user fairness,
+robustness against user mobility and channel state information
+imperfection, etc [2], [3], [4]. In a practical RSMA model with
+1-layer common stream (also known as 1-layer rate-splitting)
+[3], each user message is split into a common part and a
+private part. By encoding all common parts into a common
+stream to be decoded by all receivers and encoding the private
+parts into private streams respectively for the corresponding
+users at the transmitter while allowing each user to decode
+the common and private streams, RSMA achieves to partially
+decode the interference and partially treat the remaining in-
+terference as noise. Such powerful interference management
+capability makes RSMA more powerful than conventional
+multiple access techniques such as orthogonal multiple access
+(OMA), space-division multiple access (SDMA), and power-
+domain non-orthogonal multiple access (NOMA) [5]. RSMA
+is therefore a promising multiple access for SWIPT.
+RSMA empowered SWIPT was first studied in [6]. Under a
+sum energy constraint of ERs, RSMA empowered SWIPT is
+shown to achieve better WSR performance of IRs than SDMA
+and NOMA in a wide range of IR and ER deployments.
+Inspired by its superior performance, RSMA has been studied
+in SWIPT considering cooperative user relaying [7], cognitive
+radio [8], and imperfect channel state information at the
+transmitter (CSIT) [9], [10]. All the aforementioned works
+have shown the perfect marriage of RSMA and SWIPT. To
+further boost the performance, recent works have initiated
+the study of applying reconfigurable intelligent surface (RIS)
+to RSMA empowered SWIPT [11], [12]. Specifically, [11]
+investigates the joint transmit beamforming and RIS phase
+shifts optimization to minimize the transmit power under the
+QoS rate and energy harvesting constraint at the colocated
+IRs and ERs with a power-splitting structure. [12] extends
+the single RIS model in [11] to double RISs with reflection
+from one RIS to the other and investigates the user fairness
+beamforming design. Both [11] and [12] only investigate a
+SWIPT model with colocated IRs and ERs. There is a lack of
+investigation on the RIS empowered RSMA for SWIPT with
+separated IRs and ERs.
+In this work, we propose an RIS empowered RSMA model
+for SWIPT with separated IRs and ERs. Specifically, the
+
+RIS
+ !"
+ !#
+$!"
+$!%
+&',"
+&',(
+)',"
+)',(
+&*,"
+&*,(
+)*,"
+)*,(
++
+-',"
+-',#
+.',%
+& .',"
+.*,"
+.*,%
+-*,#
+-*,"
+H
+BS
+Fig. 1: A multi-antenna multi-user transmission network with RIS empowered
+SWIPT
+transmit beamforming, common rate allocation, and RIS phase
+shifts are jointly designed to maximize the WSR of IRs
+under the harvested energy constraint of the ERs and the
+transmit power constraint. An alternative optimization-based
+optimization algorithm is proposed to solve the problem. For
+the first time, this work illustrates the tradeoff between the
+WSR of IRs and energy harvested at ERs with or without
+RIS. For a given ER energy consumption constraint, we show
+an explicit rate region gain of RIS empowered RSMA over
+the non-RIS counterparts for SWIPT. Therefore, we draw the
+conclusion that RIS empowered RSMA is capable of further
+boosting the spectral efficiency of SWIPT with separated IRs
+and ERs. It is a promising transmission strategy for 6G.
+II. SYSTEM MODEL AND PROBLEM FORMULATION
+Consider an RIS-assisted RSMA for a multi-antenna SWIPT
+transmission network consisting of one multi-antenna base
+station (BS), one RIS with N elements, K single-antenna
+IRs, and J single-antenna ERs. The BS is equipped with
+Nt antennas, the IRs and ERs are respectively indexed by
+K = {1, 2, · · · , K} and J = {1, 2, · · · , J}. The reflecting
+elements at the RIS are indexed by N = {1, 2, · · · , N}. As per
+Fig. 1, the BS simultaneously transmits information to IRs and
+energy to ERs with the assistance of an RIS. The transmit sig-
+nal x ∈ CNt×1 contains both the information signals for IRs
+and the energy signals for ERs. RSMA is considered for infor-
+mation transmission. Specifically, the message of IR-k is split
+into a common part Wc,k and a private parts Wp,k, ∀k ∈ K.
+The common parts of all IRs {Wc,1 · · · , Wc,K} are combined
+into a common message Wc which is encoded to common
+stream sI
+c, while the private parts {Wp,1 · · · , Wp,K} are inde-
+pendently encoded to the private streams
+�
+sI
+1, · · · , sI
+K
+�
+. The
+set of streams for IRs sI =
+�
+sI
+c, sI
+1, · · · , sI
+K
+�T ∈ C(K+1)×1
+and ERs sE =
+�
+sE
+1, · · · , sE
+J
+�T ∈ CJ×1 are linearly precoded
+by P = [pc, p1, · · · , pK] and F = [f1, · · · , fJ], where
+pc ∈ CNt×1, pk ∈ CNt×1, fj ∈ CNt×1 are the respective
+precoders for the common stream, private stream for IR-k,
+and the energy signal for ER-j. The transmit signal is:
+x = pcsI
+c +
+�
+k∈K
+pksI
+k +
+�
+j∈J
+fjsE
+j.
+(1)
+We assume that E
+�
+sI �
+sI�H�
+= E
+�
+sE �
+sE�H�
+= I and
+the transmit power is limited to Pt. The transmit signal
+goes through both the direct channel between the BS and
+the IRs/ERs and the assisted channel from the BS to the
+RIS to the IRs/ERs. By controlling the reflecting phase shift
+θu ∈ [0, 2π] for each reflecting unit u, the RIS is able to
+reflect the signal towards IRs and ERs. Let θ = [θ1, · · · , θN]
+and Θ = diag(ejθ1, · · · , ejθN) be the reflecting phase shift
+vector and the corresponding phase shift matrix, respectively.
+hd,k and gd,j respectively denote the channels between the
+BS and IR-k as well as the BS and ER-j, while hr,k and
+gr,j represent the channel between the RIS and IR-k as
+well as the RIS to ER-j, respectively. H ∈ CN×Nt denotes
+the channel matrix from the BS to the RIS. The respective
+effective channels from the BS to IR-k and ER-j hH
+k and gH
+j
+are given by
+hH
+k = hH
+r,kΘH + hH
+d,k, ∀k ∈ K,
+gH
+j = gH
+r,jΘH + gH
+d,j, ∀j ∈ J .
+(2)
+Let nI
+k and nE
+j be the respective Additive White Gaussian
+Noises (AWGNs) received at IR-k and ER-j with zero mean
+and variance σ2
+k or σ2
+j , respectively. Perfect CSIT is assumed
+at all nodes. The signals received at IR-k and ER-j are
+respectively described as
+yI
+k = hH
+k x + nI
+k, ∀k ∈ K,
+yE
+j = gH
+j x + nE
+j, ∀j ∈ J .
+(3)
+The energy signal sE
+j does not carry any information, and is
+assumed to be perfectly known at the transmitter and IRs.
+Therefore, IRs are able to eliminate interference from the
+energy signals before decoding the intended signals.
+The transmission rates of the common stream sI
+c and the
+private stream sI
+k at IR-k are respectively given as
+Rc,k = log2
+�
+1 +
+��hH
+k pc
+��2
+�
+j∈K
+��hH
+k pj
+��2 + σ2
+k
+�
+, ∀k ∈ K,
+Rk = log2
+�
+1 +
+��hH
+k pk
+��2
+�
+j∈K,j̸=k
+��hH
+k pj
+��2 + σ2
+k
+�
+, ∀k ∈ K.
+(4)
+To ensure the common message can be decoded successfully
+by every IR, the achievable rate of the common stream shall
+not exceed Rc = min {R1,c, · · · , RK,c}. As Rc is shared by
+K IRs, we have �
+k∈K Ck = Rc, where Ck represents the
+part of the common rate Rc used to transmit Wc,k. The total
+achieve rate of IR-k is given by Rk,tot = Ck + Rk.
+Assuming a linear model of the energy harvester1, the
+harvested energy Qj is proportional to the power received at
+ER-j, which is given as
+Qj = ζ
+
+��gH
+j pc
+��2 +
+�
+k∈K
+��gH
+j pk
+��2 +
+�
+j′∈J
+��gH
+j fj′
+��2
+
+ , ∀j ∈ J ,
+(5)
+where ζ is the energy conversion efficiency and 0 ≤ ζ ≤ 1.
+1The linear model of the energy harvester is considered in this paper, the
+study of the nonlinear model of the energy harvester [13] will be studied in
+the future work.
+
+In this work, to find the optimal rate and energy tradeoff
+between the IRs and ERs, we jointly design the precoders
+P, F, common rate allocation c = [C1, · · · , CK] and RIS
+phase shifts θ to maximize the WSR of IRs for a certain sum
+harvested energy lower bound Eth at the ERs and the transmit
+power consumption upper bound Pt at the BS. Let uk denote
+the weight allocated to IR-k, the problem for the proposed
+RIS empowered RSMA model for SWIPT is formulated as
+max
+P,F,c,θ
+�
+k∈K
+uk(Ck + Rk)
+(6a)
+s. t.
+�
+k∈K
+Ck ≤ Rc,k, ∀k ∈ K,
+(6b)
+�
+j∈J
+Qj ≥ Eth,
+(6c)
+tr(PPH) + tr(FFH) ≤ Pt,
+(6d)
+c ≥ 0,
+(6e)
+0 ≤ θn ≤ 2π, ∀n ∈ N.
+(6f)
+Constraint (6b) ensures the successful decoding of the com-
+mon stream at all IRs. (6c) is the sum harvested energy
+constraint at the ERs and (6d) is the transmit power constraint
+at the BS. In order to solve the non-convexity of problem (6),
+we propose an optimization framework based on alternative
+optimization, which is delineated in the next section.
+III. PROPOSED OPTIMIZATION FRAMEWORK
+To solve problem (10), we first decompose the problem
+into two subproblems of beamforming {P, F, c} and RIS
+phase shifts {θ, c}, which are solved in an iterative manner.
+Since the user effective channels are reconfigured after tuning
+the RIS phase shifts or the beamformers, the common rate
+is changed accordingly and the common rate allocation is
+therefore required to be optimized in both subproblems. In
+the following two subsections, the optimization approaches to
+solve the two subproblems are specified followed by an overall
+optimization framework.
+A. Beamforming and Common Rate Optimization
+For a given RIS phase shifts θ, we follow [6] and use
+a weighted minimum mean squared error (WMMSE) and
+successive convex approximation (SCA)-based algorithm to
+find the near optimal precoders P, F and common rate al-
+location c. The augmented weighted MSEs are defined as
+ξc,k = wc,kǫc,k − log2(wc,k), ξk = wkǫk − log2(wk), where
+wc,k and wk are the weights of the MSEs of the common and
+private streams at IR-k. The optimal weights could be denoted
+as wc,k = wMMSE
+c,k
+≜ (ǫMMSE
+c,k
+)−1, wk = wMMSE
+k
+≜ (ǫMMSE
+k
+)−1
+by solving
+∂ξc,k(gMMSE
+c,k
+)
+∂wc,k
+= 0 and
+∂ξk(gMMSE
+c,k
+)
+∂wk
+= 0. Thus, the
+relationship between MSEs and rate can be established as
+ξMMSE
+c,k
+≜ 1 − Rc,k and ξMMSE
+k
+≜ 1 − Rk.
+In order to solve the non-convexity of constraint (6c), we
+carry out the first-order Taylor expansion to the harvested
+energy at each user. Based on the first-order lower bound of
+��gH
+j pk
+��2 at a given point p[n]
+k , which is given by
+��gH
+j pk
+��2 ≥ 2R((p[n]
+k )HgjgH
+j pk) −
+���gH
+j p[n]
+k
+���
+2
+≜ Ψ[n](pk, gj),
+(7)
+constraint (6c) becomes
+�
+j∈J
+ζ
+�
+Ψ[n](pc, gj) +
+�
+k∈K
+Ψ[n](pk, gj)
++
+�
+j′∈J
+Ψ[n](fj′, gj)
+
+ ≥ Eth.
+(8)
+Then, the original problem (6) is reformulated as
+min
+P,F,x,w,g
+�
+k∈K
+uk(Xk + ξk)
+(9a)
+s. t.
+�
+k∈K
+Xk + 1 ≥ ξc,k, ∀k ∈ K,
+(9b)
+tr(PPH) + tr(FFH) ≤ Pt,
+(9c)
+x ≤ 0,
+(9d)
+(8),
+where Xk = −Ck and x = [X1, · · · , XK] is the WMSE
+vector. w = [w1, · · · , wK, w1,c, · · · , wK,c] is the vector of all
+MSE weights. g = [g1, · · · , gK, g1,c, · · · , gK,c] is the vector
+containing all equalizers. Ploblem (9) is convex and can be
+solved by CVX toolbox direction [12]. The overall process of
+the SCA method to solve beamforming problem is illustrated
+in Algorithm 1, where WMMSE = �
+k∈K uk(Xk + ξk) and
+WSR = �
+k∈K uk(Ck +Rk) are the calculated WMMSE and
+WSR, respectively.
+Algorithm 1: WMMSE and SCA-based algorithm
+1 Initialize: m ← 0, n ← 0, P[n], F[n], WMMSE[n],
+WSR[m], w, g, θ;
+2 repeat
+3
+repeat
+4
+update (P[n+1], F[n+1], x[n+1]) by solving
+problem (9) using w, g and P[n], F[n];
+5
+update WMMSE[n+1] using (P[n+1], F[n+1]);
+6
+P∗ ← P[n+1], F∗ ← F[n+1], n ← n + 1;
+7
+until
+���WMMSE[n] − WMMSE[n−1]��� ≤ ǫ;
+8
+m ← m + 1, P[m] ← P∗, F[m] ← F∗,
+w ← wMMSE. g ← gMMSE;
+9
+c[m] = −x[m];
+10
+update WSR[m] using (P[m], c[m]);
+11 until
+���WSR[m] − WSR[m−1]��� ≤ ǫ;
+B. Common rate and Phase Optimization
+For given precoders P, F and common rate allocation c, we
+then optimize RIS phase shifts θ based on the SCA approach.
+By introducing the slack variable vector η = [η1, · · · , ηK]T
+
+and slack variable ηt to denote the SINRs of the private and
+common streams, and defining aki ≜ (diag(hH
+r,k)Hpi), akc ≜
+(diag(hH
+r,k)Hpc), bjj′ ≜ (diag(gH
+r,j)Hf j′), sn ≜ ejθn and
+s ≜ [s1, · · · , sN]T . With the help of aki, akc, bjj′ and s,
+we can show hH
+r,kΘHpk = aH
+kks, gH
+r,jΘHfj′ = bH
+jj′s, the
+original problem is equivalently reformulated as
+max
+θ,η,c,ηt
+�
+k∈K
+uk(Ck + log2(1 + ηk))
+(10a)
+s. t.
+���aH
+kcs + hH
+d,kpc
+���
+2
+�
+i∈K
+���aH
+kis + hH
+d,kpi
+���
+2
++ σ2
+k
+≥ ηt, ∀k ∈ K,
+(10b)
+���aH
+kks + hH
+d,kpk
+���
+2
+�
+i∈K,i̸=k
+���aH
+kis + hH
+d,kpi
+���
+2
++ σ2
+k
+≥ ηk, ∀k ∈ K,
+(10c)
+�
+j∈J
+ζ
+�
+��bH
+jcs + gH
+d,jpc
+��2 +
+�
+k∈K
+��bH
+jks + gH
+d,jpk
+��2
++
+�
+j′∈J
+��bH
+jj′s + gH
+d,jfj′
+��2
+
+ ≥ Eth,
+(10d)
+log2(1 + ηt) ≥
+�
+k∈K
+Ck,
+(10e)
+|sn| = 1, ∀n ∈ N.
+(10f)
+To deal with the non-convexity of constraint (10d), we
+carry out the first-order Taylor approximation to the harvested
+energy at each user. Based on the first-order lower bound of
+���bH
+jj′s + gH
+d,jfj′
+���
+2
+at a given point s[t−1], which is given by
+��bH
+jj′s + gH
+d,jfj′��2
+≥ 2R((bjj′s(t−1) + gH
+d,jfj′)Hbjj′s) −
+���bjj′s(t−1) + gH
+d,jfj′
+���
+2
+≜ Q′(t−1)(fj′, gd,j),
+(11)
+where s(t−1)
+n
+is the value of the variable sn at the t−1 iteration.
+With the help of (11), constraint (10d) is rewritten as
+�
+j∈J
+ζ
+�
+Q′(t−1)(pc, gd,j) +
+�
+k∈K
+Q′(t−1)(pk, gd,j)
++
+�
+j′∈J
+Q′(t−1)(fj′, gd,j)
+
+ ≥ Eth.
+(12)
+To handle the non-convexity constraint (10f), we add a
+compensation term to the objective function and problem (10)
+is rewritten as
+max
+s,η,c,ηt
+�
+k∈K
+uk(Ck + log2(1 + ηk))) + C
+�
+k∈K
+(|sn|2 − 1)
+(13a)
+s. t. |sn| ≤ 1, ∀n ∈ N,
+(13b)
+(10b), (10c), (10e), (12),
+where C is a positive constant. Constraints (10b), (10c)
+and (13a) remain non-convex. Following [14], we use the
+first-order Taylor expansion to find the upper bound of
+C �
+k∈K(|sn|2 − 1) at a given points s[t−1], which is
+C �
+k∈K(|sn|2 − 1) ≤ 2C �
+k∈K s(t−1)
+n
+(sn − s(t−1)
+n
+) and
+approximate (13a) by
+max
+s,η,c,ηt
+�
+k∈K
+uk(Ck + log2 (1 + ηk))
++ 2C
+�
+k∈K
+s(t−1)
+n
+(sn − s(t−1)
+n
+)
+(14)
+To handle the non-convexity of constraints (10b) and (10c),
+we introduce variables β1, · · · , βK, βt and constraints (10b)
+and (10c) are equivalent to:
+��akks + hH
+d,kpk
+��2 ≥ βkηk, ∀k ∈ K,
+= 1
+4((βk + ηk)2 − (βk − ηk)2),
+(15)
+��akcs + hH
+d,kpc
+��2 ≥ βtηt, ∀k ∈ K,
+= 1
+4((βt + ηt)2 − (βt − ηt)2),
+(16)
+�
+i∈K,i̸=k
+��akis + hH
+d,kpi
+��2 + σ2
+k ≤ βk, ∀k ∈ K,
+(17)
+�
+i∈K
+��akis + hH
+d,kpi
+��2 + σ2
+k ≤ βt, ∀k ∈ K.
+(18)
+By further using the first-order Taylor expansion at the given
+point {s(t−1), β(t−1)
+k
+, η(t−1)
+k
+, β(t−1)
+t
+, η(t−1)
+t
+}, constraints (15),
+(16) can be approximated by
+2R((akks(t−1) + hH
+d,kpk)Hakks) −
+���akks(t−1)���
+2
++
+��hH
+d,kpk
+��2
+≥1
+4
+�
+(βk + ηk)2 − 2(β(t−1)
+k
+− η(t−1)
+k
+)(βk − ηk)
++(β(t−1)
+k
+− η(t−1)
+k
+)2�
+, ∀k ∈ K,
+(19)
+2R((akcs(t−1) + hH
+d,kpc)Hakcs) −
+���akcs(t−1)���
+2
++
+��hH
+d,kpc
+��2
+≥1
+4
+�
+(βt + ηt)2 − 2(β(t−1)
+t
+− η(t−1)
+t
+)(βt − ηt)
++(β(t−1)
+t
+− η(t−1)
+t
+)2�
+, ∀k ∈ K,
+(20)
+where the left-hand side of inequality is the first-order lower
+bound of
+���akks + hH
+d,kpk
+���
+2
+, the right-hand side of inequality
+are the first-order upper bound of 1
+4((βk + ηk)2 − (βk − ηk)2)
+and 1
+4((βt+ηt)2−(βt−ηt)2). With the above approximations,
+problem (13) can be approximated by the following convex
+problem:
+max
+s,η,β,c,ηt
+�
+k∈K
+uk(Ck + log2 (1 + ηk))
++ 2C
+�
+k∈K
+s(t−1)
+n
+(sn − s(t−1)
+n
+)
+s. t.
+(10e), (12), (13b), (17), (18), (19), (20),
+(21)
+
+Algorithm 2: Common rate and Phase Optimization
+with SCA
+1 Initialize: t ← 0, s[t], β[t], η[t]
+t , η[t], c[t], OBJ[t];
+2 repeat
+3
+update (s[t+1], β[t+1], η[t+1], η[t+1]
+t
+, c[t+1]) by
+solving problem (21) using s[t], β[t], η[t] and the
+given P∗, F∗;
+4
+update OBJ[t+1] using η[t+1] and the c[t+1];
+5
+t ← t + 1;
+6 until
+���OBJ[t] − OBJ[t−1]��� ≤ ǫ;
+where β = [β1, · · · , βK]T . Problem (21) can be solved by
+the standard CVX toolbox. The detailed process of using the
+SCA to solve problem (21) is illustrated in Algorithm 2, where
+OBJ = �
+k∈K uk(Ck+log2(1+ηk) is the calculated objective
+function in each iteration.
+C. Alternative Optimization Algorithm
+The overall AO algorithm to jointly optimize precoders P,
+F, the common rate c and RIS phase shifts θ is illustrated
+in Algorithm 3. Specifically, the optimal precoders P, F and
+common rate c are updated with the fixed phase by Algorithm
+1. After that, the optimal phase θ and common rate c are
+jointly updated with the fixed precoders P and F by Algorithm
+2. This process continues until convergence.
+Algorithm 3: AO algorithm
+1 Initialize: p ← 0, θ[p], P[p], F[p], c[p], WSR[p];
+2 repeat
+3
+p ← p + 1;
+4
+update (P[p], F[p], c[p]) by using Algorithm 1 with
+fixed θ;
+5
+update (θ[p], c[p]) by using Algorithm 2 with fixed
+P[p], F[p];
+6
+update WSR[p] with P[p], F[p], θ[p], c[p];
+7 until
+���WSR[p] − WSR[p−1]��� ≤ ǫ;
+Convergence Analysis: The proposed Algorithm 3 contains
+two loops. In the inner loop, Algorthm 1 and Algorithm 2
+are conducted. Algorithm 1 and Algorithm 2 are conducted.
+guarantees the WSR is monotonically increasing, and the
+solution at iteration m is a feasible solution of problem (9).
+Moreover, constraint (9c) ensures that the existence of an up-
+per bound on WSR. Therefore, the convergence of Algorithm
+1 is guaranteed. The solution at iteration t by Algorithm 2 is
+also a feasible solution of problem (21). Furthermore, the SCA
+Algorithm ensures that the objective function is monotonically
+increasing. Due to the RIS phase shift constraint (13b), the
+objective function has an upper bound, thus ensuring the
+convergence of Algorithm 2. In the outer loop, the solution
+of the beamforming problem is a feasible solution of the RIS
+phase shift problem, vise versa. Furthermore, WSR has an
+(x1,0)
+(x1,0)
+BS(0,0)
+ !
+RIS (1,2)
+(1,0)
+y(m)
+x(m)
+ "
+(20,0)
+IR
+ER
+Fig. 2: The location distribution of BS, RIS, IRs and ERs
+0
+2
+4
+6
+8
+10
+12
+14
+16
+18
+20
+Eth (  W)
+5.5
+6
+6.5
+7
+7.5
+8
+8.5
+9
+9.5
+Sum Rate (bit/s/Hz)
+RSMA+RIS
+SDMA+RIS
+NOMA+RIS
+RSMA
+SDMA
+NOMA
+12.02%
+27.69%
+Fig. 3: Rate-energy region comparison of different strategies with Pt = 10dB.
+upper bound due to constraints (6c) and (13b), which ensures
+the convergence of Algorithm 3.
+IV. NUMERICAL RESULTS
+In this section, We illustrate the performance of RIS em-
+powered RSMA for SWIPT and compare it with the following
+schemes:
+1) RSMA+RIS—This is the scheme proposed in Section
+II
+2) RSMA—This is the case without RIS in case 1) as
+studied in [6].
+3) SDMA+RIS—This is a special case when common rate
+Rc in case 1) is 0.
+4) SDMA—This is the case without RIS in case 3).
+5) NOMA+RIS—This is the case that RIS empowered
+NOMA for SWIPT.
+6) NOMA—This is the case without RIS in case 5).
+RSMA+RIS, SDMA+RIS and NOMA+RIS can be solved
+by Algorithm 3, while RSMA, SDMA and NOMA can be
+solved directly using Algorithms 1.
+In the simulation, the location distribution of BS, RIS, IRs
+and ERs are shown in Fig. 2. To be specific, ERs are randomly
+distributed in a circle of radius r1 = 0.1 meters and centered
+at (1, 0), IRs are randomly distributed in a circle of radius
+r1 = 1 meter and centered at (20, 0). The path loss of all
+channels is set to P(d) = Ld−α, where d represents the
+distance between the BS and the IRs/ERs, α is the path loss
+exponents. L represents the signal attenuation at a distance
+of 1 meter, which is usually set to -30 dB. In addition, the
+
+0
+1
+2
+3
+4
+5
+6
+7
+R1,tot (bit/s/Hz)
+0
+1
+2
+3
+4
+5
+6
+R2,tot (bit/s/Hz)
+RSMA+RIS
+SDMA+RIS
+NOMA+RIS
+RSMA
+SDMA
+NOMA
+Fig. 4: IR rate region comparison of different strategies with Pt = 10dB,
+Eth = 20µW
+1
+2
+3
+4
+5
+6
+7
+8
+9
+10
+Number of Iterations
+5.5
+6
+6.5
+7
+7.5
+8
+8.5
+9
+Sum Rate (bit/s/Hz)
+RSMA
+SDMA
+NOMA
+Fig. 5: Convergence of the algorithms in one channel realization with Pt =
+10dB, Eth = 20µW
+path loss exponents of BS to all IRs, BS to all ERs, BS to
+RIS, RIS to all IRs, RIS to all ERs are set to 2, 3, 3, 3.5
+and 1.5. Besides, the number of transmit antennas at the BS
+is Nt = 2, and the number of IRs, ERs, and RIS elements
+is K = J = 2, N = 8, respectively. The energy conversion
+efficiency is ζ = 0.5, the noise of user k is σ2
+k = −80 dBm
+and the convergence tolerance is ǫ = 10−3. All simulation
+results are averaged over 100 random channel realizations.
+Fig. 3 shows the rate-energy tradeoff when Pt = 10 dB
+and the weights of IRs are set to u1 = u2 = 1. It can
+be observed that when the collected energy Eth = 20µW,
+the respective gain of RSMA+RIS over SDMA, NOMA,
+RSMA, SDMA+RIS and NOMA+RIS are 27.55%, 38.81%,
+12.78% 12.02% and 27.69%. The simulation results show that
+RSMA+RIS achieves explicit performance gain over SDMA,
+NOMA, RSMA, SDMA+RIS and NOMA+RIS especially
+when the harvested energy constraint at the ERs is high. It
+achieves a better tradeoff between the WSR of IRs and the
+harvested energy at the ERs.
+Fig. 4 shows the rate regions of IRs for different strategies,
+with the assistance of RIS, the rate regions of all strategies
+are enlarged. One interesting observation is, the rate region of
+RSMA almost coincides with that of SDMA+RIS especially
+when the two users have similar weights. It further shows
+the promising capability of RSMA to reduce the transmission
+complexity while guaranteeing the quality of service.
+Fig. 5 demonstrates the convergence of the proposed AO
+algorithm with Pt = 10dB and Eth = 20µW. This graph
+shows that the proposed algorithm converges very quickly.
+V. CONCLUSION
+In this paper, we propose a SWIPT system empowered by
+RSMA and RIS. The precoders and RIS phase shifts are jointly
+designed at the transmitter to maximize the WSR of IRs under
+the harvested energy constraint of ERs. We propose an AO
+algorithm to solve this problem, which alternately optimizes
+the phase shift matrix and the remaining optimization vari-
+ables. Numerical results shows that the proposed approach
+significantly improves the WSR and harvested energy tradeoff
+between the IRs and ERs. RIS empowered RSMA is therefore
+a promising strategy for SWIPT.
+REFERENCES
+[1] M. Latva-aho, K. Lepp¨anen, F. Clazzer, and A. Munari, “Key drivers
+and research challenges for 6G ubiquitous wireless intelligence,” White
+Paper, 2020.
+[2] Y. Mao, O. Dizdar, B. Clerckx, R. Schober, P. Popovski, and H. V.
+Poor, “Rate-splitting multiple access: Fundamentals, survey, and future
+research trends,” IEEE Commun. Surv. Tutor., vol. 24, no. 4, pp. 2073–
+2126, 2022.
+[3] B. Clerckx, H. Joudeh, C. Hao, M. Dai, and B. Rassouli, “Rate splitting
+for MIMO wireless networks: a promising PHY-layer strategy for lte
+evolution,” IEEE Commun. Mag., vol. 54, no. 5, pp. 98–105, 2016.
+[4] B. Clerckx, Y. Mao, E. A. Jorswieck, J. Yuan, D. J. Love, E. Erkip, and
+D. Niyato, “A primer on rate-splitting multiple access: Tutorial, myths,
+and frequently asked questions,” arXiv:2209.00491., 2022.
+[5] B. Clerckx, Y. Mao, R. Schober, and H. V. Poor, “Rate-splitting unifying
+SDMA, OMA, NOMA, and multicasting in MISO broadcast channel:
+A simple two-user rate analysis,” IEEE Wireless Commun. Lett., vol. 9,
+no. 3, pp. 349–353, 2020.
+[6] Y. Mao, B. Clerckx, and V. O. Li, “Rate-splitting for multi-user multi-
+antenna wireless information and power transfer,” in Proc. IEEE 20th
+Int. Workshop Signal Process. Adv. Wireless Commun. (SPAWC), 2019.
+[7] T. Li, H. Zhang, X. Zhou, and D. Yuan, “Full-duplex cooperative rate-
+splitting for multigroup multicast with SWIPT,” IEEE Trans. Wirel.
+Commun., 2021.
+[8] M. R. C. Acosta, C. E. G. Moreta, and I. Koo, “Joint power allocation
+and power splitting for MISO-RSMA cognitive radio systems with
+SWIPT and information decoder users,” IEEE Syst. J., vol. 15, no. 4,
+pp. 5289–5300, 2020.
+[9] M. R. Camana, C. E. Garcia, and I. Koo, “Deep learning-assisted power
+minimization in underlay MISO-SWIPT systems based on rate-splitting
+multiple access,” IEEE Access, vol. 10, pp. 62 137–62 156, 2022.
+[10] X. Su, L. Li, H. Yin, and P. Zhang, “Robust power-and rate-splitting-
+based transceiver design in K-user MISO SWIPT interference channel
+under imperfect csit,” IEEE Commun. Lett., vol. 23, no. 3, pp. 514–517,
+2019.
+[11] M. R. Camana, C. E. Garcia, and I. Koo, “Rate-splitting multiple access
+in a MISO SWIPT system assisted by an intelligent reflecting surface,”
+Trans. Green Commun. Networking, vol. 6, no. 4, pp. 2084–2099, 2022.
+[12] H. Pang, M. Cui, G. Zhang, and Q. Wu, “Joint beamforming design and
+resource allocation for double-IRS-assisted RSMA SWIPT systems,”
+Computer Communications, vol. 196, pp. 229–238, 2022.
+[13] B. Clerckx, R. Zhang, R. Schober, D. W. K. Ng, D. I. Kim, and H. V.
+Poor, “Fundamentals of wireless information and power transfer: From
+RF energy harvester models to signal and system designs,” IEEE J. Sel.
+Areas Commun., vol. 37, no. 1, pp. 4–33, 2019.
+[14] Z. Yang, J. Shi, Z. Li, M. Chen, W. Xu, and M. Shikh-Bahaei, “Energy
+efficient rate splitting multiple access (RSMA) with reconfigurable
+intelligent surface,” in Proc. IEEE Int. Conf. Commun. Workshops (ICC
+Workshops), 2020, pp. 1–6.
+
+This figure "location.jpg" is available in "jpg"� format from:
+http://arxiv.org/ps/2301.05648v1
+
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+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfig8Z/content/2301.05648v1.pdf,len=428
+page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfig8Z/content/2301.05648v1.pdf'}
+page_content='05648v1  [eess.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfig8Z/content/2301.05648v1.pdf'}
+page_content='SP]  13 Jan 2023  Reconfigurable Intelligent Surface Empowered  Rate-Splitting Multiple Access for Simultaneous  Wireless Information and Power Transfer  Chengzhong Tian∗,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfig8Z/content/2301.05648v1.pdf'}
+page_content=' Yijie Mao∗,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfig8Z/content/2301.05648v1.pdf'}
+page_content=' Kangchun Zhao∗,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfig8Z/content/2301.05648v1.pdf'}
+page_content=' Yuanming Shi∗,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfig8Z/content/2301.05648v1.pdf'}
+page_content=' Bruno Clerckx¶†  ∗School of Information Science and Technology,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfig8Z/content/2301.05648v1.pdf'}
+page_content=' ShanghaiTech University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfig8Z/content/2301.05648v1.pdf'}
+page_content=' Shanghai,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfig8Z/content/2301.05648v1.pdf'}
+page_content=' China  ¶Department of Electrical and Electronic Engineering,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfig8Z/content/2301.05648v1.pdf'}
+page_content=' Imperial College London,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfig8Z/content/2301.05648v1.pdf'}
+page_content=' United Kingdom  †Silicon Austria Labs (SAL),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfig8Z/content/2301.05648v1.pdf'}
+page_content=' Graz A-8010,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfig8Z/content/2301.05648v1.pdf'}
+page_content=' Austria  Email: {tianchzh,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfig8Z/content/2301.05648v1.pdf'}
+page_content=' maoyj,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfig8Z/content/2301.05648v1.pdf'}
+page_content=' zhaokch,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfig8Z/content/2301.05648v1.pdf'}
+page_content=' shiym}@shanghaitech.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfig8Z/content/2301.05648v1.pdf'}
+page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfig8Z/content/2301.05648v1.pdf'}
+page_content='cn, b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfig8Z/content/2301.05648v1.pdf'}
+page_content='clerckx@imperial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfig8Z/content/2301.05648v1.pdf'}
+page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfig8Z/content/2301.05648v1.pdf'}
+page_content='uk  Abstract—Rate-splitting multiple access (RSMA) and recon-  figurable intelligent surface (RIS) have been both recognized  as promising techniques for 6G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfig8Z/content/2301.05648v1.pdf'}
+page_content=' The benefits of combining the  two techniques to enhance the spectral and energy efficiency  have been recently exploited in communication-only networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfig8Z/content/2301.05648v1.pdf'}
+page_content='  Inspired by the recent advances on RIS empowered RSMA, in  this work we investigate the use of RIS empowered RSMA for  simultaneous wireless information and power transfer (SWIPT)  with one transmitter concurrently sending information to mul-  tiple information receivers (IRs) and transferring energy to  multiple energy receivers (ERs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfig8Z/content/2301.05648v1.pdf'}
+page_content=' Specifically, we jointly optimize  the transmit beamformers and the RIS reflection coefficients  to maximize the weighted sum-rate (WSR) of IRs under the  harvested energy constraint of ERs and the transmit power  constraint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfig8Z/content/2301.05648v1.pdf'}
+page_content=' An alternating optimization and successive convex  approximation (SCA)-based optimization framework is then  proposed to solve the problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfig8Z/content/2301.05648v1.pdf'}
+page_content=' Numerical results show that by  marrying the benefits of RSMA and RIS, the proposed RIS  empowered RSMA achieves a better tradeoff between the WSR  of IRs and energy harvested at ERs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfig8Z/content/2301.05648v1.pdf'}
+page_content=' Therefore, we conclude that  RIS empowered RSMA is a promising strategy for SWIPT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfig8Z/content/2301.05648v1.pdf'}
+page_content='  Index Terms—Simultaneous wireless information and power  transfer (SWIPT), reconfigurable intelligent surface (RIS), rate  splitting multiple access (RSMA), weighted sum-rate (WSR).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfig8Z/content/2301.05648v1.pdf'}
+page_content='  I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfig8Z/content/2301.05648v1.pdf'}
+page_content=' INTRODUCTION  In recent years, with the rise of the concept of the internet  of things (IoT), low-power communications specially catered  for IoT nodes with non-replaceable batteries have become  indispensable for 6G and beyond.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfig8Z/content/2301.05648v1.pdf'}
+page_content=' As radio frequency signals  carry both information and energy, simultaneous wireless  information and power transfer (SWIPT), which concurrently  transmits information to the information receivers (IRs) and  provides energy for wireless charging for energy receivers  (ERs), has become a feasible solution to achieve ubiqui-  tous and self-sustainable wireless communications for future  ultra-dense networks with short communication distances [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfig8Z/content/2301.05648v1.pdf'}
+page_content='  Meanwhile, as a promising interference management and  powerful multiple access technique for 6G, rate-splitting mul-  tiple access (RSMA) has significant advantages in improving  the system performance in terms of spectral efficiency (SE),  This work has been supported in part by the National Nature Science  Foundation of China under Grant 62201347;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfig8Z/content/2301.05648v1.pdf'}
+page_content=' and in part by Shanghai Sailing  Program under Grant 22YF1428400.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfig8Z/content/2301.05648v1.pdf'}
+page_content='  energy efficiency (EE), quality of service (QoS), user fairness,  robustness against user mobility and channel state information  imperfection, etc [2], [3], [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfig8Z/content/2301.05648v1.pdf'}
+page_content=' In a practical RSMA model with  1-layer common stream (also known as 1-layer rate-splitting)  [3], each user message is split into a common part and a  private part.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfig8Z/content/2301.05648v1.pdf'}
+page_content=' By encoding all common parts into a common  stream to be decoded by all receivers and encoding the private  parts into private streams respectively for the corresponding  users at the transmitter while allowing each user to decode  the common and private streams, RSMA achieves to partially  decode the interference and partially treat the remaining in-  terference as noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfig8Z/content/2301.05648v1.pdf'}
+page_content=' Such powerful interference management  capability makes RSMA more powerful than conventional  multiple access techniques such as orthogonal multiple access  (OMA), space-division multiple access (SDMA), and power-  domain non-orthogonal multiple access (NOMA) [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfig8Z/content/2301.05648v1.pdf'}
+page_content=' RSMA  is therefore a promising multiple access for SWIPT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfig8Z/content/2301.05648v1.pdf'}
+page_content='  RSMA empowered SWIPT was first studied in [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfig8Z/content/2301.05648v1.pdf'}
+page_content=' Under a  sum energy constraint of ERs, RSMA empowered SWIPT is  shown to achieve better WSR performance of IRs than SDMA  and NOMA in a wide range of IR and ER deployments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfig8Z/content/2301.05648v1.pdf'}
+page_content='  Inspired by its superior performance, RSMA has been studied  in SWIPT considering cooperative user relaying [7], cognitive  radio [8], and imperfect channel state information at the  transmitter (CSIT) [9], [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfig8Z/content/2301.05648v1.pdf'}
+page_content=' All the aforementioned works  have shown the perfect marriage of RSMA and SWIPT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfig8Z/content/2301.05648v1.pdf'}
+page_content=' To  further boost the performance, recent works have initiated  the study of applying reconfigurable intelligent surface (RIS)  to RSMA empowered SWIPT [11], [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfig8Z/content/2301.05648v1.pdf'}
+page_content=' Specifically, [11]  investigates the joint transmit beamforming and RIS phase  shifts optimization to minimize the transmit power under the  QoS rate and energy harvesting constraint at the colocated  IRs and ERs with a power-splitting structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfig8Z/content/2301.05648v1.pdf'}
+page_content=' [12] extends  the single RIS model in [11] to double RISs with reflection  from one RIS to the other and investigates the user fairness  beamforming design.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfig8Z/content/2301.05648v1.pdf'}
+page_content=' Both [11] and [12] only investigate a  SWIPT model with colocated IRs and ERs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfig8Z/content/2301.05648v1.pdf'}
+page_content=' There is a lack of  investigation on the RIS empowered RSMA for SWIPT with  separated IRs and ERs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfig8Z/content/2301.05648v1.pdf'}
+page_content='  In this work, we propose an RIS empowered RSMA model  for SWIPT with separated IRs and ERs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfig8Z/content/2301.05648v1.pdf'}
+page_content=' Specifically, the  RIS  !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfig8Z/content/2301.05648v1.pdf'}
+page_content='"  !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfig8Z/content/2301.05648v1.pdf'}
+page_content='#  $!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfig8Z/content/2301.05648v1.pdf'}
+page_content='"  $!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfig8Z/content/2301.05648v1.pdf'}
+page_content='%  &\',"  &\',(  )\',"  )\',(  &*,"  &*,(  )*,"  )*,(  +  \',"  \',#  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfig8Z/content/2301.05648v1.pdf'}
+page_content=" ',%  & ." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfig8Z/content/2301.05648v1.pdf'}
+page_content=' \',"  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfig8Z/content/2301.05648v1.pdf'}
+page_content=' *,"  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfig8Z/content/2301.05648v1.pdf'}
+page_content=' *,%  -*,#  -*,"  H  BS  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfig8Z/content/2301.05648v1.pdf'}
+page_content=' 1: A multi-antenna multi-user transmission network with RIS empowered  SWIPT  transmit beamforming, common rate allocation, and RIS phase  shifts are jointly designed to maximize the WSR of IRs  under the harvested energy constraint of the ERs and the  transmit power constraint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfig8Z/content/2301.05648v1.pdf'}
+page_content=' An alternative optimization-based  optimization algorithm is proposed to solve the problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfig8Z/content/2301.05648v1.pdf'}
+page_content=' For  the first time, this work illustrates the tradeoff between the  WSR of IRs and energy harvested at ERs with or without  RIS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfig8Z/content/2301.05648v1.pdf'}
+page_content=' For a given ER energy consumption constraint, we show  an explicit rate region gain of RIS empowered RSMA over  the non-RIS counterparts for SWIPT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfig8Z/content/2301.05648v1.pdf'}
+page_content=' Therefore, we draw the  conclusion that RIS empowered RSMA is capable of further  boosting the spectral efficiency of SWIPT with separated IRs  and ERs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfig8Z/content/2301.05648v1.pdf'}
+page_content=' It is a promising transmission strategy for 6G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfig8Z/content/2301.05648v1.pdf'}
+page_content='  II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfig8Z/content/2301.05648v1.pdf'}
+page_content=' SYSTEM MODEL AND PROBLEM FORMULATION  Consider an RIS-assisted RSMA for a multi-antenna SWIPT  transmission network consisting of one multi-antenna base  station (BS), one RIS with N elements, K single-antenna  IRs, and J single-antenna ERs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfig8Z/content/2301.05648v1.pdf'}
+page_content=' The BS is equipped with  Nt antennas, the IRs and ERs are respectively indexed by  K = {1, 2, · · · , K} and J = {1, 2, · · · , J}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfig8Z/content/2301.05648v1.pdf'}
+page_content=' The reflecting  elements at the RIS are indexed by N = {1, 2, · · · , N}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfig8Z/content/2301.05648v1.pdf'}
+page_content=' As per  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfig8Z/content/2301.05648v1.pdf'}
+page_content=' 1, the BS simultaneously transmits information to IRs and  energy to ERs with the assistance of an RIS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfig8Z/content/2301.05648v1.pdf'}
+page_content=' The transmit sig-  nal x ∈ CNt×1 contains both the information signals for IRs  and the energy signals for ERs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfig8Z/content/2301.05648v1.pdf'}
+page_content=' RSMA is considered for infor-  mation transmission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfig8Z/content/2301.05648v1.pdf'}
+page_content=' Specifically, the message of IR-k is split  into a common part Wc,k and a private parts Wp,k, ∀k ∈ K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfig8Z/content/2301.05648v1.pdf'}
+page_content='  The common parts of all IRs {Wc,1 · · · , Wc,K} are combined  into a common message Wc which is encoded to common  stream sI  c, while the private parts {Wp,1 · · · , Wp,K} are inde-  pendently encoded to the private streams  �  sI  1, · · · , sI  K  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfig8Z/content/2301.05648v1.pdf'}
+page_content=' The  set of streams for IRs sI =  �  sI  c, sI  1, · · · , sI  K  �T ∈ C(K+1)×1  and ERs sE =  �  sE  1, · · · , sE  J  �T ∈ CJ×1 are linearly precoded  by P = [pc, p1, · · · , pK] and F = [f1, · · · , fJ], where  pc ∈ CNt×1, pk ∈ CNt×1, fj ∈ CNt×1 are the respective  precoders for the common stream, private stream for IR-k,  and the energy signal for ER-j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfig8Z/content/2301.05648v1.pdf'}
+page_content=' The transmit signal is:  x = pcsI  c +  �  k∈K  pksI  k +  �  j∈J  fjsE  j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfig8Z/content/2301.05648v1.pdf'}
+page_content='  (1)  We assume that E  �  sI �  sI�H�  = E  �  sE �  sE�H�  = I and  the transmit power is limited to Pt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfig8Z/content/2301.05648v1.pdf'}
+page_content=' The transmit signal  goes through both the direct channel between the BS and  the IRs/ERs and the assisted channel from the BS to the  RIS to the IRs/ERs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfig8Z/content/2301.05648v1.pdf'}
+page_content=' By controlling the reflecting phase shift  θu ∈ [0, 2π] for each reflecting unit u, the RIS is able to  reflect the signal towards IRs and ERs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfig8Z/content/2301.05648v1.pdf'}
+page_content=' Let θ = [θ1, · · · , θN]  and Θ = diag(ejθ1, · · · , ejθN) be the reflecting phase shift  vector and the corresponding phase shift matrix, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfig8Z/content/2301.05648v1.pdf'}
+page_content='  hd,k and gd,j respectively denote the channels between the  BS and IR-k as well as the BS and ER-j, while hr,k and  gr,j represent the channel between the RIS and IR-k as  well as the RIS to ER-j, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfig8Z/content/2301.05648v1.pdf'}
+page_content=' H ∈ CN×Nt denotes  the channel matrix from the BS to the RIS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfig8Z/content/2301.05648v1.pdf'}
+page_content=' The respective  effective channels from the BS to IR-k and ER-j hH  k and gH  j  are given by  hH  k = hH  r,kΘH + hH  d,k, ∀k ∈ K,  gH  j = gH  r,jΘH + gH  d,j, ∀j ∈ J .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfig8Z/content/2301.05648v1.pdf'}
+page_content='  (2)  Let nI  k and nE  j be the respective Additive White Gaussian  Noises (AWGNs) received at IR-k and ER-j with zero mean  and variance σ2  k or σ2  j , respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfig8Z/content/2301.05648v1.pdf'}
+page_content=' Perfect CSIT is assumed  at all nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfig8Z/content/2301.05648v1.pdf'}
+page_content=' The signals received at IR-k and ER-j are  respectively described as  yI  k = hH  k x + nI  k, ∀k ∈ K,  yE  j = gH  j x + nE  j, ∀j ∈ J .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfig8Z/content/2301.05648v1.pdf'}
+page_content='  (3)  The energy signal sE  j does not carry any information, and is  assumed to be perfectly known at the transmitter and IRs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfig8Z/content/2301.05648v1.pdf'}
+page_content='  Therefore, IRs are able to eliminate interference from the  energy signals before decoding the intended signals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfig8Z/content/2301.05648v1.pdf'}
+page_content='  The transmission rates of the common stream sI  c and the  private stream sI  k at IR-k are respectively given as  Rc,k = log2  �  1 +  ��hH  k pc  ��2  �  j∈K  ��hH  k pj  ��2 + σ2  k  �  , ∀k ∈ K,  Rk = log2  �  1 +  ��hH  k pk  ��2  �  j∈K,j̸=k  ��hH  k pj  ��2 + σ2  k  �  , ∀k ∈ K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfig8Z/content/2301.05648v1.pdf'}
+page_content='  (4)  To ensure the common message can be decoded successfully  by every IR, the achievable rate of the common stream shall  not exceed Rc = min {R1,c, · · · , RK,c}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfig8Z/content/2301.05648v1.pdf'}
+page_content=' As Rc is shared by  K IRs, we have �  k∈K Ck = Rc, where Ck represents the  part of the common rate Rc used to transmit Wc,k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfig8Z/content/2301.05648v1.pdf'}
+page_content=' The total  achieve rate of IR-k is given by Rk,tot = Ck + Rk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfig8Z/content/2301.05648v1.pdf'}
+page_content='  Assuming a linear model of the energy harvester1, the  harvested energy Qj is proportional to the power received at  ER-j, which is given as  Qj = ζ  \uf8eb  \uf8ed��gH  j pc  ��2 +  �  k∈K  ��gH  j pk  ��2 +  �  j′∈J  ��gH  j fj′  ��2  \uf8f6  \uf8f8 , ∀j ∈ J ,  (5)  where ζ is the energy conversion efficiency and 0 ≤ ζ ≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfig8Z/content/2301.05648v1.pdf'}
+page_content='  1The linear model of the energy harvester is considered in this paper, the  study of the nonlinear model of the energy harvester [13] will be studied in  the future work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfig8Z/content/2301.05648v1.pdf'}
+page_content='  In this work, to find the optimal rate and energy tradeoff  between the IRs and ERs, we jointly design the precoders  P, F, common rate allocation c = [C1, · · · , CK] and RIS  phase shifts θ to maximize the WSR of IRs for a certain sum  harvested energy lower bound Eth at the ERs and the transmit  power consumption upper bound Pt at the BS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfig8Z/content/2301.05648v1.pdf'}
+page_content=' Let uk denote  the weight allocated to IR-k, the problem for the proposed  RIS empowered RSMA model for SWIPT is formulated as  max  P,F,c,θ  �  k∈K  uk(Ck + Rk)  (6a)  s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfig8Z/content/2301.05648v1.pdf'}
+page_content=' t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfig8Z/content/2301.05648v1.pdf'}
+page_content='  �  k∈K  Ck ≤ Rc,k, ∀k ∈ K,  (6b)  �  j∈J  Qj ≥ Eth,  (6c)  tr(PPH) + tr(FFH) ≤ Pt,  (6d)  c ≥ 0,  (6e)  0 ≤ θn ≤ 2π, ∀n ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfig8Z/content/2301.05648v1.pdf'}
+page_content='  (6f)  Constraint (6b) ensures the successful decoding of the com-  mon stream at all IRs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfig8Z/content/2301.05648v1.pdf'}
+page_content=' (6c) is the sum harvested energy  constraint at the ERs and (6d) is the transmit power constraint  at the BS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfig8Z/content/2301.05648v1.pdf'}
+page_content=' In order to solve the non-convexity of problem (6),  we propose an optimization framework based on alternative  optimization, which is delineated in the next section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfig8Z/content/2301.05648v1.pdf'}
+page_content='  III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfig8Z/content/2301.05648v1.pdf'}
+page_content=' PROPOSED OPTIMIZATION FRAMEWORK  To solve problem (10), we first decompose the problem  into two subproblems of beamforming {P, F, c} and RIS  phase shifts {θ, c}, which are solved in an iterative manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfig8Z/content/2301.05648v1.pdf'}
+page_content='  Since the user effective channels are reconfigured after tuning  the RIS phase shifts or the beamformers, the common rate  is changed accordingly and the common rate allocation is  therefore required to be optimized in both subproblems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfig8Z/content/2301.05648v1.pdf'}
+page_content=' In  the following two subsections, the optimization approaches to  solve the two subproblems are specified followed by an overall  optimization framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfig8Z/content/2301.05648v1.pdf'}
+page_content='  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfig8Z/content/2301.05648v1.pdf'}
+page_content=' Beamforming and Common Rate Optimization  For a given RIS phase shifts θ, we follow [6] and use  a weighted minimum mean squared error (WMMSE) and  successive convex approximation (SCA)-based algorithm to  find the near optimal precoders P, F and common rate al-  location c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfig8Z/content/2301.05648v1.pdf'}
+page_content=' The augmented weighted MSEs are defined as  ξc,k = wc,kǫc,k − log2(wc,k), ξk = wkǫk − log2(wk), where  wc,k and wk are the weights of the MSEs of the common and  private streams at IR-k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfig8Z/content/2301.05648v1.pdf'}
+page_content=' The optimal weights could be denoted  as wc,k = wMMSE  c,k  ≜ (ǫMMSE  c,k  )−1, wk = wMMSE  k  ≜ (ǫMMSE  k  )−1  by solving  ∂ξc,k(gMMSE  c,k  )  ∂wc,k  = 0 and  ∂ξk(gMMSE  c,k  )  ∂wk  = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfig8Z/content/2301.05648v1.pdf'}
+page_content=' Thus, the  relationship between MSEs and rate can be established as  ξMMSE  c,k  ≜ 1 − Rc,k and ξMMSE  k  ≜ 1 − Rk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfig8Z/content/2301.05648v1.pdf'}
+page_content='  In order to solve the non-convexity of constraint (6c), we  carry out the first-order Taylor expansion to the harvested  energy at each user.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfig8Z/content/2301.05648v1.pdf'}
+page_content=' Based on the first-order lower bound of  ��gH  j pk  ��2 at a given point p[n]  k , which is given by  ��gH  j pk  ��2 ≥ 2R((p[n]  k )HgjgH  j pk) −  ���gH  j p[n]  k  ���  2  ≜ Ψ[n](pk, gj),  (7)  constraint (6c) becomes  �  j∈J  ζ  �  Ψ[n](pc, gj) +  �  k∈K  Ψ[n](pk, gj)  +  �  j′∈J  Ψ[n](fj′, gj)  \uf8f6  \uf8f8 ≥ Eth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfig8Z/content/2301.05648v1.pdf'}
+page_content='  (8)  Then, the original problem (6) is reformulated as  min  P,F,x,w,g  �  k∈K  uk(Xk + ξk)  (9a)  s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfig8Z/content/2301.05648v1.pdf'}
+page_content=' t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfig8Z/content/2301.05648v1.pdf'}
+page_content='  �  k∈K  Xk + 1 ≥ ξc,k, ∀k ∈ K,  (9b)  tr(PPH) + tr(FFH) ≤ Pt,  (9c)  x ≤ 0,  (9d)  (8),  where Xk = −Ck and x = [X1, · · · , XK] is the WMSE  vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfig8Z/content/2301.05648v1.pdf'}
+page_content=' w = [w1, · · · , wK, w1,c, · · · , wK,c] is the vector of all  MSE weights.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfig8Z/content/2301.05648v1.pdf'}
+page_content=' g = [g1, · · · , gK, g1,c, · · · , gK,c] is the vector  containing all equalizers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfig8Z/content/2301.05648v1.pdf'}
+page_content=' Ploblem (9) is convex and can be  solved by CVX toolbox direction [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfig8Z/content/2301.05648v1.pdf'}
+page_content=' The overall process of  the SCA method to solve beamforming problem is illustrated  in Algorithm 1, where WMMSE = �  k∈K uk(Xk + ξk) and  WSR = �  k∈K uk(Ck +Rk) are the calculated WMMSE and  WSR, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfig8Z/content/2301.05648v1.pdf'}
+page_content='  Algorithm 1: WMMSE and SCA-based algorithm  1 Initialize: m ← 0, n ← 0, P[n], F[n], WMMSE[n],  WSR[m], w, g, θ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfig8Z/content/2301.05648v1.pdf'}
+page_content='  2 repeat  3  repeat  4  update (P[n+1], F[n+1], x[n+1]) by solving  problem (9) using w, g and P[n], F[n];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfig8Z/content/2301.05648v1.pdf'}
+page_content='  5  update WMMSE[n+1] using (P[n+1], F[n+1]);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfig8Z/content/2301.05648v1.pdf'}
+page_content='  6  P∗ ← P[n+1], F∗ ← F[n+1], n ← n + 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfig8Z/content/2301.05648v1.pdf'}
+page_content='  7  until  ���WMMSE[n] − WMMSE[n−1]��� ≤ ǫ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfig8Z/content/2301.05648v1.pdf'}
+page_content='  8  m ← m + 1, P[m] ← P∗, F[m] ← F∗,  w ← wMMSE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfig8Z/content/2301.05648v1.pdf'}
+page_content=' g ← gMMSE;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfig8Z/content/2301.05648v1.pdf'}
+page_content='  9  c[m] = −x[m];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfig8Z/content/2301.05648v1.pdf'}
+page_content='  10  update WSR[m] using (P[m], c[m]);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfig8Z/content/2301.05648v1.pdf'}
+page_content='  11 until  ���WSR[m] − WSR[m−1]��� ≤ ǫ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfig8Z/content/2301.05648v1.pdf'}
+page_content='  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfig8Z/content/2301.05648v1.pdf'}
+page_content=' Common rate and Phase Optimization  For given precoders P, F and common rate allocation c, we  then optimize RIS phase shifts θ based on the SCA approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfig8Z/content/2301.05648v1.pdf'}
+page_content='  By introducing the slack variable vector η = [η1, · · · , ηK]T  and slack variable ηt to denote the SINRs of the private and  common streams, and defining aki ≜ (diag(hH  r,k)Hpi), akc ≜  (diag(hH  r,k)Hpc), bjj′ ≜ (diag(gH  r,j)Hf j′), sn ≜ ejθn and  s ≜ [s1, · · · , sN]T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfig8Z/content/2301.05648v1.pdf'}
+page_content=' With the help of aki, akc, bjj′ and s,  we can show hH  r,kΘHpk = aH  kks, gH  r,jΘHfj′ = bH  jj′s, the  original problem is equivalently reformulated as  max  θ,η,c,ηt  �  k∈K  uk(Ck + log2(1 + ηk))  (10a)  s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfig8Z/content/2301.05648v1.pdf'}
+page_content=' t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfig8Z/content/2301.05648v1.pdf'}
+page_content='  ���aH  kcs + hH  d,kpc  ���  2  �  i∈K  ���aH  kis + hH  d,kpi  ���  2  + σ2  k  ≥ ηt, ∀k ∈ K,  (10b)  ���aH  kks + hH  d,kpk  ���  2  �  i∈K,i̸=k  ���aH  kis + hH  d,kpi  ���  2  + σ2  k  ≥ ηk, ∀k ∈ K,  (10c)  �  j∈J  ζ  �  ��bH  jcs + gH  d,jpc  ��2 +  �  k∈K  ��bH  jks + gH  d,jpk  ��2  +  �  j′∈J  ��bH  jj′s + gH  d,jfj′  ��2  \uf8f6  \uf8f8 ≥ Eth,  (10d)  log2(1 + ηt) ≥  �  k∈K  Ck,  (10e)  |sn| = 1, ∀n ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfig8Z/content/2301.05648v1.pdf'}
+page_content='  (10f)  To deal with the non-convexity of constraint (10d), we  carry out the first-order Taylor approximation to the harvested  energy at each user.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfig8Z/content/2301.05648v1.pdf'}
+page_content=' Based on the first-order lower bound of  ���bH  jj′s + gH  d,jfj′  ���  2  at a given point s[t−1], which is given by  ��bH  jj′s + gH  d,jfj′��2  ≥ 2R((bjj′s(t−1) + gH  d,jfj′)Hbjj′s) −  ���bjj′s(t−1) + gH  d,jfj′  ���  2  ≜ Q′(t−1)(fj′, gd,j),  (11)  where s(t−1)  n  is the value of the variable sn at the t−1 iteration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfig8Z/content/2301.05648v1.pdf'}
+page_content='  With the help of (11), constraint (10d) is rewritten as  �  j∈J  ζ  �  Q′(t−1)(pc, gd,j) +  �  k∈K  Q′(t−1)(pk, gd,j)  +  �  j′∈J  Q′(t−1)(fj′, gd,j)  \uf8f6  \uf8f8 ≥ Eth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfig8Z/content/2301.05648v1.pdf'}
+page_content='  (12)  To handle the non-convexity constraint (10f), we add a  compensation term to the objective function and problem (10)  is rewritten as  max  s,η,c,ηt  �  k∈K  uk(Ck + log2(1 + ηk))) + C  �  k∈K  (|sn|2 − 1)  (13a)  s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfig8Z/content/2301.05648v1.pdf'}
+page_content=' t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfig8Z/content/2301.05648v1.pdf'}
+page_content=' |sn| ≤ 1, ∀n ∈ N,  (13b)  (10b), (10c), (10e), (12),  where C is a positive constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfig8Z/content/2301.05648v1.pdf'}
+page_content=' Constraints (10b), (10c)  and (13a) remain non-convex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfig8Z/content/2301.05648v1.pdf'}
+page_content=' Following [14],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfig8Z/content/2301.05648v1.pdf'}
+page_content=' we use the  first-order Taylor expansion to find the upper bound of  C �  k∈K(|sn|2 − 1) at a given points s[t−1],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfig8Z/content/2301.05648v1.pdf'}
+page_content=' which is  C �  k∈K(|sn|2 − 1) ≤ 2C �  k∈K s(t−1)  n  (sn − s(t−1)  n  ) and  approximate (13a) by  max  s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfig8Z/content/2301.05648v1.pdf'}
+page_content='η,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfig8Z/content/2301.05648v1.pdf'}
+page_content='c,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfig8Z/content/2301.05648v1.pdf'}
+page_content='ηt  �  k∈K  uk(Ck + log2 (1 + ηk))  + 2C  �  k∈K  s(t−1)  n  (sn − s(t−1)  n  )  (14)  To handle the non-convexity of constraints (10b) and (10c),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfig8Z/content/2301.05648v1.pdf'}
+page_content='  we introduce variables β1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfig8Z/content/2301.05648v1.pdf'}
+page_content=' · · · ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfig8Z/content/2301.05648v1.pdf'}
+page_content=' βK,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfig8Z/content/2301.05648v1.pdf'}
+page_content=' βt and constraints (10b)  and (10c) are equivalent to:  ��akks + hH  d,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfig8Z/content/2301.05648v1.pdf'}
+page_content='kpk  ��2 ≥ βkηk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfig8Z/content/2301.05648v1.pdf'}
+page_content=' ∀k ∈ K,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfig8Z/content/2301.05648v1.pdf'}
+page_content='  = 1  4((βk + ηk)2 − (βk − ηk)2),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfig8Z/content/2301.05648v1.pdf'}
+page_content='  (15)  ��akcs + hH  d,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfig8Z/content/2301.05648v1.pdf'}
+page_content='kpc  ��2 ≥ βtηt,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfig8Z/content/2301.05648v1.pdf'}
+page_content=' ∀k ∈ K,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfig8Z/content/2301.05648v1.pdf'}
+page_content='  = 1  4((βt + ηt)2 − (βt − ηt)2),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfig8Z/content/2301.05648v1.pdf'}
+page_content='  (16)  �  i∈K,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfig8Z/content/2301.05648v1.pdf'}
+page_content='i̸=k  ��akis + hH  d,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfig8Z/content/2301.05648v1.pdf'}
+page_content='kpi  ��2 + σ2  k ≤ βk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfig8Z/content/2301.05648v1.pdf'}
+page_content=' ∀k ∈ K,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfig8Z/content/2301.05648v1.pdf'}
+page_content='  (17)  �  i∈K  ��akis + hH  d,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfig8Z/content/2301.05648v1.pdf'}
+page_content='kpi  ��2 + σ2  k ≤ βt,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfig8Z/content/2301.05648v1.pdf'}
+page_content=' ∀k ∈ K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfig8Z/content/2301.05648v1.pdf'}
+page_content='  (18)  By further using the first-order Taylor expansion at the given  point {s(t−1),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfig8Z/content/2301.05648v1.pdf'}
+page_content=' β(t−1)  k  ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfig8Z/content/2301.05648v1.pdf'}
+page_content=' η(t−1)  k  ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfig8Z/content/2301.05648v1.pdf'}
+page_content=' β(t−1)  t  ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfig8Z/content/2301.05648v1.pdf'}
+page_content=' η(t−1)  t  },' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfig8Z/content/2301.05648v1.pdf'}
+page_content=' constraints (15),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfig8Z/content/2301.05648v1.pdf'}
+page_content='  (16) can be approximated by  2R((akks(t−1) + hH  d,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfig8Z/content/2301.05648v1.pdf'}
+page_content='kpk)Hakks) −  ���akks(t−1)���  2  +  ��hH  d,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfig8Z/content/2301.05648v1.pdf'}
+page_content='kpk  ��2  ≥1  4  �  (βk + ηk)2 − 2(β(t−1)  k  − η(t−1)  k  )(βk − ηk)  +(β(t−1)  k  − η(t−1)  k  )2�  ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfig8Z/content/2301.05648v1.pdf'}
+page_content=' ∀k ∈ K,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfig8Z/content/2301.05648v1.pdf'}
+page_content='  (19)  2R((akcs(t−1) + hH  d,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfig8Z/content/2301.05648v1.pdf'}
+page_content='kpc)Hakcs) −  ���akcs(t−1)���  2  +  ��hH  d,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfig8Z/content/2301.05648v1.pdf'}
+page_content='kpc  ��2  ≥1  4  �  (βt + ηt)2 − 2(β(t−1)  t  − η(t−1)  t  )(βt − ηt)  +(β(t−1)  t  − η(t−1)  t  )2�  ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfig8Z/content/2301.05648v1.pdf'}
+page_content=' ∀k ∈ K,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfig8Z/content/2301.05648v1.pdf'}
+page_content='  (20)  where the left-hand side of inequality is the first-order lower  bound of  ���akks + hH  d,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfig8Z/content/2301.05648v1.pdf'}
+page_content='kpk  ���  2  ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfig8Z/content/2301.05648v1.pdf'}
+page_content=' the right-hand side of inequality  are the first-order upper bound of 1  4((βk + ηk)2 − (βk − ηk)2)  and 1  4((βt+ηt)2−(βt−ηt)2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfig8Z/content/2301.05648v1.pdf'}
+page_content=' With the above approximations,  problem (13) can be approximated by the following convex  problem:  max  s,η,β,c,ηt  �  k∈K  uk(Ck + log2 (1 + ηk))  + 2C  �  k∈K  s(t−1)  n  (sn − s(t−1)  n  )  s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfig8Z/content/2301.05648v1.pdf'}
+page_content=' t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfig8Z/content/2301.05648v1.pdf'}
+page_content='  (10e), (12), (13b), (17), (18), (19), (20),  (21)  Algorithm 2: Common rate and Phase Optimization  with SCA  1 Initialize: t ← 0, s[t], β[t], η[t]  t , η[t], c[t], OBJ[t];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfig8Z/content/2301.05648v1.pdf'}
+page_content='  2 repeat  3  update (s[t+1], β[t+1], η[t+1], η[t+1]  t  , c[t+1]) by  solving problem (21) using s[t], β[t], η[t] and the  given P∗, F∗;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfig8Z/content/2301.05648v1.pdf'}
+page_content='  4  update OBJ[t+1] using η[t+1] and the c[t+1];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfig8Z/content/2301.05648v1.pdf'}
+page_content='  5  t ← t + 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfig8Z/content/2301.05648v1.pdf'}
+page_content='  6 until  ���OBJ[t] − OBJ[t−1]��� ≤ ǫ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfig8Z/content/2301.05648v1.pdf'}
+page_content='  where β = [β1, · · · , βK]T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfig8Z/content/2301.05648v1.pdf'}
+page_content=' Problem (21) can be solved by  the standard CVX toolbox.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfig8Z/content/2301.05648v1.pdf'}
+page_content=' The detailed process of using the  SCA to solve problem (21) is illustrated in Algorithm 2, where  OBJ = �  k∈K uk(Ck+log2(1+ηk) is the calculated objective  function in each iteration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfig8Z/content/2301.05648v1.pdf'}
+page_content='  C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfig8Z/content/2301.05648v1.pdf'}
+page_content=' Alternative Optimization Algorithm  The overall AO algorithm to jointly optimize precoders P,  F, the common rate c and RIS phase shifts θ is illustrated  in Algorithm 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfig8Z/content/2301.05648v1.pdf'}
+page_content=' Specifically, the optimal precoders P, F and  common rate c are updated with the fixed phase by Algorithm  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfig8Z/content/2301.05648v1.pdf'}
+page_content=' After that, the optimal phase θ and common rate c are  jointly updated with the fixed precoders P and F by Algorithm  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfig8Z/content/2301.05648v1.pdf'}
+page_content=' This process continues until convergence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfig8Z/content/2301.05648v1.pdf'}
+page_content='  Algorithm 3: AO algorithm  1 Initialize: p ← 0, θ[p], P[p], F[p], c[p], WSR[p];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfig8Z/content/2301.05648v1.pdf'}
+page_content='  2 repeat  3  p ← p + 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfig8Z/content/2301.05648v1.pdf'}
+page_content='  4  update (P[p], F[p], c[p]) by using Algorithm 1 with  fixed θ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfig8Z/content/2301.05648v1.pdf'}
+page_content='  5  update (θ[p], c[p]) by using Algorithm 2 with fixed  P[p], F[p];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfig8Z/content/2301.05648v1.pdf'}
+page_content='  6  update WSR[p] with P[p], F[p], θ[p], c[p];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfig8Z/content/2301.05648v1.pdf'}
+page_content='  7 until  ���WSR[p] − WSR[p−1]��� ≤ ǫ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfig8Z/content/2301.05648v1.pdf'}
+page_content='  Convergence Analysis: The proposed Algorithm 3 contains  two loops.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfig8Z/content/2301.05648v1.pdf'}
+page_content=' In the inner loop, Algorthm 1 and Algorithm 2  are conducted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfig8Z/content/2301.05648v1.pdf'}
+page_content=' Algorithm 1 and Algorithm 2 are conducted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfig8Z/content/2301.05648v1.pdf'}
+page_content='  guarantees the WSR is monotonically increasing, and the  solution at iteration m is a feasible solution of problem (9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfig8Z/content/2301.05648v1.pdf'}
+page_content='  Moreover, constraint (9c) ensures that the existence of an up-  per bound on WSR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfig8Z/content/2301.05648v1.pdf'}
+page_content=' Therefore, the convergence of Algorithm  1 is guaranteed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfig8Z/content/2301.05648v1.pdf'}
+page_content=' The solution at iteration t by Algorithm 2 is  also a feasible solution of problem (21).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfig8Z/content/2301.05648v1.pdf'}
+page_content=' Furthermore, the SCA  Algorithm ensures that the objective function is monotonically  increasing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfig8Z/content/2301.05648v1.pdf'}
+page_content=' Due to the RIS phase shift constraint (13b), the  objective function has an upper bound, thus ensuring the  convergence of Algorithm 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfig8Z/content/2301.05648v1.pdf'}
+page_content=' In the outer loop, the solution  of the beamforming problem is a feasible solution of the RIS  phase shift problem, vise versa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfig8Z/content/2301.05648v1.pdf'}
+page_content=' Furthermore, WSR has an  (x1,0)  (x1,0)  BS(0,0)  !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfig8Z/content/2301.05648v1.pdf'}
+page_content='  RIS (1,2)  (1,0)  y(m)  x(m)  "  (20,0)  IR  ER  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfig8Z/content/2301.05648v1.pdf'}
+page_content=' 2: The location distribution of BS, RIS, IRs and ERs  0  2  4  6  8  10  12  14  16  18  20  Eth (  W)  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfig8Z/content/2301.05648v1.pdf'}
+page_content='5  6  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfig8Z/content/2301.05648v1.pdf'}
+page_content='5  7  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfig8Z/content/2301.05648v1.pdf'}
+page_content='5  8  8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfig8Z/content/2301.05648v1.pdf'}
+page_content='5  9  9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfig8Z/content/2301.05648v1.pdf'}
+page_content='5  Sum Rate (bit/s/Hz)  RSMA+RIS  SDMA+RIS  NOMA+RIS  RSMA  SDMA  NOMA  12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfig8Z/content/2301.05648v1.pdf'}
+page_content='02%  27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfig8Z/content/2301.05648v1.pdf'}
+page_content='69%  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfig8Z/content/2301.05648v1.pdf'}
+page_content=' 3: Rate-energy region comparison of different strategies with Pt = 10dB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfig8Z/content/2301.05648v1.pdf'}
+page_content='  upper bound due to constraints (6c) and (13b), which ensures  the convergence of Algorithm 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfig8Z/content/2301.05648v1.pdf'}
+page_content='  IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfig8Z/content/2301.05648v1.pdf'}
+page_content=' NUMERICAL RESULTS  In this section, We illustrate the performance of RIS em-  powered RSMA for SWIPT and compare it with the following  schemes:  1) RSMA+RIS—This is the scheme proposed in Section  II  2) RSMA—This is the case without RIS in case 1) as  studied in [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfig8Z/content/2301.05648v1.pdf'}
+page_content='  3) SDMA+RIS—This is a special case when common rate  Rc in case 1) is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfig8Z/content/2301.05648v1.pdf'}
+page_content='  4) SDMA—This is the case without RIS in case 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfig8Z/content/2301.05648v1.pdf'}
+page_content='  5) NOMA+RIS—This is the case that RIS empowered  NOMA for SWIPT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfig8Z/content/2301.05648v1.pdf'}
+page_content='  6) NOMA—This is the case without RIS in case 5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfig8Z/content/2301.05648v1.pdf'}
+page_content='  RSMA+RIS, SDMA+RIS and NOMA+RIS can be solved  by Algorithm 3, while RSMA, SDMA and NOMA can be  solved directly using Algorithms 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfig8Z/content/2301.05648v1.pdf'}
+page_content='  In the simulation, the location distribution of BS, RIS, IRs  and ERs are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfig8Z/content/2301.05648v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfig8Z/content/2301.05648v1.pdf'}
+page_content=' To be specific, ERs are randomly  distributed in a circle of radius r1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfig8Z/content/2301.05648v1.pdf'}
+page_content='1 meters and centered  at (1, 0), IRs are randomly distributed in a circle of radius  r1 = 1 meter and centered at (20, 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfig8Z/content/2301.05648v1.pdf'}
+page_content=' The path loss of all  channels is set to P(d) = Ld−α, where d represents the  distance between the BS and the IRs/ERs, α is the path loss  exponents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfig8Z/content/2301.05648v1.pdf'}
+page_content=' L represents the signal attenuation at a distance  of 1 meter, which is usually set to -30 dB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfig8Z/content/2301.05648v1.pdf'}
+page_content=' In addition, the  0  1  2  3  4  5  6  7  R1,tot (bit/s/Hz)  0  1  2  3  4  5  6  R2,tot (bit/s/Hz)  RSMA+RIS  SDMA+RIS  NOMA+RIS  RSMA  SDMA  NOMA  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfig8Z/content/2301.05648v1.pdf'}
+page_content=' 4: IR rate region comparison of different strategies with Pt = 10dB,  Eth = 20µW  1  2  3  4  5  6  7  8  9  10  Number of Iterations  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfig8Z/content/2301.05648v1.pdf'}
+page_content='5  6  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfig8Z/content/2301.05648v1.pdf'}
+page_content='5  7  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfig8Z/content/2301.05648v1.pdf'}
+page_content='5  8  8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfig8Z/content/2301.05648v1.pdf'}
+page_content='5  9  Sum Rate (bit/s/Hz)  RSMA  SDMA  NOMA  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfig8Z/content/2301.05648v1.pdf'}
+page_content=' 5: Convergence of the algorithms in one channel realization with Pt =  10dB, Eth = 20µW  path loss exponents of BS to all IRs, BS to all ERs, BS to  RIS, RIS to all IRs, RIS to all ERs are set to 2, 3, 3, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfig8Z/content/2301.05648v1.pdf'}
+page_content='5  and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfig8Z/content/2301.05648v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfig8Z/content/2301.05648v1.pdf'}
+page_content=' Besides, the number of transmit antennas at the BS  is Nt = 2, and the number of IRs, ERs, and RIS elements  is K = J = 2, N = 8, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfig8Z/content/2301.05648v1.pdf'}
+page_content=' The energy conversion  efficiency is ζ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfig8Z/content/2301.05648v1.pdf'}
+page_content='5, the noise of user k is σ2  k = −80 dBm  and the convergence tolerance is ǫ = 10−3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfig8Z/content/2301.05648v1.pdf'}
+page_content=' All simulation  results are averaged over 100 random channel realizations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfig8Z/content/2301.05648v1.pdf'}
+page_content='  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfig8Z/content/2301.05648v1.pdf'}
+page_content=' 3 shows the rate-energy tradeoff when Pt = 10 dB  and the weights of IRs are set to u1 = u2 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfig8Z/content/2301.05648v1.pdf'}
+page_content=' It can  be observed that when the collected energy Eth = 20µW,  the respective gain of RSMA+RIS over SDMA, NOMA,  RSMA, SDMA+RIS and NOMA+RIS are 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfig8Z/content/2301.05648v1.pdf'}
+page_content='55%, 38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfig8Z/content/2301.05648v1.pdf'}
+page_content='81%,  12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfig8Z/content/2301.05648v1.pdf'}
+page_content='78% 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfig8Z/content/2301.05648v1.pdf'}
+page_content='02% and 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfig8Z/content/2301.05648v1.pdf'}
+page_content='69%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfig8Z/content/2301.05648v1.pdf'}
+page_content=' The simulation results show that  RSMA+RIS achieves explicit performance gain over SDMA,  NOMA, RSMA, SDMA+RIS and NOMA+RIS especially  when the harvested energy constraint at the ERs is high.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfig8Z/content/2301.05648v1.pdf'}
+page_content=' It  achieves a better tradeoff between the WSR of IRs and the  harvested energy at the ERs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfig8Z/content/2301.05648v1.pdf'}
+page_content='  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfig8Z/content/2301.05648v1.pdf'}
+page_content=' 4 shows the rate regions of IRs for different strategies,  with the assistance of RIS, the rate regions of all strategies  are enlarged.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfig8Z/content/2301.05648v1.pdf'}
+page_content=' One interesting observation is, the rate region of  RSMA almost coincides with that of SDMA+RIS especially  when the two users have similar weights.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfig8Z/content/2301.05648v1.pdf'}
+page_content=' It further shows  the promising capability of RSMA to reduce the transmission  complexity while guaranteeing the quality of service.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfig8Z/content/2301.05648v1.pdf'}
+page_content='  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfig8Z/content/2301.05648v1.pdf'}
+page_content=' 5 demonstrates the convergence of the proposed AO  algorithm with Pt = 10dB and Eth = 20µW.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfig8Z/content/2301.05648v1.pdf'}
+page_content=' This graph  shows that the proposed algorithm converges very quickly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfig8Z/content/2301.05648v1.pdf'}
+page_content='  V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfig8Z/content/2301.05648v1.pdf'}
+page_content=' CONCLUSION  In this paper, we propose a SWIPT system empowered by  RSMA and RIS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfig8Z/content/2301.05648v1.pdf'}
+page_content=' The precoders and RIS phase shifts are jointly  designed at the transmitter to maximize the WSR of IRs under  the harvested energy constraint of ERs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfig8Z/content/2301.05648v1.pdf'}
+page_content=' We propose an AO  algorithm to solve this problem, which alternately optimizes  the phase shift matrix and the remaining optimization vari-  ables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfig8Z/content/2301.05648v1.pdf'}
+page_content=' Numerical results shows that the proposed approach  significantly improves the WSR and harvested energy tradeoff  between the IRs and ERs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfig8Z/content/2301.05648v1.pdf'}
+page_content=' RIS empowered RSMA is therefore  a promising strategy for SWIPT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE5T4oBgHgl3EQfig8Z/content/2301.05648v1.pdf'}
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+arXiv:2301.01623v1  [math.CO]  4 Jan 2023
+The signed graphs with two eigenvalues unequal
+to ±1
+Willem H. Haemers∗
+Dept. of Econometrics and O.R., Tilburg University, The Netherlands
+Hatice Topcu†
+Dept. of Mathematics, Nev¸sehir Hacı Bekta¸s Veli University, T¨urkiye
+Abstract
+We complete the determination of the signed graphs for which the adjacency matrix has all
+but at most two eigenvalues equal to ±1. The unsigned graphs and the disconnected, the
+bipartite and the complete signed graphs with this property have already been determined
+in two earlier papers. Here we deal with the remaining cases.
+Keywords: signed graph, graph spectrum, spectral characterization. AMS subject classifica-
+tion: 05C50.
+1
+Introduction
+A signed graph Gσ is a graph G = (V, E) together with a function σ : E → {−1, +1}, called
+the signature function. So, every edge is either positive or negative. The graph G is called the
+underlying graph of Gσ. The adjacency matrix A of Gσ is obtained from the adjacency matrix of
+G, by replacing 1 by −1 whenever the corresponding edge is negative. The signed graph G−σ with
+adjacency matrix −A is called the negative of Gσ. The spectrum of A is also called the spectrum
+of the signed graph Gσ. For a vertex set X ⊂ V , the operation that changes the sign of all edges
+between X and V \X is called switching. In terms of the matrix A, switching multiplies the rows
+and columns of A corresponding to X by −1. If a signed graph can be switched into an isomorphic
+copy of another signed graph, the two signed graphs are called switching isomorphic. Switching
+isomorphic signed graphs have similar adjacency matrices and therefore they are cospectral (that
+is, they have the same spectrum). For this and more background on signed graphs we refer to [2].
+In this paper we determine all signed graphs for which the adjacency matrix has all but at
+most two eigenvalues equal to ±1. The unsigned graphs with this property have been determined
+in [4]. This made it possible to find all (unsigned) graphs in this class which are determined by
+its adjacency spectrum. This result motivated us to determine the signed graphs in the class.
+The first results of this project are published in [6]. There we dealt with the disconnected, the
+∗haemers@uvt.nl
+†haticekamittopcu@gmail.com
+1
+
+bipartite and the complete signed graphs. This includes the signed graphs with just one or no
+eigenvalues unequal to ±1. Here we determine the remaining cases.
+The extension of spectral characterizations from unsigned to signed graphs has a complication
+because the spectrum is invariant under switching. Thus one can only expect a signed graph to
+be determined by the spectrum up to switching. Nevertheless, there do exist some achievements.
+For example in [1] it is determined for which n the switching class of the signed path Pn is
+determined by the spectrum.
+We define G to be the set of signed graphs for which the spectrum has all but at most two
+eigenvalues equal to 1 or −1. Then G is closed under switching, taking the negative, and adding
+or deleting isolated edges. Clearly every signed graph switching isomorphic to an unsigned graph
+in G is in G, and so is its negative. However, there are many other signed graphs in G.
+We use eigenvalue interlacing, spectral properties of equitable partitions and other techniques
+from linear algebra for which we refer to [3]. Some background on signed graphs can be found
+in [2]. As usual, J is the all-ones matrix and O the all-zeros matrix. The all-ones and all-zeros
+vector are denoted by 1 and 0 respectively. When necessary we give the size of J, O, 1 or 0 as
+a subscript. The reverse identity matrix of order m is denoted by Rm (that is, Rm(i, j) = 1 if
+i + j = m + 1, and Rm(i, j) = 0 otherwise). Note that all eigenvalues of Rm are equal to ±1.
+2
+Main result
+Here we describe the signed graphs in G. The disconnected ones, the signed complete graphs,
+and those with at most one eigenvalue unequal to ±1 were found in [6] (see Section 3) and for
+the ones switching isomorphic with an unsigned graph or its negative we refer to [4].
+Theorem 2.1. Suppose Gσ is a connected signed graph with two eigenvalues unequal to ±1,
+which is not a signed complete graph. Then Gσ or its negative G−σ is switching isomorphic with
+an unsigned graph in G, or with a signed graph represented by one of the following matrices.
+A1 =
+�
+J −Im
+J
+J
+−R2ℓ
+�
+(m, ℓ ≥ 2) with spectrum {−1ℓ+m−2, 1ℓ, 1
+2(m − 2 ±
+�
+m(m + 8ℓ) )}.
+A2 =
+�
+R2m
+J
+J
+−R2ℓ
+�
+(m, ℓ ≥ 2) with spectrum {−1m+ℓ−1, 1m+ℓ−1, ±
+√
+1 + 4mℓ}.
+A3 =
+
+
+J −Im
+1
+1
+O
+1⊤
+0
+1
+−1⊤
+1⊤
+1
+0
+1⊤
+O
+−1
+1
+Iℓ −J
+
+ (m, ℓ ≥ 1) with spectrum {−1m, 1ℓ, −ℓ − 1, m + 1}.
+A4 =
+
+
+J −Im
+J
+J
+O
+J
+Iℓ−J
+O
+J
+J
+O
+R2
+O
+O
+J
+O
+−R2
+
+
+(m, ℓ ≥ 1) with spectrum
+{−1m+1, 1ℓ+1, 1
+2(m − ℓ ±
+√
+m2 + ℓ2 + 6mℓ + 4m + 4ℓ + 4)}.
+2
+
+A5 =
+
+
+J −Im
+J
+J
+J
+J −Iℓ
+O
+J
+O
+Ik−J
+
+
+(k ≥ 2, (m, ℓ) = (3, 8), (4, 6), or (6, 5)) with spectra
+{−19, 1k, 1
+2(9 − k ±
+√
+k2 + 26k + 121)},
+{−18, 1k, 1
+2(8 − k ±
+√
+k2 + 28k + 100)},
+{−19, 1k, 1
+2(9 − k ±
+√
+k2 + 38k + 121)}.
+A6 =
+
+
+J −Im
+J
+J
+J
+Iℓ−J
+O
+J
+O
+R2k
+
+
+(m ≥ 1, (ℓ, k) = (3, 4), or (4, 3)) with spectra
+{−1m+4, 15, 1
+2(m − 1 ±
+√
+m2 + 42m + 9)},
+{−1m+3, 15, 1
+2(m − 2 ±
+√
+m2 + 40m + 16)}.
+A7 =
+
+
+R2m
+J
+J
+J
+J −Iℓ
+O
+J
+O
+−R2k
+
+
+(m ≥ 1, (ℓ, k) = (3, 3), or (4, 2)) with spectra
+{−1m+4, 1m+3, 1
+2(1 ± √8m + 1},
+{−1m+4, 1m+2, 1 ± 2√4m + 1}.
+A8 =
+
+
+R2
+J
+1
+O
+J
+Im−J
+0
+J
+1⊤
+0⊤
+0
+0⊤
+O
+J
+0
+−R4
+
+
+(m ≥ 1) with spectrum
+{−13, 1m+2, 1
+2(1 − m ±
+√
+m2 + 22m + 9)}.
+A9 =
+
+
+R2m
+J
+J
+0
+J
+R2
+O
+1
+J
+O
+−R2
+0
+0⊤
+1⊤
+0⊤
+0
+
+ (m ≥ 1) with spectrum {−1m+2, 1m+1, 1
+2(1 ± √32m + 9)}.
+A10 =
+
+
+J −Im
+J
+O
+O
+J
+O
+J
+J −Iℓ
+O
+J
+Im−J
+O
+O
+J −Iℓ
+O
+O
+
+
+((m, ℓ) = (3, 4), or (4, 3)), with spectra
+{−16, 16, ±6}, {−16, 16, ±6}.
+A11 =
+
+
+J −Im
+J
+J
+O
+J
+Im−J
+O
+J
+J
+O
+O
+J −Iℓ
+O
+J
+J −Iℓ
+O
+
+
+((m, ℓ) = (3, 4), or (4, 3)) with spectra
+{−16, 16, ±3
+√
+5)}, {−16, 16, ±2
+√
+13}.
+A12 =
+
+
+J −Im
+J
+J
+O
+J
+Iℓ−J
+O
+J
+J
+O
+J −Ik
+J
+O
+J
+J
+Ij −J
+
+
+((m, ℓ, k, j) = (6, 3, 3, 6), (6, 6, 3, 3), (6, 4, 3, 4),
+or (4, 4, 4, 4)), with spectra {−18, 18, ±
+√
+109},
+{−18, 18, ±10}, {−17, 18, 1
+2(−1 ± 3
+√
+41)},
+{−17, 17, ±9}.
+A13 =
+
+
+J −Im
+J
+O
+0
+J
+−R2ℓ
+J
+1
+O
+J
+I4−J
+0
+0⊤
+1⊤
+0⊤
+0
+
+
+((m, ℓ) = (6, 2), or (5, 3)), with spectra
+{−17, 16, 1
+2(1 ±
+√
+241)}, {−17, 17, ±6
+√
+2}.
+A14 =
+
+
+J −Im
+J
+O
+J
+−R2ℓ
+J
+O
+J
+−R4
+
+
+((m, ℓ) = (6, 2), or (5, 3)) with spectra
+{−17, 15, 1 ± 2
+√
+13}, {−17, 16, 1
+2(1 ±
+√
+249}.
+3
+
+A15 =
+
+
+J −Im
+J
+J
+O
+J
+Iℓ−J
+O
+J
+J
+O
+R2k
+J
+O
+J
+J
+−R2j
+
+
+((m, ℓ, k, j) = (3, 3, 3, 3), (4, 3, 3, 2), or (4, 4, 2, 2))
+with spectra {−18, 18, ±
+√
+85}, {−18, 17, 1
+2(1 ±
+√
+313)},
+{−17, 17, ±
+√
+73}.
+A16 =
+
+
+R2
+J
+J
+O
+J
+−R2
+O
+J
+J
+O
+O
+I3
+O
+J
+I3
+O
+
+with spectrum {−14, 14, ±
+√
+17}.
+A17 =
+
+
+R2
+J
+J
+1
+O
+0
+J
+−R2
+O
+0
+J
+1
+J
+O
+R2
+0
+J
+0
+1⊤
+0⊤
+0⊤
+0
+0⊤
+0
+O
+J
+J
+0
+−R2
+0
+0⊤
+1⊤
+0⊤
+0
+0⊤
+0
+
+
+with spectrum {−14, 14, ±2
+√
+5}.
+A18 =
+
+
+J −Im
+J
+1
+O
+J
+−R2ℓ
+0
+J
+1⊤
+0⊤
+0
+0⊤
+O
+J
+0
+−R2
+
+
+((m, ℓ) = (4, 3), or (3, 4)) with spectra
+{−16, 15, 1
+2(1 ±
+√
+177)}, {−16, 16, ±3
+√
+5}.
+A19 =
+
+
+R2
+J
+J
+1
+O
+J
+Im−J
+O
+0
+J
+J
+O
+R2ℓ
+0
+J
+1⊤
+0⊤
+0⊤
+0
+0⊤
+O
+J
+J
+0
+−R2
+
+
+((m, ℓ) = (3, 3), or (4, 2)) with spectra
+{−16, 16, ±2
+√
+10}, {−15, 16, 1
+2(−1 ± 3
+√
+17)}.
+Proof. To see that each of the above matrices has the given spectrum, we use the same method as
+used in [4] and [6]. For each matrix A the presented block structure gives an equitable partition
+(that is, each block has constant row and column sums). Part of the spectrum of A is the spectrum
+of the quotient matrix Q (containing the row sums of the blocks), and it is straightforward to
+check that each Q has just two eigenvalues different from ±1. The remaining eigenvalues of A
+remain unchanged if we add or subtract an all-one block J to some of the blocks. One easily
+verifies that this can be done in such a way that we obtain a matrix with all eigenvalues equal
+to ±1. The proof that the list is complete comprises the remainder of this paper.
+Note that the above list is not free from overlap. For example the negatives of A2, A3 and
+A4 can alo be obtained by interchanging m and ℓ, and several cases with symmetric spectrum
+are switching isomorphic with their negatives.
+Also note that A5 with k = 1 corresponds to the three unsigned graphs in G given in Theo-
+rem 1(v) of [4]. Thus these three sporadic graphs from [4] are in fact part of an infinite family of
+signed graphs in G.
+3
+Preliminaries
+We start with some results from [6].
+4
+
+Lemma 3.1. If a signed graph Gσ has smallest eigenvalue at least −1, then the underlying graph
+G is a disjoint union of complete graphs.
+The following two results are a bit more general than Proposition 2.3 and Theorem 2.4 in [6].
+The proofs, however, are basically the same.
+Proposition 3.2. If Gσ is connected and all but at most one eigenvalues are in the interval
+[−1, 1], then Gσ or G−σ is switching isomorphic with an unsigned complete graph.
+Theorem 3.3. If Gσ is disconnected and has no isolated vertices or edges, and all but two
+eigenvalues of Gσ are in the interval [−1, 1], then Gσ is the disjoint union of two signed graphs
+each of which is switching isomorphic with an unsigned complete graph or its negative.
+Theorem 3.4. If Gσ ∈ G is bipartite, then Gσ is switching isomorphic with an unsigned bipartite
+graph.
+The bipartite graphs in G are described in [5, 4, 6].
+Theorem 3.5. Assume Kσ
+n is a signed complete graph of order n for which the adjacency matrix
+has all but two eigenvalues in the interval [−1, 1]. Then Kσ
+n is switching isomorphic with a signed
+graph with adjacency matrix
+�Jm,ℓ =
+�J − Im
+J
+J
+−J + Iℓ
+�
+with n = m+ℓ, m, ℓ ≥ 2 and spectrum {−1m−1, 1ℓ−1, 1
+2(m−ℓ±
+√
+m2 + ℓ2 + 6mℓ + 4m + 4ℓ + 4)}.
+4
+Restrictions
+In the remainder we will assume that Gσ is a signed graph in G of order n which is connected,
+and not bipartite or complete. Then, by Proposition 3.2, Gσ has two eigenvalues r and s different
+from ±1, and we may assume that s < −1 and r > 1, since otherwise Gσ or its negative has
+smallest eigenvalue −1, so Gσ would be disconnected or complete by Lemma 3.1.
+An important tool is eigenvalue interlacing. It follows that every induced signed subgraph of
+a graph Gσ in G has second largest eigenvalue at most 1, and second smallest eigenvalue at least
+−1. This excludes many subgraphs, including the path Pn with n ≥ 6 and the n-cycle with n ≥ 5
+(for every signing). Some other forbidden induced signed subgraphs are presented in Figure 1.
+t
+t
+t
+t
+t
+t
+t
+t
+t
+t
+❩❩
+✚
+✚
+✑✑
+◗
+◗
+❩❩
+❩
+✚
+✚
+✚
+t
+t
+t
+t
+t
+✂
+✂
+✂
+✂
+❇
+❇❇
+✚✚✚✚
+❩
+❩
+❩
+❩
+✂
+✂✂
+❩❩
+❩
+✚
+✚
+✚
+Figure 1: forbidden signed graphs; negative edges are dashed
+Of course the signed graphs switching isomorphic with a graph in Figure 1 and their negatives
+are also forbidden, but this does not include all possible signings. For example, the unsigned
+5
+
+t
+t
+t
+t
+t
+❇
+❇❇
+✂
+✂✂
+❩❩
+❩
+✚
+✚
+✚
+t
+t
+t
+t
+t
+❇
+❇❇
+✂
+✂✂
+❩❩
+❩
+✚
+✚
+✚
+t
+t
+t
+t
+t
+✚✚✚✚
+❇
+❇❇
+✂
+✂✂
+❩❩
+❩
+✚
+✚
+✚
+��
+❅❅
+❅
+❅
+�
+�
+t
+t
+t
+t
+t
+t
+��
+❅❅
+❅
+❅
+❅
+�
+�
+�
+t
+t
+t
+t
+t
+t
+Figure 2: forbidden graphs for all signings
+4-cycle can occur. Figure 2 lists forbidden induced subgraphs for the underlying unsigned graph.
+In other words, these graphs are forbidden for all signings. Another useful fact is that if A is
+the adjacency matrix of a signed graph in G, then A2 has eigenvalues 1, r2 and s2, so A2 − I is
+positive semi-definite (psd) and has rank 2. The following Lemma is a generalization of Lemma 2
+from [4]. The proof is the same.
+Lemma 4.1. Suppose Gσ ∈ G with adjacency matrix A, then
+(i) if Gσ has no isolated edges, then there is no vertex of degree 1,
+(ii) if x and y are two columns of A such that x⊤x = x⊤y, then x and y differ in at least three
+coordinate places, in particular no two columns (or rows) of A are equal.
+Proof. (i) Suppose Gσ has a vertex v of degree 1 and a vertex w at distance 2 from v. Then the
+2×2 submatrix of A2 −I corresponding to v and w has a negative determinant which contradicts
+that A2 − I is psd. (ii) In this case A2 − I has a psd submatrix
+M =
+� x⊤x − 1
+x⊤y
+y⊤x
+y⊤y − 1
+�
+=
+� x⊤x − 1
+x⊤x
+x⊤x
+y⊤y − 1
+�
+.
+So det(M) ≥ 0 which implies that y⊤y > x⊤x + 2.
+5
+The possible formats
+Since Gσ is not bipartite and contains no odd cycle of length 5 or more, Gσ contains a triangle.
+Lemma 5.1. Let C be a maximal clique in Gσ of size c ≥ 3. Then, up to vertex ordering,
+switching and taking the negative, the c × n submatrix of A corresponding to C is one of the
+following:
+(i)
+
+ J − Ic
+Jc1×x
+−Jc2×x
+O
+Oc×z
+
+ with c1 + c2 < c, c1, c2, x > 0, z ≥ 0,
+(ii)
+�
+J − Ic
+Jc1×x
+O
+O
+Jc2×y
+Oc×z
+�
+with c1 + c2 = c, c1, c2, x > 0, y, z ≥ 0,
+(iii)
+�
+�Jc1,c2
+Jc1×x
+O
+O
+Jc2×y
+Oc×z
+�
+with c1 + c2 = c, c1, c2 ≥ 2, x > 0, y, z ≥ 0.
+6
+
+Proof. The clique C is a signed complete graph which has, by interlacing, all but at most two
+eigenvalues in the interval [−1, 1]. By Lemma 3.1 the adjacency matrix of C is, up to switching
+and taking the negative, equal to J −Ic, or �Jc1,c2, where c1 +c2 = c and c1, c2 ≥ 2. So the matrix
+N consisting of the rows of A corresponding to C takes the form [ J − Ic | L ] or [ �Jc1,c2 | �L ].
+Since NN ⊤ − I is a principal submatrix of A2 − I, the rank of NN ⊤ − I is at most 2. In case
+N = [ J − Ic | L ] we get
+NN ⊤ − I = (c − 2)J + LL⊤ = MM ⊤,
+where M = [ (√c − 2) 1 | L ]. This implies that M, and hence [ 1 | L ] has rank at most 2. Let
+k ̸= 0 be a column of L (L ̸= O, since Gσ is connected). Then k has at least one entry equal to
+0 (because C is maximal), and we may assume that k has at least one positive entry (otherwise
+we switch). From rank([ 1 | L] ) ≤ 2 it follows that every column of L is equal to k, 0. or 1 − k.
+If k has positive and negative entries, then L has the form given in (i), otherwise we get case
+(ii). In case N = [ �Jc1,c2 | �L ] we get
+NN ⊤ − I =
+�
+(c − 2)J
+(c1 − c2)Jc1×c2
+(c1 − c2)Jc2×c1
+(c − 2)J
+�
++ �L�L⊤ = MM ⊤,
+where M = [ (√c1 − 1) 1
+(√c2 − 1) �1
+�L ], with �1 = [ 1⊤
+c1 | − 1⊤
+c2 ]⊤. This implies that M, and
+hence [ 1 | �1 | �L ] has rank at most 2. Therefore every column of �L is a linear combination of 1,
+and �1, but has at least one zero entry. So, up to switching, only the columns given in case (iii)
+are allowed, and since Gσ is connected we may assume x > 0.
+6
+Format (i)
+Assume Gσ contains a clique C, which gives format (i) from Lemma 5.1. We claim that c1 =
+c2 = 1. Indeed, if c1 ≥ 2, or c2 ≥ 2, then Gσ has an induced subgraph switching isomorphic with
+the last signed graph in Figure 1. Next consider the matrix N which consists of row 1, row 3 and
+row c + i of A (1 ≤ i ≤ x). Then
+N =
+
+
+0
+1
+1
+1⊤
+1⊤
+x
+0⊤
+z
+1
+1
+0
+1⊤
+0⊤
+0⊤
+1
+−1
+0
+0⊤
+x⊤
+z⊤
+
+ , and
+NN⊤ − I =
+
+
+c − 2 + x
+c − 2
+x⊤1 − 1
+c − 2
+c − 2
+0
+x⊤1 − 1
+0
+1 + x⊤x + z⊤z
+
+
+for (0, ±1)-vectors x and z. Note that x has an entry on the diagonal of A, which is equal to 0.
+Therefore |x⊤1| ≤ x⊤x ≤ x − 1. We have that NN ⊤ − I is singular, so
+0 = det(NN⊤ − I) = (c − 2)(x(1 + x⊤x + z⊤z) − (x⊤1 − 1)2),
+which implies that −x⊤1 = x⊤x = x−1 and z⊤z = 0. Finally, z = 0 implies that z = 0 (otherwise
+Gσ is disconnected), and thus we have:
+Conclusion 6.1. If Gσ has adjacency matrix A with Format (i) then A = A3.
+7
+
+7
+Further restrictions
+In the remainder we will assume that Gσ contains no maximal clique with Format (i), and that
+the matrix A of Gσ takes the form of Format (ii) or (iii) as described in Lemma 5.1. Concerning
+the maximal clique C we will assume the following:
+Assumption 7.1.
+(i) The clique C has maximum order c,
+(ii) C has the largest number of outgoing edges among all cliques of order c.
+For both formats (ii) and (iii), C is the union of two cliques C1 and C2, such that the
+remaining vertices can be partitioned into three sets X, Y , and Z, where the vertices in X are
+adjacent to C1 and not to C2, the vertices in Y are adjacent to C2 and not to C1, and the vertices
+from Z are adjacent to C1 nor C2. Note that the sets X, Y and Z may be empty, but X and Y
+are not both empty (otherwise Gσ is disconnected or complete) and we may choose that X ̸= ∅
+(otherwise we interchange C1 with C2 and X with Y ; in case of format (iii) we also need to
+switch and take the negative). Recall that ci = |Ci| (i = 1, 2), x = |X|, y = |Y | and z = |Z|. For
+a subset V of the vertex set of Gσ we let Gσ
+V denote the subgraph of Gσ induced by V .
+Proposition 7.2.
+(i) Gσ
+X and Gσ
+Y are disjoint unions of cliques.
+(ii) Two adjacent vertices in Gσ
+X or Gσ
+Y correspond to identical rows and columns in A + I or
+A − I.
+(iii) If c1 ≥ 3 [c2 ≥ 3], then all edges of Gσ
+Y [Gσ
+X] have the same sign which is opposite to the
+sign of C1 [C2].
+(iv) Every clique in Gσ
+Y [Gσ
+X] has order at most 2, or one clique has order at least 3 and all other
+cliques have order 1 and all edges have the same sign opposite to the sign of C1 [C2].
+(v) There are no negative edges between X and Y .
+(vi) A vertex from Z cannot be adjacent to both X and Y .
+Proof. (i) Suppose Gσ
+X contains an induced path P3 then the subgraph of Gσ induced by this path
+together with one vertex of C1 and one vertex of C2 is a forbidden subgraph (no. 3 in Figure 2).
+Therefore Gσ
+X contains no P3, so Gσ
+X is a clique. (ii) Let C3 be the clique in Gσ
+A containing
+the two adjacent vertices. Then (ii) follows from Lemma 5.1 applied to to the maximal clique
+C1 ∪ C3. (iii) Take an edge e in Gσ
+X, and consider the subgraph Hσ of Gσ
+X induced by e, three
+vertices of C2 and one vertex of C1. If e has the same sign as the edges of C2, then Hσ is switching
+isomorphic with the second forbidden subgraph of Figure 1. This proves (iii). (iv) Suppose Gσ
+Y
+contains a clique C3 of order at least 3. Then c1 ≥ 3 and all edges of C3 have the same sign. If Gσ
+Y
+contains another clique of order at least 2, then Gσ contains a signed graph switching isomorphic
+with the second forbidden signed subgraph of Figure 1. This proves (iv). (v) A negative edge
+between X and Y will create a 4-cycle with one negative edge which is forbidden. (vi) A vertex
+in Z adjacent to a vertex of X and a vertex from Y creates an induced pentagon, or a pentagon
+with one chord, which are forbidden for all signings.
+Note that Assumption 7.1 implies that a clique in Gσ
+X [or Gσ
+Y ] has order at most c2 [c1], and
+for a clique C3 in Gσ
+X [Gσ
+Y ] of order c2 [c1] there are at most c2y [c1x] edges between C3 and
+Y ∪ Z [X ∪ Z].
+8
+
+8
+Format (ii) with Y = ∅
+Suppose Gσ has format (ii) and Y = ∅. Let N be the matrix consisting of three rows of A
+corresponding to a vertex from C1, a vertex from C2 and a vertex v from X. Then
+N =
+
+
+0
+1⊤
+c1−1
+1
+1⊤
+c2−1
+1⊤
+x
+0⊤
+z
+1
+1⊤
+0
+1⊤
+0⊤
+0⊤
+1
+1⊤
+0
+0⊤
+x⊤
+1
+z⊤
+1
+
+
+(for (0, ±1) vectors x1 and z1), and
+M = NN ⊤ − I =
+
+
+c + x − 2
+c − 2
+c1 + x′
+1 − 1
+c − 2
+c − 2
+c1
+c1 + x′
+1 − 1
+c1
+c1 + x1 + z1 − 1
+
+ ,
+where x′
+1 = 1⊤x, x1 = x⊤
+1x1 and z1 = z⊤
+1z1.
+Note that |x′
+1| ≤ x1 ≤ x − 1, x1 + 1 ≤ c2
+and x1 + 1 = c2 implies that z1 = 0 by Assumption 7.1. Using Gaussian elimination we get
+det(M) = x(c − 2)(c1 + x1 + z1 − 1) − (c − 2)(x′
+1 − 1)2 − xc2
+1.
+For convenience we define
+f(a, b) = ab/(a + b). We have det(M) = 0, which gives:
+x1 + z1 + f(c1, c2 − 2) = (x′
+1 − 1)2
+x
++ 1.
+(1)
+If c2 = 1, Proposition 7.2(i−iii) implies that X is a coclique and that there are no edges between
+X and Z. Then A has x + 1 equal rows, which contradicts Lemma 4.1(ii). Therefore c2 ≥ 2
+and f(c1, c2 − 2) ≥ 0. From Assumption 7.1 it follows that x1 = x′
+1 = 1, or x1 = −x′
+1. We will
+consider the different possibilities for x1.
+Proposition 8.1. Suppose x1 = 0. Then v is the only isolated vertex of Gσ
+X and one of the
+following holds:
+(i) x = 1, c2 = 2 and A = A4 with m = 1 and ℓ ≥ 1.
+(ii) A is an unsigned matrix given in case (v) of Theorem 1 from [4].
+(iii) (c1, c2, x) ∈ {(2, 5, 5), (2, 6, 3), (3, 4, 5), (4, 4, 3)}.
+Proof. Equation 1 gives z1 = 1 + 1/x − f(c1, c2 − 2), so z1 ≤ 2. If z1 = 2 then x = 1 and c2 = 2,
+which straightforwardly leads to matrix A4 with m = 1. If z1 = 1 then f(c1, c2 −2) = 1/x, which
+implies (c1, c2, x) = (2, 4, 1), or (1, 3, 2). Both of these cases cannot be completed to a signed
+graph in G. If z1 = 0 then f(c1, c2−2) = 1+1/x, which implies that (c1, c2, x) is equal to (4, 2, 1),
+(3, 8, 1), (6, 5, 1), (3, 5, 2), (4, 4, 3), (2, 6, 3), (2, 5, 5), or (3, 4, 5). The first three triples lead to case
+(v) of Theorem 1 from [4]. If Gσ
+X has a second isolated vertex then the two corresponding rows
+of A are equal which contradicts Lemma 4.1. Therefore Gσ
+X has just one isolated vertex and the
+triple (3, 5, 2) is not possible.
+Proposition 8.2. If x1 = x − 1 ≥ 2 then A = A5 with k ≥ 3.
+Proof. The underlying graph GX of Gσ
+X is complete, and from Proposition 7.2(iii) it follows that
+all signs are negative. Formula 1 becomes: z1 = 2 − f(c1, c2 − 2). If z1 = 2, then c2 = 2 and
+x1 ≤ 1, which is excluded. If z1 = 1 then f(c1, c2 − 2) = 1, so c1 = 2, c2 = 4 and x = 2 or 3
+9
+
+which has no solution by straightforward verification. If z1 = 0, then f(c1, c2 − 2) = 2, which
+gives (c1, c2) = (3, 8), (4, 6) or 6, 5). Thus we find A5 from Theorem 2 (actually, only the ones
+with k ≤ ℓ, because of Assumption 7.1).
+Proposition 8.3. If x1 = 1, then one of the following holds:
+(i) A = A5 with k = 2.
+(ii) A = −A6 with m = 1 (switched around the first vertex).
+(iii) A = A7.
+(iv) A is an unsigned matrix given in case (ii) of Theorem 1 from [4].
+Proof. If x1 = −x′
+1 = 1, then clearly x ≥ 2 and Formula 1 becomes z1 = 4/x − f(c1, c2 − 2)). It
+also follows that c2 ≥ 3, since c2 = 2 would imply z1 = 0 (by Assumption 7.1). From x ≥ 2 and
+f(c1, c2 − 2) ≥ 1 it follows that z1 ≤ 1, and that z1 = 1 implies (c1, c2, x) = (2, 4, 2), which is not
+possible. So z1 = 0 and we obtain the following solutions for Formula 1: (c1, c2, x) = (1, 3, 8),
+(1, 4, 6), (1, 6, 5), (2, 3, 6), (2, 4, 4), (2, 6, 3), (3, 8, 2), (4, 3, 5), (4, 4, 3), (4, 6, 2), or (6, 5, 2). By
+Proposition 7.2(iv), Gσ
+X has an isolated vertex if x is odd, in which case we need a solution with
+x1 = 0 for the same (c1, c2, x). Proposition 8.1 gives just two possibilities: (2, 6, 3) and (4, 4, 3).
+In both cases Gσ
+X consists of one negative edge and an isolated vertex, and it is easily checked
+that this is not possible. Thus x is even, and of the remaining seven triples the first two lead to
+−A6 with m = 1, the next two to A7, and the last three to A5 with k = 2. If x1 = x′
+1 = 1, then
+we easily have c2 = 2 and z1 = 0. None of the solutions above has c2 = 2 and x ≥ 2, therefore all
+rows corresponding to X have x1 = x′
+1 = 1, c2 = 2 and z1 = 0, which implies that Gσ is case (ii)
+of Theorem 1 from [4].
+Proposition 8.4. If 2 ≤ x1 ≤ x − 2, then there is no solution.
+Proof. Proposition 7.2(iv) implies that if there is a solution with 2 ≤ x1 ≤ x − 2, then there is
+also one with x1 = 0 for the same c1, c2 and x. From Proposition 8.1 we find that there is just
+one candidate: (c1, c2, x) = (3, 4, 5) and x1 = −x′
+1 = 2. But this does not satisfy Formula 1.
+Finally we revisit Proposition 8.1(iii).
+For each of the four cases there must be another
+solution with x1 > 0, which we haven’t found. So none of these four triples are possible, and the
+determination all graphs with Format (ii) and Y = ∅ is complete.
+Conclusion 8.5. If A has Format (ii) with Y = ∅ then A is unsigned (case (ii), or (v) from
+Theorem 1 of [4]), or A is one of the following: (i) A4 with m = 1, ℓ ≥ 1, (ii) A5 with 2 ≤ k ≤ ℓ,
+(iii) −A6 with m = 1 (switched around the first vertex), (iv) A7.
+9
+Format (iii) with Y = ∅
+Suppose Gσ has format (iii) and Y = ∅. Again we let N be the matrix consisting of row 1,
+row c1 + 1 and row c + 1 of A (so row c + 1 corresponds to a vertex in X). Then
+M = NN ⊤ − I =
+
+
+c1 + c2 + x − 2
+c1 − c2
+c1 + x′
+1 − 1
+c1 − c2
+c1 + c2
+c1
+c1 + x′
+1 − 1
+c1
+c1 + x1 + z1 − 1
+
+ ,
+10
+
+with x′
+1, x1 and z1 as before (so |x′
+1| ≤ x1 ≤ x − 1). The expression for det(M) is a bit messy
+now, so to deal with det(M) = 0 in another way. We write M = (c1 −2)J +(c2 −2)K +R, where
+K =
+
+
+1
+−1
+0
+−1
+1
+0
+0
+0
+0
+
+ , and R =
+
+
+x + 2
+0
+x′
+1 + 1
+0
+2
+2
+x′
+1 + 1
+2
+x1 + z1 + 1
+
+ .
+We easily have that (c1 − 2)J and (c2 − 2)K are positive semi-definite (psd) matrices. Thus if
+R is positive definite (pd), then M is pd and therefore nonsingular. Moreover, if R is psd, then
+M is psd and the kernel of M is the intersection of the kernels of (c1 − 2)J, (c2 − 2)K and R.
+So if this intersection is {0} then M is also nonsingular. The diagonal entries of R are positive,
+and so are the 2 × 2 principal minors. This implies that R is pd whenever det(R) > 0 and R is
+psd when det(R) = 0. Using this, and det(R) = 2((x + 2)(x1 + z1 − 1) − (x′
+1 + 1)2) we get the
+following restrictions.
+Proposition 9.1. One of the following holds:
+(i) x1 = x′
+1, z1 = 0,
+(ii) x1 = x′
+1 = 0, z1 = 1,
+(iii) x1 = −x′
+1 = 1, z1 = 0, c2 = 2.
+Proof. From Proposition 7.2(iii, iv) it follows that x1 = −x′
+1 = 1, or x1 = x′
+1. First assume
+x1 = x′
+1. Using x ≥ x1+1 we get det(R) ≥ 2(x1z1+3z1−4). This implies z1 = 0, (x1, z1) = (0, 1),
+or (1, 1). The first option is case (i), and the second one is case (ii). If x1 = z1 = 1 we find
+R =
+
+
+x + 2
+0
+2
+0
+2
+2
+2
+2
+3
+
+ .
+We have x ≥ x1 + 1 = 2 and therefore R is pd if x > 2 and psd if x = 2. If x = 2 the kernel of
+R is spanned by [1, 2, −2]⊤, which is not in the kernel of J or K. Therefore M is singular only
+if c1 = c2 = 2 and x = x1 + 1 = 2. Thus we have c2 = x1 + 1 and z1 > 0, which contradicts
+Assumption 7.1. Next we assume x1 = −x′
+1 = 1. Then det(R) = 2(x + 2)z1, so z1 = 0 and
+R =
+
+
+x + 2
+0
+0
+0
+2
+2
+0
+2
+2
+
+ .
+Again R is pd if x > 2 and psd if x = 2. If x = 2 the kernel of R is spanned by [0, 1, −1]⊤, which
+is not in the kernel of K. Therefore c2 = 2 and we have case (iii).
+We claim that Z = ∅. Suppose not, and assume u ∈ Z, then u cannot be adjacent to two (or
+more) vertices of X, because that would mean that A has two equals rows which is not possible.
+But u has degree at least 2, so there is another vertex v in Z adjacent to u. But then we create a
+forbidden subgraph from Figure 2. This implies that case (ii) of the above proposition does not
+occur. Note that Gσ
+X has at most one isolated vertex, since otherwise A has two equal rows. Also
+Gσ
+X cannot contain a positive isolated edge if c2 = 2. Indeed, interchanging C2 with the positive
+edge gives a signed graph with Format (ii) and y = 0, which has been dealt with in the previous
+section. Thus Proposition 9.1 leads to just six possible structures for Gσ
+X, being:
+11
+
+(i) Gσ
+X is a clique of order x with all edges positive,
+(ii) Gσ
+X is a clique of order x − 1 with all edges positive extended with an isolated vertex,
+(iii) c2 > 2 and Gσ
+X is the disjoint union of x/2 positive edges,
+(iv) c2 > 2 and Gσ
+X is the disjoint union of (x − 1)/2 positive edges and one isolated vertex,
+(v) c2 = 2 and Gσ
+X is the disjoint union of x/2 negative edges,
+(vi) c2 = 2 and Gσ
+X is the disjoint union of (x − 1)/2 negative edges and one isolated vertex.
+For each of these cases the matrix A of Gσ has an equitable partition with quotient matrices
+Q1 =
+
+
+c1−1
+c2
+x
+c1
+1−c2
+0
+c1
+0
+x−1
+
+ , Q2 =
+
+
+c1−1
+c2
+x−1
+1
+c1
+1−c2
+0
+0
+c1
+0
+x−2
+0
+c1
+0
+0
+0
+
+ , Q3 =
+
+
+c1−1
+c2
+x
+c1
+1−c2
+0
+c1
+0
+1
+
+ ,
+Q4 =
+
+
+c1−1
+c2
+x−1
+1
+c1
+1−c2
+0
+0
+c1
+0
+1
+0
+c1
+0
+0
+0
+
+ , Q5 =
+� c1−1 x+2
+c1
+−1
+�
+, Q6 =
+
+
+c1−1 x+1
+1
+c1
+−1
+0
+c1
+0
+0
+
+ ,
+respectively. Each of these quotient matrices must have at most two eigenvalues different from ±1.
+For Q5 this is obvious and we find matrix A1 of Theorem 2. For the other five quotient matrices
+we checked for eigenvalues ±1 (simply by computing rank(Qi +I) and rank(Qi −I)). For Q2 and
+Q4 there is at most one eigenvalue equal to ±1, and Q6 has no eigenvalue equal to ±1. However
+Q1 has an eigenvalue ±1 whenever (c1, x) = (3, 8), (4, 6), or (6, 5), which gives A5 of Theorem 2
+when k > ℓ. The quotient matrix Q3 has eigenvalue ±1 whenever (c2, x) = (3, 8), (4, 6), or (6, 5),
+but x is even so only the first two cases remain, and we find A6 of Theorem 2 (when m > 1).
+Conclusion 9.2. If Gσ has adjacency matrix A with Format (iii) with Y = ∅, then A is one of
+the matrices represented by A1, A5 (with k > ℓ), or A6 (with m > 1).
+10
+X, Y and Z nonempty
+Take v ∈ Z. By Proposition 7.2(vi) we may assume that v is adjacent to a vertex w ∈ X and
+not adjacent to Y . Consider the matrix N consisting of four rows of A corresponding to a vertex
+from C1, a vertex from C2, to w and v, respectively. Then
+N =
+
+
+0
+1⊤
+c1−1
+0
+1⊤
+c2−1
+1⊤
+x
+0⊤
+y
+0⊤
+z
+1
+1⊤
+0
+ǫ1⊤
+0⊤
+1⊤
+0⊤
+1
+1⊤
+0
+0⊤
+x⊤
+1
+y⊤
+1
+z⊤
+1
+0
+0⊤
+0
+0⊤
+x⊤
+3
+0⊤
+z⊤
+3
+
+
+for (0, ±1) vectors x1, x3, y1 z1, z3, and ǫ = ±1, where ǫ = 1 in case of format (ii) and ǫ = −1 for
+format (iii). Note that we may switch such that z1 ≥ 0 and that y1 ≥ 0 by Proposition 7.2(v).
+Define M = NN⊤ − I, then
+M =
+
+
+c1 + c2 + x − 2
+c1 − 1 + ǫ(c2 − 1)
+c1 + x′
+1 − 1
+x′
+3
+c1 − 1 + ǫ(c2 − 1)
+c1 + c2 + y − 2
+c1 + y1
+0
+c1 + x′
+1 − 1
+c1 + y1
+c1 + x1 + y1 + z1 − 1
+x1,3 + z1,3
+x′
+3
+0
+x1,3 + z1,3
+x3 + z3 − 1
+
+ ,
+12
+
+where x1 = x⊤
+1x1, x′
+1 = x⊤
+11, x1,3 = x⊤
+1x3, x3 = x⊤
+3x3 ≥ 1, x′
+3 = x⊤
+31 ̸= 0, y1 = y⊤
+1 y1 = y⊤
+1 1,
+z1 = z⊤
+1z1 ≥ 1, z3 = z⊤
+3z3 and z1,3 = z⊤
+1z3. Since M is a principal submatrix of A2 − I, we know
+rank(M) ≤ 2. Define Mi,j to be the matrix obtained from M by deletion of row i and column j.
+Then Mi,j is singular for 1 ≤ i, j ≤ 4. We treat the two considered formats seperately, and start
+with (iii).
+10.1
+Format (iii)
+In this case ǫ = −1, c1, c2 ≥ 2 and we write M3,3 = (c1 − 2)K + (c2 − 2)L + R, where
+K =
+
+
+1
+1
+0
+1
+1
+0
+0
+0
+0
+
+ ,
+L =
+
+
+1
+−1
+0
+−1
+1
+0
+0
+0
+0
+
+ ,
+R =
+
+
+x + 2
+0
+x′
+3
+0
+y + 2
+0
+x′
+3
+0
+x3 + z3 − 1
+
+ .
+Clearly K and L are psd, so if R is pd, then M3,3 is non-singular. Therefore R is not pd and
+0 ≥ det(R) = (y +2)((x+2)(x3 +z3 −1)−(x′
+3)2). This implies z3 = 0, x3 = |x′
+3| = x and x3 ≤ 2.
+We know x3 > 0, and if x3 = 1 then x′
+3 ̸= 0 and det(M3,3) = −(y + 2)(x′
+3)2 ̸= 0. Thus x3 = 2.
+Then det(R) = 0 and the kernel of R is spanned by [1, 0, −2]⊤, which is not in the kernel of K
+and L, and therefore M3,3 is singular only if c1 = c2 = 2. Thus x = x3 = c1 = c2 = 2, z3 = 0 and
+M1,1 = (y1 + 2)K +
+
+
+y − y1
+0
+0
+0
+x1 + z1 − 1
+x′
+1
+0
+x′
+1
+1
+
+ .
+One easily verifies that both terms in this formula are psd (recall that |x1| ≤ x1 ≤ 1 and z1 ≥ 1),
+and that the kernel of the sum is only nontrivial if y = y1 and z1 = 1. If x1 = 0, then the two
+rows of A corresponding to X are equal, which contradicts Lemma 4.1(ii), and x1 = 1 is ruled
+out by Proposition 7.2(iii). So we conclude that Z = ∅ in case of format (iii). (But notice that
+without Assumption 7.1 we would have found A17.)
+10.2
+Format (ii)
+In this case ǫ = 1, c = c1 + c2 ≥ 3 and M3,3 can be written as: M3,3 = (c − 2)K + R, with K as
+before and
+R =
+
+
+x
+0
+x′
+3
+0
+y + 1
+0
+x′
+3
+0
+x3 + z3 − 1
+
+ .
+With similar arguments as above we see that R, and hence M3,3 is pd if z3 > 0, if |x′
+3| < x3 and
+if x3 < x − 1. Therefore z3 = 0, |x′
+3| = x3 and x3 ∈ {x, x − 1}.
+Assume that x3 = x − 1.
+Then using Gaussian elimination we easily obtain that y(c −
+2)(x − 2) − (y + c − 2) = det(M3,3) = 0.
+Hence x − 2 = 1/(c − 2) + 1/y, which leads to
+(c, x, y) = (3, 4, 1), or (4, 3, 2). In both cases there are just a few possibilities for A, and it follows
+by straightforward verification that there are no solutions (but without Assumption 7.1 we would
+have found adjacency matrices A8 with m = 1, and A17).
+Next assume that x3 = x. Then
+M3,3 =
+
+
+x + c − 2
+c − 2
+x
+c − 2
+y + c − 2
+0
+x
+0
+x − 1
+
+ .
+13
+
+and det(M3,3) = y(c − 2)(x − 1) − x(y + c − 2) = 0, which implies c − 3 = (x + y)/(xy − x − y).
+The latter equations has only ten feasible solutions, being: (c, x, y) = (4, 4, 4), (4, 3, 6), (4, 6, 3),
+(5, 3, 3), (5, 2, 6), (5, 6, 2), (6, 2, 4), (6, 4, 2), (8, 2, 3), (8, 3, 2). Observe that if Z has a second
+vertex u then it cannot be adjacent to all vertices in X (otherwise A would have two indentaical
+rows), so u is adjacent to all vertices of Y and none of X. This shows that z ≤ 2. Using this
+observation we find that for each of the above triples there are only a few possibilities for A, and
+we check them case by case. The cases (5, 2, 6) and (4, 3, 6) lead to matrix A10, and the cases
+(5, 3, 3) and (4, 4, 4) lead to case (vi) of Theorem 1 from [4]. The other six cases have no solution.
+Thus we conclude:
+Conclusion 10.1. If A has format (ii) or (iii) with X, Y and Z nonempty, then A = A10, or
+A is the unsigned case (vi) of Theorem 1 from [4].
+11
+Format(iii) with X, Y nonempty and Z empty
+Throughout this section Gσ has adjacency matrix A with format (iii), X, Y ̸= ∅ and Z = ∅.
+Consider the matrix N consisting of four rows of A corresponding to a vertex from C1, a vertex
+from C2, a vertex v from X and a vertex from w from Y . Then
+N =
+
+
+0
+1⊤
+c1−1
+0
+1⊤
+c2−1
+1⊤
+x
+0⊤
+y
+1
+1⊤
+0
+−1⊤
+0⊤
+1⊤
+1
+1⊤
+0
+0⊤
+x⊤
+1
+y⊤
+1
+0
+0⊤
+1
+1⊤
+x⊤
+2
+y⊤
+2
+
+
+for (0, ±1) vectors x1, y2, and (0, 1) vectors y1, x2. Define M = NN⊤ − I, then
+M =
+
+
+c1 + c2 + x − 2
+c1 − c2
+c1 + x′
+1 − 1
+c2 + x2
+c1 − c2
+c1 + c2 + y − 2
+c1 + y1
+1 − c2 + y′
+2
+c1 + x′
+1 − 1
+c1 + y1
+c1 + x1 + y1 − 1
+x1,2 + y1,2
+c2 + x2
+1 − c2 + y′
+2
+x1,2 + y1,2
+c2 + x2 + y2 − 1
+
+ ,
+where x1 = x⊤
+1x1, x′
+1 = x⊤
+11, x1,2 = x⊤
+1x2, x2 = x⊤
+2x2 = x⊤
+21, y1 = y⊤
+1 y1 = y⊤
+1 1, y2 = y⊤
+2 y2,
+y′
+2 = y⊤
+2 1, and y1,2 = y⊤
+1 y2. As before we define Mi,j to be the matrix obtained from M by
+deletion of row i and column j. We have rank(M) ≤ 2, hence det(Mi,j) = 0 for 1 ≤ i, j ≤ 4.
+Proposition 11.1. Suppose y − y1 ≥ 2 for some vertex in X, then one of the following holds.
+(i): x1 = x′
+1 = 1, y1 = 0, x = y = 2 which leads to matrix A4 with m, ℓ ≥ 2.
+(ii): x1 = 0, y1 = 1, x = y = 3 and A = A16,
+(iii): x1 = y1 = 0, x2 = y2 = 0, x = y = 3 and A = A17,
+(iv): x1 = y1 = 0, x2 = x − 1, y2 = 1, which leads to A8 with m ≥ 2, A18, or A19.
+Proof. Write M4,4 = (c1 − 2)J + (c2 − 2)L + y1K + R, where
+L =
+
+
+1
+−1
+0
+−1
+1
+0
+0
+0
+0
+
+ , K =
+
+
+0
+0
+0
+0
+1
+1
+0
+1
+1
+
+ ,
+and R =
+
+
+x + 2
+0
+x′
+1 + 1
+0
+y − y1 + 2
+2
+x′
+1 + 1
+2
+x1 + 1
+
+ .
+14
+
+Clearly J, K and L are psd and R is pd, unless x1 = 0, or x1 = x′
+1 = x − 1 = 1 and y − y1 = 2.
+In the latter case R is psd with kernel spanned by [1, 1, −2]⊤, which is in the kernel of M4,4 only
+if y1 = 0. Therefore x = y = 2, and we obtain the matrices A4 with m, ℓ ≥ 2.
+So we have that x1 = 0. If y1 ≥ 1, then we use that R + y1K is not pd. By straightforward
+computation we get det(R + y1K) = (y + 2)(x + 2)(y1 + 1) − (x + 2)(y1 + 2)2 − (y + 2). Using
+y − y1 ≥ 2 this gives det(R + y1K) ≥ (x + 1)y1 − 4 and equality implies y − y1 = 2. Therefore
+y1 = 0, or y1 = 1, y = 3 and x ≤ 3, or y1 = 2, y = 4 and x = 1. If (x, x1, y, y1) = (3, 0, 3, 1)
+then R + y1K is psd and the kernel is spanned by [1, 3, −5]⊤ which is in the kernel of J nor K.
+Hence M4,4 is nonsingular, unless c1 = c2 = 2 in which case we obtain A16. It is not difficult to
+rule out the remaining cases by use of Lemma 4.1(ii), Proposition 7.2(ii), and the last forbidden
+subgraph of Figure 2.
+What remains is the case x1 = y1 = 0. Using Gaussian elimination in M2,2 we find that in this
+case det(M2,2) = (y2 − 1)(c2 + x − 1) + (x − x2 − 1)(c2 + x2). Note that y1 = 0 implies that
+x2 ≤ x − 1. Therefore M2,2 is singular only if (a): y2 = 0 and c2 + x − 1 = (x − x2 − 1)(c2 + x2),
+or (b): y2 = 1, x = x2 + 1. In case (a) we get c2 = 2, x = 3, x2 = y2 = 0. Since no two rows of A
+are equal we have that there is exactly one vertex v ∈ X for which x1 = y1 = 0, and at most one
+vertex w ∈ Y for which x2 = y2 = 0. Using this observation we find that case (a) leads to A17, and
+that otherwise (b) holds for all vertices of Y . Moreover, det(M1,2) = (1 − c1)(c2 + x − 1)(y′
+2 + 1),
+implies y′
+2 = −1. From Proposition 7.2(iii) we deduce that Gσ
+X is equal to K1, or K1 + Kx−1 (all
+signs are positive), or consists of one isolated vertex and (x − 1)/2 isolated signed edges.
+If Gσ
+X = K1 then we have an equitable partition of A with quotient matrix
+Q =
+
+
+c1 − 1
+c2
+1
+0
+c1
+1 − c2
+0
+y
+c1
+0
+0
+0
+0
+c2
+0
+−1
+
+ .
+By Gaussian elimination in Q + I and Q − I we find that Q has two eigenvalues equal to ±1 only
+if c1 = 2 and y = 4. This gives A8 with m ≥ 2.
+If Gσ
+X = K1 + Kx−1 then we have an equitable partition of A with quotient matrix
+Q =
+
+
+c1 − 1
+c2
+1
+x − 1
+0
+c1
+1 − c2
+0
+0
+y
+c1
+0
+0
+0
+0
+c1
+0
+0
+x − 2
+y
+0
+c2
+0
+x − 1
+−1
+
+
+.
+By Gaussian elimination in Q+I and Q−I we find that Q has at least three eigenvalues unequal
+to ±1.
+If Gσ
+X has an negative isolated edge e then C2 = 2 by Theorem 7.2(iv, v).
+If v is a vertex
+of e, then c2 = 2 x1 = −x′
+1 = 1, y1 = y, x2 = x − 1, y2 = −y′
+2 = 1 and det(M2,2) =
+(x + 1)(c1y − y − c1) − 4(c1 − 1) = 0. Using that x > 1 and odd, and y ≥ 2 and even, we find
+just two solutions: (c1, x, y) = (3, 5, 2) and (4, 7, 2) and obtain A18.
+Assume Gσ
+X has a positive isolated edge e. Take v ∈ e. Then x1 = x′
+1 = 1, y1 = y, x2 = x − 1,
+y2 = −y′
+2 = 1 and det(M2,2) = (c2 + x − 1)(c1y − y − c1) = 0, which implies c1 = y = 2. Since
+c1 ̸= 3 or 4, Gσ
+X has no negative isolated edge. This means that A has an equitable partition
+15
+
+with quotient matrix
+Q =
+
+
+1
+c2
+1
+x − 1
+0
+2
+1 − c2
+0
+0
+2
+2
+0
+0
+0
+0
+2
+0
+0
+1
+2
+0
+c2
+0
+x − 1
+−1
+
+
+.
+Using Gaussian elimination in Q + I and Q − I we find that Q has three eigenvalues equal to ±1
+if and only if (c2, x) = (3, 7) or (4, 5). Thus we obtain A19.
+Suppose the matrix A does not belong to one of the cases of the above proposition. Then
+each vertex from X is adjacent to y or y − 1 vertices of Y . And, by interchanging C1 with C2
+and X with Y it also follows that each vertex from Y is adjacent to x or x − 1 vertices of X. So
+it follows that the adjacency matrix of Gσ
+X∪Y takes the form:
+AX∪Y =
+� AX
+B
+B⊤
+AY
+�
+with B =
+�
+J
+J
+J − Im
+J
+�
+and 0 ≤ m ≤ min{x, y}.
+Proposition 11.2. With B as above, one of the following holds:
+(i) m = x = y, B = J − I which leads to A11,
+(ii) m = 0, B = J which leads to A2, A12, A13, A14, or A15.
+Proof. Let Xm and Ym be the subsets of X and Y corresponding to the submatrix J − Im of B.
+From Proposition 7.2(ii) it follows that Xm and Ym consists of isolated vertices in Γσ
+X and Γσ
+Y ,
+respectively. So if m = x = y then Γσ
+X and Γσ
+Y have no edges and A has the block structure of
+A12. The values of m, ℓ and k follow from the spectrum of the quotient matrix.
+If 0 < m < x then the subgraph Γσ
+X\Xm of Γσ
+X induced by X \ Xm cannot have an isolated
+edge or vertex (because of Lemma 4.1(ii)), and therefore Γσ
+X\Xm = Kx−m (because of Proposi-
+tion 7.2(iv)). Similarly, if 0 < m < y then Γσ
+Y \Ym = K−
+y−m. Thus, if 0 < m < min{x, y} then A
+has an equitable partition with quotient matrix
+Q =
+
+
+c1 − 1
+c2
+x − m
+m
+0
+0
+c1
+1 − c2
+0
+0
+m
+y − m
+c1
+0
+x − m − 1
+0
+m
+y − m
+c1
+0
+0
+0
+m − 1
+y − m
+0
+c2
+x − m
+m − 1
+0
+0
+0
+c2
+x − m
+m
+0
+m − y + 1
+
+
+.
+However Q does not have four eigenvalues equal to ±1. A quick way to verify this is by Gaussian
+elimination in Q + I with row 2 and column 6 deleted and in Q − I with row 1 and column 3
+deleted. Then we find that Q+ I and Q− I both have rank at least 5. A similar argument works
+if m = x < y, or m = y < x. Then we have an equitable partition with a 5 × 5 quotient matrix
+with at most two eigenvalues equal to ±1. Thus we can conclude that m = 0 and B = J. Now
+Γσ
+X may have an isolated vertex, but Lemma 4.1 still implies that Γσ
+X cannot have two isolated
+vertices, or an isolated vertex and an isolated edge. Therefore by use of Proposition 7.2(iii) there
+are just a few possibilities for Γσ
+X, being: (a): Γσ
+X = Kx (x ≥ 3), (b): Γσ
+X = Kx−1 + K1 (x ≥ 4),
+(c): Γσ
+X consists of x/2 isolated signed edges. For Γσ
+Y we obtain the same list with y, K−
+y and
+K−
+y−1 instead of x, Kx and Kx−1.
+16
+
+If Γσ
+X = Kx and Γσ
+Y = K−
+y , then A has an equitable partition of A11, and the quotient matrix Q
+has two eigenvalues equal to ±1 only in the given cases (taking Assumption 7.1 into account). If
+Γσ
+X = Kx and Γσ
+Y = K−
+y−1 + K1 (or Γσ
+X = Kx + K1 and Γσ
+Y = K−
+y ), then we have an equitable
+partition of A with 5 × 5 quotient matrix Q, and by Gaussian elemination it follows that Q + I
+and Q − I both have rank at least 4, and therefore Q (and A) have at least three eigenvalues
+unequal to ±1. If Γσ
+X = Kx−1 + K1 and Γσ
+Y = K−
+y−1 + K1, then the equitable partition has a
+6 × 6 quotient matrix Q, and both Q + I and Q − I both have rank at least 5. So also in this
+case we find no solution. For the remainder of the proof we may assume that Γσ
+X consists of x/2
+disjoint signed edges. We claim that all these edges have the same sign. Indeed, suppose Γσ
+X
+has a negative and a positive edge, then we consider the matrix N consisting of three rows of A
+corresponding to a vertex of the negative edge, a vertex of the positive edge and a vertex from
+C1. Then
+NN⊤ − I =
+
+
+c1 + y
+c1 + y
+c1
+c1 + y
+c1 + y
+c1 − 2
+c1
+c1 − 2
+c1 + c2 + x − 2
+
+
+is nonsingular, which proves the claim.
+Suppose that all edges of Γσ
+X have a negative sign. Then Proposition 7.2(iv) implies that c2 = 2.
+If Γσ
+Y = K−
+y then A has an equitable partition with three parts and a quotient matrix with no
+eigenvalue equal to ±1. If Γσ
+Y = K−
+y−1 + K1 then A has an equitable partition with four parts
+and quotient matrix
+Q =
+
+
+c1 − 1
+x + 2
+0
+0
+c1
+−1
+y − 1
+1
+0
+x + 2
+2 − y
+0
+0
+x + 2
+0
+0
+
+ .
+We easily obtain that Q+I is singular only if y = 5, and that Q−I is singular only if (c1, y) = (5, 4)
+or (6, 2). Thus we find A13. If Γσ
+Y consists of y/2 positive edges, then also c1 = 2, and we obtain
+A2. If Γσ
+Y consists of y/2 negative edges, then we find an equitable partition with quotient
+Q =
+
+
+c1 − 1
+x + 2
+0
+c1
+−1
+y
+0
+x + 2
+−1
+
+ .
+Now Q + I is nonsingular and Q − I is singular only if (c1, x, y) = (6, 2, 4), or (5, 4, 4). Thus we
+find A14.
+Next suppose that Γσ
+X consists of positive edges. If Γσ
+Y = K−
+y , then A has an equitable partition
+for which the quotient matrix has at least three eigenvalues unequal to ±1. If Γσ
+Y = K−
+y−1 + K1,
+then A has an equitable partition with five parts and quotient matrix Q for which it is easily seen
+that both Q + I and Q − I have rank at least 4, so we find no solution. If Γσ
+Y has y/2 disjoint
+positive edges, then we obtain A14 after by switching taking the negative. If Γσ
+Y has y/2 disjoint
+negative edges, then A has an equitable partitions with quotient matrix
+Q =
+
+
+c1 − 1
+c2
+x
+0
+c1
+1 − c2
+0
+y
+c1
+0
+1
+y
+0
+c2
+x
+−1
+
+ .
+17
+
+Using that x and y are both even we find that both Q − I and Q + I have rank 3 only if
+(c1, c2, x, y) = (3, 3, 6, 6), (4, 3, 6, 4), or (4, 4, 4, 4). Thus we find A15.
+Conclusion 11.3. If Gσ has adjacency matrix A with Format (iii) with Z = ∅ and X, Y ̸= ∅
+then A equals A2, A4, A8, A12, A13, A14, A15, A16, A17, A18, or A19.
+12
+Format(ii) with X, Y nonempty and Z empty
+In this section Gσ has adjacency matrix A with format (ii), X, Y ̸= ∅ and Z = ∅. We use the
+notation of the previous section. Again the matrix N consists of four rows of A corresponding to
+a vertex from C1, a vertex from C2, a vertex v from X and a vertex from w from Y . Then
+M = NN⊤ − I =
+
+
+c1 + c2 + x − 2
+c1 + c2 − 2
+c1 + x′
+1 − 1
+c2 + x2
+c1 + c2 − 2
+c1 + c2 + y − 2
+c1 + y1
+c2 − 1 + y′
+2
+c1 + x′
+1 − 1
+c1 + y1
+c1 + x1 + y1 − 1
+x1,2 + y1,2
+c2 + x2
+c2 − 1 + y′
+2
+x1,2 + y1,2
+c2 + x2 + y2 − 1
+
+ .
+First we consider the case that c1, c2 ≥ 2. Then we can imitate the steps of the previous section.
+12.1
+c1 ≥ 2 and c2 ≥ 2
+Proposition 12.1. Suppose y − y1 ≥ 2 for some vertex in X, then A = A9.
+Proof. From Proposition 7.2(iii, iv) it follows that x1 = −x′
+1, or x1 = x′
+1 = 1. We consider M4,4
+and write M4,4 = (c1 − 2)J + (c2 − 2)L + y1K + R, where
+L =
+
+
+1
+1
+0
+1
+1
+0
+0
+0
+0
+
+ , K =
+
+
+0
+0
+0
+0
+1
+1
+0
+1
+1
+
+ ,
+and R =
+
+
+x + 2
+2
+x′
+1 + 1
+2
+y − y1 + 2
+2
+x′
+1 + 1
+2
+x1 + 1
+
+ .
+Then J, K and L are psd. Moreover, if y − y1 ≥ 3, or y − y1 = 2 and x1 ̸= 0 then R is pd. Using
+y − y1 ≥ 2 we obtain x1 = x′
+1 = 0 and y − y1 = 2. In this case R is psd with kernel spanned
+by [0, 1, −2]⊤. So only if c1 = c2 = 2 and y1 = 0 the kernel of M4,4 is nontrivial. This implies
+that M4,4 is singular only if (c1, c2, y, x1, y1) = (2, 2, 2, 0, 0). If X contains another vertex with
+x1 = y1 = 0 then A has two equal rows, so if x ≥ 2 then there must be another solution for
+det(M4,4) = 0 with c1 = c2 = y = 2. Such a solution exists only if x1 = x′
+1 = 1 which leads to
+A9.
+As in the previous section it follows from Proposition 12.1 that if A ̸= A9 then the adjacency
+matrix of Gσ
+X∪Y takes the form:
+AX∪Y =
+� AX
+B
+B⊤
+AY
+�
+with B =
+�
+J
+J
+J − Im
+J
+�
+and 0 ≤ m ≤ min{x, y},
+and with the same arguments, but (slightly) different quotient matrices we obtain the following
+result:
+Proposition 12.2. The matrix B above has m = 0 and A equals A8 with m = 1, A12 (only
+the case k = ℓ because of Assumption 7.1), or A is an unsigned matrix given in case (iii) of
+Theorem 1 from [4].
+18
+
+12.2
+c1 = 1 and c2 ≥ 2
+Since c1 = 1, Assumption 7.1 yields that Y is a coclique, so y2 = y′
+2 = y1,2 = 0, and M becomes:
+M =
+
+
+c2 + x − 1
+c2 − 1
+x′
+1
+c2 + x2
+c2 − 1
+c2 + y − 1
+y1 + 1
+c2 − 1
+x′
+1
+y1 + 1
+x1 + y1 + z1
+x1,2
+c2 + x2
+c2 − 1
+x1,2
+c2 + x2 − 1
+
+ .
+Suppose vertex v of Gσ
+X is isolated. Then x1 = x′
+1 = x1,2 = 0 and det(M3,3) = −((c2 + x2)y +
+(x2 + 1)(c2 − 1)) ̸= 0.
+Hence Gσ
+X has no isolated vertex and from Proposition 7.2(i, iii, iv)
+it follows that there are just three possibilities: (i) Gσ
+X is the complete graph with all edges
+negative, (ii) Gσ
+X is the disjoint union of two or more negative edges, and (iii) c2 = 2 and
+Gσ
+X is the disjoint union of two or more signed edges (not all negative). Assume we have case
+(i), then x1 = −x′
+1 = x − 1 > 0 and Proposition 7.2(ii) implies that every vertex in Y is
+adjacent to all or no vertices of X. If w ∈ Y is adjacent to all of X, then x2 = x and find
+det(M3,4) = (x − 1)(c2 + y − 1) + (c2 − 1)(y1 + 1) > 0. If every vertex of Y is nonadjacent
+to every vertex of X, then y1 = x2 = 0 and we find det(M3,4) = (x − 1)yc2 > 0. So we can
+conclude that case (i) does not occur. For the remaining two cases Gσ
+X is a disjoint union of
+edges. Consider an edge e of Gσ
+X. By Proposition 7.2(ii) every vertex in Gσ
+Y is adjacent to
+none or both vertices of e. Therefore x2 and x − x2 are even. Next consider case (ii). Suppose
+{v, w} is an edge between X and Y , then x′
+1 = x1,2 = −x1 = −1 and we find det(M3,4) =
+(x − x2 − 1)(−(c2 + y − 1) − (c2 − 1)(y1 + 1)) ̸= 0. Therefore there are no edges between X and
+Y . So x2 = y1 = 0, x1 = −x′
+1 = 1 and det(M3,1) = −y(c2 − 1) − (c2 − 1), which implies y = 1.
+Now det(M1,4) = 0 gives x = 4 and c2 = 2 and thus we find that A equals A8 with m = 1. In
+case (iii) we have c2 = 2 and consider M3,3. It is straigtforward that det(M3,3) ̸= 0, except when
+x = 4, y = 1 and x2 = 0 which again leads to A8 with m = 1. Thus we conclude that the last
+case gives no new examples.
+Conclusion 12.3. If A has Format (ii) with Z = ∅ and X, Y ̸= ∅ then A is equal to A8 with
+m = 1, A9, A12 with k = ℓ, or A is an unsigned matrix given in Theorem 1(iii) of [4].
+13
+Recapitulation
+By combining the conclusions of sections 6, and 8 to 12 we have completed the proof of Theo-
+rem 2. Note that the proof doesn’t exclude the unsigned examples. Thus we rediscovered the
+unsigned characterization from [4]. As in the unsigned case, with the present characterization
+one can examine which signed graphs in G are determined, up to switching, by the spectrum.
+This however, is more involved as in the unsigned case and will be the subject of further research.
+Acknowledgement
+Part of the research was done while the first author visited the Haci Bekta¸s Veli University of
+Nev¸sehir on a grant from T¨ubitac.
+19
+
+References
+[1] S. Akbari, W.H. Haemers, H.R. Maimani and L. Parsaei Majd, Signed graphs cospectral
+with the path, Linear Algebra and its Applications 553 (2018) 104-116.
+[2] F. Berlardo, S.M. Cioab˘a, J.H. Koolen and J. Wang, Open problems in the spectral theory
+of signed graphs, The Art of Discrete and Applied Mathematics 1 (2018), #P2.10.
+[3] A.E. Brouwer and W.H. Haemers, Spectra of Graphs, Springer, 2012.
+[4] S.M. Cioab˘a, W.H. Haemers, J.R. Vermette and W. Wong, The graphs with all but two
+eigenvalues equal to ±1, J. Algebr. Comb. 41 (2015) 887-897.
+[5] E.R. van Dam and E. Spence, Combinatorial designs with two singular values-I: uniform
+multiplicative designs. J. Comb. Theory A 107 (2004), 127-142.
+[6] W.H. Haemers and H. Topcu, On signed graphs with at most two eigenvalues unequal to ±1,
+arXiv:2109.02522 (2021).
+20
+
diff --git a/mNAzT4oBgHgl3EQfp_3T/content/tmp_files/load_file.txt b/mNAzT4oBgHgl3EQfp_3T/content/tmp_files/load_file.txt
new file mode 100644
index 0000000000000000000000000000000000000000..662a3f7f2a45451a6560c22b1b292873d95628a0
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@@ -0,0 +1,605 @@
+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfp_3T/content/2301.01623v1.pdf,len=604
+page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfp_3T/content/2301.01623v1.pdf'}
+page_content='01623v1  [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfp_3T/content/2301.01623v1.pdf'}
+page_content='CO]  4 Jan 2023  The signed graphs with two eigenvalues unequal  to ±1  Willem H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfp_3T/content/2301.01623v1.pdf'}
+page_content=' Haemers∗  Dept.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfp_3T/content/2301.01623v1.pdf'}
+page_content=' of Econometrics and O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfp_3T/content/2301.01623v1.pdf'}
+page_content='R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfp_3T/content/2301.01623v1.pdf'}
+page_content=', Tilburg University, The Netherlands  Hatice Topcu†  Dept.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfp_3T/content/2301.01623v1.pdf'}
+page_content=' of Mathematics, Nev¸sehir Hacı Bekta¸s Veli University, T¨urkiye  Abstract  We complete the determination of the signed graphs for which the adjacency matrix has all  but at most two eigenvalues equal to ±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfp_3T/content/2301.01623v1.pdf'}
+page_content=' The unsigned graphs and the disconnected, the  bipartite and the complete signed graphs with this property have already been determined  in two earlier papers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfp_3T/content/2301.01623v1.pdf'}
+page_content=' Here we deal with the remaining cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfp_3T/content/2301.01623v1.pdf'}
+page_content='  Keywords: signed graph, graph spectrum, spectral characterization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfp_3T/content/2301.01623v1.pdf'}
+page_content=' AMS subject classifica-  tion: 05C50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfp_3T/content/2301.01623v1.pdf'}
+page_content='  1  Introduction  A signed graph Gσ is a graph G = (V, E) together with a function σ : E → {−1, +1}, called  the signature function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfp_3T/content/2301.01623v1.pdf'}
+page_content=' So, every edge is either positive or negative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfp_3T/content/2301.01623v1.pdf'}
+page_content=' The graph G is called the  underlying graph of Gσ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfp_3T/content/2301.01623v1.pdf'}
+page_content=' The adjacency matrix A of Gσ is obtained from the adjacency matrix of  G, by replacing 1 by −1 whenever the corresponding edge is negative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfp_3T/content/2301.01623v1.pdf'}
+page_content=' The signed graph G−σ with  adjacency matrix −A is called the negative of Gσ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfp_3T/content/2301.01623v1.pdf'}
+page_content=' The spectrum of A is also called the spectrum  of the signed graph Gσ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfp_3T/content/2301.01623v1.pdf'}
+page_content=' For a vertex set X ⊂ V , the operation that changes the sign of all edges  between X and V \\X is called switching.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfp_3T/content/2301.01623v1.pdf'}
+page_content=' In terms of the matrix A, switching multiplies the rows  and columns of A corresponding to X by −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfp_3T/content/2301.01623v1.pdf'}
+page_content=' If a signed graph can be switched into an isomorphic  copy of another signed graph, the two signed graphs are called switching isomorphic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfp_3T/content/2301.01623v1.pdf'}
+page_content=' Switching  isomorphic signed graphs have similar adjacency matrices and therefore they are cospectral (that  is, they have the same spectrum).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfp_3T/content/2301.01623v1.pdf'}
+page_content=' For this and more background on signed graphs we refer to [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfp_3T/content/2301.01623v1.pdf'}
+page_content='  In this paper we determine all signed graphs for which the adjacency matrix has all but at  most two eigenvalues equal to ±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfp_3T/content/2301.01623v1.pdf'}
+page_content=' The unsigned graphs with this property have been determined  in [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfp_3T/content/2301.01623v1.pdf'}
+page_content=' This made it possible to find all (unsigned) graphs in this class which are determined by  its adjacency spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfp_3T/content/2301.01623v1.pdf'}
+page_content=' This result motivated us to determine the signed graphs in the class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfp_3T/content/2301.01623v1.pdf'}
+page_content='  The first results of this project are published in [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfp_3T/content/2301.01623v1.pdf'}
+page_content=' There we dealt with the disconnected, the  ∗haemers@uvt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfp_3T/content/2301.01623v1.pdf'}
+page_content='nl  †haticekamittopcu@gmail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfp_3T/content/2301.01623v1.pdf'}
+page_content='com  1  bipartite and the complete signed graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfp_3T/content/2301.01623v1.pdf'}
+page_content=' This includes the signed graphs with just one or no  eigenvalues unequal to ±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfp_3T/content/2301.01623v1.pdf'}
+page_content=' Here we determine the remaining cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfp_3T/content/2301.01623v1.pdf'}
+page_content='  The extension of spectral characterizations from unsigned to signed graphs has a complication  because the spectrum is invariant under switching.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfp_3T/content/2301.01623v1.pdf'}
+page_content=' Thus one can only expect a signed graph to  be determined by the spectrum up to switching.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfp_3T/content/2301.01623v1.pdf'}
+page_content=' Nevertheless, there do exist some achievements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfp_3T/content/2301.01623v1.pdf'}
+page_content='  For example in [1] it is determined for which n the switching class of the signed path Pn is  determined by the spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfp_3T/content/2301.01623v1.pdf'}
+page_content='  We define G to be the set of signed graphs for which the spectrum has all but at most two  eigenvalues equal to 1 or −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfp_3T/content/2301.01623v1.pdf'}
+page_content=' Then G is closed under switching, taking the negative, and adding  or deleting isolated edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfp_3T/content/2301.01623v1.pdf'}
+page_content=' Clearly every signed graph switching isomorphic to an unsigned graph  in G is in G, and so is its negative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfp_3T/content/2301.01623v1.pdf'}
+page_content=' However, there are many other signed graphs in G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfp_3T/content/2301.01623v1.pdf'}
+page_content='  We use eigenvalue interlacing, spectral properties of equitable partitions and other techniques  from linear algebra for which we refer to [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfp_3T/content/2301.01623v1.pdf'}
+page_content=' Some background on signed graphs can be found  in [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfp_3T/content/2301.01623v1.pdf'}
+page_content=' As usual, J is the all-ones matrix and O the all-zeros matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfp_3T/content/2301.01623v1.pdf'}
+page_content=' The all-ones and all-zeros  vector are denoted by 1 and 0 respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfp_3T/content/2301.01623v1.pdf'}
+page_content=' When necessary we give the size of J, O, 1 or 0 as  a subscript.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfp_3T/content/2301.01623v1.pdf'}
+page_content=' The reverse identity matrix of order m is denoted by Rm (that is, Rm(i, j) = 1 if  i + j = m + 1, and Rm(i, j) = 0 otherwise).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfp_3T/content/2301.01623v1.pdf'}
+page_content=' Note that all eigenvalues of Rm are equal to ±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfp_3T/content/2301.01623v1.pdf'}
+page_content='  2  Main result  Here we describe the signed graphs in G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfp_3T/content/2301.01623v1.pdf'}
+page_content=' The disconnected ones, the signed complete graphs,  and those with at most one eigenvalue unequal to ±1 were found in [6] (see Section 3) and for  the ones switching isomorphic with an unsigned graph or its negative we refer to [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfp_3T/content/2301.01623v1.pdf'}
+page_content='  Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfp_3T/content/2301.01623v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfp_3T/content/2301.01623v1.pdf'}
+page_content=' Suppose Gσ is a connected signed graph with two eigenvalues unequal to ±1,  which is not a signed complete graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfp_3T/content/2301.01623v1.pdf'}
+page_content=' Then Gσ or its negative G−σ is switching isomorphic with  an unsigned graph in G, or with a signed graph represented by one of the following matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfp_3T/content/2301.01623v1.pdf'}
+page_content='  A1 =  �  J −Im  J  J  −R2ℓ  �  (m, ℓ ≥ 2) with spectrum {−1ℓ+m−2, 1ℓ, 1  2(m − 2 ±  �  m(m + 8ℓ) )}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfp_3T/content/2301.01623v1.pdf'}
+page_content='  A2 =  �  R2m  J  J  −R2ℓ  �  (m, ℓ ≥ 2) with spectrum {−1m+ℓ−1, 1m+ℓ−1, ±  √  1 + 4mℓ}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfp_3T/content/2301.01623v1.pdf'}
+page_content='  A3 =  \uf8ee  \uf8ef\uf8ef\uf8f0  J −Im  1  1  O  1⊤  0  1  −1⊤  1⊤  1  0  1⊤  O  −1  1  Iℓ −J  \uf8f9  \uf8fa\uf8fa\uf8fb (m, ℓ ≥ 1) with spectrum {−1m, 1ℓ, −ℓ − 1, m + 1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfp_3T/content/2301.01623v1.pdf'}
+page_content='  A4 =  \uf8ee  \uf8ef\uf8ef\uf8f0  J −Im  J  J  O  J  Iℓ−J  O  J  J  O  R2  O  O  J  O  −R2  \uf8f9  \uf8fa\uf8fa\uf8fb  (m, ℓ ≥ 1) with spectrum  {−1m+1, 1ℓ+1, 1  2(m − ℓ ±  √  m2 + ℓ2 + 6mℓ + 4m + 4ℓ + 4)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfp_3T/content/2301.01623v1.pdf'}
+page_content='  2  A5 =  \uf8ee  \uf8f0  J −Im  J  J  J  J −Iℓ  O  J  O  Ik−J  \uf8f9  \uf8fb  (k ≥ 2, (m, ℓ) = (3, 8), (4, 6), or (6, 5)) with spectra  {−19, 1k, 1  2(9 − k ±  √  k2 + 26k + 121)},  {−18, 1k, 1  2(8 − k ±  √  k2 + 28k + 100)},  {−19, 1k, 1  2(9 − k ±  √  k2 + 38k + 121)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfp_3T/content/2301.01623v1.pdf'}
+page_content='  A6 =  \uf8ee  \uf8f0  J −Im  J  J  J  Iℓ−J  O  J  O  R2k  \uf8f9  \uf8fb  (m ≥ 1, (ℓ, k) = (3, 4), or (4, 3)) with spectra  {−1m+4, 15, 1  2(m − 1 ±  √  m2 + 42m + 9)},  {−1m+3, 15, 1  2(m − 2 ±  √  m2 + 40m + 16)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfp_3T/content/2301.01623v1.pdf'}
+page_content='  A7 =  \uf8ee  \uf8f0  R2m  J  J  J  J −Iℓ  O  J  O  −R2k  \uf8f9  \uf8fb  (m ≥ 1, (ℓ, k) = (3, 3), or (4, 2)) with spectra  {−1m+4, 1m+3, 1  2(1 ± √8m + 1},  {−1m+4, 1m+2, 1 ± 2√4m + 1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfp_3T/content/2301.01623v1.pdf'}
+page_content='  A8 =  \uf8ee  \uf8ef\uf8ef\uf8f0  R2  J  1  O  J  Im−J  0  J  1⊤  0⊤  0  0⊤  O  J  0  −R4  \uf8f9  \uf8fa\uf8fa\uf8fb  (m ≥ 1) with spectrum  {−13, 1m+2, 1  2(1 − m ±  √  m2 + 22m + 9)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfp_3T/content/2301.01623v1.pdf'}
+page_content='  A9 =  \uf8ee  \uf8ef\uf8ef\uf8f0  R2m  J  J  0  J  R2  O  1  J  O  −R2  0  0⊤  1⊤  0⊤  0  \uf8f9  \uf8fa\uf8fa\uf8fb (m ≥ 1) with spectrum {−1m+2, 1m+1, 1  2(1 ± √32m + 9)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfp_3T/content/2301.01623v1.pdf'}
+page_content='  A10 =  \uf8ee  \uf8ef\uf8ef\uf8f0  J −Im  J  O  O  J  O  J  J −Iℓ  O  J  Im−J  O  O  J −Iℓ  O  O  \uf8f9  \uf8fa\uf8fa\uf8fb  ((m, ℓ) = (3, 4), or (4, 3)), with spectra  {−16, 16, ±6}, {−16, 16, ±6}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfp_3T/content/2301.01623v1.pdf'}
+page_content='  A11 =  \uf8ee  \uf8ef\uf8ef\uf8f0  J −Im  J  J  O  J  Im−J  O  J  J  O  O  J −Iℓ  O  J  J −Iℓ  O  \uf8f9  \uf8fa\uf8fa\uf8fb  ((m, ℓ) = (3, 4), or (4, 3)) with spectra  {−16, 16, ±3  √  5)}, {−16, 16, ±2  √  13}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfp_3T/content/2301.01623v1.pdf'}
+page_content='  A12 =  \uf8ee  \uf8ef\uf8ef\uf8f0  J −Im  J  J  O  J  Iℓ−J  O  J  J  O  J −Ik  J  O  J  J  Ij −J  \uf8f9  \uf8fa\uf8fa\uf8fb  ((m, ℓ, k, j) = (6, 3, 3, 6), (6, 6, 3, 3), (6, 4, 3, 4),  or (4, 4, 4, 4)), with spectra {−18, 18, ±  √  109},  {−18, 18, ±10}, {−17, 18, 1  2(−1 ± 3  √  41)},  {−17, 17, ±9}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfp_3T/content/2301.01623v1.pdf'}
+page_content='  A13 =  \uf8ee  \uf8ef\uf8ef\uf8f0  J −Im  J  O  0  J  −R2ℓ  J  1  O  J  I4−J  0  0⊤  1⊤  0⊤  0  \uf8f9  \uf8fa\uf8fa\uf8fb  ((m, ℓ) = (6, 2), or (5, 3)), with spectra  {−17, 16, 1  2(1 ±  √  241)}, {−17, 17, ±6  √  2}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfp_3T/content/2301.01623v1.pdf'}
+page_content='  A14 =  \uf8ee  \uf8f0  J −Im  J  O  J  −R2ℓ  J  O  J  −R4  \uf8f9  \uf8fb  ((m, ℓ) = (6, 2), or (5, 3)) with spectra  {−17, 15, 1 ± 2  √  13}, {−17, 16, 1  2(1 ±  √  249}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfp_3T/content/2301.01623v1.pdf'}
+page_content='  3  A15 =  \uf8ee  \uf8ef\uf8ef\uf8f0  J −Im  J  J  O  J  Iℓ−J  O  J  J  O  R2k  J  O  J  J  −R2j  \uf8f9  \uf8fa\uf8fa\uf8fb  ((m, ℓ, k, j) = (3, 3, 3, 3), (4, 3, 3, 2), or (4, 4, 2, 2))  with spectra {−18, 18, ±  √  85}, {−18, 17, 1  2(1 ±  √  313)},  {−17, 17, ±  √  73}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfp_3T/content/2301.01623v1.pdf'}
+page_content='  A16 =  \uf8ee  \uf8ef\uf8ef\uf8f0  R2  J  J  O  J  −R2  O  J  J  O  O  I3  O  J  I3  O  \uf8f9  \uf8fa\uf8fa\uf8fbwith spectrum {−14, 14, ±  √  17}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfp_3T/content/2301.01623v1.pdf'}
+page_content='  A17 =  \uf8ee  \uf8ef\uf8ef\uf8ef\uf8ef\uf8ef\uf8f0  R2  J  J  1  O  0  J  −R2  O  0  J  1  J  O  R2  0  J  0  1⊤  0⊤  0⊤  0  0⊤  0  O  J  J  0  −R2  0  0⊤  1⊤  0⊤  0  0⊤  0  \uf8f9  \uf8fa\uf8fa\uf8fa\uf8fa\uf8fa\uf8fb  with spectrum {−14, 14, ±2  √  5}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfp_3T/content/2301.01623v1.pdf'}
+page_content='  A18 =  \uf8ee  \uf8ef\uf8ef\uf8f0  J −Im  J  1  O  J  −R2ℓ  0  J  1⊤  0⊤  0  0⊤  O  J  0  −R2  \uf8f9  \uf8fa\uf8fa\uf8fb  ((m, ℓ) = (4, 3), or (3, 4)) with spectra  {−16, 15, 1  2(1 ±  √  177)}, {−16, 16, ±3  √  5}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfp_3T/content/2301.01623v1.pdf'}
+page_content='  A19 =  \uf8ee  \uf8ef\uf8ef\uf8ef\uf8ef\uf8f0  R2  J  J  1  O  J  Im−J  O  0  J  J  O  R2ℓ  0  J  1⊤  0⊤  0⊤  0  0⊤  O  J  J  0  −R2  \uf8f9  \uf8fa\uf8fa\uf8fa\uf8fa\uf8fb  ((m, ℓ) = (3, 3), or (4, 2)) with spectra  {−16, 16, ±2  √  10}, {−15, 16, 1  2(−1 ± 3  √  17)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfp_3T/content/2301.01623v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfp_3T/content/2301.01623v1.pdf'}
+page_content=' To see that each of the above matrices has the given spectrum, we use the same method as  used in [4] and [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfp_3T/content/2301.01623v1.pdf'}
+page_content=' For each matrix A the presented block structure gives an equitable partition  (that is, each block has constant row and column sums).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfp_3T/content/2301.01623v1.pdf'}
+page_content=' Part of the spectrum of A is the spectrum  of the quotient matrix Q (containing the row sums of the blocks), and it is straightforward to  check that each Q has just two eigenvalues different from ±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfp_3T/content/2301.01623v1.pdf'}
+page_content=' The remaining eigenvalues of A  remain unchanged if we add or subtract an all-one block J to some of the blocks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfp_3T/content/2301.01623v1.pdf'}
+page_content=' One easily  verifies that this can be done in such a way that we obtain a matrix with all eigenvalues equal  to ±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfp_3T/content/2301.01623v1.pdf'}
+page_content=' The proof that the list is complete comprises the remainder of this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfp_3T/content/2301.01623v1.pdf'}
+page_content='  Note that the above list is not free from overlap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfp_3T/content/2301.01623v1.pdf'}
+page_content=' For example the negatives of A2, A3 and  A4 can alo be obtained by interchanging m and ℓ, and several cases with symmetric spectrum  are switching isomorphic with their negatives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfp_3T/content/2301.01623v1.pdf'}
+page_content='  Also note that A5 with k = 1 corresponds to the three unsigned graphs in G given in Theo-  rem 1(v) of [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfp_3T/content/2301.01623v1.pdf'}
+page_content=' Thus these three sporadic graphs from [4] are in fact part of an infinite family of  signed graphs in G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfp_3T/content/2301.01623v1.pdf'}
+page_content='  3  Preliminaries  We start with some results from [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfp_3T/content/2301.01623v1.pdf'}
+page_content='  4  Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfp_3T/content/2301.01623v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfp_3T/content/2301.01623v1.pdf'}
+page_content=' If a signed graph Gσ has smallest eigenvalue at least −1, then the underlying graph  G is a disjoint union of complete graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfp_3T/content/2301.01623v1.pdf'}
+page_content='  The following two results are a bit more general than Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfp_3T/content/2301.01623v1.pdf'}
+page_content='3 and Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfp_3T/content/2301.01623v1.pdf'}
+page_content='4 in [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfp_3T/content/2301.01623v1.pdf'}
+page_content='  The proofs, however, are basically the same.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfp_3T/content/2301.01623v1.pdf'}
+page_content='  Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfp_3T/content/2301.01623v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfp_3T/content/2301.01623v1.pdf'}
+page_content=' If Gσ is connected and all but at most one eigenvalues are in the interval  [−1, 1], then Gσ or G−σ is switching isomorphic with an unsigned complete graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfp_3T/content/2301.01623v1.pdf'}
+page_content='  Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfp_3T/content/2301.01623v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfp_3T/content/2301.01623v1.pdf'}
+page_content=' If Gσ is disconnected and has no isolated vertices or edges, and all but two  eigenvalues of Gσ are in the interval [−1, 1], then Gσ is the disjoint union of two signed graphs  each of which is switching isomorphic with an unsigned complete graph or its negative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfp_3T/content/2301.01623v1.pdf'}
+page_content='  Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfp_3T/content/2301.01623v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfp_3T/content/2301.01623v1.pdf'}
+page_content=' If Gσ ∈ G is bipartite, then Gσ is switching isomorphic with an unsigned bipartite  graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfp_3T/content/2301.01623v1.pdf'}
+page_content='  The bipartite graphs in G are described in [5, 4, 6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfp_3T/content/2301.01623v1.pdf'}
+page_content='  Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfp_3T/content/2301.01623v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfp_3T/content/2301.01623v1.pdf'}
+page_content=' Assume Kσ  n is a signed complete graph of order n for which the adjacency matrix  has all but two eigenvalues in the interval [−1, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfp_3T/content/2301.01623v1.pdf'}
+page_content=' Then Kσ  n is switching isomorphic with a signed  graph with adjacency matrix  �Jm,ℓ =  �J − Im  J  J  −J + Iℓ  �  with n = m+ℓ, m, ℓ ≥ 2 and spectrum {−1m−1, 1ℓ−1, 1  2(m−ℓ±  √  m2 + ℓ2 + 6mℓ + 4m + 4ℓ + 4)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfp_3T/content/2301.01623v1.pdf'}
+page_content='  4  Restrictions  In the remainder we will assume that Gσ is a signed graph in G of order n which is connected,  and not bipartite or complete.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfp_3T/content/2301.01623v1.pdf'}
+page_content=' Then, by Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfp_3T/content/2301.01623v1.pdf'}
+page_content='2, Gσ has two eigenvalues r and s different  from ±1, and we may assume that s < −1 and r > 1, since otherwise Gσ or its negative has  smallest eigenvalue −1, so Gσ would be disconnected or complete by Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfp_3T/content/2301.01623v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfp_3T/content/2301.01623v1.pdf'}
+page_content='  An important tool is eigenvalue interlacing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfp_3T/content/2301.01623v1.pdf'}
+page_content=' It follows that every induced signed subgraph of  a graph Gσ in G has second largest eigenvalue at most 1, and second smallest eigenvalue at least  −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfp_3T/content/2301.01623v1.pdf'}
+page_content=' This excludes many subgraphs, including the path Pn with n ≥ 6 and the n-cycle with n ≥ 5  (for every signing).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfp_3T/content/2301.01623v1.pdf'}
+page_content=' Some other forbidden induced signed subgraphs are presented in Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfp_3T/content/2301.01623v1.pdf'}
+page_content='  t  t  t  t  t  t  t  t  t  t  ❩❩  ✚  ✚  ✑✑  ◗  ◗  ❩❩  ❩  ✚  ✚  ✚  t  t  t  t  t  ✂  ✂  ✂  ✂  ❇  ❇❇  ✚✚✚✚  ❩  ❩  ❩  ❩  ✂  ✂✂  ❩❩  ❩  ✚  ✚  ✚  Figure 1: forbidden signed graphs;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfp_3T/content/2301.01623v1.pdf'}
+page_content=' negative edges are dashed  Of course the signed graphs switching isomorphic with a graph in Figure 1 and their negatives  are also forbidden, but this does not include all possible signings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfp_3T/content/2301.01623v1.pdf'}
+page_content=' For example, the unsigned  5  t  t  t  t  t  ❇  ❇❇  ✂  ✂✂  ❩❩  ❩  ✚  ✚  ✚  t  t  t  t  t  ❇  ❇❇  ✂  ✂✂  ❩❩  ❩  ✚  ✚  ✚  t  t  t  t  t  ✚✚✚✚  ❇  ❇❇  ✂  ✂✂  ❩❩  ❩  ✚  ✚  ✚  ��  ❅❅  ❅  ❅  �  �  t  t  t  t  t  t  ��  ❅❅  ❅  ❅  ❅  �  �  �  t  t  t  t  t  t  Figure 2: forbidden graphs for all signings  4-cycle can occur.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfp_3T/content/2301.01623v1.pdf'}
+page_content=' Figure 2 lists forbidden induced subgraphs for the underlying unsigned graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfp_3T/content/2301.01623v1.pdf'}
+page_content='  In other words, these graphs are forbidden for all signings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfp_3T/content/2301.01623v1.pdf'}
+page_content=' Another useful fact is that if A is  the adjacency matrix of a signed graph in G, then A2 has eigenvalues 1, r2 and s2, so A2 − I is  positive semi-definite (psd) and has rank 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfp_3T/content/2301.01623v1.pdf'}
+page_content=' The following Lemma is a generalization of Lemma 2  from [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfp_3T/content/2301.01623v1.pdf'}
+page_content=' The proof is the same.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfp_3T/content/2301.01623v1.pdf'}
+page_content='  Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfp_3T/content/2301.01623v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfp_3T/content/2301.01623v1.pdf'}
+page_content=' Suppose Gσ ∈ G with adjacency matrix A, then  (i) if Gσ has no isolated edges, then there is no vertex of degree 1,  (ii) if x and y are two columns of A such that x⊤x = x⊤y, then x and y differ in at least three  coordinate places, in particular no two columns (or rows) of A are equal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfp_3T/content/2301.01623v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfp_3T/content/2301.01623v1.pdf'}
+page_content=' (i) Suppose Gσ has a vertex v of degree 1 and a vertex w at distance 2 from v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfp_3T/content/2301.01623v1.pdf'}
+page_content=' Then the  2×2 submatrix of A2 −I corresponding to v and w has a negative determinant which contradicts  that A2 − I is psd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfp_3T/content/2301.01623v1.pdf'}
+page_content=' (ii) In this case A2 − I has a psd submatrix  M =  � x⊤x − 1  x⊤y  y⊤x  y⊤y − 1  �  =  � x⊤x − 1  x⊤x  x⊤x  y⊤y − 1  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfp_3T/content/2301.01623v1.pdf'}
+page_content='  So det(M) ≥ 0 which implies that y⊤y > x⊤x + 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfp_3T/content/2301.01623v1.pdf'}
+page_content='  5  The possible formats  Since Gσ is not bipartite and contains no odd cycle of length 5 or more, Gσ contains a triangle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfp_3T/content/2301.01623v1.pdf'}
+page_content='  Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfp_3T/content/2301.01623v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfp_3T/content/2301.01623v1.pdf'}
+page_content=' Let C be a maximal clique in Gσ of size c ≥ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfp_3T/content/2301.01623v1.pdf'}
+page_content=' Then, up to vertex ordering,  switching and taking the negative, the c × n submatrix of A corresponding to C is one of the  following:  (i)  \uf8ee  \uf8f0 J − Ic  Jc1×x  −Jc2×x  O  Oc×z  \uf8f9  \uf8fb with c1 + c2 < c, c1, c2, x > 0, z ≥ 0,  (ii)  �  J − Ic  Jc1×x  O  O  Jc2×y  Oc×z  �  with c1 + c2 = c, c1, c2, x > 0, y, z ≥ 0,  (iii)  �  �Jc1,c2  Jc1×x  O  O  Jc2×y  Oc×z  �  with c1 + c2 = c, c1, c2 ≥ 2, x > 0, y, z ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfp_3T/content/2301.01623v1.pdf'}
+page_content='  6  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfp_3T/content/2301.01623v1.pdf'}
+page_content=' The clique C is a signed complete graph which has, by interlacing, all but at most two  eigenvalues in the interval [−1, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfp_3T/content/2301.01623v1.pdf'}
+page_content=' By Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfp_3T/content/2301.01623v1.pdf'}
+page_content='1 the adjacency matrix of C is, up to switching  and taking the negative, equal to J −Ic, or �Jc1,c2, where c1 +c2 = c and c1, c2 ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfp_3T/content/2301.01623v1.pdf'}
+page_content=' So the matrix  N consisting of the rows of A corresponding to C takes the form [ J − Ic | L ] or [ �Jc1,c2 | �L ].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfp_3T/content/2301.01623v1.pdf'}
+page_content='  Since NN ⊤ − I is a principal submatrix of A2 − I, the rank of NN ⊤ − I is at most 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfp_3T/content/2301.01623v1.pdf'}
+page_content=' In case  N = [ J − Ic | L ] we get  NN ⊤ − I = (c − 2)J + LL⊤ = MM ⊤,  where M = [ (√c − 2) 1 | L ].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfp_3T/content/2301.01623v1.pdf'}
+page_content=' This implies that M, and hence [ 1 | L ] has rank at most 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfp_3T/content/2301.01623v1.pdf'}
+page_content=' Let  k ̸= 0 be a column of L (L ̸= O, since Gσ is connected).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfp_3T/content/2301.01623v1.pdf'}
+page_content=' Then k has at least one entry equal to  0 (because C is maximal), and we may assume that k has at least one positive entry (otherwise  we switch).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfp_3T/content/2301.01623v1.pdf'}
+page_content=' From rank([ 1 | L] ) ≤ 2 it follows that every column of L is equal to k, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfp_3T/content/2301.01623v1.pdf'}
+page_content=' or 1 − k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfp_3T/content/2301.01623v1.pdf'}
+page_content='  If k has positive and negative entries, then L has the form given in (i), otherwise we get case  (ii).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfp_3T/content/2301.01623v1.pdf'}
+page_content=' In case N = [ �Jc1,c2 | �L ] we get  NN ⊤ − I =  �  (c − 2)J  (c1 − c2)Jc1×c2  (c1 − c2)Jc2×c1  (c − 2)J  �  + �L�L⊤ = MM ⊤,  where M = [ (√c1 − 1) 1  (√c2 − 1) �1  �L ], with �1 = [ 1⊤  c1 | − 1⊤  c2 ]⊤.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfp_3T/content/2301.01623v1.pdf'}
+page_content=' This implies that M, and  hence [ 1 | �1 | �L ] has rank at most 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfp_3T/content/2301.01623v1.pdf'}
+page_content=' Therefore every column of �L is a linear combination of 1,  and �1, but has at least one zero entry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfp_3T/content/2301.01623v1.pdf'}
+page_content=' So, up to switching, only the columns given in case (iii)  are allowed, and since Gσ is connected we may assume x > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfp_3T/content/2301.01623v1.pdf'}
+page_content='  6  Format (i)  Assume Gσ contains a clique C, which gives format (i) from Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfp_3T/content/2301.01623v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfp_3T/content/2301.01623v1.pdf'}
+page_content=' We claim that c1 =  c2 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfp_3T/content/2301.01623v1.pdf'}
+page_content=' Indeed, if c1 ≥ 2, or c2 ≥ 2, then Gσ has an induced subgraph switching isomorphic with  the last signed graph in Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfp_3T/content/2301.01623v1.pdf'}
+page_content=' Next consider the matrix N which consists of row 1, row 3 and  row c + i of A (1 ≤ i ≤ x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfp_3T/content/2301.01623v1.pdf'}
+page_content=' Then  N =  \uf8ee  \uf8ef\uf8f0  0  1  1  1⊤  1⊤  x  0⊤  z  1  1  0  1⊤  0⊤  0⊤  1  −1  0  0⊤  x⊤  z⊤  \uf8f9  \uf8fa\uf8fb , and  NN⊤ − I =  \uf8ee  \uf8f0  c − 2 + x  c − 2  x⊤1 − 1  c − 2  c − 2  0  x⊤1 − 1  0  1 + x⊤x + z⊤z  \uf8f9  \uf8fb  for (0, ±1)-vectors x and z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfp_3T/content/2301.01623v1.pdf'}
+page_content=' Note that x has an entry on the diagonal of A, which is equal to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfp_3T/content/2301.01623v1.pdf'}
+page_content='  Therefore |x⊤1| ≤ x⊤x ≤ x − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfp_3T/content/2301.01623v1.pdf'}
+page_content=' We have that NN ⊤ − I is singular, so  0 = det(NN⊤ − I) = (c − 2)(x(1 + x⊤x + z⊤z) − (x⊤1 − 1)2),  which implies that −x⊤1 = x⊤x = x−1 and z⊤z = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfp_3T/content/2301.01623v1.pdf'}
+page_content=' Finally, z = 0 implies that z = 0 (otherwise  Gσ is disconnected), and thus we have:  Conclusion 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfp_3T/content/2301.01623v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfp_3T/content/2301.01623v1.pdf'}
+page_content=' If Gσ has adjacency matrix A with Format (i) then A = A3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfp_3T/content/2301.01623v1.pdf'}
+page_content='  7  7  Further restrictions  In the remainder we will assume that Gσ contains no maximal clique with Format (i), and that  the matrix A of Gσ takes the form of Format (ii) or (iii) as described in Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfp_3T/content/2301.01623v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfp_3T/content/2301.01623v1.pdf'}
+page_content=' Concerning  the maximal clique C we will assume the following:  Assumption 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfp_3T/content/2301.01623v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfp_3T/content/2301.01623v1.pdf'}
+page_content='  (i) The clique C has maximum order c,  (ii) C has the largest number of outgoing edges among all cliques of order c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfp_3T/content/2301.01623v1.pdf'}
+page_content='  For both formats (ii) and (iii), C is the union of two cliques C1 and C2, such that the  remaining vertices can be partitioned into three sets X, Y , and Z, where the vertices in X are  adjacent to C1 and not to C2, the vertices in Y are adjacent to C2 and not to C1, and the vertices  from Z are adjacent to C1 nor C2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfp_3T/content/2301.01623v1.pdf'}
+page_content=' Note that the sets X, Y and Z may be empty, but X and Y  are not both empty (otherwise Gσ is disconnected or complete) and we may choose that X ̸= ∅  (otherwise we interchange C1 with C2 and X with Y ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfp_3T/content/2301.01623v1.pdf'}
+page_content=' in case of format (iii) we also need to  switch and take the negative).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfp_3T/content/2301.01623v1.pdf'}
+page_content=' Recall that ci = |Ci| (i = 1, 2), x = |X|, y = |Y | and z = |Z|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfp_3T/content/2301.01623v1.pdf'}
+page_content=' For  a subset V of the vertex set of Gσ we let Gσ  V denote the subgraph of Gσ induced by V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfp_3T/content/2301.01623v1.pdf'}
+page_content='  Proposition 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfp_3T/content/2301.01623v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfp_3T/content/2301.01623v1.pdf'}
+page_content='  (i) Gσ  X and Gσ  Y are disjoint unions of cliques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfp_3T/content/2301.01623v1.pdf'}
+page_content='  (ii) Two adjacent vertices in Gσ  X or Gσ  Y correspond to identical rows and columns in A + I or  A − I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfp_3T/content/2301.01623v1.pdf'}
+page_content='  (iii) If c1 ≥ 3 [c2 ≥ 3], then all edges of Gσ  Y [Gσ  X] have the same sign which is opposite to the  sign of C1 [C2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfp_3T/content/2301.01623v1.pdf'}
+page_content='  (iv) Every clique in Gσ  Y [Gσ  X] has order at most 2, or one clique has order at least 3 and all other  cliques have order 1 and all edges have the same sign opposite to the sign of C1 [C2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfp_3T/content/2301.01623v1.pdf'}
+page_content='  (v) There are no negative edges between X and Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfp_3T/content/2301.01623v1.pdf'}
+page_content='  (vi) A vertex from Z cannot be adjacent to both X and Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfp_3T/content/2301.01623v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfp_3T/content/2301.01623v1.pdf'}
+page_content=' (i) Suppose Gσ  X contains an induced path P3 then the subgraph of Gσ induced by this path  together with one vertex of C1 and one vertex of C2 is a forbidden subgraph (no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfp_3T/content/2301.01623v1.pdf'}
+page_content=' 3 in Figure 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfp_3T/content/2301.01623v1.pdf'}
+page_content='  Therefore Gσ  X contains no P3, so Gσ  X is a clique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfp_3T/content/2301.01623v1.pdf'}
+page_content=' (ii) Let C3 be the clique in Gσ  A containing  the two adjacent vertices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfp_3T/content/2301.01623v1.pdf'}
+page_content=' Then (ii) follows from Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfp_3T/content/2301.01623v1.pdf'}
+page_content='1 applied to to the maximal clique  C1 ∪ C3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfp_3T/content/2301.01623v1.pdf'}
+page_content=' (iii) Take an edge e in Gσ  X, and consider the subgraph Hσ of Gσ  X induced by e, three  vertices of C2 and one vertex of C1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfp_3T/content/2301.01623v1.pdf'}
+page_content=' If e has the same sign as the edges of C2, then Hσ is switching  isomorphic with the second forbidden subgraph of Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfp_3T/content/2301.01623v1.pdf'}
+page_content=' This proves (iii).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfp_3T/content/2301.01623v1.pdf'}
+page_content=' (iv) Suppose Gσ  Y  contains a clique C3 of order at least 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfp_3T/content/2301.01623v1.pdf'}
+page_content=' Then c1 ≥ 3 and all edges of C3 have the same sign.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfp_3T/content/2301.01623v1.pdf'}
+page_content=' If Gσ  Y  contains another clique of order at least 2, then Gσ contains a signed graph switching isomorphic  with the second forbidden signed subgraph of Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfp_3T/content/2301.01623v1.pdf'}
+page_content=' This proves (iv).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfp_3T/content/2301.01623v1.pdf'}
+page_content=' (v) A negative edge  between X and Y will create a 4-cycle with one negative edge which is forbidden.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfp_3T/content/2301.01623v1.pdf'}
+page_content=' (vi) A vertex  in Z adjacent to a vertex of X and a vertex from Y creates an induced pentagon, or a pentagon  with one chord, which are forbidden for all signings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfp_3T/content/2301.01623v1.pdf'}
+page_content='  Note that Assumption 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfp_3T/content/2301.01623v1.pdf'}
+page_content='1 implies that a clique in Gσ  X [or Gσ  Y ] has order at most c2 [c1], and  for a clique C3 in Gσ  X [Gσ  Y ] of order c2 [c1] there are at most c2y [c1x] edges between C3 and  Y ∪ Z [X ∪ Z].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfp_3T/content/2301.01623v1.pdf'}
+page_content='  8  8  Format (ii) with Y = ∅  Suppose Gσ has format (ii) and Y = ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfp_3T/content/2301.01623v1.pdf'}
+page_content=' Let N be the matrix consisting of three rows of A  corresponding to a vertex from C1, a vertex from C2 and a vertex v from X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfp_3T/content/2301.01623v1.pdf'}
+page_content=' Then  N =  \uf8ee  \uf8ef\uf8f0  0  1⊤  c1−1  1  1⊤  c2−1  1⊤  x  0⊤  z  1  1⊤  0  1⊤  0⊤  0⊤  1  1⊤  0  0⊤  x⊤  1  z⊤  1  \uf8f9  \uf8fa\uf8fb  (for (0, ±1) vectors x1 and z1), and  M = NN ⊤ − I =  \uf8ee  \uf8ef\uf8f0  c + x − 2  c − 2  c1 + x′  1 − 1  c − 2  c − 2  c1  c1 + x′  1 − 1  c1  c1 + x1 + z1 − 1  \uf8f9  \uf8fa\uf8fb ,  where x′  1 = 1⊤x, x1 = x⊤  1x1 and z1 = z⊤  1z1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfp_3T/content/2301.01623v1.pdf'}
+page_content='  Note that |x′  1| ≤ x1 ≤ x − 1, x1 + 1 ≤ c2  and x1 + 1 = c2 implies that z1 = 0 by Assumption 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfp_3T/content/2301.01623v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfp_3T/content/2301.01623v1.pdf'}
+page_content=' Using Gaussian elimination we get  det(M) = x(c − 2)(c1 + x1 + z1 − 1) − (c − 2)(x′  1 − 1)2 − xc2  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfp_3T/content/2301.01623v1.pdf'}
+page_content='  For convenience we define  f(a, b) = ab/(a + b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfp_3T/content/2301.01623v1.pdf'}
+page_content=' We have det(M) = 0, which gives:  x1 + z1 + f(c1, c2 − 2) = (x′  1 − 1)2  x  + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfp_3T/content/2301.01623v1.pdf'}
+page_content='  (1)  If c2 = 1, Proposition 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfp_3T/content/2301.01623v1.pdf'}
+page_content='2(i−iii) implies that X is a coclique and that there are no edges between  X and Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfp_3T/content/2301.01623v1.pdf'}
+page_content=' Then A has x + 1 equal rows, which contradicts Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfp_3T/content/2301.01623v1.pdf'}
+page_content='1(ii).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfp_3T/content/2301.01623v1.pdf'}
+page_content=' Therefore c2 ≥ 2  and f(c1, c2 − 2) ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfp_3T/content/2301.01623v1.pdf'}
+page_content=' From Assumption 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfp_3T/content/2301.01623v1.pdf'}
+page_content='1 it follows that x1 = x′  1 = 1, or x1 = −x′  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfp_3T/content/2301.01623v1.pdf'}
+page_content=' We will  consider the different possibilities for x1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfp_3T/content/2301.01623v1.pdf'}
+page_content='  Proposition 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfp_3T/content/2301.01623v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfp_3T/content/2301.01623v1.pdf'}
+page_content=' Suppose x1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfp_3T/content/2301.01623v1.pdf'}
+page_content=' Then v is the only isolated vertex of Gσ  X and one of the  following holds:  (i) x = 1, c2 = 2 and A = A4 with m = 1 and ℓ ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfp_3T/content/2301.01623v1.pdf'}
+page_content='  (ii) A is an unsigned matrix given in case (v) of Theorem 1 from [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfp_3T/content/2301.01623v1.pdf'}
+page_content='  (iii) (c1, c2, x) ∈ {(2, 5, 5), (2, 6, 3), (3, 4, 5), (4, 4, 3)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfp_3T/content/2301.01623v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfp_3T/content/2301.01623v1.pdf'}
+page_content=' Equation 1 gives z1 = 1 + 1/x − f(c1, c2 − 2), so z1 ≤ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfp_3T/content/2301.01623v1.pdf'}
+page_content=' If z1 = 2 then x = 1 and c2 = 2,  which straightforwardly leads to matrix A4 with m = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfp_3T/content/2301.01623v1.pdf'}
+page_content=' If z1 = 1 then f(c1, c2 −2) = 1/x, which  implies (c1, c2, x) = (2, 4, 1), or (1, 3, 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfp_3T/content/2301.01623v1.pdf'}
+page_content=' Both of these cases cannot be completed to a signed  graph in G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfp_3T/content/2301.01623v1.pdf'}
+page_content=' If z1 = 0 then f(c1, c2−2) = 1+1/x, which implies that (c1, c2, x) is equal to (4, 2, 1),  (3, 8, 1), (6, 5, 1), (3, 5, 2), (4, 4, 3), (2, 6, 3), (2, 5, 5), or (3, 4, 5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfp_3T/content/2301.01623v1.pdf'}
+page_content=' The first three triples lead to case  (v) of Theorem 1 from [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfp_3T/content/2301.01623v1.pdf'}
+page_content=' If Gσ  X has a second isolated vertex then the two corresponding rows  of A are equal which contradicts Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfp_3T/content/2301.01623v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfp_3T/content/2301.01623v1.pdf'}
+page_content=' Therefore Gσ  X has just one isolated vertex and the  triple (3, 5, 2) is not possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfp_3T/content/2301.01623v1.pdf'}
+page_content='  Proposition 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfp_3T/content/2301.01623v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfp_3T/content/2301.01623v1.pdf'}
+page_content=' If x1 = x − 1 ≥ 2 then A = A5 with k ≥ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfp_3T/content/2301.01623v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfp_3T/content/2301.01623v1.pdf'}
+page_content=' The underlying graph GX of Gσ  X is complete, and from Proposition 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfp_3T/content/2301.01623v1.pdf'}
+page_content='2(iii) it follows that  all signs are negative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfp_3T/content/2301.01623v1.pdf'}
+page_content=' Formula 1 becomes: z1 = 2 − f(c1, c2 − 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfp_3T/content/2301.01623v1.pdf'}
+page_content=' If z1 = 2, then c2 = 2 and  x1 ≤ 1, which is excluded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfp_3T/content/2301.01623v1.pdf'}
+page_content=' If z1 = 1 then f(c1, c2 − 2) = 1, so c1 = 2, c2 = 4 and x = 2 or 3  9  which has no solution by straightforward verification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfp_3T/content/2301.01623v1.pdf'}
+page_content=' If z1 = 0, then f(c1, c2 − 2) = 2, which  gives (c1, c2) = (3, 8), (4, 6) or 6, 5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfp_3T/content/2301.01623v1.pdf'}
+page_content=' Thus we find A5 from Theorem 2 (actually, only the ones  with k ≤ ℓ, because of Assumption 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfp_3T/content/2301.01623v1.pdf'}
+page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfp_3T/content/2301.01623v1.pdf'}
+page_content='  Proposition 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfp_3T/content/2301.01623v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfp_3T/content/2301.01623v1.pdf'}
+page_content=' If x1 = 1, then one of the following holds:  (i) A = A5 with k = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfp_3T/content/2301.01623v1.pdf'}
+page_content='  (ii) A = −A6 with m = 1 (switched around the first vertex).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfp_3T/content/2301.01623v1.pdf'}
+page_content='  (iii) A = A7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfp_3T/content/2301.01623v1.pdf'}
+page_content='  (iv) A is an unsigned matrix given in case (ii) of Theorem 1 from [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfp_3T/content/2301.01623v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfp_3T/content/2301.01623v1.pdf'}
+page_content=' If x1 = −x′  1 = 1, then clearly x ≥ 2 and Formula 1 becomes z1 = 4/x − f(c1, c2 − 2)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfp_3T/content/2301.01623v1.pdf'}
+page_content=' It  also follows that c2 ≥ 3, since c2 = 2 would imply z1 = 0 (by Assumption 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfp_3T/content/2301.01623v1.pdf'}
+page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfp_3T/content/2301.01623v1.pdf'}
+page_content=' From x ≥ 2 and  f(c1, c2 − 2) ≥ 1 it follows that z1 ≤ 1, and that z1 = 1 implies (c1, c2, x) = (2, 4, 2), which is not  possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfp_3T/content/2301.01623v1.pdf'}
+page_content=' So z1 = 0 and we obtain the following solutions for Formula 1: (c1, c2, x) = (1, 3, 8),  (1, 4, 6), (1, 6, 5), (2, 3, 6), (2, 4, 4), (2, 6, 3), (3, 8, 2), (4, 3, 5), (4, 4, 3), (4, 6, 2), or (6, 5, 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfp_3T/content/2301.01623v1.pdf'}
+page_content=' By  Proposition 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfp_3T/content/2301.01623v1.pdf'}
+page_content='2(iv), Gσ  X has an isolated vertex if x is odd, in which case we need a solution with  x1 = 0 for the same (c1, c2, x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfp_3T/content/2301.01623v1.pdf'}
+page_content=' Proposition 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfp_3T/content/2301.01623v1.pdf'}
+page_content='1 gives just two possibilities: (2, 6, 3) and (4, 4, 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfp_3T/content/2301.01623v1.pdf'}
+page_content='  In both cases Gσ  X consists of one negative edge and an isolated vertex, and it is easily checked  that this is not possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfp_3T/content/2301.01623v1.pdf'}
+page_content=' Thus x is even, and of the remaining seven triples the first two lead to  −A6 with m = 1, the next two to A7, and the last three to A5 with k = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfp_3T/content/2301.01623v1.pdf'}
+page_content=' If x1 = x′  1 = 1, then  we easily have c2 = 2 and z1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfp_3T/content/2301.01623v1.pdf'}
+page_content=' None of the solutions above has c2 = 2 and x ≥ 2, therefore all  rows corresponding to X have x1 = x′  1 = 1, c2 = 2 and z1 = 0, which implies that Gσ is case (ii)  of Theorem 1 from [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfp_3T/content/2301.01623v1.pdf'}
+page_content='  Proposition 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfp_3T/content/2301.01623v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfp_3T/content/2301.01623v1.pdf'}
+page_content=' If 2 ≤ x1 ≤ x − 2, then there is no solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfp_3T/content/2301.01623v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfp_3T/content/2301.01623v1.pdf'}
+page_content=' Proposition 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfp_3T/content/2301.01623v1.pdf'}
+page_content='2(iv) implies that if there is a solution with 2 ≤ x1 ≤ x − 2, then there is  also one with x1 = 0 for the same c1, c2 and x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfp_3T/content/2301.01623v1.pdf'}
+page_content=' From Proposition 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfp_3T/content/2301.01623v1.pdf'}
+page_content='1 we find that there is just  one candidate: (c1, c2, x) = (3, 4, 5) and x1 = −x′  1 = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfp_3T/content/2301.01623v1.pdf'}
+page_content=' But this does not satisfy Formula 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfp_3T/content/2301.01623v1.pdf'}
+page_content='  Finally we revisit Proposition 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfp_3T/content/2301.01623v1.pdf'}
+page_content='1(iii).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfp_3T/content/2301.01623v1.pdf'}
+page_content='  For each of the four cases there must be another  solution with x1 > 0, which we haven’t found.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfp_3T/content/2301.01623v1.pdf'}
+page_content=' So none of these four triples are possible, and the  determination all graphs with Format (ii) and Y = ∅ is complete.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfp_3T/content/2301.01623v1.pdf'}
+page_content='  Conclusion 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfp_3T/content/2301.01623v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfp_3T/content/2301.01623v1.pdf'}
+page_content=' If A has Format (ii) with Y = ∅ then A is unsigned (case (ii), or (v) from  Theorem 1 of [4]), or A is one of the following: (i) A4 with m = 1, ℓ ≥ 1, (ii) A5 with 2 ≤ k ≤ ℓ,  (iii) −A6 with m = 1 (switched around the first vertex), (iv) A7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfp_3T/content/2301.01623v1.pdf'}
+page_content='  9  Format (iii) with Y = ∅  Suppose Gσ has format (iii) and Y = ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfp_3T/content/2301.01623v1.pdf'}
+page_content=' Again we let N be the matrix consisting of row 1,  row c1 + 1 and row c + 1 of A (so row c + 1 corresponds to a vertex in X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfp_3T/content/2301.01623v1.pdf'}
+page_content=' Then  M = NN ⊤ − I =  \uf8ee  \uf8ef\uf8f0  c1 + c2 + x − 2  c1 − c2  c1 + x′  1 − 1  c1 − c2  c1 + c2  c1  c1 + x′  1 − 1  c1  c1 + x1 + z1 − 1  \uf8f9  \uf8fa\uf8fb ,  10  with x′  1, x1 and z1 as before (so |x′  1| ≤ x1 ≤ x − 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfp_3T/content/2301.01623v1.pdf'}
+page_content=' The expression for det(M) is a bit messy  now, so to deal with det(M) = 0 in another way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfp_3T/content/2301.01623v1.pdf'}
+page_content=' We write M = (c1 −2)J +(c2 −2)K +R, where  K =  \uf8ee  \uf8f0  1  −1  0  −1  1  0  0  0  0  \uf8f9  \uf8fb , and R =  \uf8ee  \uf8f0  x + 2  0  x′  1 + 1  0  2  2  x′  1 + 1  2  x1 + z1 + 1  \uf8f9  \uf8fb .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfp_3T/content/2301.01623v1.pdf'}
+page_content='  We easily have that (c1 − 2)J and (c2 − 2)K are positive semi-definite (psd) matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfp_3T/content/2301.01623v1.pdf'}
+page_content=' Thus if  R is positive definite (pd), then M is pd and therefore nonsingular.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfp_3T/content/2301.01623v1.pdf'}
+page_content=' Moreover, if R is psd, then  M is psd and the kernel of M is the intersection of the kernels of (c1 − 2)J, (c2 − 2)K and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfp_3T/content/2301.01623v1.pdf'}
+page_content='  So if this intersection is {0} then M is also nonsingular.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfp_3T/content/2301.01623v1.pdf'}
+page_content=' The diagonal entries of R are positive,  and so are the 2 × 2 principal minors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfp_3T/content/2301.01623v1.pdf'}
+page_content=' This implies that R is pd whenever det(R) > 0 and R is  psd when det(R) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfp_3T/content/2301.01623v1.pdf'}
+page_content=' Using this, and det(R) = 2((x + 2)(x1 + z1 − 1) − (x′  1 + 1)2) we get the  following restrictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfp_3T/content/2301.01623v1.pdf'}
+page_content='  Proposition 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfp_3T/content/2301.01623v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfp_3T/content/2301.01623v1.pdf'}
+page_content=' One of the following holds:  (i) x1 = x′  1, z1 = 0,  (ii) x1 = x′  1 = 0, z1 = 1,  (iii) x1 = −x′  1 = 1, z1 = 0, c2 = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfp_3T/content/2301.01623v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfp_3T/content/2301.01623v1.pdf'}
+page_content=' From Proposition 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfp_3T/content/2301.01623v1.pdf'}
+page_content='2(iii, iv) it follows that x1 = −x′  1 = 1, or x1 = x′  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfp_3T/content/2301.01623v1.pdf'}
+page_content=' First assume  x1 = x′  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfp_3T/content/2301.01623v1.pdf'}
+page_content=' Using x ≥ x1+1 we get det(R) ≥ 2(x1z1+3z1−4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfp_3T/content/2301.01623v1.pdf'}
+page_content=' This implies z1 = 0, (x1, z1) = (0, 1),  or (1, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfp_3T/content/2301.01623v1.pdf'}
+page_content=' The first option is case (i), and the second one is case (ii).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfp_3T/content/2301.01623v1.pdf'}
+page_content=' If x1 = z1 = 1 we find  R =  \uf8ee  \uf8f0  x + 2  0  2  0  2  2  2  2  3  \uf8f9  \uf8fb .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfp_3T/content/2301.01623v1.pdf'}
+page_content='  We have x ≥ x1 + 1 = 2 and therefore R is pd if x > 2 and psd if x = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfp_3T/content/2301.01623v1.pdf'}
+page_content=' If x = 2 the kernel of  R is spanned by [1, 2, −2]⊤, which is not in the kernel of J or K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfp_3T/content/2301.01623v1.pdf'}
+page_content=' Therefore M is singular only  if c1 = c2 = 2 and x = x1 + 1 = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfp_3T/content/2301.01623v1.pdf'}
+page_content=' Thus we have c2 = x1 + 1 and z1 > 0, which contradicts  Assumption 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfp_3T/content/2301.01623v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfp_3T/content/2301.01623v1.pdf'}
+page_content=' Next we assume x1 = −x′  1 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfp_3T/content/2301.01623v1.pdf'}
+page_content=' Then det(R) = 2(x + 2)z1, so z1 = 0 and  R =  \uf8ee  \uf8f0  x + 2  0  0  0  2  2  0  2  2  \uf8f9  \uf8fb .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfp_3T/content/2301.01623v1.pdf'}
+page_content='  Again R is pd if x > 2 and psd if x = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfp_3T/content/2301.01623v1.pdf'}
+page_content=' If x = 2 the kernel of R is spanned by [0, 1, −1]⊤, which  is not in the kernel of K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfp_3T/content/2301.01623v1.pdf'}
+page_content=' Therefore c2 = 2 and we have case (iii).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfp_3T/content/2301.01623v1.pdf'}
+page_content='  We claim that Z = ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfp_3T/content/2301.01623v1.pdf'}
+page_content=' Suppose not, and assume u ∈ Z, then u cannot be adjacent to two (or  more) vertices of X, because that would mean that A has two equals rows which is not possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfp_3T/content/2301.01623v1.pdf'}
+page_content='  But u has degree at least 2, so there is another vertex v in Z adjacent to u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfp_3T/content/2301.01623v1.pdf'}
+page_content=' But then we create a  forbidden subgraph from Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfp_3T/content/2301.01623v1.pdf'}
+page_content=' This implies that case (ii) of the above proposition does not  occur.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfp_3T/content/2301.01623v1.pdf'}
+page_content=' Note that Gσ  X has at most one isolated vertex, since otherwise A has two equal rows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfp_3T/content/2301.01623v1.pdf'}
+page_content=' Also  Gσ  X cannot contain a positive isolated edge if c2 = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfp_3T/content/2301.01623v1.pdf'}
+page_content=' Indeed, interchanging C2 with the positive  edge gives a signed graph with Format (ii) and y = 0, which has been dealt with in the previous  section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfp_3T/content/2301.01623v1.pdf'}
+page_content=' Thus Proposition 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfp_3T/content/2301.01623v1.pdf'}
+page_content='1 leads to just six possible structures for Gσ  X,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfp_3T/content/2301.01623v1.pdf'}
+page_content=' being:  11  (i) Gσ  X is a clique of order x with all edges positive,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfp_3T/content/2301.01623v1.pdf'}
+page_content='  (ii) Gσ  X is a clique of order x − 1 with all edges positive extended with an isolated vertex,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfp_3T/content/2301.01623v1.pdf'}
+page_content='  (iii) c2 > 2 and Gσ  X is the disjoint union of x/2 positive edges,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfp_3T/content/2301.01623v1.pdf'}
+page_content='  (iv) c2 > 2 and Gσ  X is the disjoint union of (x − 1)/2 positive edges and one isolated vertex,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfp_3T/content/2301.01623v1.pdf'}
+page_content='  (v) c2 = 2 and Gσ  X is the disjoint union of x/2 negative edges,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfp_3T/content/2301.01623v1.pdf'}
+page_content='  (vi) c2 = 2 and Gσ  X is the disjoint union of (x − 1)/2 negative edges and one isolated vertex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfp_3T/content/2301.01623v1.pdf'}
+page_content='  For each of these cases the matrix A of Gσ has an equitable partition with quotient matrices  Q1 =  \uf8ee  \uf8f0  c1−1  c2  x  c1  1−c2  0  c1  0  x−1  \uf8f9  \uf8fb , Q2 =  \uf8ee  \uf8ef\uf8ef\uf8f0  c1−1  c2  x−1  1  c1  1−c2  0  0  c1  0  x−2  0  c1  0  0  0  \uf8f9  \uf8fa\uf8fa\uf8fb , Q3 =  \uf8ee  \uf8f0  c1−1  c2  x  c1  1−c2  0  c1  0  1  \uf8f9  \uf8fb ,  Q4 =  \uf8ee  \uf8ef\uf8ef\uf8f0  c1−1  c2  x−1  1  c1  1−c2  0  0  c1  0  1  0  c1  0  0  0  \uf8f9  \uf8fa\uf8fa\uf8fb , Q5 =  � c1−1 x+2  c1  −1  �  , Q6 =  \uf8ee  \uf8f0  c1−1 x+1  1  c1  −1  0  c1  0  0  \uf8f9  \uf8fb ,  respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfp_3T/content/2301.01623v1.pdf'}
+page_content=' Each of these quotient matrices must have at most two eigenvalues different from ±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfp_3T/content/2301.01623v1.pdf'}
+page_content='  For Q5 this is obvious and we find matrix A1 of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfp_3T/content/2301.01623v1.pdf'}
+page_content=' For the other five quotient matrices  we checked for eigenvalues ±1 (simply by computing rank(Qi +I) and rank(Qi −I)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfp_3T/content/2301.01623v1.pdf'}
+page_content=' For Q2 and  Q4 there is at most one eigenvalue equal to ±1, and Q6 has no eigenvalue equal to ±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfp_3T/content/2301.01623v1.pdf'}
+page_content=' However  Q1 has an eigenvalue ±1 whenever (c1, x) = (3, 8), (4, 6), or (6, 5), which gives A5 of Theorem 2  when k > ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfp_3T/content/2301.01623v1.pdf'}
+page_content=' The quotient matrix Q3 has eigenvalue ±1 whenever (c2, x) = (3, 8), (4, 6), or (6, 5),  but x is even so only the first two cases remain, and we find A6 of Theorem 2 (when m > 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfp_3T/content/2301.01623v1.pdf'}
+page_content='  Conclusion 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfp_3T/content/2301.01623v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfp_3T/content/2301.01623v1.pdf'}
+page_content=' If Gσ has adjacency matrix A with Format (iii) with Y = ∅, then A is one of  the matrices represented by A1, A5 (with k > ℓ), or A6 (with m > 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfp_3T/content/2301.01623v1.pdf'}
+page_content='  10  X, Y and Z nonempty  Take v ∈ Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfp_3T/content/2301.01623v1.pdf'}
+page_content=' By Proposition 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfp_3T/content/2301.01623v1.pdf'}
+page_content='2(vi) we may assume that v is adjacent to a vertex w ∈ X and  not adjacent to Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfp_3T/content/2301.01623v1.pdf'}
+page_content=' Consider the matrix N consisting of four rows of A corresponding to a vertex  from C1, a vertex from C2, to w and v, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfp_3T/content/2301.01623v1.pdf'}
+page_content=' Then  N =  \uf8ee  \uf8ef\uf8ef\uf8ef\uf8f0  0  1⊤  c1−1  0  1⊤  c2−1  1⊤  x  0⊤  y  0⊤  z  1  1⊤  0  ǫ1⊤  0⊤  1⊤  0⊤  1  1⊤  0  0⊤  x⊤  1  y⊤  1  z⊤  1  0  0⊤  0  0⊤  x⊤  3  0⊤  z⊤  3  \uf8f9  \uf8fa\uf8fa\uf8fa\uf8fb  for (0, ±1) vectors x1, x3, y1 z1, z3, and ǫ = ±1, where ǫ = 1 in case of format (ii) and ǫ = −1 for  format (iii).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfp_3T/content/2301.01623v1.pdf'}
+page_content=' Note that we may switch such that z1 ≥ 0 and that y1 ≥ 0 by Proposition 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfp_3T/content/2301.01623v1.pdf'}
+page_content='2(v).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfp_3T/content/2301.01623v1.pdf'}
+page_content='  Define M = NN⊤ − I, then  M =  \uf8ee  \uf8ef\uf8ef\uf8ef\uf8f0  c1 + c2 + x − 2  c1 − 1 + ǫ(c2 − 1)  c1 + x′  1 − 1  x′  3  c1 − 1 + ǫ(c2 − 1)  c1 + c2 + y − 2  c1 + y1  0  c1 + x′  1 − 1  c1 + y1  c1 + x1 + y1 + z1 − 1  x1,3 + z1,3  x′  3  0  x1,3 + z1,3  x3 + z3 − 1  \uf8f9  \uf8fa\uf8fa\uf8fa\uf8fb ,  12  where x1 = x⊤  1x1, x′  1 = x⊤  11, x1,3 = x⊤  1x3, x3 = x⊤  3x3 ≥ 1, x′  3 = x⊤  31 ̸= 0, y1 = y⊤  1 y1 = y⊤  1 1,  z1 = z⊤  1z1 ≥ 1, z3 = z⊤  3z3 and z1,3 = z⊤  1z3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfp_3T/content/2301.01623v1.pdf'}
+page_content=' Since M is a principal submatrix of A2 − I, we know  rank(M) ≤ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfp_3T/content/2301.01623v1.pdf'}
+page_content=' Define Mi,j to be the matrix obtained from M by deletion of row i and column j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfp_3T/content/2301.01623v1.pdf'}
+page_content='  Then Mi,j is singular for 1 ≤ i, j ≤ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfp_3T/content/2301.01623v1.pdf'}
+page_content=' We treat the two considered formats seperately, and start  with (iii).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfp_3T/content/2301.01623v1.pdf'}
+page_content='  10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfp_3T/content/2301.01623v1.pdf'}
+page_content='1  Format (iii)  In this case ǫ = −1, c1, c2 ≥ 2 and we write M3,3 = (c1 − 2)K + (c2 − 2)L + R, where  K =  \uf8ee  \uf8f0  1  1  0  1  1  0  0  0  0  \uf8f9  \uf8fb ,  L =  \uf8ee  \uf8f0  1  −1  0  −1  1  0  0  0  0  \uf8f9  \uf8fb ,  R =  \uf8ee  \uf8f0  x + 2  0  x′  3  0  y + 2  0  x′  3  0  x3 + z3 − 1  \uf8f9  \uf8fb .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfp_3T/content/2301.01623v1.pdf'}
+page_content='  Clearly K and L are psd, so if R is pd, then M3,3 is non-singular.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfp_3T/content/2301.01623v1.pdf'}
+page_content=' Therefore R is not pd and  0 ≥ det(R) = (y +2)((x+2)(x3 +z3 −1)−(x′  3)2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfp_3T/content/2301.01623v1.pdf'}
+page_content=' This implies z3 = 0, x3 = |x′  3| = x and x3 ≤ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfp_3T/content/2301.01623v1.pdf'}
+page_content='  We know x3 > 0, and if x3 = 1 then x′  3 ̸= 0 and det(M3,3) = −(y + 2)(x′  3)2 ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfp_3T/content/2301.01623v1.pdf'}
+page_content=' Thus x3 = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfp_3T/content/2301.01623v1.pdf'}
+page_content='  Then det(R) = 0 and the kernel of R is spanned by [1, 0, −2]⊤, which is not in the kernel of K  and L, and therefore M3,3 is singular only if c1 = c2 = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfp_3T/content/2301.01623v1.pdf'}
+page_content=' Thus x = x3 = c1 = c2 = 2, z3 = 0 and  M1,1 = (y1 + 2)K +  \uf8ee  \uf8f0  y − y1  0  0  0  x1 + z1 − 1  x′  1  0  x′  1  1  \uf8f9  \uf8fb .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfp_3T/content/2301.01623v1.pdf'}
+page_content='  One easily verifies that both terms in this formula are psd (recall that |x1| ≤ x1 ≤ 1 and z1 ≥ 1),  and that the kernel of the sum is only nontrivial if y = y1 and z1 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfp_3T/content/2301.01623v1.pdf'}
+page_content=' If x1 = 0, then the two  rows of A corresponding to X are equal, which contradicts Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfp_3T/content/2301.01623v1.pdf'}
+page_content='1(ii), and x1 = 1 is ruled  out by Proposition 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfp_3T/content/2301.01623v1.pdf'}
+page_content='2(iii).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfp_3T/content/2301.01623v1.pdf'}
+page_content=' So we conclude that Z = ∅ in case of format (iii).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfp_3T/content/2301.01623v1.pdf'}
+page_content=' (But notice that  without Assumption 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfp_3T/content/2301.01623v1.pdf'}
+page_content='1 we would have found A17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfp_3T/content/2301.01623v1.pdf'}
+page_content=')  10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfp_3T/content/2301.01623v1.pdf'}
+page_content='2  Format (ii)  In this case ǫ = 1, c = c1 + c2 ≥ 3 and M3,3 can be written as: M3,3 = (c − 2)K + R, with K as  before and  R =  \uf8ee  \uf8f0  x  0  x′  3  0  y + 1  0  x′  3  0  x3 + z3 − 1  \uf8f9  \uf8fb .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfp_3T/content/2301.01623v1.pdf'}
+page_content='  With similar arguments as above we see that R, and hence M3,3 is pd if z3 > 0, if |x′  3| < x3 and  if x3 < x − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfp_3T/content/2301.01623v1.pdf'}
+page_content=' Therefore z3 = 0, |x′  3| = x3 and x3 ∈ {x, x − 1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfp_3T/content/2301.01623v1.pdf'}
+page_content='  Assume that x3 = x − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfp_3T/content/2301.01623v1.pdf'}
+page_content='  Then using Gaussian elimination we easily obtain that y(c −  2)(x − 2) − (y + c − 2) = det(M3,3) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfp_3T/content/2301.01623v1.pdf'}
+page_content='  Hence x − 2 = 1/(c − 2) + 1/y, which leads to  (c, x, y) = (3, 4, 1), or (4, 3, 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfp_3T/content/2301.01623v1.pdf'}
+page_content=' In both cases there are just a few possibilities for A, and it follows  by straightforward verification that there are no solutions (but without Assumption 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfp_3T/content/2301.01623v1.pdf'}
+page_content='1 we would  have found adjacency matrices A8 with m = 1, and A17).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfp_3T/content/2301.01623v1.pdf'}
+page_content='  Next assume that x3 = x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfp_3T/content/2301.01623v1.pdf'}
+page_content=' Then  M3,3 =  \uf8ee  \uf8f0  x + c − 2  c − 2  x  c − 2  y + c − 2  0  x  0  x − 1  \uf8f9  \uf8fb .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfp_3T/content/2301.01623v1.pdf'}
+page_content='  13  and det(M3,3) = y(c − 2)(x − 1) − x(y + c − 2) = 0, which implies c − 3 = (x + y)/(xy − x − y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfp_3T/content/2301.01623v1.pdf'}
+page_content='  The latter equations has only ten feasible solutions, being: (c, x, y) = (4, 4, 4), (4, 3, 6), (4, 6, 3),  (5, 3, 3), (5, 2, 6), (5, 6, 2), (6, 2, 4), (6, 4, 2), (8, 2, 3), (8, 3, 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfp_3T/content/2301.01623v1.pdf'}
+page_content=' Observe that if Z has a second  vertex u then it cannot be adjacent to all vertices in X (otherwise A would have two indentaical  rows), so u is adjacent to all vertices of Y and none of X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfp_3T/content/2301.01623v1.pdf'}
+page_content=' This shows that z ≤ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfp_3T/content/2301.01623v1.pdf'}
+page_content=' Using this  observation we find that for each of the above triples there are only a few possibilities for A, and  we check them case by case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfp_3T/content/2301.01623v1.pdf'}
+page_content=' The cases (5, 2, 6) and (4, 3, 6) lead to matrix A10, and the cases  (5, 3, 3) and (4, 4, 4) lead to case (vi) of Theorem 1 from [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfp_3T/content/2301.01623v1.pdf'}
+page_content=' The other six cases have no solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfp_3T/content/2301.01623v1.pdf'}
+page_content='  Thus we conclude:  Conclusion 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfp_3T/content/2301.01623v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfp_3T/content/2301.01623v1.pdf'}
+page_content=' If A has format (ii) or (iii) with X, Y and Z nonempty, then A = A10, or  A is the unsigned case (vi) of Theorem 1 from [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfp_3T/content/2301.01623v1.pdf'}
+page_content='  11  Format(iii) with X, Y nonempty and Z empty  Throughout this section Gσ has adjacency matrix A with format (iii), X, Y ̸= ∅ and Z = ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfp_3T/content/2301.01623v1.pdf'}
+page_content='  Consider the matrix N consisting of four rows of A corresponding to a vertex from C1, a vertex  from C2, a vertex v from X and a vertex from w from Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfp_3T/content/2301.01623v1.pdf'}
+page_content=' Then  N =  \uf8ee  \uf8ef\uf8ef\uf8ef\uf8f0  0  1⊤  c1−1  0  1⊤  c2−1  1⊤  x  0⊤  y  1  1⊤  0  −1⊤  0⊤  1⊤  1  1⊤  0  0⊤  x⊤  1  y⊤  1  0  0⊤  1  1⊤  x⊤  2  y⊤  2  \uf8f9  \uf8fa\uf8fa\uf8fa\uf8fb  for (0, ±1) vectors x1, y2, and (0, 1) vectors y1, x2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfp_3T/content/2301.01623v1.pdf'}
+page_content=' Define M = NN⊤ − I, then  M =  \uf8ee  \uf8ef\uf8ef\uf8ef\uf8f0  c1 + c2 + x − 2  c1 − c2  c1 + x′  1 − 1  c2 + x2  c1 − c2  c1 + c2 + y − 2  c1 + y1  1 − c2 + y′  2  c1 + x′  1 − 1  c1 + y1  c1 + x1 + y1 − 1  x1,2 + y1,2  c2 + x2  1 − c2 + y′  2  x1,2 + y1,2  c2 + x2 + y2 − 1  \uf8f9  \uf8fa\uf8fa\uf8fa\uf8fb ,  where x1 = x⊤  1x1, x′  1 = x⊤  11, x1,2 = x⊤  1x2, x2 = x⊤  2x2 = x⊤  21, y1 = y⊤  1 y1 = y⊤  1 1, y2 = y⊤  2 y2,  y′  2 = y⊤  2 1, and y1,2 = y⊤  1 y2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfp_3T/content/2301.01623v1.pdf'}
+page_content=' As before we define Mi,j to be the matrix obtained from M by  deletion of row i and column j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfp_3T/content/2301.01623v1.pdf'}
+page_content=' We have rank(M) ≤ 2, hence det(Mi,j) = 0 for 1 ≤ i, j ≤ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfp_3T/content/2301.01623v1.pdf'}
+page_content='  Proposition 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfp_3T/content/2301.01623v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfp_3T/content/2301.01623v1.pdf'}
+page_content=' Suppose y − y1 ≥ 2 for some vertex in X, then one of the following holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfp_3T/content/2301.01623v1.pdf'}
+page_content='  (i): x1 = x′  1 = 1, y1 = 0, x = y = 2 which leads to matrix A4 with m, ℓ ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfp_3T/content/2301.01623v1.pdf'}
+page_content='  (ii): x1 = 0, y1 = 1, x = y = 3 and A = A16,  (iii): x1 = y1 = 0, x2 = y2 = 0, x = y = 3 and A = A17,  (iv): x1 = y1 = 0, x2 = x − 1, y2 = 1, which leads to A8 with m ≥ 2, A18, or A19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfp_3T/content/2301.01623v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfp_3T/content/2301.01623v1.pdf'}
+page_content=' Write M4,4 = (c1 − 2)J + (c2 − 2)L + y1K + R, where  L =  \uf8ee  \uf8f0  1  −1  0  −1  1  0  0  0  0  \uf8f9  \uf8fb , K =  \uf8ee  \uf8f0  0  0  0  0  1  1  0  1  1  \uf8f9  \uf8fb ,  and R =  \uf8ee  \uf8f0  x + 2  0  x′  1 + 1  0  y − y1 + 2  2  x′  1 + 1  2  x1 + 1  \uf8f9  \uf8fb .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfp_3T/content/2301.01623v1.pdf'}
+page_content='  14  Clearly J, K and L are psd and R is pd, unless x1 = 0, or x1 = x′  1 = x − 1 = 1 and y − y1 = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfp_3T/content/2301.01623v1.pdf'}
+page_content='  In the latter case R is psd with kernel spanned by [1, 1, −2]⊤, which is in the kernel of M4,4 only  if y1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfp_3T/content/2301.01623v1.pdf'}
+page_content=' Therefore x = y = 2, and we obtain the matrices A4 with m, ℓ ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfp_3T/content/2301.01623v1.pdf'}
+page_content='  So we have that x1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfp_3T/content/2301.01623v1.pdf'}
+page_content=' If y1 ≥ 1, then we use that R + y1K is not pd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfp_3T/content/2301.01623v1.pdf'}
+page_content=' By straightforward  computation we get det(R + y1K) = (y + 2)(x + 2)(y1 + 1) − (x + 2)(y1 + 2)2 − (y + 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfp_3T/content/2301.01623v1.pdf'}
+page_content=' Using  y − y1 ≥ 2 this gives det(R + y1K) ≥ (x + 1)y1 − 4 and equality implies y − y1 = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfp_3T/content/2301.01623v1.pdf'}
+page_content=' Therefore  y1 = 0, or y1 = 1, y = 3 and x ≤ 3, or y1 = 2, y = 4 and x = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfp_3T/content/2301.01623v1.pdf'}
+page_content=' If (x, x1, y, y1) = (3, 0, 3, 1)  then R + y1K is psd and the kernel is spanned by [1, 3, −5]⊤ which is in the kernel of J nor K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfp_3T/content/2301.01623v1.pdf'}
+page_content='  Hence M4,4 is nonsingular, unless c1 = c2 = 2 in which case we obtain A16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfp_3T/content/2301.01623v1.pdf'}
+page_content=' It is not difficult to  rule out the remaining cases by use of Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfp_3T/content/2301.01623v1.pdf'}
+page_content='1(ii), Proposition 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfp_3T/content/2301.01623v1.pdf'}
+page_content='2(ii), and the last forbidden  subgraph of Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfp_3T/content/2301.01623v1.pdf'}
+page_content='  What remains is the case x1 = y1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfp_3T/content/2301.01623v1.pdf'}
+page_content=' Using Gaussian elimination in M2,2 we find that in this  case det(M2,2) = (y2 − 1)(c2 + x − 1) + (x − x2 − 1)(c2 + x2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfp_3T/content/2301.01623v1.pdf'}
+page_content=' Note that y1 = 0 implies that  x2 ≤ x − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfp_3T/content/2301.01623v1.pdf'}
+page_content=' Therefore M2,2 is singular only if (a): y2 = 0 and c2 + x − 1 = (x − x2 − 1)(c2 + x2),  or (b): y2 = 1, x = x2 + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfp_3T/content/2301.01623v1.pdf'}
+page_content=' In case (a) we get c2 = 2, x = 3, x2 = y2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfp_3T/content/2301.01623v1.pdf'}
+page_content=' Since no two rows of A  are equal we have that there is exactly one vertex v ∈ X for which x1 = y1 = 0, and at most one  vertex w ∈ Y for which x2 = y2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfp_3T/content/2301.01623v1.pdf'}
+page_content=' Using this observation we find that case (a) leads to A17, and  that otherwise (b) holds for all vertices of Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfp_3T/content/2301.01623v1.pdf'}
+page_content=' Moreover, det(M1,2) = (1 − c1)(c2 + x − 1)(y′  2 + 1),  implies y′  2 = −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfp_3T/content/2301.01623v1.pdf'}
+page_content=' From Proposition 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfp_3T/content/2301.01623v1.pdf'}
+page_content='2(iii) we deduce that Gσ  X is equal to K1, or K1 + Kx−1 (all  signs are positive), or consists of one isolated vertex and (x − 1)/2 isolated signed edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfp_3T/content/2301.01623v1.pdf'}
+page_content='  If Gσ  X = K1 then we have an equitable partition of A with quotient matrix  Q =  \uf8ee  \uf8ef\uf8ef\uf8f0  c1 − 1  c2  1  0  c1  1 − c2  0  y  c1  0  0  0  0  c2  0  −1  \uf8f9  \uf8fa\uf8fa\uf8fb .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfp_3T/content/2301.01623v1.pdf'}
+page_content='  By Gaussian elimination in Q + I and Q − I we find that Q has two eigenvalues equal to ±1 only  if c1 = 2 and y = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfp_3T/content/2301.01623v1.pdf'}
+page_content=' This gives A8 with m ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfp_3T/content/2301.01623v1.pdf'}
+page_content='  If Gσ  X = K1 + Kx−1 then we have an equitable partition of A with quotient matrix  Q =  \uf8ee  \uf8ef\uf8ef\uf8ef\uf8ef\uf8f0  c1 − 1  c2  1  x − 1  0  c1  1 − c2  0  0  y  c1  0  0  0  0  c1  0  0  x − 2  y  0  c2  0  x − 1  −1  \uf8f9  \uf8fa\uf8fa\uf8fa\uf8fa\uf8fb  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfp_3T/content/2301.01623v1.pdf'}
+page_content='  By Gaussian elimination in Q+I and Q−I we find that Q has at least three eigenvalues unequal  to ±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfp_3T/content/2301.01623v1.pdf'}
+page_content='  If Gσ  X has an negative isolated edge e then C2 = 2 by Theorem 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfp_3T/content/2301.01623v1.pdf'}
+page_content='2(iv, v).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfp_3T/content/2301.01623v1.pdf'}
+page_content='  If v is a vertex  of e, then c2 = 2 x1 = −x′  1 = 1, y1 = y, x2 = x − 1, y2 = −y′  2 = 1 and det(M2,2) =  (x + 1)(c1y − y − c1) − 4(c1 − 1) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfp_3T/content/2301.01623v1.pdf'}
+page_content=' Using that x > 1 and odd, and y ≥ 2 and even, we find  just two solutions: (c1, x, y) = (3, 5, 2) and (4, 7, 2) and obtain A18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfp_3T/content/2301.01623v1.pdf'}
+page_content='  Assume Gσ  X has a positive isolated edge e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfp_3T/content/2301.01623v1.pdf'}
+page_content=' Take v ∈ e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfp_3T/content/2301.01623v1.pdf'}
+page_content=' Then x1 = x′  1 = 1, y1 = y, x2 = x − 1,  y2 = −y′  2 = 1 and det(M2,2) = (c2 + x − 1)(c1y − y − c1) = 0, which implies c1 = y = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfp_3T/content/2301.01623v1.pdf'}
+page_content=' Since  c1 ̸= 3 or 4, Gσ  X has no negative isolated edge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfp_3T/content/2301.01623v1.pdf'}
+page_content=' This means that A has an equitable partition  15  with quotient matrix  Q =  \uf8ee  \uf8ef\uf8ef\uf8ef\uf8ef\uf8f0  1  c2  1  x − 1  0  2  1 − c2  0  0  2  2  0  0  0  0  2  0  0  1  2  0  c2  0  x − 1  −1  \uf8f9  \uf8fa\uf8fa\uf8fa\uf8fa\uf8fb  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfp_3T/content/2301.01623v1.pdf'}
+page_content='  Using Gaussian elimination in Q + I and Q − I we find that Q has three eigenvalues equal to ±1  if and only if (c2, x) = (3, 7) or (4, 5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfp_3T/content/2301.01623v1.pdf'}
+page_content=' Thus we obtain A19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfp_3T/content/2301.01623v1.pdf'}
+page_content='  Suppose the matrix A does not belong to one of the cases of the above proposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfp_3T/content/2301.01623v1.pdf'}
+page_content=' Then  each vertex from X is adjacent to y or y − 1 vertices of Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfp_3T/content/2301.01623v1.pdf'}
+page_content=' And, by interchanging C1 with C2  and X with Y it also follows that each vertex from Y is adjacent to x or x − 1 vertices of X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfp_3T/content/2301.01623v1.pdf'}
+page_content=' So  it follows that the adjacency matrix of Gσ  X∪Y takes the form:  AX∪Y =  � AX  B  B⊤  AY  �  with B =  �  J  J  J − Im  J  �  and 0 ≤ m ≤ min{x, y}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfp_3T/content/2301.01623v1.pdf'}
+page_content='  Proposition 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfp_3T/content/2301.01623v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfp_3T/content/2301.01623v1.pdf'}
+page_content=' With B as above, one of the following holds:  (i) m = x = y, B = J − I which leads to A11,  (ii) m = 0, B = J which leads to A2, A12, A13, A14, or A15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfp_3T/content/2301.01623v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfp_3T/content/2301.01623v1.pdf'}
+page_content=' Let Xm and Ym be the subsets of X and Y corresponding to the submatrix J − Im of B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfp_3T/content/2301.01623v1.pdf'}
+page_content='  From Proposition 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfp_3T/content/2301.01623v1.pdf'}
+page_content='2(ii) it follows that Xm and Ym consists of isolated vertices in Γσ  X and Γσ  Y ,  respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfp_3T/content/2301.01623v1.pdf'}
+page_content=' So if m = x = y then Γσ  X and Γσ  Y have no edges and A has the block structure of  A12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfp_3T/content/2301.01623v1.pdf'}
+page_content=' The values of m, ℓ and k follow from the spectrum of the quotient matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfp_3T/content/2301.01623v1.pdf'}
+page_content='  If 0 < m < x then the subgraph Γσ  X\\Xm of Γσ  X induced by X \\ Xm cannot have an isolated  edge or vertex (because of Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfp_3T/content/2301.01623v1.pdf'}
+page_content='1(ii)), and therefore Γσ  X\\Xm = Kx−m (because of Proposi-  tion 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfp_3T/content/2301.01623v1.pdf'}
+page_content='2(iv)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfp_3T/content/2301.01623v1.pdf'}
+page_content=' Similarly, if 0 < m < y then Γσ  Y \\Ym = K−  y−m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfp_3T/content/2301.01623v1.pdf'}
+page_content=' Thus, if 0 < m < min{x, y} then A  has an equitable partition with quotient matrix  Q =  \uf8ee  \uf8ef\uf8ef\uf8ef\uf8ef\uf8ef\uf8ef\uf8f0  c1 − 1  c2  x − m  m  0  0  c1  1 − c2  0  0  m  y − m  c1  0  x − m − 1  0  m  y − m  c1  0  0  0  m − 1  y − m  0  c2  x − m  m − 1  0  0  0  c2  x − m  m  0  m − y + 1  \uf8f9  \uf8fa\uf8fa\uf8fa\uf8fa\uf8fa\uf8fa\uf8fb  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfp_3T/content/2301.01623v1.pdf'}
+page_content='  However Q does not have four eigenvalues equal to ±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfp_3T/content/2301.01623v1.pdf'}
+page_content=' A quick way to verify this is by Gaussian  elimination in Q + I with row 2 and column 6 deleted and in Q − I with row 1 and column 3  deleted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfp_3T/content/2301.01623v1.pdf'}
+page_content=' Then we find that Q+ I and Q− I both have rank at least 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfp_3T/content/2301.01623v1.pdf'}
+page_content=' A similar argument works  if m = x < y, or m = y < x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfp_3T/content/2301.01623v1.pdf'}
+page_content=' Then we have an equitable partition with a 5 × 5 quotient matrix  with at most two eigenvalues equal to ±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfp_3T/content/2301.01623v1.pdf'}
+page_content=' Thus we can conclude that m = 0 and B = J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfp_3T/content/2301.01623v1.pdf'}
+page_content=' Now  Γσ  X may have an isolated vertex, but Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfp_3T/content/2301.01623v1.pdf'}
+page_content='1 still implies that Γσ  X cannot have two isolated  vertices, or an isolated vertex and an isolated edge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfp_3T/content/2301.01623v1.pdf'}
+page_content=' Therefore by use of Proposition 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfp_3T/content/2301.01623v1.pdf'}
+page_content='2(iii) there  are just a few possibilities for Γσ  X, being: (a): Γσ  X = Kx (x ≥ 3), (b): Γσ  X = Kx−1 + K1 (x ≥ 4),  (c): Γσ  X consists of x/2 isolated signed edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfp_3T/content/2301.01623v1.pdf'}
+page_content=' For Γσ  Y we obtain the same list with y, K−  y and  K−  y−1 instead of x, Kx and Kx−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfp_3T/content/2301.01623v1.pdf'}
+page_content='  16  If Γσ  X = Kx and Γσ  Y = K−  y , then A has an equitable partition of A11, and the quotient matrix Q  has two eigenvalues equal to ±1 only in the given cases (taking Assumption 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfp_3T/content/2301.01623v1.pdf'}
+page_content='1 into account).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfp_3T/content/2301.01623v1.pdf'}
+page_content=' If  Γσ  X = Kx and Γσ  Y = K−  y−1 + K1 (or Γσ  X = Kx + K1 and Γσ  Y = K−  y ), then we have an equitable  partition of A with 5 × 5 quotient matrix Q, and by Gaussian elemination it follows that Q + I  and Q − I both have rank at least 4, and therefore Q (and A) have at least three eigenvalues  unequal to ±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfp_3T/content/2301.01623v1.pdf'}
+page_content=' If Γσ  X = Kx−1 + K1 and Γσ  Y = K−  y−1 + K1, then the equitable partition has a  6 × 6 quotient matrix Q, and both Q + I and Q − I both have rank at least 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfp_3T/content/2301.01623v1.pdf'}
+page_content=' So also in this  case we find no solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfp_3T/content/2301.01623v1.pdf'}
+page_content=' For the remainder of the proof we may assume that Γσ  X consists of x/2  disjoint signed edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfp_3T/content/2301.01623v1.pdf'}
+page_content=' We claim that all these edges have the same sign.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfp_3T/content/2301.01623v1.pdf'}
+page_content=' Indeed, suppose Γσ  X  has a negative and a positive edge, then we consider the matrix N consisting of three rows of A  corresponding to a vertex of the negative edge, a vertex of the positive edge and a vertex from  C1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfp_3T/content/2301.01623v1.pdf'}
+page_content=' Then  NN⊤ − I =  \uf8ee  \uf8f0  c1 + y  c1 + y  c1  c1 + y  c1 + y  c1 − 2  c1  c1 − 2  c1 + c2 + x − 2  \uf8f9  \uf8fb  is nonsingular, which proves the claim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfp_3T/content/2301.01623v1.pdf'}
+page_content='  Suppose that all edges of Γσ  X have a negative sign.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfp_3T/content/2301.01623v1.pdf'}
+page_content=' Then Proposition 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfp_3T/content/2301.01623v1.pdf'}
+page_content='2(iv) implies that c2 = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfp_3T/content/2301.01623v1.pdf'}
+page_content='  If Γσ  Y = K−  y then A has an equitable partition with three parts and a quotient matrix with no  eigenvalue equal to ±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfp_3T/content/2301.01623v1.pdf'}
+page_content=' If Γσ  Y = K−  y−1 + K1 then A has an equitable partition with four parts  and quotient matrix  Q =  \uf8ee  \uf8ef\uf8ef\uf8f0  c1 − 1  x + 2  0  0  c1  −1  y − 1  1  0  x + 2  2 − y  0  0  x + 2  0  0  \uf8f9  \uf8fa\uf8fa\uf8fb .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfp_3T/content/2301.01623v1.pdf'}
+page_content='  We easily obtain that Q+I is singular only if y = 5, and that Q−I is singular only if (c1, y) = (5, 4)  or (6, 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfp_3T/content/2301.01623v1.pdf'}
+page_content=' Thus we find A13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfp_3T/content/2301.01623v1.pdf'}
+page_content=' If Γσ  Y consists of y/2 positive edges, then also c1 = 2, and we obtain  A2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfp_3T/content/2301.01623v1.pdf'}
+page_content=' If Γσ  Y consists of y/2 negative edges, then we find an equitable partition with quotient  Q =  \uf8ee  \uf8f0  c1 − 1  x + 2  0  c1  −1  y  0  x + 2  −1  \uf8f9  \uf8fb .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfp_3T/content/2301.01623v1.pdf'}
+page_content='  Now Q + I is nonsingular and Q − I is singular only if (c1, x, y) = (6, 2, 4), or (5, 4, 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfp_3T/content/2301.01623v1.pdf'}
+page_content=' Thus we  find A14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfp_3T/content/2301.01623v1.pdf'}
+page_content='  Next suppose that Γσ  X consists of positive edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfp_3T/content/2301.01623v1.pdf'}
+page_content=' If Γσ  Y = K−  y , then A has an equitable partition  for which the quotient matrix has at least three eigenvalues unequal to ±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfp_3T/content/2301.01623v1.pdf'}
+page_content=' If Γσ  Y = K−  y−1 + K1,  then A has an equitable partition with five parts and quotient matrix Q for which it is easily seen  that both Q + I and Q − I have rank at least 4, so we find no solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfp_3T/content/2301.01623v1.pdf'}
+page_content=' If Γσ  Y has y/2 disjoint  positive edges, then we obtain A14 after by switching taking the negative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfp_3T/content/2301.01623v1.pdf'}
+page_content=' If Γσ  Y has y/2 disjoint  negative edges, then A has an equitable partitions with quotient matrix  Q =  \uf8ee  \uf8ef\uf8ef\uf8f0  c1 − 1  c2  x  0  c1  1 − c2  0  y  c1  0  1  y  0  c2  x  −1  \uf8f9  \uf8fa\uf8fa\uf8fb .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfp_3T/content/2301.01623v1.pdf'}
+page_content='  17  Using that x and y are both even we find that both Q − I and Q + I have rank 3 only if  (c1, c2, x, y) = (3, 3, 6, 6), (4, 3, 6, 4), or (4, 4, 4, 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfp_3T/content/2301.01623v1.pdf'}
+page_content=' Thus we find A15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfp_3T/content/2301.01623v1.pdf'}
+page_content='  Conclusion 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfp_3T/content/2301.01623v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfp_3T/content/2301.01623v1.pdf'}
+page_content=' If Gσ has adjacency matrix A with Format (iii) with Z = ∅ and X, Y ̸= ∅  then A equals A2, A4, A8, A12, A13, A14, A15, A16, A17, A18, or A19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfp_3T/content/2301.01623v1.pdf'}
+page_content='  12  Format(ii) with X, Y nonempty and Z empty  In this section Gσ has adjacency matrix A with format (ii), X, Y ̸= ∅ and Z = ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfp_3T/content/2301.01623v1.pdf'}
+page_content=' We use the  notation of the previous section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfp_3T/content/2301.01623v1.pdf'}
+page_content=' Again the matrix N consists of four rows of A corresponding to  a vertex from C1, a vertex from C2, a vertex v from X and a vertex from w from Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfp_3T/content/2301.01623v1.pdf'}
+page_content=' Then  M = NN⊤ − I =  \uf8ee  \uf8ef\uf8ef\uf8ef\uf8f0  c1 + c2 + x − 2  c1 + c2 − 2  c1 + x′  1 − 1  c2 + x2  c1 + c2 − 2  c1 + c2 + y − 2  c1 + y1  c2 − 1 + y′  2  c1 + x′  1 − 1  c1 + y1  c1 + x1 + y1 − 1  x1,2 + y1,2  c2 + x2  c2 − 1 + y′  2  x1,2 + y1,2  c2 + x2 + y2 − 1  \uf8f9  \uf8fa\uf8fa\uf8fa\uf8fb .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfp_3T/content/2301.01623v1.pdf'}
+page_content='  First we consider the case that c1, c2 ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfp_3T/content/2301.01623v1.pdf'}
+page_content=' Then we can imitate the steps of the previous section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfp_3T/content/2301.01623v1.pdf'}
+page_content='  12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfp_3T/content/2301.01623v1.pdf'}
+page_content='1  c1 ≥ 2 and c2 ≥ 2  Proposition 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfp_3T/content/2301.01623v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfp_3T/content/2301.01623v1.pdf'}
+page_content=' Suppose y − y1 ≥ 2 for some vertex in X, then A = A9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfp_3T/content/2301.01623v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfp_3T/content/2301.01623v1.pdf'}
+page_content=' From Proposition 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfp_3T/content/2301.01623v1.pdf'}
+page_content='2(iii, iv) it follows that x1 = −x′  1, or x1 = x′  1 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfp_3T/content/2301.01623v1.pdf'}
+page_content=' We consider M4,4  and write M4,4 = (c1 − 2)J + (c2 − 2)L + y1K + R, where  L =  \uf8ee  \uf8f0  1  1  0  1  1  0  0  0  0  \uf8f9  \uf8fb , K =  \uf8ee  \uf8f0  0  0  0  0  1  1  0  1  1  \uf8f9  \uf8fb ,  and R =  \uf8ee  \uf8f0  x + 2  2  x′  1 + 1  2  y − y1 + 2  2  x′  1 + 1  2  x1 + 1  \uf8f9  \uf8fb .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfp_3T/content/2301.01623v1.pdf'}
+page_content='  Then J, K and L are psd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfp_3T/content/2301.01623v1.pdf'}
+page_content=' Moreover, if y − y1 ≥ 3, or y − y1 = 2 and x1 ̸= 0 then R is pd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfp_3T/content/2301.01623v1.pdf'}
+page_content=' Using  y − y1 ≥ 2 we obtain x1 = x′  1 = 0 and y − y1 = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfp_3T/content/2301.01623v1.pdf'}
+page_content=' In this case R is psd with kernel spanned  by [0, 1, −2]⊤.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfp_3T/content/2301.01623v1.pdf'}
+page_content=' So only if c1 = c2 = 2 and y1 = 0 the kernel of M4,4 is nontrivial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfp_3T/content/2301.01623v1.pdf'}
+page_content=' This implies  that M4,4 is singular only if (c1, c2, y, x1, y1) = (2, 2, 2, 0, 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfp_3T/content/2301.01623v1.pdf'}
+page_content=' If X contains another vertex with  x1 = y1 = 0 then A has two equal rows, so if x ≥ 2 then there must be another solution for  det(M4,4) = 0 with c1 = c2 = y = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfp_3T/content/2301.01623v1.pdf'}
+page_content=' Such a solution exists only if x1 = x′  1 = 1 which leads to  A9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfp_3T/content/2301.01623v1.pdf'}
+page_content='  As in the previous section it follows from Proposition 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfp_3T/content/2301.01623v1.pdf'}
+page_content='1 that if A ̸= A9 then the adjacency  matrix of Gσ  X∪Y takes the form:  AX∪Y =  � AX  B  B⊤  AY  �  with B =  �  J  J  J − Im  J  �  and 0 ≤ m ≤ min{x, y},  and with the same arguments, but (slightly) different quotient matrices we obtain the following  result:  Proposition 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfp_3T/content/2301.01623v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfp_3T/content/2301.01623v1.pdf'}
+page_content=' The matrix B above has m = 0 and A equals A8 with m = 1, A12 (only  the case k = ℓ because of Assumption 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfp_3T/content/2301.01623v1.pdf'}
+page_content='1), or A is an unsigned matrix given in case (iii) of  Theorem 1 from [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfp_3T/content/2301.01623v1.pdf'}
+page_content='  18  12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfp_3T/content/2301.01623v1.pdf'}
+page_content='2  c1 = 1 and c2 ≥ 2  Since c1 = 1, Assumption 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfp_3T/content/2301.01623v1.pdf'}
+page_content='1 yields that Y is a coclique, so y2 = y′  2 = y1,2 = 0, and M becomes:  M =  \uf8ee  \uf8ef\uf8ef\uf8ef\uf8f0  c2 + x − 1  c2 − 1  x′  1  c2 + x2  c2 − 1  c2 + y − 1  y1 + 1  c2 − 1  x′  1  y1 + 1  x1 + y1 + z1  x1,2  c2 + x2  c2 − 1  x1,2  c2 + x2 − 1  \uf8f9  \uf8fa\uf8fa\uf8fa\uf8fb .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfp_3T/content/2301.01623v1.pdf'}
+page_content='  Suppose vertex v of Gσ  X is isolated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfp_3T/content/2301.01623v1.pdf'}
+page_content=' Then x1 = x′  1 = x1,2 = 0 and det(M3,3) = −((c2 + x2)y +  (x2 + 1)(c2 − 1)) ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfp_3T/content/2301.01623v1.pdf'}
+page_content='  Hence Gσ  X has no isolated vertex and from Proposition 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfp_3T/content/2301.01623v1.pdf'}
+page_content='2(i, iii, iv)  it follows that there are just three possibilities: (i) Gσ  X is the complete graph with all edges  negative, (ii) Gσ  X is the disjoint union of two or more negative edges, and (iii) c2 = 2 and  Gσ  X is the disjoint union of two or more signed edges (not all negative).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfp_3T/content/2301.01623v1.pdf'}
+page_content=' Assume we have case  (i), then x1 = −x′  1 = x − 1 > 0 and Proposition 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfp_3T/content/2301.01623v1.pdf'}
+page_content='2(ii) implies that every vertex in Y is  adjacent to all or no vertices of X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfp_3T/content/2301.01623v1.pdf'}
+page_content=' If w ∈ Y is adjacent to all of X, then x2 = x and find  det(M3,4) = (x − 1)(c2 + y − 1) + (c2 − 1)(y1 + 1) > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfp_3T/content/2301.01623v1.pdf'}
+page_content=' If every vertex of Y is nonadjacent  to every vertex of X, then y1 = x2 = 0 and we find det(M3,4) = (x − 1)yc2 > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfp_3T/content/2301.01623v1.pdf'}
+page_content=' So we can  conclude that case (i) does not occur.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfp_3T/content/2301.01623v1.pdf'}
+page_content=' For the remaining two cases Gσ  X is a disjoint union of  edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfp_3T/content/2301.01623v1.pdf'}
+page_content=' Consider an edge e of Gσ  X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfp_3T/content/2301.01623v1.pdf'}
+page_content=' By Proposition 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfp_3T/content/2301.01623v1.pdf'}
+page_content='2(ii) every vertex in Gσ  Y is adjacent to  none or both vertices of e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfp_3T/content/2301.01623v1.pdf'}
+page_content=' Therefore x2 and x − x2 are even.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfp_3T/content/2301.01623v1.pdf'}
+page_content=' Next consider case (ii).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfp_3T/content/2301.01623v1.pdf'}
+page_content=' Suppose  {v, w} is an edge between X and Y , then x′  1 = x1,2 = −x1 = −1 and we find det(M3,4) =  (x − x2 − 1)(−(c2 + y − 1) − (c2 − 1)(y1 + 1)) ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfp_3T/content/2301.01623v1.pdf'}
+page_content=' Therefore there are no edges between X and  Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfp_3T/content/2301.01623v1.pdf'}
+page_content=' So x2 = y1 = 0, x1 = −x′  1 = 1 and det(M3,1) = −y(c2 − 1) − (c2 − 1), which implies y = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfp_3T/content/2301.01623v1.pdf'}
+page_content='  Now det(M1,4) = 0 gives x = 4 and c2 = 2 and thus we find that A equals A8 with m = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfp_3T/content/2301.01623v1.pdf'}
+page_content=' In  case (iii) we have c2 = 2 and consider M3,3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfp_3T/content/2301.01623v1.pdf'}
+page_content=' It is straigtforward that det(M3,3) ̸= 0, except when  x = 4, y = 1 and x2 = 0 which again leads to A8 with m = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfp_3T/content/2301.01623v1.pdf'}
+page_content=' Thus we conclude that the last  case gives no new examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfp_3T/content/2301.01623v1.pdf'}
+page_content='  Conclusion 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfp_3T/content/2301.01623v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfp_3T/content/2301.01623v1.pdf'}
+page_content=' If A has Format (ii) with Z = ∅ and X, Y ̸= ∅ then A is equal to A8 with  m = 1, A9, A12 with k = ℓ, or A is an unsigned matrix given in Theorem 1(iii) of [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfp_3T/content/2301.01623v1.pdf'}
+page_content='  13  Recapitulation  By combining the conclusions of sections 6, and 8 to 12 we have completed the proof of Theo-  rem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfp_3T/content/2301.01623v1.pdf'}
+page_content=' Note that the proof doesn’t exclude the unsigned examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfp_3T/content/2301.01623v1.pdf'}
+page_content=' Thus we rediscovered the  unsigned characterization from [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfp_3T/content/2301.01623v1.pdf'}
+page_content=' As in the unsigned case, with the present characterization  one can examine which signed graphs in G are determined, up to switching, by the spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfp_3T/content/2301.01623v1.pdf'}
+page_content='  This however, is more involved as in the unsigned case and will be the subject of further research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfp_3T/content/2301.01623v1.pdf'}
+page_content='  Acknowledgement  Part of the research was done while the first author visited the Haci Bekta¸s Veli University of  Nev¸sehir on a grant from T¨ubitac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfp_3T/content/2301.01623v1.pdf'}
+page_content='  19  References  [1] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfp_3T/content/2301.01623v1.pdf'}
+page_content=' Akbari, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfp_3T/content/2301.01623v1.pdf'}
+page_content='H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfp_3T/content/2301.01623v1.pdf'}
+page_content=' Haemers, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfp_3T/content/2301.01623v1.pdf'}
+page_content='R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfp_3T/content/2301.01623v1.pdf'}
+page_content=' Maimani and L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfp_3T/content/2301.01623v1.pdf'}
+page_content=' Parsaei Majd, Signed graphs cospectral  with the path, Linear Algebra and its Applications 553 (2018) 104-116.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNAzT4oBgHgl3EQfp_3T/content/2301.01623v1.pdf'}
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+Physics–guided neural networks for feedforward control
+with input–to–state stability guarantees∗
+Max Bolderman†,1, Hans Butler1,2, Sjirk Koekebakker3, Eelco van Horssen4, Ramidin Kamidi2,
+Theresa Spaan–Burke5, Nard Strijbosch5, and Mircea Lazar1
+Abstract— Currently, there is an increasing interest in merg-
+ing physics–based methods and artificial intelligence to push
+performance of feedforward controllers for high–precision
+mechatronics beyond what is achievable with linear feedforward
+control. In this paper, we develop a systematic design procedure
+for feedforward control using physics–guided neural networks
+(PGNNs) that can handle nonlinear and unknown dynamics.
+PGNNs effectively merge physics–based and NN–based models,
+and thereby result in nonlinear feedforward controllers with
+higher performance and the same reliability as classical, linear
+feedforward controllers. In particular, conditions are presented
+to validate (after training) and impose (before training) input–
+to–state stability (ISS) of PGNN feedforward controllers. The
+developed PGNN feedforward control framework is validated
+on a real–life, high–precision industrial linear motor used in
+lithography machines, where it reaches a factor 2 improvement
+with respect to conventional mass–friction feedforward.
+Index Terms— Feedforward control, neural networks, non-
+linear system identification, high–precision mechatronics, linear
+motors.
+I. INTRODUCTION
+The field of high–precision mechatronics requires con-
+tinuously innovating control methods to facilitate the ever–
+increasing demands on both throughput as well as accuracy.
+For example, wafer scanners in lithography machines used
+for semiconductor manufacturing [1] require sub–nanometer
+position accuracy at velocities and accelerations exceeding
+1
+m
+s
+and 30
+m
+s2 , respectively, see [2, Chapter 9]. On a
+similar note, the ability to increase the throughput while
+decreasing the position error can allow for the use of
+components that are manufactured with larger tolerances and
+thereby improving the cost effectiveness when no further
+accuracy and throughput improvements are required. This
+is an objective in, for example, the manufacturing industry
+of printing applications using relatively low–cost stepping
+motors [3].
+Feedforward control is a dominant actor in achieving this
+high position accuracy, while feedback control predomi-
+nantly concerns closed–loop stability and disturbance rejec-
+*This work is supported by the NWO research project PGN Mechatronics,
+project number 17973.
+†Corresponding author: m.bolderman@tue.nl
+1Eindhoven University of Technology, Groene Loper 19, Eindhoven,
+5612 AP, The Netherlands.
+2ASML, Veldhoven, The Netherlands.
+3Canon Production Printing, St. Urbanusweg 43, Venlo, 5900 AE, The
+Netherlands.
+4Philips Engineering Solutions, High Tech Campus 34, Eindhoven, 5656
+AE, The Netherlands.
+5IBS Precision Engineering, ESP 201, Eindhoven, 5633 AD, The Nether-
+lands.
+tion [4]. Robustness against varying references is achieved
+by generating the feedforward input by passing the refer-
+ence through a model of the inverse system dynamics, see,
+e.g., [5]–[7]. As a result, performance of these so–called
+inversion–based feedforward controllers is limited by the
+accuracy of the model of the inverse dynamics [8].
+Conventional feedforward control methods employ para-
+metric models of the inverse system derived from physical
+knowledge, which are often linear [5], [7], but can also be
+linear–in–the–parameters (LIP) [6], [9] or nonlinear [10],
+[11]. Advantages of using physics–based models are the
+interpretable nature of the models with robust performance
+due to the reliance on physical knowledge, as well as a
+systematic stability analysis based on linear system theory.
+Moreover, methods exist for constructing stable inverses
+of linear nonminimum phase systems, i.e., systems that
+have an unstable inverse, using approximate solutions or
+non–causal filtering [12], [13]. Additionally, the training
+of linear and LIP models, i.e., offline optimization to find
+the model parameters for which the model fits the data, is
+typically a convex optimization problem. On the downside
+however, using models derived from physical knowledge
+yields structural model errors due to the intrinsic simplifi-
+cations when deriving such models, e.g., when neglecting
+parasitic effects present in real–life applications such as
+electromagnetic distortions and complex friction forces [14].
+Therefore, more general model structures that can learn
+nonlinear and unknown parasitic effects are needed for
+improving performance of feedforward controllers.
+Black–box neural network (NN) models are a good
+candidate because of their universal approximation capa-
+bilities [15], and have been already used in feedforward
+control in, e.g., [16], [17]. The efficient backpropagation
+algorithm [18] enables training using large amounts of data,
+which is required for high–precision mechatronic systems
+due to their high sampling rates (> O(103) Hz). Currently,
+however, the use of NN–based feedforward controllers in
+practical applications remains limited, see, e.g., [19], [20]
+for experimental results. This is explained by the fact that
+using NNs comes with more difficult design challenges
+compared to linear physics–based models. For example,
+NNs require a tedious hyperparameter tuning [21]; the NN
+training can get stuck in local minima; and NNs suffer from
+poor extrapolation outside the training data [22]. Moreover,
+validating stability of NN–based feedforward controllers is
+non–trivial [23], [24], and the linear stable inversion tools
+do not apply for the nonlinear NN models.
+arXiv:2301.08568v1  [eess.SY]  20 Jan 2023
+
+The above analysis suggests that a systematic method for
+effectively merging physics–based models and NNs would
+enable feedforward controllers with higher performance for
+mechatronic systems with complex nonlinear dynamics due
+to the presence of parasitic effects. In this paper we develop
+a generalized feedforward control design methodology based
+on physics–guided neural networks (PGNNs), which effec-
+tively merge physics–based and NN–based models within
+a common model class. PGNNs, as described in this paper,
+were originally developed and employed in feedforward con-
+trol design in [25], and can be regarded as a generalization of
+the parallel linear–NN model that has been used in nonlinear
+system identification, see, e.g., [26, Chapter 21]. The main
+contribution of this paper is the development of a generalized
+feedforward control design method based on PGNNs, with
+the following key features:
+1) Regularized training of PGNNs based on identi-
+fied physical model parameters, to avoid competition
+among the NN and physical layer in the PGNN;
+2) Regularized training of PGNNs based on the physical
+model output to promote physical model compliance,
+and enhance robustness to non–training data;
+3) Input–to–state stability (ISS) guarantees for nonlinear
+PGNN–based feedforward controllers;
+4) Experimental validation on a real–life high–precision
+industrial linear motor for lithography machines.
+Remark 1.1: Compared
+to
+previous
+conference
+pa-
+pers [25], [27], we present the following original contri-
+butions in this journal paper: (i) methods for regularized
+training, including the NN weights, and methods for opti-
+mal selection of the regularization parameter; (ii) graceful
+degradation of the PGNN feedforward controller when it is
+operated on conditions that were not present in the training
+data; (iii) ISS guarantees for PGNN feedforward controllers,
+which, in combination with stable inversion methods for
+linear systems, enable the design of a stable nonlinear
+feedforward controller for nonminimum phase systems; (iv)
+novel, real–life experimental results.
+The remainder of this paper is organized as follows:
+Section II introduces inversion–based feedforward control,
+followed by the problem statement in Section III. The
+methodology for PGNN feedforward controller design with
+regularized training and graceful degradation is presented in
+Section IV. Methods for testing and imposing ISS of PGNN
+feedforward controllers are developed in Section V. Sec-
+tion VI demonstrates effectiveness of the PGNN feedforward
+controller on a real–life coreless linear motor and a nonmin-
+imum phase simulation example. The main conclusions are
+summarized in Section VII.
+For streamlining exposition of the results, in this paper all
+proofs are reported in Appendices.
+II. INVERSION–BASED FEEDFORWARD CONTROL
+A. Feedback–feedforward control architecture
+Fig. 1 displays a standard feedback–feedforward control
+scheme, where u(t) is the control input, and y(t) the system
+output, with time t ∈ R≥0. A mechatronic motion system
+typically has a dominant linear part, denoted by G(s) with
+s the Laplace variable, and possible higher order modes
+�
+i Gi(s). Additionally, g(·) represents nonlinear parasitic
+effects that typically arise from manufacturing tolerances.
+The goal is to minimize the tracking error e(k) := r(k) −
+y(k), where r(k) is the reference, and y(k) is the measured
+at discrete–time indices k ∈ Z≥0.
+The control input u(t) is computed at discrete–time in-
+stances k according to
+u(k) = ufb(k) + uff(k),
+(1)
+where ufb(k) is the feedback, and uff(k) the feedforward
+input. The zero–order–hold (ZOH) in Fig. 1 is a discrete–
+to–continous (D2C) operator, which lets u(t) = u(k) for
+t ∈ [kTs, (k + 1)Ts), with sampling time Ts ∈ R>0. The
+feedback input is of the form
+ufb(k) = C(q)e(k),
+(2)
+where q is the forward–shift operator, e.g., e(k) = qe(k −
+1), and C(q) a rational transfer function. The design of the
+feedforward controller is presented in the following sections.
+Remark 2.1: In this paper we focus on the tracking prob-
+lem, i.e., minimizing the output tracking error e(k) with
+respect to a desired reference r(k). Nevertheless, the methods
+proposed in this work can be extended to reject known
+disturbances acting on the closed–loop system.
+B. Linear feedforward controller design
+As an introductory example to feedforward controller
+design, we consider a moving mass experiencing viscous
+friction, i.e.,
+m¨y(t) = u(t) − fv ˙y(t),
+t ∈ R≥0.
+(3)
+In (3), m ∈ R>0 is the mass, fv ∈ R>0 the viscous friction
+coefficient, and u(t) and y(t) are respectively the force input
+and position output. Application of the Laplace transform
+to (3) gives
+Y (s) = G(s)U(s) :=
+1
+ms2 + fvsU(s).
+(4)
+The moving mass in (4) is a simple case of the general sys-
+tem displayed in Fig. 1, i.e., without higher order dynamics
+�
+i Gi(s) and nonlinear parasitic effects g(·).
+In order to derive a discrete–time feedforward controller,
+we discretize the dynamics (4) using ZOH, which gives
+y(k) = G(q)u(k) :=
+b1q−1 + b2q−2
+1 + a1q−1 + a2q−2 u(k),
+(5)
+where a1, a2, b1, b2
+∈
+R are the discrete–time transfer
+function coefficients. Suppose that the feedforward yields
+perfect tracking, i.e., e(k) = r(k) − y(k) = 0, then (2)
+gives ufb(k) = 0, such that u(k) = uff(k) from (1). Corre-
+spondingly, from (5) we compute the feedforward controller
+
+Fig. 1: Feedback–feedforward control scheme for a system that is described by a nominal linear model G(s), possible higher
+order dynamics �
+i Gi(s), and nonlinear parasitic effects g(·) which can depend on derivatives of u and y.
+as
+uff(k) = G−1(q)r(k) = 1 + a1q−1 + a2q−2
+b1q−1 + b2q−2
+r(k)
+= 1
+b1
+r(k + 1) + a1
+b1
+r(k) + a2
+b2
+r(k − 1) − b2
+b1
+uff(k − 1).
+(6)
+The linear feedforward controller design directly extends
+to systems with higher order dynamics. Then, (5) becomes
+y(k) = G(q)u(k) := q−nk �nb
+i=1 biq−i
+1 + �na
+i=1 aiq−i u(k),
+(7)
+with na ∈ Z≥0, nb ∈ Z>0 the order of the dynamics, and
+nk ∈ Z≥0 the number of pure input delays. We denote a0 =
+1 and, following the same reasoning as for (6), obtain the
+feedforward controller corresponding to (7) as
+uff(k) =
+na
+�
+i=0
+ai
+b1
+r(k+nk +1−i)−
+nb
+�
+i=2
+bi
+b1
+uff(k+1−i). (8)
+In order to implement the feedforward controller (8) we
+require two assumptions:
+1) Reference preview: future reference values up to r(k+
+nk + 1) are known at time instant k;
+2) Stable inverse dynamics: G−1(q) is stable, i.e., has
+poles within the unit circle, or, equivalently, G(q)
+is minimum phase. This requires that the zeros of
+�nb
+i=1 biqnb−i are within the unit circle.
+Remark 2.2: When G(q) is nonminimum phase, it is com-
+mon practice to either approximate G(q) using a minimum
+phase transfer function or to use non–causal filtering to
+obtain a stable exact inverse of G(q), see, e.g., [12] for an
+overview.
+C. Nonlinear feedforward controller design
+The linear feedforward control framework can be extended
+to deal with the nonlinearities g(·), by starting from a non-
+linear correspondent of (7), i.e., a nonlinear autoregressive
+exogeneous (NARX) representation, which is given as
+y(k) = h
+�
+[y(k − 1), . . . , y(k − na),
+u(k − nk − 1), . . . , u(k − nk − nb)]T �
+,
+(9)
+where h : Rna+nb → R is a nonlinear function that describes
+the complete dynamics. We assume that there exists an
+inverse representation of (9), such that, with a slight abuse
+of notation, we can write
+u(k) = h−1�
+φ(k)
+�
+,
+(10)
+where φ(k) := [y(k+nk +1), . . . , y(k+nk −na +1), u(k−
+1), . . . , u(k − nb + 1)]T . Following the same reasoning as
+for the linear feedforward (5), we obtain the ideal nonlinear
+feedforward controller as
+uff(k) = h−1�
+φff(k)
+�
+,
+(11)
+with φff(k) := [r(k+nk +1), . . . , r(k+nk −na+1), uff(k−
+1), . . . , uff(k − nb + 1)]T .
+In general, the function h−1 in (11) is unknown. Moreover,
+the precision demanded by industry exceeds manufactur-
+ing tolerances, which implies that a machine–specific h−1
+needs to be found. For these reasons, a systematic data–
+based approach for finding a model of the inverse system
+dynamics h−1 would be desirable.
+D. Identification for feedforward control
+We discuss the three main aspects to perform a direct
+identification of the inverse dynamics (10): the data set, the
+model class, and the identification criterion.
+Data set: we consider the availability of a data set that is
+generated on the system displayed in Fig. 1, i.e., satisfying
+dynamics (9). As a result, with a slight abuse of notation,
+we have
+ZN = {φ0, u0, . . . , φN−1, uN−1},
+(12)
+where φi := φ(i) and ui := u(i) for i ∈ {0, . . . , N − 1}
+in (10) during the data generating experiment, where N ∈
+Z>0 is the number of data points.
+
+Model class: we require a model parametrization of the
+inverse dynamics.
+Definition 2.1: A model parametrization of the inverse
+dynamics is given as
+ˆu
+�
+θ, φ(k)
+�
+= f
+�
+θ, φ(k)
+�
+,
+(13)
+where ˆu
+�
+θ, φ(k)
+�
+is the prediction of the input u(k), θ ∈
+Rnθ denotes the vector of parameters with nθ ∈ Z>0, and
+f : Rnθ × Rna+nb → R is a user–defined function.
+Identification criterion: the identification criterion defines
+the best choice of parameters θ to have the model (13) fit the
+inverse system dynamics (10). Typically, the identification
+criterion aims to minimize a cost function, i.e.,
+ˆθ = arg min
+θ
+V (θ, ZN),
+(14)
+such as the mean–squared error (MSE)
+V (θ, ZN) = VMSE(θ, ZN) := 1
+N
+N−1
+�
+i=0
+�
+ui − ˆu(θ, φi)
+�2.
+(15)
+Similar to (8) and (11), the feedforward controller is obtained
+by computing the input u(k) that yields the output y(k) =
+r(k) for the identified model, i.e., (13) with θ = ˆθ from (14),
+such that
+uff(k) = ˆu
+�ˆθ, φff(k)
+�
+= f
+�ˆθ, φff(k)
+�
+.
+(16)
+The model parametrization (13) is a crucial choice made
+by the user. Conventional methods employ a model that is
+derived from physical knowledge of the system, such that
+f
+�
+θ, φ(k)
+�
+= fphy
+�
+θphy, φ(k)
+�
+,
+(17)
+where θphy ∈ Rnθphy are the physical parameters. For reasons
+that become apparent in Section IV, we denote θ∗
+phy as the
+parameters of the physical model (17) identified according to
+the MSE identification criterion (14), (15). As an alternative
+to physics–based models (17), black–box NNs can be used
+to parametrize the inverse dynamics (10), i.e.,
+ˆu
+�
+θNN, φ(k)
+�
+= fNN
+�
+θNN, φ(k)
+�
+= WL+1αL
+�
+. . . α1
+�
+W1φ(k) + B1
+��
++ BL+1,
+(18)
+where αl
+:
+Rnl
+→
+Rnl denotes the aggregation of
+activation functions with nl
+∈
+Z>0
+the number of
+neurons
+in
+layer
+l
+∈
+{0, . . . , L},
+and
+L
+∈
+Z>0
+the number of hidden layers. The parameters θNN
+:=
+[col(W1)T , BT
+1 , . . . , col(WL+1)T , BT
+L+1]T are the concate-
+nation of all weights Wl ∈ Rnl×nl−1 and biases Bl ∈ Rnl,
+where col(Wl) stacks the columns of Wl. A NN model (18)
+can approximate any continuous nonlinear mapping on a
+bounded set with arbitrary accuracy, provided that it consists
+of a sufficient number of neurons and/or hidden layers [15].
+This concludes the required preliminary knowledge to define
+the problem considered in this paper.
+III. PROBLEM STATEMENT
+To formally characterize the inherent limitation of a physi-
+cal model (17), we define the corresponding structural model
+errors.
+Definition 3.1: Given the physical model (17), we define
+the structural model error as
+g
+�
+φ(k)
+�
+:= h−1�
+φ(k)
+�
+− fphy
+�
+θ∗
+phy, φ(k)
+�
+.
+(19)
+Notice that Fig. 1 displays the structural model error g
+�
+φ(k)
+�
+for the situation in which the forward dynamics are consid-
+ered and the physical model is linear, i.e., fphy
+�
+θphy, φ(k)
+�
+=
+θT
+phyφ(k). The inverse dynamics (10) can be rewritten into
+u(k) = fphy
+�
+θ∗
+phy, φ(k)
+�
++ g
+�
+φ(k)
+�
+.
+(20)
+In order to derive conditions for which the feedforward
+controller (11) achieves zero tracking error, we define a set
+of operating conditions R as follows.
+Definition 3.2: Denote Φff ⊆ Rna+nb as all regressor
+points φff(k) supplied to the feedforward controller (16) for
+all references r(k) and all k. Then, the set of operating
+conditions R is defined as
+R := Φff ∪ {φ0, ..., φN−1}.
+(21)
+As an example, it is possible to obtain the operating condi-
+tions R for the moving mass (3) by considering maxima and
+minima on the position, velocity, and acceleration.
+By comparing the inverse dynamics (10) with the feed-
+forward input (11), we obtain the following condition for
+perfect tracking
+uff(k) = f
+�ˆθ, φff(k)
+�
+= h−1�
+φff(k)
+�
+,
+∀ φff(k) ∈ R.
+(22)
+Consequently, due to the structural model errors (20),
+condition (22) cannot be satisfied when using a physical
+model (17). On the other hand, when using a NN model (18),
+it is possible to find a sufficient number of hidden layers and
+neurons per layer in combination with a set of parameters
+θ∗
+NN such that (22) is satisfied. In practice however, the
+tedious hyperparameter tuning and non–convex optimization
+can return suboptimal identified parameters ˆθNN that do not
+even reach performance of the physics–based feedforward.
+Additionally, dangerous situations can arise when the data
+set ZN does not cover R sufficiently well, e.g., as a result
+of limitations on either time or safety there can be no training
+data available for all relevant operating conditions. Finally,
+there is only limited research available for validating stability
+of NN–based feedforward controllers, see, e.g., [23], [24],
+and no tools to enforce stability a priori.
+To solve the above issues of NN–based feedforward con-
+trollers, in this paper, we develop a generalized feedforward
+control design methodology based on PGNNs. To this end,
+we introduce the formal definition of a PGNN model class.
+Definition 3.3: The PGNN model parameterization of the
+inverse dynamics is given as
+ˆu
+�
+θ, φ(k)
+�
+= fphy
+�
+θphy, φ(k)
+�
++ fNN
+�
+θNN, T
+�
+φ(k)
+��
+, (23)
+
+Fig. 2: Schematic overview of the physics–guided neural
+network.
+where θ = [θT
+NN, θT
+phy]T are the PGNN parameters, and T :
+Rna+nb → Rn0 is an input transformation, with n0 ∈ Z>0
+the number of NN inputs.
+Remark 3.1: The input transformation T can be used to
+improve the numerical properties of the PGNN training, as
+well as to create physically relevant inputs. As a practical
+example, consider a moving mass and let φi = [yi+1, yi]T
+with yi the measured position. Then, choosing T(φi) =
+� yi+1−yi
+Ts
+, yi
+�T with Ts ∈ R>0 the sampling time, can
+improve numerical properties of the training when Ts is
+small, such that yi+1 ≈ yi. As a second example, let yi
+be the measured rotation of a rotary motor. Then, choosing
+T(φi) =
+� yi+1−yi
+Ts
+, mod(yi)
+�T with mod(yi) the remainder
+after division of yi by 2π can improve the training conver-
+gence and extrapolation in terms of the rotation yi, since
+T(·) imposes the rotational reproducible behaviour.
+Remark 3.2: The PGNN (23) imposes physical knowl-
+edge by embedding a known physical model within a NN.
+Note that this is different from physics–informed neural net-
+works (PINNs) [28], [29], which impose physical knowledge
+via a regularization term in the cost function that penalizes
+deviations of the NN output with respect to the output of the
+available physical model. The use of PINNs for feedforward
+control was explored in, e.g., [30]. It is also worth to point
+out that, conforming to this definition, the PGNNs developed
+in [31] are PINNs, i.e., the physical model is not embedded
+within a NN therein.
+IV. PHYSICS–GUIDED NEURAL NETWORKS FOR
+FEEDFORWARD CONTROL
+In this section, we introduce a regularized cost function
+for training the PGNN (24) to avoid competition between
+the physical and NN layers. We demonstrate that, under
+suitable assumptions on the model class, training data, and
+training convergence, the PGNN recovers the inverse system
+dynamics. Moreover, we consider the situation in which
+these assumptions are violated and introduce methods to
+ensure that i) the PGNN improves with respect to using
+only the physical model in terms of data fit via an optimized
+parameter initialization, and ii) we impose a form of graceful
+degradation for the PGNN when operated at conditions that
+were not present in the training data.
+For ease of presentation, in the remainder of this paper we
+Fig. 3: L–curve obtained by training the PGNN (24) accord-
+ing to (14), (25) with Λphy = ΛNN = I for 20 values of
+λ logarithmically spaced in [10−5, 10], and optimal choice
+λ = 0.07 (red circle).
+consider T
+�
+φ(k)
+�
+= φ(k), such that the PGNN (23) reduces
+to
+ˆu
+�
+θ, φ(k)
+�
+= fphy
+�
+θphy, φ(k)
+�
++ fNN
+�
+θNN, φ(k)
+�
+.
+(24)
+A. Regularized PGNN training and inverse system identifi-
+cation
+The PGNN (24) is generally overparameterized, i.e., the
+NN can identify parts of the physical model which results
+in a parameter drift during training. Therefore, we employ a
+regularized cost function
+V (θ, ZN) = VMSE(θ, ZN) + λVreg(θ),
+(25)
+with λ > 0, and the regularization cost is
+Vreg(θ) :=
+����
+�ΛNN
+0
+0
+Λphy
+� �
+θ −
+� 0
+θ∗
+phy
+������
+2
+2
+.
+(26)
+In (26), ΛNN, Λphy are matrices that define the relative
+importance of the different regularization terms. Note that,
+ΛNN relates to the standard L2 regularization for the network
+weights and biases [26, Chapter 7], while Λphy solves the
+overparameterization of the PGNN model (24).
+Remark 4.1: The hyperparameter λ in (25) relates the
+importance of the data fit with the regularization costs, and
+can be tuned using, e.g., the L–curve method. The L–curve
+displays the data fit (x–axis) and regularization cost (y–axis)
+for varying values of λ. The optimal choice for λ is the point
+where the curvature is largest [32]. An example is shown in
+Fig. 3 with Λphy = ΛNN = I using data that is generated on a
+coreless linear motor, which is discussed in Section VI. The
+optimal choice is λ = 0.07. Although the differences in data
+fit VMSE in Fig. 3 seem minor, the minimal data fit is limited
+by the noise that is present in the data set. Correspondingly,
+the effect of λ is enlarged when the noise is subtracted.
+
+fphy
+phy, Φ(k)0,b(k)
+nT
+b(k)NN, T(Φ(k)
+fNNPhysics
+network
+gtlcec
+nettratThe PGNN (24) recovers the inverse dynamics (10) when
+the following assumptions are satisfied.
+Assumption 4.1: There
+exists
+a
+θ∗
+NN
+such
+that
+fNN
+�
+θ∗
+NN, φ(k)
+�
+= g
+�
+φ(k)
+�
+for all φ(k) ∈ R.
+Assumption 4.2: For θA
+NN ̸= θB
+NN with fNN
+�
+θA
+NN, φ(k)
+�
+̸=
+fNN
+�
+θB
+NN, φ(k)
+�
+for some φ(k) ∈ R, it holds that
+1
+N
+N−1
+�
+i=0
+�
+fNN(θA
+NN, φi) − fNN(θB
+NN, φi)
+�2 > 0.
+(27)
+Assumption 4.3: The minimization of (25) over θ yields
+a global optimum.
+Assumption 4.1 dictates that there should exist a choice of
+parameters for which the PGNN (24) recovers the original
+inverse system (10), i.e., the system should be in the model
+set. Assumption 4.2 relates to the persistence of excitation.
+Proposition 4.1 (PGNN consistency): Consider
+the
+PGNN
+(24)
+that
+is
+used
+to
+identify
+the
+inverse
+dynamics (10) according to identification criterion (14)
+with cost function (25) using ΛNN
+= 0. Suppose that
+Assumptions 4.1, 4.2, and 4.3 hold. Then, the identified
+PGNN parameters satisfy ˆθ = [ˆθT
+phy, ˆθT
+NN]T = [θ∗T
+phy, θ∗T
+NN]T .
+Proof: See Appendix I.
+From Assumption 4.1 and (20), it is observed that the iden-
+tified PGNN recovers the inverse dynamics for all operating
+conditions, i.e.,
+fphy
+�ˆθphy, φ(k)
+�
++fNN
+�ˆθNN, φ(k)
+�
+= h−1�
+φ(k)
+�
+, ∀ φ(k) ∈ R,
+(28)
+such that the zero tracking error condition (22) is satisfied.
+Remark 4.2: A parallel linear–NN model structure was
+also employed for feedforward control design in [33].
+Therein, an alternative regularization method based on or-
+thogonal projection was developed to avoid the competition
+between the linear and NN layers under the assumption that
+g(·) is identically zero outside R.
+B. Optimized PGNN parameter selection
+When Assumption 4.1 or 4.3 is violated, recovering the
+inverse dynamics is not guaranteed, or the training does
+not find the optimal parameters. The training, i.e., the op-
+timization over θ of (25), typically employs a nonlinear
+optimization scheme, that attains parameter estimates θ(j)
+where j ∈ {0, . . . , J}, with J ∈ Z≥0 the number of solver
+iterations. The estimated parameter vector is chosen as
+ˆθ = θ(j),
+j = arg
+min
+j∈{0,...,J} V
+�
+θ(j), ZN�
+.
+(29)
+In what follows, we will derive a specific choice for the
+parameters θ(j) which achieve a smaller value of the cost
+function (25) compared to using only the physical model.
+For simplicity, we consider a LIP physical model (17),
+i.e., fphy
+�
+θphy, φ(k)
+�
+= θT
+phyTphy
+�
+φ(k)
+�
+, and rewrite the
+PGNN (24) according to
+ˆu
+�
+θ, φ(k)
+�
+= θT
+L φL
+�
+θNL, φ(k)
+�
+:= θT
+L
+�
+�
+αl
+�
+θNL, φ(k)
+�
+1
+Tphy
+�
+φ(k)
+�
+�
+� ,
+(30)
+where θL
+:=
+[col(WL+1)T , BL+1, θT
+phy]T
+are the pa-
+rameters
+in
+which
+the
+PGNN
+(30)
+is
+linear,
+and
+θNL := [col(W1)T , BT
+1 , . . . , col(WL)T , BT
+L]T , such that θ =
+[θT
+NL, θT
+L ]T . From (30), we observe that an equivalent of the
+physical model (17) is obtained by selecting θL = θL :=
+[0, 0, θ∗T
+phy]T . We rewrite the regularization term (26) into
+Vreg(θ) =
+����
+�ΛNL
+Λ
+ΛT
+ΛL
+� ��
+θNL
+θL
+�
+−
+� 0
+θL
+������
+2
+2
+,
+(31)
+where Λ, ΛT
+NL = ΛNL, and ΛT
+L
+= ΛL are obtained by
+selecting rows and columns of ΛNN and Λphy accordingly.
+Then, given any parameter set θ(j)
+NL, choose θ(j)
+L
+according to
+θ(j)
+L
+=M(θ(j)
+NL)−1
+�
+1
+N
+N−1
+�
+i=0
+uiφL(θ(j)
+NL, φi)
++ λ
+��
+Λ2
+L + ΛT Λ
+�
+θL −
+�
+ΛT ΛNL + ΛLΛT �
+θ(j)
+NL
+��
+,
+(32)
+where M(θ(j)
+NL) is given as
+M(θ(j)
+NL) := 1
+N
+N−1
+�
+i=0
+φL(θ(j)
+NL, φi)φL(θ(j)
+NL, φi)T
++ λ
+�
+Λ2
+L + ΛT Λ
+�
+.
+(33)
+Note that (32) is valid only if M(θ(j)
+NL) is nonsingular, which
+yields a unique solution. Nonsingularity of M(θ(j)
+NL) is also
+referred to as the persistence of excitation.
+Proposition 4.2 (PGNN improves over physics):
+Consider the PGNN (30) with θ(j)
+NL given, e.g., initialized
+randomly for j = 0 or attained during training for j ̸= 0.
+Suppose that M(θ(j)
+NL) is nonsingular, and choose θ(j)
+L
+according to (32). Then, for the cost function (25), we have
+V
+��
+θ(j)
+NL
+θ(j)
+L
+�
+, ZN
+�
+≤ V
+��
+θ(j)
+NL
+θL
+�
+, ZN
+�
+,
+(34)
+with strict inequality if and only if
+1
+N
+N−1
+�
+i=0
+�
+uiφL
+�
+θ(j)
+NL, φi
+�
+− φL
+�
+θ(j)
+NL, φi
+�
+φL
+�
+θ(j)
+NL, φi
+�T θL
+�
+− λ(ΛT ΛNL + ΛLΛT )θ(j)
+NL ̸= 0.
+(35)
+Moreover, if ΛNN is such that Λ = 0, i.e., the cross terms
+between θNL and [W T
+L+1, BL+1]T are not penalized, the
+PGNN achieves a better data fit compared to the physical
+model, i.e.,
+VMSE
+��
+θ(j)
+NL
+θ(j)
+L
+�
+, ZN
+�
+≤ VMSE(θ∗
+phy, ZN),
+(36)
+with strict inequality if (35) holds.
+Proof: See Appendix II.
+Condition (35) is obtained by rewriting θ(j)
+L
+̸= θL, i.e.,
+W (j)
+L+1 and B(j)
+L+1 are nonzero, and possibly θ(j)
+phy ̸= θ∗
+phy as
+well. This is satisfied if θ(j)
+NL is such that there is a correlation
+
+between the output of the final hidden layer αL(·) and the
+structural model errors present in the data g(φi).
+Remark 4.3: Proposition 4.2 can be directly extended for
+PGNNs with physical models that are not LIP. To see this,
+rewrite the physical model
+fphy
+�
+θphy, φ(k)
+�
+= θT
+L,phyTphy
+�
+θNL,phy, φ(k)
+�
+,
+(37)
+and
+choose
+θ(j)
+L
+=
+[col(W (j)
+L+1)T , B(j)
+L+1, θ(j)T
+L,phy]T
+according
+to
+(32)
+using
+θ(j)
+NL
+=
+[col(W (j)
+1 )T , B(j)T
+1
+, ..., col(W (j)
+L )T , B(j)T
+L
+, θ∗
+NL,phy
+T ]T .
+In practice, the optimized parameter selection is used after
+training, i.e., update θ(j)
+L
+for j = J, during training for
+each j = {0, ..., J}, or as an initialization for j = 0. Note
+that, from (29) and Proposition 4.2, if (35) holds, we have
+that
+V (ˆθ, ZN) ≤ V (θ(0), ZN) < V (θ, ZN),
+(38)
+when (32) is used for initialization of θ(0)
+L .
+C. Enhancing PGNN robustness via regularized training
+In general, there is no systematic method to validate
+Assumption 4.2 in practice. For this reason, [34] gives as a
+user guideline to sample the complete domain of interest, i.e.,
+the operating conditions R. For reasons of time and safety, it
+can be infeasible to design an experiment that achieves this.
+Correspondingly, we require a form of graceful degradation
+for situations in which the PGNN is operated on conditions
+that were not present in the training data. Since there is no
+data available for these conditions, the physical model is the
+only source of information. Therefore, we aim to have the
+PGNN comply with the physical model via a regularization
+term in the cost function, such that (25) becomes
+V (θ, ZN) = VMSE(θ, ZN) + λVreg(θ) + γVphy(θ, ZE). (39)
+In (39), γ ∈ R>0 is a regularization parameter, and the
+regularization cost is
+Vphy(θ, ZE) := 1
+E
+E−1
+�
+i=0
+�
+fphy(θ∗
+phy, φE
+i ) − ˆu(θ, φE
+i )
+�2 . (40)
+The set ZE = {φE
+0 , . . . , φE
+E−1} contains user–defined re-
+gressor points for which we aim to have the PGNN comply
+with the physical model, e.g., due to the absence of reliable
+data in ZN. Thereby, the aim is to have ZN ∪ ZE cover
+the operating conditions R up to high accuracy, which
+is done automatically by Algorithm 1. In Algorithm 1,
+C(ζ, ZN, ZE) is an objective function that specifies the goal
+of the optimization. For example, choosing φE
+i to maximize
+the minimum squared Euclidean distance with respect to all
+regressor points, is achieved by choosing
+C(ζ, ZN, ZE) :=
+min
+φ ∈ZN,ZE ∥φ − ζ∥2
+2 .
+(41)
+Algorithm 1 iterates until a stopping criterion is met, e.g., by
+fixing a maximum number of points, or a minimum threshold
+ϵ for the objective function (41).
+Remark 4.4: The results of Proposition 4.2 can be ex-
+tended directly to the cost function (39) by appropriately
+Algorithm 1 Design algorithm for ZE.
+Initialize ZE = {}, i = 0,
+while i < E − 1 ∧ C
+�
+φ(i − 1), ZN, ZE�
+> ϵ do
+φE
+i = arg maxζ∈R C
+�
+ζ, ZN, ZE�
+,
+ZE = ZE ∪ φE
+i ,
+i = i + 1.
+end while
+-0.15
+-0.1
+-0.05
+0
+0.05
+0.1
+0.15
+-0.3
+-0.2
+-0.1
+0
+0.1
+0.2
+0.3
+Fig. 4: Two–dimensional illustration of the regressor points
+in the CLM data set ZN discussed in Section VI, and the
+regressor points ZE designed following Algorithm 1.
+revising the computation of θ(j)
+L
+in (32), i.e., compute the
+least squares solution of (39) instead of (25).
+Consider a mechanical system as in (3), where the operat-
+ing conditions R are represented by a maximum position of
+0.15 m and velocity of 0.2 m
+s , and neglect the acceleration
+for the sake of simplicity. Since it was deemed unsafe to
+travel the full stroke during the data generating experiment,
+the training data ZN describes R only in a limited range of
+the position, see the blue crosses in Fig. 4. Consequently,
+application of Algorithm 1 generates the set ZE, see the
+orange crosses in Fig. 4, such that ZN ∪ ZE cover the
+complete operating conditions R.
+V. ISS OF PGNN–BASED FEEDFORWARD CONTROLLERS
+Stability of a feedforward controller (16) is a prereq-
+uisite for safe operation of the closed–loop system, i.e.,
+for bounded reference values r(k) the feedforward input
+uff(k) should remain bounded. For linear feedforward con-
+trollers, stability is tested by ensuring that the poles of the
+feedforward controller transfer function are within the unit
+circle. However, the nonlinear NN in the PGNN feedfor-
+ward (16), (24) complicates the stability assessment. For
+the sake of presentation, we consider a PGNN (24) with
+linear physical model, i.e., fphy
+�
+θphy, φ(k)
+�
+= θT
+phyφ(k), and
+highlight some further generalizations. Next, to facilitate the
+stability analysis of PGNN–based feedforward controllers,
+
+we define
+ˆa := [ˆa1, . . . , ˆana+1]T ,
+ˆb := [ˆb1, . . . ,ˆbnb−1]T ,
+(42)
+such that ˆθphy = [ˆaT ,ˆbT ]T . Moreover, we define
+φr(k) := [r(k + nk + 1), . . . , r(k + nk − na + 1)]T ,
+φuff(k) := [uff(k − 1), . . . , uff(k − nb + 1)]T ,
+(43)
+such that φff(k) = [φr(k)T , φuff(k)T ]T . Note that, for a
+linear feedforward controller (8), stability is determined only
+by ˆb. Then, by choosing x(k) := φuff(k), we obtain a state–
+space representation of the PGNN (24) as
+x(k + 1) =A(ˆb)x(k)
++ B
+�
+ˆaT φr(k) + fNN
+�
+ˆθNN,
+�
+φr(k)
+x(k)
+���
+,
+(44)
+where B
+=
+[1, O]T
+∈
+Rnb−1, A(ˆb)
+=
+� ˆbT
+I
+O
+�
+∈
+R(nb−1)×(nb−1), and φr(k) is the external input.
+Because the state–space PGNN feedforward (44) has
+φr(k) as an exogeneous input, it makes sense to pursue ISS
+guarantees [35], which is a stronger property than nominal
+stability and guarantees boundedness of the state x(k) for
+any bounded reference φr(k). First, we provide a general ISS
+result for the PGNN state–space representation (44). After-
+wards, we show how ISS of the resulting PGNN feedforward
+controllers can be imposed a priori during the PGNN training
+by adjusting the identification criterion.
+We recall the following notions: a function κ : R≥0 →
+R≥0 is a K–function if it is continuous, strictly increasing
+and κ(0) = 0. It is a K∞–function if it is a K–function and
+κ(s) → ∞ as s → ∞. A function η : R≥0 × R≥0 is a KL–
+function if, for each fixed k ≥ 0, the function η(·, k) is a
+K–function and for each fixed s ≥ 0, the function η(s, ·) is
+decreasing and η(s, k) → 0 as k → ∞.
+Definition 5.1 (ISS [35]): A
+system
+x(k + 1)
+=
+f
+�
+x(k), φr(k + 1)
+�
+is ISS if there exist a KL–function
+η : R≥0 × R≥0 → R≥0 and a K–function κ such that,
+for each finite input sequence φr and each initial state
+x(0) = ξ ∈ Rnb−1, it holds that
+|x(k)| ≤ η
+�
+|ξ|, k
+�
++ κ
+�
+∥φr∥
+�
+(45)
+for each k ∈ Z+.
+Remark 5.1: If the state–space PGNN representation (44)
+is ISS with respect to the external input φr(k + 1), we have
+that:
+1) x(k) remains bounded for bounded φr(k) and, conse-
+quently, uff(k) remains bounded;
+2) x(k) → 0 for φr(k) → 0 and, consequently, uff(k) →
+0.
+In order to analyze ISS of the PGNN feedforward con-
+troller (44), we will use a continuous ISS–Lyapunov func-
+tion, which implies ISS as shown in [35].
+Definition 5.2 (ISS–Lyapunov [35]): A continuous func-
+tion V : Rn → R≥0 is called an ISS–Lyapunov function for a
+discrete–time state–space system x(k+1) = f
+�
+x(k), φr(k)
+�
+if the following conditions hold:
+1) There exists K∞–functions κ1, κ2 such that
+κ1
+�
+|x(k)|
+�
+≤ V
+�
+x(k)
+�
+≤ κ2
+�
+|x(k)|
+�
+, ∀
+x(k) ∈ Rnb−1.
+(46)
+2) There exists a K∞–function κ3 and a K–function σ,
+such that
+V
+�
+x(k + 1)
+�
+− V
+�
+x(k)
+�
+≤ −κ3(|x(k)|) + σ(|φr(k + 1)|),
+∀ x(k) ∈ Rnb−1, ∀ φr(k) ∈ Rna+1.
+(47)
+Remark 5.2: For all commonly applied activation func-
+tions, the NN (18) is globally Lipschitz, i.e., there exists an
+L := [LT
+r , LT
+uff]T such that
+���fNN
+�ˆθNN, φA
+ff (k)
+�
+− fNN
+�ˆθNN, φB
+ff (k)
+����
+≤ LT ��φA
+ff (k) − φB
+ff (k)
+�� .
+(48)
+Using backpropagation, it is possible to find values for L,
+e.g.,
+LT = max
+φff(k)
+∂fNN
+�ˆθNN, φff(k)
+�
+∂φff(k)
+= max
+φff(k)
+ˆWl+1diag(α′
+l) . . . ˆW2diag(α′
+1) ˆW1 ≤ Πl+1
+i=1| ˆWi|,
+(49)
+where α′
+l = ∂αl(x)
+∂x
+��
+x=xl, with xl the input to NN layer l. The
+upperbound in (49) is obtained by substitution of diag(α′
+l) =
+I, which holds for tanh, ReLU, and several other activation
+functions.
+Assumption 5.1: The origin x(k) = 0 is an equilibrium
+for the unexcited PGNN (44), i.e., φr(k) = 0, such that (44)
+gives
+fNN
+�ˆθNN, 0
+�
+= 0.
+(50)
+Note that, we can introduce a coordinate transformation, e.g.,
+ζ(k) = x(k) + ε to satisfy Assumption 5.1.
+Assumption 5.2: There exists a P ≻ 0 such that Q :=
+P − A(ˆb)T PA(ˆb) ≻ 0.
+Assumption 5.2 requires that the parameters ˆb are such that
+the physical model is stable, i.e., A(ˆb) is a Schur matrix. A
+(P, Q) pair satisfying Assumption 5.2 is typically obtained
+by choosing Q ≻ 0 and solving the discrete–time Lyapunov
+equation to obtain P ≻ 0. We denote λmin(Q) and λmax(Q)
+as the smallest and largest eigenvalue of Q, respectively.
+Theorem 5.1 (PGNN feedforward ISS): Consider
+the
+PGNN
+feedforward
+(16),
+(24)
+with
+linear
+physical
+model, and its state–space representation (44). Suppose
+that Assumptions 5.1 and 5.2 hold. Let (P, Q) satisfy
+Assumption 5.2, and define
+cβ := BT P
+�
+I +
+1
+βλmin(Q)AAT P
+�
+B.
+(51)
+Then, if there exists a β > 0 such that
+LT
+uffLuff < (1 − β)λmin(Q)
+cβ
+,
+(52)
+
+the PGNN feedforward state–space representation (44) is
+ISS.
+Proof: See Appendix III.
+Remark 5.3: Most NN stability research is conducted for
+systems operating in closed–loop with a NN–based feedback
+controller, see, e.g., [24], [36] which provide local stability
+conditions via linearization and integral quadratic constraints,
+respectively. A global ISS result for a black box NN with
+no physical model embedded is presented in [23].
+Remark 5.4: The value of β for which the right–hand side
+in (52) is maximal, is
+β =
+1
+λmin(Q)BT PB
+�
+− BT PAAT PB+
+�
+BT P
+�
+λmin(Q)I + AAT P
+�
+BBT PAAT PB
+�
+.
+(53)
+Note that, since B is a column, the square root and division
+are scalar operations. Eq. (53) is obtained by setting the
+derivative w.r.t. β of the right hand side of (52) equal to
+zero, and choosing the option for which β > 0.
+Remark 5.5: A similar result can be obtained for a PGNN
+with a general nonlinear physical model, when a quadratic
+Lyapunov function is available for the physical model.
+The ISS condition (52) in Theorem 5.1 can be validated
+after training by using β in (53), the upperbound Luff =
+�O,
+I�
+L in (49) for some (P, Q) pair.
+Recall that θ = [θT
+NN, θT
+phy]T , θphy = [aT , bT ]T for the
+linear physical model, and θ∗
+phy are the parameters θphy
+identified using only the physical model (17) with MSE cost
+function (15). With the aim to ensure before training that
+the PGNN is ISS, we fix ˆb = b∗, and constrain the network
+weights to satisfy the ISS condition (52).
+Lemma 5.1 (Training imposed ISS – stable physical model):
+Consider the PGNN feedforward controller with linear
+physical
+model,
+such
+that
+it
+admits
+a
+state–space
+representation of the form (44). Suppose that Assumption 5.1
+holds, that a (P, Q) pair satisfying Assumption 5.2 is
+available, and choose β as in (53). Define the set
+Θ :=
+�
+θ ∈ Rnθ
+�����
+�
+b = b∗�
+∧
+�����
+�
+ΠL
+l=1|Wl|
+� �0
+I
+� ����
+2
+2
+< (1 − β)λmin(Q)
+cβ
+� �
+,
+(54)
+and train the PGNN according to identification criterion
+ˆθ = arg min
+θ ∈ Θ V
+�
+θ, ZN�
+.
+(55)
+Then, the training returns an ISS PGNN feedforward con-
+troller (16), (24).
+Proof: See Appendix IV.
+Assumption 5.2 is violated when A(ˆb) is not Schur, i.e.,
+the system is nonminimum phase. There are two main
+approaches to obtain a stable feedforward controller for
+linear systems, see, e.g., [12]:
+1) Non–causal feedforward controller design;
+2) Stable approximate inversion techniques, such as
+ZPETC, ZMETC, NPZ–ignore.
+In the remainder of this section, we demonstrate approaches
+to assess and impose ISS of PGNN feedforward controllers
+for nonminimum phase systems.
+The nonminimum phase behaviour can be dealt with by
+extending the preview window of the PGNN, i.e., we adjust
+the PGNN (24) to
+ˆu
+�
+θ, ˜φ(k)
+�
+= [aT , bT ]
+� ˜φy(k)
+˜φu(k)
+�
++ fNN
+�
+θNN,
+� ˜φy(k)
+˜φu(k)
+��
+,
+˜φy(k) := [r(k + nk + npw + 1), ..., r(k + nk − na + 1)]T ,
+˜φu(k) := [u(k − 1), ..., u(k − nb + nus + 1)]T ,
+(56)
+where nus ∈ Z>0 are the number of unstable eigenvalues
+of A(ˆb) of the original PGNN (24), and npw ∈ Z>0 is
+the number of extended preview samples. The non–causal
+PGNN (56) is then trained following Lemma 5.1.
+Remark 5.6: In order to satisfy Assumption 5.2, the pa-
+rameters b∗ should be such that A(b∗) is Schur. This is
+achieved either by choosing npw large enough, such that the
+identification of θ∗
+phy = [a∗T , b∗T ]T returns stable parameters
+b∗, or by fixing b∗ to retain the stable eigenvalues of the
+original A(b∗).
+As an alternative approach, it is possible to employ linear
+stable approximation inversion tools on the linear part of the
+dynamics. In order to see this, we rewrite the state–space
+PGNN (44) into
+uff(k) = G(ˆb, q−1)
+�
+ˆaT φr(k) + fNN
+�
+ˆθ,
+�
+φr(k)
+φuff(k)
+���
+,
+(57)
+where G(ˆb, q−1)
+:=
+1
+1−�nb
+i=1 ˆbiq−i . Let
+˜G(˜b, q−1)
+:=
+�˜nn
+b
+i=0 ˜bn
+i q−i
+1−�˜nd
+b
+i=1 ˜bd
+i q−i be a stable approximate of G(ˆb, q−1) obtained
+via a user–preferred method which gives ˜nn
+b , ˜nd
+b ∈ Z≥0 and
+˜bn
+i ,˜bd
+i ∈ R. Then, a PGNN with stable physical model is
+obtained from (57) as
+uff(k) = ˜G(˜b, q−1)
+�
+ˆaT φr(k) + fNN
+�
+ˆθ,
+� φr(k)
+φuff(k)
+���
+.
+(58)
+If the stable approximation method creates terms with qd,
+d ∈ Z>0 in the numerator of G(˜b, q−1) in (58), φuff(k) in the
+NN can only contain feedforward inputs up until uff(k−d−1)
+in order to compute the feedforward according (58).
+ISS of the stable approximated PGNN (58) is imposed
+by training the unstable PGNN (57) according a revised
+version of Lemma 5.1. To do so, we rewrite the stably
+inverted PGNN (58) into state–space form using x(k) :=
+[uff(k), ..., uff(k − nb − ˜nn
+b + 1)]T , and use the triangular
+inequality with Lipschitz bound (48) to construct Lr+ and
+Lx such that
+�����
+˜nn
+b
+�
+i=0
+˜bn
+i fNN
+�ˆθNN, φff(k − i)
+�
+����� ≤ [LT
+r+, LT
+x ]
+�φr+(k)
+x(k)
+�
+, (59)
+
+Fig. 5: Complete industrial coreless linear motor setup (top
+window), with an enlarged view of the coreless linear motor
+(bottom window).
+with φr+(k) = [r(k+nk +1), ..., r(k+nk −na − ˜nn
+b +1)]T .
+Lemma 5.2 (Training imposed ISS – unstable physical model):
+Consider the PGNN (24) with linear physical model.
+Suppose that Assumption 5.1 holds, and choose β as
+in (53). Define
+Θ :=
+�
+θ ∈ Rnθ
+�����
+�
+b = b∗�
+∧
+�
+∥Lx ∥2
+2 < (1 − β)λmin( ˜Q)
+cβ
+� �
+,
+(60)
+with ˜Q = P − A(˜bd)T PA(˜bd) ≻ 0 for some P ≻ 0. Then,
+the stable inverted PGNN feedforward controller (58) trained
+according to identification criterion (55) is ISS.
+Proof: See Appendix V.
+VI. EXPERIMENTAL AND SIMULATION VALIDATION
+A. Real–life industrial coreless linear motor
+Experimental setup: we consider the problem of position
+control of the industrial coreless linear motor (CLM) dis-
+played in Fig. 5, which was formerly part of the longstroke
+actuation in a lithography machine. For research purposes,
+the CLM is limited to exert forces up to 500 N. The
+CLM contains three coil sets, each consisting of three
+coils connected in star configuration that are powered by
+a three–phase power amplifier. The system is controlled
+by a dSPACE MicroLabBox that receives encoder position
+measurements, computes the control input, and converts them
+into current setpoints via a commutation algorithm to be
+send to the power amplifiers. The PC is used to program
+software in Matlab/Simulink that is uploaded to the dSPACE
+MicroLabBox. Relevant signals are accessed during run–
+time using the dSPACE ControlDesk software. The coils are
+cooled using water and the cooling unit. Finally, the safety
+programmable logic controller (PLC) cuts of the power to the
+amplifiers when safety is at risk, e.g., when the stop button
+is pushed, or when the CLM access doors are opened.
+System modelling: a physical model of the CLM displayed
+in Fig. 5 is derived based on Newton’s second law, such that
+m¨y(t) = u(t) − fv ˙y(t) − fcsign
+�
+˙y(t)
+�
+,
+(61)
+with m ∈ R>0 the mass, and fv ∈ R>0 and fc ∈ R>0
+the viscous and Coulomb friction coefficient, respectively.
+The CLM is controlled in closed–loop by a ZOH discretized
+version of
+C(s) = 1.1 · 107 s + 5
+32π
+s
+s + 5
+62π
+s + 30π
+1
+s + 60π ,
+(62)
+which was tuned to achieve a 5 Hz bandwidth for the linear
+model, i.e., (61) with m = 20 kg, fv = 150 kg
+s , and fc = 0
+N.
+Training data generation: data is generated in closed–
+loop by sampling u(k) and y(k) at a frequency of 1 kHz,
+while exciting the system with the third order reference r1(k)
+shown in the top window of Fig. 8 using 0.1 m
+s velocity, 4 m
+s2
+acceleration and 1000 m
+s3 jerk, and a zero–mean white noise
+with variance 50 N 2 added to the input. The data points are
+visualized in Fig. 4.
+Feedforward controllers: we discretize (61) using ˙y(t) =
+q−q−1
+2T
+y(k) and assume the ZOH to yield half a sample delay,
+i.e., u(t) =
+2
+q+1u(k), to obtain na = 5, nb = 1, and nk = 2
+as the orders in φ(k). An encoder with precision d = 5 ·
+10−6 m is used to measure position. Therefore, discrete–
+time velocity approximations are measured at ˙y(k) = n d
+2Ts ,
+compared to ˙y(k) = n d
+Ts when using backward or forward
+Euler.
+In order to demonstrate the importance of the regulariza-
+tion term in (25), we train two PGNN–based feedforward
+controllers according to (14) with NN dimensions n1 = 16,
+l = 1 (18) using ΛNN = 0, Λphy = I and λ = 0 and λ = 0.07.
+Fig. 6 displays the resulting feedforward signals for a given
+reference, including the parts created by the physical model
+and the NN. It is clear that, for λ = 0, there is a competition
+between the physical and NN layer in the PGNN, which
+is not present when the PGNN is trained with λ = 0.07.
+Such a competition among layers is dangerous (not desired),
+because it implies that the physical model does not learn the
+underlying physics, and the resulting feedforward controller
+will not have physics–induced extrapolation properties.
+In order to test tracking performance achieved by the
+PGNN, we consider the following feedforward controllers:
+1) Physics–based feedforward using physical model (61)
+with the parameters identified according to (14), (15);
+2) PGNN–based feedforward with NN dimensions n1 =
+16, l = 1 (18), using physical model (61) identified
+according to (14), (25) with λ = 0.07, and Λphy =
+ΛNN = I;
+
+dSpace
+MicroLabBox
+Safety PLC
+CLM
+Power amplifiers
+Cooling airco
+PCTABLE I: MAE in [m] of the tracking errors in Fig. 7.
+Physics
+PGNN γ = 0
+PGNN γ = 0.1
+MAE
+2.16 · 10−4
+1.35 · 10−4
+1.58 · 10−4
+TABLE II: Parameter values of the rotating–translating mass
+displayed in Fig. 9.
+m
+lx, ly
+M
+fv
+k
+d
+lm
+c
+20
+1
+40
+3
+50
+25·103
+3
+575
+3
+0.05
+1
+kg
+m
+kgm2
+kg
+s
+kg
+s2
+kg
+s
+m
+kgm
+s2
+3) PGNN–based feedforward with graceful degradation
+with NN dimensions n1 = 16, l = 1 (18), using
+physical model (61) identified according to (14), (39)
+with λ = 0.07, Λphy = ΛNN = I, and γ = 0.1.
+Results: Fig. 7 the tracking error e(k) resulting achieved
+by the different feedforward controllers for r1(k) in Fig. 6.
+In order to test robustness of the PGNN, we compute the
+mean–absolute error (MAE)
+MAE = 1
+N
+N−1
+�
+k=0
+|e(k)|,
+(63)
+for two references r1(k) and r2(k) in Fig. 6 using different
+velocities max
+�
+˙ri(k)
+�
+= n 0.025
+m
+s , i = {1, 2}, n =
+{1, ..., 6}. Note that, since max
+�
+r2(k)
+�
+= 0.15 m, the
+feedforward requires to extrapolate for r2(k), see Fig. 4.
+Fig. 8 displays the resulting MAE values, and Table I lists
+the MAE for r1(k).
+For r1(k), both the PGNNs outperform the physics–based
+feedforward in terms of the MAE for all tested velocities.
+A significant drop in performance is observed when using
+the PGNN with γ = 0 on reference r2(k), which is caused
+by the PGNN extrapolating badly for r(k) > 0.1 m. The
+physics–based regularization term (40) prevents this, as seen
+by the MAE of the PGNN with γ = 0.1.
+B. Nonminimum phase rotating–translating mass
+System dynamics: we consider a translating–rotating mass
+with force input u(k) and position output y(k) at opposite
+sides of the centre of mass, see Fig. 9. The dynamics are
+given as
+y(t) =
+�
+1
+ms2 + fvs −
+l2
+y
+Ms2 + 2lxds + 2lxk
+�
+·
+�
+u(t) − g
+�
+y(t)
+��
+,
+g
+�
+y(t)
+�
+=c sin
+�2π
+lm
+y(t)
+�
+,
+(64)
+where lx, ly ∈ R≥0 are the width and height of the mass
+m ∈ R>0, M =
+1
+3m(l2
+x + l2
+y) is the moment of intertia,
+fv ∈ R>0 the viscous friction, and d, k ∈ R>0 the damping
+and spring constant counteracting rotation at both ends of
+the mass. The function g
+�
+y(t)
+�
+is the cogging force and is
+assumed unknown, with lm ∈ R>0 the magnet pole pitch and
+c ∈ R>0 the cogging magnitude. Parameter values are listed
+0
+0.5
+1
+1.5
+2
+2.5
+3
+3.5
+4
+4.5
+-0.1
+0
+0.1
+0.2
+0
+0.5
+1
+1.5
+2
+2.5
+3
+3.5
+4
+4.5
+-100
+-50
+0
+50
+100
+0
+0.5
+1
+1.5
+2
+2.5
+3
+3.5
+4
+4.5
+-100
+-50
+0
+50
+100
+Fig. 6: References r1(k) and r2(k) (top window), and the
+generated feedforward input for r1(k) using PGNNs with
+λ = 0 (middle window) and λ = 0.07 (bottom window).
+0
+0.5
+1
+1.5
+2
+2.5
+3
+3.5
+4
+4.5
+-6
+-4
+-2
+0
+2
+4
+6
+10-4
+Fig. 7: Tracking error for reference r1(k) resulting from the
+different feedforward controllers.
+in Table II. The system (64) is controlled in closed–loop by
+a ZOH–discretized version of
+C(s) = 5 · 103 s + 4π
+s + 20π ,
+(65)
+which achieves a 1.22 Hz bandwidth.
+Training data generation: data is generated in closed–
+loop by sampling u(k) and y(k) at a frequency of 1 kHz,
+while exciting the system with the reference r(k) in the top
+window of Fig. 10, and adding a white noise with variance
+50 N 2 to the input u(k).
+Feedforward controllers: ZOH discretization of the trans-
+fer function in (64) gives na = nb = 4, nk = 0. The
+identified feedforward controllers are unstable, due to the
+nonminimum phase transfer function in the system (64). We
+test the following situations:
+1) No feedforward, i.e., uff(k) = 0;
+2) Physics–based feedforward with ZPETC stable in-
+
+0.02
+0.04
+0.06
+0.08
+0.1
+0.12
+0.14
+0.16
+0
+1
+2
+10-4
+0.02
+0.04
+0.06
+0.08
+0.1
+0.12
+0.14
+0.16
+0
+0.5
+1
+10-3
+Fig. 8: MAE of the tracking errors for r1(k) and r2(k)
+displayed in Fig. 7 with different velocities max
+�
+˙r(k)
+�
+.
+⋅
+Fig. 9: Rotating–translating mass with actuation and sensing
+on opposite sides of the centre of mass.
+version, i.e., (16), (17) with linear physical model
+fphy
+�
+θphy, φ(k)
+�
+= θT
+phyφ(k) and parameters identified
+according to (14) with MSE cost function (15);
+3) PGNN–based feedforward with NN dimensions n1 =
+16 and l = 1, using ZPETC stable inversion, i.e., (58)
+identified according to Lemma 5.2 which guarantees
+that the feedforward control input remains bounded;
+4) Physics–based feedforward with extended preview,
+i.e.,
+(16),
+(17)
+with
+linear
+physical
+model
+fphy
+�
+θphy, ˜φ(k)
+�
+=
+θT
+phy ˜φ(k),
+npw
+=
+20,
+and
+parameters identified according to (14) with MSE cost
+function (15);
+5) PGNN–based feedforward with NN dimensions n1 =
+16 and l = 1 and extended preview window, i.e., (56)
+with npw = 20 identified according to Lemma 5.1.
+We choose ˜Q = Q = I to find Θ in (54) for the extended
+preview PGNN (56) or in (60) for the stably inverted
+0
+0.5
+1
+1.5
+0
+0.05
+0.1
+0
+0.5
+1
+1.5
+2
+2.5
+3
+-1
+0
+1
+10-3
+2
+2.5
+3
+Fig. 10: Tracking error resulting from different feedforward
+controllers.
+TABLE III: MSE in [m2] of the tracking errors in Fig. 10.
+ 
+No FF 
+Lin.+ZPETC 
+PGNN+ZPETC 
+Lin.+PW 
+PGNN+PW 
+MSE 
+4.50 ⋅ 10−5 
+1.44 ⋅ 10−7 
+2.10 ⋅ 10−8 
+1.59 ⋅ 10−7 
+4.74 ⋅ 10−9 
+ 
+PGNN (58). Consider Lemma 5.2 as an example. We find P
+by solving A(˜bd)T PA(˜bd) − P + ˜Q = 0, which gives β =
+0.4982 from (53). Consequently, we use ˜bn in (59) obtained
+from ZPETC approximation and combine it with (60), to
+obtain ∥Lx∥2
+2 ≤ ∥403Luff∥2
+2 < 1.96 · 10−4, which directly
+translates into NN parameter constraints when using (49).
+Results: Fig. 10 shows the tracking error resulting from
+the aforementioned feedforward controllers for the reference
+shown in the top window. It is clear that the PGNN manages
+to significantly outperform the physics–based feedforward
+controller in terms of tracking error, as is confirmed by the
+mean–squared error (MSE) values listed in Table III.
+VII. CONCLUSIONS
+In this paper, we presented a generalized framework for
+nonlinear feedforward control using physics–guided neural
+networks. It was demonstrated that, under suitable assump-
+tions, the PGNNs can recover the inverse system dynamics
+exactly. Moreover, it was shown that the use of physical
+knowledge within the PGNN framework enabled tools to
+ensure improved training convergence with respect to the
+physical model, as well as providing a form of graceful
+degradation. A systematic method for analyzing input–to–
+state stability of PGNN–based feedforward controllers was
+provided, based on which the identification criterion was re-
+vised to impose ISS before training. The PGNN feedforward
+showed superior performance compared to physics–based
+feedforward controllers on a real–life industrial linear motor
+and a nonminimum phase simulation example. This shows
+that indeed, the developed PGNN–based feedforward con-
+troller design methodology offers a systematic and reliable
+solution for increasing performance of mechatronic systems
+with unknown, complex nonlinearities and parasitic effects.
+
+APPENDIX I
+PROOF OF PROPOSITION 4.1
+Proof:
+Substitution of the inverse dynamics (20) and
+PGNN (24) into the cost function (25) with ΛNN = 0 gives
+1
+N
+N−1
+�
+i=0
+�
+fphy(θ∗
+phy, φi) + g(φi) − fphy(θphy, φi)
+− fNN(θNN, φi)
+�2
++ λ
+����
+�0
+0
+0
+Λphy
+� �
+θ −
+� 0
+θ∗
+phy
+������
+2
+2
+.
+(66)
+Both terms in (66) are non–negative, such that the global
+minimum is attained for ˆθphy = θ∗
+phy (regularization term)
+and ˆθNN = θ∗
+NN (MSE term, after substitution of θphy = θ∗
+phy).
+APPENDIX II
+PROOF OF PROPOSITION 4.2
+Proof: The proof of (34) follows directly by observing
+that θ(j)
+L
+in (32) is the least squares solution of (14), (25)
+for the PGNN (30) given θ(j)
+NL. Since M(θ(j)
+NL) is non–
+singular, θ(j)
+L
+is unique, such that (34) holds with strict
+inequality if and only if θ(j)
+L
+̸= θL. Observe that, θ(j)
+L
+−
+θL = θ(j)
+L
+− M
+�
+θ(j)
+NL
+�−1M
+�
+θ(j)
+NL
+�
+θL ̸= 0 gives condition (35)
+after substitution of (32) and (33) and using nonsingularity
+of M(θ(j)
+NL).
+Secondly, (36) follows by observing that choosing θ(j)
+L
+̸=
+θL cannot decrease Vreg when Λ = 0. Correspondingly, (34)
+states that VMSE must decrease if (35) holds, such that
+VMSE
+��
+θ(j)
+NL
+θ(j)
+L
+�
+, ZN
+�
+≤ V
+��
+θ(j)
+NL
+θL
+�
+, ZN
+�
+= VMSE(θ∗
+phy, ZN),
+(67)
+and with strict inequality if (35) holds.
+APPENDIX III
+PROOF OF THEOREM 5.1
+Proof: The proof follows by showing that V
+�
+x(k)
+�
+=
+x(k)T Px(k) is an ISS–Lyapunov function as in Defini-
+tion 5.2. Condition (46) is satisfied with κ1
+�
+x(k)
+�
+=
+λmin(P)x(k)T x(k) and κ2
+�
+x(k)
+�
+= λmax(P)x(k)T x(k). We
+compute the difference V
+�
+x(k + 1)
+�
+− V
+�
+x(k)
+�
+to obtain
+V
+�
+x(k + 1)
+�
+− V
+�
+x(k)
+�
+= −x(k)T Qx(k)
++ 2C1BT PAx(k) + BT PBC2
+1,
+C1 := ˆaT φr(k + 1) + fNN
+�
+ˆθNN,
+�φr(k + 1)
+x(k)
+��
+.
+(68)
+For a term 2pT q and a ε ∈ R>0, we can complete the squares
+as:
+2pT q = εpT p−ε(p− 1
+εq)T (p− 1
+εq)+ 1
+εqT q ≤ εpT p+ 1
+εqT q.
+(69)
+We complete the squares (69) of 2C1BT PAx(k) in (68)
+using ε = βλmin(Q), β ∈ R>0, p = x(k), and q = AT PBC1
+to obtain
+V
+�
+x(k + 1)
+�
+− V
+�
+x(k)
+�
+≤ − (1 − β)λmin(Q)x(k)T x(k)
++ cβC2
+1,
+(70)
+with cβ := BT PB
+�
+I +
+1
+βλmin(Q)AAT P
+�
+B. Similarly, by
+substituting C1 in (70) and completing the squares (69) for
+2ˆaT φr(k)fNN using ε = β1, β1 ∈ R>0, p = fNN and q =
+ˆaT φr(k + 1), we obtain
+V
+�
+x(k + 1)
+�
+− V
+�
+x(k)
+�
+≤ −(1 − β)λmin(Q)x(k)T x(k)
++ cβ(1 + β1)fNN
+�
+ˆθNN,
+�φr(k + 1)
+x(k)
+��2
++ cβ(1 + 1
+β1
+)
+�
+ˆaT φr(k + 1)
+�2,
+(71)
+with β1
+∈
+R>0. Finally, substitution of the Lipschitz
+condition (48) with φB
+ff (k) = 0 and (50), and completing
+the squares (69) for 2LT
+uffx(k)LT
+r φr(k + 1) using ε = β2,
+β2 ∈ R>0, p = LT
+uffx(k), and q = LT
+r φr(k + 1), gives
+V
+�
+x(k + 1)
+�
+− V
+�
+x(k)
+�
+≤ −x(k)T �
+(1 − β)λmin(Q) − cβ·
+(1 + β1)(1 + β2)LuffLT
+uff
+�
+x(k) + φr(k + 1)T cβ·
+�
+(1 + β1)(1 + 1
+β2
+)LrLT
+r + (1 + 1
+β1
+)ˆaˆaT �
+φr(k + 1)
+=: −κ3
+�
+|x(k)|
+�
++ σ
+�
+|φr(k + 1)|
+�
+.
+(72)
+It is clear that κ3
+�
+|x(k)
+�
+is a K∞–function if the matrix
+between x(k)T and x(k) is positive definite, which reduces
+to the scalar condition (52) when using the Cauchy–Schwartz
+inequality,
+i.e.,
+�
+LT
+uffx(k)
+�2
+=
+x(k)T LuffLT
+uffx(k)
+≤
+LT
+uffLuffx(k)T x(k), and choosing β1, β2 arbitrary small.
+APPENDIX IV
+PROOF OF LEMMA 5.1
+Proof: The proof follows directly from Proposition 5.1
+and observing that limiting θ ∈ Θ ensures that (52) is
+satisfied by using the NN Lipschitz bound in (49).
+APPENDIX V
+PROOF OF LEMMA 5.2
+Proof:
+The proof follows by rewriting the stable
+inverted PGNN (58) into state–space form, and following the
+proof of Proposition 5.1 with the Lipschitz bound in (59).
+A (P, ˜Q) pair exists, since the stable inversion of G(˜b, q−1)
+gives a Schur A(˜b).
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+
diff --git a/ntFAT4oBgHgl3EQfdB3_/content/tmp_files/load_file.txt b/ntFAT4oBgHgl3EQfdB3_/content/tmp_files/load_file.txt
new file mode 100644
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+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf,len=1023
+page_content='Physics–guided neural networks for feedforward control  with input–to–state stability guarantees∗  Max Bolderman†,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content='1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content=' Hans Butler1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content='2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content=' Sjirk Koekebakker3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content=' Eelco van Horssen4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content=' Ramidin Kamidi2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content='  Theresa Spaan–Burke5,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content=' Nard Strijbosch5,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content=' and Mircea Lazar1  Abstract— Currently,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content=' there is an increasing interest in merg-  ing physics–based methods and artificial intelligence to push  performance of feedforward controllers for high–precision  mechatronics beyond what is achievable with linear feedforward  control.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content=' In this paper, we develop a systematic design procedure  for feedforward control using physics–guided neural networks  (PGNNs) that can handle nonlinear and unknown dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content='  PGNNs effectively merge physics–based and NN–based models,  and thereby result in nonlinear feedforward controllers with  higher performance and the same reliability as classical, linear  feedforward controllers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content=' In particular, conditions are presented  to validate (after training) and impose (before training) input–  to–state stability (ISS) of PGNN feedforward controllers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content=' The  developed PGNN feedforward control framework is validated  on a real–life, high–precision industrial linear motor used in  lithography machines, where it reaches a factor 2 improvement  with respect to conventional mass–friction feedforward.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content='  Index Terms— Feedforward control, neural networks, non-  linear system identification, high–precision mechatronics, linear  motors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content='  I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content=' INTRODUCTION  The field of high–precision mechatronics requires con-  tinuously innovating control methods to facilitate the ever–  increasing demands on both throughput as well as accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content='  For example, wafer scanners in lithography machines used  for semiconductor manufacturing [1] require sub–nanometer  position accuracy at velocities and accelerations exceeding  1  m  s  and 30  m  s2 , respectively, see [2, Chapter 9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content=' On a  similar note, the ability to increase the throughput while  decreasing the position error can allow for the use of  components that are manufactured with larger tolerances and  thereby improving the cost effectiveness when no further  accuracy and throughput improvements are required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content=' This  is an objective in, for example, the manufacturing industry  of printing applications using relatively low–cost stepping  motors [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content='  Feedforward control is a dominant actor in achieving this  high position accuracy, while feedback control predomi-  nantly concerns closed–loop stability and disturbance rejec-  This work is supported by the NWO research project PGN Mechatronics,  project number 17973.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content='  †Corresponding author: m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content='bolderman@tue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content='nl  1Eindhoven University of Technology, Groene Loper 19, Eindhoven,  5612 AP, The Netherlands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content='  2ASML, Veldhoven, The Netherlands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content='  3Canon Production Printing, St.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content=' Urbanusweg 43, Venlo, 5900 AE, The  Netherlands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content='  4Philips Engineering Solutions, High Tech Campus 34, Eindhoven, 5656  AE, The Netherlands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content='  5IBS Precision Engineering, ESP 201, Eindhoven, 5633 AD, The Nether-  lands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content='  tion [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content=' Robustness against varying references is achieved  by generating the feedforward input by passing the refer-  ence through a model of the inverse system dynamics, see,  e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content=', [5]–[7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content=' As a result, performance of these so–called  inversion–based feedforward controllers is limited by the  accuracy of the model of the inverse dynamics [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content='  Conventional feedforward control methods employ para-  metric models of the inverse system derived from physical  knowledge, which are often linear [5], [7], but can also be  linear–in–the–parameters (LIP) [6], [9] or nonlinear [10],  [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content=' Advantages of using physics–based models are the  interpretable nature of the models with robust performance  due to the reliance on physical knowledge, as well as a  systematic stability analysis based on linear system theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content='  Moreover, methods exist for constructing stable inverses  of linear nonminimum phase systems, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content=', systems that  have an unstable inverse, using approximate solutions or  non–causal filtering [12], [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content=' Additionally, the training  of linear and LIP models, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content=', offline optimization to find  the model parameters for which the model fits the data, is  typically a convex optimization problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content=' On the downside  however, using models derived from physical knowledge  yields structural model errors due to the intrinsic simplifi-  cations when deriving such models, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content=', when neglecting  parasitic effects present in real–life applications such as  electromagnetic distortions and complex friction forces [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content='  Therefore, more general model structures that can learn  nonlinear and unknown parasitic effects are needed for  improving performance of feedforward controllers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content='  Black–box neural network (NN) models are a good  candidate because of their universal approximation capa-  bilities [15], and have been already used in feedforward  control in, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content=', [16], [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content=' The efficient backpropagation  algorithm [18] enables training using large amounts of data,  which is required for high–precision mechatronic systems  due to their high sampling rates (> O(103) Hz).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content=' Currently,  however, the use of NN–based feedforward controllers in  practical applications remains limited, see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content=', [19], [20]  for experimental results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content=' This is explained by the fact that  using NNs comes with more difficult design challenges  compared to linear physics–based models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content=' For example,  NNs require a tedious hyperparameter tuning [21];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content=' the NN  training can get stuck in local minima;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content=' and NNs suffer from  poor extrapolation outside the training data [22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content=' Moreover,  validating stability of NN–based feedforward controllers is  non–trivial [23], [24], and the linear stable inversion tools  do not apply for the nonlinear NN models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content='  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content='08568v1  [eess.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content='SY]  20 Jan 2023  The above analysis suggests that a systematic method for  effectively merging physics–based models and NNs would  enable feedforward controllers with higher performance for  mechatronic systems with complex nonlinear dynamics due  to the presence of parasitic effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content=' In this paper we develop  a generalized feedforward control design methodology based  on physics–guided neural networks (PGNNs), which effec-  tively merge physics–based and NN–based models within  a common model class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content=' PGNNs, as described in this paper,  were originally developed and employed in feedforward con-  trol design in [25], and can be regarded as a generalization of  the parallel linear–NN model that has been used in nonlinear  system identification, see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content=', [26, Chapter 21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content=' The main  contribution of this paper is the development of a generalized  feedforward control design method based on PGNNs, with  the following key features:  1) Regularized training of PGNNs based on identi-  fied physical model parameters, to avoid competition  among the NN and physical layer in the PGNN;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content='  2) Regularized training of PGNNs based on the physical  model output to promote physical model compliance,  and enhance robustness to non–training data;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content='  3) Input–to–state stability (ISS) guarantees for nonlinear  PGNN–based feedforward controllers;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content='  4) Experimental validation on a real–life high–precision  industrial linear motor for lithography machines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content='  Remark 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content='1: Compared  to  previous  conference  pa-  pers [25], [27], we present the following original contri-  butions in this journal paper: (i) methods for regularized  training, including the NN weights, and methods for opti-  mal selection of the regularization parameter;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content=' (ii) graceful  degradation of the PGNN feedforward controller when it is  operated on conditions that were not present in the training  data;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content=' (iii) ISS guarantees for PGNN feedforward controllers,  which, in combination with stable inversion methods for  linear systems, enable the design of a stable nonlinear  feedforward controller for nonminimum phase systems;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content=' (iv)  novel, real–life experimental results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content='  The remainder of this paper is organized as follows:  Section II introduces inversion–based feedforward control,  followed by the problem statement in Section III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content=' The  methodology for PGNN feedforward controller design with  regularized training and graceful degradation is presented in  Section IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content=' Methods for testing and imposing ISS of PGNN  feedforward controllers are developed in Section V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content=' Sec-  tion VI demonstrates effectiveness of the PGNN feedforward  controller on a real–life coreless linear motor and a nonmin-  imum phase simulation example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content=' The main conclusions are  summarized in Section VII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content='  For streamlining exposition of the results, in this paper all  proofs are reported in Appendices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content='  II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content=' INVERSION–BASED FEEDFORWARD CONTROL  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content=' Feedback–feedforward control architecture  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content=' 1 displays a standard feedback–feedforward control  scheme, where u(t) is the control input, and y(t) the system  output, with time t ∈ R≥0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content=' A mechatronic motion system  typically has a dominant linear part, denoted by G(s) with  s the Laplace variable, and possible higher order modes  �  i Gi(s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content=' Additionally, g(·) represents nonlinear parasitic  effects that typically arise from manufacturing tolerances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content='  The goal is to minimize the tracking error e(k) := r(k) −  y(k), where r(k) is the reference, and y(k) is the measured  at discrete–time indices k ∈ Z≥0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content='  The control input u(t) is computed at discrete–time in-  stances k according to  u(k) = ufb(k) + uff(k),  (1)  where ufb(k) is the feedback, and uff(k) the feedforward  input.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content=' The zero–order–hold (ZOH) in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content=' 1 is a discrete–  to–continous (D2C) operator, which lets u(t) = u(k) for  t ∈ [kTs, (k + 1)Ts), with sampling time Ts ∈ R>0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content=' The  feedback input is of the form  ufb(k) = C(q)e(k),  (2)  where q is the forward–shift operator, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content=', e(k) = qe(k −  1), and C(q) a rational transfer function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content=' The design of the  feedforward controller is presented in the following sections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content='  Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content='1: In this paper we focus on the tracking prob-  lem, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content=', minimizing the output tracking error e(k) with  respect to a desired reference r(k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content=' Nevertheless, the methods  proposed in this work can be extended to reject known  disturbances acting on the closed–loop system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content='  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content=' Linear feedforward controller design  As an introductory example to feedforward controller  design, we consider a moving mass experiencing viscous  friction, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content=',  m¨y(t) = u(t) − fv ˙y(t),  t ∈ R≥0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content='  (3)  In (3), m ∈ R>0 is the mass, fv ∈ R>0 the viscous friction  coefficient, and u(t) and y(t) are respectively the force input  and position output.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content=' Application of the Laplace transform  to (3) gives  Y (s) = G(s)U(s) :=  1  ms2 + fvsU(s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content='  (4)  The moving mass in (4) is a simple case of the general sys-  tem displayed in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content=' 1, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content=', without higher order dynamics  �  i Gi(s) and nonlinear parasitic effects g(·).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content='  In order to derive a discrete–time feedforward controller,  we discretize the dynamics (4) using ZOH, which gives  y(k) = G(q)u(k) :=  b1q−1 + b2q−2  1 + a1q−1 + a2q−2 u(k),  (5)  where a1, a2, b1, b2  ∈  R are the discrete–time transfer  function coefficients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content=' Suppose that the feedforward yields  perfect tracking, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content=', e(k) = r(k) − y(k) = 0, then (2)  gives ufb(k) = 0, such that u(k) = uff(k) from (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content=' Corre-  spondingly, from (5) we compute the feedforward controller  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content=' 1: Feedback–feedforward control scheme for a system that is described by a nominal linear model G(s), possible higher  order dynamics �  i Gi(s), and nonlinear parasitic effects g(·) which can depend on derivatives of u and y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content='  as  uff(k) = G−1(q)r(k) = 1 + a1q−1 + a2q−2  b1q−1 + b2q−2  r(k)  = 1  b1  r(k + 1) + a1  b1  r(k) + a2  b2  r(k − 1) − b2  b1  uff(k − 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content='  (6)  The linear feedforward controller design directly extends  to systems with higher order dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content=' Then, (5) becomes  y(k) = G(q)u(k) := q−nk �nb  i=1 biq−i  1 + �na  i=1 aiq−i u(k),  (7)  with na ∈ Z≥0, nb ∈ Z>0 the order of the dynamics, and  nk ∈ Z≥0 the number of pure input delays.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content=' We denote a0 =  1 and, following the same reasoning as for (6), obtain the  feedforward controller corresponding to (7) as  uff(k) =  na  �  i=0  ai  b1  r(k+nk +1−i)−  nb  �  i=2  bi  b1  uff(k+1−i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content=' (8)  In order to implement the feedforward controller (8) we  require two assumptions:  1) Reference preview: future reference values up to r(k+  nk + 1) are known at time instant k;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content='  2) Stable inverse dynamics: G−1(q) is stable, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content=', has  poles within the unit circle, or, equivalently, G(q)  is minimum phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content=' This requires that the zeros of  �nb  i=1 biqnb−i are within the unit circle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content='  Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content='2: When G(q) is nonminimum phase, it is com-  mon practice to either approximate G(q) using a minimum  phase transfer function or to use non–causal filtering to  obtain a stable exact inverse of G(q), see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content=', [12] for an  overview.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content='  C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content=' Nonlinear feedforward controller design  The linear feedforward control framework can be extended  to deal with the nonlinearities g(·), by starting from a non-  linear correspondent of (7), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content=', a nonlinear autoregressive  exogeneous (NARX) representation, which is given as  y(k) = h  �  [y(k − 1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content=' , y(k − na),  u(k − nk − 1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content=' , u(k − nk − nb)]T �  ,  (9)  where h : Rna+nb → R is a nonlinear function that describes  the complete dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content=' We assume that there exists an  inverse representation of (9), such that, with a slight abuse  of notation, we can write  u(k) = h−1�  φ(k)  �  ,  (10)  where φ(k) := [y(k+nk +1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content=' , y(k+nk −na +1), u(k−  1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content=' , u(k − nb + 1)]T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content=' Following the same reasoning as  for the linear feedforward (5), we obtain the ideal nonlinear  feedforward controller as  uff(k) = h−1�  φff(k)  �  ,  (11)  with φff(k) := [r(k+nk +1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content=' , r(k+nk −na+1), uff(k−  1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content=' , uff(k − nb + 1)]T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content='  In general, the function h−1 in (11) is unknown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content=' Moreover,  the precision demanded by industry exceeds manufactur-  ing tolerances, which implies that a machine–specific h−1  needs to be found.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content=' For these reasons, a systematic data–  based approach for finding a model of the inverse system  dynamics h−1 would be desirable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content='  D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content=' Identification for feedforward control  We discuss the three main aspects to perform a direct  identification of the inverse dynamics (10): the data set, the  model class, and the identification criterion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content='  Data set: we consider the availability of a data set that is  generated on the system displayed in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content=' 1, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content=', satisfying  dynamics (9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content=' As a result, with a slight abuse of notation,  we have  ZN = {φ0, u0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content=' , φN−1, uN−1},  (12)  where φi := φ(i) and ui := u(i) for i ∈ {0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content=' , N − 1}  in (10) during the data generating experiment, where N ∈  Z>0 is the number of data points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content='  Model class: we require a model parametrization of the  inverse dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content='  Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content='1: A model parametrization of the inverse  dynamics is given as  ˆu  �  θ, φ(k)  �  = f  �  θ, φ(k)  �  ,  (13)  where ˆu  �  θ, φ(k)  �  is the prediction of the input u(k), θ ∈  Rnθ denotes the vector of parameters with nθ ∈ Z>0, and  f : Rnθ × Rna+nb → R is a user–defined function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content='  Identification criterion: the identification criterion defines  the best choice of parameters θ to have the model (13) fit the  inverse system dynamics (10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content=' Typically, the identification  criterion aims to minimize a cost function, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content=',  ˆθ = arg min  θ  V (θ, ZN),  (14)  such as the mean–squared error (MSE)  V (θ, ZN) = VMSE(θ, ZN) := 1  N  N−1  �  i=0  �  ui − ˆu(θ, φi)  �2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content='  (15)  Similar to (8) and (11), the feedforward controller is obtained  by computing the input u(k) that yields the output y(k) =  r(k) for the identified model, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content=', (13) with θ = ˆθ from (14),  such that  uff(k) = ˆu  �ˆθ, φff(k)  �  = f  �ˆθ, φff(k)  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content='  (16)  The model parametrization (13) is a crucial choice made  by the user.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content=' Conventional methods employ a model that is  derived from physical knowledge of the system, such that  f  �  θ, φ(k)  �  = fphy  �  θphy, φ(k)  �  ,  (17)  where θphy ∈ Rnθphy are the physical parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content=' For reasons  that become apparent in Section IV, we denote θ∗  phy as the  parameters of the physical model (17) identified according to  the MSE identification criterion (14), (15).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content=' As an alternative  to physics–based models (17), black–box NNs can be used  to parametrize the inverse dynamics (10), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content=',  ˆu  �  θNN, φ(k)  �  = fNN  �  θNN, φ(k)  �  = WL+1αL  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content=' α1  �  W1φ(k) + B1  ��  + BL+1,  (18)  where αl  :  Rnl  →  Rnl denotes the aggregation of  activation functions with nl  ∈  Z>0  the number of  neurons  in  layer  l  ∈  {0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content=' , L},  and  L  ∈  Z>0  the number of hidden layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content=' The parameters θNN  :=  [col(W1)T , BT  1 , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content=' , col(WL+1)T , BT  L+1]T are the concate-  nation of all weights Wl ∈ Rnl×nl−1 and biases Bl ∈ Rnl,  where col(Wl) stacks the columns of Wl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content=' A NN model (18)  can approximate any continuous nonlinear mapping on a  bounded set with arbitrary accuracy, provided that it consists  of a sufficient number of neurons and/or hidden layers [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content='  This concludes the required preliminary knowledge to define  the problem considered in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content='  III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content=' PROBLEM STATEMENT  To formally characterize the inherent limitation of a physi-  cal model (17), we define the corresponding structural model  errors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content='  Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content='1: Given the physical model (17), we define  the structural model error as  g  �  φ(k)  �  := h−1�  φ(k)  �  − fphy  �  θ∗  phy, φ(k)  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content='  (19)  Notice that Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content=' 1 displays the structural model error g  �  φ(k)  �  for the situation in which the forward dynamics are consid-  ered and the physical model is linear, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content=', fphy  �  θphy, φ(k)  �  =  θT  phyφ(k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content=' The inverse dynamics (10) can be rewritten into  u(k) = fphy  �  θ∗  phy, φ(k)  �  + g  �  φ(k)  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content='  (20)  In order to derive conditions for which the feedforward  controller (11) achieves zero tracking error, we define a set  of operating conditions R as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content='  Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content='2: Denote Φff ⊆ Rna+nb as all regressor  points φff(k) supplied to the feedforward controller (16) for  all references r(k) and all k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content=' Then, the set of operating  conditions R is defined as  R := Φff ∪ {φ0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content=', φN−1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content='  (21)  As an example, it is possible to obtain the operating condi-  tions R for the moving mass (3) by considering maxima and  minima on the position, velocity, and acceleration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content='  By comparing the inverse dynamics (10) with the feed-  forward input (11), we obtain the following condition for  perfect tracking  uff(k) = f  �ˆθ, φff(k)  �  = h−1�  φff(k)  �  ,  ∀ φff(k) ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content='  (22)  Consequently, due to the structural model errors (20),  condition (22) cannot be satisfied when using a physical  model (17).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content=' On the other hand, when using a NN model (18),  it is possible to find a sufficient number of hidden layers and  neurons per layer in combination with a set of parameters  θ∗  NN such that (22) is satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content=' In practice however, the  tedious hyperparameter tuning and non–convex optimization  can return suboptimal identified parameters ˆθNN that do not  even reach performance of the physics–based feedforward.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content='  Additionally, dangerous situations can arise when the data  set ZN does not cover R sufficiently well, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content=', as a result  of limitations on either time or safety there can be no training  data available for all relevant operating conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content=' Finally,  there is only limited research available for validating stability  of NN–based feedforward controllers, see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content=', [23], [24],  and no tools to enforce stability a priori.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content='  To solve the above issues of NN–based feedforward con-  trollers, in this paper, we develop a generalized feedforward  control design methodology based on PGNNs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content=' To this end,  we introduce the formal definition of a PGNN model class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content='  Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content='3: The PGNN model parameterization of the  inverse dynamics is given as  ˆu  �  θ, φ(k)  �  = fphy  �  θphy, φ(k)  �  + fNN  �  θNN, T  �  φ(k)  ��  , (23)  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content=' 2: Schematic overview of the physics–guided neural  network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content='  where θ = [θT  NN, θT  phy]T are the PGNN parameters, and T :  Rna+nb → Rn0 is an input transformation, with n0 ∈ Z>0  the number of NN inputs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content='  Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content='1: The input transformation T can be used to  improve the numerical properties of the PGNN training, as  well as to create physically relevant inputs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content=' As a practical  example, consider a moving mass and let φi = [yi+1, yi]T  with yi the measured position.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content=' Then, choosing T(φi) =  � yi+1−yi  Ts  , yi  �T with Ts ∈ R>0 the sampling time, can  improve numerical properties of the training when Ts is  small, such that yi+1 ≈ yi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content=' As a second example, let yi  be the measured rotation of a rotary motor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content=' Then, choosing  T(φi) =  � yi+1−yi  Ts  , mod(yi)  �T with mod(yi) the remainder  after division of yi by 2π can improve the training conver-  gence and extrapolation in terms of the rotation yi, since  T(·) imposes the rotational reproducible behaviour.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content='  Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content='2: The PGNN (23) imposes physical knowl-  edge by embedding a known physical model within a NN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content='  Note that this is different from physics–informed neural net-  works (PINNs) [28], [29], which impose physical knowledge  via a regularization term in the cost function that penalizes  deviations of the NN output with respect to the output of the  available physical model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content=' The use of PINNs for feedforward  control was explored in, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content=', [30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content=' It is also worth to point  out that, conforming to this definition, the PGNNs developed  in [31] are PINNs, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content=', the physical model is not embedded  within a NN therein.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content='  IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content=' PHYSICS–GUIDED NEURAL NETWORKS FOR  FEEDFORWARD CONTROL  In this section, we introduce a regularized cost function  for training the PGNN (24) to avoid competition between  the physical and NN layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content=' We demonstrate that, under  suitable assumptions on the model class, training data, and  training convergence, the PGNN recovers the inverse system  dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content=' Moreover, we consider the situation in which  these assumptions are violated and introduce methods to  ensure that i) the PGNN improves with respect to using  only the physical model in terms of data fit via an optimized  parameter initialization, and ii) we impose a form of graceful  degradation for the PGNN when operated at conditions that  were not present in the training data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content='  For ease of presentation, in the remainder of this paper we  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content=' 3: L–curve obtained by training the PGNN (24) accord-  ing to (14), (25) with Λphy = ΛNN = I for 20 values of  λ logarithmically spaced in [10−5, 10], and optimal choice  λ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content='07 (red circle).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content='  consider T  �  φ(k)  �  = φ(k), such that the PGNN (23) reduces  to  ˆu  �  θ, φ(k)  �  = fphy  �  θphy, φ(k)  �  + fNN  �  θNN, φ(k)  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content='  (24)  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content=' Regularized PGNN training and inverse system identifi-  cation  The PGNN (24) is generally overparameterized, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content=', the  NN can identify parts of the physical model which results  in a parameter drift during training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content=' Therefore, we employ a  regularized cost function  V (θ, ZN) = VMSE(θ, ZN) + λVreg(θ),  (25)  with λ > 0, and the regularization cost is  Vreg(θ) :=  ����  �ΛNN  0  0  Λphy  � �  θ −  � 0  θ∗  phy  ������  2  2  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content='  (26)  In (26), ΛNN, Λphy are matrices that define the relative  importance of the different regularization terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content=' Note that,  ΛNN relates to the standard L2 regularization for the network  weights and biases [26, Chapter 7], while Λphy solves the  overparameterization of the PGNN model (24).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content='  Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content='1: The hyperparameter λ in (25) relates the  importance of the data fit with the regularization costs, and  can be tuned using, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content=', the L–curve method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content=' The L–curve  displays the data fit (x–axis) and regularization cost (y–axis)  for varying values of λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content=' The optimal choice for λ is the point  where the curvature is largest [32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content=' An example is shown in  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content=' 3 with Λphy = ΛNN = I using data that is generated on a  coreless linear motor, which is discussed in Section VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content=' The  optimal choice is λ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content='07.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content=' Although the differences in data  fit VMSE in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content=' 3 seem minor, the minimal data fit is limited  by the noise that is present in the data set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content=' Correspondingly,  the effect of λ is enlarged when the noise is subtracted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content='  fphy  phy, Φ(k)0,b(k)  nT  b(k)NN, T(Φ(k)  fNNPhysics  network  gtlcec  nettratThe PGNN (24) recovers the inverse dynamics (10) when  the following assumptions are satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content='  Assumption 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content='1: There  exists  a  θ∗  NN  such  that  fNN  �  θ∗  NN, φ(k)  �  = g  �  φ(k)  �  for all φ(k) ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content='  Assumption 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content='2: For θA  NN ̸= θB  NN with fNN  �  θA  NN, φ(k)  �  ̸=  fNN  �  θB  NN, φ(k)  �  for some φ(k) ∈ R, it holds that  1  N  N−1  �  i=0  �  fNN(θA  NN, φi) − fNN(θB  NN, φi)  �2 > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content='  (27)  Assumption 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content='3: The minimization of (25) over θ yields  a global optimum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content='  Assumption 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content='1 dictates that there should exist a choice of  parameters for which the PGNN (24) recovers the original  inverse system (10), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content=', the system should be in the model  set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content=' Assumption 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content='2 relates to the persistence of excitation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content='  Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content='1 (PGNN consistency): Consider  the  PGNN  (24)  that  is  used  to  identify  the  inverse  dynamics (10) according to identification criterion (14)  with cost function (25) using ΛNN  = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content=' Suppose that  Assumptions 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content='1, 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content='2, and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content='3 hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content=' Then, the identified  PGNN parameters satisfy ˆθ = [ˆθT  phy, ˆθT  NN]T = [θ∗T  phy, θ∗T  NN]T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content='  Proof: See Appendix I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content='  From Assumption 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content='1 and (20), it is observed that the iden-  tified PGNN recovers the inverse dynamics for all operating  conditions, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content=',  fphy  �ˆθphy, φ(k)  �  +fNN  �ˆθNN, φ(k)  �  = h−1�  φ(k)  �  , ∀ φ(k) ∈ R,  (28)  such that the zero tracking error condition (22) is satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content='  Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content='2: A parallel linear–NN model structure was  also employed for feedforward control design in [33].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content='  Therein, an alternative regularization method based on or-  thogonal projection was developed to avoid the competition  between the linear and NN layers under the assumption that  g(·) is identically zero outside R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content='  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content=' Optimized PGNN parameter selection  When Assumption 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content='1 or 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content='3 is violated, recovering the  inverse dynamics is not guaranteed, or the training does  not find the optimal parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content=' The training, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content=', the op-  timization over θ of (25), typically employs a nonlinear  optimization scheme, that attains parameter estimates θ(j)  where j ∈ {0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content=' , J}, with J ∈ Z≥0 the number of solver  iterations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content=' The estimated parameter vector is chosen as  ˆθ = θ(j),  j = arg  min  j∈{0,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content=',J} V  �  θ(j), ZN�  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content='  (29)  In what follows, we will derive a specific choice for the  parameters θ(j) which achieve a smaller value of the cost  function (25) compared to using only the physical model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content='  For simplicity, we consider a LIP physical model (17),  i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content=', fphy  �  θphy, φ(k)  �  = θT  phyTphy  �  φ(k)  �  , and rewrite the  PGNN (24) according to  ˆu  �  θ, φ(k)  �  = θT  L φL  �  θNL, φ(k)  �  := θT  L  �  �  αl  �  θNL, φ(k)  �  1  Tphy  �  φ(k)  �  �  � ,  (30)  where θL  :=  [col(WL+1)T , BL+1, θT  phy]T  are the pa-  rameters  in  which  the  PGNN  (30)  is  linear,  and  θNL := [col(W1)T , BT  1 , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content=' , col(WL)T , BT  L]T , such that θ =  [θT  NL, θT  L ]T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content=' From (30), we observe that an equivalent of the  physical model (17) is obtained by selecting θL = θL :=  [0, 0, θ∗T  phy]T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content=' We rewrite the regularization term (26) into  Vreg(θ) =  ����  �ΛNL  Λ  ΛT  ΛL  � ��  θNL  θL  �  −  � 0  θL  ������  2  2  ,  (31)  where Λ, ΛT  NL = ΛNL, and ΛT  L  = ΛL are obtained by  selecting rows and columns of ΛNN and Λphy accordingly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content='  Then, given any parameter set θ(j)  NL, choose θ(j)  L  according to  θ(j)  L  =M(θ(j)  NL)−1  �  1  N  N−1  �  i=0  uiφL(θ(j)  NL, φi)  + λ  ��  Λ2  L + ΛT Λ  �  θL −  �  ΛT ΛNL + ΛLΛT �  θ(j)  NL  ��  ,  (32)  where M(θ(j)  NL) is given as  M(θ(j)  NL) := 1  N  N−1  �  i=0  φL(θ(j)  NL, φi)φL(θ(j)  NL, φi)T  + λ  �  Λ2  L + ΛT Λ  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content='  (33)  Note that (32) is valid only if M(θ(j)  NL) is nonsingular, which  yields a unique solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content=' Nonsingularity of M(θ(j)  NL) is also  referred to as the persistence of excitation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content='  Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content='2 (PGNN improves over physics):  Consider the PGNN (30) with θ(j)  NL given, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content=', initialized  randomly for j = 0 or attained during training for j ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content='  Suppose that M(θ(j)  NL) is nonsingular, and choose θ(j)  L  according to (32).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content=' Then, for the cost function (25), we have  V  ��  θ(j)  NL  θ(j)  L  �  , ZN  �  ≤ V  ��  θ(j)  NL  θL  �  , ZN  �  ,  (34)  with strict inequality if and only if  1  N  N−1  �  i=0  �  uiφL  �  θ(j)  NL, φi  �  − φL  �  θ(j)  NL, φi  �  φL  �  θ(j)  NL, φi  �T θL  �  − λ(ΛT ΛNL + ΛLΛT )θ(j)  NL ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content='  (35)  Moreover, if ΛNN is such that Λ = 0, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content=', the cross terms  between θNL and [W T  L+1, BL+1]T are not penalized, the  PGNN achieves a better data fit compared to the physical  model, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content=',  VMSE  ��  θ(j)  NL  θ(j)  L  �  , ZN  �  ≤ VMSE(θ∗  phy, ZN),  (36)  with strict inequality if (35) holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content='  Proof: See Appendix II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content='  Condition (35) is obtained by rewriting θ(j)  L  ̸= θL, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content=',  W (j)  L+1 and B(j)  L+1 are nonzero, and possibly θ(j)  phy ̸= θ∗  phy as  well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content=' This is satisfied if θ(j)  NL is such that there is a correlation  between the output of the final hidden layer αL(·) and the  structural model errors present in the data g(φi).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content='  Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content='3: Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content='2 can be directly extended for  PGNNs with physical models that are not LIP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content=' To see this,  rewrite the physical model  fphy  �  θphy, φ(k)  �  = θT  L,phyTphy  �  θNL,phy, φ(k)  �  ,  (37)  and  choose  θ(j)  L  =  [col(W (j)  L+1)T , B(j)  L+1, θ(j)T  L,phy]T  according  to  (32)  using  θ(j)  NL  =  [col(W (j)  1 )T , B(j)T  1  , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content=', col(W (j)  L )T , B(j)T  L  , θ∗  NL,phy  T ]T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content='  In practice, the optimized parameter selection is used after  training, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content=', update θ(j)  L  for j = J, during training for  each j = {0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content=', J}, or as an initialization for j = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content=' Note  that, from (29) and Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content='2, if (35) holds, we have  that  V (ˆθ, ZN) ≤ V (θ(0), ZN) < V (θ, ZN),  (38)  when (32) is used for initialization of θ(0)  L .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content='  C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content=' Enhancing PGNN robustness via regularized training  In general, there is no systematic method to validate  Assumption 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content='2 in practice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content=' For this reason, [34] gives as a  user guideline to sample the complete domain of interest, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content=',  the operating conditions R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content=' For reasons of time and safety, it  can be infeasible to design an experiment that achieves this.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content='  Correspondingly, we require a form of graceful degradation  for situations in which the PGNN is operated on conditions  that were not present in the training data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content=' Since there is no  data available for these conditions, the physical model is the  only source of information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content=' Therefore, we aim to have the  PGNN comply with the physical model via a regularization  term in the cost function, such that (25) becomes  V (θ, ZN) = VMSE(θ, ZN) + λVreg(θ) + γVphy(θ, ZE).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content=' (39)  In (39), γ ∈ R>0 is a regularization parameter, and the  regularization cost is  Vphy(θ, ZE) := 1  E  E−1  �  i=0  �  fphy(θ∗  phy, φE  i ) − ˆu(θ, φE  i )  �2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content=' (40)  The set ZE = {φE  0 , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content=' , φE  E−1} contains user–defined re-  gressor points for which we aim to have the PGNN comply  with the physical model, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content=', due to the absence of reliable  data in ZN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content=' Thereby, the aim is to have ZN ∪ ZE cover  the operating conditions R up to high accuracy, which  is done automatically by Algorithm 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content=' In Algorithm 1,  C(ζ, ZN, ZE) is an objective function that specifies the goal  of the optimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content=' For example, choosing φE  i to maximize  the minimum squared Euclidean distance with respect to all  regressor points, is achieved by choosing  C(ζ, ZN, ZE) :=  min  φ ∈ZN,ZE ∥φ − ζ∥2  2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content='  (41)  Algorithm 1 iterates until a stopping criterion is met, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content=', by  fixing a maximum number of points, or a minimum threshold  ϵ for the objective function (41).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content='  Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content='4: The results of Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content='2 can be ex-  tended directly to the cost function (39) by appropriately  Algorithm 1 Design algorithm for ZE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content='  Initialize ZE = {}, i = 0,  while i < E − 1 ∧ C  �  φ(i − 1), ZN, ZE�  > ϵ do  φE  i = arg maxζ∈R C  �  ζ, ZN, ZE�  ,  ZE = ZE ∪ φE  i ,  i = i + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content='  end while  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content='15  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content='1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content='05  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content='05  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content='1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content='15  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content='3  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content='1  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content='1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content='3  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content=' 4: Two–dimensional illustration of the regressor points  in the CLM data set ZN discussed in Section VI, and the  regressor points ZE designed following Algorithm 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content='  revising the computation of θ(j)  L  in (32), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content=', compute the  least squares solution of (39) instead of (25).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content='  Consider a mechanical system as in (3), where the operat-  ing conditions R are represented by a maximum position of  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content='15 m and velocity of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content='2 m  s , and neglect the acceleration  for the sake of simplicity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content=' Since it was deemed unsafe to  travel the full stroke during the data generating experiment,  the training data ZN describes R only in a limited range of  the position, see the blue crosses in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content=' Consequently,  application of Algorithm 1 generates the set ZE, see the  orange crosses in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content=' 4, such that ZN ∪ ZE cover the  complete operating conditions R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content='  V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content=' ISS OF PGNN–BASED FEEDFORWARD CONTROLLERS  Stability of a feedforward controller (16) is a prereq-  uisite for safe operation of the closed–loop system, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content=',  for bounded reference values r(k) the feedforward input  uff(k) should remain bounded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content=' For linear feedforward con-  trollers, stability is tested by ensuring that the poles of the  feedforward controller transfer function are within the unit  circle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content=' However, the nonlinear NN in the PGNN feedfor-  ward (16), (24) complicates the stability assessment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content=' For  the sake of presentation, we consider a PGNN (24) with  linear physical model, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content=', fphy  �  θphy, φ(k)  �  = θT  phyφ(k), and  highlight some further generalizations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content=' Next, to facilitate the  stability analysis of PGNN–based feedforward controllers,  we define  ˆa := [ˆa1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content=' , ˆana+1]T ,  ˆb := [ˆb1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content=' ,ˆbnb−1]T ,  (42)  such that ˆθphy = [ˆaT ,ˆbT ]T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content=' Moreover, we define  φr(k) := [r(k + nk + 1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content=' , r(k + nk − na + 1)]T ,  φuff(k) := [uff(k − 1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content=' , uff(k − nb + 1)]T ,  (43)  such that φff(k) = [φr(k)T , φuff(k)T ]T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content=' Note that, for a  linear feedforward controller (8), stability is determined only  by ˆb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content=' Then, by choosing x(k) := φuff(k), we obtain a state–  space representation of the PGNN (24) as  x(k + 1) =A(ˆb)x(k)  + B  �  ˆaT φr(k) + fNN  �  ˆθNN,  �  φr(k)  x(k)  ���  ,  (44)  where B  =  [1, O]T  ∈  Rnb−1, A(ˆb)  =  � ˆbT  I  O  �  ∈  R(nb−1)×(nb−1), and φr(k) is the external input.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content='  Because the state–space PGNN feedforward (44) has  φr(k) as an exogeneous input, it makes sense to pursue ISS  guarantees [35], which is a stronger property than nominal  stability and guarantees boundedness of the state x(k) for  any bounded reference φr(k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content=' First, we provide a general ISS  result for the PGNN state–space representation (44).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content=' After-  wards, we show how ISS of the resulting PGNN feedforward  controllers can be imposed a priori during the PGNN training  by adjusting the identification criterion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content='  We recall the following notions: a function κ : R≥0 →  R≥0 is a K–function if it is continuous, strictly increasing  and κ(0) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content=' It is a K∞–function if it is a K–function and  κ(s) → ∞ as s → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content=' A function η : R≥0 × R≥0 is a KL–  function if, for each fixed k ≥ 0, the function η(·, k) is a  K–function and for each fixed s ≥ 0, the function η(s, ·) is  decreasing and η(s, k) → 0 as k → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content='  Definition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content='1 (ISS [35]): A  system  x(k + 1)  =  f  �  x(k), φr(k + 1)  �  is ISS if there exist a KL–function  η : R≥0 × R≥0 → R≥0 and a K–function κ such that,  for each finite input sequence φr and each initial state  x(0) = ξ ∈ Rnb−1, it holds that  |x(k)| ≤ η  �  |ξ|, k  �  + κ  �  ∥φr∥  �  (45)  for each k ∈ Z+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content='  Remark 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content='1: If the state–space PGNN representation (44)  is ISS with respect to the external input φr(k + 1), we have  that:  1) x(k) remains bounded for bounded φr(k) and, conse-  quently, uff(k) remains bounded;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content='  2) x(k) → 0 for φr(k) → 0 and, consequently, uff(k) →  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content='  In order to analyze ISS of the PGNN feedforward con-  troller (44), we will use a continuous ISS–Lyapunov func-  tion, which implies ISS as shown in [35].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content='  Definition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content='2 (ISS–Lyapunov [35]): A continuous func-  tion V : Rn → R≥0 is called an ISS–Lyapunov function for a  discrete–time state–space system x(k+1) = f  �  x(k), φr(k)  �  if the following conditions hold:  1) There exists K∞–functions κ1, κ2 such that  κ1  �  |x(k)|  �  ≤ V  �  x(k)  �  ≤ κ2  �  |x(k)|  �  , ∀  x(k) ∈ Rnb−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content='  (46)  2) There exists a K∞–function κ3 and a K–function σ,  such that  V  �  x(k + 1)  �  − V  �  x(k)  �  ≤ −κ3(|x(k)|) + σ(|φr(k + 1)|),  ∀ x(k) ∈ Rnb−1, ∀ φr(k) ∈ Rna+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content='  (47)  Remark 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content='2: For all commonly applied activation func-  tions, the NN (18) is globally Lipschitz, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content=', there exists an  L := [LT  r , LT  uff]T such that  ���fNN  �ˆθNN, φA  ff (k)  �  − fNN  �ˆθNN, φB  ff (k)  ����  ≤ LT ��φA  ff (k) − φB  ff (k)  �� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content='  (48)  Using backpropagation, it is possible to find values for L,  e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content=',  LT = max  φff(k)  ∂fNN  �ˆθNN, φff(k)  �  ∂φff(k)  = max  φff(k)  ˆWl+1diag(α′  l) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content=' ˆW2diag(α′  1) ˆW1 ≤ Πl+1  i=1| ˆWi|,  (49)  where α′  l = ∂αl(x)  ∂x  ��  x=xl, with xl the input to NN layer l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content=' The  upperbound in (49) is obtained by substitution of diag(α′  l) =  I, which holds for tanh, ReLU, and several other activation  functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content='  Assumption 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content='1: The origin x(k) = 0 is an equilibrium  for the unexcited PGNN (44), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content=', φr(k) = 0, such that (44)  gives  fNN  �ˆθNN, 0  �  = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content='  (50)  Note that, we can introduce a coordinate transformation, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content=',  ζ(k) = x(k) + ε to satisfy Assumption 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content='  Assumption 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content='2: There exists a P ≻ 0 such that Q :=  P − A(ˆb)T PA(ˆb) ≻ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content='  Assumption 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content='2 requires that the parameters ˆb are such that  the physical model is stable, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content=', A(ˆb) is a Schur matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content=' A  (P, Q) pair satisfying Assumption 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content='2 is typically obtained  by choosing Q ≻ 0 and solving the discrete–time Lyapunov  equation to obtain P ≻ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content=' We denote λmin(Q) and λmax(Q)  as the smallest and largest eigenvalue of Q, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content='  Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content='1 (PGNN feedforward ISS): Consider  the  PGNN  feedforward  (16),  (24)  with  linear  physical  model, and its state–space representation (44).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content=' Suppose  that Assumptions 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content='1 and 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content='2 hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content=' Let (P, Q) satisfy  Assumption 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content='2, and define  cβ := BT P  �  I +  1  βλmin(Q)AAT P  �  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content='  (51)  Then, if there exists a β > 0 such that  LT  uffLuff < (1 − β)λmin(Q)  cβ  ,  (52)  the PGNN feedforward state–space representation (44) is  ISS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content='  Proof: See Appendix III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content='  Remark 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content='3: Most NN stability research is conducted for  systems operating in closed–loop with a NN–based feedback  controller, see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content=', [24], [36] which provide local stability  conditions via linearization and integral quadratic constraints,  respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content=' A global ISS result for a black box NN with  no physical model embedded is presented in [23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content='  Remark 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content='4: The value of β for which the right–hand side  in (52) is maximal, is  β =  1  λmin(Q)BT PB  �  − BT PAAT PB+  �  BT P  �  λmin(Q)I + AAT P  �  BBT PAAT PB  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content='  (53)  Note that, since B is a column, the square root and division  are scalar operations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content=' Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content=' (53) is obtained by setting the  derivative w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content=' β of the right hand side of (52) equal to  zero, and choosing the option for which β > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content='  Remark 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content='5: A similar result can be obtained for a PGNN  with a general nonlinear physical model, when a quadratic  Lyapunov function is available for the physical model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content='  The ISS condition (52) in Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content='1 can be validated  after training by using β in (53), the upperbound Luff =  �O,  I�  L in (49) for some (P, Q) pair.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content='  Recall that θ = [θT  NN, θT  phy]T , θphy = [aT , bT ]T for the  linear physical model, and θ∗  phy are the parameters θphy  identified using only the physical model (17) with MSE cost  function (15).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content=' With the aim to ensure before training that  the PGNN is ISS, we fix ˆb = b∗, and constrain the network  weights to satisfy the ISS condition (52).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content='  Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content='1 (Training imposed ISS – stable physical model):  Consider the PGNN feedforward controller with linear  physical  model,  such  that  it  admits  a  state–space  representation of the form (44).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content=' Suppose that Assumption 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content='1  holds, that a (P, Q) pair satisfying Assumption 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content='2 is  available, and choose β as in (53).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content=' Define the set  Θ :=  �  θ ∈ Rnθ  �����  �  b = b∗�  ∧  �����  �  ΠL  l=1|Wl|  � �0  I  � ����  2  2  < (1 − β)λmin(Q)  cβ  � �  ,  (54)  and train the PGNN according to identification criterion  ˆθ = arg min  θ ∈ Θ V  �  θ, ZN�  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content='  (55)  Then, the training returns an ISS PGNN feedforward con-  troller (16), (24).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content='  Proof: See Appendix IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content='  Assumption 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content='2 is violated when A(ˆb) is not Schur, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content=',  the system is nonminimum phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content=' There are two main  approaches to obtain a stable feedforward controller for  linear systems, see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content=', [12]:  1) Non–causal feedforward controller design;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content='  2) Stable approximate inversion techniques, such as  ZPETC, ZMETC, NPZ–ignore.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content='  In the remainder of this section, we demonstrate approaches  to assess and impose ISS of PGNN feedforward controllers  for nonminimum phase systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content='  The nonminimum phase behaviour can be dealt with by  extending the preview window of the PGNN, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content=', we adjust  the PGNN (24) to  ˆu  �  θ, ˜φ(k)  �  = [aT , bT ]  � ˜φy(k)  ˜φu(k)  �  + fNN  �  θNN,  � ˜φy(k)  ˜φu(k)  ��  ,  ˜φy(k) := [r(k + nk + npw + 1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content=', r(k + nk − na + 1)]T ,  ˜φu(k) := [u(k − 1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content=', u(k − nb + nus + 1)]T ,  (56)  where nus ∈ Z>0 are the number of unstable eigenvalues  of A(ˆb) of the original PGNN (24), and npw ∈ Z>0 is  the number of extended preview samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content=' The non–causal  PGNN (56) is then trained following Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content='  Remark 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content='6: In order to satisfy Assumption 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content='2, the pa-  rameters b∗ should be such that A(b∗) is Schur.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content=' This is  achieved either by choosing npw large enough, such that the  identification of θ∗  phy = [a∗T , b∗T ]T returns stable parameters  b∗, or by fixing b∗ to retain the stable eigenvalues of the  original A(b∗).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content='  As an alternative approach, it is possible to employ linear  stable approximation inversion tools on the linear part of the  dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content=' In order to see this, we rewrite the state–space  PGNN (44) into  uff(k) = G(ˆb, q−1)  �  ˆaT φr(k) + fNN  �  ˆθ,  �  φr(k)  φuff(k)  ���  ,  (57)  where G(ˆb, q−1)  :=  1  1−�nb  i=1 ˆbiq−i .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content=' Let  ˜G(˜b, q−1)  :=  �˜nn  b  i=0 ˜bn  i q−i  1−�˜nd  b  i=1 ˜bd  i q−i be a stable approximate of G(ˆb, q−1) obtained  via a user–preferred method which gives ˜nn  b , ˜nd  b ∈ Z≥0 and  ˜bn  i ,˜bd  i ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content=' Then, a PGNN with stable physical model is  obtained from (57) as  uff(k) = ˜G(˜b, q−1)  �  ˆaT φr(k) + fNN  �  ˆθ,  � φr(k)  φuff(k)  ���  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content='  (58)  If the stable approximation method creates terms with qd,  d ∈ Z>0 in the numerator of G(˜b, q−1) in (58), φuff(k) in the  NN can only contain feedforward inputs up until uff(k−d−1)  in order to compute the feedforward according (58).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content='  ISS of the stable approximated PGNN (58) is imposed  by training the unstable PGNN (57) according a revised  version of Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content=' To do so, we rewrite the stably  inverted PGNN (58) into state–space form using x(k) :=  [uff(k), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content=', uff(k − nb − ˜nn  b + 1)]T , and use the triangular  inequality with Lipschitz bound (48) to construct Lr+ and  Lx such that  �����  ˜nn  b  �  i=0  ˜bn  i fNN  �ˆθNN, φff(k − i)  �  ����� ≤ [LT  r+, LT  x ]  �φr+(k)  x(k)  �  , (59)  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content=' 5: Complete industrial coreless linear motor setup (top  window), with an enlarged view of the coreless linear motor  (bottom window).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content='  with φr+(k) = [r(k+nk +1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content=', r(k+nk −na − ˜nn  b +1)]T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content='  Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content='2 (Training imposed ISS – unstable physical model):  Consider the PGNN (24) with linear physical model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content='  Suppose that Assumption 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content='1 holds, and choose β as  in (53).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content=' Define  Θ :=  �  θ ∈ Rnθ  �����  �  b = b∗�  ∧  �  ∥Lx ∥2  2 < (1 − β)λmin( ˜Q)  cβ  � �  ,  (60)  with ˜Q = P − A(˜bd)T PA(˜bd) ≻ 0 for some P ≻ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content=' Then,  the stable inverted PGNN feedforward controller (58) trained  according to identification criterion (55) is ISS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content='  Proof: See Appendix V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content='  VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content=' EXPERIMENTAL AND SIMULATION VALIDATION  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content=' Real–life industrial coreless linear motor  Experimental setup: we consider the problem of position  control of the industrial coreless linear motor (CLM) dis-  played in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content=' 5, which was formerly part of the longstroke  actuation in a lithography machine.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content=' For research purposes,  the CLM is limited to exert forces up to 500 N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content=' The  CLM contains three coil sets, each consisting of three  coils connected in star configuration that are powered by  a three–phase power amplifier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content=' The system is controlled  by a dSPACE MicroLabBox that receives encoder position  measurements, computes the control input, and converts them  into current setpoints via a commutation algorithm to be  send to the power amplifiers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content=' The PC is used to program  software in Matlab/Simulink that is uploaded to the dSPACE  MicroLabBox.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content=' Relevant signals are accessed during run–  time using the dSPACE ControlDesk software.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content=' The coils are  cooled using water and the cooling unit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content=' Finally, the safety  programmable logic controller (PLC) cuts of the power to the  amplifiers when safety is at risk, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content=', when the stop button  is pushed, or when the CLM access doors are opened.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content='  System modelling: a physical model of the CLM displayed  in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content=' 5 is derived based on Newton’s second law, such that  m¨y(t) = u(t) − fv ˙y(t) − fcsign  �  ˙y(t)  �  ,  (61)  with m ∈ R>0 the mass, and fv ∈ R>0 and fc ∈ R>0  the viscous and Coulomb friction coefficient, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content='  The CLM is controlled in closed–loop by a ZOH discretized  version of  C(s) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content='1 · 107 s + 5  32π  s  s + 5  62π  s + 30π  1  s + 60π ,  (62)  which was tuned to achieve a 5 Hz bandwidth for the linear  model, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content=', (61) with m = 20 kg, fv = 150 kg  s , and fc = 0  N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content='  Training data generation: data is generated in closed–  loop by sampling u(k) and y(k) at a frequency of 1 kHz,  while exciting the system with the third order reference r1(k)  shown in the top window of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content=' 8 using 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content='1 m  s velocity, 4 m  s2  acceleration and 1000 m  s3 jerk, and a zero–mean white noise  with variance 50 N 2 added to the input.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content=' The data points are  visualized in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content='  Feedforward controllers: we discretize (61) using ˙y(t) =  q−q−1  2T  y(k) and assume the ZOH to yield half a sample delay,  i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content=', u(t) =  2  q+1u(k), to obtain na = 5, nb = 1, and nk = 2  as the orders in φ(k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content=' An encoder with precision d = 5 ·  10−6 m is used to measure position.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content=' Therefore, discrete–  time velocity approximations are measured at ˙y(k) = n d  2Ts ,  compared to ˙y(k) = n d  Ts when using backward or forward  Euler.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content='  In order to demonstrate the importance of the regulariza-  tion term in (25), we train two PGNN–based feedforward  controllers according to (14) with NN dimensions n1 = 16,  l = 1 (18) using ΛNN = 0, Λphy = I and λ = 0 and λ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content='07.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content='  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content=' 6 displays the resulting feedforward signals for a given  reference, including the parts created by the physical model  and the NN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content=' It is clear that, for λ = 0, there is a competition  between the physical and NN layer in the PGNN, which  is not present when the PGNN is trained with λ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content='07.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content='  Such a competition among layers is dangerous (not desired),  because it implies that the physical model does not learn the  underlying physics, and the resulting feedforward controller  will not have physics–induced extrapolation properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content='  In order to test tracking performance achieved by the  PGNN, we consider the following feedforward controllers:  1) Physics–based feedforward using physical model (61)  with the parameters identified according to (14), (15);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content='  2) PGNN–based feedforward with NN dimensions n1 =  16, l = 1 (18), using physical model (61) identified  according to (14), (25) with λ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content='07, and Λphy =  ΛNN = I;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content='  dSpace  MicroLabBox  Safety PLC  CLM  Power amplifiers  Cooling airco  PCTABLE I: MAE in [m] of the tracking errors in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content='  Physics  PGNN γ = 0  PGNN γ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content='1  MAE  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content='16 · 10−4  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content='35 · 10−4  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content='58 · 10−4  TABLE II: Parameter values of the rotating–translating mass  displayed in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content='  m  lx, ly  M  fv  k  d  lm  c  20  1  40  3  50  25·103  3  575  3  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content='05  1  kg  m  kgm2  kg  s  kg  s2  kg  s  m  kgm  s2  3) PGNN–based feedforward with graceful degradation  with NN dimensions n1 = 16, l = 1 (18), using  physical model (61) identified according to (14), (39)  with λ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content='07, Λphy = ΛNN = I, and γ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content='  Results: Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content=' 7 the tracking error e(k) resulting achieved  by the different feedforward controllers for r1(k) in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content='  In order to test robustness of the PGNN, we compute the  mean–absolute error (MAE)  MAE = 1  N  N−1  �  k=0  |e(k)|,  (63)  for two references r1(k) and r2(k) in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content=' 6 using different  velocities max  �  ˙ri(k)  �  = n 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content='025  m  s , i = {1, 2}, n =  {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content=', 6}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content=' Note that, since max  �  r2(k)  �  = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content='15 m, the  feedforward requires to extrapolate for r2(k), see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content='  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content=' 8 displays the resulting MAE values, and Table I lists  the MAE for r1(k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content='  For r1(k), both the PGNNs outperform the physics–based  feedforward in terms of the MAE for all tested velocities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content='  A significant drop in performance is observed when using  the PGNN with γ = 0 on reference r2(k), which is caused  by the PGNN extrapolating badly for r(k) > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content='1 m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content=' The  physics–based regularization term (40) prevents this, as seen  by the MAE of the PGNN with γ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content='  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content=' Nonminimum phase rotating–translating mass  System dynamics: we consider a translating–rotating mass  with force input u(k) and position output y(k) at opposite  sides of the centre of mass, see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content=' The dynamics are  given as  y(t) =  �  1  ms2 + fvs −  l2  y  Ms2 + 2lxds + 2lxk  � �  u(t) − g  �  y(t)  ��  ,  g  �  y(t)  �  =c sin  �2π  lm  y(t)  �  ,  (64)  where lx, ly ∈ R≥0 are the width and height of the mass  m ∈ R>0, M =  1  3m(l2  x + l2  y) is the moment of intertia,  fv ∈ R>0 the viscous friction, and d, k ∈ R>0 the damping  and spring constant counteracting rotation at both ends of  the mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content=' The function g  �  y(t)  �  is the cogging force and is  assumed unknown, with lm ∈ R>0 the magnet pole pitch and  c ∈ R>0 the cogging magnitude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content=' Parameter values are listed  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content='5  1  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content='5  2  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content='5  3  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content='5  4  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content='1  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content='1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content='2  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content='5  1  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content='5  2  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content='5  3  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content='5  4  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content='5  100  50  0  50  100  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content='5  1  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content='5  2  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content='5  3  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content='5  4  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content='5  100  50  0  50  100  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content=' 6: References r1(k) and r2(k) (top window), and the  generated feedforward input for r1(k) using PGNNs with  λ = 0 (middle window) and λ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content='07 (bottom window).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content='  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content='5  1  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content='5  2  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content='5  3  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content='5  4  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content='5  6  4  2  0  2  4  6  10-4  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content=' 7: Tracking error for reference r1(k) resulting from the  different feedforward controllers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content='  in Table II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content=' The system (64) is controlled in closed–loop by  a ZOH–discretized version of  C(s) = 5 · 103 s + 4π  s + 20π ,  (65)  which achieves a 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content='22 Hz bandwidth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content='  Training data generation: data is generated in closed–  loop by sampling u(k) and y(k) at a frequency of 1 kHz,  while exciting the system with the reference r(k) in the top  window of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content=' 10, and adding a white noise with variance  50 N 2 to the input u(k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content='  Feedforward controllers: ZOH discretization of the trans-  fer function in (64) gives na = nb = 4, nk = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content=' The  identified feedforward controllers are unstable, due to the  nonminimum phase transfer function in the system (64).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content=' We  test the following situations:  1) No feedforward, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content=', uff(k) = 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content='  2) Physics–based feedforward with ZPETC stable in-  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content='02  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content='04  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content='06  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content='08  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content='1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content='12  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content='14  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content='16  0  1  2  10-4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content='02  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content='04  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content='06  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content='08  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content='1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content='12  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content='14  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content='16  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content='5  1  10-3  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content=' 8: MAE of the tracking errors for r1(k) and r2(k)  displayed in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content=' 7 with different velocities max  �  ˙r(k)  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content='  ⋅  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content=' 9: Rotating–translating mass with actuation and sensing  on opposite sides of the centre of mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content='  version, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content=', (16), (17) with linear physical model  fphy  �  θphy, φ(k)  �  = θT  phyφ(k) and parameters identified  according to (14) with MSE cost function (15);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content='  3) PGNN–based feedforward with NN dimensions n1 =  16 and l = 1, using ZPETC stable inversion, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content=', (58)  identified according to Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content='2 which guarantees  that the feedforward control input remains bounded;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content='  4) Physics–based feedforward with extended preview,  i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content=',  (16),  (17)  with  linear  physical  model  fphy  �  θphy, ˜φ(k)  �  =  θT  phy ˜φ(k),  npw  =  20,  and  parameters identified according to (14) with MSE cost  function (15);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content='  5) PGNN–based feedforward with NN dimensions n1 =  16 and l = 1 and extended preview window, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content=', (56)  with npw = 20 identified according to Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content='  We choose ˜Q = Q = I to find Θ in (54) for the extended  preview PGNN (56) or in (60) for the stably inverted  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content='5  1  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content='5  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content='05  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content='1  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content='5  1  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content='5  2  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content='5  3  1  0  1  10-3  2  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content='5  3  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content=' 10: Tracking error resulting from different feedforward  controllers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content='  TABLE III: MSE in [m2] of the tracking errors in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content=' 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content='  No FF  Lin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content='+ZPETC  PGNN+ZPETC  Lin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content='+PW  PGNN+PW  MSE  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content='50 ⋅ 10−5  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content='44 ⋅ 10−7  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content='10 ⋅ 10−8  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content='59 ⋅ 10−7  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content='74 ⋅ 10−9  PGNN (58).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content=' Consider Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content='2 as an example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content=' We find P  by solving A(˜bd)T PA(˜bd) − P + ˜Q = 0, which gives β =  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content='4982 from (53).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content=' Consequently, we use ˜bn in (59) obtained  from ZPETC approximation and combine it with (60), to  obtain ∥Lx∥2  2 ≤ ∥403Luff∥2  2 < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content='96 · 10−4, which directly  translates into NN parameter constraints when using (49).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content='  Results: Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content=' 10 shows the tracking error resulting from  the aforementioned feedforward controllers for the reference  shown in the top window.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content=' It is clear that the PGNN manages  to significantly outperform the physics–based feedforward  controller in terms of tracking error, as is confirmed by the  mean–squared error (MSE) values listed in Table III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content='  VII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content=' CONCLUSIONS  In this paper, we presented a generalized framework for  nonlinear feedforward control using physics–guided neural  networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content=' It was demonstrated that, under suitable assump-  tions, the PGNNs can recover the inverse system dynamics  exactly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content=' Moreover, it was shown that the use of physical  knowledge within the PGNN framework enabled tools to  ensure improved training convergence with respect to the  physical model, as well as providing a form of graceful  degradation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content=' A systematic method for analyzing input–to–  state stability of PGNN–based feedforward controllers was  provided, based on which the identification criterion was re-  vised to impose ISS before training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content=' The PGNN feedforward  showed superior performance compared to physics–based  feedforward controllers on a real–life industrial linear motor  and a nonminimum phase simulation example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content=' This shows  that indeed, the developed PGNN–based feedforward con-  troller design methodology offers a systematic and reliable  solution for increasing performance of mechatronic systems  with unknown, complex nonlinearities and parasitic effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content='  APPENDIX I  PROOF OF PROPOSITION 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content='1  Proof:  Substitution of the inverse dynamics (20) and  PGNN (24) into the cost function (25) with ΛNN = 0 gives  1  N  N−1  �  i=0  �  fphy(θ∗  phy, φi) + g(φi) − fphy(θphy, φi)  − fNN(θNN, φi)  �2  + λ  ����  �0  0  0  Λphy  � �  θ −  � 0  θ∗  phy  ������  2  2  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content='  (66)  Both terms in (66) are non–negative, such that the global  minimum is attained for ˆθphy = θ∗  phy (regularization term)  and ˆθNN = θ∗  NN (MSE term, after substitution of θphy = θ∗  phy).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content='  APPENDIX II  PROOF OF PROPOSITION 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content='2  Proof: The proof of (34) follows directly by observing  that θ(j)  L  in (32) is the least squares solution of (14), (25)  for the PGNN (30) given θ(j)  NL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content=' Since M(θ(j)  NL) is non–  singular, θ(j)  L  is unique, such that (34) holds with strict  inequality if and only if θ(j)  L  ̸= θL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content=' Observe that, θ(j)  L  −  θL = θ(j)  L  − M  �  θ(j)  NL  �−1M  �  θ(j)  NL  �  θL ̸= 0 gives condition (35)  after substitution of (32) and (33) and using nonsingularity  of M(θ(j)  NL).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content='  Secondly, (36) follows by observing that choosing θ(j)  L  ̸=  θL cannot decrease Vreg when Λ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content=' Correspondingly, (34)  states that VMSE must decrease if (35) holds, such that  VMSE  ��  θ(j)  NL  θ(j)  L  �  , ZN  �  ≤ V  ��  θ(j)  NL  θL  �  , ZN  �  = VMSE(θ∗  phy, ZN),  (67)  and with strict inequality if (35) holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content='  APPENDIX III  PROOF OF THEOREM 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content='1  Proof: The proof follows by showing that V  �  x(k)  �  =  x(k)T Px(k) is an ISS–Lyapunov function as in Defini-  tion 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content=' Condition (46) is satisfied with κ1  �  x(k)  �  =  λmin(P)x(k)T x(k) and κ2  �  x(k)  �  = λmax(P)x(k)T x(k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content=' We  compute the difference V  �  x(k + 1)  �  − V  �  x(k)  �  to obtain  V  �  x(k + 1)  �  − V  �  x(k)  �  = −x(k)T Qx(k)  + 2C1BT PAx(k) + BT PBC2  1,  C1 := ˆaT φr(k + 1) + fNN  �  ˆθNN,  �φr(k + 1)  x(k)  ��  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content='  (68)  For a term 2pT q and a ε ∈ R>0, we can complete the squares  as:  2pT q = εpT p−ε(p− 1  εq)T (p− 1  εq)+ 1  εqT q ≤ εpT p+ 1  εqT q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content='  (69)  We complete the squares (69) of 2C1BT PAx(k) in (68)  using ε = βλmin(Q), β ∈ R>0, p = x(k), and q = AT PBC1  to obtain  V  �  x(k + 1)  �  − V  �  x(k)  �  ≤ − (1 − β)λmin(Q)x(k)T x(k)  + cβC2  1,  (70)  with cβ := BT PB  �  I +  1  βλmin(Q)AAT P  �  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content=' Similarly, by  substituting C1 in (70) and completing the squares (69) for  2ˆaT φr(k)fNN using ε = β1, β1 ∈ R>0, p = fNN and q =  ˆaT φr(k + 1), we obtain  V  �  x(k + 1)  �  − V  �  x(k)  �  ≤ −(1 − β)λmin(Q)x(k)T x(k)  + cβ(1 + β1)fNN  �  ˆθNN,  �φr(k + 1)  x(k)  ��2  + cβ(1 + 1  β1  )  �  ˆaT φr(k + 1)  �2,  (71)  with β1  ∈  R>0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content=' Finally, substitution of the Lipschitz  condition (48) with φB  ff (k) = 0 and (50), and completing  the squares (69) for 2LT  uffx(k)LT  r φr(k + 1) using ε = β2,  β2 ∈ R>0, p = LT  uffx(k), and q = LT  r φr(k + 1), gives  V  �  x(k + 1)  �  − V  �  x(k)  �  ≤ −x(k)T �  (1 − β)λmin(Q) − cβ·  (1 + β1)(1 + β2)LuffLT  uff  �  x(k) + φr(k + 1)T cβ·  �  (1 + β1)(1 + 1  β2  )LrLT  r + (1 + 1  β1  )ˆaˆaT �  φr(k + 1)  =: −κ3  �  |x(k)|  �  + σ  �  |φr(k + 1)|  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content='  (72)  It is clear that κ3  �  |x(k)  �  is a K∞–function if the matrix  between x(k)T and x(k) is positive definite, which reduces  to the scalar condition (52) when using the Cauchy–Schwartz  inequality,  i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content=',  �  LT  uffx(k)  �2  =  x(k)T LuffLT  uffx(k)  ≤  LT  uffLuffx(k)T x(k), and choosing β1, β2 arbitrary small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content='  APPENDIX IV  PROOF OF LEMMA 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content='1  Proof: The proof follows directly from Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content='1  and observing that limiting θ ∈ Θ ensures that (52) is  satisfied by using the NN Lipschitz bound in (49).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content='  APPENDIX V  PROOF OF LEMMA 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content='2  Proof:  The proof follows by rewriting the stable  inverted PGNN (58) into state–space form, and following the  proof of Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content='1 with the Lipschitz bound in (59).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content='  A (P, ˜Q) pair exists, since the stable inversion of G(˜b, q−1)  gives a Schur A(˜b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
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+page_content=' Ahlawat, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content=' O’Brien, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content=' Simonelli, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content='-T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
+page_content=' Teng, and Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFAT4oBgHgl3EQfdB3_/content/2301.08568v1.pdf'}
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+1
+VQNet 2.0: A New Generation Machine Learning
+Framework that Unifies Classical and Quantum
+Huanyu Bian, 1,7 Zhilong Jia,1,4,6 Menghan Dou, 1,4 Yuan Fang, 1,4 Lei Li, 1,4 Yiming Zhao, 1,4 Hanchao Wang,
+1,4 Zhaohui Zhou, 1,4 Wei Wang,1,4 Wenyu Zhu,1,4 Ye Li, 1,4 Yang Yang, 3,5 Weiming Zhang,2,∗ Nenghai Yu, 2,∗
+Zhaoyun Chen,5,∗ Guoping Guo5,6,∗
+1Origin Quantum Computing Company Limited, Hefei 230026, China
+2CAS Key Laboratory of Electromagnetic Space Information(University of Science and Technology of China),
+Hefei 230026, China
+3School of Electronics and Information Engineering(Anhui University), Hefei, Anhui 230601, China
+4Anhui Engineering Research Center of Quantum Computing, Hefei 230026, China
+5Institute of Artificial Intelligence( Hefei Comprehensive National Science Center), Hefei, Anhui 230026, China
+6CAS Key Laboratory of Quantum Information (University of Science and Technology of China), Hefei 230026,
+China
+7Aerospace Information Research Institute, Chinese Academy of Sciences, Beijing 100190, China
+Abstract—With the rapid development of classical and quan-
+tum machine learning, a large number of machine learning
+frameworks have been proposed. However, existing machine
+learning frameworks usually only focus on classical or quantum,
+rather than both. Therefore, based on VQNet 1.0, we further
+propose VQNet 2.0, a new generation of unified classical and
+quantum machine learning framework that supports hybrid
+optimization. The core library of the framework is implemented
+in C++, and the user level is implemented in Python, and
+it supports deployment on quantum and classical hardware.
+In this article, we analyze the development trend of the new
+generation machine learning framework and introduce the design
+principles of VQNet 2.0 in detail: unity, practicality, efficiency,
+and compatibility, as well as full particulars of implementation.
+We illustrate the functions of VQNet 2.0 through several basic
+applications, including classical convolutional neural networks,
+quantum autoencoders, hybrid classical-quantum networks, etc.
+After that, through extensive experiments, we demonstrate that
+the operation speed of VQNet 2.0 is higher than the comparison
+method. Finally, through extensive experiments, we demonstrate
+that VQNet 2.0 can deploy on different hardware platforms, the
+overall calculation speed is faster than the comparison method. It
+also can be mixed and optimized with quantum circuits composed
+of multiple quantum computing libraries.
+Index Terms—Machine Learning Framework; Quantum Ma-
+chine Learning; Classical Machine Learning
+I. INTRODUCTION
+With the rapid development of quantum computing, quan-
+tum machine learning, which combines quantum computing
+with classical machine learning, has also developed rapidly [1],
+[2], [3], [4], [5], [6], [7], [8]. Quantum computing is a
+new computing method that uses quantum properties such
+as superposition and entangled states to perform calculations.
+Compared with classical computing, quantum computing has
+two advantages: 1) Theoretically, quantum computers far
+surpass classical computers in calculation speed and storage
+efficiency. 2) With the continuous size reduction of classical
+electronic components, the development of classical computers
+Table I: Classification of machine learning frameworks.
+Classical machine learning
+Quantum machine learning
+Caffe
+�
+�
+TensorFlow
+�
+�
+Theano
+�
+�
+PyTorch
+�
+�
+TensorFlow Quantum
+�
+�
+PennyLane
+�
+�
+Paddle Quantum
+�
+�
+VQNet 2.0
+�
+�
+is about to face the "Size Effect" and other issues that hinder
+continued rapid developments. Quantum machine learning
+will further improve the processing capabilities of big data
+by using the computational efficiency of quantum computers
+far surpassing classical computers, combined with machine
+learning algorithms that have developed rapidly in the era of
+big data.
+Meanwhile, the increasing interest in classical machine
+learning and quantum machine learning has led to an emer-
+gence of quantum machine learning frameworks. Although
+many previous works have developed a variety of machine
+learning frameworks, there is still a lack of a new generation
+of machine learning frameworks to support both classical
+machine learning and quantum machine learning, as shown
+in Tab. I. To solve the above-mentioned lack of the new gen-
+eration machine learning framework, based on VQNet 1.0 [9],
+we propose VQNet 2.0, a Python framework that unifies clas-
+sical machine learning and quantum machine learning, which
+means 1) unifying classical and quantum machine learning
+algorithms, and 2) unifying classical and quantum computers.
+Its software stack is shown in Fig.2. In addition to the above
+unification of machine learning algorithms, VQNet 2.0 also
+achieves the unification of classical computers and quantum
+computers: 1) Classical nodes are deployed on classical com-
+arXiv:2301.03251v1  [quant-ph]  9 Jan 2023
+
+2
+Figure 1: VQNet 2.0’s software stack. At the top of the stack
+is the data to be processed. VQNet 2.0 has the ability to
+process classical data (images, text, etc.) and quantum data
+(quantum circuits, quantum operators, etc.). Next is the model
+API of VQNet 2.0. Then comes the computing layers and
+the differentiators. The computing layers include the classical
+layers and the quantum layers, and the differentiators can
+perform hybrid automatic differentiation for networks built
+on any computing layer. After that, the quantum circuit is
+accomplished by QPanda, and the entire network can be
+operated on a classical computer, quantum simulator, or QPU
+according to its function.
+puters, similar to classical machine learning frameworks. 2)
+We design Qpanda [10], a software package that connects
+hardware and quantum computing. Through Qpanda, quantum
+nodes are deployed on the quantum processing unit (QPU) or
+quantum simulator.
+The rest of the paper is organized as follows. In Section II,
+we describe the background of our proposed framework and
+the trend of the machine learning framework. In Section III,
+we describe our design principles. In Section IV, we give the
+pipeline of our VQNet 2.0, which can support both classical
+machine learning and quantum machine learning. Section V
+shows the implementation details of our proposed framework.
+Section VI and Section VII give the extensibility of case
+studies which contain both classical and quantum applications.
+Section VIII shows the experimental setup and experimental
+results of our VQNet 2.0 compared with other state-of-the-art
+frameworks. Section IX draws the conclusion.
+II. BACKGROUND
+A. VQNet 1.0
+Quantum computing is a completely new method that can
+revolutionize computers. In order to explore the possibility of
+hybrid implementation of the variable quantum algorithm and
+classical machine learning framework, Chen et al.
+[9] pro-
+posed VQNet 1.0, a quantum-classical hybrid machine learn-
+ing architecture. VQNet 1.0 designed two kinds of quantum
+operators (QOPs): qop and qop_pmeasure . Through making
+these two QOPs support forward and backward propagation
+at the same time, VQNet 1.0 could achieve model training.
+Furthermore, it is possible to use a gradient-based optimizer
+for network optimization in VQNet 1.0. Chen et al. used
+VQNet 1.0 to construct a variety of quantum machine learning
+algorithms, including QAOA, VQE algorithm, quantum classi-
+fier, and quantum circuit learning, and they all achieved good
+performance.
+Although VQNet 1.0 effectively connected machine learn-
+ing and quantum algorithms, it only supports a few types
+of quantum machine learning algorithms like QAOA, VQE
+algorithm, quantum classifier and quantum circuit learning.
+With the rapid development of machine learning algorithms,
+it is urgent to propose a new generation of machine learning
+frameworks based on VQNet 1.0.
+B. Trend of Machine Learning Framework
+Compared with the previous single-class machine learning
+frameworks, we summarize the development trends of the
+next-generation machine learning frameworks:
+First of all, classical machine learning become popular in re-
+cent years, and the boom period of quantum machine learning
+begins. So far, a large number of classical machine learning
+frameworks, such as Caffe[11], TensorFlow[12], Theano[13],
+PyTorch[14], and quantum machine learning frameworks,
+such as TensorFlow Quantum[15], PennyLane[16], Paddle
+Quantum[17], have emerged. These frameworks meet the
+needs of researchers and industrial production for mature
+frameworks, lower the threshold for research, and further
+promote the development of classical quantum machine learn-
+ing. However, with the maturity of classical machine learning
+algorithms and the continuous progress of quantum machine
+learning, quantum machine learning will inevitably become a
+supplement to classical machine learning. The interaction of
+these two kinds of algorithms make researchers and developers
+an urgent need for a new generation of machine learning
+framework that can unify classical and quantum.
+Secondly, various gradient-based optimization methods play
+an extremely important role in the training and conver-
+gence of classical deep learning models. Because automatic
+differentiation[18] greatly reduces the programming difficulty
+of classical machine learning, it has become the core software
+design paradigm of today’s machine learning frameworks.
+However, the existing automatic optimization[19] algorithms
+are difficult to directly apply to the training of quantum ma-
+chine learning models: the gradients of parameterized quantum
+circuits in quantum machine learning must be evaluated using
+quantum algorithms, so compared to the gradients of clas-
+sical machine learning, the existing automatic differentiation
+algorithm may not work. One type of algorithm in quantum
+machine learning is hybrid quantum-classical networks, which
+requires quantum computing and classical computing to coop-
+erate with each other, so hybrid optimization[16] that can
+output hybrid calculation results is the core function of the
+new machine learning framework design.
+Third, Python[20], [21] is an object-oriented, interpreted,
+general, and open source scripting programming language. It
+has the characteristics of simple and easy to use, low learning
+cost, and high writing efficiency, but it has the corresponding
+disadvantage of slower running speed. Python has been used
+in a large number of machine learning frameworks such as
+
+Classical Data
+Quantum Data
+VONet2.0 Models
+VQNet 2.0
+Classical Layers
+Quantum Layers
+Differentiators
+VQNet 2.0 Classical Execution Engine
+QPanda
+Classical Computer
+QPU
+Quantum Simulator3
+TensorFlow, PyTorch, etc. Meanwhile, C++[22], [23] is an
+object-oriented compiled language, which is convenient for the
+direct operation of the memory. C++ not only has the practical
+characteristics of the efficient operation of computers, but also
+is committed to improving the programming quality of large-
+scale programs and the problem description ability of pro-
+gramming languages. It is often used in industrial applications
+for image processing. Correspondingly, it has shortcomings
+such as high code complexity and being unfriendly to users.
+Therefore, in order to avoid these two language shortcomings
+while inheriting the advantages of both, it is inevitable to
+separate the user level in various languages from the
+core library[12]. The core library is implemented in C++,
+which improves operating efficiency and portability; while the
+user-level implementation is implemented in Python, which
+improves user-friendliness and reduces learning costs and
+difficulties.
+Finally, in order to use general-purpose massively parallel
+hardware (such as GPU) for computational acceleration, exist-
+ing machine learning frameworks can be deployed on classical
+hardware such as CPU, GPU, and TPU[24], [25], [26]. The
+rapid increase in the computing power of quantum computers
+has provided a solid foundation for the development of quan-
+tum machine learning. At the same time, the current shortage
+of quantum computing power makes quantum simulation on
+classical computers the first choice for most researchers to
+conduct verification experiments. Therefore, it is inevitable
+that the new generation of machine learning frameworks can
+be deployed on more types of hardware such as classical
+computer, QPU, and quantum simulator.
+Based on the above trends, VQNet 2.0 provides a unified
+classical and quantum machine learning framework that sup-
+ports hybrid optimization. The core library of the framework is
+implemented in C++, the user-level is implemented in Python,
+and it supports deployment on both quantum and classical
+hardware.
+III. DESIGN PRINCIPLES
+There are four main principles behind the design of VQNet
+2.0:
+A. Be United
+VQNet 2.0 aims to achieve multiple unities of classical and
+quantum: 1) The unification of classical machine learning and
+quantum machine learning, that is, compared with frameworks
+such as Qiskit and Pennylane, VQNet 2.0 does not need to
+import Torch, Tensorflow, or other huge classical machine
+learning libraries. VQNet 2.0 comes with a complete neural
+network module, including classical and quantum machine
+learning, optimizer, loss function, etc. It is easy to quickly
+build hybrid quantum-classical machine learning models. 2)
+The deployment of classical computers and quantum com-
+puters is unified, that is, compared with frameworks such as
+TensorFlow and TensorFlow quantum, VQNet 2.0 can deploy
+classical machine learning models on classical computers,
+and deploy quantum machine learning models on QPU and
+quantum simulators.
+B. Put Practicality First
+VQNet 2.0 strives to use the framework to build models,
+train networks, and deploy multiple machines as simple and
+efficient as possible. In order to bring maximum practicality,
+it is necessary to realize the following functions including but
+not limited to: 1) Friendly interface. VQNet 2.0 implements a
+wealth of quantum machine learning models and interfaces for
+quickly building quantum machine learning models. VQNet
+2.0 performs well in constructing these quantum machine
+learning algorithms. The interface definition is similar to the
+syntax of commonly used machine learning frameworks, and
+users can quickly get started. 2) Automatic differentiation. The
+realization of automatic differentiation can help VQNet 2.0’s
+users get rid of tedious mathematical derivation and code,
+further improving the degree of automation and efficiency of
+using VQNet 2.0. 3)
+Dynamic computation graph. VQNet 2.0 uses the dynamic
+computation graph mechanism, that is, the construction and
+calculation of the calculation graph are performed at the same
+time, which reduces the difficulty of debugging and which also
+improves the friendliness of programming.
+C. Provide Sufficient Performance
+As a general machine learning framework, VQNet 2.0 needs
+to provide sufficient performance, but not at the expense
+of practicality. Compared with quantum machine learning
+frameworks such as Paddle Quantum, Mindspore[27], etc.,
+VQNet 2.0 uses QPanda as the quantum computing module,
+it is possible to build more fully functional and more efficient
+quantum circuits on classical and quantum computers, and
+achieve faster forward and backward speed.
+D. A Good Framework Embraces the Rest
+As a new generation of machine learning framework,
+VQNet 2.0 is designed to be compatible with other quantum
+computing libraries. In order to maximize compatibility with
+Python-based frameworks that can have any configuration
+system, VQNet 2.0 is designed with compatibility: A com-
+patibility layer is designed to call other frameworks, which
+is convenient for users to take the quantum circuits described
+by other frameworks as the black box for optimization calcu-
+lation, facilitate cross model cooperation, and is suitable for
+different business and research applications.
+IV. PIPELINE OF VQNET 2.0
+Here, we give an abstract overview of the computational
+steps involved in the general training and testing of machine
+learning models in VQNet 2.0.
+A. Dataset Preprocessing
+1) Classical Data: Usually, classical data comes from
+Internet or user-built datasets. In VQNet 2.0, it is easy to
+download commonly used public data sets by using the func-
+tions in
+pyvqnet.data , similar to the operations in PyTorch
+and TensorFlow.
+
+4
+……
+……
+…
+N1
+1
+𝑁2
+1
+𝑁m1
+1
+…
+Nm2
+2
+…
+Nmn
+n
+N1
+2
+N2
+2
+N2
+n
+N1
+n
+Dataset 
+Preprocessing
+data
+Result
+Calculation Node
+Evaluate
+Cost Function
+Optimization
+Figure 2: Abstract pipeline for VQNet 2.0.
+2) Quantum Data:
+Since it is currently impossible to
+import quantum data from the outside into a QPU, users
+need to specify the quantum circuit that generates quantum
+data, while encoding the classical data into the quantum
+state. There are mainly two existing encoding methods: 1)
+Encoding features into the basic state, which means that
+after the classical data is converted into integer form, the
+binary expansion is expressed as a direct product state, and
+this encoding method can be realized by using function
+pyvqnet.qnn.template.BasicEmbeddingCircuit .2) Encod-
+ing features into the rotation angles, which refers to the
+rotation angle of encoding classical data into a qubit, and
+this encoding method can be realized by using the function
+pyvqnet.qnn.template.AngleEmbeddingCircuit . 3) Encod-
+ing features into the amplitude vector, which refers to
+the amplitude vector of converting classical data into a
+quantum state through a quantum gate operation, and this
+encoding method can be realized by using the function
+pyvqnet.qnn.template.AmplitudeEmbeddingCircuit .
+B. Calculation Node
+After the data is processed, it can be sent to calculation
+nodes. In VQNet 2.0, calculation nodes include classical nodes
+and quantum nodes. In the network built by VQNet 2.0, any
+number of classical nodes and quantum nodes can be used to
+connect in any way. It is worth noting that if the number of
+quantum nodes selected is 0, the network built is a classical
+machine learning model, and if the number of classical nodes
+and quantum nodes is not 0, it is a hybrid quantum-classical
+network.
+1) Classical
+Node:
+Classical
+node
+refers
+to
+various
+network components that process classical data. VQNet
+2.0
+includes
+a
+variety
+of
+common
+basic
+building
+blocks,
+including
+convolutional
+layers pyvqnet.nn.conv ,
+pooling
+layers pyvqnet.nn.pooling ,
+and
+normlization
+layers pyvqnet.nn.batch_norm ,
+etc.
+Various
+layers
+can
+extract features of different dimensions from the original
+data, and the combination of different features can finally
+achieve amazing performance.
+2) Quantum Node: The quantum node is responsible for
+calculating quantum data.
+QuantumLayer simulate a param-
+eterized quantum circuit and get the measurement result. It
+inherits from
+Module , so that it can calculate gradients of
+circuits parameters, and trains variational quantum circuits
+model or embeds variational quantum circuits into hybrid
+quantum and classical model. The user defines the function
+qprog_with_meansure as a parameter, it needs to include
+the quantum circuit defined by QPanda: generally includes
+the quantum circuit’s coding circuit, evolution circuit, and
+measurement operation. This class can be embedded in the
+hybrid quantum-classical machine learning model, and the
+objective function or loss function of the hybrid quantum-
+classical model can be minimized through the gradient descent
+method.
+It is particularly worth noting that, in addition to the
+different calculations, there is another huge difference between
+quantum nodes and classical nodes. If quantum nodes are
+required to output classical data, measurement operations are
+required. Quantum measurement refers to the interference of
+the outside world to the quantum system to obtain the required
+information. The measurement gate uses the Monte Carlo
+method of measurement. After the qubit is measured, the
+measurement result will be stored in the classical register.
+VQNet 2.0 uses
+pyvqnet.qnn.measure
+for measurement
+operations.
+C. Loss Function
+After passing through the calculation node, the loss function
+needs to be calculated next. The loss function selection de-
+pends on the completion of the task. It is especially important
+to note that the calculation of the existing loss function needs
+to be performed on a classical computer but not a quantum
+
+5
+device, so if the network is built with quantum nodes as
+the last network layer, it must be measured before the loss
+calculation. VQNet 2.0 builds common loss functions in class
+pyvqnet.nn.loss .
+D. Optimization
+After setting the loss function, the network parameters are
+updated through back propagation, that is, model training is
+performed. Common optimization methods are mostly based
+on gradient descent. VQNet 2.0 constructs common opti-
+mization strategies in the class
+pyvqnet.optim.optimizer .
+As mentioned earlier, if there is no quantum node in the
+model involved in the calculation, automatic differentiation
+can be directly used to calculate the gradient and perform
+back propagation. If the model is a hybrid quantum-classical
+network, the hybrid optimization strategy[16] needs to be used
+to train the network.
+V. IMPLEMENTATION OF VQNET 2.0
+A. Be United
+1) Unification of Classical and Quantum Machine Learn-
+ing: In order to achieve the unity of classical and quan-
+tum in one machine learning framework, VQNet 2.0 de-
+fines an abstract class
+pyvqnet.nn.module.Module , which
+is the base class of all neural network modules. All
+calculation nodes are designed to be sub-class of class
+pyvqnet.nn.module.Module , so that they can be used for
+automatic differential calculation.
+Module can also contain
+other Modules, allowing to nest them in a tree structure. In
+VQNet 2.0, it is easy to assign the sub-modules as regular
+attributes:
+class Model(Module):
+def __init__(self):
+super(Model, self).__init__()
+self.conv1 = pyvqnet.nn.Conv2d(1, 20, (5,5))
+self.conv2 = pyvqnet.nn.Conv2d(20, 20, (5,5)
+)
+def forward(self, x):
+x = pyvqnet.nn.activation.relu(self.conv1(x)
+)
+return pyvqnet.nn.activation.relu(self.conv2
+(x))
+Sub-modules assigned in this way will be registered.
+2) Unification of Classical and Quantum Machine: VQNet
+2.0 not only supports the unification of quantum and classical
+machine learning but also supports operations on classical
+computers and quantum computers. It should be noted that dif-
+ferent calculation nodes have different hardware requirements:
+1) For classical nodes, the tools provided by VQNet 2.0 can
+be directly operated on the classical computer. 2) For quantum
+nodes, VQNet 2.0 can accomplish real quantum hardware
+interaction and quantum simulator interaction through QPanda.
+For the classical node, since the calculation of classical
+modules can only be implemented on the classical computer,
+there is no need for additional specifications in VQNet 2.0.
+By default, all classical nodes are run in a classical computer
+and the gradient is obtained through classical methods.
+Due to the current lack of quantum computer resources,
+quantum simulator that simulates the operating environment
+of QPU on classical computers has become one choice of
+most scientific researchers. In order to improve the general
+capabilities of the framework, for quantum node, VQNet 2.0
+defines a class
+QuantumLayer . Users can set the parameter
+machine_type
+in
+QuantumLayer
+according to their own
+computing resources and call the corresponding interface pro-
+vided by QPanda to use QPU or quantum simulator to conduct
+quantum calculations. Because different quantum simulators
+have different simulation effects, VQNet 2.0 provides five
+different kinds of simulators to realize different simulation en-
+vironments, namely full-amplitude quantum simulator, single-
+amplitude quantum simulator, partial-amplitude quantum sim-
+ulator, tensor network quantum simulator, and noise inclusive
+quantum simulator.
+At the same time, in order to enable researchers to better
+simulate the running results of QPU on the simulator when the
+calculation resources of QPU are short, VQNet 2.0 adds the
+noise of QPU to the quantum simulator. VQNet 2.0 adds quan-
+tum mapping based on the topology between QPU’s qubits to
+the noise simulator, so as to simulate more real experimental
+results. The simulation of the noise-containing quantum simu-
+lator is closer to QPU. VQNet 2.0 can customize the supported
+logic gate types and customize the noise model supported by
+the logic gates. The existing supported quantum noise models
+are defined in QPanda. VQNet 2.0 uses NoiseQuantumLayer
+to define an automatic micro-classification of QPU. The
+user defines function qprog_with_measure as the parameter,
+which needs to include the quantum circuit defined by QPanda,
+and also needs to pass in the parameter
+noise_set_config ,
+using the QPanda interface to set the noise model.
+B. Practicality Centric Design
+1) Friendly Interface: The core element to achieve interface
+friendliness is to design for the interface. The friendliness of
+the interface definition not only facilitates the maintenance
+of VQNet 2.0 itself, but also avoids excessive changes to
+the upper application layer, reducing the cost and difficulty
+of users. First, in order to provide a more convenient and
+concise interface, VQNet 2.0 shields the coordination details
+of multiple interfaces, and provides a coarse-grained interface
+for upper-layer applications. That is, the user only needs
+to define the quantum circuit and network structure. After
+constructing the forward propagation path, VQNet 2.0 can
+automatically constitute the back propagation and automatic
+optimization. Secondly, VQNet 2.0 follows the conventions
+of most frameworks and uses scene-oriented interface nam-
+ing methods, such as naming the class that performs one-
+dimensional convolution operations as class Conv1D to further
+enhance the friendliness of the interface. Finally, for interface
+parameters, VQNet 2.0 implements the principle of minimum
+granularity, that is, to ensure that all parameters passed are
+used by the interface.
+2) Dynamic Computation Graph:
+Similar to PyTorch,
+VQNet 2.0 also uses the dynamic computation graph. Compu-
+tation graphs are directed acyclic computation graphs used to
+
+6
+describe operations. They are composed of two main elements:
+Node which represents data, and Edge which represents oper-
+ations. In VQNet 2.0, predefined classical and quantum nodes,
+such as convolutional layer and quantum layer, are also nodes
+in the computation graph. Edge is the operation performed by
+each node, QTensor is the basis for constructing the computa-
+tion graph and the computation graph forms the structural basis
+of forward and backward propagation. Dynamic graph means
+that the operation and construction of machine learning model
+are established at the same time, that is, the value of nodes can
+be calculated according to the order of forward propagation,
+and then the computation graph is constructed on the basis of
+these nodes, and at last the graph constructed by the classical
+and quantum nodes is used to get the gradient of
+QTensor .
+The advantage of the dynamic computation graph is that it is
+flexible and easy to debug.
+C. Implementation with Fabulous Performance
+How to implement an efficient machine learning framework
+has always been a great challenge for researchers and devel-
+opers. In order to improve the efficiency of the new generation
+machine learning framework, in addition to optimizing classi-
+cal algorithms and classical computers, two aspects need to be
+considered: 1) the interaction between classical and quantum
+machine learning algorithms, and 2) the interaction between
+classical and quantum computers. In order to solve the above
+problems, VQNet 2.0 makes improvements in two aspects: 1)
+open-source QPanda, which is a software package specially
+designed to perform QPU operations and also simulate quan-
+tum circuits on classical computers. 2)
+Unified structure.
+The classical and quantum nodes are unified in design.
+QPanda. Three high-performance simulations are proposed
+in QPanda, namely 1) OpenMP multi-thread optimization.
+Using OpenMP instructions to execute functions in parallel can
+effectively execute the iterations in the function and effectively
+reduce the calculation time. In QPanda, the backends of all
+simulators running large quantum circuit calculations are par-
+allelized to maximize the workload of each core and minimize
+the overhead caused by multithreading. 2) GPU optimization.
+GPU has higher memory bandwidth than CPU. Because GPU
+has a large number of processing cores, it also has higher peak
+performance than CPU. In QPanda, when the GPU is detected,
+the GPU is used to accelerate the quantum simulation. 3)
+Supercomputer simulation. Supercomputers represent the peak
+of classical computing power in the world, and quantum com-
+puting simulations are based on classical computers. QPanda
+proposes to distribute tasks on multiple calculation nodes on
+the supercomputer platform ("Shenwei Light of Taihu Lake").
+The operation control core runs the program instruction set one
+by one and calls the operation core to perform core computing
+tasks, that is, to improve computing efficiency through a two-
+level parallel computing method.
+Unified structure. 1)Unified data structure. Similar to
+machine learning frameworks such as Torch and Paddle,
+in VQNet 2.0, the data structure is uniformly defined as
+QTensor .
+QTensor is the basic data structure of quantum
+circuits and neural networks. It supports multiple operations
+and realizes their derivative functions. When performing var-
+ious operations required for quantum and classical machine
+learning calculations, data can be stored by creating QTensor .
+It can be created using the interface of class
+Creation . In
+addition, QTensor can use Numpy, a popular matrix scientific
+computing package, to load data conveniently. 2) Unified class
+inheritance. In VQNet 2.0, the classes of quantum and classical
+machine learning models need to be inherited from the abstract
+class
+Module to perform autograd calculations.
+D. A Good Framework Embraces the Rest
+VQNet 2.0 hybrid quantum-classical optimization can
+support not only the quantum circuits of QPanda but
+also
+the
+quantum
+circuits
+composed
+of
+other
+quan-
+tum computing frameworks (such as Cirq, Qiskit, etc.).
+VQNet 2.0 provides a quantum circuit computing inter-
+face
+Compatiblelayer
+for automatic differentiation. The
+parameters of
+Compatiblelayer
+need to be passed in a
+class, which defines the third-party library quantum circuit,
+as well as its operation and measurement function
+run .
+By using
+Compatiblelayer , the input of quantum circuits
+and automatic differentiation of parameters can be imple-
+mented by VQNet 2.0. Class Compatiblelayer is an abstract
+wrapper to use other framework’s quantum circuits(such as
+qiskit.QuantumCircuit in Qiskit,
+cirq.Circuit in Cirq)
+to forward and backward in the form of VQNet 2.0. The
+users need to define quantum circuits in the
+forward() and
+backward() functions.
+VI. EVALUATION
+In this section, we first introduce the experimental settings,
+and then give the deployment results of VQNet 2.0 on different
+hardware to support its heterogeneous capabilities. After that,
+we support its efficiency by comparing computing efficiency
+with different frameworks under different conditions. Finally,
+an example of VQNet 2.0’s support for Qiskit is given to show
+the compatibility of the framework.
+A. Experiment Setup
+Dataset. For all subsequent experiments, we use the common
+MNIST data set in computer vision as the experimental data
+set. The MNIST data set is the handwritten character library.
+There are 70,000 images, of which 60,000 are the training set
+and 10,000 are the test set.
+Model. For classical machine learning deployment, we use
+the most popular neural network: convolutional neural net-
+works(CNN). The CNN model consists of two convolution
+layers and one fully connected layer. Each convolution layer
+performs the max-pooling operation after the relu activa-
+tion function. For quantum machine learning deployment,
+we use quantum autoencoder for efficient compression of
+quantum data as pure quantum networks, and hybrid quantum-
+classical neural networks as hybrid networks. To verify the
+compatibility of VQNet 2.0, we use IBM’s Qiskit quantum
+computing library for quantum machine learning tasks. All
+running instances are shown in the Appendix.
+
+7
+B. Deployment
+In order to show that our proposed framework can be de-
+ployed normally on different hardware devices, the following
+shows the deployment results of VQNet 2.0 on the classical
+computer and QPU.
+1) Classical Machine Learning Deployment: Firstly, we
+give the deployment results of the classical convolutional
+neural network implemented by VQNet 2.0 on the classical
+computer. The running instance is as described in Sec.A. As
+shown in Fig.3, we can find that the classical convolutional
+neural network implemented by VQNet 2.0 converges after 10
+epochs when training on the classical computer, and the test
+accuracy converges after the same number of epochs, which
+proves the deployment ability of VQNet 2.0 on the classical
+computer.
+2) Quantum Machine Learning Deployment: This section
+shows the deployment results of VQNet 2.0 on the Wuyuan2
+QPU which is a quantum computing hardware platform with
+64 superconducting qubits created by Origin Quantum com-
+pany based on Xmon architecture. According to whether there
+are quantum nodes or not, quantum machine learning can
+be divided into pure quantum networks and hybrid quantum-
+classical networks. Their running instances are shown in Sec.B
+and Sec.C respectively. It is worth noting that in this section,
+the quantum nodes of the network operate on the quantum
+simulator, and the classical nodes are still deployed on the
+classical computer.
+C. Operation Efficiency
+By comparing our proposed VQNet 2.0 with the most
+popular quantum machine learning framework, we can fully
+understand the performance of VQNet 2.0. It is worth noting
+that since the VQNet 2.0 we proposed is the first unified
+framework for quantum and classical machine learning, the
+comparison methods in this section are the combination of the
+classical machine learning framework and quantum computing
+library. For example, PyTorch + Qiskit means using PyTorch
+to call a quantum circuit generated by Qiskit composes the
+completed HQCNN. For the fairness of comparison, we use
+different frameworks to train the same network model on
+the MNIST dataset: HQCNN, the loss function is cross-
+entropy loss, the optimizer is Adam, the learning rate reduction
+strategy is the same, and they are all trained for 20 epochs
+After stopping.
+We first show the relationship between the loss of VQNet
+2.0 and different comparison methods training the same net-
+work and the training epoch in Fig.3. Obviously, the loss of
+VQNet 2.0 decreases more smoothly and converges faster. As
+shown in Tab.V, the training and testing time of VQNet 2.0
+is much shorter than the comparison method. For example,
+for the overall training time, PyTorch + Qiskit requires 62.5s,
+PyTorch+QPanda requires 19.7s, and VQNet 2.0 only requires
+12.9s, which is 20% of PyTorch + Qiskit, 65% of PyTorch
++ QPanda. Thus, the efficient performance of our proposed
+framework is verified.
+a
+b
+c
+d
+Figure 3: Loss and accuracy comparison of each machine
+learning framework in training and testing process.
+
+8
+Table II: Classical Machine Learning Deployment Loss
+Mxnet
+Paddle
+PyTorch
+tensorflow
+VQNet 2.0
+LeNet
+0.077
+0.001
+0.00
+0.002
+0.046
+AlexNet
+0.050
+0.230
+0.00
+0.059
+0.040
+VGG
+0.022
+0.014
+0.004
+0.176
+0.137
+ResNet-18
+0.568
+0.112
+0.082
+0.116
+0.118
+Table III: Accuracy of Quantum Machine Learning Deployment
+Mxnet
+Paddle
+PyTorch
+tensorflow
+VQNet 2.0
+LeNet
+0.975
+1.00
+0.990
+0.989
+0.988
+AlexNet
+0.983
+0.927
+1.00
+0.991
+0.999
+VGG
+0.978
+0.976
+0.984
+0.98
+0.997
+ResNet-18
+0.845
+0.979
+0.973
+0.975
+0.971
+Table IV: Loss and Accuracy of Quantum Machine Learning Deployment on Quantum simulator
+Paddle Quantum
+PyTorch+Pennylane
+TensorFlow Quantum
+VQNet 2.0
+HQCNN
+1.305/0.733
+1.682/0.767
+1.726/0.767
+1.209/0.733
+Table V: The duration of HQCNN is composed of different machine learning frameworks. BP stands for back propagation, FP
+stands for forward propagation. All data are in seconds.
+PyTorch + Qiskit
+PyTorch + QPanda
+VQNet 2.0
+Total training duration
+62.55
+19.75
+12.91
+Single data training duration (×10−3)
+15.38
+4.66
+3.36
+Quantum node BP duration(×10−4)
+80.39
+8.68
+7.07
+Quantum node FP duration(×10−4)
+41.83
+5.35
+4.66
+Network FP duration(×10−3)
+4.92
+1.29
+1.04
+Single data testing duration(×10−3)
+5.01
+1.31
+1.19
+Dataset testing duration(×10−1)
+5.19
+1.61
+1.19
+D. Compatibility
+In order to show the compatibility of VQNet 2.0, we choose
+the quantum circuit built by Qiskit as a black box and send it
+to VQNet 2.0 for optimization. The running instance is shown
+in Sec.D.
+The relationship between training and testing loss is shown
+in Fig.4. We can find that although VQNet 2.0 does not need
+to know the details of the quantum circuit composed by Qiskit,
+it can also carry out hybrid optimization and converge quickly.
+Experiments verify the compatibility of VQNet 2.0 with other
+quantum computing libraries.
+VII. CONCLUSION AND FUTURE WORK
+In this paper, on the basis of VQNet 1.0, we propose the first
+classical and quantum unified machine learning framework
+named VQNet 2.0. It supports both classical and quantum
+machine learning, and supports deployment on classical com-
+puters and QPU. VQNet 2.0 puts practicability in the first place
+through the design of the friendly interface, automatic differ-
+entiation, dynamic computation graph, and so on. Secondly,
+it provides superb performance by accomplishing a unified
+structure and quantum computing function based on QPanda.
+Finally, it can be used with other quantum computing libraries
+to provide better compatibility. Experiments prove that VQNet
+2.0 can achieve better operating speed compared with the
+comparison method.
+Figure 4: Loss of VQNet 2.0 uses the quantum circuit com-
+posed of Qiskit.
+In the future, we plan to continue to improve the speed
+and practicality of VQNet 2.0 while supporting the latest
+developments in classical and quantum machine learning.
+VIII. ACKNOWLEDGMENTS
+This work was supported in part by Anhui provin-
+cial major science and technology projects under Grant
+202203a13010003, the National Natural Science Founda-
+
+9
+tion of
+China under
+Grant 12034018,
+Innovation
+Pro-
+gram for Quantum Science and Technology under Grant
+2021ZD0302300 and by the Natural Science Foundation of
+China under Grant 62272003.
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+APPENDIX
+RUNNING INSTANCES
+A. Classical Convolutional Neural Network for Image Recog-
+nition
+1) Background: Neural networks are part of artificial intel-
+ligence research. The most popular neural networks are con-
+volutional neural networks(CNN). CNNs have achieved great
+success in many research fields, such as speech recognition,
+image recognition, image segmentation, natural language pro-
+cessing. Convolutional neural networks are mainly composed
+of these types of layers: input layer, convolutional layer, ReLU
+layer, pooling layer, and fully connected layer, etc.
+2) Implementations: As mentioned earlier, the first step
+of the VQNet 2.0 process is data set preprocessing. In this
+example, we use the common MNIST data set in computer
+vision as the experimental data set. The MNIST data set is
+the handwritten character library. There are 70,000 images, of
+which 60,000 are the training set and 10,000 are the test set:
+def load_mnist(dataset="training_data", digits=np.
+arange(2), path="..//..//data//MNIST_data"):
+import os, struct
+from array import array as pyarray
+if dataset == "training_data":
+fname_image = os.path.join(path, ’train-
+images.idx3-ubyte’).replace(’\\’, ’/’)
+fname_label = os.path.join(path, ’train-
+labels.idx1-ubyte’).replace(’\\’, ’/’)
+elif dataset == "testing_data":
+fname_image = os.path.join(path, ’t10k-
+images.idx3-ubyte’).replace(’\\’, ’/’)
+fname_label = os.path.join(path, ’t10k-
+labels.idx1-ubyte’).replace(’\\’, ’/’)
+else:
+raise ValueError("dataset must be ’
+training_data’ or ’testing_data’")
+Once the dataset is ready, we start building the model.
+The model consists of two convolution layers and one full
+connection layer. Each convolution layer performs the max-
+pooling operation after the relu activation function.
+class CNN(Module):
+def __init__(self):
+super(CNN, self).__init__()
+self.maxpool4 = MaxPool2D([2, 2], [2, 2],
+padding="valid")
+self.maxpool2 = MaxPool2D([2, 2], [2, 2],
+padding="valid")
+self.conv5 = Conv2D(input_channels=1,
+output_channels=32, kernel_size=(3, 3), stride
+=(1, 1), padding="valid")
+self.conv6 = Conv2D(input_channels=32,
+output_channels=32, kernel_size=(3, 3), stride
+=(1, 1), padding="valid")
+self.fc3 = Linear(input_channels=800,
+output_channels=10)
+self.Relu2 = F.ReLu()
+def forward(self, x):
+x = self.maxpool2(self.Relu2(self.conv5(x)))
+x = self.maxpool4(self.Relu2(self.conv6(x)))
+x = tensor.flatten(x,1)
+x = self.fc3(x)
+return x
+In this example, we set the loss function as cross-entropy
+loss, and the optimizer is a common Adam optimization. If
+
+11
+we set the maximum number of epochs and batch size, we
+can start training the whole model.
+model = CNN()
+model.train()
+loss_func = SoftmaxCrossEntropy()
+model_hybrid = model
+optimizer_hybrid = Adam(model_hybrid.parameters(),
+lr=0.001)
+for epoch in range(1, epochs):
+total_loss = []
+train_acc = 0
+for x, y in data_generator(x_train, y_train,
+batch_size=16, shuffle=True):
+x = x.reshape(-1, 1, 28, 28)
+optimizer_hybrid.zero_grad()
+# Forward pass
+output = model_hybrid(x)
+np_output = np.array(output.data)
+loss = loss_func(y, output)
+loss_np = np.array(loss.data)
+# Backward pass
+loss.backward()
+optimizer_hybrid._step()
+total_loss.append(loss_np)
+B. Quantum Autoencoder for Efficient Compression of Quan-
+tum Data
+1) Background: Inspired by the classical autoencoder, the
+model of the quantum autoencoder is used to perform similar
+tasks on quantum data. Quantum autoencoder is trained to
+compress specific data sets of quantum states, while classical
+compression algorithms cannot be used. The parameters of the
+quantum autoencoder are trained by the classical optimization
+algorithm. We show an example of a simple programmable
+circuit, which can be trained into an efficient autoencoder.
+In the context of quantum simulation, we apply our model
+to compress the ground state of the Hubbard model and
+molecular Hamiltonian.
+2) Implementations: We first define
+QAElayer , which is
+parameterized quantum circuit Layer. It contains parameters
+that can be trained.
+class QAElayer(Module):
+def __init__(self,trash_qubits_number:int=2,
+total_qubits_number:int=7,machine:str="cpu"):
+super().__init__()
+self.machine = machine
+if machine!="cpu":
+raise ValueError("machine only tested on
+cpu simulation")
+self.machine = pq.CPUQVM()
+self.machine.init_qvm()
+self.qlist = self.machine.qAlloc_many(
+total_qubits_number)
+aux_qubits_number = 1
+self.clist = self.machine.cAlloc_many(
+aux_qubits_number)
+self.history_prob = []
+self.n_qubits = total_qubits_number
+self.n_aux_qubits = aux_qubits_number
+self.n_trash_qubits = trash_qubits_number
+training_qubits_size = self.n_qubits - self.
+n_aux_qubits - self.n_trash_qubits
+weight_shapes = {"params_rot_begin":
+training_qubits_size * 3,
+"params_crot":
+training_qubits_size * (training_qubits_size -
+1) * 3,
+"params_rot_end":
+training_qubits_size * 3}
+self.weights = Parameter(weight_shapes[’
+params_rot_begin’]
++ weight_shapes[’
+params_crot’]
++ weight_shapes[’
+params_rot_end’])
+def forward(self, x):
+self.history_prob = []
+batchsize = x.shape[0]
+batch_measure = np.zeros([batchsize])
+prog = pqc(x.data,
+self.weights.data,
+self.qlist,
+self.clist,
+self.n_qubits,
+self.n_aux_qubits,
+self.n_trash_qubits)
+result = self.machine.run_with_configuration
+(prog, self.clist, 100)
+counts = result[’0’]
+probabilities = counts / 100
+print("probabilities", probabilities)
+batch_measure[0] = probabilities
+requires_grad = (x.requires_grad or self.
+weights.requires_grad) and not QTensor.NO_GRAD
+nodes = []
+if x.requires_grad:
+nodes.append(QTensor.GraphNode(tensor=x,
+df=lambda g: 1))
+if self.weights.requires_grad:
+nodes.append(QTensor.GraphNode(
+tensor = self.weights,
+df=lambda g: _grad(g,
+pqc,
+x.data,
+self.weights.data,
+self.machine,
+self.qlist,self.clist,
+self.n_qubits,
+self.n_aux_qubits,
+self.n_trash_qubits
+)))
+return QTensor(data = [batch_measure],
+requires_grad = requires_grad,nodes =nodes)
+Finally, the forward of the whole network is set, and the
+network training can be conducted with common training
+modules:
+class Model(Module):
+def __init__(self, trash_num: int = 2, total_num
+: int = 7):
+super().__init__()
+self.pqc = QAElayer(trash_num, total_num)
+def forward(self, x):
+x = self.pqc(x)
+return x
+
+12
+C. Hybrid Quantum-Classical Neural Networks in VQNet 2.0
+1) Background: We explore how a classical neural network
+can be partially quantized to create a hybrid quantum-classical
+neural network. We will code up a simple example that
+integrates QPanda with VQNet 2.0.
+2) Implementations: We create a simple hybrid neural
+network from MNIST datasets, and two types of digital (0
+or 1) images are classified. We first load MNIST and pick out
+the images containing 0 and 1. These are used as the input of
+the neural network for classification.
+def load_mnist(dataset="training_data", digits=np.
+arange(2), path="../../../dataset/MNIST_data"):
+import os, struct
+from array import array as pyarray
+if dataset == "training_data":
+fname_image = os.path.join(path, ’train-
+images.idx3-ubyte’).replace(’\\’, ’/’)
+fname_label = os.path.join(path, ’train-
+labels.idx1-ubyte’).replace(’\\’, ’/’)
+elif dataset == "testing_data":
+fname_image = os.path.join(path, ’t10k-
+images.idx3-ubyte’).replace(’\\’, ’/’)
+fname_label = os.path.join(path, ’t10k-
+labels.idx1-ubyte’).replace(’\\’, ’/’)
+else:
+raise ValueError("dataset must be ’
+training_data’ or ’testing_data’")
+flbl = open(fname_label, ’rb’)
+magic_nr, size = struct.unpack(">II", flbl.read
+(8))
+lbl = pyarray("b", flbl.read())
+flbl.close()
+fimg = open(fname_image, ’rb’)
+magic_nr, size, rows, cols = struct.unpack(">
+IIII", fimg.read(16))
+img = pyarray("B", fimg.read())
+fimg.close()
+ind = [k for k in range(size) if lbl[k] in
+digits]
+N = len(ind)
+images = np.zeros((N, rows, cols))
+labels = np.zeros((N, 1), dtype=int)
+for i in range(len(ind)):
+images[i] = np.array(img[ind[i] * rows *
+cols: (ind[i] + 1) * rows * cols]).reshape((rows
+, cols))
+labels[i] = lbl[ind[i]]
+return images, labels
+This experiment is a binary classification task, so one qubit
+is used.
+def circuit(weights):
+num_qubits = 1
+machine = pq.CPUQVM()
+machine.init_qvm()
+qubits = machine.qAlloc_many(num_qubits)
+cbits = machine.cAlloc_many(num_qubits)
+circuit = pq.QCircuit()
+circuit.insert(pq.H(qubits[0]))
+circuit.insert(pq.RY(qubits[0], weights[0]))
+prog = pq.QProg()
+prog.insert(circuit)
+prog << measure_all(qubits, cbits)
+result = machine.run_with_configuration(prog,
+cbits, 100)
+counts = np.array(list(result.values()))
+states = np.array(list(result.keys())).astype(
+float)
+# Compute probabilities for each state
+probabilities = counts / 100
+# Get state expectation
+expectation = np.sum(states * probabilities)
+return expectation
+Here, we first reduce the data’s dimension to 2D through
+the pooling layer and the full connection layer, then through
+the single qubit circuit, and finally into 2D output through the
+full connection layer to adapt to the 2-classification task.
+class Hybrid(Module):
+def __init__(self, shift):
+super(Hybrid, self).__init__()
+self.shift = shift
+def forward(self, input):
+self.input = input
+expectation_z = circuit(np.array(input.data)
+)
+result = [[expectation_z]]
+requires_grad = input.requires_grad and not
+QTensor.NO_GRAD
+def _backward_mnist(g, input):
+input_list = np.array(input.data)
+shift_right = input_list + np.ones(
+input_list.shape) * self.shift
+shift_left = input_list - np.ones(
+input_list.shape) * self.shift
+gradients = []
+for i in range(len(input_list)):
+expectation_right = circuit(
+shift_right[i])
+expectation_left = circuit(
+shift_left[i])
+gradient = expectation_right -
+expectation_left
+gradients.append(gradient)
+gradients = np.array([gradients]).T
+return gradients * np.array(g)
+nodes = []
+if input.requires_grad:
+nodes.append(QTensor.
+ComputationalGraphNode(tensor=input, df=lambda g
+: _backward_mnist(g, input)))
+return QTensor(data=result, requires_grad=
+requires_grad, nodes=nodes)
+class Net(Module):
+def __init__(self):
+super(Net, self).__init__()
+self.conv1 = Conv2D(input_channels=1,
+output_channels=6, kernel_size=(5, 5), stride
+=(1, 1), padding="valid")
+self.maxpool1 = MaxPool2D([2, 2], [2, 2],
+padding="valid")
+self.conv2 = Conv2D(input_channels=6,
+output_channels=16, kernel_size=(5, 5), stride
+=(1, 1), padding="valid")
+self.maxpool2 = MaxPool2D([2, 2], [2, 2],
+padding="valid")
+self.fc1 = Linear(input_channels=256,
+output_channels=64)
+self.fc2 = Linear(input_channels=64,
+output_channels=1)
+
+13
+self.hybrid = Hybrid(np.pi / 2)
+self.fc3 = Linear(input_channels=1,
+output_channels=2)
+def forward(self, x):
+x = F.ReLu()(self.conv1(x))
+x = self.maxpool1(x)
+x = F.ReLu()(self.conv2(x))
+x = self.maxpool2(x)
+x = tensor.flatten(x, 1)
+x = F.ReLu()(self.fc1(x))
+x = self.fc2(x)
+x = self.hybrid(x)
+x = self.fc3(x)
+return x
+D. Using Qiskit in VQNet 2.0
+1) Background: In VQNet 2.0, we can also use IBM’s
+Qiskit quantum computing library for quantum machine learn-
+ing tasks.
+VQNet 2.0 implements the automatic differential Qiskit
+quantum circuit operation class QiskitLayer , which inherits
+from
+Compatiblelayer .
+Compatiblelayer is a class that
+is compatible with other framework circuits to VQNet 2.0.
+The parameters of constructing
+QiskitLayer
+need to be
+passed in a class, which defines the Qiskit quantum cir-
+cuit
+qiskit.QuantumCircuit , as well as its operation and
+measurement function
+run . The
+run
+function needs to
+bind the input value to the quantum circuit of Qiskit. Using
+QiskitLayer , the input of quantum circuits and automatic
+differentiation can be implemented by VQNet 2.0.
+2) Implementations: We use Qiskit to construct a class
+QISKI_VQC . The variable
+self._circuit
+is a quantum
+circuit. self.input is a variable input parameter. In the run
+function, you need to use Qiskit’s
+assign_parameters to
+bind the parameters and use
+self.backend.run to run.
+class QISKIT_VQC:
+def __init__(self, n_qubits, backend, shots)
+:
+# --- Circuit definition ---
+self._circuit = qiskit.
+QuantumCircuit(n_qubits)
+all_qubits = [i for i in range(
+n_qubits)]
+self.input = [qiskit.circuit.
+Parameter(’input’)]
+self._circuit.h(all_qubits)
+self._circuit.barrier()
+self._circuit.ry(self.input[0],
+all_qubits)
+self._circuit.measure_all()
+self.backend = backend
+self.shots = shots
+def run(self,x):
+params = dict(zip(self.input, x))
+c1 = self._circuit.assign_parameters
+(params)
+job = self.backend.run(c1,shots=self
+.shots)
+result = job.result().get_counts()
+counts = np.array(list(result.values
+()))
+states = np.array(list(result.keys()
+)).astype(float)
+# Compute probabilities for each
+state
+probabilities = counts / self.shots
+# Get state expectation
+expectation = np.sum(states *
+probabilities)
+return expectation
+The next step is to use VQNet 2.0 to define the model and
+training process. Use
+QiskitLayer to add the Qiskit circuit
+to the model.
+#define Qiskit circuits class
+circuit = QISKIT_VQC(1, simulator, 100)
+class Net(Module):
+def __init__(self):
+super(Net, self).__init__()
+self.conv1 = Conv2D(input_channels
+=1, output_channels=6, kernel_size=(5, 5),
+stride=(1, 1), padding="valid")
+self.maxpool1 = MaxPool2D([2, 2],
+[2, 2], padding="valid")
+self.conv2 = Conv2D(input_channels
+=6, output_channels=16, kernel_size=(5, 5),
+stride=(1, 1), padding="valid")
+self.maxpool2 = MaxPool2D([2, 2],
+[2, 2], padding="valid")
+self.fc1 = Linear(input_channels
+=256, output_channels=64)
+self.fc2 = Linear(input_channels=64,
+output_channels=1)
+self.hybrid = QiskitLayer(circuit,0)
+self.fc3 = Linear(input_channels=1,
+output_channels=2)
+def forward(self, x):
+x = F.ReLu()(self.conv1(x))
+x = self.maxpool1(x)
+x = F.ReLu()(self.conv2(x))
+x = self.maxpool2(x)
+x = tensor.flatten(x, 1)
+x = F.ReLu()(self.fc1(x))
+x = self.fc2(x)
+x = self.hybrid(x)
+x = self.fc3(x)
+return x
+
diff --git a/p9E1T4oBgHgl3EQfigQB/content/tmp_files/load_file.txt b/p9E1T4oBgHgl3EQfigQB/content/tmp_files/load_file.txt
new file mode 100644
index 0000000000000000000000000000000000000000..2eb1abfcd41b1a894d36c735894bd64209a09c6b
--- /dev/null
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+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf,len=1105
+page_content='1  VQNet 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content='0: A New Generation Machine Learning  Framework that Unifies Classical and Quantum  Huanyu Bian,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content=' 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content='7 Zhilong Jia,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content='1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content='4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content='6 Menghan Dou,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content=' 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content='4 Yuan Fang,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content=' 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content='4 Lei Li,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content=' 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content='4 Yiming Zhao,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content=' 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content='4 Hanchao Wang,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content='  1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content='4 Zhaohui Zhou,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content=' 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content='4 Wei Wang,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content='1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content='4 Wenyu Zhu,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content='1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content='4 Ye Li,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content=' 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content='4 Yang Yang,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content=' 3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content='5 Weiming Zhang,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content='2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content='∗ Nenghai Yu,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content=' 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content='∗  Zhaoyun Chen,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content='5,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content='∗ Guoping Guo5,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content='6,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content='∗  1Origin Quantum Computing Company Limited,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content=' Hefei 230026,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content=' China  2CAS Key Laboratory of Electromagnetic Space Information(University of Science and Technology of China),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content='  Hefei 230026,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content=' China  3School of Electronics and Information Engineering(Anhui University),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content=' Hefei,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content=' Anhui 230601,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content=' China  4Anhui Engineering Research Center of Quantum Computing,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content=' Hefei 230026,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content=' China  5Institute of Artificial Intelligence( Hefei Comprehensive National Science Center),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content=' Hefei,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content=' Anhui 230026,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content=' China  6CAS Key Laboratory of Quantum Information (University of Science and Technology of China),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content=' Hefei 230026,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content='  China  7Aerospace Information Research Institute,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content=' Chinese Academy of Sciences,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content=' Beijing 100190,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content=' China  Abstract—With the rapid development of classical and quan-  tum machine learning,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content=' a large number of machine learning  frameworks have been proposed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content=' However, existing machine  learning frameworks usually only focus on classical or quantum,  rather than both.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content=' Therefore, based on VQNet 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content='0, we further  propose VQNet 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content='0, a new generation of unified classical and  quantum machine learning framework that supports hybrid  optimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content=' The core library of the framework is implemented  in C++, and the user level is implemented in Python, and  it supports deployment on quantum and classical hardware.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content='  In this article, we analyze the development trend of the new  generation machine learning framework and introduce the design  principles of VQNet 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content='0 in detail: unity, practicality, efficiency,  and compatibility, as well as full particulars of implementation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content='  We illustrate the functions of VQNet 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content='0 through several basic  applications, including classical convolutional neural networks,  quantum autoencoders, hybrid classical-quantum networks, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content='  After that, through extensive experiments, we demonstrate that  the operation speed of VQNet 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content='0 is higher than the comparison  method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content=' Finally, through extensive experiments, we demonstrate  that VQNet 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content='0 can deploy on different hardware platforms, the  overall calculation speed is faster than the comparison method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content=' It  also can be mixed and optimized with quantum circuits composed  of multiple quantum computing libraries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content='  Index Terms—Machine Learning Framework;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content=' Quantum Ma-  chine Learning;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content=' Classical Machine Learning  I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content=' INTRODUCTION  With the rapid development of quantum computing, quan-  tum machine learning, which combines quantum computing  with classical machine learning, has also developed rapidly [1],  [2], [3], [4], [5], [6], [7], [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content=' Quantum computing is a  new computing method that uses quantum properties such  as superposition and entangled states to perform calculations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content='  Compared with classical computing, quantum computing has  two advantages: 1) Theoretically, quantum computers far  surpass classical computers in calculation speed and storage  efficiency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content=' 2) With the continuous size reduction of classical  electronic components, the development of classical computers  Table I: Classification of machine learning frameworks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content='  Classical machine learning  Quantum machine learning  Caffe  �  �  TensorFlow  �  �  Theano  �  �  PyTorch  �  �  TensorFlow Quantum  �  �  PennyLane  �  �  Paddle Quantum  �  �  VQNet 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content='0  �  �  is about to face the "Size Effect" and other issues that hinder  continued rapid developments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content=' Quantum machine learning  will further improve the processing capabilities of big data  by using the computational efficiency of quantum computers  far surpassing classical computers, combined with machine  learning algorithms that have developed rapidly in the era of  big data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content='  Meanwhile, the increasing interest in classical machine  learning and quantum machine learning has led to an emer-  gence of quantum machine learning frameworks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content=' Although  many previous works have developed a variety of machine  learning frameworks, there is still a lack of a new generation  of machine learning frameworks to support both classical  machine learning and quantum machine learning, as shown  in Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content=' To solve the above-mentioned lack of the new gen-  eration machine learning framework, based on VQNet 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content='0 [9],  we propose VQNet 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content='0, a Python framework that unifies clas-  sical machine learning and quantum machine learning, which  means 1) unifying classical and quantum machine learning  algorithms, and 2) unifying classical and quantum computers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content='  Its software stack is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content=' In addition to the above  unification of machine learning algorithms, VQNet 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content='0 also  achieves the unification of classical computers and quantum  computers: 1) Classical nodes are deployed on classical com-  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content='03251v1  [quant-ph]  9 Jan 2023  2  Figure 1: VQNet 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content='0’s software stack.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content=' At the top of the stack  is the data to be processed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content=' VQNet 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content='0 has the ability to  process classical data (images, text, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content=') and quantum data  (quantum circuits, quantum operators, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content=').' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content=' Next is the model  API of VQNet 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content=' Then comes the computing layers and  the differentiators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content=' The computing layers include the classical  layers and the quantum layers, and the differentiators can  perform hybrid automatic differentiation for networks built  on any computing layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content=' After that, the quantum circuit is  accomplished by QPanda, and the entire network can be  operated on a classical computer, quantum simulator, or QPU  according to its function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content='  puters, similar to classical machine learning frameworks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content=' 2)  We design Qpanda [10], a software package that connects  hardware and quantum computing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content=' Through Qpanda, quantum  nodes are deployed on the quantum processing unit (QPU) or  quantum simulator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content='  The rest of the paper is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content=' In Section II,  we describe the background of our proposed framework and  the trend of the machine learning framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content=' In Section III,  we describe our design principles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content=' In Section IV, we give the  pipeline of our VQNet 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content='0, which can support both classical  machine learning and quantum machine learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content=' Section V  shows the implementation details of our proposed framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content='  Section VI and Section VII give the extensibility of case  studies which contain both classical and quantum applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content='  Section VIII shows the experimental setup and experimental  results of our VQNet 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content='0 compared with other state-of-the-art  frameworks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content=' Section IX draws the conclusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content='  II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content=' BACKGROUND  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content=' VQNet 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content='0  Quantum computing is a completely new method that can  revolutionize computers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content=' In order to explore the possibility of  hybrid implementation of the variable quantum algorithm and  classical machine learning framework, Chen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content='  [9] pro-  posed VQNet 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content='0, a quantum-classical hybrid machine learn-  ing architecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content=' VQNet 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content='0 designed two kinds of quantum  operators (QOPs): qop and qop_pmeasure .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content=' Through making  these two QOPs support forward and backward propagation  at the same time, VQNet 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content='0 could achieve model training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content='  Furthermore, it is possible to use a gradient-based optimizer  for network optimization in VQNet 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content=' Chen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content=' used  VQNet 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content='0 to construct a variety of quantum machine learning  algorithms, including QAOA, VQE algorithm, quantum classi-  fier, and quantum circuit learning, and they all achieved good  performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content='  Although VQNet 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content='0 effectively connected machine learn-  ing and quantum algorithms, it only supports a few types  of quantum machine learning algorithms like QAOA, VQE  algorithm, quantum classifier and quantum circuit learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content='  With the rapid development of machine learning algorithms,  it is urgent to propose a new generation of machine learning  frameworks based on VQNet 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content='  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content=' Trend of Machine Learning Framework  Compared with the previous single-class machine learning  frameworks, we summarize the development trends of the  next-generation machine learning frameworks:  First of all, classical machine learning become popular in re-  cent years, and the boom period of quantum machine learning  begins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content=' So far, a large number of classical machine learning  frameworks, such as Caffe[11], TensorFlow[12], Theano[13],  PyTorch[14], and quantum machine learning frameworks,  such as TensorFlow Quantum[15], PennyLane[16], Paddle  Quantum[17], have emerged.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content=' These frameworks meet the  needs of researchers and industrial production for mature  frameworks, lower the threshold for research, and further  promote the development of classical quantum machine learn-  ing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content=' However, with the maturity of classical machine learning  algorithms and the continuous progress of quantum machine  learning, quantum machine learning will inevitably become a  supplement to classical machine learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content=' The interaction of  these two kinds of algorithms make researchers and developers  an urgent need for a new generation of machine learning  framework that can unify classical and quantum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content='  Secondly, various gradient-based optimization methods play  an extremely important role in the training and conver-  gence of classical deep learning models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content=' Because automatic  differentiation[18] greatly reduces the programming difficulty  of classical machine learning, it has become the core software  design paradigm of today’s machine learning frameworks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content='  However, the existing automatic optimization[19] algorithms  are difficult to directly apply to the training of quantum ma-  chine learning models: the gradients of parameterized quantum  circuits in quantum machine learning must be evaluated using  quantum algorithms, so compared to the gradients of clas-  sical machine learning, the existing automatic differentiation  algorithm may not work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content=' One type of algorithm in quantum  machine learning is hybrid quantum-classical networks, which  requires quantum computing and classical computing to coop-  erate with each other, so hybrid optimization[16] that can  output hybrid calculation results is the core function of the  new machine learning framework design.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content='  Third, Python[20], [21] is an object-oriented, interpreted,  general, and open source scripting programming language.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content=' It  has the characteristics of simple and easy to use, low learning  cost, and high writing efficiency, but it has the corresponding  disadvantage of slower running speed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content=' Python has been used  in a large number of machine learning frameworks such as  Classical Data  Quantum Data  VONet2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content='0 Models  VQNet 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content='0  Classical Layers  Quantum Layers  Differentiators  VQNet 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content='0 Classical Execution Engine  QPanda  Classical Computer  QPU  Quantum Simulator3  TensorFlow, PyTorch, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content=' Meanwhile, C++[22], [23] is an  object-oriented compiled language, which is convenient for the  direct operation of the memory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content=' C++ not only has the practical  characteristics of the efficient operation of computers, but also  is committed to improving the programming quality of large-  scale programs and the problem description ability of pro-  gramming languages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content=' It is often used in industrial applications  for image processing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content=' Correspondingly, it has shortcomings  such as high code complexity and being unfriendly to users.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content='  Therefore, in order to avoid these two language shortcomings  while inheriting the advantages of both, it is inevitable to  separate the user level in various languages from the  core library[12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content=' The core library is implemented in C++,  which improves operating efficiency and portability;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content=' while the  user-level implementation is implemented in Python, which  improves user-friendliness and reduces learning costs and  difficulties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content='  Finally, in order to use general-purpose massively parallel  hardware (such as GPU) for computational acceleration, exist-  ing machine learning frameworks can be deployed on classical  hardware such as CPU, GPU, and TPU[24], [25], [26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content=' The  rapid increase in the computing power of quantum computers  has provided a solid foundation for the development of quan-  tum machine learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content=' At the same time, the current shortage  of quantum computing power makes quantum simulation on  classical computers the first choice for most researchers to  conduct verification experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content=' Therefore, it is inevitable  that the new generation of machine learning frameworks can  be deployed on more types of hardware such as classical  computer, QPU, and quantum simulator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content='  Based on the above trends, VQNet 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content='0 provides a unified  classical and quantum machine learning framework that sup-  ports hybrid optimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content=' The core library of the framework is  implemented in C++, the user-level is implemented in Python,  and it supports deployment on both quantum and classical  hardware.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content='  III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content=' DESIGN PRINCIPLES  There are four main principles behind the design of VQNet  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content='0:  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content=' Be United  VQNet 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content='0 aims to achieve multiple unities of classical and  quantum: 1) The unification of classical machine learning and  quantum machine learning, that is, compared with frameworks  such as Qiskit and Pennylane, VQNet 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content='0 does not need to  import Torch, Tensorflow, or other huge classical machine  learning libraries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content=' VQNet 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content='0 comes with a complete neural  network module, including classical and quantum machine  learning, optimizer, loss function, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content=' It is easy to quickly  build hybrid quantum-classical machine learning models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content=' 2)  The deployment of classical computers and quantum com-  puters is unified, that is, compared with frameworks such as  TensorFlow and TensorFlow quantum, VQNet 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content='0 can deploy  classical machine learning models on classical computers,  and deploy quantum machine learning models on QPU and  quantum simulators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content='  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content=' Put Practicality First  VQNet 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content='0 strives to use the framework to build models,  train networks, and deploy multiple machines as simple and  efficient as possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content=' In order to bring maximum practicality,  it is necessary to realize the following functions including but  not limited to: 1) Friendly interface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content=' VQNet 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content='0 implements a  wealth of quantum machine learning models and interfaces for  quickly building quantum machine learning models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content=' VQNet  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content='0 performs well in constructing these quantum machine  learning algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content=' The interface definition is similar to the  syntax of commonly used machine learning frameworks, and  users can quickly get started.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content=' 2) Automatic differentiation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content=' The  realization of automatic differentiation can help VQNet 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content='0’s  users get rid of tedious mathematical derivation and code,  further improving the degree of automation and efficiency of  using VQNet 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content=' 3)  Dynamic computation graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content=' VQNet 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content='0 uses the dynamic  computation graph mechanism, that is, the construction and  calculation of the calculation graph are performed at the same  time, which reduces the difficulty of debugging and which also  improves the friendliness of programming.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content='  C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content=' Provide Sufficient Performance  As a general machine learning framework, VQNet 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content='0 needs  to provide sufficient performance, but not at the expense  of practicality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content=' Compared with quantum machine learning  frameworks such as Paddle Quantum, Mindspore[27], etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content=',  VQNet 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content='0 uses QPanda as the quantum computing module,  it is possible to build more fully functional and more efficient  quantum circuits on classical and quantum computers, and  achieve faster forward and backward speed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content='  D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content=' A Good Framework Embraces the Rest  As a new generation of machine learning framework,  VQNet 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content='0 is designed to be compatible with other quantum  computing libraries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content=' In order to maximize compatibility with  Python-based frameworks that can have any configuration  system, VQNet 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content='0 is designed with compatibility: A com-  patibility layer is designed to call other frameworks, which  is convenient for users to take the quantum circuits described  by other frameworks as the black box for optimization calcu-  lation, facilitate cross model cooperation, and is suitable for  different business and research applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content='  IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content=' PIPELINE OF VQNET 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content='0  Here, we give an abstract overview of the computational  steps involved in the general training and testing of machine  learning models in VQNet 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content='  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content=' Dataset Preprocessing  1) Classical Data: Usually, classical data comes from  Internet or user-built datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content=' In VQNet 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content='0, it is easy to  download commonly used public data sets by using the func-  tions in  pyvqnet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content='data , similar to the operations in PyTorch  and TensorFlow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content='  4  ……' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content='  ……' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content='  …  N1  1  𝑁2  1  𝑁m1  1  …  Nm2  2  …  Nmn  n  N1  2  N2  2  N2  n  N1  n  Dataset  Preprocessing  data  Result  Calculation Node  Evaluate  Cost Function  Optimization  Figure 2: Abstract pipeline for VQNet 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content='  2) Quantum Data:  Since it is currently impossible to  import quantum data from the outside into a QPU, users  need to specify the quantum circuit that generates quantum  data, while encoding the classical data into the quantum  state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content=' There are mainly two existing encoding methods: 1)  Encoding features into the basic state, which means that  after the classical data is converted into integer form, the  binary expansion is expressed as a direct product state, and  this encoding method can be realized by using function  pyvqnet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content='qnn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content='template.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content='BasicEmbeddingCircuit .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content='2) Encod-  ing features into the rotation angles, which refers to the  rotation angle of encoding classical data into a qubit, and  this encoding method can be realized by using the function  pyvqnet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content='qnn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content='template.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content='AngleEmbeddingCircuit .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content=' 3) Encod-  ing features into the amplitude vector, which refers to  the amplitude vector of converting classical data into a  quantum state through a quantum gate operation, and this  encoding method can be realized by using the function  pyvqnet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content='qnn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content='template.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content='AmplitudeEmbeddingCircuit .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content='  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content=' Calculation Node  After the data is processed, it can be sent to calculation  nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content=' In VQNet 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content='0, calculation nodes include classical nodes  and quantum nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content=' In the network built by VQNet 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content='0, any  number of classical nodes and quantum nodes can be used to  connect in any way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content=' It is worth noting that if the number of  quantum nodes selected is 0, the network built is a classical  machine learning model, and if the number of classical nodes  and quantum nodes is not 0, it is a hybrid quantum-classical  network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content='  1) Classical  Node:  Classical  node  refers  to  various  network components that process classical data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content=' VQNet  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content='0  includes  a  variety  of  common  basic  building  blocks,  including  convolutional  layers pyvqnet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content='nn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content='conv ,  pooling  layers pyvqnet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content='nn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content='pooling ,  and  normlization  layers pyvqnet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content='nn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content='batch_norm ,  etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content='  Various  layers  can  extract features of different dimensions from the original  data, and the combination of different features can finally  achieve amazing performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content='  2) Quantum Node: The quantum node is responsible for  calculating quantum data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content='  QuantumLayer simulate a param-  eterized quantum circuit and get the measurement result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content=' It  inherits from  Module , so that it can calculate gradients of  circuits parameters, and trains variational quantum circuits  model or embeds variational quantum circuits into hybrid  quantum and classical model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content=' The user defines the function  qprog_with_meansure as a parameter, it needs to include  the quantum circuit defined by QPanda: generally includes  the quantum circuit’s coding circuit, evolution circuit, and  measurement operation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content=' This class can be embedded in the  hybrid quantum-classical machine learning model, and the  objective function or loss function of the hybrid quantum-  classical model can be minimized through the gradient descent  method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content='  It is particularly worth noting that, in addition to the  different calculations, there is another huge difference between  quantum nodes and classical nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content=' If quantum nodes are  required to output classical data, measurement operations are  required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content=' Quantum measurement refers to the interference of  the outside world to the quantum system to obtain the required  information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content=' The measurement gate uses the Monte Carlo  method of measurement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content=' After the qubit is measured, the  measurement result will be stored in the classical register.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content='  VQNet 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content='0 uses  pyvqnet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content='qnn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content='measure  for measurement  operations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content='  C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content=' Loss Function  After passing through the calculation node, the loss function  needs to be calculated next.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content=' The loss function selection de-  pends on the completion of the task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content=' It is especially important  to note that the calculation of the existing loss function needs  to be performed on a classical computer but not a quantum  5  device, so if the network is built with quantum nodes as  the last network layer, it must be measured before the loss  calculation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content=' VQNet 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content='0 builds common loss functions in class  pyvqnet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content='nn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content='loss .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content='  D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content=' Optimization  After setting the loss function, the network parameters are  updated through back propagation, that is, model training is  performed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content=' Common optimization methods are mostly based  on gradient descent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content=' VQNet 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content='0 constructs common opti-  mization strategies in the class  pyvqnet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content='optim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content='optimizer .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content='  As mentioned earlier, if there is no quantum node in the  model involved in the calculation, automatic differentiation  can be directly used to calculate the gradient and perform  back propagation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content=' If the model is a hybrid quantum-classical  network, the hybrid optimization strategy[16] needs to be used  to train the network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content='  V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content=' IMPLEMENTATION OF VQNET 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content='0  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content=' Be United  1) Unification of Classical and Quantum Machine Learn-  ing: In order to achieve the unity of classical and quan-  tum in one machine learning framework, VQNet 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content='0 de-  fines an abstract class  pyvqnet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content='nn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content='module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content='Module , which  is the base class of all neural network modules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content=' All  calculation nodes are designed to be sub-class of class  pyvqnet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content='nn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content='module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content='Module , so that they can be used for  automatic differential calculation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content='  Module can also contain  other Modules, allowing to nest them in a tree structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content=' In  VQNet 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content='0, it is easy to assign the sub-modules as regular  attributes:  class Model(Module):  def __init__(self):  super(Model, self).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content='__init__()  self.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content='conv1 = pyvqnet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content='nn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content='Conv2d(1, 20, (5,5))  self.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content='conv2 = pyvqnet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content='nn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content='Conv2d(20, 20, (5,5)  )  def forward(self, x):  x = pyvqnet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content='nn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content='activation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content='relu(self.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content='conv1(x)  )  return pyvqnet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content='nn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content='activation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content='relu(self.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content='conv2  (x))  Sub-modules assigned in this way will be registered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content='  2) Unification of Classical and Quantum Machine: VQNet  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content='0 not only supports the unification of quantum and classical  machine learning but also supports operations on classical  computers and quantum computers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content=' It should be noted that dif-  ferent calculation nodes have different hardware requirements:  1) For classical nodes, the tools provided by VQNet 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content='0 can  be directly operated on the classical computer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content=' 2) For quantum  nodes, VQNet 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content='0 can accomplish real quantum hardware  interaction and quantum simulator interaction through QPanda.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content='  For the classical node, since the calculation of classical  modules can only be implemented on the classical computer,  there is no need for additional specifications in VQNet 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content='  By default, all classical nodes are run in a classical computer  and the gradient is obtained through classical methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content='  Due to the current lack of quantum computer resources,  quantum simulator that simulates the operating environment  of QPU on classical computers has become one choice of  most scientific researchers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content=' In order to improve the general  capabilities of the framework, for quantum node, VQNet 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content='0  defines a class  QuantumLayer .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content=' Users can set the parameter  machine_type  in  QuantumLayer  according to their own  computing resources and call the corresponding interface pro-  vided by QPanda to use QPU or quantum simulator to conduct  quantum calculations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content=' Because different quantum simulators  have different simulation effects, VQNet 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content='0 provides five  different kinds of simulators to realize different simulation en-  vironments, namely full-amplitude quantum simulator, single-  amplitude quantum simulator, partial-amplitude quantum sim-  ulator, tensor network quantum simulator, and noise inclusive  quantum simulator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content='  At the same time, in order to enable researchers to better  simulate the running results of QPU on the simulator when the  calculation resources of QPU are short, VQNet 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content='0 adds the  noise of QPU to the quantum simulator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content=' VQNet 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content='0 adds quan-  tum mapping based on the topology between QPU’s qubits to  the noise simulator, so as to simulate more real experimental  results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content=' The simulation of the noise-containing quantum simu-  lator is closer to QPU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content=' VQNet 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content='0 can customize the supported  logic gate types and customize the noise model supported by  the logic gates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content=' The existing supported quantum noise models  are defined in QPanda.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content=' VQNet 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content='0 uses NoiseQuantumLayer  to define an automatic micro-classification of QPU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content=' The  user defines function qprog_with_measure as the parameter,  which needs to include the quantum circuit defined by QPanda,  and also needs to pass in the parameter  noise_set_config ,  using the QPanda interface to set the noise model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content='  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content=' Practicality Centric Design  1) Friendly Interface: The core element to achieve interface  friendliness is to design for the interface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content=' The friendliness of  the interface definition not only facilitates the maintenance  of VQNet 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content='0 itself, but also avoids excessive changes to  the upper application layer, reducing the cost and difficulty  of users.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content=' First, in order to provide a more convenient and  concise interface, VQNet 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content='0 shields the coordination details  of multiple interfaces, and provides a coarse-grained interface  for upper-layer applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content=' That is, the user only needs  to define the quantum circuit and network structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content=' After  constructing the forward propagation path, VQNet 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content='0 can  automatically constitute the back propagation and automatic  optimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content=' Secondly, VQNet 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content='0 follows the conventions  of most frameworks and uses scene-oriented interface nam-  ing methods, such as naming the class that performs one-  dimensional convolution operations as class Conv1D to further  enhance the friendliness of the interface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content=' Finally, for interface  parameters, VQNet 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content='0 implements the principle of minimum  granularity, that is, to ensure that all parameters passed are  used by the interface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content='  2) Dynamic Computation Graph:  Similar to PyTorch,  VQNet 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content='0 also uses the dynamic computation graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content=' Compu-  tation graphs are directed acyclic computation graphs used to  6  describe operations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content=' They are composed of two main elements:  Node which represents data, and Edge which represents oper-  ations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content=' In VQNet 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content='0, predefined classical and quantum nodes,  such as convolutional layer and quantum layer, are also nodes  in the computation graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content=' Edge is the operation performed by  each node, QTensor is the basis for constructing the computa-  tion graph and the computation graph forms the structural basis  of forward and backward propagation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content=' Dynamic graph means  that the operation and construction of machine learning model  are established at the same time, that is, the value of nodes can  be calculated according to the order of forward propagation,  and then the computation graph is constructed on the basis of  these nodes, and at last the graph constructed by the classical  and quantum nodes is used to get the gradient of  QTensor .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content='  The advantage of the dynamic computation graph is that it is  flexible and easy to debug.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content='  C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content=' Implementation with Fabulous Performance  How to implement an efficient machine learning framework  has always been a great challenge for researchers and devel-  opers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content=' In order to improve the efficiency of the new generation  machine learning framework, in addition to optimizing classi-  cal algorithms and classical computers, two aspects need to be  considered: 1) the interaction between classical and quantum  machine learning algorithms, and 2) the interaction between  classical and quantum computers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content=' In order to solve the above  problems, VQNet 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content='0 makes improvements in two aspects: 1)  open-source QPanda, which is a software package specially  designed to perform QPU operations and also simulate quan-  tum circuits on classical computers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content=' 2)  Unified structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content='  The classical and quantum nodes are unified in design.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content='  QPanda.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content=' Three high-performance simulations are proposed  in QPanda, namely 1) OpenMP multi-thread optimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content='  Using OpenMP instructions to execute functions in parallel can  effectively execute the iterations in the function and effectively  reduce the calculation time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content=' In QPanda, the backends of all  simulators running large quantum circuit calculations are par-  allelized to maximize the workload of each core and minimize  the overhead caused by multithreading.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content=' 2) GPU optimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content='  GPU has higher memory bandwidth than CPU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content=' Because GPU  has a large number of processing cores, it also has higher peak  performance than CPU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content=' In QPanda, when the GPU is detected,  the GPU is used to accelerate the quantum simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content=' 3)  Supercomputer simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content=' Supercomputers represent the peak  of classical computing power in the world, and quantum com-  puting simulations are based on classical computers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content=' QPanda  proposes to distribute tasks on multiple calculation nodes on  the supercomputer platform ("Shenwei Light of Taihu Lake").' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content='  The operation control core runs the program instruction set one  by one and calls the operation core to perform core computing  tasks, that is, to improve computing efficiency through a two-  level parallel computing method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content='  Unified structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content=' 1)Unified data structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content=' Similar to  machine learning frameworks such as Torch and Paddle,  in VQNet 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content='0, the data structure is uniformly defined as  QTensor .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content='  QTensor is the basic data structure of quantum  circuits and neural networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content=' It supports multiple operations  and realizes their derivative functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content=' When performing var-  ious operations required for quantum and classical machine  learning calculations, data can be stored by creating QTensor .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content='  It can be created using the interface of class  Creation .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content=' In  addition, QTensor can use Numpy, a popular matrix scientific  computing package, to load data conveniently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content=' 2) Unified class  inheritance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content=' In VQNet 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content='0, the classes of quantum and classical  machine learning models need to be inherited from the abstract  class  Module to perform autograd calculations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content='  D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content=' A Good Framework Embraces the Rest  VQNet 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content='0 hybrid quantum-classical optimization can  support not only the quantum circuits of QPanda but  also  the  quantum  circuits  composed  of  other  quan-  tum computing frameworks (such as Cirq, Qiskit, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content=' ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content='  VQNet 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content='0 provides a quantum circuit computing inter-  face  Compatiblelayer  for automatic differentiation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content=' The  parameters of  Compatiblelayer  need to be passed in a  class, which defines the third-party library quantum circuit,  as well as its operation and measurement function  run .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content='  By using  Compatiblelayer , the input of quantum circuits  and automatic differentiation of parameters can be imple-  mented by VQNet 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content=' Class Compatiblelayer is an abstract  wrapper to use other framework’s quantum circuits(such as  qiskit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content='QuantumCircuit in Qiskit,  cirq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content='Circuit in Cirq)  to forward and backward in the form of VQNet 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content=' The  users need to define quantum circuits in the  forward() and  backward() functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content='  VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content=' EVALUATION  In this section, we first introduce the experimental settings,  and then give the deployment results of VQNet 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content='0 on different  hardware to support its heterogeneous capabilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content=' After that,  we support its efficiency by comparing computing efficiency  with different frameworks under different conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content=' Finally,  an example of VQNet 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content='0’s support for Qiskit is given to show  the compatibility of the framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content='  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content=' Experiment Setup  Dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content=' For all subsequent experiments, we use the common  MNIST data set in computer vision as the experimental data  set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content=' The MNIST data set is the handwritten character library.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content='  There are 70,000 images, of which 60,000 are the training set  and 10,000 are the test set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content='  Model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content=' For classical machine learning deployment, we use  the most popular neural network: convolutional neural net-  works(CNN).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content=' The CNN model consists of two convolution  layers and one fully connected layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content=' Each convolution layer  performs the max-pooling operation after the relu activa-  tion function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content=' For quantum machine learning deployment,  we use quantum autoencoder for efficient compression of  quantum data as pure quantum networks, and hybrid quantum-  classical neural networks as hybrid networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content=' To verify the  compatibility of VQNet 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content='0, we use IBM’s Qiskit quantum  computing library for quantum machine learning tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content=' All  running instances are shown in the Appendix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content='  7  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content=' Deployment  In order to show that our proposed framework can be de-  ployed normally on different hardware devices, the following  shows the deployment results of VQNet 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content='0 on the classical  computer and QPU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content='  1) Classical Machine Learning Deployment: Firstly, we  give the deployment results of the classical convolutional  neural network implemented by VQNet 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content='0 on the classical  computer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content=' The running instance is as described in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content=' As  shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content='3, we can find that the classical convolutional  neural network implemented by VQNet 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content='0 converges after 10  epochs when training on the classical computer, and the test  accuracy converges after the same number of epochs, which  proves the deployment ability of VQNet 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content='0 on the classical  computer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content='  2) Quantum Machine Learning Deployment: This section  shows the deployment results of VQNet 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content='0 on the Wuyuan2  QPU which is a quantum computing hardware platform with  64 superconducting qubits created by Origin Quantum com-  pany based on Xmon architecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content=' According to whether there  are quantum nodes or not, quantum machine learning can  be divided into pure quantum networks and hybrid quantum-  classical networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content=' Their running instances are shown in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content='B  and Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content='C respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content=' It is worth noting that in this section,  the quantum nodes of the network operate on the quantum  simulator, and the classical nodes are still deployed on the  classical computer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content='  C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content=' Operation Efficiency  By comparing our proposed VQNet 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content='0 with the most  popular quantum machine learning framework, we can fully  understand the performance of VQNet 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content=' It is worth noting  that since the VQNet 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content='0 we proposed is the first unified  framework for quantum and classical machine learning, the  comparison methods in this section are the combination of the  classical machine learning framework and quantum computing  library.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content=' For example, PyTorch + Qiskit means using PyTorch  to call a quantum circuit generated by Qiskit composes the  completed HQCNN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content=' For the fairness of comparison, we use  different frameworks to train the same network model on  the MNIST dataset: HQCNN, the loss function is cross-  entropy loss, the optimizer is Adam, the learning rate reduction  strategy is the same, and they are all trained for 20 epochs  After stopping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content='  We first show the relationship between the loss of VQNet  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content='0 and different comparison methods training the same net-  work and the training epoch in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content=' Obviously, the loss of  VQNet 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content='0 decreases more smoothly and converges faster.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content=' As  shown in Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content='V, the training and testing time of VQNet 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content='0  is much shorter than the comparison method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content=' For example,  for the overall training time, PyTorch + Qiskit requires 62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content='5s,  PyTorch+QPanda requires 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content='7s, and VQNet 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content='0 only requires  12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content='9s, which is 20% of PyTorch + Qiskit, 65% of PyTorch  + QPanda.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content=' Thus, the efficient performance of our proposed  framework is verified.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content='  a  b  c  d  Figure 3: Loss and accuracy comparison of each machine  learning framework in training and testing process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content='  8  Table II: Classical Machine Learning Deployment Loss  Mxnet  Paddle  PyTorch  tensorflow  VQNet 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content='0  LeNet  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content='077  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content='001  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content='00  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content='002  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content='046  AlexNet  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content='050  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content='230  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content='00  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content='059  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content='040  VGG  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content='022  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content='014  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content='004  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content='176  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content='137  ResNet-18  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content='568  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
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+page_content='971  Table IV: Loss and Accuracy of Quantum Machine Learning Deployment on Quantum simulator  Paddle Quantum  PyTorch+Pennylane  TensorFlow Quantum  VQNet 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content='0  HQCNN  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content='305/0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content='733  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
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+page_content='726/0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content='767  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content='209/0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content='733  Table V: The duration of HQCNN is composed of different machine learning frameworks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content=' BP stands for back propagation, FP  stands for forward propagation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content=' All data are in seconds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content='  PyTorch + Qiskit  PyTorch + QPanda  VQNet 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content='0  Total training duration  62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content='55  19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content='75  12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content='91  Single data training duration (×10−3)  15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content='38  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content='66  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content='36  Quantum node BP duration(×10−4)  80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content='39  8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content='68  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content='07  Quantum node FP duration(×10−4)  41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content='83  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content='35  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content='66  Network FP duration(×10−3)  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content='92  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content='29  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content='04  Single data testing duration(×10−3)  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content='01  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content='31  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content='19  Dataset testing duration(×10−1)  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content='19  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content='61  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content='19  D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content=' Compatibility  In order to show the compatibility of VQNet 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content='0, we choose  the quantum circuit built by Qiskit as a black box and send it  to VQNet 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content='0 for optimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content=' The running instance is shown  in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content='D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content='  The relationship between training and testing loss is shown  in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content=' We can find that although VQNet 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content='0 does not need  to know the details of the quantum circuit composed by Qiskit,  it can also carry out hybrid optimization and converge quickly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content='  Experiments verify the compatibility of VQNet 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content='0 with other  quantum computing libraries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content='  VII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content=' CONCLUSION AND FUTURE WORK  In this paper, on the basis of VQNet 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content='0, we propose the first  classical and quantum unified machine learning framework  named VQNet 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content=' It supports both classical and quantum  machine learning, and supports deployment on classical com-  puters and QPU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content=' VQNet 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content='0 puts practicability in the first place  through the design of the friendly interface, automatic differ-  entiation, dynamic computation graph, and so on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content=' Secondly,  it provides superb performance by accomplishing a unified  structure and quantum computing function based on QPanda.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content='  Finally, it can be used with other quantum computing libraries  to provide better compatibility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content=' Experiments prove that VQNet  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content='0 can achieve better operating speed compared with the  comparison method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content='  Figure 4: Loss of VQNet 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content='0 uses the quantum circuit com-  posed of Qiskit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content='  In the future, we plan to continue to improve the speed  and practicality of VQNet 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content='0 while supporting the latest  developments in classical and quantum machine learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content='  VIII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content=' ACKNOWLEDGMENTS  This work was supported in part by Anhui provin-  cial major science and technology projects under Grant  202203a13010003, the National Natural Science Founda-  9  tion of  China under  Grant 12034018,  Innovation  Pro-  gram for Quantum Science and Technology under Grant  2021ZD0302300 and by the Natural Science Foundation of  China under Grant 62272003.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
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+page_content=' team, “Mindspore,” 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content=' [Online].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content=' Available: https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content='  com/mindspore-ai/mindspore  APPENDIX  RUNNING INSTANCES  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content=' Classical Convolutional Neural Network for Image Recog-  nition  1) Background: Neural networks are part of artificial intel-  ligence research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content=' The most popular neural networks are con-  volutional neural networks(CNN).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content=' CNNs have achieved great  success in many research fields, such as speech recognition,  image recognition, image segmentation, natural language pro-  cessing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content=' Convolutional neural networks are mainly composed  of these types of layers: input layer, convolutional layer, ReLU  layer, pooling layer, and fully connected layer, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content='  2) Implementations: As mentioned earlier, the first step  of the VQNet 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content='0 process is data set preprocessing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content=' In this  example, we use the common MNIST data set in computer  vision as the experimental data set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content=' The MNIST data set is  the handwritten character library.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content=' There are 70,000 images, of  which 60,000 are the training set and 10,000 are the test set:  def load_mnist(dataset="training_data", digits=np.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content='  arange(2), path=".' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content='.//.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content='.//data//MNIST_data"):  import os, struct  from array import array as pyarray  if dataset == "training_data":  fname_image = os.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
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+page_content='join(path, ’t10k-  images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
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+page_content='join(path, ’t10k-  labels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content='idx1-ubyte’).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content='replace(’\\\\’, ’/’)  else:  raise ValueError("dataset must be ’  training_data’ or ’testing_data’")  Once the dataset is ready, we start building the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content='  The model consists of two convolution layers and one full  connection layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content=' Each convolution layer performs the max-  pooling operation after the relu activation function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content='  class CNN(Module):  def __init__(self):  super(CNN, self).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content='__init__()  self.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content='maxpool4 = MaxPool2D([2, 2], [2, 2],  padding="valid")  self.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content='maxpool2 = MaxPool2D([2, 2], [2, 2],  padding="valid")  self.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content='conv5 = Conv2D(input_channels=1,  output_channels=32, kernel_size=(3, 3), stride  =(1, 1), padding="valid")  self.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content='conv6 = Conv2D(input_channels=32,  output_channels=32, kernel_size=(3, 3), stride  =(1, 1), padding="valid")  self.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content='fc3 = Linear(input_channels=800,  output_channels=10)  self.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content='Relu2 = F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content='ReLu()  def forward(self, x):  x = self.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content='maxpool2(self.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content='Relu2(self.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content='conv5(x)))  x = self.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content='maxpool4(self.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content='Relu2(self.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content='conv6(x)))  x = tensor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content='flatten(x,1)  x = self.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content='fc3(x)  return x  In this example, we set the loss function as cross-entropy  loss, and the optimizer is a common Adam optimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content=' If  11  we set the maximum number of epochs and batch size, we  can start training the whole model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content='  model = CNN()  model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content='train()  loss_func = SoftmaxCrossEntropy()  model_hybrid = model  optimizer_hybrid = Adam(model_hybrid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content='parameters(),  lr=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content='001)  for epoch in range(1, epochs):  total_loss = []  train_acc = 0  for x, y in data_generator(x_train, y_train,  batch_size=16, shuffle=True):  x = x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content='reshape(-1, 1, 28, 28)  optimizer_hybrid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content='zero_grad()  # Forward pass  output = model_hybrid(x)  np_output = np.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content='array(output.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content='data)  loss = loss_func(y, output)  loss_np = np.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content='array(loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content='data)  # Backward pass  loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content='backward()  optimizer_hybrid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content='_step()  total_loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content='append(loss_np)  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content=' Quantum Autoencoder for Efficient Compression of Quan-  tum Data  1) Background: Inspired by the classical autoencoder, the  model of the quantum autoencoder is used to perform similar  tasks on quantum data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content=' Quantum autoencoder is trained to  compress specific data sets of quantum states, while classical  compression algorithms cannot be used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content=' The parameters of the  quantum autoencoder are trained by the classical optimization  algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content=' We show an example of a simple programmable  circuit, which can be trained into an efficient autoencoder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content='  In the context of quantum simulation, we apply our model  to compress the ground state of the Hubbard model and  molecular Hamiltonian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content='  2) Implementations: We first define  QAElayer , which is  parameterized quantum circuit Layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content=' It contains parameters  that can be trained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content='  class QAElayer(Module):  def __init__(self,trash_qubits_number:int=2,  total_qubits_number:int=7,machine:str="cpu"):  super().' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content='__init__()  self.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content='machine = machine  if machine!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content='="cpu":  raise ValueError("machine only tested on  cpu simulation")  self.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content='machine = pq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content='CPUQVM()  self.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content='machine.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content='init_qvm()  self.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content='qlist = self.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content='machine.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content='qAlloc_many(  total_qubits_number)  aux_qubits_number = 1  self.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content='clist = self.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content='machine.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content='cAlloc_many(  aux_qubits_number)  self.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content='history_prob = []  self.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content='n_qubits = total_qubits_number  self.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content='n_aux_qubits = aux_qubits_number  self.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content='n_trash_qubits = trash_qubits_number  training_qubits_size = self.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content='n_qubits - self.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content='  n_aux_qubits - self.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content='n_trash_qubits  weight_shapes = {"params_rot_begin":  training_qubits_size * 3,  "params_crot":  training_qubits_size * (training_qubits_size -  1) * 3,  "params_rot_end":  training_qubits_size * 3}  self.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content='weights = Parameter(weight_shapes[’  params_rot_begin’]  + weight_shapes[’  params_crot’]  + weight_shapes[’  params_rot_end’])  def forward(self, x):  self.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content='history_prob = []  batchsize = x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content='shape[0]  batch_measure = np.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content='zeros([batchsize])  prog = pqc(x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content='data,  self.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content='weights.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content='data,  self.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content='qlist,  self.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content='clist,  self.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content='n_qubits,  self.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content='n_aux_qubits,  self.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content='n_trash_qubits)  result = self.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content='machine.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content='run_with_configuration  (prog, self.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content='clist, 100)  counts = result[’0’]  probabilities = counts / 100  print("probabilities", probabilities)  batch_measure[0] = probabilities  requires_grad = (x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content='requires_grad or self.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content='  weights.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content='requires_grad) and not QTensor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content='NO_GRAD  nodes = []  if x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content='requires_grad:  nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content='append(QTensor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content='GraphNode(tensor=x,  df=lambda g: 1))  if self.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content='weights.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content='requires_grad:  nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content='append(QTensor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content='GraphNode(  tensor = self.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content='weights,  df=lambda g: _grad(g,  pqc,  x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content='data,  self.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content='weights.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content='data,  self.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content='machine,  self.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content='qlist,self.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content='clist,  self.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content='n_qubits,  self.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content='n_aux_qubits,  self.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content='n_trash_qubits  )))  return QTensor(data = [batch_measure],  requires_grad = requires_grad,nodes =nodes)  Finally, the forward of the whole network is set, and the  network training can be conducted with common training  modules:  class Model(Module):  def __init__(self, trash_num: int = 2, total_num  : int = 7):  super().' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content='__init__()  self.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content='pqc = QAElayer(trash_num, total_num)  def forward(self, x):  x = self.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content='pqc(x)  return x  12  C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content=' Hybrid Quantum-Classical Neural Networks in VQNet 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content='0  1) Background: We explore how a classical neural network  can be partially quantized to create a hybrid quantum-classical  neural network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content=' We will code up a simple example that  integrates QPanda with VQNet 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content='  2) Implementations: We create a simple hybrid neural  network from MNIST datasets, and two types of digital (0  or 1) images are classified.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content=' We first load MNIST and pick out  the images containing 0 and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content=' These are used as the input of  the neural network for classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content='  def load_mnist(dataset="training_data", digits=np.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content='  arange(2), path=".' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content='./.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content='./.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content='./dataset/MNIST_data"):  import os, struct  from array import array as pyarray  if dataset == "training_data":  fname_image = os.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content='path.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content='join(path, ’train-  images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content='idx3-ubyte’).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content='replace(’\\\\’, ’/’)  fname_label = os.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content='path.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content='join(path, ’train-  labels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content='idx1-ubyte’).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content='replace(’\\\\’, ’/’)  elif dataset == "testing_data":  fname_image = os.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content='path.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content='join(path, ’t10k-  images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content='idx3-ubyte’).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content='replace(’\\\\’, ’/’)  fname_label = os.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content='path.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content='join(path, ’t10k-  labels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content='idx1-ubyte’).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content='replace(’\\\\’, ’/’)  else:  raise ValueError("dataset must be ’  training_data’ or ’testing_data’")  flbl = open(fname_label, ’rb’)  magic_nr, size = struct.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content='unpack(">II", flbl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content='read  (8))  lbl = pyarray("b", flbl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content='read())  flbl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content='close()  fimg = open(fname_image, ’rb’)  magic_nr, size, rows, cols = struct.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content='unpack(">  IIII", fimg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content='read(16))  img = pyarray("B", fimg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content='read())  fimg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content='close()  ind = [k for k in range(size) if lbl[k] in  digits]  N = len(ind)  images = np.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content='zeros((N, rows, cols))  labels = np.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content='zeros((N, 1), dtype=int)  for i in range(len(ind)):  images[i] = np.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content='array(img[ind[i] * rows *  cols: (ind[i] + 1) * rows * cols]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content='reshape((rows  , cols))  labels[i] = lbl[ind[i]]  return images, labels  This experiment is a binary classification task, so one qubit  is used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content='  def circuit(weights):  num_qubits = 1  machine = pq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content='CPUQVM()  machine.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content='init_qvm()  qubits = machine.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content='qAlloc_many(num_qubits)  cbits = machine.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content='cAlloc_many(num_qubits)  circuit = pq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content='QCircuit()  circuit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content='insert(pq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content='H(qubits[0]))  circuit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content='insert(pq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content='RY(qubits[0], weights[0]))  prog = pq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content='QProg()  prog.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content='insert(circuit)  prog << measure_all(qubits, cbits)  result = machine.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content='run_with_configuration(prog,  cbits, 100)  counts = np.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content='array(list(result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content='values()))  states = np.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content='array(list(result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content='keys())).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content='astype(  float)  # Compute probabilities for each state  probabilities = counts / 100  # Get state expectation  expectation = np.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content='sum(states * probabilities)  return expectation  Here, we first reduce the data’s dimension to 2D through  the pooling layer and the full connection layer, then through  the single qubit circuit, and finally into 2D output through the  full connection layer to adapt to the 2-classification task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content='  class Hybrid(Module):  def __init__(self, shift):  super(Hybrid, self).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content='__init__()  self.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content='shift = shift  def forward(self, input):  self.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content='input = input  expectation_z = circuit(np.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content='array(input.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content='data)  )  result = [[expectation_z]]  requires_grad = input.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content='requires_grad and not  QTensor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content='NO_GRAD  def _backward_mnist(g, input):  input_list = np.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content='array(input.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content='data)  shift_right = input_list + np.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content='ones(  input_list.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content='shape) * self.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content='shift  shift_left = input_list - np.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content='ones(  input_list.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content='shape) * self.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content='shift  gradients = []  for i in range(len(input_list)):  expectation_right = circuit(  shift_right[i])  expectation_left = circuit(  shift_left[i])  gradient = expectation_right -  expectation_left  gradients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content='append(gradient)  gradients = np.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content='array([gradients]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content='T  return gradients * np.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content='array(g)  nodes = []  if input.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content='requires_grad:  nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content='append(QTensor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content='  ComputationalGraphNode(tensor=input, df=lambda g  : _backward_mnist(g, input)))  return QTensor(data=result, requires_grad=  requires_grad, nodes=nodes)  class Net(Module):  def __init__(self):  super(Net, self).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content='__init__()  self.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content='conv1 = Conv2D(input_channels=1,  output_channels=6, kernel_size=(5, 5), stride  =(1, 1), padding="valid")  self.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content='maxpool1 = MaxPool2D([2, 2], [2, 2],  padding="valid")  self.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content='conv2 = Conv2D(input_channels=6,  output_channels=16, kernel_size=(5, 5), stride  =(1, 1), padding="valid")  self.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content='maxpool2 = MaxPool2D([2, 2], [2, 2],  padding="valid")  self.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content='fc1 = Linear(input_channels=256,  output_channels=64)  self.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content='fc2 = Linear(input_channels=64,  output_channels=1)  13  self.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content='hybrid = Hybrid(np.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content='pi / 2)  self.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content='fc3 = Linear(input_channels=1,  output_channels=2)  def forward(self, x):  x = F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content='ReLu()(self.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content='conv1(x))  x = self.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content='maxpool1(x)  x = F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content='ReLu()(self.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content='conv2(x))  x = self.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content='maxpool2(x)  x = tensor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content='flatten(x, 1)  x = F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content='ReLu()(self.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content='fc1(x))  x = self.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content='fc2(x)  x = self.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content='hybrid(x)  x = self.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content='fc3(x)  return x  D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content=' Using Qiskit in VQNet 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content='0  1) Background: In VQNet 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content='0, we can also use IBM’s  Qiskit quantum computing library for quantum machine learn-  ing tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content='  VQNet 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content='0 implements the automatic differential Qiskit  quantum circuit operation class QiskitLayer , which inherits  from  Compatiblelayer .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content='  Compatiblelayer is a class that  is compatible with other framework circuits to VQNet 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content='  The parameters of constructing  QiskitLayer  need to be  passed in a class, which defines the Qiskit quantum cir-  cuit  qiskit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content='QuantumCircuit , as well as its operation and  measurement function  run .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content=' The  run  function needs to  bind the input value to the quantum circuit of Qiskit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content=' Using  QiskitLayer , the input of quantum circuits and automatic  differentiation can be implemented by VQNet 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content='  2) Implementations: We use Qiskit to construct a class  QISKI_VQC .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content=' The variable  self.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content='_circuit  is a quantum  circuit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content=' self.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content='input is a variable input parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content=' In the run  function, you need to use Qiskit’s  assign_parameters to  bind the parameters and use  self.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content='backend.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content='run to run.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content='  class QISKIT_VQC:  def __init__(self, n_qubits, backend, shots)  :  # --- Circuit definition ---  self.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content='_circuit = qiskit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content='  QuantumCircuit(n_qubits)  all_qubits = [i for i in range(  n_qubits)]  self.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content='input = [qiskit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content='circuit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content='  Parameter(’input’)]  self.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content='_circuit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content='h(all_qubits)  self.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content='_circuit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content='barrier()  self.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content='_circuit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content='ry(self.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content='input[0],  all_qubits)  self.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content='_circuit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content='measure_all()  self.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content='backend = backend  self.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content='shots = shots  def run(self,x):  params = dict(zip(self.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content='input, x))  c1 = self.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content='_circuit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content='assign_parameters  (params)  job = self.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content='backend.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content='run(c1,shots=self  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content='shots)  result = job.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content='result().' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content='get_counts()  counts = np.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content='array(list(result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content='values  ()))  states = np.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content='array(list(result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content='keys()  )).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content='astype(float)  # Compute probabilities for each  state  probabilities = counts / self.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content='shots  # Get state expectation  expectation = np.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content='sum(states *  probabilities)  return expectation  The next step is to use VQNet 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content='0 to define the model and  training process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content=' Use  QiskitLayer to add the Qiskit circuit  to the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content='  #define Qiskit circuits class  circuit = QISKIT_VQC(1, simulator, 100)  class Net(Module):  def __init__(self):  super(Net, self).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content='__init__()  self.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content='conv1 = Conv2D(input_channels  =1, output_channels=6, kernel_size=(5, 5),  stride=(1, 1), padding="valid")  self.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content='maxpool1 = MaxPool2D([2, 2],  [2, 2], padding="valid")  self.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content='conv2 = Conv2D(input_channels  =6, output_channels=16, kernel_size=(5, 5),  stride=(1, 1), padding="valid")  self.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content='maxpool2 = MaxPool2D([2, 2],  [2, 2], padding="valid")  self.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content='fc1 = Linear(input_channels  =256, output_channels=64)  self.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content='fc2 = Linear(input_channels=64,  output_channels=1)  self.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content='hybrid = QiskitLayer(circuit,0)  self.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content='fc3 = Linear(input_channels=1,  output_channels=2)  def forward(self, x):  x = F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content='ReLu()(self.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content='conv1(x))  x = self.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content='maxpool1(x)  x = F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content='ReLu()(self.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content='conv2(x))  x = self.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content='maxpool2(x)  x = tensor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content='flatten(x, 1)  x = F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content='ReLu()(self.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content='fc1(x))  x = self.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content='fc2(x)  x = self.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content='hybrid(x)  x = self.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
+page_content='fc3(x)  return x' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E1T4oBgHgl3EQfigQB/content/2301.03251v1.pdf'}
diff --git a/r9FAT4oBgHgl3EQfgB2V/content/tmp_files/2301.08585v1.pdf.txt b/r9FAT4oBgHgl3EQfgB2V/content/tmp_files/2301.08585v1.pdf.txt
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@@ -0,0 +1,1525 @@
+arXiv:2301.08585v1  [hep-ph]  20 Jan 2023
+Probe of a Randall-Sundrum-like model from
+muon pair production at high energy muon
+collider
+S.C. ˙Inan∗
+Department of Physics, Sivas Cumhuriyet University, 58140, Sivas, Turkey
+and
+A.V. Kisselev†
+A.A. Logunov Institute for High Energy Physics, NRC “Kurchatov Institute”,
+142281, Protvino, Russian Federation
+Abstract
+We have examined µ+µ− → 2(µ+µ−) and µ+µ− → µ+µ− colli-
+sions at future high energy muon colliders in the framework of the
+Randall-Sundrum-like model with a small curvature of space-time.
+The collision energies of 3 TeV, 14 TeV and, 100 TeV are addressed.
+Both differential and total cross sections are calculated, and excluded
+bounds on a 5-dimensional gravity scale are obtained depending on
+collision energy and integrated luminosity of the muon collider.
+1
+Introduction
+The Standard Model (SM) has been proven in a lot number of collider exper-
+iments. Nevertheless, we are still searching for solutions for many problems
+that SM cannot give a satisfactory solution.
+One of such problem is the
+so-called hierarchy problem which means the large energy gap between the
+∗Electronic address: sceminan@cumhuriyet.edu.tr
+†Electronic address: alexandre.kisselev@ihep.ru
+1
+
+electroweak scale and gravity scale. The most elegant answer to this phe-
+nomenon has been given in the framework of the Randall-Sundrum (RS)
+model [1] which is based on a 5D theory with one extra dimension compact-
+ified in an orbifold S1/Z2. The main parameters of the RS model are the
+compactification radius rc and AdS5 curvature parameter k (hereinafter re-
+ferred to as the curvature k). The model predicts Kaluza-Klein (KK) gravi-
+tons which are heavy resonances with masses around the TeV scale. The
+most stringent limits on KK graviton masses come from the LHC searches
+for heavy resonances.
+The experimental limits depend on a ratio k/MP,
+where MP is the Planck mass. The CMS collaboration have excluded KK
+graviton masses below 2.3 to 4.0 TeV for the diphoton final state [2]. For the
+dilepton final state the CMS have excluded the RS graviton masses in the
+region 2.47-4.78 TeV [3]. The best lower limit of the ATLAS collaboration,
+4.6 TeV, has been obtained in searching for the diphoton final state [4].
+In papers [5, 6] the Randall-Sundrum-like model with a small curvature
+of the 5-dimensional space-time (RSSC model) has been proposed. In par-
+ticular, a general solution for the warped metric has been obtained [7]. In
+contrast to the original RS model, the RSSC model has an almost continuous
+graviton mass spectrum which is similar to the spectrum of the ADD model
+[8]-[10], if k ≪ MP. Thus, the above mentioned experimental bounds are
+not applied to the RS scenario with a small value of k. A probe of the RSSC
+model at the LHC can be found in [11, 12]. A detailed comparison of the
+RSSC model with the RS model is given in Section 2.
+In the present paper, we intend to examine the RSSC model through the
+µ+µ− → 2(µ+µ−) and µ+µ− → µ+µ− processes at a future muon collider.
+The idea of the muon collider was proposed by F. Tikhonin and G. Budker in
+the late 1960’s [13, 14], and it was also discussed in the early 1980’s [15, 16].
+At present, a great physical potential of the muon collider for collisions of
+elementary particles at very high energies is being actively examined. Its
+advantage lies in the fact that muons can be accelerated in a ring without
+limitation from synchrotron radiation compared to linear or circular electron-
+positron colliders [17]-[22]. For instance, the muon collider may provide a
+determination of the electroweak couplings of the Higgs boson which is sig-
+nificantly better than what is considered attainable at other future colliders
+[23]-[29]. Interest in designing and building a muon collider is also based on
+its capability of probing the physics beyond the SM. In a number of recent
+papers searches for SUSY particles [30], WIMPs [31], vector boson fusion
+[32], leptoquarks [33], lepton flavor violation [34], and physics of (g −2)µ [35]
+2
+
+at the muon colliders are presented. For more details on a spectacular oppor-
+tunity of the muon collider in the direct exploration of the energy frontier,
+see [36].
+In the µ+µ− → 2(µ+µ−) scattering one pair of the outgoing muons with
+large transverse momenta and large dimuon invariant mass is detected, while
+the other two scattered muons escape a detector. Our goal is to obtain bounds
+on the 5-dimensional gravity scale M5 which can be probed at TeV and multi-
+TeV muon colliders. The gravity contribution to this process comes from the
+subprocess V V → G → µ+µ−, where V is the γ or Z boson, G is the KK
+graviton, and a summation over all KK gravitons is assumed. We will also
+study the µ+µ− → µ+µ− scattering, taking into account contributions from
+s- and t-channel graviton exchanges, to derive the bounds on M5. Note that
+the processes we are interested in can be easily distinguished experimentally
+from each other, since they have quite different distributions in invariant
+mass of the detected dimuon pair.
+The paper is organized as follows. In the next section, the detailed de-
+scription of the RSSC model is presented. The production of four muons
+via vector boson fusion at the muon collider is examined in section 3. The
+bounds on 5-dimensional Planck scale M5 are obtained. In section 4 we study
+the µ+µ− → µ+µ− process and we calculate the values of M5 which can be
+probed at the muon collider.
+2
+Model of warped extra dimension with small
+curvature
+In this section, we describe the RSSC model in detail and compare it with
+the original RS model. The RS scenario with one extra dimension and two
+branes [1] was proposed as an alternative to the ADD scenario with large flat
+extra dimensions [8]-[10]. It has the following background warped metric
+ds2 = e−2σ(y) ηµν dxµ dxν − dy2 ,
+(1)
+where ηµν is the Minkowski tensor with the signature (+, −, −, −), y is an
+extra coordinate, and σ(y) is the warp factor. The periodicity condition y =
+y +2πrc is imposed, and the points (xµ, y) and (xµ, −y) are identified. Thus,
+we have a model of gravity in a slice of the AdS5 space-time compactified to
+the orbifold S1/Z2. This orbifold has two fixed points, y = 0, and y = πrc.
+3
+
+There are two branes located at these points (called Planck and TeV brane,
+respectively). The SM fields are confined to the TeV brane.
+The classical action of the RS model is given by [1]
+S =
+�
+d4x
+� πrc
+−πrc
+dy
+√
+G (2 ¯
+M3
+5R − Λ)
++
+�
+d4x
+�
+|g(1)| (L1 − Λ1) +
+�
+d4x
+�
+|g(2)| (L2 − Λ2) ,
+(2)
+where GMN(x, y) is the 5-dimensional metric, with M, N = 0, 1, 2, 3, 4, µ =
+0, 1, 2, 3. The quantities
+g(1)
+µν (x) = Gµν(x, y = 0) ,
+g(2)
+µν (x) = Gµν(x, y = πrc)
+(3)
+are induced metrics on the branes, L1 and L2 are brane Lagrangians, G =
+det(GMN), g(i) = det(g(i)
+µν). The parameter ¯
+M5 is a reduced 5-dimensional
+Planck scale related to the fundamental gravity scale M5, namely,
+¯
+M5 =
+M5/(2π)1/3. The parameter Λ is a 5-dimensional cosmological constant, and
+Λ1,2 are brane tensions.
+For the first time, the solution for σ(y) in (1) has been obtained in [1],
+σRS(y) = k|y| ,
+(4)
+where k is a parameter with a dimension of mass. It defines the curvature
+of the 5-dimensional space-time. Later on in [7] the generalized solution for
+σ(y) was derived
+σ(y) = krc
+2
+�����Arccos
+�
+cos y
+rc
+����� −
+����π − Arccos
+�
+cos y
+rc
+�����
+�
++ π |k|rc
+2
+−C , (5)
+where C is a y-independent quantity. Note that the brane tensions obey the
+fine-tuning relations
+Λ = −24 ¯
+M3
+5k2 ,
+Λ1 = − Λ2 = 24 ¯
+M3
+5k .
+(6)
+Here Arccos(z) is a principal value of the multivalued inverse trigonometric
+function arccos(z).
+Let us underline that the generalized solution (5) (i)
+obeys the orbifold symmetry y → −y; (ii) makes the jumps of σ′(y) on both
+branes; (iii) has explicit symmetry with respect to the branes. More details
+can be found in [7].
+4
+
+By taking C = 0 in (5), we get the RS model (4), while taking C = πkrc,
+we come to the RS-like scenario with the small curvature of the space-time
+(RSSC model, see [5]-[7]).
+It is worth to remind the main features of the RSSC model in comparison
+with those of the RS model.
+The interactions of the Kaluza-Klein (KK)
+gravitons h(n)
+µν with the SM fields on the TeV brane are given by the effective
+Lagrangian density
+Lint = − 1
+¯
+MPl
+h(0)
+µν (x) Tαβ(x) ηµαηνβ − 1
+Λπ
+∞
+�
+n=1
+h(n)
+µν (x) Tαβ(x) ηµαηνβ ,
+(7)
+were
+¯
+MPl = MPl/
+√
+8π is the reduced Planck mass, T µν(x) is the energy-
+momentum tensor of the SM fields. The coupling constant is equal to
+Λπ = ¯
+M5
+� ¯
+M5
+k
+.
+(8)
+The hierarchy relation looks like
+¯
+M2
+Pl =
+¯
+M3
+5
+k
+�
+e2πkrc − 1
+�
+.
+(9)
+To compare, in the original RS model it is defined as ¯
+M2
+Pl = ( ¯
+M3
+5/k)
+�
+1 − e−2πkrc�
+.
+It usually assumed that kπrc ≫ 1.
+The masses of the KK gravitons are proportional to the curvature k [6]
+mn = xnk ,
+n = 1, 2, . . . ,
+(10)
+where xn are zeros of the Bessel function J1(x). Should we take k ≪ ¯
+M5 ∼ 1
+TeV, the mass splitting ∆m will be very small, ∆m ≃ πk, and we come to
+an almost continuous mass spectrum, similar to the mass spectrum of the
+ADD model [8]. On the contrary, in the RS model the gravitons are heavy
+resonances with masses above one-few TeV.
+As was shown in a number of phenomenological papers on the RSSC
+model [11, 12], the cross sections weakly depend on the parameter k, if k ≪
+¯
+M5. That is why, in what follows we will put k = 1 GeV.
+5
+
+3
+Production of two muon pairs in muon col-
+lisions
+Let us consider the process µ−µ+ → µ−V1V2µ+ → 2(µ+µ−) shown in Fig. 1.
+Our goal is to estimate a contribution of the KK gravitons to the gauge
+boson fusion V V → l−l+ in the framework of the RSSC model described in
+the previous section. The amplitude of this subprocess is defined as M =
+MSM + MKK, where MSM is the SM term, and MKK is given by the sum of
+s-channel KK gravitons
+MKK =
+1
+2Λ2
+π
+∞
+�
+n=1
+�
+¯u(p1)Γµν
+2 v(p2)
+Bµναβ
+s − m2
+n + i mnΓn
+Γαβρσ
+1
+eρ(k1)eσ(k2)
+�
+. (11)
+Here k1, k2, p1, p2 and eρ(k1), eσ(k2) are, respectively, incoming photon mo-
+G
+µ+
+l+
+l−
+V2
+V1
+µ−
+µ−
+µ+
+Figure 1: The Feynman diagrams describing contribution of the KK graviton
+G to the collision of two vector bosons V1, V2 = γ or Z, with two outgoing
+charged leptons at the muon collider.
+menta, outgoing lepton momenta and polarization vectors of photons. Γn is
+the total width of the graviton with the mass mn. The coherent sum in (11)
+is over all massive KK modes. The Feynman rules for the KK graviton were
+derived in [37, 38] (see also [39]). In particular, the KK–V V vertex looks
+like
+Γαβρσ
+1
+= − i
+2 {[m2
+V + (k1 · k2)] Cαβρσ + Dαβρσ} ,
+(12)
+6
+
+where
+Cαβρσ = ηαρηβσ + ηασηβρ − ηαβηρσ ,
+(13)
+Dαβρσ = ηαβkσ
+1kρ
+2 − (ηασkβ
+1kρ
+2 + ηαρkσ
+1kβ
+2 − ηρσkα
+1 kβ
+2 )
+− (ηβσkα
+1 kρ
+2 + ηβρkσ
+1kα
+2 − ηρσkβ
+1kα
+2 ) .
+(14)
+The KK–l−l+ vertex is defined as
+Γµν
+2 = − i
+8 [γµ(pν
+1 − pν
+2) + γν(pµ
+1 − pµ
+2)] .
+(15)
+Finally, Bµναβ in (11) is a tensor part of the KK graviton propagator. Its
+explicit expression was derived in [37, 38]. We can safely omit terms in Bµναβ
+which give zero contribution to eq. (11). Then we find
+Bµναβ = ηµαηνβ + ηµβηνα − 2
+3 ηµνηαβ .
+(16)
+The s-channel contribution of the KK gravitons is equal to
+S(s) = 1
+Λ2π
+∞
+�
+n=1
+1
+s − m2n + i mnΓn
+.
+(17)
+This sum has been calculated in ref. [40],
+S(s) = −
+1
+4 ¯
+M3
+5
+√s
+sin(2A) + i sinh(2ε)
+cos2A + sinh2ε
+,
+(18)
+where
+A =
+√s
+k ,
+ε = 0.045
+�√s
+¯
+M5
+�3
+.
+(19)
+The squared amplitude of the subprocess V V → l−l+ is a sum of three
+terms,
+|M|2 = |MSM|2 + |MKK|2 + |Mint|2 ,
+(20)
+where MSM denotes the SM amplitude, while MKK and Mint denote pure KK
+graviton and interference terms. In [39] the quantities |MSM(γγ → l−l+)|2,
+|MKK(γγ → l−l+)|2, and |Mint(γγ → l−l+)|2 were calculated for massless
+leptons. The results of our calculations of the squared amplitudes |M(V V →
+l−l+)|2 for nonzero ml and mV (V = γ, Z) are presented in Appendix A.
+7
+
+The virtual KK graviton production should lead to deviations from the
+SM predictions in a magnitude of the cross section. The cross section of our
+process µ−µ+ → µ−V1V2µ+ → 2(µ+µ−) is defined by the formula
+dσ =
+τmax
+�
+τmin
+dτ
+xmax
+�
+xmin
+dx
+x
+�
+V1,V2=γ,ZT ,ZL
+fV1/µ+(x, Q2)fV2/µ−(τ/x, Q2) dˆσ(V1V2 → µ+µ−) .
+(21)
+Here
+xmax = 1 − mµ
+Eµ
+, τmax =
+�
+1 − mµ
+Eµ
+�2
+, xmin = τ/xmax , τmin = p2
+⊥
+E2µ
+,
+(22)
+and p⊥ is the transverse momenta of the outgoing photons. The boson distri-
+butions inside the muon beam, fγ/µ±(x, Q2), fZT /µ±(x, Q2), and fZL/µ±(x, Q2)
+are [41]
+fγ/µ±(x, Q2) = α
+2π
+1 + (1 − x)2
+x
+ln Q2
+m2
+µ
+,
+(23)
+and [42, 43]
+fZT /µ±(x, Q2) = α±
+Z
+2π
+1 + (1 − x)2
+x
+ln Q2
+m2
+Z
+,
+fZL/µ±(x, Q2) = α±
+Z
+π
+(1 − x)
+x
+,
+(24)
+where
+α±
+Z =
+α
+(cos θW sin θW)2
+�
+(g±
+V )2 + (g±
+A)2�
+,
+(25)
+g±
+V = −1/4∓sin2 θW, g±
+A = 1/4, and mµ is the muon mass. The variable x in
+(23), (24) is the ratio of the boson energy and energy of the incoming muon
+Eµ. Note that the Z boson has different distributions for its transverse (T)
+and longitudinal (L) polarizations (24).
+The differential cross section of the subprocess V1V2 → µ+µ−, where
+V1,2 = γ or Z, is a sum of helicity amplitudes squared
+dˆσ
+dΩ =
+1
+64π2ˆs
+�
+λ1,λ2,λ3,λ4
+|MV1V2
+λ1λ2λ3λ4|2,
+(26)
+8
+
+where
+√
+ˆs is a collision energy of this subprocess, and λ1,2 (λ3,4) are boson
+(muon) helicities. In boson distributions (23), (24) we put Q2 = ˆs, where
+√
+ˆs = 2Eµ
+√τ is the invariant energy of the subprocess V1V2 → µ+µ−.
+As it was mentioned in [44], the scattering angle for high energy initial
+muons is peaked near θµ ≈ 0.02◦ − 1.2◦. These very forward muons would
+most likely escape a muon detector away from colliding beams. Thus, only
+muons produced in the boson fusion (as those shown in Fig. 1) will be de-
+tected. We apply the cuts to the final muons, pt > 50 GeV, |η| < 2.5. The
+main purpose of these cuts is to ensure that only two muons are detected in
+the final state. As was already mentioned at the end of section 2, we take
+k = 1 GeV.
+The results of our numerical calculations of the differential cross sections
+for the µ+µ− → 2(µ+µ−) scattering at the future muon collider are presented
+in Fig. 2. The predictions for three collision invariant energies of the muon
+collider are shown. As one can see, for each energy the cross sections rise as
+the invariant mass of the detected muons grows, while the SM cross sections
+decrease rapidly with an increase of mµ+µ−. We have also calculated the
+differential cross sections via transverse momentum of the detected muons,
+see Fig. 3.
+The total cross sections as functions of the minimal invariant mass of two
+detected muons at the muon collider mµ+µ−,min are shown in Fig. 4. The cross
+sections strongly depend on the fundamental gravity scale ¯
+M5. If ¯
+M5 = 1
+TeV, the cross section exceeds the SM one for all three collision energies. For
+larger values of ¯
+M5 the total cross section strongly dominates over the SM
+cross section for √s = 14 TeV and √s = 100 TeV.
+All this enables us to derive the excluded bounds on the 5-dimensional
+reduced Planck scale ¯
+M5. To derive them, we apply the following formula
+for the statistical significance SS [45]
+SS =
+�
+2[(S − B ln(1 + S/B)] ,
+(27)
+where S is the number of signal events and B is the number of background
+(SM) events. We define the regions SS ⩽ 1.645 as the regions that can be
+excluded at the 95% C.L. To reduce the SM background, we used the cuts
+mµ−µ+ > 1 TeV, mµ−µ+ > 5 TeV, and mµ−µ+ > 50 TeV for the colliding
+energy of 3 TeV, 14 TeV, and 100 TeV, respectively. The results are shown
+in Fig. 5. Our best limits for √s = 3 TeV, 14 TeV and 100 TeV are ¯
+M5 = 3.8
+TeV, 13.1 TeV, and 106.4 TeV, respectively.
+9
+
+500
+1000
+1500
+2000
+10
+-6
+10
+-5
+10
+-4
+10
+-3
+10
+-2
+10
+-1
+10
+0
+10
+1
+10
+2
+10
+3
+2000
+4000
+6000
+8000
+15000
+30000
+45000
+60000
+s
+TeV
+d
+dm
+(fb/GeV )
+m
+ (GeV)
+s
+TeV
+ M
+5
+TeV
+ M
+5
+TeV
+ M
+5
+TeV
+ SM
+m
+ (GeV)
+s
+TeV
+m
+ (GeV)
+Figure 2: The differential cross sections for the process µ+µ− → 2(µ+µ−)
+via invariant mass of two detected muons at the muon collider. The left,
+middle and right panels correspond to the colliding energy of 3 TeV, 14 TeV,
+and 100 TeV. The curves (from the top down) correspond to ¯
+M5 = 1 TeV,
+¯
+M5 = 2 TeV, and ¯
+M5 = 3 TeV, respectively. The SM cross sections (low
+curves) are also shown.
+4
+Production of one muon pair in muon col-
+lisions
+As was said in the previous section, we expect that in the µ+µ− → 2(µ+µ−)
+process only two final muons are detected, while two scattered muons escape
+the detector. In the process µ−µ+ → µ+µ− two outgoing muons with high
+transverse momenta are also detected. However, in such a case the invariant
+mass of the dimuon system mµ+µ− is fixed and equal to the collision energy
+√s, that does not occur for the µ−µ+ → 2(µ+µ−) scattering. That is why
+one can easily discriminate between these two process experimentally by
+measuring the invariant mass of the detected muon pair.
+The virtual KK graviton exchanges have to give a contribution to the cross
+sections of the µ−µ+ → µ+µ− scattering. It is shown in Fig. 6. The analytical
+10
+
+200
+400
+600
+800
+1000
+10
+-6
+10
+-5
+10
+-4
+10
+-3
+10
+-2
+10
+-1
+10
+0
+10
+1
+10
+2
+10
+3
+10
+4
+10
+5
+10
+6
+10
+7
+1000
+2000
+3000
+4000
+5000
+10000
+20000
+30000
+40000
+d
+/dp
+t
+ (fb/GeV)
+p
+t
+ (GeV)
+ M
+5
+TeV
+ M
+5
+TeV
+ M
+5
+TeV
+ SM
+p
+t
+ (GeV)
+p
+t
+ (GeV)
+Figure 3: The differential cross sections for the process µ+µ− → 2(µ+µ−) via
+transverse momentum of the detected muons at the muon collider.
+expressions for amplitudes squared of this collision are given in Appendix B.
+Using them we have calculated the differential cross sections for the µ+µ− →
+µ+µ− scattering at the muon collider depending on transverse momentum of
+the final muons pt, taking into account gravity contribution. Our result are
+presented in Fig. 7 for three values of the collision energy √s and different
+values of the reduced 5-dimensional Planck scale
+¯
+M5. As we can see, for
+√s = 14 TeV and √s = 100 TeV, the cross section significantly dominates
+the SM one, especially for large pt. It is worth to compare this figure with
+the pt-distribution in Fig. 3. The oscillations of the curves in Fig. 3 originate
+from the function S(s) (18) which describes the s-channel contribution of
+all KK gravitons, since the invariant energy of the detected dimuon pair
+is not a constant. On the other hand, in the process µ−µ+ → µ+µ− the
+invariant energy of the dimuon pair is fixed, and we have no oscillations.
+The differential cross sections integrated in pt from the minimal transverse
+momentum of the detected muons pt,min are presented in Fig. 8.
+As before, we have calculated the excluded bounds on ¯
+M5 which can be
+probed in the process µ+µ− → µ+µ− depending on the integrated luminosity
+11
+
+500
+1000
+1500
+2000
+10
+-6
+10
+-4
+10
+-2
+10
+0
+10
+2
+10
+4
+10
+6
+10
+8
+2000
+4000
+6000
+8000
+15000
+30000
+45000
+60000
+10
+-6
+10
+-4
+10
+-2
+10
+0
+10
+2
+10
+4
+10
+6
+10
+8
+s
+TeV
+(fb)
+m
+min
+ (GeV )
+s
+TeV
+ M
+5
+TeV
+ M
+5
+TeV
+ M
+5
+TeV
+ SM
+m
+min
+ (GeV )
+s
+TeV
+m
+min
+ (GeV )
+Figure 4: The total cross sections for the process µ+µ− → 2(µ+µ−) via min-
+imal invariant mass of two detected muons at the muon collider mµ+µ−,min.
+of the future muon collider, see Fig. 9. We have used eq. (27) for the statistical
+significance. In doing so, the cuts pt,min = 0.5 TeV, pt,min = 2.5 TeV, and
+pt,min = 25 TeV were applied for the 3 TeV, 14 TeV, and 100 TeV center-
+of-mass energies, respectively. We see that in the process µ+µ− → µ+µ−
+the scales up to ¯
+M5 = 3.85 TeV, 17.8 TeV and 126.3 TeV can be probed for
+√s = 3 TeV, 14 TeV, and 100 TeV, respectively. We conclude that these
+limits on ¯
+M5 are stronger than the limits obtained in the previous section for
+the µ−µ+ → 2(µ+µ−) scattering.
+5
+Conclusions
+We have examined two collisions at future TeV and multi-TeV muon colliders
+in the Randall-Sundrum-like model with the small curvature (RSSC model)
+[5, 6]. It is the model with one extra dimension and warped metric whose
+5-dimensional space-time curvature k is about one GeV. The other main
+parameter of the RSSC model, the 5-dimensional Planck scale ¯
+M5, is equal
+to (larger than) one TeV.
+12
+
+10
+0
+10
+1
+10
+2
+10
+3
+10
+0
+10
+1
+10
+2
+10
+0
+10
+1
+10
+2
+10
+3
+10
+4
+10
+0
+10
+1
+10
+2
+10
+3
+10
+4
+10
+5
+10
+6
+ 
+s
+TeV  
+M
+5
+ (TeV)
+L (fb
+)
+ 
+s
+TeV  
+L (fb
+)
+L (fb
+)
+ 
+s
+TeV  
+Figure 5: The excluded bounds on the reduced fundamental gravity scale
+¯
+M5 via integrated luminosity of the muon collider for the process µ+µ− →
+2(µ+µ−). The left, middle and right panels correspond to the colliding energy
+of 3 TeV, 14 TeV, and 100 TeV.
+We have studied the µ−µ+ → 2(µ+µ−) scattering first. The collision goes
+via the V V → µ−µ+ scattering, where V = γ, Z. For this scattering, the ana-
+lytical expressions for the squared amplitudes, including the gravity, SM and,
+interference terms, are derived for the first time for massive leptons. They
+are presented in Appendix A. Then the differential cross sections depending
+on the invariant mass of the detected muons mµ+µ− are calculated for three
+values of the reduced 5-dimensional Planck scale ¯
+M5 for 3 TeV, 14 TeV and
+100 TeV muon colliders. The total cross section is calculated as a function of
+the minimal value of mµ+µ−. As a result, the excluded bounds on the scale
+¯
+M5 are obtained. They are ¯
+M5 = 3.80 TeV, 13.1 TeV and 106.4 TeV, for the
+collision energy of √s = 3 TeV, 14 TeV and 100 TeV, respectively.
+The µ−µ+ → µ+µ− scattering is also studied. As in the previous case, we
+have calculated the gravity, SM and interference squared amplitudes analyt-
+ically, see Appendix B. It enabled us to estimate numerically the differential
+cross sections as functions of the transverse momenta of the outgoing muons.
+The total cross sections are also calculated. Finally, the excluded bounds on
+13
+
+µ+
+G
+µ−
+µ+
+µ−
+G
+µ+
+µ+
+µ−
+µ−
+µ+
+Figure 6: The Feynman diagrams describing contribution of the KK graviton
+G to the µ−µ+ → µ+µ− scattering.
+the main parameter of the RSSC model, the scale ¯
+M5, have been obtained.
+We have shown that the values of ¯
+M5 = 3.85 TeV, 17.8 TeV and 126.3 TeV
+can be probed at 3 TeV, 14 TeV, and 100 TeV muon colliders.
+Let us remember that
+¯
+M5 = M5/(2π)1/3 ≈ 0.54M5, where M5 is the
+fundamental 5-dimensional gravity scale M5 in the RSSC model. It means
+that the bounds on the scale M5 are approximately twice stronger.
+Let us stress that our bounds on ¯
+M5 should not be directly compared
+with the current experimental bound on the 5-dimensional Planck scale of
+the original RS model [1], since the mass spectra of the KK gravitons and,
+correspondingly, experimental signatures are quite different in the RSSC and
+RS models [7].
+Appendix A. Squared amplitudes for V V →
+l−l+ scattering
+Our calculations give the following analytical expressions for the squared
+amplitudes for the γγ → l−l+ collision in eq. (20)
+|MSM|2 =
+8e4
+(t − m2
+l )2(s + t − m2
+l )2[−34m8
+l + m6
+l (60s + 64t)
+− m4
+l (31s2 + 52st + 28t2) + m2
+l s(s2 − 2st − 4t2)
+− t(s + t)(s2 + 2st + 2t2)] ,
+(A.1)
+14
+
+200
+400
+600
+800
+1000
+10
+-6
+10
+-5
+10
+-4
+10
+-3
+10
+-2
+10
+-1
+10
+0
+10
+1
+10
+2
+10
+3
+10
+4
+10
+5
+10
+6
+10
+7
+1000
+2000
+3000
+4000
+5000
+10000
+20000
+30000
+40000
+s
+TeV
+d
+dp
+t
+ (fb/GeV)
+p
+t
+ (GeV)
+s
+TeV
+ M
+5
+TeV
+ M
+5
+TeV
+ M
+5
+TeV
+ SM
+p
+t
+ (GeV)
+s
+TeV
+p
+t
+ (GeV)
+Figure 7: The differential cross sections for the process µ+µ− → µ+µ− via
+transverse momentum of the detected muons at the muon collider. The left,
+middle and right panels correspond to the colliding energy of 3 TeV, 14 TeV,
+and 100 TeV. The curves (from the top down) correspond to ¯
+M5 = 1 TeV,
+¯
+M5 = 2 TeV, and ¯
+M5 = 3 TeV, respectively. The SM cross sections (low
+curves) are also shown.
+|MKK|2 = −1
+8|S(s)|2[2m8
+l − 8m6
+l t + m4
+l (s2 + 4st + 12t2) − 2m2
+l t(s + 2t)2
++ t(s + t)(s2 + 2st + 2t2)] ,
+(A.2)
+|Mint|2 = −
+e2[S(s) + S⋆(s)]
+2(t − m2
+l )(s + t − m2
+l )[−2m8
+l + m6
+l (3s + 4t) + m4
+l s(3s − 4t)
+− m2
+l (s3 + 2s2t + 3st2 + 4t3) + t(s + t)(s2 + 2st + 2t2)] ,
+(A.3)
+where s, t are Mandelstam variables, ml is the lepton mass, and S(s) is
+defined in the text (17)-(19). If we take ml = 0 we get known results obtained
+in [39].
+15
+
+200
+400
+600
+800
+1000
+10
+-2
+10
+-1
+10
+0
+10
+1
+10
+2
+10
+3
+10
+4
+10
+5
+10
+6
+10
+7
+10
+8
+10
+9
+10
+10
+10
+11
+1000
+2000
+3000
+4000
+5000
+10000
+20000
+30000
+40000
+s
+TeV
+ (fb)
+p
+t, min
+ (GeV)
+s
+TeV
+ M
+5
+TeV
+ M
+5
+TeV
+ M
+5
+TeV
+ SM
+p
+t, min
+ (GeV)
+s
+TeV
+p
+t, min
+ (GeV)
+Figure 8: The total cross sections for the process µ+µ− → µ+µ− via minimal
+transverse momentum of the outgoing muons pt,min.
+16
+
+10
+0
+10
+1
+10
+2
+10
+3
+10
+0
+10
+1
+10
+2
+10
+0
+10
+1
+10
+2
+10
+3
+10
+4
+10
+0
+10
+1
+10
+2
+10
+3
+10
+4
+10
+5
+10
+6
+ 
+s
+TeV  
+M
+5
+(TeV)
+L (fb
+ 
+)
+ 
+s
+TeV  
+L (fb
+)
+ 
+s
+TeV  
+L (fb
+)
+Figure 9: The excluded bounds on the reduced fundamental gravity scale ¯
+M5
+via integrated luminosity of the muon collider for the process µ+µ− → µ+µ−.
+The left, middle and right panels correspond to the colliding energy of 3 TeV,
+14 TeV, and 100 TeV.
+17
+
+For the ZZ → l−l+ collision our calculations result in the following for-
+mulas
+|MSM|2 =
+g4
+Z
+(t − m2
+l )2(s + t − m2
+l − 2m2
+Z)2{−2m8
+l [−300 cos(2θw) + 184 cos(4θw)
+− 68 cos(6θw) + 17 cos(8θw) + 195]
++ 4m6
+l [−4m2
+Z(−94 cos(2θw) + 59 cos(4θw) − 24 cos(6θw)
++ 6 cos(8θw) + 56) + 163s + 196t − 8(33s + 37t) cos(2θw)
++ 18(9s + 10t) cos(4θw) + (15s + 16t)(cos(8θw) − 4 cos(6θw))]
++ m4
+l [−2m4
+Z(−164 cos(2θw) + 128 cos(4θw)
++ 23(−4 cos(6θw) + cos(8θw) + 3))
++ 8m2
+Z(72s + 65t − 4(32s + 27t) cos(2θw) + (84s + 68t) cos(4θw)
++ (10s + 7t)(cos(8θw) − 4 cos(6θw))) − 301s2 − 436t2 − 668st
++ 4(125s2 + 260st + 156t2) cos(2θw) − 8(39s2 + 78st + 46t2) cos(4θw)
+− (31s2 + 52st + 28t2)(cos(8θw) − 4 cos(6θw))]
++ m2
+l [7s3 + 50s2t + 92st2 + s(s2 − 2st − 4t2)(cos(8θw) − 4 cos(6θw))
++ 80t3 − 4(3s3 + 14s2t + 24st2 + 24t3) cos(2θw)
++ 8(s + 2t)(s2 + st + 3t2) cos(4θw) + 2m2
+Z(79s2 + 252st + 112t2
+− 4(31s2 + 108st + 52t2) cos(2θw) + 8(9s2 + 34st + 17t2) cos(4θw)
++ (5s2 + 28st + 16t2)(cos(8θw) − 4 cos(6θw)))
++ 2m4
+Z(−263s − 414t + (428s + 696t) cos(2θw) − 16(16s + 27t) cos(4θw)
+− 21(s + 2t)(cos(8θw) − 4 cos(6θw)))
++ 2m6
+Z(−156 cos(2θw) + 96 cos(4θw) − 36 cos(6θw) + 9 cos(8θw) + 91)]
+− [−28 cos(2θw) + 16 cos(4θw) − 4 cos(6θw) + cos(8θw) + 19]
+× [4m8
+Z − 4m6
+Z(s + 3t) + m4
+Z(s2 + 6st + 14t2) − 2m2
+Zt(s + 2t)2
++ t(s + t)(s2 + 2st + 2t2)]} ,
+(A.4)
+|MKK|2 =
+1
+288|S(s)|2{−72m8
+Z + 6m6
+Z(−40m2
+l + 9s + 48t)
+− 4m4
+Z[−2m2
+l (5s + 96t) + 136m4
+l + 9t(7s + 12t)]
++ 3m2
+Z[−80m6
+l + m4
+l (256t − 14s) − 4m2
+l (s2 + 29st + 68t2)
++ 9s3 + 42s2t + 114st2 + 96t3)] − 36[m4
+l − 2m2
+l t + t(s + t)]
+× (2m4
+l − 4m2
+l t + s2 + 2st + 2t2)} ,
+(A.5)
+18
+
+|Mint|2 = −
+g2
+Z[S(s) + S⋆(s)]
+96(t − m2
+l )(s + t − m2
+l − 2m2
+Z){−12m8
+l [cos(4θw) − 2 cos(2θw)]
++ 2m6
+l [(16m2
+Z + 3(3s + 4t)(cos(4θw) − 2 cos(2θw)) + 6(s − 2t)]
++ 2m4
+l [−m2
+Z((9s + 4t)(cos(4θw) − 2 cos(2θw))
++ 24s + 60t) + 8m4
+Z(−2 cos(2θw) + cos(4θw) + 4) + 6(3s2 + st + 6t2)
++ 3s(3s − 4t)(cos(4θw) − 2 cos(2θw))]
++ m2
+l [4m2
+Z((2s2 − 3st + 14t2)(cos(4θw) − 2 cos(2θw))
++ 2s2 + 5st + 46t2) + 4m4
+Z((s − 10t)(cos(4θw) − 2 cos(2θw)) − 38t)
++ 8m6
+Z(−2 cos(2θw) + cos(4θw) + 5)
+− 3(3s3 + 10s2t + 2(s3 + 2s2t + 3st2 + 4t3)
+× (cos(4θw) − 2 cos(2θw)) + 24st(s + t))]
++ 6[−2 cos(2θw) + cos(4θw) + 2][2m8
+Z + m6
+Z(s − 8t)
++ m4
+Z(−s2 + 2st + 12t2) − m2
+Zt(s2 + 7st + 8t2)
++ t(s + t)(s2 + 2st + 2t2)]} ,
+(A.6)
+where θw is the Weinberg angle, gZ = e/[sin(θw) cos(θw)] is the weak coupling
+constant, and mZ is the mass of the Z boson. Because of conservation of
+helicity, in the massless limit ml = mZ = 0 s-channel graviton amplitudes
+squared (A.2) and (A.5) are proportional to the factor t(s + t) = s(sin θ)2/4,
+where θ is the scattering angle.
+19
+
+Finally, the SM squared amplitude for the γZ → l+l− collision looks like
+|MSM|2 =
+4g4
+Z
+(t − m2
+l )2(s + t − m2
+l − m2
+Z)2
+× {−2m8
+l [184 cos(4θw) + 17 cos(8θw) − 300 cos(2θw)
+− 68 cos(6θw) + 195] + 4m6
+l [−2(59 cos(4θw) + 6 cos(8θw) − 94 cos(2θw)
+− 24 cos(6θw) + 56)m2
+Z + 163s + 196t − 8(33s + 37t) cos(2θw)
++ 18(9s + 10t) cos(4θw) + (15s + 16t)(cos(8θw) − 4 cos(6θw))]
++ m4
+l [(68 cos(2θw) + 44 cos(6θw) − 56 cos(4θw) − 11 cos(8θw) − 25)m4
+Z
++ 4[72s + 65t − 4(32s + 27t) cos(2θw) + (84s + 68t) cos(4θw)
++ (10s + 7t)(cos(8θw) − 4 cos(6θw))]m2
+Z − 301s2 − 436t2 − 668st
++ 4(125s2 + 260ts + 156t2) cos(2θw) − 8(3s2 + 78st + 46t2) cos(4θw)
+− (31s2 + 52st + 28t2)(cos(8θw) − 4 cos(6θw))]
++ m2
+l [7s3 + 50s2t + 92st2 + (s2 − 2st − 4t2)(cos(8θw)
+− 4 cos(6θw))s + 80t3 + (79s2 + 252st + 112t2
++ ((5(cos(8θw) − 4 cos(6θw) + 11) + 56 cos(4θw)
+− 92(cos(2θw)))m2
+Z − 141s − 226t + 4(57s + 94t)(cos(2θw))
+− 8(17s + 29t)(cos(4θw)) − 11(s + 2t)(cos(8θw) − 4 cos(6θw)))m2
+Z
+− 4(31s2 + 108st + 52t2) cos(2θw) + 8(9s2 + 34st + 17t2) cos(4θw)
++ (5s2 + 28st + 16t2)(cos(8θw) − 4 cos(6θw)))m2
+Z
+− 4(3s3 + 14s2t + 24st2 + 24t3) cos(2θw)
++ 8(s + 2t)(s2 + st + 3t2) cos(4θw)]
+− t[16 cos(4θw) + cos(8θw) − 28 cos(2θw) − 4 cos(6θw) + 19]
+× (s + t − m2
+Z)(s2 + 2t2 + 2st − 2tm2
+Z + m4
+Z)} .
+(A.7)
+The contribution to the γZ → l+l− collision from the gravitons G is zero,
+since there is no γZG vertex.
+Note that in the limit ml = mZ = 0 all
+squared amplitudes depend on variables s and t(s + t) = tu, where u is the
+Mandelstam variable.
+20
+
+Appendix B. Squared amplitudes for l+l− →
+l+l− scattering
+Here we present the result of our calculations of the squared amplitudes
+for the l+l− → l+l− process (both incoming and outgoing leptons have the
+same flavor). The SM squared amplitude has both the Z boson and photon
+contributions. The latter one is given by the formula
+|MSM|2 = 16e4
+s2t2 [m4
+l (5s2 + 11st + 5t2) + 4m2
+l (s + t)(s2 + 5st + t2)
++ (s2 + st + t2)2] ,
+(B.1)
+where ml is the lepton mass. Note that the photon contribution to |MSM|2
+is dominant. That is why, we do not present (rather complicated) analytical
+expression for the Z boson contribution to |MSM|2. The graviton squared
+amplitude is defined by KK graviton exchanges in s-, and t-channels,
+|MKK|2 =
+1
+4608{|S(s)|2F1(s, t) + |S(t)|2F1(t, s)
++ [S(s)S(t)⋆ + S(s)⋆S(t)]F2(s, t)} .
+(B.2)
+Finally, the interference term of |M|2 is equal to (neglecting small Z boson
+contribution)
+|Mint|2 = − e2
+24st{[S(s) + S(t)⋆]F3(s, t) + [S(s)⋆ + S(t)]F3(t, s)} .
+(B.3)
+Here S(s) is defined by eqs. (18), (19), and the following functions are intro-
+duced
+F1(s, t) = 6656m8
+l + m6
+l (8576s + 10752t) + m4
+l (3440s2 + 7296t2 + 10752st)
++ m2
+l (360s3 + 2376s2t + 4320st2 + 2304t3)
++ 9s4 + 90s3t + 378s2t2 + 576st3 + 288t4 ,
+(B.4)
+F2(s, t) = 7552m8
+l + 15168m6
+l (s + t) + m4
+l (7248(s2 + t2) + 14968st)
++ m2
+l (1032(s3 + t3) + 3690st(s + t))
++ 36(s4 + t4) + 225st(s + t) + 378s2t2 ,
+(B.5)
+21
+
+F3(s, t) = m6
+l (576s + 512t) + m4
+l (552s2 + 448t2 + 1120st)
++ m2
+l (86s3 + 360s2t + 432st2 + 144t3)
++ 3s4 + 21s3t + 45s2t2 + 48st3 + 24t4 ,
+(B.6)
+where ml is the lepton mass. Note that F2(s, t) = F2(t, s).
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diff --git a/r9FAT4oBgHgl3EQfgB2V/content/tmp_files/load_file.txt b/r9FAT4oBgHgl3EQfgB2V/content/tmp_files/load_file.txt
new file mode 100644
index 0000000000000000000000000000000000000000..9747d7f84a4df505c3b64463e26986e2ea8e6077
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@@ -0,0 +1,775 @@
+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FAT4oBgHgl3EQfgB2V/content/2301.08585v1.pdf,len=774
+page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FAT4oBgHgl3EQfgB2V/content/2301.08585v1.pdf'}
+page_content='08585v1  [hep-ph]  20 Jan 2023  Probe of a Randall-Sundrum-like model from  muon pair production at high energy muon  collider  S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FAT4oBgHgl3EQfgB2V/content/2301.08585v1.pdf'}
+page_content='C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FAT4oBgHgl3EQfgB2V/content/2301.08585v1.pdf'}
+page_content=' ˙Inan∗  Department of Physics, Sivas Cumhuriyet University, 58140, Sivas, Turkey  and  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FAT4oBgHgl3EQfgB2V/content/2301.08585v1.pdf'}
+page_content='V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FAT4oBgHgl3EQfgB2V/content/2301.08585v1.pdf'}
+page_content=' Kisselev†  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FAT4oBgHgl3EQfgB2V/content/2301.08585v1.pdf'}
+page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FAT4oBgHgl3EQfgB2V/content/2301.08585v1.pdf'}
+page_content=' Logunov Institute for High Energy Physics, NRC “Kurchatov Institute”,  142281, Protvino, Russian Federation  Abstract  We have examined µ+µ− → 2(µ+µ−) and µ+µ− → µ+µ− colli-  sions at future high energy muon colliders in the framework of the  Randall-Sundrum-like model with a small curvature of space-time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FAT4oBgHgl3EQfgB2V/content/2301.08585v1.pdf'}
+page_content='  The collision energies of 3 TeV, 14 TeV and, 100 TeV are addressed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FAT4oBgHgl3EQfgB2V/content/2301.08585v1.pdf'}
+page_content='  Both differential and total cross sections are calculated, and excluded  bounds on a 5-dimensional gravity scale are obtained depending on  collision energy and integrated luminosity of the muon collider.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FAT4oBgHgl3EQfgB2V/content/2301.08585v1.pdf'}
+page_content='  1  Introduction  The Standard Model (SM) has been proven in a lot number of collider exper-  iments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FAT4oBgHgl3EQfgB2V/content/2301.08585v1.pdf'}
+page_content=' Nevertheless, we are still searching for solutions for many problems  that SM cannot give a satisfactory solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FAT4oBgHgl3EQfgB2V/content/2301.08585v1.pdf'}
+page_content='  One of such problem is the  so-called hierarchy problem which means the large energy gap between the  ∗Electronic address: sceminan@cumhuriyet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FAT4oBgHgl3EQfgB2V/content/2301.08585v1.pdf'}
+page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FAT4oBgHgl3EQfgB2V/content/2301.08585v1.pdf'}
+page_content='tr  †Electronic address: alexandre.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FAT4oBgHgl3EQfgB2V/content/2301.08585v1.pdf'}
+page_content='kisselev@ihep.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FAT4oBgHgl3EQfgB2V/content/2301.08585v1.pdf'}
+page_content='ru  1  electroweak scale and gravity scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FAT4oBgHgl3EQfgB2V/content/2301.08585v1.pdf'}
+page_content=' The most elegant answer to this phe-  nomenon has been given in the framework of the Randall-Sundrum (RS)  model [1] which is based on a 5D theory with one extra dimension compact-  ified in an orbifold S1/Z2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FAT4oBgHgl3EQfgB2V/content/2301.08585v1.pdf'}
+page_content=' The main parameters of the RS model are the  compactification radius rc and AdS5 curvature parameter k (hereinafter re-  ferred to as the curvature k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FAT4oBgHgl3EQfgB2V/content/2301.08585v1.pdf'}
+page_content=' The model predicts Kaluza-Klein (KK) gravi-  tons which are heavy resonances with masses around the TeV scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FAT4oBgHgl3EQfgB2V/content/2301.08585v1.pdf'}
+page_content=' The  most stringent limits on KK graviton masses come from the LHC searches  for heavy resonances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FAT4oBgHgl3EQfgB2V/content/2301.08585v1.pdf'}
+page_content='  The experimental limits depend on a ratio k/MP,  where MP is the Planck mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FAT4oBgHgl3EQfgB2V/content/2301.08585v1.pdf'}
+page_content=' The CMS collaboration have excluded KK  graviton masses below 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FAT4oBgHgl3EQfgB2V/content/2301.08585v1.pdf'}
+page_content='3 to 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FAT4oBgHgl3EQfgB2V/content/2301.08585v1.pdf'}
+page_content='0 TeV for the diphoton final state [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FAT4oBgHgl3EQfgB2V/content/2301.08585v1.pdf'}
+page_content=' For the  dilepton final state the CMS have excluded the RS graviton masses in the  region 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FAT4oBgHgl3EQfgB2V/content/2301.08585v1.pdf'}
+page_content='47-4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FAT4oBgHgl3EQfgB2V/content/2301.08585v1.pdf'}
+page_content='78 TeV [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FAT4oBgHgl3EQfgB2V/content/2301.08585v1.pdf'}
+page_content=' The best lower limit of the ATLAS collaboration,  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FAT4oBgHgl3EQfgB2V/content/2301.08585v1.pdf'}
+page_content='6 TeV, has been obtained in searching for the diphoton final state [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FAT4oBgHgl3EQfgB2V/content/2301.08585v1.pdf'}
+page_content='  In papers [5, 6] the Randall-Sundrum-like model with a small curvature  of the 5-dimensional space-time (RSSC model) has been proposed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FAT4oBgHgl3EQfgB2V/content/2301.08585v1.pdf'}
+page_content=' In par-  ticular, a general solution for the warped metric has been obtained [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FAT4oBgHgl3EQfgB2V/content/2301.08585v1.pdf'}
+page_content=' In  contrast to the original RS model, the RSSC model has an almost continuous  graviton mass spectrum which is similar to the spectrum of the ADD model  [8]-[10], if k ≪ MP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FAT4oBgHgl3EQfgB2V/content/2301.08585v1.pdf'}
+page_content=' Thus, the above mentioned experimental bounds are  not applied to the RS scenario with a small value of k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FAT4oBgHgl3EQfgB2V/content/2301.08585v1.pdf'}
+page_content=' A probe of the RSSC  model at the LHC can be found in [11, 12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FAT4oBgHgl3EQfgB2V/content/2301.08585v1.pdf'}
+page_content=' A detailed comparison of the  RSSC model with the RS model is given in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FAT4oBgHgl3EQfgB2V/content/2301.08585v1.pdf'}
+page_content='  In the present paper, we intend to examine the RSSC model through the  µ+µ− → 2(µ+µ−) and µ+µ− → µ+µ− processes at a future muon collider.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FAT4oBgHgl3EQfgB2V/content/2301.08585v1.pdf'}
+page_content='  The idea of the muon collider was proposed by F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FAT4oBgHgl3EQfgB2V/content/2301.08585v1.pdf'}
+page_content=' Tikhonin and G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FAT4oBgHgl3EQfgB2V/content/2301.08585v1.pdf'}
+page_content=' Budker in  the late 1960’s [13, 14], and it was also discussed in the early 1980’s [15, 16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FAT4oBgHgl3EQfgB2V/content/2301.08585v1.pdf'}
+page_content='  At present, a great physical potential of the muon collider for collisions of  elementary particles at very high energies is being actively examined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FAT4oBgHgl3EQfgB2V/content/2301.08585v1.pdf'}
+page_content=' Its  advantage lies in the fact that muons can be accelerated in a ring without  limitation from synchrotron radiation compared to linear or circular electron-  positron colliders [17]-[22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FAT4oBgHgl3EQfgB2V/content/2301.08585v1.pdf'}
+page_content=' For instance, the muon collider may provide a  determination of the electroweak couplings of the Higgs boson which is sig-  nificantly better than what is considered attainable at other future colliders  [23]-[29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FAT4oBgHgl3EQfgB2V/content/2301.08585v1.pdf'}
+page_content=' Interest in designing and building a muon collider is also based on  its capability of probing the physics beyond the SM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FAT4oBgHgl3EQfgB2V/content/2301.08585v1.pdf'}
+page_content=' In a number of recent  papers searches for SUSY particles [30], WIMPs [31], vector boson fusion  [32], leptoquarks [33], lepton flavor violation [34], and physics of (g −2)µ [35]  2  at the muon colliders are presented.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FAT4oBgHgl3EQfgB2V/content/2301.08585v1.pdf'}
+page_content=' For more details on a spectacular oppor-  tunity of the muon collider in the direct exploration of the energy frontier,  see [36].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FAT4oBgHgl3EQfgB2V/content/2301.08585v1.pdf'}
+page_content='  In the µ+µ− → 2(µ+µ−) scattering one pair of the outgoing muons with  large transverse momenta and large dimuon invariant mass is detected, while  the other two scattered muons escape a detector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FAT4oBgHgl3EQfgB2V/content/2301.08585v1.pdf'}
+page_content=' Our goal is to obtain bounds  on the 5-dimensional gravity scale M5 which can be probed at TeV and multi-  TeV muon colliders.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FAT4oBgHgl3EQfgB2V/content/2301.08585v1.pdf'}
+page_content=' The gravity contribution to this process comes from the  subprocess V V → G → µ+µ−, where V is the γ or Z boson, G is the KK  graviton, and a summation over all KK gravitons is assumed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FAT4oBgHgl3EQfgB2V/content/2301.08585v1.pdf'}
+page_content=' We will also  study the µ+µ− → µ+µ− scattering, taking into account contributions from  s- and t-channel graviton exchanges, to derive the bounds on M5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FAT4oBgHgl3EQfgB2V/content/2301.08585v1.pdf'}
+page_content=' Note that  the processes we are interested in can be easily distinguished experimentally  from each other, since they have quite different distributions in invariant  mass of the detected dimuon pair.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FAT4oBgHgl3EQfgB2V/content/2301.08585v1.pdf'}
+page_content='  The paper is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FAT4oBgHgl3EQfgB2V/content/2301.08585v1.pdf'}
+page_content=' In the next section, the detailed de-  scription of the RSSC model is presented.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FAT4oBgHgl3EQfgB2V/content/2301.08585v1.pdf'}
+page_content=' The production of four muons  via vector boson fusion at the muon collider is examined in section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FAT4oBgHgl3EQfgB2V/content/2301.08585v1.pdf'}
+page_content=' The  bounds on 5-dimensional Planck scale M5 are obtained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FAT4oBgHgl3EQfgB2V/content/2301.08585v1.pdf'}
+page_content=' In section 4 we study  the µ+µ− → µ+µ− process and we calculate the values of M5 which can be  probed at the muon collider.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FAT4oBgHgl3EQfgB2V/content/2301.08585v1.pdf'}
+page_content='  2  Model of warped extra dimension with small  curvature  In this section, we describe the RSSC model in detail and compare it with  the original RS model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FAT4oBgHgl3EQfgB2V/content/2301.08585v1.pdf'}
+page_content=' The RS scenario with one extra dimension and two  branes [1] was proposed as an alternative to the ADD scenario with large flat  extra dimensions [8]-[10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FAT4oBgHgl3EQfgB2V/content/2301.08585v1.pdf'}
+page_content=' It has the following background warped metric  ds2 = e−2σ(y) ηµν dxµ dxν − dy2 ,  (1)  where ηµν is the Minkowski tensor with the signature (+, −, −, −), y is an  extra coordinate, and σ(y) is the warp factor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FAT4oBgHgl3EQfgB2V/content/2301.08585v1.pdf'}
+page_content=' The periodicity condition y =  y +2πrc is imposed, and the points (xµ, y) and (xµ, −y) are identified.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FAT4oBgHgl3EQfgB2V/content/2301.08585v1.pdf'}
+page_content=' Thus,  we have a model of gravity in a slice of the AdS5 space-time compactified to  the orbifold S1/Z2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FAT4oBgHgl3EQfgB2V/content/2301.08585v1.pdf'}
+page_content=' This orbifold has two fixed points, y = 0, and y = πrc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FAT4oBgHgl3EQfgB2V/content/2301.08585v1.pdf'}
+page_content='  3  There are two branes located at these points (called Planck and TeV brane,  respectively).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FAT4oBgHgl3EQfgB2V/content/2301.08585v1.pdf'}
+page_content=' The SM fields are confined to the TeV brane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FAT4oBgHgl3EQfgB2V/content/2301.08585v1.pdf'}
+page_content='  The classical action of the RS model is given by [1]  S =  �  d4x  � πrc  −πrc  dy  √  G (2 ¯  M3  5R − Λ)  +  �  d4x  �  |g(1)| (L1 − Λ1) +  �  d4x  �  |g(2)| (L2 − Λ2) ,  (2)  where GMN(x, y) is the 5-dimensional metric, with M, N = 0, 1, 2, 3, 4, µ =  0, 1, 2, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FAT4oBgHgl3EQfgB2V/content/2301.08585v1.pdf'}
+page_content=' The quantities  g(1)  µν (x) = Gµν(x, y = 0) ,  g(2)  µν (x) = Gµν(x, y = πrc)  (3)  are induced metrics on the branes, L1 and L2 are brane Lagrangians, G =  det(GMN), g(i) = det(g(i)  µν).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FAT4oBgHgl3EQfgB2V/content/2301.08585v1.pdf'}
+page_content=' The parameter ¯  M5 is a reduced 5-dimensional  Planck scale related to the fundamental gravity scale M5, namely,  ¯  M5 =  M5/(2π)1/3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FAT4oBgHgl3EQfgB2V/content/2301.08585v1.pdf'}
+page_content=' The parameter Λ is a 5-dimensional cosmological constant, and  Λ1,2 are brane tensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FAT4oBgHgl3EQfgB2V/content/2301.08585v1.pdf'}
+page_content='  For the first time, the solution for σ(y) in (1) has been obtained in [1],  σRS(y) = k|y| ,  (4)  where k is a parameter with a dimension of mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FAT4oBgHgl3EQfgB2V/content/2301.08585v1.pdf'}
+page_content=' It defines the curvature  of the 5-dimensional space-time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FAT4oBgHgl3EQfgB2V/content/2301.08585v1.pdf'}
+page_content=' Later on in [7] the generalized solution for  σ(y) was derived  σ(y) = krc  2  �����Arccos  �  cos y  rc  ����� −  ����π − Arccos  �  cos y  rc  �����  �  + π |k|rc  2  −C , (5)  where C is a y-independent quantity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FAT4oBgHgl3EQfgB2V/content/2301.08585v1.pdf'}
+page_content=' Note that the brane tensions obey the  fine-tuning relations  Λ = −24 ¯  M3  5k2 ,  Λ1 = − Λ2 = 24 ¯  M3  5k .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FAT4oBgHgl3EQfgB2V/content/2301.08585v1.pdf'}
+page_content='  (6)  Here Arccos(z) is a principal value of the multivalued inverse trigonometric  function arccos(z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FAT4oBgHgl3EQfgB2V/content/2301.08585v1.pdf'}
+page_content='  Let us underline that the generalized solution (5) (i)  obeys the orbifold symmetry y → −y;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FAT4oBgHgl3EQfgB2V/content/2301.08585v1.pdf'}
+page_content=' (ii) makes the jumps of σ′(y) on both  branes;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FAT4oBgHgl3EQfgB2V/content/2301.08585v1.pdf'}
+page_content=' (iii) has explicit symmetry with respect to the branes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FAT4oBgHgl3EQfgB2V/content/2301.08585v1.pdf'}
+page_content=' More details  can be found in [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FAT4oBgHgl3EQfgB2V/content/2301.08585v1.pdf'}
+page_content='  4  By taking C = 0 in (5), we get the RS model (4), while taking C = πkrc,  we come to the RS-like scenario with the small curvature of the space-time  (RSSC model, see [5]-[7]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FAT4oBgHgl3EQfgB2V/content/2301.08585v1.pdf'}
+page_content='  It is worth to remind the main features of the RSSC model in comparison  with those of the RS model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FAT4oBgHgl3EQfgB2V/content/2301.08585v1.pdf'}
+page_content='  The interactions of the Kaluza-Klein (KK)  gravitons h(n)  µν with the SM fields on the TeV brane are given by the effective  Lagrangian density  Lint = − 1  ¯  MPl  h(0)  µν (x) Tαβ(x) ηµαηνβ − 1  Λπ  ∞  �  n=1  h(n)  µν (x) Tαβ(x) ηµαηνβ ,  (7)  were  ¯  MPl = MPl/  √  8π is the reduced Planck mass, T µν(x) is the energy-  momentum tensor of the SM fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FAT4oBgHgl3EQfgB2V/content/2301.08585v1.pdf'}
+page_content=' The coupling constant is equal to  Λπ = ¯  M5  � ¯  M5  k  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FAT4oBgHgl3EQfgB2V/content/2301.08585v1.pdf'}
+page_content='  (8)  The hierarchy relation looks like  ¯  M2  Pl =  ¯  M3  5  k  �  e2πkrc − 1  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FAT4oBgHgl3EQfgB2V/content/2301.08585v1.pdf'}
+page_content='  (9)  To compare, in the original RS model it is defined as ¯  M2  Pl = ( ¯  M3  5/k)  �  1 − e−2πkrc�  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FAT4oBgHgl3EQfgB2V/content/2301.08585v1.pdf'}
+page_content='  It usually assumed that kπrc ≫ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FAT4oBgHgl3EQfgB2V/content/2301.08585v1.pdf'}
+page_content='  The masses of the KK gravitons are proportional to the curvature k [6]  mn = xnk ,  n = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FAT4oBgHgl3EQfgB2V/content/2301.08585v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FAT4oBgHgl3EQfgB2V/content/2301.08585v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FAT4oBgHgl3EQfgB2V/content/2301.08585v1.pdf'}
+page_content=' ,  (10)  where xn are zeros of the Bessel function J1(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FAT4oBgHgl3EQfgB2V/content/2301.08585v1.pdf'}
+page_content=' Should we take k ≪ ¯  M5 ∼ 1  TeV, the mass splitting ∆m will be very small, ∆m ≃ πk, and we come to  an almost continuous mass spectrum, similar to the mass spectrum of the  ADD model [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FAT4oBgHgl3EQfgB2V/content/2301.08585v1.pdf'}
+page_content=' On the contrary, in the RS model the gravitons are heavy  resonances with masses above one-few TeV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FAT4oBgHgl3EQfgB2V/content/2301.08585v1.pdf'}
+page_content='  As was shown in a number of phenomenological papers on the RSSC  model [11, 12], the cross sections weakly depend on the parameter k, if k ≪  ¯  M5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FAT4oBgHgl3EQfgB2V/content/2301.08585v1.pdf'}
+page_content=' That is why, in what follows we will put k = 1 GeV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FAT4oBgHgl3EQfgB2V/content/2301.08585v1.pdf'}
+page_content='  5  3  Production of two muon pairs in muon col-  lisions  Let us consider the process µ−µ+ → µ−V1V2µ+ → 2(µ+µ−) shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FAT4oBgHgl3EQfgB2V/content/2301.08585v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FAT4oBgHgl3EQfgB2V/content/2301.08585v1.pdf'}
+page_content='  Our goal is to estimate a contribution of the KK gravitons to the gauge  boson fusion V V → l−l+ in the framework of the RSSC model described in  the previous section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FAT4oBgHgl3EQfgB2V/content/2301.08585v1.pdf'}
+page_content=' The amplitude of this subprocess is defined as M =  MSM + MKK, where MSM is the SM term, and MKK is given by the sum of  s-channel KK gravitons  MKK =  1  2Λ2  π  ∞  �  n=1  �  ¯u(p1)Γµν  2 v(p2)  Bµναβ  s − m2  n + i mnΓn  Γαβρσ  1  eρ(k1)eσ(k2)  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FAT4oBgHgl3EQfgB2V/content/2301.08585v1.pdf'}
+page_content=' (11)  Here k1, k2, p1, p2 and eρ(k1), eσ(k2) are, respectively, incoming photon mo-  G  µ+  l+  l−  V2  V1  µ−  µ−  µ+  Figure 1: The Feynman diagrams describing contribution of the KK graviton  G to the collision of two vector bosons V1, V2 = γ or Z, with two outgoing  charged leptons at the muon collider.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FAT4oBgHgl3EQfgB2V/content/2301.08585v1.pdf'}
+page_content='  menta, outgoing lepton momenta and polarization vectors of photons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FAT4oBgHgl3EQfgB2V/content/2301.08585v1.pdf'}
+page_content=' Γn is  the total width of the graviton with the mass mn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FAT4oBgHgl3EQfgB2V/content/2301.08585v1.pdf'}
+page_content=' The coherent sum in (11)  is over all massive KK modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FAT4oBgHgl3EQfgB2V/content/2301.08585v1.pdf'}
+page_content=' The Feynman rules for the KK graviton were  derived in [37, 38] (see also [39]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FAT4oBgHgl3EQfgB2V/content/2301.08585v1.pdf'}
+page_content=' In particular, the KK–V V vertex looks  like  Γαβρσ  1  = − i  2 {[m2  V + (k1 · k2)] Cαβρσ + Dαβρσ} ,  (12)  6  where  Cαβρσ = ηαρηβσ + ηασηβρ − ηαβηρσ ,  (13)  Dαβρσ = ηαβkσ  1kρ  2 − (ηασkβ  1kρ  2 + ηαρkσ  1kβ  2 − ηρσkα  1 kβ  2 )  − (ηβσkα  1 kρ  2 + ηβρkσ  1kα  2 − ηρσkβ  1kα  2 ) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FAT4oBgHgl3EQfgB2V/content/2301.08585v1.pdf'}
+page_content='  (14)  The KK–l−l+ vertex is defined as  Γµν  2 = − i  8 [γµ(pν  1 − pν  2) + γν(pµ  1 − pµ  2)] .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FAT4oBgHgl3EQfgB2V/content/2301.08585v1.pdf'}
+page_content='  (15)  Finally, Bµναβ in (11) is a tensor part of the KK graviton propagator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FAT4oBgHgl3EQfgB2V/content/2301.08585v1.pdf'}
+page_content=' Its  explicit expression was derived in [37, 38].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FAT4oBgHgl3EQfgB2V/content/2301.08585v1.pdf'}
+page_content=' We can safely omit terms in Bµναβ  which give zero contribution to eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FAT4oBgHgl3EQfgB2V/content/2301.08585v1.pdf'}
+page_content=' (11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FAT4oBgHgl3EQfgB2V/content/2301.08585v1.pdf'}
+page_content=' Then we find  Bµναβ = ηµαηνβ + ηµβηνα − 2  3 ηµνηαβ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FAT4oBgHgl3EQfgB2V/content/2301.08585v1.pdf'}
+page_content='  (16)  The s-channel contribution of the KK gravitons is equal to  S(s) = 1  Λ2π  ∞  �  n=1  1  s − m2n + i mnΓn  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FAT4oBgHgl3EQfgB2V/content/2301.08585v1.pdf'}
+page_content='  (17)  This sum has been calculated in ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FAT4oBgHgl3EQfgB2V/content/2301.08585v1.pdf'}
+page_content=' [40],  S(s) = −  1  4 ¯  M3  5  √s  sin(2A) + i sinh(2ε)  cos2A + sinh2ε  ,  (18)  where  A =  √s  k ,  ε = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FAT4oBgHgl3EQfgB2V/content/2301.08585v1.pdf'}
+page_content='045  �√s  ¯  M5  �3  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FAT4oBgHgl3EQfgB2V/content/2301.08585v1.pdf'}
+page_content='  (19)  The squared amplitude of the subprocess V V → l−l+ is a sum of three  terms,  |M|2 = |MSM|2 + |MKK|2 + |Mint|2 ,  (20)  where MSM denotes the SM amplitude, while MKK and Mint denote pure KK  graviton and interference terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FAT4oBgHgl3EQfgB2V/content/2301.08585v1.pdf'}
+page_content=' In [39] the quantities |MSM(γγ → l−l+)|2,  |MKK(γγ → l−l+)|2, and |Mint(γγ → l−l+)|2 were calculated for massless  leptons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FAT4oBgHgl3EQfgB2V/content/2301.08585v1.pdf'}
+page_content=' The results of our calculations of the squared amplitudes |M(V V →  l−l+)|2 for nonzero ml and mV (V = γ, Z) are presented in Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FAT4oBgHgl3EQfgB2V/content/2301.08585v1.pdf'}
+page_content='  7  The virtual KK graviton production should lead to deviations from the  SM predictions in a magnitude of the cross section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FAT4oBgHgl3EQfgB2V/content/2301.08585v1.pdf'}
+page_content=' The cross section of our  process µ−µ+ → µ−V1V2µ+ → 2(µ+µ−) is defined by the formula  dσ =  τmax  �  τmin  dτ  xmax  �  xmin  dx  x  �  V1,V2=γ,ZT ,ZL  fV1/µ+(x, Q2)fV2/µ−(τ/x, Q2) dˆσ(V1V2 → µ+µ−) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FAT4oBgHgl3EQfgB2V/content/2301.08585v1.pdf'}
+page_content='  (21)  Here  xmax = 1 − mµ  Eµ  , τmax =  �  1 − mµ  Eµ  �2  , xmin = τ/xmax , τmin = p2  ⊥  E2µ  ,  (22)  and p⊥ is the transverse momenta of the outgoing photons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FAT4oBgHgl3EQfgB2V/content/2301.08585v1.pdf'}
+page_content=' The boson distri-  butions inside the muon beam, fγ/µ±(x, Q2), fZT /µ±(x, Q2), and fZL/µ±(x, Q2)  are [41]  fγ/µ±(x, Q2) = α  2π  1 + (1 − x)2  x  ln Q2  m2  µ  ,  (23)  and [42, 43]  fZT /µ±(x, Q2) = α±  Z  2π  1 + (1 − x)2  x  ln Q2  m2  Z  ,  fZL/µ±(x, Q2) = α±  Z  π  (1 − x)  x  ,  (24)  where  α±  Z =  α  (cos θW sin θW)2  �  (g±  V )2 + (g±  A)2�  ,  (25)  g±  V = −1/4∓sin2 θW, g±  A = 1/4, and mµ is the muon mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FAT4oBgHgl3EQfgB2V/content/2301.08585v1.pdf'}
+page_content=' The variable x in  (23), (24) is the ratio of the boson energy and energy of the incoming muon  Eµ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FAT4oBgHgl3EQfgB2V/content/2301.08585v1.pdf'}
+page_content=' Note that the Z boson has different distributions for its transverse (T)  and longitudinal (L) polarizations (24).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FAT4oBgHgl3EQfgB2V/content/2301.08585v1.pdf'}
+page_content='  The differential cross section of the subprocess V1V2 → µ+µ−, where  V1,2 = γ or Z, is a sum of helicity amplitudes squared  dˆσ  dΩ =  1  64π2ˆs  �  λ1,λ2,λ3,λ4  |MV1V2  λ1λ2λ3λ4|2,  (26)  8  where  √  ˆs is a collision energy of this subprocess, and λ1,2 (λ3,4) are boson  (muon) helicities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FAT4oBgHgl3EQfgB2V/content/2301.08585v1.pdf'}
+page_content=' In boson distributions (23), (24) we put Q2 = ˆs, where  √  ˆs = 2Eµ  √τ is the invariant energy of the subprocess V1V2 → µ+µ−.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FAT4oBgHgl3EQfgB2V/content/2301.08585v1.pdf'}
+page_content='  As it was mentioned in [44], the scattering angle for high energy initial  muons is peaked near θµ ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FAT4oBgHgl3EQfgB2V/content/2301.08585v1.pdf'}
+page_content='02◦ − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FAT4oBgHgl3EQfgB2V/content/2301.08585v1.pdf'}
+page_content='2◦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FAT4oBgHgl3EQfgB2V/content/2301.08585v1.pdf'}
+page_content=' These very forward muons would  most likely escape a muon detector away from colliding beams.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FAT4oBgHgl3EQfgB2V/content/2301.08585v1.pdf'}
+page_content=' Thus, only  muons produced in the boson fusion (as those shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FAT4oBgHgl3EQfgB2V/content/2301.08585v1.pdf'}
+page_content=' 1) will be de-  tected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FAT4oBgHgl3EQfgB2V/content/2301.08585v1.pdf'}
+page_content=' We apply the cuts to the final muons, pt > 50 GeV, |η| < 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FAT4oBgHgl3EQfgB2V/content/2301.08585v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FAT4oBgHgl3EQfgB2V/content/2301.08585v1.pdf'}
+page_content=' The  main purpose of these cuts is to ensure that only two muons are detected in  the final state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FAT4oBgHgl3EQfgB2V/content/2301.08585v1.pdf'}
+page_content=' As was already mentioned at the end of section 2, we take  k = 1 GeV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FAT4oBgHgl3EQfgB2V/content/2301.08585v1.pdf'}
+page_content='  The results of our numerical calculations of the differential cross sections  for the µ+µ− → 2(µ+µ−) scattering at the future muon collider are presented  in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FAT4oBgHgl3EQfgB2V/content/2301.08585v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FAT4oBgHgl3EQfgB2V/content/2301.08585v1.pdf'}
+page_content=' The predictions for three collision invariant energies of the muon  collider are shown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FAT4oBgHgl3EQfgB2V/content/2301.08585v1.pdf'}
+page_content=' As one can see, for each energy the cross sections rise as  the invariant mass of the detected muons grows, while the SM cross sections  decrease rapidly with an increase of mµ+µ−.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FAT4oBgHgl3EQfgB2V/content/2301.08585v1.pdf'}
+page_content=' We have also calculated the  differential cross sections via transverse momentum of the detected muons,  see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FAT4oBgHgl3EQfgB2V/content/2301.08585v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FAT4oBgHgl3EQfgB2V/content/2301.08585v1.pdf'}
+page_content='  The total cross sections as functions of the minimal invariant mass of two  detected muons at the muon collider mµ+µ−,min are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FAT4oBgHgl3EQfgB2V/content/2301.08585v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FAT4oBgHgl3EQfgB2V/content/2301.08585v1.pdf'}
+page_content=' The cross  sections strongly depend on the fundamental gravity scale ¯  M5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FAT4oBgHgl3EQfgB2V/content/2301.08585v1.pdf'}
+page_content=' If ¯  M5 = 1  TeV, the cross section exceeds the SM one for all three collision energies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FAT4oBgHgl3EQfgB2V/content/2301.08585v1.pdf'}
+page_content=' For  larger values of ¯  M5 the total cross section strongly dominates over the SM  cross section for √s = 14 TeV and √s = 100 TeV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FAT4oBgHgl3EQfgB2V/content/2301.08585v1.pdf'}
+page_content='  All this enables us to derive the excluded bounds on the 5-dimensional  reduced Planck scale ¯  M5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FAT4oBgHgl3EQfgB2V/content/2301.08585v1.pdf'}
+page_content=' To derive them, we apply the following formula  for the statistical significance SS [45]  SS =  �  2[(S − B ln(1 + S/B)] ,  (27)  where S is the number of signal events and B is the number of background  (SM) events.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FAT4oBgHgl3EQfgB2V/content/2301.08585v1.pdf'}
+page_content=' We define the regions SS ⩽ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FAT4oBgHgl3EQfgB2V/content/2301.08585v1.pdf'}
+page_content='645 as the regions that can be  excluded at the 95% C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FAT4oBgHgl3EQfgB2V/content/2301.08585v1.pdf'}
+page_content='L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FAT4oBgHgl3EQfgB2V/content/2301.08585v1.pdf'}
+page_content=' To reduce the SM background, we used the cuts  mµ−µ+ > 1 TeV, mµ−µ+ > 5 TeV, and mµ−µ+ > 50 TeV for the colliding  energy of 3 TeV, 14 TeV, and 100 TeV, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FAT4oBgHgl3EQfgB2V/content/2301.08585v1.pdf'}
+page_content=' The results are shown  in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FAT4oBgHgl3EQfgB2V/content/2301.08585v1.pdf'}
+page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FAT4oBgHgl3EQfgB2V/content/2301.08585v1.pdf'}
+page_content=' Our best limits for √s = 3 TeV, 14 TeV and 100 TeV are ¯  M5 = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FAT4oBgHgl3EQfgB2V/content/2301.08585v1.pdf'}
+page_content='8  TeV, 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FAT4oBgHgl3EQfgB2V/content/2301.08585v1.pdf'}
+page_content='1 TeV, and 106.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FAT4oBgHgl3EQfgB2V/content/2301.08585v1.pdf'}
+page_content='4 TeV, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FAT4oBgHgl3EQfgB2V/content/2301.08585v1.pdf'}
+page_content='  9  500  1000  1500  2000  10  6  10  5  10  4  10  3  10  2  10  1  10  0  10  1  10  2  10  3  2000  4000  6000  8000  15000  30000  45000  60000  s  TeV  d  dm  (fb/GeV )  m  (GeV)  s  TeV  M  5  TeV  M  5  TeV  M  5  TeV  SM  m  (GeV)  s  TeV  m  (GeV)  Figure 2: The differential cross sections for the process µ+µ− → 2(µ+µ−)  via invariant mass of two detected muons at the muon collider.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FAT4oBgHgl3EQfgB2V/content/2301.08585v1.pdf'}
+page_content=' The left,  middle and right panels correspond to the colliding energy of 3 TeV, 14 TeV,  and 100 TeV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FAT4oBgHgl3EQfgB2V/content/2301.08585v1.pdf'}
+page_content=' The curves (from the top down) correspond to ¯  M5 = 1 TeV,  ¯  M5 = 2 TeV, and ¯  M5 = 3 TeV, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FAT4oBgHgl3EQfgB2V/content/2301.08585v1.pdf'}
+page_content=' The SM cross sections (low  curves) are also shown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FAT4oBgHgl3EQfgB2V/content/2301.08585v1.pdf'}
+page_content='  4  Production of one muon pair in muon col-  lisions  As was said in the previous section, we expect that in the µ+µ− → 2(µ+µ−)  process only two final muons are detected, while two scattered muons escape  the detector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FAT4oBgHgl3EQfgB2V/content/2301.08585v1.pdf'}
+page_content=' In the process µ−µ+ → µ+µ− two outgoing muons with high  transverse momenta are also detected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FAT4oBgHgl3EQfgB2V/content/2301.08585v1.pdf'}
+page_content=' However, in such a case the invariant  mass of the dimuon system mµ+µ− is fixed and equal to the collision energy  √s, that does not occur for the µ−µ+ → 2(µ+µ−) scattering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FAT4oBgHgl3EQfgB2V/content/2301.08585v1.pdf'}
+page_content=' That is why  one can easily discriminate between these two process experimentally by  measuring the invariant mass of the detected muon pair.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FAT4oBgHgl3EQfgB2V/content/2301.08585v1.pdf'}
+page_content='  The virtual KK graviton exchanges have to give a contribution to the cross  sections of the µ−µ+ → µ+µ− scattering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FAT4oBgHgl3EQfgB2V/content/2301.08585v1.pdf'}
+page_content=' It is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FAT4oBgHgl3EQfgB2V/content/2301.08585v1.pdf'}
+page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FAT4oBgHgl3EQfgB2V/content/2301.08585v1.pdf'}
+page_content=' The analytical  10  200  400  600  800  1000  10  6  10  5  10  4  10  3  10  2  10  1  10  0  10  1  10  2  10  3  10  4  10  5  10  6  10  7  1000  2000  3000  4000  5000  10000  20000  30000  40000  d  /dp  t  (fb/GeV)  p  t  (GeV)  M  5  TeV  M  5  TeV  M  5  TeV  SM  p  t  (GeV)  p  t  (GeV)  Figure 3: The differential cross sections for the process µ+µ− → 2(µ+µ−) via  transverse momentum of the detected muons at the muon collider.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FAT4oBgHgl3EQfgB2V/content/2301.08585v1.pdf'}
+page_content='  expressions for amplitudes squared of this collision are given in Appendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FAT4oBgHgl3EQfgB2V/content/2301.08585v1.pdf'}
+page_content='  Using them we have calculated the differential cross sections for the µ+µ− →  µ+µ− scattering at the muon collider depending on transverse momentum of  the final muons pt, taking into account gravity contribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FAT4oBgHgl3EQfgB2V/content/2301.08585v1.pdf'}
+page_content=' Our result are  presented in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FAT4oBgHgl3EQfgB2V/content/2301.08585v1.pdf'}
+page_content=' 7 for three values of the collision energy √s and different  values of the reduced 5-dimensional Planck scale  ¯  M5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FAT4oBgHgl3EQfgB2V/content/2301.08585v1.pdf'}
+page_content=' As we can see, for  √s = 14 TeV and √s = 100 TeV, the cross section significantly dominates  the SM one, especially for large pt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FAT4oBgHgl3EQfgB2V/content/2301.08585v1.pdf'}
+page_content=' It is worth to compare this figure with  the pt-distribution in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FAT4oBgHgl3EQfgB2V/content/2301.08585v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FAT4oBgHgl3EQfgB2V/content/2301.08585v1.pdf'}
+page_content=' The oscillations of the curves in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FAT4oBgHgl3EQfgB2V/content/2301.08585v1.pdf'}
+page_content=' 3 originate  from the function S(s) (18) which describes the s-channel contribution of  all KK gravitons, since the invariant energy of the detected dimuon pair  is not a constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FAT4oBgHgl3EQfgB2V/content/2301.08585v1.pdf'}
+page_content=' On the other hand, in the process µ−µ+ → µ+µ− the  invariant energy of the dimuon pair is fixed, and we have no oscillations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FAT4oBgHgl3EQfgB2V/content/2301.08585v1.pdf'}
+page_content='  The differential cross sections integrated in pt from the minimal transverse  momentum of the detected muons pt,min are presented in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FAT4oBgHgl3EQfgB2V/content/2301.08585v1.pdf'}
+page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FAT4oBgHgl3EQfgB2V/content/2301.08585v1.pdf'}
+page_content='  As before,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FAT4oBgHgl3EQfgB2V/content/2301.08585v1.pdf'}
+page_content=' we have calculated the excluded bounds on ¯  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FAT4oBgHgl3EQfgB2V/content/2301.08585v1.pdf'}
+page_content='M5 which can be  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FAT4oBgHgl3EQfgB2V/content/2301.08585v1.pdf'}
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+page_content='Figure 4: The total cross sections for the process µ+µ− → 2(µ+µ−) via min-  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FAT4oBgHgl3EQfgB2V/content/2301.08585v1.pdf'}
+page_content='imal invariant mass of two detected muons at the muon collider mµ+µ−,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FAT4oBgHgl3EQfgB2V/content/2301.08585v1.pdf'}
+page_content='min.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FAT4oBgHgl3EQfgB2V/content/2301.08585v1.pdf'}
+page_content='  of the future muon collider, see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FAT4oBgHgl3EQfgB2V/content/2301.08585v1.pdf'}
+page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FAT4oBgHgl3EQfgB2V/content/2301.08585v1.pdf'}
+page_content=' We have used eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FAT4oBgHgl3EQfgB2V/content/2301.08585v1.pdf'}
+page_content=' (27) for the statistical  significance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FAT4oBgHgl3EQfgB2V/content/2301.08585v1.pdf'}
+page_content=' In doing so, the cuts pt,min = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FAT4oBgHgl3EQfgB2V/content/2301.08585v1.pdf'}
+page_content='5 TeV, pt,min = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FAT4oBgHgl3EQfgB2V/content/2301.08585v1.pdf'}
+page_content='5 TeV, and  pt,min = 25 TeV were applied for the 3 TeV, 14 TeV, and 100 TeV center-  of-mass energies, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FAT4oBgHgl3EQfgB2V/content/2301.08585v1.pdf'}
+page_content=' We see that in the process µ+µ− → µ+µ−  the scales up to ¯  M5 = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FAT4oBgHgl3EQfgB2V/content/2301.08585v1.pdf'}
+page_content='85 TeV, 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FAT4oBgHgl3EQfgB2V/content/2301.08585v1.pdf'}
+page_content='8 TeV and 126.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FAT4oBgHgl3EQfgB2V/content/2301.08585v1.pdf'}
+page_content='3 TeV can be probed for  √s = 3 TeV, 14 TeV, and 100 TeV, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FAT4oBgHgl3EQfgB2V/content/2301.08585v1.pdf'}
+page_content=' We conclude that these  limits on ¯  M5 are stronger than the limits obtained in the previous section for  the µ−µ+ → 2(µ+µ−) scattering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FAT4oBgHgl3EQfgB2V/content/2301.08585v1.pdf'}
+page_content='  5  Conclusions  We have examined two collisions at future TeV and multi-TeV muon colliders  in the Randall-Sundrum-like model with the small curvature (RSSC model)  [5, 6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FAT4oBgHgl3EQfgB2V/content/2301.08585v1.pdf'}
+page_content=' It is the model with one extra dimension and warped metric whose  5-dimensional space-time curvature k is about one GeV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FAT4oBgHgl3EQfgB2V/content/2301.08585v1.pdf'}
+page_content=' The other main  parameter of the RSSC model, the 5-dimensional Planck scale ¯  M5, is equal  to (larger than) one TeV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FAT4oBgHgl3EQfgB2V/content/2301.08585v1.pdf'}
+page_content='  12  10  0  10  1  10  2  10  3  10  0  10  1  10  2  10  0  10  1  10  2  10  3  10  4  10  0  10  1  10  2  10  3  10  4  10  5  10  6  s  TeV  M  5  (TeV)  L (fb  )  s  TeV  L (fb  )  L (fb  )  s  TeV  Figure 5: The excluded bounds on the reduced fundamental gravity scale  ¯  M5 via integrated luminosity of the muon collider for the process µ+µ− →  2(µ+µ−).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FAT4oBgHgl3EQfgB2V/content/2301.08585v1.pdf'}
+page_content=' The left, middle and right panels correspond to the colliding energy  of 3 TeV, 14 TeV, and 100 TeV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FAT4oBgHgl3EQfgB2V/content/2301.08585v1.pdf'}
+page_content='  We have studied the µ−µ+ → 2(µ+µ−) scattering first.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FAT4oBgHgl3EQfgB2V/content/2301.08585v1.pdf'}
+page_content=' The collision goes  via the V V → µ−µ+ scattering, where V = γ, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FAT4oBgHgl3EQfgB2V/content/2301.08585v1.pdf'}
+page_content=' For this scattering, the ana-  lytical expressions for the squared amplitudes, including the gravity, SM and,  interference terms, are derived for the first time for massive leptons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FAT4oBgHgl3EQfgB2V/content/2301.08585v1.pdf'}
+page_content=' They  are presented in Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FAT4oBgHgl3EQfgB2V/content/2301.08585v1.pdf'}
+page_content=' Then the differential cross sections depending  on the invariant mass of the detected muons mµ+µ− are calculated for three  values of the reduced 5-dimensional Planck scale ¯  M5 for 3 TeV, 14 TeV and  100 TeV muon colliders.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FAT4oBgHgl3EQfgB2V/content/2301.08585v1.pdf'}
+page_content=' The total cross section is calculated as a function of  the minimal value of mµ+µ−.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FAT4oBgHgl3EQfgB2V/content/2301.08585v1.pdf'}
+page_content=' As a result, the excluded bounds on the scale  ¯  M5 are obtained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FAT4oBgHgl3EQfgB2V/content/2301.08585v1.pdf'}
+page_content=' They are ¯  M5 = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FAT4oBgHgl3EQfgB2V/content/2301.08585v1.pdf'}
+page_content='80 TeV, 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FAT4oBgHgl3EQfgB2V/content/2301.08585v1.pdf'}
+page_content='1 TeV and 106.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FAT4oBgHgl3EQfgB2V/content/2301.08585v1.pdf'}
+page_content='4 TeV, for the  collision energy of √s = 3 TeV, 14 TeV and 100 TeV, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FAT4oBgHgl3EQfgB2V/content/2301.08585v1.pdf'}
+page_content='  The µ−µ+ → µ+µ− scattering is also studied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FAT4oBgHgl3EQfgB2V/content/2301.08585v1.pdf'}
+page_content=' As in the previous case, we  have calculated the gravity, SM and interference squared amplitudes analyt-  ically, see Appendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FAT4oBgHgl3EQfgB2V/content/2301.08585v1.pdf'}
+page_content=' It enabled us to estimate numerically the differential  cross sections as functions of the transverse momenta of the outgoing muons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FAT4oBgHgl3EQfgB2V/content/2301.08585v1.pdf'}
+page_content='  The total cross sections are also calculated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FAT4oBgHgl3EQfgB2V/content/2301.08585v1.pdf'}
+page_content=' Finally, the excluded bounds on  13  µ+  G  µ−  µ+  µ−  G  µ+  µ+  µ−  µ−  µ+  Figure 6: The Feynman diagrams describing contribution of the KK graviton  G to the µ−µ+ → µ+µ− scattering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FAT4oBgHgl3EQfgB2V/content/2301.08585v1.pdf'}
+page_content='  the main parameter of the RSSC model, the scale ¯  M5, have been obtained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FAT4oBgHgl3EQfgB2V/content/2301.08585v1.pdf'}
+page_content='  We have shown that the values of ¯  M5 = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FAT4oBgHgl3EQfgB2V/content/2301.08585v1.pdf'}
+page_content='85 TeV, 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FAT4oBgHgl3EQfgB2V/content/2301.08585v1.pdf'}
+page_content='8 TeV and 126.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FAT4oBgHgl3EQfgB2V/content/2301.08585v1.pdf'}
+page_content='3 TeV  can be probed at 3 TeV, 14 TeV, and 100 TeV muon colliders.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FAT4oBgHgl3EQfgB2V/content/2301.08585v1.pdf'}
+page_content='  Let us remember that  ¯  M5 = M5/(2π)1/3 ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FAT4oBgHgl3EQfgB2V/content/2301.08585v1.pdf'}
+page_content='54M5, where M5 is the  fundamental 5-dimensional gravity scale M5 in the RSSC model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FAT4oBgHgl3EQfgB2V/content/2301.08585v1.pdf'}
+page_content=' It means  that the bounds on the scale M5 are approximately twice stronger.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FAT4oBgHgl3EQfgB2V/content/2301.08585v1.pdf'}
+page_content='  Let us stress that our bounds on ¯  M5 should not be directly compared  with the current experimental bound on the 5-dimensional Planck scale of  the original RS model [1], since the mass spectra of the KK gravitons and,  correspondingly, experimental signatures are quite different in the RSSC and  RS models [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FAT4oBgHgl3EQfgB2V/content/2301.08585v1.pdf'}
+page_content='  Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FAT4oBgHgl3EQfgB2V/content/2301.08585v1.pdf'}
+page_content=' Squared amplitudes for V V →  l−l+ scattering  Our calculations give the following analytical expressions for the squared  amplitudes for the γγ → l−l+ collision in eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FAT4oBgHgl3EQfgB2V/content/2301.08585v1.pdf'}
+page_content=' (20)  |MSM|2 =  8e4  (t − m2  l )2(s + t − m2  l )2[−34m8  l + m6  l (60s + 64t)  − m4  l (31s2 + 52st + 28t2) + m2  l s(s2 − 2st − 4t2)  − t(s + t)(s2 + 2st + 2t2)] ,  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FAT4oBgHgl3EQfgB2V/content/2301.08585v1.pdf'}
+page_content='1)  14  200  400  600  800  1000  10  6  10  5  10  4  10  3  10  2  10  1  10  0  10  1  10  2  10  3  10  4  10  5  10  6  10  7  1000  2000  3000  4000  5000  10000  20000  30000  40000  s  TeV  d  dp  t  (fb/GeV)  p  t  (GeV)  s  TeV  M  5  TeV  M  5  TeV  M  5  TeV  SM  p  t  (GeV)  s  TeV  p  t  (GeV)  Figure 7: The differential cross sections for the process µ+µ− → µ+µ− via  transverse momentum of the detected muons at the muon collider.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FAT4oBgHgl3EQfgB2V/content/2301.08585v1.pdf'}
+page_content=' The left,  middle and right panels correspond to the colliding energy of 3 TeV, 14 TeV,  and 100 TeV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FAT4oBgHgl3EQfgB2V/content/2301.08585v1.pdf'}
+page_content=' The curves (from the top down) correspond to ¯  M5 = 1 TeV,  ¯  M5 = 2 TeV, and ¯  M5 = 3 TeV, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FAT4oBgHgl3EQfgB2V/content/2301.08585v1.pdf'}
+page_content=' The SM cross sections (low  curves) are also shown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FAT4oBgHgl3EQfgB2V/content/2301.08585v1.pdf'}
+page_content='  |MKK|2 = −1  8|S(s)|2[2m8  l − 8m6  l t + m4  l (s2 + 4st + 12t2) − 2m2  l t(s + 2t)2  + t(s + t)(s2 + 2st + 2t2)] ,  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FAT4oBgHgl3EQfgB2V/content/2301.08585v1.pdf'}
+page_content='2)  |Mint|2 = −  e2[S(s) + S⋆(s)]  2(t − m2  l )(s + t − m2  l )[−2m8  l + m6  l (3s + 4t) + m4  l s(3s − 4t)  − m2  l (s3 + 2s2t + 3st2 + 4t3) + t(s + t)(s2 + 2st + 2t2)] ,  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FAT4oBgHgl3EQfgB2V/content/2301.08585v1.pdf'}
+page_content='3)  where s, t are Mandelstam variables, ml is the lepton mass, and S(s) is  defined in the text (17)-(19).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FAT4oBgHgl3EQfgB2V/content/2301.08585v1.pdf'}
+page_content=' If we take ml = 0 we get known results obtained  in [39].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FAT4oBgHgl3EQfgB2V/content/2301.08585v1.pdf'}
+page_content='  15  200  400  600  800  1000  10  2  10  1  10  0  10  1  10  2  10  3  10  4  10  5  10  6  10  7  10  8  10  9  10  10  10  11  1000  2000  3000  4000  5000  10000  20000  30000  40000  s  TeV  (fb)  p  t, min  (GeV)  s  TeV  M  5  TeV  M  5  TeV  M  5  TeV  SM  p  t, min  (GeV)  s  TeV  p  t, min  (GeV)  Figure 8: The total cross sections for the process µ+µ− → µ+µ− via minimal  transverse momentum of the outgoing muons pt,min.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FAT4oBgHgl3EQfgB2V/content/2301.08585v1.pdf'}
+page_content='  16  10  0  10  1  10  2  10  3  10  0  10  1  10  2  10  0  10  1  10  2  10  3  10  4  10  0  10  1  10  2  10  3  10  4  10  5  10  6  s  TeV  M  5  (TeV)  L (fb  )  s  TeV  L (fb  )  s  TeV  L (fb  )  Figure 9: The excluded bounds on the reduced fundamental gravity scale ¯  M5  via integrated luminosity of the muon collider for the process µ+µ− → µ+µ−.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FAT4oBgHgl3EQfgB2V/content/2301.08585v1.pdf'}
+page_content='  The left, middle and right panels correspond to the colliding energy of 3 TeV,  14 TeV, and 100 TeV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FAT4oBgHgl3EQfgB2V/content/2301.08585v1.pdf'}
+page_content='  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FAT4oBgHgl3EQfgB2V/content/2301.08585v1.pdf'}
+page_content='17  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FAT4oBgHgl3EQfgB2V/content/2301.08585v1.pdf'}
+page_content='For the ZZ → l−l+ collision our calculations result in the following for-  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FAT4oBgHgl3EQfgB2V/content/2301.08585v1.pdf'}
+page_content='mulas  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FAT4oBgHgl3EQfgB2V/content/2301.08585v1.pdf'}
+page_content='|MSM|2 =  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FAT4oBgHgl3EQfgB2V/content/2301.08585v1.pdf'}
+page_content='g4  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FAT4oBgHgl3EQfgB2V/content/2301.08585v1.pdf'}
+page_content='Z  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FAT4oBgHgl3EQfgB2V/content/2301.08585v1.pdf'}
+page_content='(t − m2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FAT4oBgHgl3EQfgB2V/content/2301.08585v1.pdf'}
+page_content='l )2(s + t − m2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FAT4oBgHgl3EQfgB2V/content/2301.08585v1.pdf'}
+page_content='l − 2m2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FAT4oBgHgl3EQfgB2V/content/2301.08585v1.pdf'}
+page_content='Z)2{−2m8  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FAT4oBgHgl3EQfgB2V/content/2301.08585v1.pdf'}
+page_content='l [−300 cos(2θw) + 184 cos(4θw)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FAT4oBgHgl3EQfgB2V/content/2301.08585v1.pdf'}
+page_content='− 68 cos(6θw) + 17 cos(8θw) + 195]  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FAT4oBgHgl3EQfgB2V/content/2301.08585v1.pdf'}
+page_content='+ 4m6  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FAT4oBgHgl3EQfgB2V/content/2301.08585v1.pdf'}
+page_content='l [−4m2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FAT4oBgHgl3EQfgB2V/content/2301.08585v1.pdf'}
+page_content='Z(−94 cos(2θw) + 59 cos(4θw) − 24 cos(6θw)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FAT4oBgHgl3EQfgB2V/content/2301.08585v1.pdf'}
+page_content='+ 6 cos(8θw) + 56) + 163s + 196t − 8(33s + 37t) cos(2θw)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FAT4oBgHgl3EQfgB2V/content/2301.08585v1.pdf'}
+page_content='+ 18(9s + 10t) cos(4θw) + (15s + 16t)(cos(8θw) − 4 cos(6θw))]  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FAT4oBgHgl3EQfgB2V/content/2301.08585v1.pdf'}
+page_content='+ m4  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FAT4oBgHgl3EQfgB2V/content/2301.08585v1.pdf'}
+page_content='l [−2m4  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FAT4oBgHgl3EQfgB2V/content/2301.08585v1.pdf'}
+page_content='Z(−164 cos(2θw) + 128 cos(4θw)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FAT4oBgHgl3EQfgB2V/content/2301.08585v1.pdf'}
+page_content='+ 23(−4 cos(6θw) + cos(8θw) + 3))  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FAT4oBgHgl3EQfgB2V/content/2301.08585v1.pdf'}
+page_content='+ 8m2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FAT4oBgHgl3EQfgB2V/content/2301.08585v1.pdf'}
+page_content='Z(72s + 65t − 4(32s + 27t) cos(2θw) + (84s + 68t) cos(4θw)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FAT4oBgHgl3EQfgB2V/content/2301.08585v1.pdf'}
+page_content='+ (10s + 7t)(cos(8θw) − 4 cos(6θw))) − 301s2 − 436t2 − 668st  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FAT4oBgHgl3EQfgB2V/content/2301.08585v1.pdf'}
+page_content='+ 4(125s2 + 260st + 156t2) cos(2θw) − 8(39s2 + 78st + 46t2) cos(4θw)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FAT4oBgHgl3EQfgB2V/content/2301.08585v1.pdf'}
+page_content='− (31s2 + 52st + 28t2)(cos(8θw) − 4 cos(6θw))]  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FAT4oBgHgl3EQfgB2V/content/2301.08585v1.pdf'}
+page_content='+ m2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FAT4oBgHgl3EQfgB2V/content/2301.08585v1.pdf'}
+page_content='l [7s3 + 50s2t + 92st2 + s(s2 − 2st − 4t2)(cos(8θw) − 4 cos(6θw))  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FAT4oBgHgl3EQfgB2V/content/2301.08585v1.pdf'}
+page_content='+ 80t3 − 4(3s3 + 14s2t + 24st2 + 24t3) cos(2θw)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FAT4oBgHgl3EQfgB2V/content/2301.08585v1.pdf'}
+page_content='+ 8(s + 2t)(s2 + st + 3t2) cos(4θw) + 2m2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FAT4oBgHgl3EQfgB2V/content/2301.08585v1.pdf'}
+page_content='Z(79s2 + 252st + 112t2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FAT4oBgHgl3EQfgB2V/content/2301.08585v1.pdf'}
+page_content='− 4(31s2 + 108st + 52t2) cos(2θw) + 8(9s2 + 34st + 17t2) cos(4θw)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FAT4oBgHgl3EQfgB2V/content/2301.08585v1.pdf'}
+page_content='+ (5s2 + 28st + 16t2)(cos(8θw) − 4 cos(6θw)))  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FAT4oBgHgl3EQfgB2V/content/2301.08585v1.pdf'}
+page_content='+ 2m4  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FAT4oBgHgl3EQfgB2V/content/2301.08585v1.pdf'}
+page_content='Z(−263s − 414t + (428s + 696t) cos(2θw) − 16(16s + 27t) cos(4θw)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FAT4oBgHgl3EQfgB2V/content/2301.08585v1.pdf'}
+page_content='− 21(s + 2t)(cos(8θw) − 4 cos(6θw)))  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FAT4oBgHgl3EQfgB2V/content/2301.08585v1.pdf'}
+page_content='+ 2m6  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FAT4oBgHgl3EQfgB2V/content/2301.08585v1.pdf'}
+page_content='Z(−156 cos(2θw) + 96 cos(4θw) − 36 cos(6θw) + 9 cos(8θw) + 91)]  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FAT4oBgHgl3EQfgB2V/content/2301.08585v1.pdf'}
+page_content='− [−28 cos(2θw) + 16 cos(4θw) − 4 cos(6θw) + cos(8θw) + 19]  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FAT4oBgHgl3EQfgB2V/content/2301.08585v1.pdf'}
+page_content='× [4m8  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FAT4oBgHgl3EQfgB2V/content/2301.08585v1.pdf'}
+page_content='Z − 4m6  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FAT4oBgHgl3EQfgB2V/content/2301.08585v1.pdf'}
+page_content='Z(s + 3t) + m4  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FAT4oBgHgl3EQfgB2V/content/2301.08585v1.pdf'}
+page_content='Z(s2 + 6st + 14t2) − 2m2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FAT4oBgHgl3EQfgB2V/content/2301.08585v1.pdf'}
+page_content='Zt(s + 2t)2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FAT4oBgHgl3EQfgB2V/content/2301.08585v1.pdf'}
+page_content='+ t(s + t)(s2 + 2st + 2t2)]} ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FAT4oBgHgl3EQfgB2V/content/2301.08585v1.pdf'}
+page_content='  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FAT4oBgHgl3EQfgB2V/content/2301.08585v1.pdf'}
+page_content='4)  |MKK|2 =  1  288|S(s)|2{−72m8  Z + 6m6  Z(−40m2  l + 9s + 48t)  − 4m4  Z[−2m2  l (5s + 96t) + 136m4  l + 9t(7s + 12t)]  + 3m2  Z[−80m6  l + m4  l (256t − 14s) − 4m2  l (s2 + 29st + 68t2)  + 9s3 + 42s2t + 114st2 + 96t3)] − 36[m4  l − 2m2  l t + t(s + t)]  × (2m4  l − 4m2  l t + s2 + 2st + 2t2)} ,  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FAT4oBgHgl3EQfgB2V/content/2301.08585v1.pdf'}
+page_content='5)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FAT4oBgHgl3EQfgB2V/content/2301.08585v1.pdf'}
+page_content='18  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FAT4oBgHgl3EQfgB2V/content/2301.08585v1.pdf'}
+page_content='|Mint|2 = −  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FAT4oBgHgl3EQfgB2V/content/2301.08585v1.pdf'}
+page_content='g2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FAT4oBgHgl3EQfgB2V/content/2301.08585v1.pdf'}
+page_content='Z[S(s) + S⋆(s)]  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FAT4oBgHgl3EQfgB2V/content/2301.08585v1.pdf'}
+page_content='96(t − m2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FAT4oBgHgl3EQfgB2V/content/2301.08585v1.pdf'}
+page_content='l )(s + t − m2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FAT4oBgHgl3EQfgB2V/content/2301.08585v1.pdf'}
+page_content='l − 2m2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FAT4oBgHgl3EQfgB2V/content/2301.08585v1.pdf'}
+page_content='Z){−12m8  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FAT4oBgHgl3EQfgB2V/content/2301.08585v1.pdf'}
+page_content='l [cos(4θw) − 2 cos(2θw)]  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FAT4oBgHgl3EQfgB2V/content/2301.08585v1.pdf'}
+page_content='+ 2m6  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FAT4oBgHgl3EQfgB2V/content/2301.08585v1.pdf'}
+page_content='l [(16m2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FAT4oBgHgl3EQfgB2V/content/2301.08585v1.pdf'}
+page_content='Z + 3(3s + 4t)(cos(4θw) − 2 cos(2θw)) + 6(s − 2t)]  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FAT4oBgHgl3EQfgB2V/content/2301.08585v1.pdf'}
+page_content='+ 2m4  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FAT4oBgHgl3EQfgB2V/content/2301.08585v1.pdf'}
+page_content='l [−m2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FAT4oBgHgl3EQfgB2V/content/2301.08585v1.pdf'}
+page_content='Z((9s + 4t)(cos(4θw) − 2 cos(2θw))  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FAT4oBgHgl3EQfgB2V/content/2301.08585v1.pdf'}
+page_content='+ 24s + 60t) + 8m4  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FAT4oBgHgl3EQfgB2V/content/2301.08585v1.pdf'}
+page_content='Z(−2 cos(2θw) + cos(4θw) + 4) + 6(3s2 + st + 6t2)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FAT4oBgHgl3EQfgB2V/content/2301.08585v1.pdf'}
+page_content='+ 3s(3s − 4t)(cos(4θw) − 2 cos(2θw))]  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FAT4oBgHgl3EQfgB2V/content/2301.08585v1.pdf'}
+page_content='+ m2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FAT4oBgHgl3EQfgB2V/content/2301.08585v1.pdf'}
+page_content='l [4m2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FAT4oBgHgl3EQfgB2V/content/2301.08585v1.pdf'}
+page_content='Z((2s2 − 3st + 14t2)(cos(4θw) − 2 cos(2θw))  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FAT4oBgHgl3EQfgB2V/content/2301.08585v1.pdf'}
+page_content='+ 2s2 + 5st + 46t2) + 4m4  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FAT4oBgHgl3EQfgB2V/content/2301.08585v1.pdf'}
+page_content='Z((s − 10t)(cos(4θw) − 2 cos(2θw)) − 38t)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FAT4oBgHgl3EQfgB2V/content/2301.08585v1.pdf'}
+page_content='+ 8m6  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FAT4oBgHgl3EQfgB2V/content/2301.08585v1.pdf'}
+page_content='Z(−2 cos(2θw) + cos(4θw) + 5)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FAT4oBgHgl3EQfgB2V/content/2301.08585v1.pdf'}
+page_content='− 3(3s3 + 10s2t + 2(s3 + 2s2t + 3st2 + 4t3)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FAT4oBgHgl3EQfgB2V/content/2301.08585v1.pdf'}
+page_content='× (cos(4θw) − 2 cos(2θw)) + 24st(s + t))]  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FAT4oBgHgl3EQfgB2V/content/2301.08585v1.pdf'}
+page_content='+ 6[−2 cos(2θw) + cos(4θw) + 2][2m8  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FAT4oBgHgl3EQfgB2V/content/2301.08585v1.pdf'}
+page_content='Z + m6  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FAT4oBgHgl3EQfgB2V/content/2301.08585v1.pdf'}
+page_content='Z(s − 8t)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FAT4oBgHgl3EQfgB2V/content/2301.08585v1.pdf'}
+page_content='+ m4  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FAT4oBgHgl3EQfgB2V/content/2301.08585v1.pdf'}
+page_content='Z(−s2 + 2st + 12t2) − m2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FAT4oBgHgl3EQfgB2V/content/2301.08585v1.pdf'}
+page_content='Zt(s2 + 7st + 8t2)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FAT4oBgHgl3EQfgB2V/content/2301.08585v1.pdf'}
+page_content='+ t(s + t)(s2 + 2st + 2t2)]} ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FAT4oBgHgl3EQfgB2V/content/2301.08585v1.pdf'}
+page_content='  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FAT4oBgHgl3EQfgB2V/content/2301.08585v1.pdf'}
+page_content='6)  where θw is the Weinberg angle, gZ = e/[sin(θw) cos(θw)] is the weak coupling  constant, and mZ is the mass of the Z boson.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FAT4oBgHgl3EQfgB2V/content/2301.08585v1.pdf'}
+page_content=' Because of conservation of  helicity, in the massless limit ml = mZ = 0 s-channel graviton amplitudes  squared (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FAT4oBgHgl3EQfgB2V/content/2301.08585v1.pdf'}
+page_content='2) and (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FAT4oBgHgl3EQfgB2V/content/2301.08585v1.pdf'}
+page_content='5) are proportional to the factor t(s + t) = s(sin θ)2/4,  where θ is the scattering angle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FAT4oBgHgl3EQfgB2V/content/2301.08585v1.pdf'}
+page_content='  19  Finally,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FAT4oBgHgl3EQfgB2V/content/2301.08585v1.pdf'}
+page_content=' the SM squared amplitude for the γZ → l+l− collision looks like  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FAT4oBgHgl3EQfgB2V/content/2301.08585v1.pdf'}
+page_content='|MSM|2 =  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FAT4oBgHgl3EQfgB2V/content/2301.08585v1.pdf'}
+page_content='4g4  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FAT4oBgHgl3EQfgB2V/content/2301.08585v1.pdf'}
+page_content='Z  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FAT4oBgHgl3EQfgB2V/content/2301.08585v1.pdf'}
+page_content='(t − m2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FAT4oBgHgl3EQfgB2V/content/2301.08585v1.pdf'}
+page_content='l )2(s + t − m2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FAT4oBgHgl3EQfgB2V/content/2301.08585v1.pdf'}
+page_content='l − m2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FAT4oBgHgl3EQfgB2V/content/2301.08585v1.pdf'}
+page_content='Z)2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FAT4oBgHgl3EQfgB2V/content/2301.08585v1.pdf'}
+page_content='× {−2m8  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FAT4oBgHgl3EQfgB2V/content/2301.08585v1.pdf'}
+page_content='l [184 cos(4θw) + 17 cos(8θw) − 300 cos(2θw)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FAT4oBgHgl3EQfgB2V/content/2301.08585v1.pdf'}
+page_content='− 68 cos(6θw) + 195] + 4m6  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FAT4oBgHgl3EQfgB2V/content/2301.08585v1.pdf'}
+page_content='l [−2(59 cos(4θw) + 6 cos(8θw) − 94 cos(2θw)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FAT4oBgHgl3EQfgB2V/content/2301.08585v1.pdf'}
+page_content='− 24 cos(6θw) + 56)m2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FAT4oBgHgl3EQfgB2V/content/2301.08585v1.pdf'}
+page_content='Z + 163s + 196t − 8(33s + 37t) cos(2θw)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FAT4oBgHgl3EQfgB2V/content/2301.08585v1.pdf'}
+page_content='+ 18(9s + 10t) cos(4θw) + (15s + 16t)(cos(8θw) − 4 cos(6θw))]  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FAT4oBgHgl3EQfgB2V/content/2301.08585v1.pdf'}
+page_content='+ m4  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FAT4oBgHgl3EQfgB2V/content/2301.08585v1.pdf'}
+page_content='l [(68 cos(2θw) + 44 cos(6θw) − 56 cos(4θw) − 11 cos(8θw) − 25)m4  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FAT4oBgHgl3EQfgB2V/content/2301.08585v1.pdf'}
+page_content='Z  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FAT4oBgHgl3EQfgB2V/content/2301.08585v1.pdf'}
+page_content='+ 4[72s + 65t − 4(32s + 27t) cos(2θw) + (84s + 68t) cos(4θw)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FAT4oBgHgl3EQfgB2V/content/2301.08585v1.pdf'}
+page_content='+ (10s + 7t)(cos(8θw) − 4 cos(6θw))]m2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FAT4oBgHgl3EQfgB2V/content/2301.08585v1.pdf'}
+page_content='Z − 301s2 − 436t2 − 668st  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FAT4oBgHgl3EQfgB2V/content/2301.08585v1.pdf'}
+page_content='+ 4(125s2 + 260ts + 156t2) cos(2θw) − 8(3s2 + 78st + 46t2) cos(4θw)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FAT4oBgHgl3EQfgB2V/content/2301.08585v1.pdf'}
+page_content='− (31s2 + 52st + 28t2)(cos(8θw) − 4 cos(6θw))]  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FAT4oBgHgl3EQfgB2V/content/2301.08585v1.pdf'}
+page_content='+ m2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FAT4oBgHgl3EQfgB2V/content/2301.08585v1.pdf'}
+page_content='l [7s3 + 50s2t + 92st2 + (s2 − 2st − 4t2)(cos(8θw)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FAT4oBgHgl3EQfgB2V/content/2301.08585v1.pdf'}
+page_content='− 4 cos(6θw))s + 80t3 + (79s2 + 252st + 112t2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FAT4oBgHgl3EQfgB2V/content/2301.08585v1.pdf'}
+page_content='+ ((5(cos(8θw) − 4 cos(6θw) + 11) + 56 cos(4θw)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FAT4oBgHgl3EQfgB2V/content/2301.08585v1.pdf'}
+page_content='− 92(cos(2θw)))m2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FAT4oBgHgl3EQfgB2V/content/2301.08585v1.pdf'}
+page_content='Z − 141s − 226t + 4(57s + 94t)(cos(2θw))  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FAT4oBgHgl3EQfgB2V/content/2301.08585v1.pdf'}
+page_content='− 8(17s + 29t)(cos(4θw)) − 11(s + 2t)(cos(8θw) − 4 cos(6θw)))m2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FAT4oBgHgl3EQfgB2V/content/2301.08585v1.pdf'}
+page_content='Z  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FAT4oBgHgl3EQfgB2V/content/2301.08585v1.pdf'}
+page_content='− 4(31s2 + 108st + 52t2) cos(2θw) + 8(9s2 + 34st + 17t2) cos(4θw)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FAT4oBgHgl3EQfgB2V/content/2301.08585v1.pdf'}
+page_content='+ (5s2 + 28st + 16t2)(cos(8θw) − 4 cos(6θw)))m2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FAT4oBgHgl3EQfgB2V/content/2301.08585v1.pdf'}
+page_content='Z  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FAT4oBgHgl3EQfgB2V/content/2301.08585v1.pdf'}
+page_content='− 4(3s3 + 14s2t + 24st2 + 24t3) cos(2θw)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FAT4oBgHgl3EQfgB2V/content/2301.08585v1.pdf'}
+page_content='+ 8(s + 2t)(s2 + st + 3t2) cos(4θw)]  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FAT4oBgHgl3EQfgB2V/content/2301.08585v1.pdf'}
+page_content='− t[16 cos(4θw) + cos(8θw) − 28 cos(2θw) − 4 cos(6θw) + 19]  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FAT4oBgHgl3EQfgB2V/content/2301.08585v1.pdf'}
+page_content='× (s + t − m2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FAT4oBgHgl3EQfgB2V/content/2301.08585v1.pdf'}
+page_content='Z)(s2 + 2t2 + 2st − 2tm2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FAT4oBgHgl3EQfgB2V/content/2301.08585v1.pdf'}
+page_content='Z + m4  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FAT4oBgHgl3EQfgB2V/content/2301.08585v1.pdf'}
+page_content='Z)} .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FAT4oBgHgl3EQfgB2V/content/2301.08585v1.pdf'}
+page_content='  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FAT4oBgHgl3EQfgB2V/content/2301.08585v1.pdf'}
+page_content='7)  The contribution to the γZ → l+l− collision from the gravitons G is zero,  since there is no γZG vertex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FAT4oBgHgl3EQfgB2V/content/2301.08585v1.pdf'}
+page_content='  Note that in the limit ml = mZ = 0 all  squared amplitudes depend on variables s and t(s + t) = tu, where u is the  Mandelstam variable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FAT4oBgHgl3EQfgB2V/content/2301.08585v1.pdf'}
+page_content='  20  Appendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FAT4oBgHgl3EQfgB2V/content/2301.08585v1.pdf'}
+page_content=' Squared amplitudes for l+l− →  l+l− scattering  Here we present the result of our calculations of the squared amplitudes  for the l+l− → l+l− process (both incoming and outgoing leptons have the  same flavor).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FAT4oBgHgl3EQfgB2V/content/2301.08585v1.pdf'}
+page_content=' The SM squared amplitude has both the Z boson and photon  contributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FAT4oBgHgl3EQfgB2V/content/2301.08585v1.pdf'}
+page_content=' The latter one is given by the formula  |MSM|2 = 16e4  s2t2 [m4  l (5s2 + 11st + 5t2) + 4m2  l (s + t)(s2 + 5st + t2)  + (s2 + st + t2)2] ,  (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FAT4oBgHgl3EQfgB2V/content/2301.08585v1.pdf'}
+page_content='1)  where ml is the lepton mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FAT4oBgHgl3EQfgB2V/content/2301.08585v1.pdf'}
+page_content=' Note that the photon contribution to |MSM|2  is dominant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FAT4oBgHgl3EQfgB2V/content/2301.08585v1.pdf'}
+page_content=' That is why, we do not present (rather complicated) analytical  expression for the Z boson contribution to |MSM|2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FAT4oBgHgl3EQfgB2V/content/2301.08585v1.pdf'}
+page_content=' The graviton squared  amplitude is defined by KK graviton exchanges in s-, and t-channels,  |MKK|2 =  1  4608{|S(s)|2F1(s, t) + |S(t)|2F1(t, s)  + [S(s)S(t)⋆ + S(s)⋆S(t)]F2(s, t)} .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FAT4oBgHgl3EQfgB2V/content/2301.08585v1.pdf'}
+page_content='  (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FAT4oBgHgl3EQfgB2V/content/2301.08585v1.pdf'}
+page_content='2)  Finally, the interference term of |M|2 is equal to (neglecting small Z boson  contribution)  |Mint|2 = − e2  24st{[S(s) + S(t)⋆]F3(s, t) + [S(s)⋆ + S(t)]F3(t, s)} .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FAT4oBgHgl3EQfgB2V/content/2301.08585v1.pdf'}
+page_content='  (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FAT4oBgHgl3EQfgB2V/content/2301.08585v1.pdf'}
+page_content='3)  Here S(s) is defined by eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FAT4oBgHgl3EQfgB2V/content/2301.08585v1.pdf'}
+page_content=' (18), (19), and the following functions are intro-  duced  F1(s, t) = 6656m8  l + m6  l (8576s + 10752t) + m4  l (3440s2 + 7296t2 + 10752st)  + m2  l (360s3 + 2376s2t + 4320st2 + 2304t3)  + 9s4 + 90s3t + 378s2t2 + 576st3 + 288t4 ,  (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FAT4oBgHgl3EQfgB2V/content/2301.08585v1.pdf'}
+page_content='4)  F2(s, t) = 7552m8  l + 15168m6  l (s + t) + m4  l (7248(s2 + t2) + 14968st)  + m2  l (1032(s3 + t3) + 3690st(s + t))  + 36(s4 + t4) + 225st(s + t) + 378s2t2 ,  (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FAT4oBgHgl3EQfgB2V/content/2301.08585v1.pdf'}
+page_content='5)  21  F3(s, t) = m6  l (576s + 512t) + m4  l (552s2 + 448t2 + 1120st)  + m2  l (86s3 + 360s2t + 432st2 + 144t3)  + 3s4 + 21s3t + 45s2t2 + 48st3 + 24t4 ,  (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FAT4oBgHgl3EQfgB2V/content/2301.08585v1.pdf'}
+page_content='6)  where ml is the lepton mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FAT4oBgHgl3EQfgB2V/content/2301.08585v1.pdf'}
+page_content=' Note that F2(s, t) = F2(t, s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FAT4oBgHgl3EQfgB2V/content/2301.08585v1.pdf'}
+page_content='  References  [1] L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FAT4oBgHgl3EQfgB2V/content/2301.08585v1.pdf'}
+page_content=' Randall and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FAT4oBgHgl3EQfgB2V/content/2301.08585v1.pdf'}
+page_content=' Sundrum, Large mass hierarchy from a small extra  dimension, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FAT4oBgHgl3EQfgB2V/content/2301.08585v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FAT4oBgHgl3EQfgB2V/content/2301.08585v1.pdf'}
+page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FAT4oBgHgl3EQfgB2V/content/2301.08585v1.pdf'}
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+page_content='  [2] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FAT4oBgHgl3EQfgB2V/content/2301.08585v1.pdf'}
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+page_content=' Sirunyan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FAT4oBgHgl3EQfgB2V/content/2301.08585v1.pdf'}
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+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FAT4oBgHgl3EQfgB2V/content/2301.08585v1.pdf'}
+page_content=' D 98, 092001 (2018)  [arXiv:1809.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FAT4oBgHgl3EQfgB2V/content/2301.08585v1.pdf'}
+page_content='00327].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FAT4oBgHgl3EQfgB2V/content/2301.08585v1.pdf'}
+page_content='  [3] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FAT4oBgHgl3EQfgB2V/content/2301.08585v1.pdf'}
+page_content='M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FAT4oBgHgl3EQfgB2V/content/2301.08585v1.pdf'}
+page_content=' Sirunyan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FAT4oBgHgl3EQfgB2V/content/2301.08585v1.pdf'}
+page_content=' (CMS Collaboration), Search for resonant and non-  resonant new phenomena in high-mass dilepton final states at √s = 13  TeV, JHEP 07, 7 (2021) [arXiv:2103.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FAT4oBgHgl3EQfgB2V/content/2301.08585v1.pdf'}
+page_content='02708].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FAT4oBgHgl3EQfgB2V/content/2301.08585v1.pdf'}
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+page_content='  26' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FAT4oBgHgl3EQfgB2V/content/2301.08585v1.pdf'}
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+ 
+1 
+ 
+ 
+ 
+ 
+A double-hybrid density functional based on good local physics 
+with outstanding performance on the GMTKN55 database 
+ 
+ 
+ 
+Axel D. Becke1,a, Golokesh Santra2, and Jan M. L. Martin2 
+ 
+ 
+ 
+1Department of Chemistry, Dalhousie University, 6274 Coburg Road, P.O. Box 15000, 
+Halifax, Nova Scotia B3H 4R2, Canada 
+ 
+2Department of Molecular Chemistry and Materials Science, Weizmann Institute of 
+Science, 7610001 Reḥovot, Israel 
+ 
+a)Corresponding author: axel.becke@dal.ca 
+ 
+ 
+ 
+Abstract 
+ 
+In two recent papers [A. D. Becke, J. Chem. Phys. 156, 214101 (2022) and 157, 234102 
+(2022)] we compared two Kohn-Sham density functionals based on physical modelling 
+and theory with the best density-functional power-series fits in the literature.  The best 
+error statistics reported to date for a hybrid functional on the GMTKN55 chemical 
+database of Goerigk, Grimme, and coworkers [Phys. Chem. Chem. Phys. 19, 32184 
+(2017)] were obtained.  In the present work, additional second-order perturbation-theory 
+terms are considered.  The result is a 12-parameter double-hybrid (DH) density functional 
+with the lowest GMTKN55 “WTMAD2” error yet seen for a DH functional.  We call it 
+“DH23”. 
+ 
+
+ 
+2 
+ 
+The GMTKN55 (“general main-group thermochemistry, kinetics, and noncovalent 
+interactions”) database of Goerigk, Grimme and coworkers1 contains about 1500 
+chemical reaction and barrier-height reference energies obtained from high-level 
+wavefunction computations from over fifty sources.  GMTKN55 and its predecessors, 
+GMTKN242 and GMTKN30,3 were compiled in order to assess the ever-growing number 
+of density-functional theory (DFT) approximations in the literature.  The database is 
+composed of 55 subsets organized into five categories representing different chemical 
+reaction types: basic properties, reactions between larger systems and isomerizations, 
+barrier heights, and inter- and intra-molecular noncovalent interactions.  Table S1 in the 
+Supplementary Material describes the subsets in each category and Table S2 gives 
+references to the original literature sources. 
+ 
+ 
+Although the intension of the GMTKN55 authors was the assessment of DFTs, its 
+complexity and diversity make GMTKN55 highly attractive as a database for fitting of 
+DFTs as well.  The first (we believe) such application was by Santra, Sylvetsky, and 
+Martin,4 who trained double-hybrid (DH) functionals on the full GMTKN55 data.  Very 
+recently, Becke5,6 trained hybrid functionals on GMTKN55 also.  The focus of Becke’s 
+work was to test a suite of physically and theoretically motivated density-functional 
+models against heavily fit functionals such as ωB97X-V7 and ωB97M-V8 from 
+Mardirossian and Head-Gordon.  Until Becke’s work, ωB97X-V and ωB97M-V, trained 
+on the massive MGCDB849 “main-group chemistry database,” were the best-performing 
+hybrid functionals tested on GMTKN55.1,10  Since Santra and Martin had GMTKN55 
+
+ 
+3 
+second-order perturbation theory (PT2) correlation energies in hand from their DH 
+computations, they suggested to Becke that it might be interesting to combine their PT2 
+energies with his DFT models.  This Communication describes the outcome of that 
+suggestion.  In the end, we obtain a double-hybrid density functional with the lowest 
+GMTKN55 error statistics reported to date. 
+ 
+ 
+First, we briefly review the exchange and correlation models used by Becke in his 
+GMTKN55 trials.  Full details can be found in Refs. 5 and 6.  Our parent functional,6 
+which will be appended by power-series and PT2 terms later, is 
+par
+DC
+par
+DC
+opp
+DC
+opp
+DC
+par
+NDC
+par
+NDC
+opp
+NDC
+opp
+NDC
+exact
+X
+XC
+E
+a
+E
+a
+a
+a
+E
+E
++
++
++
++
+=
+U
+U
+0
+                               (1) 
+                        
+)
+(
+)
+(
+10
+8
+6
+XDM
+C
+XDM
+C
+RC
+XDM
+C
+exact
+X
+BR
+X
+BR
+X
+E
+E
+s
+E
+E
+E
+a
++
++
++
+−
++
+   . 
+The opposite-spin (opp) and parallel-spin (par) NDC terms are explicit models of non-
+dynamical potential energy of correlation11 (“B05”).  B05 is highly nonlocal, requiring 
+the exact-exchange energy density at every grid point.  The DC terms are total energies of 
+dynamical correlation modelled as in Refs. 12 and 13.  The term 
+)
+(
+exact
+X
+BR
+X
+BR
+X
+E
+E
+a
+−
+, 
+involving the BR exchange model,14 simulates non-dynamical correlation by virtue of the 
+locality of BR exchange.  The C6, C8, and C10 dispersion terms are from the XDM 
+(exchange-hole dipole moment) model of Becke and Johnson.15-18  The five “a” pre-
+factors and the 
+RC
+s
+ pre-factor allow fine tuning to GMTKN55 data by least-squares 
+fitting.  The “repulsion correction” factor 
+RC
+s
+ controls the interplay between the 
+correlation terms and the dispersion terms to give correct Pauli repulsion when atomic 
+densities overlap.  All terms in the parent functional are fully described in the Appendices 
+of Ref. 5. 
+
+ 
+4 
+ 
+ 
+Each term in Eq. (1) is based on an underlying reference-point-by-reference-point 
+model of its associated exchange/correlation hole.  Each model is governed by exact 
+short-range behavior and normalization constraints at every point.  The BR, NDC, and 
+XDM models rely on the Taylor expansion of the exact spherically-averaged exchange 
+hole to second order19 as follows: 
+...
+)
+,
+(
+2 +
+−
+−
+=
+s
+Q
+s
+hexact
+X
+
+
+
+
+r
+                                                                                (2) 
+where 
+(
+)
+
+
+
+
+D
+Q
+2
+6
+1
+2
+−
+
+=
+                                                                                           (3) 
+and 
+(
+)
+
+
+
+
+
+
+
+2
+4
+1 
+−
+=
+D
+                                                                                            (4) 
+and 
+
+  is the following kinetic-energy density (without a ½ factor): 
+(
+)
+ 
+=
+i
+i
+2
+
+
+
+
+   .                                                                                               (5) 
+In the above, r is the reference point, s is the interelectronic distance, and σ is the electron 
+spin.  
+
+Q  is called the exchange-charge curvature.  The DC dynamical correlation holes 
+satisfy well-known interelectronic cusp conditions as discussed in Ref. 12.  Thus, each 
+term in Eq. (1) reflects “good local physics” (as asserted in the title) at every reference 
+point. 
+ 
+ 
+ 
+
+ 
+5 
+ 
+In Ref. 6, exchange and correlation power series were added to Eq. (1) to 
+investigate whether significant improvements could thereby be made.  We invoked a 
+dimensionless power-series variable 
+|
+|
+UEG
+UEG
+Q
+Q
+Q
+q
+
+
+
+
+−
+=
+                                                                                                   (6) 
+where 
+3
+/
+5
+3
+/
+2
+2)
+6
+(
+5
+1
+
+
+
+
+−
+=
+UEG
+Q
+                                                                                       (7) 
+is the exchange-charge curvature of the uniform electron gas (UEG).  This variable, too, 
+reflects “good local physics”.  If 
+
+q  is positive (negative), then the local exchange-
+energy density is enhanced (attenuated) with respect to the UEG.  Conversely, for 
+dynamical correlation, positive (negative) 
+
+q  implies attenuation (enhancement) of the 
+local correlation-energy density with respect to its UEG counterpart.  
+
+q  is 
+predominantly positive in valence regions, with negative value inside atomic cores. 
+ 
+ 
+As 
+
+q  has infinite range, 
++
+
+
+
+−
+
+q
+, remapping into a finite domain is 
+desirable.  We use the same mapping for exchange, opposite-spin correlation, and 
+parallel-spin correlation, but with different constants 
+X
+
+, 
+opp
+C
+
+, and 
+par
+C
+
+ for each: 
+2
+2
+1
+
+
+
+
+
+q
+q
+w
+X
+X
+X
++
+=
+   ,   
+2
+2
+1
+
+
+
+
+
+q
+q
+w
+Copp
+opp
+C
+C
++
+=
+   ,   
+2
+2
+1
+
+
+
+
+
+q
+q
+w
+Cpar
+par
+C
+C
++
+=
+              (8) 
+with 
+|
+|
+UEG
+UEG
+UEG
+UEG
+Q
+Q
+Q
+Q
+Q
+Q
+q
+
+
+
+
+
+
+
++
+−
+−
++
+=
+                                                                             (9) 
+
+ 
+6 
+and 
+11
+.0
+=
+X
+
+, 
+0.14
+=
+Copp
+
+, and 
+1.6
+=
+Cpar
+
+.  The range of the w variable(s) is 
+1
+1
++
+
+
+−
+w
+, and 
+0
+=
+w
+ corresponds to the UEG limit.  We then add exchange, opposite-
+spin, and parallel-spin power series to the parent functional, Eq. (1), as follows: 
+
+
+=
++
+−
++
+=
+
+
+
+r
+3
+1
+,
+0
+,
+0
+)
+(
+d
+w
+e
+c
+E
+E
+c
+E
+E
+i
+X
+UEG
+X
+m
+i
+i
+X
+exact
+X
+UEG
+X
+X
+XC
+XC
+X
+                               (10) 
+            
+
+
+
+
+=
+=
++
++
+
+
+
+
+
+
+r
+r
+3
+0
+,
+3
+0
+,
+d
+w
+f
+e
+c
+d
+w
+e
+c
+i
+C
+SCC
+UEG
+C
+m
+i
+i
+Cpar
+i
+C
+UEG
+C
+m
+i
+i
+Copp
+Cpar
+Copp
+ 
+with respective expansion orders 
+X
+m , 
+Copp
+m
+, and 
+Cpar
+m
+.  An additional self-correlation 
+correction 
+
+
+
+
+D
+f SCC =
+                                                                                                         (11) 
+is inserted into the parallel-spin correlation terms to guarantee zero correlation energy in 
+all one-electron systems.  We subtract 
+exact
+X
+E
+ from the 
+0
+=
+i
+ term in the exchange series 
+to enforce the UEG limit.  The UEG limit is not, however, enforced for correlation.  Full 
+details are available in Ref. 6, where this functional is called “B22plus”. 
+ 
+Overall performance of a functional on GMTKN55 will be assessed1 by the 
+Weighted Total Mean Absolute Deviation “WTMAD2”: 
+i
+i
+i
+i
+i
+i
+E
+N
+N
+MAD
+|
+|
+kcal/mol
+ 
+84
+.
+56
+1
+WTMAD2
+,
+avg
+55
+1
+55
+1
+
+
+
+=
+=
+=
+   .                                           (12) 
+The MAD (mean absolute deviation) of each subset i is weighted by the energy ratio 
+56.84/|ΔE|avg,i and the number of data Ni in the set, as given in Table S1.  The constant 
+56.84 kcal/mol is the average of |ΔE|avg,i over all the sets.  Our highly nonlocal B05 
+
+ 
+7 
+functional has never been tested in the presence of core pseudopotentials.  We therefore 
+limit ourselves to atoms up to Kr, beyond which common basis sets employ 
+pseudopotentials.  Hence we omit5,6 the HEAVY28 subset from GMTKN55, and a few 
+reactions in the HEAVYSB11 and HAL59 sets.  This leaves 1443 reactions altogether in 
+our “GMTKN54” database.  As the HEAVY28 set is excluded, the summations in Eq. 
+(12) are over 54 sets instead of 55, but we retain the 56.84 kcal/mol constant for 
+compatibility with the literature. 
+ 
+Pre-factors and expansion coefficients are determined by least squares fitting.  We 
+fit to our GMTKN54 data, taking the weight of each data point from Eq. (12) except for 
+the W4-11 atomization energies, which are weighted by 1 (see Ref. 5).  Note that this is 
+not the same as minimizing the WTMAD2 directly, as has been done by the Martin group 
+in Ref. 4 and later papers.  For “B22plus” in Ref. 6, we chose optimum expansion orders 
+2
+=
+X
+m
+, 
+3
+=
+Copp
+m
+, and 
+4
+=
+Cpar
+m
+, for a total of 12 series expansion coefficients.  With 
+the six pre-factors in 
+0
+XC
+E
+ included, B22plus has 18 linear parameters altogether.  We 
+used BHandHLYP orbitals computed with the Gaussian 16 program20 and the Dunning 
+aug-cc-pVTZ basis set21 (def2-TZVP22 for K and Ca, and cc-pVTZ for the C60 isomers).  
+All correlation and power-series terms were computed using an in-house “postG” code, 
+with the analytical 
+exact
+X
+E
+ obtained from Gaussian 16 using keyword “Extralink=L608”.  
+Grids are the same as in Refs. 5 and 6.  The resulting B22plus WTMAD2 on GMTKN54 
+was 2.76 kcal/mol, substantially better than the previous best hybrid functionals, “B22” 
+of Ref. 5 at 3.31 kcal/mol, and ωB97M-V of Mardirossian and Head-Gordon8 at 3.37 
+kcal/mol.  See Ref. 6 for additional information and statistics. 
+
+ 
+8 
+ 
+ 
+We now consider the further addition of second-order perturbation theory (PT2) 
+terms in the double-hybrid DFT framework.  Contemporary double hybrids originated 
+with Grimme23 and employ an MP2-like (Møller-Plesset perturbation theory) correlation 
+component evaluated using Kohn-Sham DFT orbitals instead of Hartree-Fock (HF) 
+orbitals.  This is known as in Görling-Levy perturbation theory, GLPT2.24  The benefits 
+of using GLPT2 versus HF-MP2 as originally proposed in Ref. 25, are analyzed in Ref. 
+26.  The authors of the GMTKN55 paper1 clearly showed that double hybrids are 
+superior to hybrid functionals which use only occupied orbitals.  The literature on double 
+hybrids is vast.  A recent review by Martin and Santra27 provides an overview of the 
+current status of the field.  Earlier reviews may be found in Refs. 28-30. 
+ 
+ 
+ωB97M(2) from Mardirossian and Head-Gordon31 is considered by many to be 
+the leading double hybrid today.  It has 14 independent parameters and is a “meta”-GGA 
+(generalized gradient approximation) power series functional obtained after prescreening 
+trillions of candidates.  The WTMAD2 of ωB97M(2) on GMTKN55 is 2.13 kcal/mol.32  
+Santra, Semidalas, and Martin33 have reported WTMAD2s as low as 1.85 kcal/mol by 
+experimenting with an “XYG8” functional form that includes separate LDA (local 
+density approximation) and GGA coefficients as inspired by Ref. 34, and a sequence of 
+input orbitals with varying amounts of exact-exchange fraction.  These are not 
+constrained to the correct asymptotic C6 limit, as is ωB97M(2).  They recover about 50-
+60% of the correct C6. 
+ 
+
+ 
+9 
+ 
+In the present work, we add PT2 terms to Eqs. (1) and (10) in a manner that 
+respects the correct asymptotic C6 limit.  Our functional therefore reflects not only good 
+local physics, but good long-range physics as well.  Independent opposite-spin and 
+parallel-spin PT2 correlation energies are appended to Eq. (10) as follows: 
+
+
+=
++
+−
++
+=
+
+
+
+r
+3
+1
+,
+0
+,
+0
+)
+(
+d
+w
+e
+c
+E
+E
+c
+E
+E
+i
+X
+UEG
+X
+m
+i
+i
+X
+exact
+X
+UEG
+X
+X
+XC
+XC
+X
+                               (13) 
+             
+
+
+
+
+=
+=
++
++
+
+
+
+
+
+
+r
+r
+3
+0
+,
+3
+0
+,
+d
+w
+f
+e
+c
+d
+w
+e
+c
+i
+C
+SCC
+UEG
+C
+m
+i
+i
+Cpar
+i
+C
+UEG
+C
+m
+i
+i
+Copp
+Cpar
+Copp
+ 
+         
+)
+2
+1
+(
+)
+2
+1
+(
+6
+2
+2
+6
+2
+2
+XDM
+C
+par
+PT
+par
+PT
+XDM
+C
+opp
+PT
+opp
+PT
+E
+E
+a
+E
+E
+a
+−
++
+−
++
+ 
+where 
+
+
+2
+2
+2
+PT
+PT
+par
+PT
+E
+E
+E
++
+=
+   .                                                                                         (14) 
+ 
+The logic in the added PT2 terms is simple.  Lochan, Jung, and Head-Gordon35 have 
+stated that, at long range for two non-overlapping closed-shell systems, the opposite-spin 
+and parallel-spin parts of the inter-system MP2 correlation energy are equal.  Assuming 
+that MP2 (PT2) and XDM give the correct asymptotic dispersion energy, we therefore 
+have that 
+XDM
+C
+par
+PT
+opp
+PT
+E
+E
+E
+6
+2
+2
+2
+1
+=
+=
+                                                                                        (15) 
+at long range.  The parent functional, Eq. (1), contains a full 
+XDM
+C
+E
+6
+ term.  Hence the PT2 
+terms in Eq. (13) must vanish asymptotically.  By Eq. (15), they do. 
+ 
+ 
+PT2 energies are computed by the Q-Chem 6.0 program36 using “resolution of the 
+identity” (RI-MP2) speed-ups.  As is common practice in DH applications, single-
+
+ 
+10 
+excitation terms, non-zero in principle in GLPT2, have been omitted.  For the PT2 
+correlation energies, the same frozen-core settings were used as in Ref. 4.  These 
+calculations were run on the Weizmann Faculty of Chemistry’s HPC facility 
+“ChemFarm”. 
+ 
+In our initial explorations, we discovered some welcome simplifications of Eq. 
+(13).  Power-series orders 
+X
+m , 
+Copp
+m
+, and 
+Cpar
+m
+ beyond quadratic offered virtually no 
+improvements.  Furthermore, the entire exchange series can be removed with negligible 
+degradation of the WTMAD2.  Most significantly, the B05 non-dynamical correlation 
+terms in the parent functional, Eq. (1), are not important in a DH context despite being 
+critical to the success of B22plus.6  PT2 correlation handily compensates B05, with 
+WTMAD2 affected by less than ~ 0.05 kcal/mol if the latter is omitted.  This is a major 
+win, as the exact-exchange energy density required by B05 at every grid point is 
+available in very few quantum chemistry programs.  Hereafter, then, all results pertain to 
+the simplified functional 
+ 
+)
+(
+exact
+X
+BR
+X
+BR
+X
+par
+DC
+par
+DC
+opp
+DC
+opp
+DC
+exact
+X
+XC
+E
+E
+a
+E
+a
+E
+a
+E
+E
+−
++
++
++
+=
+                                      (16) 
+                                  
+)
+(
+10
+8
+6
+XDM
+C
+XDM
+C
+RC
+XDM
+C
+E
+E
+s
+E
++
++
++
+ 
+             
+
+
+
+
+=
+=
++
++
+
+
+
+
+
+
+r
+r
+3
+2
+0
+,
+3
+2
+0
+,
+d
+w
+f
+e
+c
+d
+w
+e
+c
+i
+C
+SCC
+UEG
+C
+i
+i
+Cpar
+i
+C
+UEG
+C
+i
+i
+Copp
+ 
+         
+)
+2
+1
+(
+)
+2
+1
+(
+6
+2
+2
+6
+2
+2
+XDM
+C
+par
+PT
+par
+PT
+XDM
+C
+opp
+PT
+opp
+PT
+E
+E
+a
+E
+E
+a
+−
++
+−
++
+ 
+ 
+
+ 
+11 
+with 12 linear coefficients altogether.  This is two fewer than the number of parameters in 
+ωB97M(2).  We also note that, after eliminating the exchange series from Eq. (13), Eq. 
+(16) exactly models the hydrogen atom at every reference point.  Only BR-based 
+functionals have this nice property.14  We shall call the functional of Eq. (16) “DH23”. 
+ 
+To make contact with our previous work,5,6 tests have been carried out with 
+BHandHLYP orbitals and the Dunning aug-cc-pVTZ basis set21 (def2-TZVP22 for K and 
+Ca, and cc-pVTZ for the C60 isomers).  The resulting WTMAD2 on GMTKN54 is 2.18 
+kcal/mol, slightly larger than for ωB97M(2), yet about 0.6 kcal/mol lower than for our 
+B22plus hybrid.  Next, we expand from the Dunning triple-zeta basis to the quadruple-
+zeta def2-QZVPP basis.22  For subsets involving anions (AHB21, BH76 and BH76RC, 
+G21EA, IL16, WATER27) or rare-gas interactions (RG18), diffuse functions are added 
+using def2-QZVPPD.37  We drop down, however, to def2-TZVPP for the very large 
+UPU23 and C60ISO systems and the two largest isomers (1 and 4) in the ISOL24 subset.  
+The BHandHLYP quadruple-zeta WTMAD2 is 1.76 kcal/mol, a substantial reduction 
+from the Dunning triple-zeta 2.18 kcal/mol value. 
+ 
+BHandHLYP orbitals incorporate 50% Hartree-Fock (HF) exchange.  There is 
+evidence33 that orbitals with 20-30% HF may offer optimal DH performance.  We 
+explore this by using “B1LYP” orbitals38 with 25% HF exchange.  We observe a slight 
+decrease in the GMTKN54 WTMAD2 from 1.76 (BHandHLYP) to 1.72 (B1LYP) 
+kcal/mol, employing the same def2 basis sets described above.  For completeness, 
+B1LYP computations have been performed using diffuse basis functions everywhere 
+
+ 
+12 
+(def2-QZVPPD, except def2-TZVPPD for UPU23, C60ISO, and the two largest ISOL24 
+isomers, 1 and 4).  Universal diffuse functions have a minor effect on the WTMAD2, 
+giving a value of 1.74 kcal/mol.  Similar observations were made in Ref. 39. 
+ 
+The 1.72 kcal/mol GMTKN54 WTMAD2 achieved here is more than 1 kcal/mol 
+lower than the 2.76 kcal/mol of B22plus, our best hybrid functional of Ref. 6, and does 
+not depend on the cumbersome B05 non-dynamical correlation energy.  Readily available 
+and economical (thanks to resolution of the identity RI-MP2 technology40-42) PT2 
+correlation is used instead.  How important are the power-series terms in Eq. (16)?  
+Omitting them loses six parameters, but raises our best B1LYP def2-QZVPP WTMAD2 
+from 1.72 kcal/mol to 2.11 kcal/mol, almost 0.4 kcal/mol higher.  This is nevertheless 
+rather good for a DH with only six parameters.  Table I summarizes our GMTKN54 
+WTMAD2s for BHandHLYP vs. B1LYP, and aug-cc-pVTZ vs. def2-QZVPP. 
+ 
+ 
+The coefficients of B1LYP DH23 are given in Table II.  The exact exchange 
+fraction is a sizable 82.6%.  The UEG limit for opposite-spin correlation is 0.882, not far 
+from 1, and for parallel-spin correlation is roughly 0.85 (using that the UEG limit of 
+par
+DC
+E
+ 
+is about half the correct value12).  The negative value, -0.326, of the repulsion-correction 
+scale factor 
+RC
+s
+ implies that PT2 overestimates dispersion energy at intermediate range.  
+This echoes previous findings of the Martin group.27,43   
+ 
+ 
+In Table III, we list WTMAD2s for each of the GMTKN chemical categories and 
+for the total GMTKN54 set.  For comparison, we include data for four additional leading 
+
+ 
+13 
+DH functionals: ωB97M(2)31 from Mardirossian and Head-Gordon, and 
+XYG8[f1]@B20LYP,33 revDSD-PBEP86-D4,32 and xDSD75-PBEP86-D432 from the 
+Martin group.  The number of parameters in these functionals is 14 for ωB97M(2) and 8 
+for the last three, versus 12 for DH23.  All computations employ the def2-QZVPP basis 
+set with exceptions as noted earlier.  DH23 has the smallest WTMAD2 on the total 
+GMTKN54 set at 1.72 kcal/mol, followed by XYG8[f1]@B20LYP at 1.81 kcal/mol.  The 
+categories “basic properties of small systems” and “reaction and isomerization energies 
+of large systems” are clearly led by DH23, but XYG8[f1]@B20LYP leads for barrier 
+heights and (all) noncovalent interactions.  The WTMAD2s in Tables I and III are on 
+GMTKN54, not the full GMTKN55.  For our four comparative functionals, the 
+difference varies from 0.04 to 0.08 kcal/mol (higher for GMTKN55).  Therefore, ~ 1.78 
+kcal/mol would be a reasonable guess for the DH23 WTMAD2 on the full GMTKN55 
+database.  See Table S3 in the Supplementary Material for MADs of all the considered 
+functionals on all the GMTKN54 subsets. 
+ 
+Let us comment, finally, on the asymptotic C6 limit.  DH23 and ωB97M(2) 
+recover the correct limit by design.  XYG8[f1]@B20LYP, revDSD-PBEP86-D4, and 
+xDSD75-PBEP86-D4 recover about 50%, 90%, and 80% respectively.  The importance of 
+enforcing the correct C6 limit is debatable in molecules near equilibrium structures.  See 
+Wu and Truhlar44 for an assessment on very large systems containing 200 to 910 atoms.  
+Nevertheless, good physics is our goal for DH23. 
+ 
+
+ 
+14 
+DH23 is evidently the best-performing hybrid or double-hybrid DFT on the 
+GMTKN55 database at present.  This is gratifying given the simple manner in which it 
+was derived.  ωB97M(2) was derived from trillions of prescreening fits.  The functionals 
+XYG8[f1]@B20LYP, revDSD-PBEP86-D4, and xDSD75-PBEP86-D4 are the result of 
+many trials with the zoo1 of known local-density, GGA, and meta-GGA exchange and 
+correlation functionals.  DH23, on the other hand, is based on explicit exchange-
+correlation hole models satisfying exact short-range-behavior and normalization 
+constraints at every point, and a power-series variable 
+
+q  governing the 
+enhancement/attenuation of exchange and correlation energy densities with respect to the 
+UEG, again, at every point.  With minimal experimentation, DH23 was quickly and 
+straightforwardly derived from its hybrid-functional parents.5,6  Although the theory is 
+already quite tight, we will explore possible improvements of DH23 in future work. 
+ 
+ 
+Supplementary Material 
+See Supplementary Material for descriptions of the 55 subsets of the GMTKN55 
+database, the original literature references, and GMTKN54 MADs for the five DH 
+functionals compared in this work. 
+ 
+ 
+Acknowledgments 
+A.D.B. gratefully acknowledges the support of the Natural Sciences and Engineering 
+Research Council of Canada (NSERC) and Prof. Erin Johnson (Dalhousie University) for 
+
+ 
+15 
+providing computing access through the Digital Research Alliance of Canada.  G.S. 
+acknowledges a doctoral fellowship from the Feinberg Graduate School (Weizmann 
+Institute of Science).  Research at Weizmann was supported by the Israel Science 
+Foundation (grant 1969/20), by the Minerva Foundation (grant 2020/05), and by a 
+research grant from the Dr. Uriel Arnon Memorial Fund for Artificial Intelligence and 
+Smart Materials.  
+ 
+ 
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+Y. Zhu, B. Alam, B. J. Albrecht, A. Aldossary, E. Alguire, J. H. Andersen, V. Athavale, 
+D. Barton, K. Begam, A. Behn, N. Bellonzi, Y. A. Bernard, E. J. Berquist, H. G. A. 
+Burton, A. Carreras, K. Carter-Fenk, R. Chakraborty, A. D. Chien, K. D. Closser, V. 
+Cofer-Shabica, S. Dasgupta, M. de Wergifosse, J. Deng, M. Diedenhofen, H. Do, S. 
+Ehlert, P.-T. Fang, S. Fatehi, Q. Feng, T. Friedhoff, J. Gayvert, Q. Ge, G. Gidofalvi, M. 
+Goldey, J. Gomes, C. E. González-Espinoza, S. Gulania, A. O. Gunina, M. W. D. 
+Hanson-Heine, P. H. P. Harbach, A. Hauser, M. F. Herbst, M. Hernández Vera, M. 
+Hodecker, Z. C. Holden, S. Houck, X. Huang, K. Hui, B. C. Huynh, M. Ivanov, Á. Jász, 
+H. Ji, H. Jiang, B. Kaduk, S. Kähler, K. Khistyaev, J. Kim, G. Kis, P. Klunzinger, Z. 
+Koczor-Benda, J. H. Koh, D. Kosenkov, L. Koulias, T. Kowalczyk, C. M. Krauter, K. 
+Kue, A. Kunitsa, T. Kus, I. Ladjánszki, A. Landau, K. V. Lawler, D. Lefrancois, S. 
+Lehtola, R. R. Li, Y.-P. Li, J. Liang, M. Liebenthal, H.-H. Lin, Y.-S. Lin, F. Liu, K.-Y. 
+Liu, M. Loipersberger, A. Luenser, A. Manjanath, P. Manohar, E. Mansoor, S. F. 
+Manzer, S.-P. Mao, A. V. Marenich, T. Markovich, S. Mason, S. A. Maurer, P. F. 
+McLaughlin, M. F. S. J. Menger, J.-M. Mewes, S. A. Mewes, P. Morgante, J. W. 
+Mullinax, K. J. Oosterbaan, G. Paran, A. C. Paul, S. K. Paul, F. Pavošević, Z. Pei, S. 
+Prager, E. I. Proynov, Á. Rák, E. Ramos-Cordoba, B. Rana, A. E. Rask, A. Rettig, R. M. 
+Richard, F. Rob, E. Rossomme, T. Scheele, M. Scheurer, M. Schneider, N. Sergueev, S. 
+M. Sharada, W. Skomorowski, D. W. Small, C. J. Stein, Y.-C. Su, E. J. Sundstrom, Z. 
+Tao, J. Thirman, G. J. Tornai, T. Tsuchimochi, N. M. Tubman, S. P. Veccham, O. 
+Vydrov, J. Wenzel, J. Witte, A. Yamada, K. Yao, S. Yeganeh, S. R. Yost, A. Zech, I. Y. 
+Zhang, X. Zhang, Y. Zhang, D. Zuev, A. Aspuru-Guzik, A. T. Bell, N. A. Besley, K. B. 
+Bravaya, B. R. Brooks, D. Casanova, J.-D. Chai, S. Coriani, C. J. Cramer, G. Cserey, A. 
+E. DePrince, R. A. DiStasio, A. Dreuw, B. D. Dunietz, T. R. Furlani, W. A. Goddard, S. 
+Hammes-Schiffer, T. Head-Gordon, W. J. Hehre, C.-P. Hsu, T.-C. Jagau, Y. Jung, A. 
+Klamt, J. Kong, D. S. Lambrecht, W. Liang, N. J. Mayhall, C. W. McCurdy, J. B. 
+Neaton, C. Ochsenfeld, J. A. Parkhill, R. Peverati, V. A. Rassolov, Y. Shao, L. V. 
+Slipchenko, T. Stauch, R. P. Steele, J. E. Subotnik, A. J. W. Thom, A. Tkatchenko, D. G. 
+Truhlar, T. Van Voorhis, T. A. Wesolowski, K. B. Whaley, H. L. Woodcock, P. M. 
+Zimmerman, S. Faraji, P. M. W. Gill, M. Head-Gordon, J. M. Herbert, and A. I. Krylov, 
+J. Chem. Phys. 155, 084801 (2021). 
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+38)  C. Adamo and V. Barone, Chem. Phys. Lett. 274, 242 (1997). 
+39)  N. Mehta and J. M. L. Martin, J. Chem. Theory Comput. 18, 5978 (2022). 
+
+ 
+18 
+40)  O. Vahtras, J. Almlöf, and M. W. Feyereisen, Chem. Phys. Lett. 213, 514 (1993). 
+41)  F. Weigend and M. Häser, Theor. Chem. Acc. 97, 331 (1997). 
+42)  F. Weigend, M. Häser, H. Patzelt, and R. Ahlrichs, Chem. Phys. Lett. 294, 143 
+(1998). 
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+18, 20905 (2016). 
+ 
+44)  D. Wu and D. G. Truhlar, J. Chem. Theory Comput. 17, 3967 (2021). 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+
+ 
+19 
+Table I:  WTMAD2 on GMTKN54, in kcal/mol. 
+ 
+                                                  Eq. (16)                   Eq. (16) 
+                                                                         without power series 
+BHandHLYP/aVTZa                   2.18                          2.24 
+BHandHLYP/QZVPPb               1.76                          2.05 
+B1LYP/QZVPPb                         1.72                          2.11 
+B1LYP/QZVPPDc                      1.74                          2.14 
+                                             12 parameters           6 parameters 
+ 
+a)  aug-cc-pVTZ with exceptions noted in text. 
+b)  def2-QZVPP with exceptions noted in text. 
+c)  def2-QZVPPD with exceptions noted in text.           
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+
+ 
+20 
+Table II:  Eq. (16) coefficients for B1LYP orbitals and QZVPPa basis set. 
+ 
+opp
+DC
+a
+    0.063              
+0
+,
+Copp
+c
+    0.418              
+opp
+PT
+a
+2     0.401 
+par
+DC
+a
+    1.464               
+1,
+Copp
+c
+   -0.253              
+par
+PT
+a
+2    0.171 
+ 
+BR
+X
+a
+    0.174              
+2
+,
+Copp
+c
+    0.556         
+ 
+RC
+s
+   -0.326               
+0
+,
+Cpar
+c
+  -0.052                   
+                                   
+1,
+Cpar
+c
+   -0.503                              
+                                   
+2
+,
+Cpar
+c
+    0.852 
+ 
+a)  def2-QZVPP with exceptions noted in text. 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+
+ 
+21 
+Table III:  WTMAD2 (kcal/mol) for GMTKN54 categories.a 
+ 
+   DH23b     ωB97M(2)c    XYG8d    revDSDe    xDSDf 
+ 
+Basic properties and reaction energies of small systems: 
+     1.22             1.38           1.43           1.76          1.64 
+Reaction energies for large systems and isomerization reactions: 
+     1.76             2.61           2.32           3.59          3.07 
+Reaction barrier heights: 
+     1.79             1.66           1.65           2.02          1.95 
+Intermolecular noncovalent interactions: 
+     1.78             1.99           1.75           1.92          1.88 
+Intramolecular noncovalent interactions: 
+     2.39             3.00           2.17           2.08          2.11 
+All noncovalent interactions: 
+     2.09             2.51           1.96           2.00          2.00 
+ 
+All GMTKN54: 
+     1.72             2.05           1.81           2.19          2.05 
+ 
+a)  See Table S1 in the Supplementary Material. 
+b)  Present work,  Eq. (16). 
+c)  Ref. 31. 
+d)  XYG8[f1]@B20LYP,  Ref. 33 (see line 5 in Table 1). 
+e)  revDSD-PBEP86-D4,  Ref. 32 (see line 6 in Table S2). 
+f)  xDSD75-PBEP86-D4,  Ref. 32 (see line 1 in Table 1). 
+ 
+ 
+
+ 
+ 
+                                   Supplementary Material 
+ 
+ 
+A double-hybrid density functional based on good local physics 
+with outstanding performance on the GMTKN55 database 
+ 
+ 
+ 
+ 
+Axel D. Becke1,a, Golokesh Santra2, and Jan M. L. Martin2 
+ 
+ 
+ 
+ 
+1Department of Chemistry, Dalhousie University, 6274 Coburg Road, P.O. Box 15000, Halifax, 
+Nova Scotia B3H 4R2, Canada 
+ 
+2Department of Molecular Chemistry and Materials Science, Weizmann Institute of Science, 
+7610001 Reḥovot, Israel 
+ 
+a)Corresponding author: axel.becke@dal.ca 
+ 
+ 
+ 
+ 
+ 
+ 
+                   Table S1:  GMTKN55 database subsets. 
+                   Table S2:  GMTKN55 database references. 
+                   Table S3:  MADs of DH functionals on GMTKN54 subsets. 
+ 
+ 
+ 
+ 
+
+Table S1:  GMTKN55 database subsets. 
+                                                                                                                             Na   |ΔE|avgb 
+Basic properties and reaction energies of small systems: 
+ 
+W4-11              Total atomization energies                                                                    140   306.91 
+G21EA             Adiabatic electron affinities                                                                    25    33.62 
+G21IP               Adiabatic ionization potentials                                                               36   257.61 
+DIPCS10          Double-ionization potentials of closed-shell systems                            10   654.26 
+PA26                 Adiabatic proton affinities (incl. of amino acids)                                  26   189.05 
+SIE4x4              Self-interaction-error related problems                                                  16   33.72 
+ALKBDE10      Dissociation energies of group-1 and -2 diatomics                               10   100.69 
+YBDE18           Bond-dissociation energies in ylides                                                      18   49.28 
+AL2x6              Dimerization energies of AlX3 compounds                                             6    35.88 
+HEAVYSB11   Dissociation energies of heavy-element compounds                              11   58.02 
+NBPRC             Oligomerizations and H2 fragmentations of NH3/BH3 systems             12    27.71 
+                          H2 activation reactions with PH3/BH3 systems 
+ALK8                Dissociation and other reactions of alkaline compounds                         8    62.60 
+RC21                 Fragmentations and rearrangements in radical cations                           21   35.70 
+G2RC                Reaction energies of selected G2/97 systems                                         25   51.26 
+BH76RC           Reaction energies of the BH76 set                                                          30   21.39 
+FH51                 Reaction energies in various (in-)organic systems                                 51    31.01 
+TAUT15           Relative energies of tautomers                                                                15     3.05 
+DC13                13 difficult cases for DFT methods                                                         13   54.98 
+ 
+Reaction energies for large systems and isomerization reactions: 
+ 
+MB16-43          Decomposition energies of artificial molecules                                      43   414.73 
+DARC               Reaction energies of Diels–Alder reactions                                            14   32.47 
+RSE43               Radical-stabilization energies                                                                 43     7.60 
+BSR36               Bond-separation reactions of saturated hydrocarbons                            36   16.20 
+CDIE20             Double-bond isomerization energies in cyclic systems                           20    4.06 
+
+ISO34               Isomerization energies of small and medium-sized organic molecules   34   14.57 
+ISOL24             Isomerization energies of large organic molecules                                  24   21.92 
+C60ISO             Relative energies of C60 isomers                                                               9    98.25 
+PArel                 Relative energies of protonated isomers                                                  20    4.63 
+ 
+Reaction barrier heights: 
+ 
+BH76                Barrier heights of hydrogen transfer, heavy atom transfer,                     76   18.61 
+                          nucleophilic substitution, unimolecular and association reactions 
+BHPERI           Barrier heights of pericyclic reactions                                                     26   20.87 
+BHDIV10         Diverse reaction barrier heights                                                              10   45.33 
+INV24               Inversion/racemization barrier heights                                                   24   31.85 
+BHROT27        Barrier heights for rotation around single bonds                                     27    6.27 
+PX13                 Proton-exchange barriers in H2O, NH3, and HF clusters                        13   33.36 
+WCPT18          Proton-transfer barriers in uncatalyzed and water-catalyzed reactions   18   34.99 
+ 
+Intermolecular noncovalent interactions: 
+ 
+RG18                Interaction energies of rare-gas complexes                                             18    0.58 
+ADIM6             Interaction energies of n-alkane dimers                                                    6    3.36 
+S22                   Binding energies of noncovalently bound dimers                                   22    7.30 
+S66                   Binding energies of noncovalently bound dimers                                   66    5.47 
+HEAVY28       Noncovalent interaction energies between heavy element hydrides       28    1.24 
+WATER27       Binding energies in (H2O)n, H+(H2O)n and OH-(H2O)n                           27   81.14 
+CARBHB12     Hydrogen-bonded complexes between carbene analogs                         12    6.04 
+                          and H2O, NH3, or HCl 
+PNICO23          Interaction energies in pnicogen-containing dimers                               23    4.27 
+HAL59              Binding energies in halogenated dimers (incl. halogen bonds)              59    4.59 
+AHB21              Interaction energies in anion–neutral dimers                                         21   22.49 
+CHB6                Interaction energies in cation–neutral dimers                                          6   26.79 
+IL16                  Interaction energies in anion–cation dimers                                          16   109.04 
+
+ 
+Intramolecular noncovalent interactions: 
+ 
+IDISP               Intramolecular dispersion interactions                                                      6    14.22 
+ICONF             Relative energies of conformers of inorganic systems                            17     3.27 
+ACONF           Relative energies of alkane conformers                                                   15     1.83 
+AMINO20x4   Relative energies of amino acid conformers                                            80     2.44 
+PCONF21        Relative energies of tri- and tetrapeptide conformers                              18     1.62 
+MCONF           Relative energies of melatonin conformers                                              51    4.97 
+SCONF            Relative energies of sugar conformers                                                     17    4.60 
+UPU23             Relative energies between RNA-backbone conformers                           23    5.72 
+BUT14DIOL   Relative energies of butane-1,4-diol conformers                                      64    2.80 
+ 
+a)  Number of reactions in subset. 
+b)  Averaged absolute reaction energy (kcal/mol). 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+
+Table S2:  GMTKN55 database references. 
+ 
+ACONF1 Relative energies of alkane conformers 
+ADIM62 Interaction energies of n-alkane dimers 
+AHB213 Interaction energies in anion–neutral dimers 
+AL2X64 Dimerization energies of AlX3 compounds 
+ALK84 Dissociation and other reactions of alkaline compounds 
+ALKBDE105 Dissociation energies in group-1 and -2 diatomics 
+AMINO20x46 Relative energies in amino acid conformers 
+BH76RC7 Reaction energies of the BH767-9 set 
+BH767-9 Barrier heights of hydrogen transfer, heavy atom transfer, nucleophilic substitution, 
+unimolecular and association reactions 
+BHDIV104 Diverse reaction barrier heights 
+BHPERI4,10-12 Barrier heights of pericyclic reactions 
+BHROT274 Barrier heights for rotation around single bonds 
+BSR3613,14 Bond-separation reactions of saturated hydrocarbons 
+BUT14DIOL15 Relative energies in butane-1,4-diol conformers 
+C60ISO16 Relative energies between C60 isomers 
+CARBHB124 Hydrogen-bonded complexes between carbene analogues and H2O, NH3, or HCl 
+CDIE2017 Double-bond isomerization energies in cyclic systems 
+CHB63 Interaction energies in cation–neutral dimers 
+DARC7,18 Reaction energies of Diels–Alder reactions 
+DC1319,7,20–29 13 difficult cases for DFT methods 
+DIPCS104 Double-ionization potentials of closed-shell systems 
+FH5130,31 Reaction energies in various (in-)organic systems 
+G21EA7,32 Adiabatic electron affinities 
+G21IP7,32 Adiabatic ionization potentials 
+G2RC1,33 Reaction energies of selected G2/97 systems 
+
+HAL5934,35 Binding energies in halogenated dimers (incl. halogen bonds) 
+HEAVY289 Noncovalent interaction energies between heavy element hydrides 
+HEAVYSB114 Dissociation energies in heavy-element compounds 
+ICONF4 Relative energies in conformers of inorganic systems 
+IDISP7,36–39 Intramolecular dispersion interactions 
+IL163 Interaction energies in anion–cation dimers 
+INV2440 Inversion/racemization barrier heights 
+ISO3436 Isomerization energies of small and medium-sized organic molecules 
+ISOL2441 Isomerization energies of large organic molecules 
+MB16-434 Decomposition energies of artificial molecules 
+MCONF42 Relative energies in melatonin conformers 
+NBPRC7,38,43 Oligomerizations and H2 fragmentations of NH3/BH3 systems; H2 activation 
+reactions with PH3/BH3 systems 
+PA264 Adiabatic proton affinities (incl. of amino acids) 
+PArel4 Relative energies in protonated isomers 
+PCONF214 Relative energies in tri- and tetrapeptide conformers 
+PNICO2344 Interaction energies in pnicogen-containing dimers 
+PX1345 Proton-exchange barriers in H2O, NH3, and HF clusters 
+RC214 Fragmentations and rearrangements in radical cations 
+RG184 Interaction energies in rare-gas complexes 
+RSE4346 Radical-stabilization energies 
+S2247 Binding energies of noncovalently bound dimers 
+S6648 Binding energies of noncovalently bound dimers 
+SCONF7,49 Relative energies of sugar conformers 
+SIE4x450 Self-interaction-error related problems 
+TAUT154 Relative energies in tautomers 
+UPU2351 Relative energies between RNA-backbone conformers 
+W4-1152 Total atomization energies 
+WATER2753 Binding energies in (H2O)n, H+(H2O)n and OH−(H2O)n 
+
+WCPT1854 Proton-transfer barriers in uncatalyzed and water-catalyzed reactions 
+YBDE1855 Bond-dissociation energies in ylides 
+ 
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+ 
+
+Table S3:  MADs (kcal/mol) of DH functionals on GMTKN54 subsets. 
+ 
+                             DH23a   ωB97M(2)b   XYG8c     revDSDd     xDSDe  
+W4-11       
+ 
+ 2.19  
+     1.38           2.30          2.49          2.07 
+G21EA    
+ 
+ 1.46  
+     1.50           1.91          2.57          2.70 
+G21IP       
+ 
+ 1.18  
+     1.60           1.77          2.27          2.12 
+DIPCS10     
+ 
+ 1.08  
+     2.53           4.48          4.47          4.78 
+PA26       
+ 
+ 0.72  
+     1.06           0.85          1.37          1.29 
+SIE4x4     
+ 
+ 2.07  
+     5.25           1.38          4.94          3.85 
+ALKBDE10    
+ 
+ 3.45  
+     3.54           4.88          2.95          2.80 
+YBDE18     
+ 
+ 0.89  
+     0.98           0.60          0.73          0.84 
+AL2X6      
+ 
+ 0.55  
+     1.41           0.49          1.69          1.41 
+HEAVYSB11   
+ 
+ 1.31  
+     0.65           1.19          1.72          1.42 
+NBPRC      
+ 
+ 0.94  
+     0.73           0.61          0.73          0.60 
+ALK8       
+ 
+ 0.74  
+     2.12           1.13          2.08          1.83 
+RC21       
+ 
+ 0.65  
+     1.08           0.82          1.50          1.77 
+G2RC       
+ 
+ 0.87  
+     1.36           1.33          1.51          1.34 
+BH76RC    
+ 
+ 0.97  
+     0.81           0.98          0.79          0.72 
+FH51      
+ 
+ 0.68  
+     0.65           0.72          0.72          0.65 
+TAUT15     
+ 
+ 0.30  
+     0.27           0.48          0.53          0.51 
+DC13       
+ 
+ 2.32  
+     1.59           3.54          1.74          1.85 
+MB16-43    
+ 
+ 4.71  
+    13.62         11.62        14.37        10.46 
+DARC      
+ 
+ 0.46  
+     1.56           0.55          0.49          0.37 
+RSE43     
+ 
+ 0.24  
+     0.44           0.15          0.83          0.76 
+BSR36     
+ 
+ 0.33  
+     0.64           0.70          1.34          0.77 
+CDIE20    
+ 
+ 0.13  
+     0.22           0.23          0.35          0.33 
+ISO34     
+ 
+ 0.27  
+     0.38           0.42          0.37          0.38 
+ISOL24     
+ 
+ 0.85  
+     1.02           1.30          1.13          1.02 
+C60ISO     
+ 
+ 9.18  
+     2.44           9.94          6.24          7.47 
+
+PArel      
+ 
+ 0.39  
+     0.43           0.38          0.36          0.36 
+BH76      
+ 
+ 0.79  
+     0.65           0.68          0.93          0.95 
+BHPERI    
+ 
+ 0.33  
+     0.51           0.42          0.75          0.72 
+BHDIV10    
+ 
+ 0.66  
+     0.77           0.93          0.80          0.66 
+INV24     
+ 
+ 0.58  
+     0.94           0.58          0.67          0.70 
+BHROT27     
+ 
+ 0.13  
+     0.17           0.16          0.10          0.09 
+PX13       
+ 
+ 2.13  
+     0.55           0.95          1.50          1.17 
+WCPT18     
+ 
+ 0.99  
+     1.07           1.21          0.89          0.73 
+RG18       
+ 
+ 0.05  
+     0.07           0.05          0.08          0.08 
+ADIM6      
+ 
+ 0.27  
+     0.43           0.34          0.26          0.36 
+S22       
+ 
+ 0.18  
+     0.14           0.12          0.14          0.11 
+S66        
+ 
+ 0.17  
+     0.15           0.16          0.16          0.17 
+WATER27    
+ 
+ 0.91  
+     0.37           1.45          0.53          0.69 
+CARBHB12    
+ 
+ 0.26  
+     0.22           0.25          0.36          0.31 
+PNICO23    
+ 
+ 0.09  
+     0.21           0.06          0.10          0.08 
+HAL59     
+ 
+ 0.16  
+     0.18           0.16          0.17          0.15 
+AHB21      
+ 
+ 0.37  
+     0.25           0.35          0.23          0.24 
+CHB6      
+  
+0.92  
+     0.87           1.14          0.70          0.72 
+IL16       
+ 
+ 0.65  
+     0.26           0.70          0.36          0.24 
+IDISP       
+ 
+ 0.59  
+     1.29           0.33          0.58          0.66 
+ICONF     
+ 
+ 0.11  
+     0.12           0.12          0.11          0.11 
+ACONF    
+ 
+ 0.11  
+     0.11           0.08          0.03          0.05 
+Amino20x4   
+ 
+ 0.08  
+     0.08           0.09          0.12          0.12 
+PCONF21    
+ 
+ 0.16  
+     0.33           0.17          0.12          0.11 
+MCONF     
+ 
+ 0.30  
+     0.32           0.26          0.09          0.09 
+SCONF    
+ 
+ 0.08  
+     0.22           0.04          0.07          0.06 
+UPU23     
+ 
+ 0.40  
+     0.44           0.39          0.49          0.47 
+BUT14DIOL   
+ 
+ 0.05  
+     0.05           0.02          0.05          0.06 
+ 
+
+a)  Present work,  Eq. (16). 
+b)  Ref. 31 in text. 
+c)  XYG8[f1]@B20LYP,  Ref. 33 in text (see line 5 in Table 1). 
+d)  revDSD-PBEP86-D4,  Ref. 32 in text (see line 6 in Table S2). 
+e)  xDSD75-PBEP86-D4,  Ref. 32 in text (see line 1 in Table 1). 
+ 
+ 
+
diff --git a/t9AzT4oBgHgl3EQfPvsd/content/tmp_files/load_file.txt b/t9AzT4oBgHgl3EQfPvsd/content/tmp_files/load_file.txt
new file mode 100644
index 0000000000000000000000000000000000000000..20f2bf2eb8cbf358e598990c96c0b8951b973f45
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+++ b/t9AzT4oBgHgl3EQfPvsd/content/tmp_files/load_file.txt
@@ -0,0 +1,1929 @@
+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf,len=1928
+page_content='1  A double-hybrid density functional based on good local physics with outstanding performance on the GMTKN55 database  Axel D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' Becke1,a, Golokesh Santra2, and Jan M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' Martin2  1Department of Chemistry, Dalhousie University, 6274 Coburg Road, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content='O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' Box 15000, Halifax, Nova Scotia B3H 4R2, Canada  2Department of Molecular Chemistry and Materials Science, Weizmann Institute of Science, 7610001 Reḥovot, Israel  a)Corresponding author: axel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content='becke@dal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content='ca  Abstract  In two recent papers [A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' Becke, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' Chem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' 156, 214101 (2022) and 157, 234102 (2022)] we compared two Kohn-Sham density functionals based on physical modelling and theory with the best density-functional power-series fits in the literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' The best error statistics reported to date for a hybrid functional on the GMTKN55 chemical database of Goerigk, Grimme, and coworkers [Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' Chem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' Chem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' 19, 32184 (2017)] were obtained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' In the present work, additional second-order perturbation-theory terms are considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' The result is a 12-parameter double-hybrid (DH) density functional with the lowest GMTKN55 “WTMAD2” error yet seen for a DH functional.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' We call it “DH23”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content='  2  The GMTKN55 (“general main-group thermochemistry, kinetics, and noncovalent interactions”) database of Goerigk, Grimme and coworkers1 contains about 1500 chemical reaction and barrier-height reference energies obtained from high-level wavefunction computations from over fifty sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' GMTKN55 and its predecessors, GMTKN242 and GMTKN30,3 were compiled in order to assess the ever-growing number of density-functional theory (DFT) approximations in the literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' The database is composed of 55 subsets organized into five categories representing different chemical reaction types: basic properties, reactions between larger systems and isomerizations, barrier heights, and inter- and intra-molecular noncovalent interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' Table S1 in the Supplementary Material describes the subsets in each category and Table S2 gives references to the original literature sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content='  Although the intension of the GMTKN55 authors was the assessment of DFTs, its complexity and diversity make GMTKN55 highly attractive as a database for fitting of DFTs as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' The first (we believe) such application was by Santra, Sylvetsky, and Martin,4 who trained double-hybrid (DH) functionals on the full GMTKN55 data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' Very recently, Becke5,6 trained hybrid functionals on GMTKN55 also.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' The focus of Becke’s work was to test a suite of physically and theoretically motivated density-functional models against heavily fit functionals such as ωB97X-V7 and ωB97M-V8 from Mardirossian and Head-Gordon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' Until Becke’s work, ωB97X-V and ωB97M-V, trained on the massive MGCDB849 “main-group chemistry database,” were the best-performing hybrid functionals tested on GMTKN55.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content='1,10  Since Santra and Martin had GMTKN55  3 second-order perturbation theory (PT2) correlation energies in hand from their DH computations, they suggested to Becke that it might be interesting to combine their PT2 energies with his DFT models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' This Communication describes the outcome of that suggestion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' In the end, we obtain a double-hybrid density functional with the lowest GMTKN55 error statistics reported to date.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content='  First, we briefly review the exchange and correlation models used by Becke in his GMTKN55 trials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' Full details can be found in Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' 5 and 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' Our parent functional,6 which will be appended by power-series and PT2 terms later, is par DC par DC opp DC opp DC par NDC par NDC opp NDC opp NDC exact X XC E a E a a a E E + + + + = U U 0 (1)  ) ( ) ( 10 8 6 XDM C XDM C RC XDM C exact X BR X BR X E E s E E E a + + + − + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' The opposite-spin (opp) and parallel-spin (par) NDC terms are explicit models of non- dynamical potential energy of correlation11 (“B05”).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' B05 is highly nonlocal, requiring the exact-exchange energy density at every grid point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' The DC terms are total energies of dynamical correlation modelled as in Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' 12 and 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' The term ) ( exact X BR X BR X E E a − , involving the BR exchange model,14 simulates non-dynamical correlation by virtue of the locality of BR exchange.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' The C6, C8, and C10 dispersion terms are from the XDM (exchange-hole dipole moment) model of Becke and Johnson.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content='15-18  The five “a” pre- factors and the RC s pre-factor allow fine tuning to GMTKN55 data by least-squares fitting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' The “repulsion correction” factor RC s controls the interplay between the correlation terms and the dispersion terms to give correct Pauli repulsion when atomic densities overlap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' All terms in the parent functional are fully described in the Appendices of Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content='  4  Each term in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' (1) is based on an underlying reference-point-by-reference-point model of its associated exchange/correlation hole.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' Each model is governed by exact short-range behavior and normalization constraints at every point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' The BR, NDC, and XDM models rely on the Taylor expansion of the exact spherically-averaged exchange hole to second order19 as follows: .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' ) , ( 2 + − − = s Q s hexact X \uf073 \uf073 \uf073 \uf072 r (2) where ( ) \uf073 \uf073 \uf073 \uf072 D Q 2 6 1 2 − \uf0d1 = (3) and ( ) \uf073 \uf073 \uf073 \uf073 \uf072 \uf072 \uf074 2 4 1 \uf0d1 − = D (4) and \uf073 \uf074  is the following kinetic-energy density (without a ½ factor): ( ) \uf0e5 \uf0d1 = i i 2 \uf073 \uf073 \uf079 \uf074 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' (5) In the above, r is the reference point, s is the interelectronic distance, and σ is the electron spin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' \uf073 Q  is called the exchange-charge curvature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' The DC dynamical correlation holes satisfy well-known interelectronic cusp conditions as discussed in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' Thus, each term in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' (1) reflects “good local physics” (as asserted in the title) at every reference point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content='  5  In Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' 6, exchange and correlation power series were added to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' (1) to investigate whether significant improvements could thereby be made.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' We invoked a dimensionless power-series variable | | UEG UEG Q Q Q q \uf073 \uf073 \uf073 \uf073 − = (6) where 3 / 5 3 / 2 2) 6 ( 5 1 \uf073 \uf073 \uf072 \uf070 − = UEG Q (7) is the exchange-charge curvature of the uniform electron gas (UEG).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' This variable, too, reflects “good local physics”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' If \uf073 q  is positive (negative), then the local exchange- energy density is enhanced (attenuated) with respect to the UEG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' Conversely, for dynamical correlation, positive (negative) \uf073 q  implies attenuation (enhancement) of the local correlation-energy density with respect to its UEG counterpart.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' \uf073 q  is predominantly positive in valence regions, with negative value inside atomic cores.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content='  As \uf073 q  has infinite range, +\uf0a5 \uf03c \uf03c \uf0a5 − \uf073 q , remapping into a finite domain is desirable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' We use the same mapping for exchange, opposite-spin correlation, and parallel-spin correlation, but with different constants X \uf067 , opp C \uf067 , and par C \uf067 for each: 2 2 1 \uf073 \uf073 \uf073 \uf067 \uf067 q q w X X X + = , 2 2 1 \uf061\uf062 \uf061\uf062 \uf061\uf062 \uf067 \uf067 q q w Copp opp C C + = , 2 2 1 \uf073 \uf073 \uf073\uf073 \uf067 \uf067 q q w Cpar par C C + = (8) with | | UEG UEG UEG UEG Q Q Q Q Q Q q \uf062 \uf061 \uf062 \uf061 \uf062 \uf061 \uf061\uf062 + − − + = (9)  6 and 11 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content='0 = X \uf067 , 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content='14 = Copp \uf067 , and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content='6 = Cpar \uf067 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' The range of the w variable(s) is 1 1 + \uf0a3 \uf0a3 − w , and 0 = w corresponds to the UEG limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' We then add exchange, opposite- spin, and parallel-spin power series to the parent functional, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' (1), as follows: \uf0e5\uf0f2 \uf0e5 = + − + = \uf073 \uf073 \uf073 r 3 1 , 0 , 0 ) ( d w e c E E c E E i X UEG X m i i X exact X UEG X X XC XC X (10)  \uf0e5\uf0f2  \uf0e5  \uf0f2  \uf0e5  =  =  +  +  \uf073  \uf073\uf073  \uf073  \uf073\uf073  \uf061\uf062  \uf061\uf062  r  r  3  0  ,  3  0  ,  d  w  f  e  c  d  w  e  c  i  C  SCC  UEG  C  m  i  i  Cpar  i  C  UEG  C  m  i  i  Copp  Cpar  Copp  with respective expansion orders X m , Copp m , and Cpar m .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' An additional self-correlation correction \uf073 \uf073 \uf073 \uf074 D f SCC = (11) is inserted into the parallel-spin correlation terms to guarantee zero correlation energy in all one-electron systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' We subtract exact X E from the 0 = i term in the exchange series to enforce the UEG limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' The UEG limit is not, however, enforced for correlation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' Full details are available in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' 6, where this functional is called “B22plus”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content='  Overall performance of a functional on GMTKN55 will be assessed1 by the Weighted Total Mean Absolute Deviation “WTMAD2”: i i i i i i E N N MAD | | kcal/mol  84 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' 56 1 WTMAD2 , avg 55 1 55 1 \uf044 \uf0e5 \uf0e5 = = = .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' (12) The MAD (mean absolute deviation) of each subset i is weighted by the energy ratio 56.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content='84/|ΔE|avg,i and the number of data Ni in the set, as given in Table S1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' The constant 56.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content='84 kcal/mol is the average of |ΔE|avg,i over all the sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' Our highly nonlocal B05  7 functional has never been tested in the presence of core pseudopotentials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' We therefore limit ourselves to atoms up to Kr, beyond which common basis sets employ pseudopotentials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' Hence we omit5,6 the HEAVY28 subset from GMTKN55, and a few reactions in the HEAVYSB11 and HAL59 sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' This leaves 1443 reactions altogether in our “GMTKN54” database.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' As the HEAVY28 set is excluded, the summations in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' (12) are over 54 sets instead of 55, but we retain the 56.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content='84 kcal/mol constant for compatibility with the literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content='  Pre-factors and expansion coefficients are determined by least squares fitting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' We fit to our GMTKN54 data, taking the weight of each data point from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' (12) except for the W4-11 atomization energies, which are weighted by 1 (see Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' 5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' Note that this is not the same as minimizing the WTMAD2 directly, as has been done by the Martin group in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' 4 and later papers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' For “B22plus” in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' 6, we chose optimum expansion orders 2 = X m , 3 = Copp m , and 4 = Cpar m , for a total of 12 series expansion coefficients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' With the six pre-factors in 0 XC E included, B22plus has 18 linear parameters altogether.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' We used BHandHLYP orbitals computed with the Gaussian 16 program20 and the Dunning aug-cc-pVTZ basis set21 (def2-TZVP22 for K and Ca, and cc-pVTZ for the C60 isomers).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' All correlation and power-series terms were computed using an in-house “postG” code, with the analytical exact X E obtained from Gaussian 16 using keyword “Extralink=L608”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' Grids are the same as in Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' 5 and 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' The resulting B22plus WTMAD2 on GMTKN54 was 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content='76 kcal/mol, substantially better than the previous best hybrid functionals, “B22” of Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' 5 at 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content='31 kcal/mol, and ωB97M-V of Mardirossian and Head-Gordon8 at 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content='37 kcal/mol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' See Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' 6 for additional information and statistics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content='  8  We now consider the further addition of second-order perturbation theory (PT2) terms in the double-hybrid DFT framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' Contemporary double hybrids originated with Grimme23 and employ an MP2-like (Møller-Plesset perturbation theory) correlation component evaluated using Kohn-Sham DFT orbitals instead of Hartree-Fock (HF) orbitals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' This is known as in Görling-Levy perturbation theory, GLPT2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content='24  The benefits of using GLPT2 versus HF-MP2 as originally proposed in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' 25, are analyzed in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' The authors of the GMTKN55 paper1 clearly showed that double hybrids are superior to hybrid functionals which use only occupied orbitals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' The literature on double hybrids is vast.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' A recent review by Martin and Santra27 provides an overview of the current status of the field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' Earlier reviews may be found in Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' 28-30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content='  ωB97M(2) from Mardirossian and Head-Gordon31 is considered by many to be the leading double hybrid today.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' It has 14 independent parameters and is a “meta”-GGA (generalized gradient approximation) power series functional obtained after prescreening trillions of candidates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' The WTMAD2 of ωB97M(2) on GMTKN55 is 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content='13 kcal/mol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content='32 Santra, Semidalas, and Martin33 have reported WTMAD2s as low as 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content='85 kcal/mol by experimenting with an “XYG8” functional form that includes separate LDA (local density approximation) and GGA coefficients as inspired by Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' 34, and a sequence of input orbitals with varying amounts of exact-exchange fraction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' These are not constrained to the correct asymptotic C6 limit, as is ωB97M(2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' They recover about 50- 60% of the correct C6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content='  9  In the present work, we add PT2 terms to Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' (1) and (10) in a manner that respects the correct asymptotic C6 limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' Our functional therefore reflects not only good local physics, but good long-range physics as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' Independent opposite-spin and parallel-spin PT2 correlation energies are appended to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' (10) as follows: \uf0e5\uf0f2 \uf0e5 = + − + = \uf073 \uf073 \uf073 r 3 1 , 0 , 0 ) ( d w e c E E c E E i X UEG X m i i X exact X UEG X X XC XC X (13)  \uf0e5\uf0f2  \uf0e5  \uf0f2  \uf0e5  =  =  +  +  \uf073  \uf073\uf073  \uf073  \uf073\uf073  \uf061\uf062  \uf061\uf062  r  r  3  0  ,  3  0  ,  d  w  f  e  c  d  w  e  c  i  C  SCC  UEG  C  m  i  i  Cpar  i  C  UEG  C  m  i  i  Copp  Cpar  Copp  )  2  1  (  )  2  1  (  6  2  2  6  2  2  XDM  C  par  PT  par  PT  XDM  C  opp  PT  opp  PT  E  E  a  E  E  a  −  +  −  +  where \uf062\uf062 \uf061\uf061 2 2 2 PT PT par PT E E E + = .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' (14)  The logic in the added PT2 terms is simple.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' Lochan, Jung, and Head-Gordon35 have stated that, at long range for two non-overlapping closed-shell systems, the opposite-spin and parallel-spin parts of the inter-system MP2 correlation energy are equal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' Assuming that MP2 (PT2) and XDM give the correct asymptotic dispersion energy, we therefore have that XDM C par PT opp PT E E E 6 2 2 2 1 = = (15) at long range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' The parent functional, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' (1), contains a full XDM C E 6 term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' Hence the PT2 terms in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' (13) must vanish asymptotically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' By Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' (15), they do.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content='  PT2 energies are computed by the Q-Chem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content='0 program36 using “resolution of the identity” (RI-MP2) speed-ups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' As is common practice in DH applications, single-  10 excitation terms, non-zero in principle in GLPT2, have been omitted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' For the PT2 correlation energies, the same frozen-core settings were used as in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' These calculations were run on the Weizmann Faculty of Chemistry’s HPC facility “ChemFarm”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content='  In our initial explorations, we discovered some welcome simplifications of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' (13).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' Power-series orders X m , Copp m , and Cpar m beyond quadratic offered virtually no improvements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' Furthermore, the entire exchange series can be removed with negligible degradation of the WTMAD2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' Most significantly, the B05 non-dynamical correlation terms in the parent functional, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' (1), are not important in a DH context despite being critical to the success of B22plus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content='6  PT2 correlation handily compensates B05, with WTMAD2 affected by less than ~ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content='05 kcal/mol if the latter is omitted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' This is a major win, as the exact-exchange energy density required by B05 at every grid point is available in very few quantum chemistry programs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' Hereafter,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' then,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' all results pertain to the simplified functional  )  (  exact  X  BR  X  BR  X  par  DC  par  DC  opp  DC  opp  DC  exact  X  XC  E  E  a  E  a  E  a  E  E  −  +  +  +  =  (16)  )  (  10  8  6  XDM  C  XDM  C  RC  XDM  C  E  E  s  E  +  +  +  \uf0e5\uf0f2  \uf0e5  \uf0f2  \uf0e5  =  =  +  +  \uf073  \uf073\uf073  \uf073  \uf073\uf073  \uf061\uf062  \uf061\uf062  r  r  3  2  0  ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content='  3  2  0  ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content='  d  w  f  e  c  d  w  e  c  i  C  SCC  UEG  C  i  i  Cpar  i  C  UEG  C  i  i  Copp  )  2  1  (  )  2  1  (  6  2  2  6  2  2  XDM  C  par  PT  par  PT  XDM  C  opp  PT  opp  PT  E  E  a  E  E  a  −  +  −  +  11 with 12 linear coefficients altogether.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' This is two fewer than the number of parameters in ωB97M(2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' We also note that, after eliminating the exchange series from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' (13), Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' (16) exactly models the hydrogen atom at every reference point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' Only BR-based functionals have this nice property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content='14  We shall call the functional of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' (16) “DH23”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content='  To make contact with our previous work,5,6 tests have been carried out with BHandHLYP orbitals and the Dunning aug-cc-pVTZ basis set21 (def2-TZVP22 for K and Ca, and cc-pVTZ for the C60 isomers).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' The resulting WTMAD2 on GMTKN54 is 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content='18 kcal/mol, slightly larger than for ωB97M(2), yet about 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content='6 kcal/mol lower than for our B22plus hybrid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' Next, we expand from the Dunning triple-zeta basis to the quadruple- zeta def2-QZVPP basis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content='22  For subsets involving anions (AHB21, BH76 and BH76RC, G21EA, IL16, WATER27) or rare-gas interactions (RG18), diffuse functions are added using def2-QZVPPD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content='37  We drop down, however, to def2-TZVPP for the very large UPU23 and C60ISO systems and the two largest isomers (1 and 4) in the ISOL24 subset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' The BHandHLYP quadruple-zeta WTMAD2 is 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content='76 kcal/mol, a substantial reduction from the Dunning triple-zeta 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content='18 kcal/mol value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content='  BHandHLYP orbitals incorporate 50% Hartree-Fock (HF) exchange.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' There is evidence33 that orbitals with 20-30% HF may offer optimal DH performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' We explore this by using “B1LYP” orbitals38 with 25% HF exchange.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' We observe a slight decrease in the GMTKN54 WTMAD2 from 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content='76 (BHandHLYP) to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content='72 (B1LYP) kcal/mol, employing the same def2 basis sets described above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' For completeness, B1LYP computations have been performed using diffuse basis functions everywhere  12 (def2-QZVPPD, except def2-TZVPPD for UPU23, C60ISO, and the two largest ISOL24 isomers, 1 and 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' Universal diffuse functions have a minor effect on the WTMAD2, giving a value of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content='74 kcal/mol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' Similar observations were made in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' 39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content='  The 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content='72 kcal/mol GMTKN54 WTMAD2 achieved here is more than 1 kcal/mol lower than the 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content='76 kcal/mol of B22plus, our best hybrid functional of Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' 6, and does not depend on the cumbersome B05 non-dynamical correlation energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' Readily available and economical (thanks to resolution of the identity RI-MP2 technology40-42) PT2 correlation is used instead.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' How important are the power-series terms in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' (16)?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' Omitting them loses six parameters, but raises our best B1LYP def2-QZVPP WTMAD2 from 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content='72 kcal/mol to 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content='11 kcal/mol, almost 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content='4 kcal/mol higher.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' This is nevertheless rather good for a DH with only six parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' Table I summarizes our GMTKN54 WTMAD2s for BHandHLYP vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' B1LYP, and aug-cc-pVTZ vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' def2-QZVPP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content='  The coefficients of B1LYP DH23 are given in Table II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' The exact exchange fraction is a sizable 82.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content='6%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' The UEG limit for opposite-spin correlation is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content='882, not far from 1, and for parallel-spin correlation is roughly 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content='85 (using that the UEG limit of par DC E  is about half the correct value12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' The negative value, -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content='326, of the repulsion-correction scale factor RC s implies that PT2 overestimates dispersion energy at intermediate range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' This echoes previous findings of the Martin group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content='27,43  In Table III, we list WTMAD2s for each of the GMTKN chemical categories and for the total GMTKN54 set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' For comparison, we include data for four additional leading  13 DH functionals: ωB97M(2)31 from Mardirossian and Head-Gordon, and XYG8[f1]@B20LYP,33 revDSD-PBEP86-D4,32 and xDSD75-PBEP86-D432 from the Martin group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' The number of parameters in these functionals is 14 for ωB97M(2) and 8 for the last three, versus 12 for DH23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' All computations employ the def2-QZVPP basis set with exceptions as noted earlier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' DH23 has the smallest WTMAD2 on the total GMTKN54 set at 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content='72 kcal/mol, followed by XYG8[f1]@B20LYP at 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content='81 kcal/mol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' The categories “basic properties of small systems” and “reaction and isomerization energies of large systems” are clearly led by DH23, but XYG8[f1]@B20LYP leads for barrier heights and (all) noncovalent interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' The WTMAD2s in Tables I and III are on GMTKN54, not the full GMTKN55.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' For our four comparative functionals, the difference varies from 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content='04 to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content='08 kcal/mol (higher for GMTKN55).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' Therefore, ~ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content='78 kcal/mol would be a reasonable guess for the DH23 WTMAD2 on the full GMTKN55 database.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' See Table S3 in the Supplementary Material for MADs of all the considered functionals on all the GMTKN54 subsets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content='  Let us comment, finally, on the asymptotic C6 limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' DH23 and ωB97M(2) recover the correct limit by design.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' XYG8[f1]@B20LYP, revDSD-PBEP86-D4, and xDSD75-PBEP86-D4 recover about 50%, 90%, and 80% respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' The importance of enforcing the correct C6 limit is debatable in molecules near equilibrium structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' See Wu and Truhlar44 for an assessment on very large systems containing 200 to 910 atoms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' Nevertheless, good physics is our goal for DH23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content='  14 DH23 is evidently the best-performing hybrid or double-hybrid DFT on the GMTKN55 database at present.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' This is gratifying given the simple manner in which it was derived.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' ωB97M(2) was derived from trillions of prescreening fits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' The functionals XYG8[f1]@B20LYP, revDSD-PBEP86-D4, and xDSD75-PBEP86-D4 are the result of many trials with the zoo1 of known local-density, GGA, and meta-GGA exchange and correlation functionals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' DH23, on the other hand, is based on explicit exchange- correlation hole models satisfying exact short-range-behavior and normalization constraints at every point, and a power-series variable \uf073 q  governing the enhancement/attenuation of exchange and correlation energy densities with respect to the UEG, again, at every point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' With minimal experimentation, DH23 was quickly and straightforwardly derived from its hybrid-functional parents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content='5,6  Although the theory is already quite tight, we will explore possible improvements of DH23 in future work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content='  Supplementary Material See Supplementary Material for descriptions of the 55 subsets of the GMTKN55 database, the original literature references, and GMTKN54 MADs for the five DH functionals compared in this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content='  Acknowledgments A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content='D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content='B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' gratefully acknowledges the support of the Natural Sciences and Engineering Research Council of Canada (NSERC) and Prof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' Erin Johnson (Dalhousie University) for  15 providing computing access through the Digital Research Alliance of Canada.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' acknowledges a doctoral fellowship from the Feinberg Graduate School (Weizmann Institute of Science).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' Research at Weizmann was supported by the Israel Science Foundation (grant 1969/20), by the Minerva Foundation (grant 2020/05), and by a research grant from the Dr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' Uriel Arnon Memorial Fund for Artificial Intelligence and Smart Materials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content='  References 1)  L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' Goerigk, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' Hansen, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' Bauer, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' Ehrlich, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' Najibi, and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' Grimme, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' Chem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' Chem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' 19, 32184 (2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' 2)  L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' Goerigk and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' Grimme, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' Chem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' Theory Comput.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' 6, 107 (2010).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' 3)  L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' Goerigk and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' Grimme, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' Chem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' Theory Comput.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' 7, 291 (2011).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' 4)  G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' Santra, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' Sylvetsky, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' Martin, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' Chem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' A 123, 5129 (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' 5)  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' Becke, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' Chem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' 156, 214101 (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' 6)  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' Becke, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' Chem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' 157, 234102 (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' 7)  N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' Mardirossian and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' Head-Gordon, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' Chem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' Chem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' 16, 9904 (2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' 8)  N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' Mardirossian and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' Head-Gordon, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' Chem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' 144, 214110 (2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' 9)  N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' Mardirossian and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' Head-Gordon, Mol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' 115, 2315 (2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' 10)  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' Najibi and L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' Goerigk, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' Chem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' Theory Comput.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' 14, 5725 (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' 11)  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' Becke, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' Chem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' 119, 2972 (2003);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' 122, 064101 (2005).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' 12)  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' Becke, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' Chem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' 88, 1053 (1988).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' 13)  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' Becke, Int.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' Quantum Chem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
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+page_content=' 14)  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' Becke and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' Roussel, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
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+page_content='  16 15)  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' Becke and E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' Johnson, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' Chem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' 124, 014104 (2006).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' 16)  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' Becke and E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' Johnson, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' Chem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' 127, 154108 (2007).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' 17)  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' Becke and E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' Johnson, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' Chem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' 123, 154101 (2005).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' 18)  E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' Johnson and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
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+page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
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+page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' Martin, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
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+page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
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+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
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+page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
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+page_content=' Weigend and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' Ahlrichs, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' Chem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' Chem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
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+page_content=' 23)  S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
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+page_content=' Chem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
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+page_content=' 24)  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
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+page_content=' Levy, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' A 50, 196 (1994).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' 25)  Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' Zhao, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' Lynch, and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' Truhlar, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
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+page_content=' A 108, 4786 (2004).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' 26)  G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' Santra and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
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+page_content=' Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' 2611, 020015 (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
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+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' Martin and G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' Santra, Isr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' Chem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' 60, 787 (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' 28)  I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' Zhang and X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' Xu, Int.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' Chem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' 30, 115 (2011).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' 29)  L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' Goerigk and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' Grimme, Wiley Interdisc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' Comput.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' Mol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' Sci 4, 576 (2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' 30)  E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' Brémond, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
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+page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' Sancho-Garcia, and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' Adamo, Acc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
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+page_content=' Res.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
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+page_content=' Mardirossian and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' Head-Gordon, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
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+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
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+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
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+page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
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+page_content=' Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' Zhang and X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
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+page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
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+page_content=' Hehre, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
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+page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
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+page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
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+page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
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+page_content=' Ochsenfeld, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content='  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
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+page_content=' Peverati, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' Rassolov, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' Shao, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' Slipchenko, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' Stauch, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' Steele, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' Subotnik, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' Thom, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' Tkatchenko, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' Truhlar, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' Van Voorhis, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' Wesolowski, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' Whaley, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' Woodcock, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' Zimmerman, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' Faraji, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' Gill, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' Head-Gordon, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' Herbert, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' Krylov, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' Chem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' 155, 084801 (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' 37)  D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' Rappoport and F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' Furche, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' Chem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' 133, 134105 (2010).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' 38)  C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' Adamo and V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' Barone, Chem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' 274, 242 (1997).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' 39)  N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' Mehta and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' Martin, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' Chem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' Theory Comput.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' 18, 5978 (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content='  18 40)  O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' Vahtras, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' Almlöf, and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' Feyereisen, Chem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
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+page_content=' 41)  F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' Weigend and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' Häser, Theor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' Chem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' Acc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' 97, 331 (1997).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' 42)  F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' Weigend, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' Häser, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' Patzelt, and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' Ahlrichs, Chem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' 294, 143 (1998).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' 43)  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' Brauer, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' Kesharwani, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' Kozuch, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' Martin, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' Chem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' Chem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' 18, 20905 (2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content='  44)  D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' Wu and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' Truhlar, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' Chem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' Theory Comput.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' 17, 3967 (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content='  19 Table I:  WTMAD2 on GMTKN54, in kcal/mol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content='  Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' (16)                   Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' (16) without power series BHandHLYP/aVTZa                   2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content='18                          2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content='24 BHandHLYP/QZVPPb               1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content='76                          2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content='05 B1LYP/QZVPPb                         1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content='72                          2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content='11 B1LYP/QZVPPDc                      1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content='74                          2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content='14 12 parameters           6 parameters  a)  aug-cc-pVTZ with exceptions noted in text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' b)  def2-QZVPP with exceptions noted in text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' c)  def2-QZVPPD with exceptions noted in text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content='  20 Table II:  Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' (16) coefficients for B1LYP orbitals and QZVPPa basis set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content='  opp DC a 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content='063 0 , Copp c 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content='418 opp PT a 2     0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content='401 par DC a 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content='464 1, Copp c -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content='253 par PT a 2    0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content='171  BR  X  a  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content='174  2  ,  Copp  c  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content='556  RC  s 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content='326  0  ,  Cpar  c 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content='052  1,  Cpar  c 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content='503  2  ,  Cpar  c  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content='852  a)  def2-QZVPP with exceptions noted in text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content='  21 Table III:  WTMAD2 (kcal/mol) for GMTKN54 categories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content='a  DH23b     ωB97M(2)c    XYG8d    revDSDe    xDSDf  Basic properties and reaction energies of small systems: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content='22             1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content='38           1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content='43           1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content='76          1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content='64 Reaction energies for large systems and isomerization reactions: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content='76             2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content='61           2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content='32           3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content='59          3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content='07 Reaction barrier heights: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content='79             1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content='66           1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content='65           2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content='02          1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content='95 Intermolecular noncovalent interactions: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content='78             1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content='99           1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content='75           1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content='92          1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content='88 Intramolecular noncovalent interactions: 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content='39             3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content='00           2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content='17           2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content='08          2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content='11 All noncovalent interactions: 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content='09             2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content='51           1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content='96           2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content='00          2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content='00  All GMTKN54: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content='72             2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content='05           1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content='81           2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content='19          2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content='05  a)  See Table S1 in the Supplementary Material.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' b)  Present work,  Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' (16).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' c)  Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' 31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' d)  XYG8[f1]@B20LYP,  Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' 33 (see line 5 in Table 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' e)  revDSD-PBEP86-D4,  Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' 32 (see line 6 in Table S2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' f)  xDSD75-PBEP86-D4,  Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' 32 (see line 1 in Table 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content='  Supplementary Material  A double-hybrid density functional based on good local physics with outstanding performance on the GMTKN55 database  Axel D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' Becke1,a, Golokesh Santra2, and Jan M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' Martin2  1Department of Chemistry, Dalhousie University, 6274 Coburg Road, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content='O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' Box 15000, Halifax, Nova Scotia B3H 4R2, Canada  2Department of Molecular Chemistry and Materials Science, Weizmann Institute of Science, 7610001 Reḥovot, Israel  a)Corresponding author: axel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content='becke@dal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content='ca  Table S1:  GMTKN55 database subsets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' Table S2:  GMTKN55 database references.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' Table S3:  MADs of DH functionals on GMTKN54 subsets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content='  Table S1:  GMTKN55 database subsets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' Na   |ΔE|avgb Basic properties and reaction energies of small systems:  W4-11              Total atomization energies                                                                    140   306.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content='91 G21EA             Adiabatic electron affinities                                                                    25    33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content='62 G21IP               Adiabatic ionization potentials                                                               36   257.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content='61 DIPCS10          Double-ionization potentials of closed-shell systems                            10   654.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content='26 PA26                 Adiabatic proton affinities (incl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content='  of amino acids)                                  26   189.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content='05 SIE4x4              Self-interaction-error related problems                                                  16   33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content='72 ALKBDE10      Dissociation energies of group-1 and -2 diatomics                               10   100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content='69 YBDE18           Bond-dissociation energies in ylides                                                      18   49.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content='28 AL2x6              Dimerization energies of AlX3 compounds                                             6    35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content='88 HEAVYSB11   Dissociation energies of heavy-element compounds                              11   58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content='02 NBPRC             Oligomerizations and H2 fragmentations of NH3/BH3 systems             12    27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content='71 H2 activation reactions with PH3/BH3 systems ALK8  Dissociation and other reactions of alkaline compounds                         8    62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content='60 RC21                 Fragmentations and rearrangements in radical cations                           21   35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content='70 G2RC                Reaction energies of selected G2/97 systems                                         25   51.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content='26 BH76RC           Reaction energies of the BH76 set                                                          30   21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content='39 FH51                 Reaction energies in various (in-)organic systems                                 51    31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content='01 TAUT15           Relative energies of tautomers                                                                15     3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content='05 DC13                13 difficult cases for DFT methods                                                         13   54.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content='98  Reaction energies for large systems and isomerization reactions:  MB16-43          Decomposition energies of artificial molecules                                      43   414.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content='73 DARC               Reaction energies of Diels–Alder reactions                                            14   32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content='47 RSE43               Radical-stabilization energies                                                                 43     7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content='60 BSR36               Bond-separation reactions of saturated hydrocarbons                            36   16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content='20 CDIE20             Double-bond isomerization energies in cyclic systems                           20    4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content='06  ISO34               Isomerization energies of small and medium-sized organic molecules   34   14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content='57 ISOL24             Isomerization energies of large organic molecules                                  24   21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content='92 C60ISO             Relative energies of C60 isomers                                                               9    98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content='25 PArel                 Relative energies of protonated isomers                                                  20    4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content='63  Reaction barrier heights:  BH76                Barrier heights of hydrogen transfer, heavy atom transfer,                     76   18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content='61 nucleophilic substitution, unimolecular and association reactions BHPERI           Barrier heights of pericyclic reactions                                                     26   20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content='87 BHDIV10         Diverse reaction barrier heights                                                              10   45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content='33 INV24               Inversion/racemization barrier heights                                                   24   31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content='85 BHROT27        Barrier heights for rotation around single bonds                                     27    6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content='27 PX13                 Proton-exchange barriers in H2O, NH3, and HF clusters                        13   33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content='36 WCPT18          Proton-transfer barriers in uncatalyzed and water-catalyzed reactions   18   34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content='99  Intermolecular noncovalent interactions:  RG18                Interaction energies of rare-gas complexes                                             18    0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content='58 ADIM6             Interaction energies of n-alkane dimers                                                    6    3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content='36 S22                   Binding energies of noncovalently bound dimers                                   22    7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content='30 S66                   Binding energies of noncovalently bound dimers                                   66    5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content='47 HEAVY28       Noncovalent interaction energies between heavy element hydrides       28    1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content='24 WATER27       Binding energies in (H2O)n, H+(H2O)n and OH-(H2O)n                           27   81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content='14 CARBHB12     Hydrogen-bonded complexes between carbene analogs                         12    6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content='04 and H2O, NH3, or HCl PNICO23          Interaction energies in pnicogen-containing dimers                               23    4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content='27 HAL59              Binding energies in halogenated dimers (incl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' halogen bonds)              59    4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content='59 AHB21              Interaction energies in anion–neutral dimers                                         21   22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content='49 CHB6                Interaction energies in cation–neutral dimers                                          6   26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content='79 IL16                  Interaction energies in anion–cation dimers                                          16   109.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content='04  Intramolecular noncovalent interactions:  IDISP               Intramolecular dispersion interactions                                                      6    14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content='22 ICONF             Relative energies of conformers of inorganic systems                            17     3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content='27 ACONF           Relative energies of alkane conformers                                                   15     1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content='83 AMINO20x4   Relative energies of amino acid conformers                                            80     2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content='44 PCONF21        Relative energies of tri- and tetrapeptide conformers                              18     1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content='62 MCONF           Relative energies of melatonin conformers                                              51    4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content='97 SCONF            Relative energies of sugar conformers                                                     17    4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content='60 UPU23             Relative energies between RNA-backbone conformers                           23    5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content='72 BUT14DIOL   Relative energies of butane-1,4-diol conformers                                      64    2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content='80  a)  Number of reactions in subset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' b)  Averaged absolute reaction energy (kcal/mol).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content='  Table S2:  GMTKN55 database references.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content='  ACONF1 Relative energies of alkane conformers ADIM62 Interaction energies of n-alkane dimers AHB213 Interaction energies in anion–neutral dimers AL2X64 Dimerization energies of AlX3 compounds ALK84 Dissociation and other reactions of alkaline compounds ALKBDE105 Dissociation energies in group-1 and -2 diatomics AMINO20x46 Relative energies in amino acid conformers BH76RC7 Reaction energies of the BH767-9 set BH767-9 Barrier heights of hydrogen transfer,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' heavy atom transfer,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' nucleophilic substitution,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' unimolecular and association reactions BHDIV104 Diverse reaction barrier heights BHPERI4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content='10-12 Barrier heights of pericyclic reactions BHROT274 Barrier heights for rotation around single bonds BSR3613,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content='14 Bond-separation reactions of saturated hydrocarbons BUT14DIOL15 Relative energies in butane-1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content='4-diol conformers C60ISO16 Relative energies between C60 isomers CARBHB124 Hydrogen-bonded complexes between carbene analogues and H2O,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' NH3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' or HCl CDIE2017 Double-bond isomerization energies in cyclic systems CHB63 Interaction energies in cation–neutral dimers DARC7,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content='18 Reaction energies of Diels–Alder reactions DC1319,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content='7,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content='20–29 13 difficult cases for DFT methods DIPCS104 Double-ionization potentials of closed-shell systems FH5130,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content='31 Reaction energies in various (in-)organic systems G21EA7,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content='32 Adiabatic electron affinities G21IP7,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content='32 Adiabatic ionization potentials G2RC1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content='33 Reaction energies of selected G2/97 systems  HAL5934,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content='35 Binding energies in halogenated dimers (incl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' halogen bonds) HEAVY289 Noncovalent interaction energies between heavy element hydrides HEAVYSB114 Dissociation energies in heavy-element compounds ICONF4 Relative energies in conformers of inorganic systems IDISP7,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content='36–39 Intramolecular dispersion interactions IL163 Interaction energies in anion–cation dimers INV2440 Inversion/racemization barrier heights ISO3436 Isomerization energies of small and medium-sized organic molecules ISOL2441 Isomerization energies of large organic molecules MB16-434 Decomposition energies of artificial molecules MCONF42 Relative energies in melatonin conformers NBPRC7,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content='38,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content='43 Oligomerizations and H2 fragmentations of NH3/BH3 systems;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' H2 activation reactions with PH3/BH3 systems PA264 Adiabatic proton affinities (incl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content='  of amino acids) PArel4 Relative energies in protonated isomers PCONF214 Relative energies in tri- and tetrapeptide conformers PNICO2344 Interaction energies in pnicogen-containing dimers PX1345 Proton-exchange barriers in H2O,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' NH3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' and HF clusters RC214 Fragmentations and rearrangements in radical cations RG184 Interaction energies in rare-gas complexes RSE4346 Radical-stabilization energies S2247 Binding energies of noncovalently bound dimers S6648 Binding energies of noncovalently bound dimers SCONF7,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content='49 Relative energies of sugar conformers SIE4x450 Self-interaction-error related problems TAUT154 Relative energies in tautomers UPU2351 Relative energies between RNA-backbone conformers W4-1152 Total atomization energies WATER2753 Binding energies in (H2O)n,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' H+(H2O)n and OH−(H2O)n  WCPT1854 Proton-transfer barriers in uncatalyzed and water-catalyzed reactions YBDE1855 Bond-dissociation energies in ylides  1)  D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' Gruzman, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' Karton, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' Martin, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' Chem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' A 113, 11974 (2009).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content='  2)  S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' Grimme, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' Antony, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' Ehrlich, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' Krieg, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' Chem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' 132, 154104 (2010).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content='  3)  K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' Lao, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' Schäffer, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' Jansen, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' Herbert, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' Chem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' Theory Comput.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' 11, 2473 (2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content='  4)  L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' Goerigk, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' Hansen, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' Bauer, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' Ehrlich, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' Najibi, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' Grimme, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' Chem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' Chem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' 19, 32184 (2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content='  5)  H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' Yu, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' Truhlar, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' Chem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' Theory Comput.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' 11, 2968 (2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content='  6)  M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' Kesharwani, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
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+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
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+page_content=' Goerigk, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
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+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' Lynch, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
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+page_content=' Chem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
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+page_content=' Zhao, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' González-García, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
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+page_content=' A 109, 2012 (2005).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
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+page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' Khuong, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' Leach, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' Lee, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' Bartberger, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' Houk, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
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+page_content=' A 107, 11445 (2003).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
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+page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' Ess, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' Houk, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
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+page_content=' A 109, 9542 (2005).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
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+page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
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+page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' Sastry, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
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+page_content=' A 106, 1627 (2002).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
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+page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content='Steinmann, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
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+page_content=' Chem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
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+page_content=' 5, 2950 (2009).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content='  14)  H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
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+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
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+page_content=' Cohen, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
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+page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
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+page_content=' Grimme, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' Mück-Lichtenfeld, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content='-U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' Würthwein, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' Ehlers, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' Goumans, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' Lammertsma, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
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+page_content=' Chem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' A 110, 2583 (2006).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
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+page_content=' Chem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' 25, 83 (2004).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content='  23)  H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' Woodcock, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' Schaefer, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' Schreiner, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' Chem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' A 106, 11923 (2002).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content='  24)  P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' Schreiner, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' Fokin, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' Pascal, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' de Meijere, Org.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' 8, 3635 (2006).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content='  25)  C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' Lepetit, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' Chermette, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' Gicquel, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' Heully, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' Chauvin, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' Chem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' A 111, 136 (2007).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content='  26)  J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' Lee, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' Chem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' A 109, 11927 (2005).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content='  27)  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' Karton, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' Martin, Mol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' 110, 2477 (2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content='  28)  Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' Zhao, O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' Tishchenko, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' Gour, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' Li, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' Lutz, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
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+page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' Truhlar, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' Chem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
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+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
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+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' Chem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' A 120, 153 (2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
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+page_content=' Hänchen, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' Chem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' Theory Comput.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
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+page_content=' Chem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' Theory Comput.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
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+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
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+page_content=' Curtiss, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' Raghavachari, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
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+page_content=' Pople, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' Chem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
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+page_content=' Chem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
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+page_content=' 9, 1918 (2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content='  35)  J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' Řezáč, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' Riley, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' Hobza, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' Chem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' Theory Comput.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' 8, 4285 (2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content='  36)  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' Schwabe, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' Grimme, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' Chem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' Chem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' 9, 3397 (2007).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content='  37)  S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' Grimme, Angew.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' Chemie Int.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' Ed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' 45, 4460 (2006).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content='  38)  L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' Goerigk, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' Grimme, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' Chem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' Theory Comput.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' 7, 291 (2011).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content='  39)  S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' Grimme, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' Steinmetz, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' Korth, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' Org.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' Chem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' 72, 2118 (2007).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content='  40)  L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' Goerigk, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' Sharma, Can.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' Chem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' 94, 1133 (2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content='  41)  R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' Huenerbein, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' Schirmer, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' Moellmann, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' Grimme, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' Chem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' Chem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' 12, 6940 (2010).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content='  42)  U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' Fogueri, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' Kozuch, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' Karton, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' Martin, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' Chem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' A 117, 2269 (2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content='  43)  S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' Grimme, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' Kruse, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' Goerigk, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' Erker, Angew.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' Chemie Int.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' Ed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' 49, 1402 (2010).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content='  44)  D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' Setiawan, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' Kraka, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' Cremer, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' Chem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' A 119, 1642 (2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content='  45)  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' Karton, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' O’Reilly, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' Chan, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' Radom, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' Chem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' Theory Comput.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' 8, 3128 (2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content='  46)  F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' Neese, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' Schwabe, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' Kossmann, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' Schirmer, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' Grimme, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' Chem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' Theory Comput.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' 5, 3060 (2009).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content='  47)  P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' Jurečka, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' Šponer, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' Černy, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' Hobza, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' Chem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' Chem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' 8, 1985 (2006).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content='  48)  J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' Rezáč, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' Riley, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' Hobza, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' Chem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' Theory Comput.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' 7, 2427 (2011).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content='  49)  G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' Csonka, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' French, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
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+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
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+page_content=' Chem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' Theory Comput.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' 5, 679 (2009).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
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+page_content=' Karton, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
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+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' O’Reilly, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
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+page_content=' A 116, 4211 (2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
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+page_content=' Ng, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
+page_content=' Peverati, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9AzT4oBgHgl3EQfPvsd/content/2301.01187v1.pdf'}
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+First-principles thermal equation of state of fcc iridium
+Kai Luo∗ and Ruifeng Lu
+Institute of Ultrafast Optical Physics, Department of Applied Physics,
+Nanjing University of Science and Technology, Nanjing 210094, China
+R. E. Cohen
+Extreme Materials Initiative, Earth and Planets Laboratory, Carnegie Institution for Science,
+5241 Broad Branch Rd., N.W., Washington, DC 20015, USA
+(Dated: Modified on Jan 5, 2023)
+The thermal equation of states for fcc iridium (Ir) is obtained from first-principles molecular
+dynamics up to 3000 K and 540 GPa. The equation of state (EoS) is globally fitted to a simpli-
+fied free energy model and various parameters are derived. The theoretical principal Hugoniot is
+compared with shockwave experiments, where discrepancy suggests formation of new Ir phases. A
+few representative EoS parameters, such as bulk modulus KT , thermal expansivity α, Gr¨uneisen
+parameter γ, and constant pressure capacity CP , Debye temperature, ΘD are computed to compare
+with experimental data.
+I.
+INTRODUCTION
+Iridium (Ir) is a 5d transition metal of the platinum
+group. It is the second-densest metal with a density of
+22.56 g/cm3 at ambient condition, only slightly lower by
+about 0.12% than the densest metal osmium (Os). It has
+the largest shear modulus, G=210 GPa among the face-
+centered cubic (fcc) metals. The solid Ir remains in the
+fcc structure up to the melting point of 2719 K[1]. Due to
+its prominent thermophysical and mechanical properties
+and high corrosion resistance, it is used in many tech-
+nological applications, such as crucibles, thermocouples,
+spark plugs, aircraft engine parts, and deep water pipes.
+The lack of phase transitions, simple fcc structure, high
+melting temperature, and non-reactivity, makes it ideal
+for experiments as a heater, absorber, or standard for
+example in diamond-anvil cell (DAC) experiments, and
+ideal for studying effects of compression on noble metals.
+Our understanding of the properties of Ir is still limited,
+and fundamental research on it remains of great interest.
+With the advances in lab technologies, extreme con-
+ditions (P > 200 GPa, T > 2000 K) become more and
+more amenable to study. Fundamental to all studies at
+extreme conditions is the equation of state (EoS) that
+relates P, V, T, and U or F, where symbols of P, V, T, U,
+and F stand for pressure, volume, temperature, internal
+energy, and Helmholtz free energy. The earliest investiga-
+tion of iridium EoS dates back to 1937 by P.W. Bridgman
+up to 7 GPa[2, 3], followed by work of Schock and John-
+son [4] and then of Akella[5] up to 30 GPa. Cerenius and
+Dubrovinsky [6] measured the compressibility of Ir using
+DAC up to 65 GPa. Later, Cynn et al. found that Ir
+has the second-lowest compressibility of any element af-
+ter Os from their DAC experiment up to 65 GPa, which
+was corroborated by first-principles calculations[7].
+For
+the
+EoS
+diagrams,
+zero-temperature
+first-
+∗ kluo@njust.edu.cn
+principles
+EoS
+can
+be
+supplemented
+with
+finite-
+temperature vibrational entropies from the phonon dis-
+persions. Phonon frequencies can be calculated from fi-
+nite differences, or with the density-functional perturba-
+tion theory (DFPT) [8]. Thanks to the development in
+the density functional theory toolkit, theoretical EoS for
+Ir appeared in several experimental work [7, 9–12]. How-
+ever, these theoretical EoS’s were limited to low tem-
+peratures (around 300 K) using static calculations fitted
+to Birch-Murnaghan (BM) EoS [13]. Anharmonic lattice
+vibrations were considered in Ref.[9], but the focus was
+the phase diagram and phase stability. Anzellini et al.
+studied Ir up to 80 GPa and 3100K combining in situ
+synchrotron X-ray diffraction using laser-heating DACs
+and density functional theory calculations [14]. A com-
+prehensive study covering a larger range of temperatures
+and pressures has not been performed. Indeed, studying
+other phases would be interesting, but in applications as
+a standard in experiments, we focus on the fcc phase.
+In this work, we aim to provide the EoS for fcc Ir up to
+3000 K and 540 GPa in first-principles molecular dynam-
+ics (FPMD).
+II.
+THEORETICAL EOS FROM FPMD
+First principles methods have been widely adopted in
+the simulation of condensed phases where no phenomeno-
+logical parameters are needed.
+They give access to a
+space of thermodynamic conditions, which are hard to
+reach for experimental efforts and can be used to help cal-
+ibrate experiments, where, for example temperature data
+are not sometimes available at the desired conditions.
+FPMD takes into account of anharmonic vibrations of
+ions directly at finite temperatures through thermostat-
+ting. The electronic free energy is given by the Mermin-
+Kohn-Sham density functional theory (DFT) [15, 16].
+FPMD becomes the most used tool for predicting the
+thermal EoS, subject to the exchange-correlation free en-
+ergy functional approximations[17–19]. Classical molec-
+arXiv:2301.04825v1  [cond-mat.mtrl-sci]  12 Jan 2023
+
+2
+ 0
+ 100
+ 200
+ 300
+ 400
+ 500
+ 60
+ 65
+ 70
+ 75
+ 80
+ 85
+ 90
+ 95
+ 100
+P (GPa)
+V (bohr3)
+elk
+QE
+ 0
+ 2
+ 4
+ 6
+ 8
+ 10
+ 60 65 70 75 80 85 90 95 100
+∆P (GPa)
+V (bohr3)
+PQE-Pelk
+FIG. 1.
+Static equations of state from GBRV pseudopoten-
+tial planewaves (QE) and LAPW (elk) calculation are com-
+pared.
+The Vinet EoS was used to fit the energy-volume
+curve. Inset figure shows the pressure difference and the max-
+imum is less than 10 GPa.
+ular dynamics is suitable for high temperatures above the
+Debye temperature as it includes anharmonicity exactly,
+unlike other approaches.
+At lower temperature devia-
+tions from the P −V −T equation of state are small, but
+heat capacities and high order properties such as thermal
+expansivity which show strong quantum effects at tem-
+perature below the Debye temperature are indeed less
+accurate.
+A.
+FPMD details
+First, we computed the static EoS of Ir at zero temper-
+ature. We used Quantum Espresso ver. 6.7 throughout
+this work [20]. We used the scalar-relativisitic (Garrity-
+Bennett-Rabe-Vanderbilt) GBRV ultrasoft pseudopoten-
+tial [21] with Perdew-Burke-Erzernhorf (PBE) exchange-
+correlation (xc) functional [22, 23]. The electronic con-
+figuration for the pseudopotential is [Xe]5p6.05d8.5. EoS
+was derived by fitting energy-volume curve in the 3rd-
+order BM equation. To validate the range of applicabil-
+ity of the pseudopotential, we performed similar calcula-
+tions in linearized augmented planewave (LAPW) code
+Elk [24] and above two P −V curves agree well up to 550
+GPa (see Fig.1).
+For the FPMD calculation, we prepared a cubic box
+containing 108 atoms in the fcc structure.
+The en-
+ergy cutoffs for planewaves and density are 80 Ry and
+320 Ry, respectively.
+Energy is converged within 5
+meV per atom.
+Only Γ point was sampled.
+The
+bands are occupied according to the Fermi-Dirac dis-
+tribution at each temperature,
+and the number of
+bands are large enough to guarantee the occupation
+number is smaller than 10−7 for the highest occupied
+state.
+Early studies showed that the spin-orbit cou-
+pling does not affect the EoS and hence we used spin-
+unpolarized DFT neglecting spin-orbit coupling. Then
+conditions at a combination of lattice constants a/a0 =
+0.86, 0.88, 0.90, 0.92, 0.96, 1.00 (a0 = 3.801 ˚A) and tem-
+peratures T = 300, 1000, 1500, 2000, 2500, 3000 K were
+used in the simulations (see conditions in Table I). The
+time step is 20 a.u. (0.9676 fs). The equilibrated time
+steps are more than 2000 to get the statistical means
+and standard deviations, which give less than 1% stan-
+dard deviation. The ionic temperature is regulated by
+the stochastic-velocity rescaling thermostat [25] and no
+quantum corrections to the ionic motion are included.
+B.
+Free energy model
+We fit the Helmholtz free energy (F) as a function of V
+and T, F(V, T). In FPMD, we have direct access to the
+variables of volume (V ), temperature (T), pressure (P),
+and internal energy (U). Cohen and G¨ulseren [26] stud-
+ied the thermal EoS of tantalum (Ta) in full potential
+LAPW and mixed-basis pseudopotential methods.
+An
+accurate high-temperature global EoS was formed from
+the T = 0 K Vinet isotherm and the thermal free-energy
+was fitted by the polynomial expansion in V and T (see
+Eq. (11) in Ref. [26]). de Koker and Stixrude [27] com-
+puted the free energy of MgO periclase and MgSiO3 per-
+ovskite using FPMD, where the excess free energy was
+fitted in a similar expansion. Incorporating the Debye
+model [28], the total free energy is approximated by the
+polynomial expansion up to order Ni, Nj,
+F(V, T) =
+Ni,Nj
+�
+i,j=0
+AijT i(V − 2
+3 )j + F0 .
+(1)
+Neglecting
+the
+zero-point
+motion,
+F0
+=
+kBT [−D3(x) + 3 ln(1 − e−x)]
+where
+a
+dimensionless
+parameter x =
+ΘD
+T
+with Debye temperature ΘD.
+kB
+is the Boltzmann constant.
+D3(x) is the third order
+Debye function (see Appendix B). Aij are fitting co-
+efficients yet to be determined.
+For comparison, we
+mention the classical model, where F0 = −3kBT ln T,
+with T ln T giving the proper classical behavior at low
+temperatures. That is CV = 3kB and S = −∞ at 0 K.
+The Debye temperature ΘD cannot be determined from
+the U(T, V ), P(T, V ) data from the classical molecular
+dynamics, so we obtain ΘD from the RMS displacements
+(see below).
+For simplicity, the Debye temperature at
+P = 0 GPa, T=300 K, is used.
+III.
+RESULTS
+We obtained the equilibrated quantities from FPMD,
+where U, P includes the ionic kinetic energy and ideal
+gas pressure, respectively. We subtracted each internal
+energy by the global minimum, as only the energy dif-
+ference matters. The pressure and internal energy are
+P = −
+� ∂F
+∂V
+�
+T , U = F + TS = F − T
+� ∂F
+∂T
+�
+V .
+(U, P)
+
+3
+TABLE I. Pressure P and internal energy per atom U and its standard deviation of the means Perr and Uerr are extracted from
+FPMD simulations of fcc Ir for a given temperature T and atomic volume V (or the mass density ρ). The global minimum of
+U is underlined.
+T (K)
+V (bohr3)
+ρ (g/cm3)
+P (GPa)
+Perr (GPa)
+U (Ry)
+Uerr (Ry)
+300
+61.01
+35.31
+527.6
+0.01
+-181.353
+0.00004
+300
+63.14
+34.12
+451.6
+0.01
+-181.423
+0.00004
+300
+67.54
+31.89
+326.1
+0.02
+-181.538
+0.00005
+300
+72.14
+29.86
+229.5
+0.01
+-181.624
+0.00004
+300
+81.97
+26.28
+99.3
+0.02
+-181.729
+0.00005
+300
+92.65
+23.25
+25.0
+0.03
+-181.770
+0.00006
+1000
+61.01
+35.31
+531.3
+0.04
+-181.339
+0.00012
+1000
+63.14
+34.12
+455.3
+0.06
+-181.410
+0.00021
+1000
+67.54
+31.89
+330.0
+0.04
+-181.524
+0.00014
+1000
+72.14
+29.86
+233.4
+0.06
+-181.611
+0.00017
+1000
+81.97
+26.28
+103.5
+0.06
+-181.715
+0.00016
+1000
+92.65
+23.25
+29.5
+0.07
+-181.756
+0.00018
+1500
+61.01
+35.31
+533.9
+0.08
+-181.329
+0.00039
+1500
+63.14
+34.12
+458.0
+0.06
+-181.400
+0.00022
+1500
+67.54
+31.89
+332.8
+0.07
+-181.514
+0.00021
+1500
+72.14
+29.86
+236.6
+0.09
+-181.600
+0.00033
+1500
+81.97
+26.28
+106.4
+0.09
+-181.705
+0.00027
+1500
+92.65
+23.25
+32.6
+0.13
+-181.746
+0.00034
+2000
+61.01
+35.31
+536.9
+0.10
+-181.318
+0.00037
+2000
+63.14
+34.12
+461.0
+0.08
+-181.389
+0.00027
+2000
+67.54
+31.89
+335.6
+0.08
+-181.504
+0.00027
+2000
+72.14
+29.86
+239.2
+0.11
+-181.591
+0.00034
+2000
+81.97
+26.28
+109.4
+0.17
+-181.694
+0.00054
+2000
+92.65
+23.25
+35.6
+0.22
+-181.735
+0.00076
+2500
+61.01
+35.31
+539.5
+0.14
+-181.308
+0.00038
+2500
+63.14
+34.12
+463.7
+0.14
+-181.379
+0.00051
+2500
+67.54
+31.89
+338.6
+0.10
+-181.493
+0.00032
+2500
+72.14
+29.86
+242.3
+0.17
+-181.579
+0.00058
+2500
+81.97
+26.28
+112.5
+0.15
+-181.683
+0.00046
+2500
+92.65
+23.25
+38.7
+0.19
+-181.725
+0.00053
+3000
+61.01
+35.31
+542.5
+0.11
+-181.298
+0.00037
+3000
+63.14
+34.12
+466.6
+0.20
+-181.368
+0.00072
+3000
+67.54
+31.89
+341.1
+0.16
+-181.484
+0.00057
+3000
+72.14
+29.86
+245.6
+0.22
+-181.568
+0.00075
+3000
+81.97
+26.28
+115.1
+0.23
+-181.674
+0.00068
+3000
+92.65
+23.25
+41.7
+0.45
+-181.714
+0.00141
+data are grouped as a pair and fitted together to avoid
+bias between these two quantities. The fitting was per-
+formed using the weighted least-square fit with the lm
+function including offset in R language. Internal energy
+U and pressure P were fitted simultaneously to F(V, T).
+w = 1/∆2 is set for the weight, where ∆ is the stan-
+dard deviation of U and P.
+We fitted Eq. (1) with
+Ni = 2, Nj = 3. We analyzed the MD trajectories us-
+ing the code VMD, and computed the root-mean-square
+displacement (RMSD) for each run. From this we can
+obtain the effective Debye temperature ΘD using:
+⟨u2⟩ =
+3h2
+4π2MkBΘD
+�D1(ΘD/T)
+ΘD/T
++ 1
+4
+�
+,
+(2)
+where ⃗u, M, h are the displacement vector, the ion mass,
+the Planck constant, and D1 is the first order Debye func-
+tion. The quantum correction term 1
+4 shall be omitted
+in the classical treatment. Since the phonon density of
+states is not exactly Debye-like, this is the effective De-
+bye temperature for the second moment of the vibrational
+density of states (VDOS) , not exactly equal to the ther-
+modynamic Debye temperature [29].
+The residual is the deviation between the target func-
+tion and the sample mean. From Fig. 2, we observe that
+the residuals are randomly distributed across the volume
+range. The absolute value of residuals for U and P are
+less than 0.002 Ry and 1.0 GPa (except for data point at
+
+4
+60
+65
+70
+75
+80
+85
+90
+−0.002
+−0.001
+0.000
+0.001
+Volume/atom (bohr3)
+Residuals U (Ry)
+300K
+1000K
+1500K
+2000K
+2500K
+3000K
+60
+65
+70
+75
+80
+85
+90
+−1.0
+−0.5
+0.0
+0.5
+1.0
+Volume/atom (bohr3)
+Residuals P (GPa)
+60
+65
+70
+75
+80
+85
+90
+0.0
+0.1
+0.2
+0.3
+0.4
+0.5
+Volume/atom (bohr3)
+∆U (Ry)
+60
+65
+70
+75
+80
+85
+90
+0
+5
+10
+15
+Volume/atom (bohr3)
+P − P300K (GPa)
+FIG. 2.
+The residuals of the fit to Eq. (1) for the EoS of
+fcc iridium are shown in a) and b). ∆U = U − Umin, where
+Umin is the minimum in the dataset underlined in Table I.
+The fitted curves are compared against the dataset in c) and
+d). ∆U and thermal pressure P − P300K from the fit align
+well against the dataset.
+3000 K). For the internal energy, a global minimum Umin
+is subtracted from the dataset. On the scale of half Ry,
+the internal energy is well represented. As for the pres-
+sure, we computed the pressure differences with respect
+to the T = 300 K reference and the fitted curves aligned
+with the dataset. Only the T = 3000 K fit is slightly off.
+The resultant fitting coefficients in atomic unit for both
+the Debye model and the classical model are tabulated
+(see Table II). The statistical summary from lm function
+is included in the Appendix A (see Fig. 12).
+A.
+P − V − T EoS
+The equilibrium atomic volume (P = 0 GPa) at 300
+K is 14.559 ˚A3, 2.9% larger than the experimental value
+14.145 ˚A3. The overestimation of the lattice constants
+is expected for the PBE exchange-correlation functional.
+Experimental P − V curves of 300 K isotherm are read-
+ily compared with our theoretical predictions. Pressures
+measured by Akella et al.
+[5] are underestimated for
+compression (see Fig. 3), ∆V/V0 larger than 0.05 with
+∆V = V0 − V . Overall the theoretical 300 K isotherm
+agrees well with the experiments within the uncertainty
+especially when the compression is smaller than 0.15
+(P < 70 GPa) [6, 10, 30]. In contrast, the 3rd order BM
+fit done by Monteseguro et al. [10] sits along our 1000 K
+isotherm for compression > 0.15, and reflects the inade-
+quecy of BM EoS at high compression. For comparison,
+we have also included the FPMD and experimental study
+of Anzellini et al. [14]. Their P −V −T curves (both the-
+ory and experiments) below 80 GPa are obtained using
+the EoSFit7 package with ingredients such as the third-
+order BM EoS for the isothermal part. Their FPMD used
+the local density approximations and smaller energy cut-
+off (300 eV). Isotherm of 0 K compared well against our
+300 K curve at low compression but not at high com-
+pression (compression > 0.90). Similar for the isotherms
+of 1000 K and 3000 K. The shock-wave experiment by
+Al’tshuler et al. [31–33] exhibits quite distinct behavior
+in the P −V curve. Around compression of 0.1, the tem-
+perature is pinned slightly above the isotherm of 1000
+K and at compression of 0.22 the temperature is close
+to the 3000 K isotherm. The high compression pressure
+(≈ 600 GPa) of Al’tshuler et al. was mistakenly reported
+in Ref. [10]. The recent shockwave experimental work by
+Khishchenko [34] is also compared. We observe the room
+temperature isotherm of recent work by Khishchenko et
+al. align almost perfectly with our EoS data. The data
+by Monteseguro et al. runs along our 1000 K isotherm for
+compression over 0.1. It is well-known that dynamic com-
+pression experiment lead to a temperature rise. Contrary
+to the claim that the temperature effect is negligible by
+Monteseguro et al., [10] we believe the temperatures in-
+crease (not measured) along the shock compression P −V
+curve is significant from our predicted EoS.
+Thermal pressure measures the pressure change upon
+temperature increase at constant volume, Pth(V, T) =
+P(V, T) − P(V, T0). The thermal pressure is quite linear
+in T given that αKT (α and KT are the thermal expan-
+sivity and the bulk modulus) is constant in the classical
+regime (above the Debye temperature), expressed as
+Pth(V, T) =
+� T
+T0
+dT
+�∂P
+∂T
+�
+V
+=
+� T
+T0
+dT αKT .
+(3)
+An oversimplified linear equation (see Fig.4) could be
+given to the thermal pressure Pth(T) = λT, with λ =
+0.0056 GPa/K for the equilibrium volume.
+One could
+also see the volume dependence is weak from the bottom
+right panel of Fig. 2.
+The equilibrium bulk modulus B0 (or inverse com-
+pressibility at room temperature and zero pressure) is
+an important parameter in the EoS formula, such as the
+Vinet EoS [35, 36].
+The fitted bulk modulus is com-
+pared against earlier studies (see Table. III). We note
+that Cerenius and Dubrovinsky [6] obtained similar bulk
+modulus, 354 GPa versus 306 GPa, by fitting the second
+order BM EoS with constraint B′
+0 = 4, or third-order BM
+EoS without constraint both using experimental equilib-
+rium volume. Park et al. obtained the bulk modulus of
+399 GPa and 344 GPa for the LDA and GGA functional
+in DFT, respectively [37].
+We note that B0 from our
+fit is close to the accepted value of about 365 GPa and
+evidently smaller than Cynn’s value 383 GPa [7]. The
+parameter B′
+0 from our model is 5.3.
+
+5
+TABLE II. Coefficient matrix A in atomic units for the choice of F0, the Debye model and the classical model. The root mean
+squared errors (RMS) in the fitting for the pressure P (in GPa) and the internal energy U (in mRy) are listed.
+Aij
+P RMS (GPa)
+U RMS (mRy)
+Debye model
+1.904
+-62.84
+88.81
+8175
+0
+0.008698
+-0.1175
+0.5574
+6.681 × 10−09 −3.752 × 10−07 6.321 × 10−06 −3.439 × 10−05
+0.5380
+0.706
+Classical model
+1.902
+-62.83
+88.68
+8176
+0
+0.008676
+-0.1171
+0.5549
+6.567 × 10−09 −3.667 × 10−07 6.166 × 10−06 −3.347 × 10−05
+0.5378
+0.661
+TABLE III. Experimental equilibrium volume V0 (˚A3 per atom), bulk modulus B0 (GPa), and B′
+0 at room temperature are
+compared against reported theoretical results. Data and method are briefly summarized, and the original references are given.
+Method description
+V0
+B0
+B′
+0
+References
+(˚A3/at)
+(GPa)
+Exp. data fitted to 3rd-order BM EoS
+14.120
+339
+5.3
+Monteseguro et al. [10]
+Exp. data fitted to 3rd-order BM EoS
+14.145
+383
+3.1
+Cynn et al. [7]
+Exp. data fitted to
+Cerenius and Dubrovinsky[6]
+2nd-order BM EoS, with B′
+0 = 4
+14.173 (exp. value)
+354
+4.0
+3rd-order BM EoS, without constraint
+14.173 (exp. value)
+306
+6.8
+DFT data fitted to BM EoS
+Park et al. [37], Table 1 and 2
+PAW LDA
+13.925
+399
+PAW GGA
+14.524
+344
+FPMD data fitted to 3rd-order BM EoS
+14.150
+366
+5.0
+Burakovsky et al. [9]
+FPMD data fitted to our EoS
+14.559
+361
+5.3
+This work
+B.
+Shock compression
+High pressure high temperature conditions are gener-
+ated by laser heating [38] or resistive heating [39] in a
+DAC or by laser or gas gun [40] driven shock compres-
+sion. Strong shocks obey the Rankine-Hugoniot,
+U − U0 + 1
+2(P + P0)(V − V0) = 0 .
+(4)
+Since the analytical expression for U, P as a function of
+V, T is known, for each volume V , we solve Eq. (4) by
+searching its root T given the experimental value V0, T0.
+We compared our predicted principle Hugoniot with
+available shock experimental data from several facilities
+[41, 43] (Fig.
+5).
+Our theoretical principle Hugoniot
+agrees well with that from earlier data of Al’tshuler and
+LANL March, as well as more recent data of STAR Hugo-
+niot, for P < 200 GPa. Above 200 GPa, our predicted
+pressure is higher than that of LANL and STAR but
+lower than Al’tshuler’s. Our theoretical Hugoniot below
+3000 K (shock temperature) are fairly reliable which cor-
+respond to pressure less than 200 GPa. The shock tem-
+perature, calculated as the solution to Eq. (4), is shown.
+C.
+Equation of state parameters
+Thermal EoS parameters such as thermal expansivity
+α, isothermal compressibility βT , Gr¨uneisen parameter
+γ, and the heat capacity CV and CP are obtained by
+differentiation and algebraic manipulation of Eq. (1). We
+now discuss some of these parameters. As expected from
+the thermal pressure, αKT is weakly dependent on the
+volume and temperature (see Fig. 6). The Gr¨uneisen
+parameter
+γ = V
+�∂P
+∂U
+�
+V
+= V αKT
+CV
+(5)
+is another important parameter. It is used in the Mie-
+Gr¨uneisen EoS, where γ is assumed independent of tem-
+perature. The span of γ as a function of temperature
+reduces when the pressure increases (see Fig. 7). For
+high compressions, it is indeed fairly temperature inde-
+pendent.
+The
+volumetric
+thermal
+expansivity
+α
+=
+−1/V (∂V/∂T)P for isotropic materials is three times the
+linear thermal expansivity coefficient αL, α = 3αL. α in
+Fig. 8 is essentially temperature independent but rather
+volume sensitive. Our theoretical prediction is below the
+reported experimental value [44], but it is noted that
+around room temperature, our theory prediction gives
+the right thermal expansion coefficient. The expansivity
+has downsized by a factor of 4 when the pressure goes
+to 300 GPa.
+The heat capacity at high pressures are
+almost linear above 500 K, see Fig. 9. The higher the
+temperature, the slope of CP is smaller. At 0 GPa, the
+predicted value 25.68 J K−1 mol−1is fairly accurate and
+only 2.3% larger than the experimental heat capacity
+25.10 J K−1 mol−1[45], given that we used the formula,
+CP = CV (1 + Tαγ) (see Appendix A) with errors in
+
+6
+0.75
+0.80
+0.85
+0.90
+0.95
+1.00
+V/V0
+0
+25
+50
+75
+100
+125
+150
+175
+200
+Pressure (GPa)
+This work from FPMD
+300 K
+1000 K
+2000 K
+3000 K
+T=0K DFT
+T=0K Exp.
+T=1000K DFT
+T=1000K Exp.
+T=3000K DFT
+T=3000K Exp.
+Khishchenko
+Yusenko
+Monteseguro
+Cerenius
+Akella
+Al'tshuler
+FIG. 3.
+Theoretical and experimental EoS’s of fcc iridium
+are compared. The DAC experimental data for Yusenko [30]
+Monteseguro[10], Cerenius[6], and Akella[5] were compared
+at 300 K, where the BM EoS’s were available. Experimental
+and theoretical data from Anzellini [14] are included (pink,
+green, and yellow lines).
+The shockwave data (red cross)
+of Al’tshuler were taken from Ref. [10]. Room temperature
+isotherm of Khishchenko shock experiments [34] agrees very
+well against our FPMD results.
+0
+500
+1000
+1500
+2000
+2500
+3000
+3500
+4000
+Temperature (K)
+0
+5
+10
+15
+20
+25
+Thermal Pressure (GPa)
+volume/bohr3
+70
+80
+90
+100
+110
+FIG. 4.
+The thermal pressure of fcc iridium is roughly linear
+in T.
+CV , α, and γ.
+The RMSD is a critical quantity for the analysis
+of the phonon vibrations.
+Moseley et al.
+[46] pre-
+sented temperature-dependent inelastic neutron scatter-
+ing (INS) experiments as well as quasi-harmonic density
+functional theory calculations to study the thermody-
+namic properties of Ir.
+Our FPMD RMSD at 300 K
+20.0
+22.5
+25.0
+27.5
+30.0
+32.5
+35.0
+37.5
+40.0
+Density (g/cm3)
+0
+100
+200
+300
+400
+500
+600
+700
+800
+Pressure (GPa)
+This work, FPMD
+STAR Hugoniot
+LANL Hugoniot (Marsh)
+Al'tshuler
+0
+1
+2
+3
+4
+5
+Temperature (kK)
+Shock temperature
+FIG. 5.
+The principle Hugoniot curve from our theoretical
+EoS (solid black) of fcc Ir is compared against the shockwave
+experimental data (symbols). Round green is for the STAR
+Hugoniot [41], square blue for the LANL Hugoniot [42], and
+diamond orange for Al’tshuler [31, 32], respectively.
+Com-
+puted temperature from the Eq. (4) is shown in red.
+The
+dashed extension is beyond the simulation domain.
+0
+500
+1000
+1500
+2000
+2500
+3000
+3500
+4000
+T (K)
+4.5
+5.0
+5.5
+6.0
+6.5
+7.0
+αKT (10−3GPa/K)
+0 GPa
+50 GPa
+100 GPa
+200 GPa
+300 GPa
+FIG. 6.
+Parameter αKT as a function of temperature for
+various pressures.
+0
+500
+1000
+1500
+2000
+2500
+3000
+3500
+4000
+Temperature (K)
+1.0
+1.2
+1.4
+1.6
+1.8
+2.0
+2.2
+2.4
+Gr ̈uneisen ̈arameter
+0 G̈a
+50 G̈a
+100 G̈a
+200 G̈a
+300 G̈a
+FIG. 7.
+Gr¨uneisen parameter γ as a function of temperature
+for various pressures.
+
+7
+0
+50
+100
+150
+200
+250
+300
+Pressure (GPa)
+0.50
+0.75
+1.00
+1.25
+1.50
+1.75
+2.00
+α (10−5K−1)
+300 K
+1000 K
+2000 K
+3000 K
+0
+1000
+2000
+3000
+Temperature (K)
+0
+1
+2
+3
+4
+5
+α (10−5K−1)
+0 GPa
+50 GPa
+100 GPa
+200 GPa
+300 GPa
+Exp.
+FIG. 8.
+The volumetric thermal expansivity α of fcc iridium
+as a function of pressure at various temperatures.
+Halvor-
+son and Wimber measured the linear thermal expansion as
+αt = a0 + a1t + a2t2 + a3t3 with a0 = 6.167 × 10−6, a1 =
+3.038×10−9, a2 = −0.8448×10−12, a3 = 0.5852×10−15, for t
+expressed in ◦C [44] at ambient pressure (see inset solid line),
+where one can show α = 3(L0/Lt)αt for isotropic materials
+with reference length L0.
+agrees particularly well with their experimental findings.
+Their reported ⟨u2⟩ at higher temperatures (T=673 K
+and 823 K, see Table I of Ref. [46]) however, are higher
+than our FPMD predictions.
+This is reasonable since
+our NVT ensembles at these temperatures lead to higher
+pressure and confined vibrations.
+It is worth to note
+that their phonon density of states (PDOS) integrates
+to 1, and is not fitted well at higher energies. Further
+investigation of PDOS with a FPMD simulation to com-
+pare against the experiments may give insight to the an-
+harmonic effects. Debye temperatures of isochores using
+Eq. (2) exhibit weak temperature dependence(see Fig.
+11).
+IV.
+CONCLUSIONS
+We have performed a series of FPMD simulations for
+the fcc Ir at conditions up to 3000 K and 540 GPa. By
+using a simplified model for the free-energy as a func-
+tion of temperature and volume and the statistical av-
+erage quantities internal energy and pressure (U, P), the
+thermal EoS is obtained by globally fitting to the model.
+We have compared previous experimental EoS’s and pro-
+vided the thermal EoS up to 3000 K, and 540 GPa. The
+P − V − T curve is reasonably agreeing with the fitted
+BM EoS at low compression but differs at high com-
+pression. Our first-principles EoS accords with the most
+recent shockwave experiment work by Khishchenko We
+find that αKT and the thermal pressure is quite constant
+from its dependence in temperature and which turns out
+to be true for a wide range of materials. We have shown
+some representative derived thermal parameters against
+available experiments and found agreements and discrep-
+0
+500
+1000
+1500
+2000
+2500
+3000
+Temperature (K)
+0.0
+0.5
+1.0
+1.5
+2.0
+2.5
+3.0
+3.5
+4.0
+CP (kB)
+0 GPa
+50 GPa
+100 GPa
+200 GPa
+300 GPa
+Exp. value at 0 GPa
+Classical model at 0 GPa
+0
+50
+100
+150
+200
+250
+300
+P (GPa)
+2.8
+3.0
+3.2
+3.4
+3.6
+3.8
+4.0
+CP (kB)
+300 K
+1000 K
+2000 K
+3000 K
+0
+500
+1000
+1500
+2000
+2500
+3000
+Temperature (K)
+2.5
+3.0
+3.5
+CV (kB)
+FIG. 9.
+Constant pressure heat capacity CP as a function of
+temperature (top panel) and pressure (bottom panel). Exper-
+imental value at ambient condition is 25.10 J K−1 mol−1[45].
+The classical model (blue dash-dotted) for 0 GPa starts to
+deviate for T below 500 K, and approaches the classical limit
+as T → 0 K. Inset shows the constant volume heat capacity
+CV . Heat capacity largely reduces when pressure goes up.
+0
+500
+1000
+1500
+2000
+2500
+3000
+3500
+4000
+T (K)
+0.05
+0.10
+0.15
+0.20
+0.25
+RMSD (Å)
+a/a0
+1.00
+0.96
+0.92
+0.90
+0.88
+0.87
+Moseley 2020, from DPS
+FIG. 10.
+RMSD as a function of temperature. Experimental
+data was obtained by Moseley et al. [46] for constant pressure.
+ancies. Further work might resolve these discrepancies.
+V.
+ACKNOWLEDGMENTS
+The work was done under the auspices of the US Na-
+tional Science Foundation CSEDI grant EAR-1901813 to
+
+8
+0
+500
+1000
+1500
+2000
+2500
+3000
+3500
+4000
+T (K)
+300
+400
+500
+600
+700
+ΘD (K)
+a/a0
+1.00
+0.96
+0.92
+0.90
+0.88
+0.87
+FIG. 11.
+Debye temperature ΘD as a function of tempera-
+ture. Zero point motion is not included to obtain the Debye
+temperature. In the high temperature, classical region, our
+RMSD are classical from classical FPMD (Fig. 10) but when
+an effective classical Debye temperature is derived to model
+the RMSD, there is a large change with decreasing tempera-
+ture into to quantum regime in the Debye model. A 4th order
+polynomial fit is done only for higher temperature due to the
+classical treatment to the ions.
+R.E.C. and the National Natural Science Foundation of
+China (Grant No.
+12104230) to K.L.; R.E.C. is sup-
+ported by the Carnegie Institution for Science and grate-
+fully acknowledges the Gauss Centre for Supercomputing
+e.V. for funding this project by providing computing time
+on the GCS Supercomputer Supermuc-NG at Leibniz Su-
+percomputing Centre. All the FPMD calculations were
+performed on Supermuc-NG.
+Appendix A: Thermodynamic relations
+Once the Helmholtz free energy F = F(V, T) is known
+as a function of volume (V ) and temperature (T), the fol-
+lowing thermodynamical quantities can be obtained from
+it [47]:
+P = −
+�∂F
+∂V
+�
+T
+,
+(A1)
+S = −
+�∂F
+∂T
+�
+V
+,
+(A2)
+KT = β−1
+T
+= −V
+�∂P
+∂V
+�
+T
+= V
+�∂2F
+∂V 2
+�
+T
+,
+(A3)
+CV = T
+�∂S
+∂T
+�
+V
+= −T
+�∂2F
+∂T 2
+�
+V
+,
+(A4)
+αKT = −
+� ∂2F
+∂T∂V
+�
+,
+(A5)
+γ = V
+�∂P
+∂U
+�
+V
+= V αKT
+CV
+,
+(A6)
+CP
+CV
+= KS
+KT
+= 1 + Tαγ ,
+(A7)
+U = F + TS ,
+(A8)
+where pressure, entropy, isothermal compressibility (its
+inverse is the bulk modulus KT ), constant volume
+molar heat capacity, volumetric expansion coefficient,
+Gr¨uneisen parameter, internal energy are denoted by
+P, S, βT , CV , α, γ, U. CP is the constant pressure molar
+heat capacity. The parameter B′
+0 (or K′
+0 = ∂K
+∂P
+��
+P =0) can
+thus be computed using above relations:
+K′ = ∂K
+∂P
+=
+�∂K
+∂T
+�
+V
+�∂T
+∂P
+�
+V
++
+�∂K
+∂V
+�
+T
+�∂V
+∂P
+�
+T
+=
+�∂K
+∂T
+�
+V
+1
+� ∂P
+∂T
+�
+V
++
+�∂K
+∂V
+�
+T
+1
+� ∂P
+∂V
+�
+T
+=
+�∂K
+∂T
+�
+V
+1
+� ∂P
+∂T
+�
+V
+−
+�∂K
+∂V
+�
+T
+V
+KT
+.
+(A9)
+Appendix B: Debye Model
+The Helmholtz free energy F of a vibrating lattice at
+volume V and temperature T, can be approximated as
+F(V, T) = E(V ) + Fvib(V, T) + Fel(V, T) ,
+(B1)
+where Fvib is the vibrating energy of the lattice and Fel
+is the thermal electronic free energy which is typically
+negligible.
+Moruzzi et al.
+[28] proposed an empirical
+Debye model with
+Fvib = kBT
+�
+−D3(x) + 3 ln(1 − e−x)
+�
++ 9
+8kBΘD , (B2)
+with Debye temperature ΘD and dimensionless param-
+eter x =
+ΘD
+T .
+The last term is the zero-point energy.
+
+9
+0.0
+0.1
+0.2
+0.3
+0.4
+0.5
+−0.002
+0.000
+0.001
+Fitted values
+Residuals
+G
+G
+G
+G
+G
+G
+G
+G
+G
+G
+G
+G
+G
+G
+G
+G
+G
+G
+G
+G
+G
+G
+G
+G
+G
+G
+G
+G
+G
+G
+G
+G
+G
+G
+G
+G
+G
+G
+G
+G
+G
+G
+G
+G
+G
+G
+G
+G
+G
+G
+G
+G
+G
+G
+G
+G
+G
+G
+G
+G
+G
+G
+G
+G
+G
+G
+G
+G
+G
+G
+G
+G
+G
+G
+G
+G
+G
+G
+G
+G
+G
+G
+G
+G
+Residuals vs Fitted
+16
+11
+9
+G
+G G
+GG
+G GG
+G
+G
+G
+GG
+G
+G
+G
+G
+G
+G
+G G
+GG
+G
+G G
+G
+G
+G
+G
+G
+G
+G
+G
+GG
+G
+G
+G
+GG
+G
+G
+G
+G
+G
+G
+G
+G
+G
+G
+G
+G
+G
+G
+G
+G
+G
+G
+G
+G
+G
+G
+G
+G
+G
+G
+G
+G
+G
+G
+G
+G
+G
+G
+G
+G
+G
+G
+G
+G
+G
+G
+G
+−2
+−1
+0
+1
+2
+−4
+−2
+0
+2
+4
+6
+Theoretical Quantiles
+Standardized residuals
+Normal Q−Q
+84
+83
+79
+0.0
+0.1
+0.2
+0.3
+0.4
+0.5
+0.0
+0.5
+1.0
+1.5
+2.0
+Fitted values
+Standardized residuals
+G
+G
+G
+G
+G
+G
+G
+G
+G
+G
+G
+G
+G
+G
+G
+G
+G
+G
+G
+G
+G
+G
+G
+G
+G
+G
+G
+G
+G
+G
+G
+G
+G
+G
+G
+G
+G
+G
+G
+G
+G
+G
+G
+G
+G
+G
+G
+G
+G
+G
+G
+G
+G
+G
+G
+G
+G
+G
+G
+G
+G
+G
+G
+G
+G
+G
+G
+G
+G
+G
+G
+G
+G
+G
+G
+G
+G
+G
+G
+G
+G
+G
+G
+G
+Scale−Location
+84
+83
+79
+0.0
+0.2
+0.4
+0.6
+−6
+−4
+−2
+0
+2
+4
+6
+Leverage
+Standardized residuals
+G
+G G
+G G G G
+G
+G
+G
+G
+G
+G
+G
+G
+G GG
+G
+GG
+GGG
+GGG
+GGG
+GG
+G
+G
+GG
+G
+G
+G
+G
+G
+G
+G
+G
+G
+G
+G
+G
+G
+G
+G
+G
+G
+G
+G
+G
+G
+G
+G
+G
+G
+G
+G
+G
+G
+G
+G
+G
+G
+G
+G
+G
+G
+GG
+G
+G
+G
+G
+G
+G
+G
+G
+G
+Cook's distance
+1
+0.5
+0.5
+1
+Residuals vs Leverage
+18
+1611
+FIG. 12.
+The fit summary plot from the lm function in R.
+D3(x) is the third order Debye function. The nth order
+Debye function is defined as
+Dn(x) =
+� x
+0
+tn
+et − 1dt, x ≥ 0 ,
+(B3)
+where n, a non-negative integer, is the order of the Debye
+function. The vibrational entropy is
+Svib = 4kBD(x) − 3kB ln(1 − e−x) .
+(B4)
+Neglecting the zero-point motion, the vibrational internal
+energy Uvib thus can be obtained
+Uvib = Fvib + TSvib = 3kBTD(x) .
+(B5)
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+page_content='First-principles thermal equation of state of fcc iridium  Kai Luo∗ and Ruifeng Lu  Institute of Ultrafast Optical Physics, Department of Applied Physics,  Nanjing University of Science and Technology, Nanjing 210094, China  R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQf-QvH/content/2301.04825v1.pdf'}
+page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQf-QvH/content/2301.04825v1.pdf'}
+page_content=' Cohen  Extreme Materials Initiative, Earth and Planets Laboratory, Carnegie Institution for Science,  5241 Broad Branch Rd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQf-QvH/content/2301.04825v1.pdf'}
+page_content=', N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQf-QvH/content/2301.04825v1.pdf'}
+page_content='W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQf-QvH/content/2301.04825v1.pdf'}
+page_content=', Washington, DC 20015, USA  (Dated: Modified on Jan 5, 2023)  The thermal equation of states for fcc iridium (Ir) is obtained from first-principles molecular  dynamics up to 3000 K and 540 GPa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQf-QvH/content/2301.04825v1.pdf'}
+page_content=' The equation of state (EoS) is globally fitted to a simpli-  fied free energy model and various parameters are derived.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQf-QvH/content/2301.04825v1.pdf'}
+page_content=' The theoretical principal Hugoniot is  compared with shockwave experiments, where discrepancy suggests formation of new Ir phases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQf-QvH/content/2301.04825v1.pdf'}
+page_content=' A  few representative EoS parameters, such as bulk modulus KT , thermal expansivity α, Gr¨uneisen  parameter γ, and constant pressure capacity CP , Debye temperature, ΘD are computed to compare  with experimental data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQf-QvH/content/2301.04825v1.pdf'}
+page_content='  I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQf-QvH/content/2301.04825v1.pdf'}
+page_content='  INTRODUCTION  Iridium (Ir) is a 5d transition metal of the platinum  group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQf-QvH/content/2301.04825v1.pdf'}
+page_content=' It is the second-densest metal with a density of  22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQf-QvH/content/2301.04825v1.pdf'}
+page_content='56 g/cm3 at ambient condition, only slightly lower by  about 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQf-QvH/content/2301.04825v1.pdf'}
+page_content='12% than the densest metal osmium (Os).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQf-QvH/content/2301.04825v1.pdf'}
+page_content=' It has  the largest shear modulus, G=210 GPa among the face-  centered cubic (fcc) metals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQf-QvH/content/2301.04825v1.pdf'}
+page_content=' The solid Ir remains in the  fcc structure up to the melting point of 2719 K[1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQf-QvH/content/2301.04825v1.pdf'}
+page_content=' Due to  its prominent thermophysical and mechanical properties  and high corrosion resistance, it is used in many tech-  nological applications, such as crucibles, thermocouples,  spark plugs, aircraft engine parts, and deep water pipes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQf-QvH/content/2301.04825v1.pdf'}
+page_content='  The lack of phase transitions, simple fcc structure, high  melting temperature, and non-reactivity, makes it ideal  for experiments as a heater, absorber, or standard for  example in diamond-anvil cell (DAC) experiments, and  ideal for studying effects of compression on noble metals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQf-QvH/content/2301.04825v1.pdf'}
+page_content='  Our understanding of the properties of Ir is still limited,  and fundamental research on it remains of great interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQf-QvH/content/2301.04825v1.pdf'}
+page_content='  With the advances in lab technologies, extreme con-  ditions (P > 200 GPa, T > 2000 K) become more and  more amenable to study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQf-QvH/content/2301.04825v1.pdf'}
+page_content=' Fundamental to all studies at  extreme conditions is the equation of state (EoS) that  relates P, V, T, and U or F, where symbols of P, V, T, U,  and F stand for pressure, volume, temperature, internal  energy, and Helmholtz free energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQf-QvH/content/2301.04825v1.pdf'}
+page_content=' The earliest investiga-  tion of iridium EoS dates back to 1937 by P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQf-QvH/content/2301.04825v1.pdf'}
+page_content='W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQf-QvH/content/2301.04825v1.pdf'}
+page_content=' Bridgman  up to 7 GPa[2, 3], followed by work of Schock and John-  son [4] and then of Akella[5] up to 30 GPa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQf-QvH/content/2301.04825v1.pdf'}
+page_content=' Cerenius and  Dubrovinsky [6] measured the compressibility of Ir using  DAC up to 65 GPa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQf-QvH/content/2301.04825v1.pdf'}
+page_content=' Later, Cynn et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQf-QvH/content/2301.04825v1.pdf'}
+page_content=' found that Ir  has the second-lowest compressibility of any element af-  ter Os from their DAC experiment up to 65 GPa, which  was corroborated by first-principles calculations[7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQf-QvH/content/2301.04825v1.pdf'}
+page_content='  For  the  EoS  diagrams,  zero-temperature  first-  ∗ kluo@njust.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQf-QvH/content/2301.04825v1.pdf'}
+page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQf-QvH/content/2301.04825v1.pdf'}
+page_content='cn  principles  EoS  can  be  supplemented  with  finite-  temperature vibrational entropies from the phonon dis-  persions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQf-QvH/content/2301.04825v1.pdf'}
+page_content=' Phonon frequencies can be calculated from fi-  nite differences, or with the density-functional perturba-  tion theory (DFPT) [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQf-QvH/content/2301.04825v1.pdf'}
+page_content=' Thanks to the development in  the density functional theory toolkit, theoretical EoS for  Ir appeared in several experimental work [7, 9–12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQf-QvH/content/2301.04825v1.pdf'}
+page_content=' How-  ever, these theoretical EoS’s were limited to low tem-  peratures (around 300 K) using static calculations fitted  to Birch-Murnaghan (BM) EoS [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQf-QvH/content/2301.04825v1.pdf'}
+page_content=' Anharmonic lattice  vibrations were considered in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQf-QvH/content/2301.04825v1.pdf'}
+page_content=' [9], but the focus was  the phase diagram and phase stability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQf-QvH/content/2301.04825v1.pdf'}
+page_content=' Anzellini et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQf-QvH/content/2301.04825v1.pdf'}
+page_content='  studied Ir up to 80 GPa and 3100K combining in situ  synchrotron X-ray diffraction using laser-heating DACs  and density functional theory calculations [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQf-QvH/content/2301.04825v1.pdf'}
+page_content=' A com-  prehensive study covering a larger range of temperatures  and pressures has not been performed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQf-QvH/content/2301.04825v1.pdf'}
+page_content=' Indeed, studying  other phases would be interesting, but in applications as  a standard in experiments, we focus on the fcc phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQf-QvH/content/2301.04825v1.pdf'}
+page_content='  In this work, we aim to provide the EoS for fcc Ir up to  3000 K and 540 GPa in first-principles molecular dynam-  ics (FPMD).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQf-QvH/content/2301.04825v1.pdf'}
+page_content='  II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQf-QvH/content/2301.04825v1.pdf'}
+page_content='  THEORETICAL EOS FROM FPMD  First principles methods have been widely adopted in  the simulation of condensed phases where no phenomeno-  logical parameters are needed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQf-QvH/content/2301.04825v1.pdf'}
+page_content='  They give access to a  space of thermodynamic conditions, which are hard to  reach for experimental efforts and can be used to help cal-  ibrate experiments, where, for example temperature data  are not sometimes available at the desired conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQf-QvH/content/2301.04825v1.pdf'}
+page_content='  FPMD takes into account of anharmonic vibrations of  ions directly at finite temperatures through thermostat-  ting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQf-QvH/content/2301.04825v1.pdf'}
+page_content=' The electronic free energy is given by the Mermin-  Kohn-Sham density functional theory (DFT) [15, 16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQf-QvH/content/2301.04825v1.pdf'}
+page_content='  FPMD becomes the most used tool for predicting the  thermal EoS, subject to the exchange-correlation free en-  ergy functional approximations[17–19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQf-QvH/content/2301.04825v1.pdf'}
+page_content=' Classical molec-  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQf-QvH/content/2301.04825v1.pdf'}
+page_content='04825v1  [cond-mat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQf-QvH/content/2301.04825v1.pdf'}
+page_content='mtrl-sci]  12 Jan 2023  2  0  100  200  300  400  500  60  65  70  75  80  85  90  95  100  P (GPa)  V (bohr3)  elk  QE  0  2  4  6  8  10  60 65 70 75 80 85 90 95 100  ∆P (GPa)  V (bohr3)  PQE-Pelk  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQf-QvH/content/2301.04825v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQf-QvH/content/2301.04825v1.pdf'}
+page_content='  Static equations of state from GBRV pseudopoten-  tial planewaves (QE) and LAPW (elk) calculation are com-  pared.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQf-QvH/content/2301.04825v1.pdf'}
+page_content='  The Vinet EoS was used to fit the energy-volume  curve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQf-QvH/content/2301.04825v1.pdf'}
+page_content=' Inset figure shows the pressure difference and the max-  imum is less than 10 GPa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQf-QvH/content/2301.04825v1.pdf'}
+page_content='  ular dynamics is suitable for high temperatures above the  Debye temperature as it includes anharmonicity exactly,  unlike other approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQf-QvH/content/2301.04825v1.pdf'}
+page_content='  At lower temperature devia-  tions from the P −V −T equation of state are small, but  heat capacities and high order properties such as thermal  expansivity which show strong quantum effects at tem-  perature below the Debye temperature are indeed less  accurate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQf-QvH/content/2301.04825v1.pdf'}
+page_content='  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQf-QvH/content/2301.04825v1.pdf'}
+page_content='  FPMD details  First, we computed the static EoS of Ir at zero temper-  ature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQf-QvH/content/2301.04825v1.pdf'}
+page_content=' We used Quantum Espresso ver.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQf-QvH/content/2301.04825v1.pdf'}
+page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQf-QvH/content/2301.04825v1.pdf'}
+page_content='7 throughout  this work [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQf-QvH/content/2301.04825v1.pdf'}
+page_content=' We used the scalar-relativisitic (Garrity-  Bennett-Rabe-Vanderbilt) GBRV ultrasoft pseudopoten-  tial [21] with Perdew-Burke-Erzernhorf (PBE) exchange-  correlation (xc) functional [22, 23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQf-QvH/content/2301.04825v1.pdf'}
+page_content=' The electronic con-  figuration for the pseudopotential is [Xe]5p6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQf-QvH/content/2301.04825v1.pdf'}
+page_content='05d8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQf-QvH/content/2301.04825v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQf-QvH/content/2301.04825v1.pdf'}
+page_content=' EoS  was derived by fitting energy-volume curve in the 3rd-  order BM equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQf-QvH/content/2301.04825v1.pdf'}
+page_content=' To validate the range of applicabil-  ity of the pseudopotential, we performed similar calcula-  tions in linearized augmented planewave (LAPW) code  Elk [24] and above two P −V curves agree well up to 550  GPa (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQf-QvH/content/2301.04825v1.pdf'}
+page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQf-QvH/content/2301.04825v1.pdf'}
+page_content='  For the FPMD calculation, we prepared a cubic box  containing 108 atoms in the fcc structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQf-QvH/content/2301.04825v1.pdf'}
+page_content='  The en-  ergy cutoffs for planewaves and density are 80 Ry and  320 Ry, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQf-QvH/content/2301.04825v1.pdf'}
+page_content='  Energy is converged within 5  meV per atom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQf-QvH/content/2301.04825v1.pdf'}
+page_content='  Only Γ point was sampled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQf-QvH/content/2301.04825v1.pdf'}
+page_content='  The  bands are occupied according to the Fermi-Dirac dis-  tribution at each temperature,  and the number of  bands are large enough to guarantee the occupation  number is smaller than 10−7 for the highest occupied  state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQf-QvH/content/2301.04825v1.pdf'}
+page_content='  Early studies showed that the spin-orbit cou-  pling does not affect the EoS and hence we used spin-  unpolarized DFT neglecting spin-orbit coupling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQf-QvH/content/2301.04825v1.pdf'}
+page_content=' Then  conditions at a combination of lattice constants a/a0 =  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQf-QvH/content/2301.04825v1.pdf'}
+page_content='86, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQf-QvH/content/2301.04825v1.pdf'}
+page_content='88, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQf-QvH/content/2301.04825v1.pdf'}
+page_content='90, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQf-QvH/content/2301.04825v1.pdf'}
+page_content='92, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQf-QvH/content/2301.04825v1.pdf'}
+page_content='96, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQf-QvH/content/2301.04825v1.pdf'}
+page_content='00 (a0 = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQf-QvH/content/2301.04825v1.pdf'}
+page_content='801 ˚A) and tem-  peratures T = 300, 1000, 1500, 2000, 2500, 3000 K were  used in the simulations (see conditions in Table I).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQf-QvH/content/2301.04825v1.pdf'}
+page_content=' The  time step is 20 a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQf-QvH/content/2301.04825v1.pdf'}
+page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQf-QvH/content/2301.04825v1.pdf'}
+page_content=' (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQf-QvH/content/2301.04825v1.pdf'}
+page_content='9676 fs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQf-QvH/content/2301.04825v1.pdf'}
+page_content=' The equilibrated time  steps are more than 2000 to get the statistical means  and standard deviations, which give less than 1% stan-  dard deviation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQf-QvH/content/2301.04825v1.pdf'}
+page_content=' The ionic temperature is regulated by  the stochastic-velocity rescaling thermostat [25] and no  quantum corrections to the ionic motion are included.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQf-QvH/content/2301.04825v1.pdf'}
+page_content='  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQf-QvH/content/2301.04825v1.pdf'}
+page_content='  Free energy model  We fit the Helmholtz free energy (F) as a function of V  and T, F(V, T).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQf-QvH/content/2301.04825v1.pdf'}
+page_content=' In FPMD, we have direct access to the  variables of volume (V ), temperature (T), pressure (P),  and internal energy (U).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQf-QvH/content/2301.04825v1.pdf'}
+page_content=' Cohen and G¨ulseren [26] stud-  ied the thermal EoS of tantalum (Ta) in full potential  LAPW and mixed-basis pseudopotential methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQf-QvH/content/2301.04825v1.pdf'}
+page_content='  An  accurate high-temperature global EoS was formed from  the T = 0 K Vinet isotherm and the thermal free-energy  was fitted by the polynomial expansion in V and T (see  Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQf-QvH/content/2301.04825v1.pdf'}
+page_content=' (11) in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQf-QvH/content/2301.04825v1.pdf'}
+page_content=' [26]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQf-QvH/content/2301.04825v1.pdf'}
+page_content=' de Koker and Stixrude [27] com-  puted the free energy of MgO periclase and MgSiO3 per-  ovskite using FPMD, where the excess free energy was  fitted in a similar expansion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQf-QvH/content/2301.04825v1.pdf'}
+page_content=' Incorporating the Debye  model [28], the total free energy is approximated by the  polynomial expansion up to order Ni, Nj,  F(V, T) =  Ni,Nj  �  i,j=0  AijT i(V − 2  3 )j + F0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQf-QvH/content/2301.04825v1.pdf'}
+page_content='  (1)  Neglecting  the  zero-point  motion,  F0  =  kBT [−D3(x) + 3 ln(1 − e−x)]  where  a  dimensionless  parameter x =  ΘD  T  with Debye temperature ΘD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQf-QvH/content/2301.04825v1.pdf'}
+page_content='  kB  is the Boltzmann constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQf-QvH/content/2301.04825v1.pdf'}
+page_content='  D3(x) is the third order  Debye function (see Appendix B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQf-QvH/content/2301.04825v1.pdf'}
+page_content=' Aij are fitting co-  efficients yet to be determined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQf-QvH/content/2301.04825v1.pdf'}
+page_content='  For comparison, we  mention the classical model, where F0 = −3kBT ln T,  with T ln T giving the proper classical behavior at low  temperatures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQf-QvH/content/2301.04825v1.pdf'}
+page_content=' That is CV = 3kB and S = −∞ at 0 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQf-QvH/content/2301.04825v1.pdf'}
+page_content='  The Debye temperature ΘD cannot be determined from  the U(T, V ), P(T, V ) data from the classical molecular  dynamics, so we obtain ΘD from the RMS displacements  (see below).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQf-QvH/content/2301.04825v1.pdf'}
+page_content='  For simplicity, the Debye temperature at  P = 0 GPa, T=300 K, is used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQf-QvH/content/2301.04825v1.pdf'}
+page_content='  III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQf-QvH/content/2301.04825v1.pdf'}
+page_content='  RESULTS  We obtained the equilibrated quantities from FPMD,  where U, P includes the ionic kinetic energy and ideal  gas pressure, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQf-QvH/content/2301.04825v1.pdf'}
+page_content=' We subtracted each internal  energy by the global minimum, as only the energy dif-  ference matters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQf-QvH/content/2301.04825v1.pdf'}
+page_content=' The pressure and internal energy are  P = −  � ∂F  ∂V  �  T , U = F + TS = F − T  � ∂F  ∂T  �  V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQf-QvH/content/2301.04825v1.pdf'}
+page_content='  (U, P)  3  TABLE I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQf-QvH/content/2301.04825v1.pdf'}
+page_content=' Pressure P and internal energy per atom U and its standard deviation of the means Perr and Uerr are extracted from  FPMD simulations of fcc Ir for a given temperature T and atomic volume V (or the mass density ρ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQf-QvH/content/2301.04825v1.pdf'}
+page_content=' The global minimum of  U is underlined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQf-QvH/content/2301.04825v1.pdf'}
+page_content='  T (K)  V (bohr3)  ρ (g/cm3)  P (GPa)  Perr (GPa)  U (Ry)  Uerr (Ry)  300  61.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQf-QvH/content/2301.04825v1.pdf'}
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+page_content=' The fitting was per-  formed using the weighted least-square fit with the lm  function including offset in R language.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQf-QvH/content/2301.04825v1.pdf'}
+page_content=' Internal energy  U and pressure P were fitted simultaneously to F(V, T).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQf-QvH/content/2301.04825v1.pdf'}
+page_content='  w = 1/∆2 is set for the weight, where ∆ is the stan-  dard deviation of U and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQf-QvH/content/2301.04825v1.pdf'}
+page_content='  We fitted Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQf-QvH/content/2301.04825v1.pdf'}
+page_content=' (1) with  Ni = 2, Nj = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQf-QvH/content/2301.04825v1.pdf'}
+page_content=' We analyzed the MD trajectories us-  ing the code VMD, and computed the root-mean-square  displacement (RMSD) for each run.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQf-QvH/content/2301.04825v1.pdf'}
+page_content=' From this we can  obtain the effective Debye temperature ΘD using:  ⟨u2⟩ =  3h2  4π2MkBΘD  �D1(ΘD/T)  ΘD/T  + 1  4  �  ,  (2)  where ⃗u, M, h are the displacement vector, the ion mass,  the Planck constant, and D1 is the first order Debye func-  tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQf-QvH/content/2301.04825v1.pdf'}
+page_content=' The quantum correction term 1  4 shall be omitted  in the classical treatment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQf-QvH/content/2301.04825v1.pdf'}
+page_content=' Since the phonon density of  states is not exactly Debye-like, this is the effective De-  bye temperature for the second moment of the vibrational  density of states (VDOS) , not exactly equal to the ther-  modynamic Debye temperature [29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQf-QvH/content/2301.04825v1.pdf'}
+page_content='  The residual is the deviation between the target func-  tion and the sample mean.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQf-QvH/content/2301.04825v1.pdf'}
+page_content=' From Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQf-QvH/content/2301.04825v1.pdf'}
+page_content=' 2, we observe that  the residuals are randomly distributed across the volume  range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQf-QvH/content/2301.04825v1.pdf'}
+page_content=' The absolute value of residuals for U and P are  less than 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQf-QvH/content/2301.04825v1.pdf'}
+page_content='002 Ry and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQf-QvH/content/2301.04825v1.pdf'}
+page_content='0 GPa (except for data point at  4  60  65  70  75  80  85  90  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQf-QvH/content/2301.04825v1.pdf'}
+page_content='002  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQf-QvH/content/2301.04825v1.pdf'}
+page_content='001  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQf-QvH/content/2301.04825v1.pdf'}
+page_content='000  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQf-QvH/content/2301.04825v1.pdf'}
+page_content='001  Volume/atom (bohr3)  Residuals U (Ry)  300K  1000K  1500K  2000K  2500K  3000K  60  65  70  75  80  85  90  −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQf-QvH/content/2301.04825v1.pdf'}
+page_content='0  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQf-QvH/content/2301.04825v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQf-QvH/content/2301.04825v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQf-QvH/content/2301.04825v1.pdf'}
+page_content='5  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQf-QvH/content/2301.04825v1.pdf'}
+page_content='0  Volume/atom (bohr3)  Residuals P (GPa)  60  65  70  75  80  85  90  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQf-QvH/content/2301.04825v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQf-QvH/content/2301.04825v1.pdf'}
+page_content='1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQf-QvH/content/2301.04825v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQf-QvH/content/2301.04825v1.pdf'}
+page_content='3  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQf-QvH/content/2301.04825v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQf-QvH/content/2301.04825v1.pdf'}
+page_content='5  Volume/atom (bohr3)  ∆U (Ry)  60  65  70  75  80  85  90  0  5  10  15  Volume/atom (bohr3)  P − P300K (GPa)  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQf-QvH/content/2301.04825v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQf-QvH/content/2301.04825v1.pdf'}
+page_content='  The residuals of the fit to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQf-QvH/content/2301.04825v1.pdf'}
+page_content=' (1) for the EoS of  fcc iridium are shown in a) and b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQf-QvH/content/2301.04825v1.pdf'}
+page_content=' ∆U = U − Umin, where  Umin is the minimum in the dataset underlined in Table I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQf-QvH/content/2301.04825v1.pdf'}
+page_content='  The fitted curves are compared against the dataset in c) and  d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQf-QvH/content/2301.04825v1.pdf'}
+page_content=' ∆U and thermal pressure P − P300K from the fit align  well against the dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQf-QvH/content/2301.04825v1.pdf'}
+page_content='  3000 K).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQf-QvH/content/2301.04825v1.pdf'}
+page_content=' For the internal energy, a global minimum Umin  is subtracted from the dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQf-QvH/content/2301.04825v1.pdf'}
+page_content=' On the scale of half Ry,  the internal energy is well represented.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQf-QvH/content/2301.04825v1.pdf'}
+page_content=' As for the pres-  sure, we computed the pressure differences with respect  to the T = 300 K reference and the fitted curves aligned  with the dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQf-QvH/content/2301.04825v1.pdf'}
+page_content=' Only the T = 3000 K fit is slightly off.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQf-QvH/content/2301.04825v1.pdf'}
+page_content='  The resultant fitting coefficients in atomic unit for both  the Debye model and the classical model are tabulated  (see Table II).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQf-QvH/content/2301.04825v1.pdf'}
+page_content=' The statistical summary from lm function  is included in the Appendix A (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQf-QvH/content/2301.04825v1.pdf'}
+page_content=' 12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQf-QvH/content/2301.04825v1.pdf'}
+page_content='  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQf-QvH/content/2301.04825v1.pdf'}
+page_content='  P − V − T EoS  The equilibrium atomic volume (P = 0 GPa) at 300  K is 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQf-QvH/content/2301.04825v1.pdf'}
+page_content='559 ˚A3, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQf-QvH/content/2301.04825v1.pdf'}
+page_content='9% larger than the experimental value  14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQf-QvH/content/2301.04825v1.pdf'}
+page_content='145 ˚A3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQf-QvH/content/2301.04825v1.pdf'}
+page_content=' The overestimation of the lattice constants  is expected for the PBE exchange-correlation functional.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQf-QvH/content/2301.04825v1.pdf'}
+page_content='  Experimental P − V curves of 300 K isotherm are read-  ily compared with our theoretical predictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQf-QvH/content/2301.04825v1.pdf'}
+page_content=' Pressures  measured by Akella et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQf-QvH/content/2301.04825v1.pdf'}
+page_content='  [5] are underestimated for  compression (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQf-QvH/content/2301.04825v1.pdf'}
+page_content=' 3), ∆V/V0 larger than 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQf-QvH/content/2301.04825v1.pdf'}
+page_content='05 with  ∆V = V0 − V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQf-QvH/content/2301.04825v1.pdf'}
+page_content=' Overall the theoretical 300 K isotherm  agrees well with the experiments within the uncertainty  especially when the compression is smaller than 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQf-QvH/content/2301.04825v1.pdf'}
+page_content='15  (P < 70 GPa) [6, 10, 30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQf-QvH/content/2301.04825v1.pdf'}
+page_content=' In contrast, the 3rd order BM  fit done by Monteseguro et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQf-QvH/content/2301.04825v1.pdf'}
+page_content=' [10] sits along our 1000 K  isotherm for compression > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQf-QvH/content/2301.04825v1.pdf'}
+page_content='15, and reflects the inade-  quecy of BM EoS at high compression.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQf-QvH/content/2301.04825v1.pdf'}
+page_content=' For comparison,  we have also included the FPMD and experimental study  of Anzellini et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQf-QvH/content/2301.04825v1.pdf'}
+page_content=' [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQf-QvH/content/2301.04825v1.pdf'}
+page_content=' Their P −V −T curves (both the-  ory and experiments) below 80 GPa are obtained using  the EoSFit7 package with ingredients such as the third-  order BM EoS for the isothermal part.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQf-QvH/content/2301.04825v1.pdf'}
+page_content=' Their FPMD used  the local density approximations and smaller energy cut-  off (300 eV).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQf-QvH/content/2301.04825v1.pdf'}
+page_content=' Isotherm of 0 K compared well against our  300 K curve at low compression but not at high com-  pression (compression > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQf-QvH/content/2301.04825v1.pdf'}
+page_content='90).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQf-QvH/content/2301.04825v1.pdf'}
+page_content=' Similar for the isotherms  of 1000 K and 3000 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQf-QvH/content/2301.04825v1.pdf'}
+page_content=' The shock-wave experiment by  Al’tshuler et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQf-QvH/content/2301.04825v1.pdf'}
+page_content=' [31–33] exhibits quite distinct behavior  in the P −V curve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQf-QvH/content/2301.04825v1.pdf'}
+page_content=' Around compression of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQf-QvH/content/2301.04825v1.pdf'}
+page_content='1, the tem-  perature is pinned slightly above the isotherm of 1000  K and at compression of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQf-QvH/content/2301.04825v1.pdf'}
+page_content='22 the temperature is close  to the 3000 K isotherm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQf-QvH/content/2301.04825v1.pdf'}
+page_content=' The high compression pressure  (≈ 600 GPa) of Al’tshuler et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQf-QvH/content/2301.04825v1.pdf'}
+page_content=' was mistakenly reported  in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQf-QvH/content/2301.04825v1.pdf'}
+page_content=' [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQf-QvH/content/2301.04825v1.pdf'}
+page_content=' The recent shockwave experimental work by  Khishchenko [34] is also compared.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQf-QvH/content/2301.04825v1.pdf'}
+page_content=' We observe the room  temperature isotherm of recent work by Khishchenko et  al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQf-QvH/content/2301.04825v1.pdf'}
+page_content=' align almost perfectly with our EoS data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQf-QvH/content/2301.04825v1.pdf'}
+page_content=' The data  by Monteseguro et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQf-QvH/content/2301.04825v1.pdf'}
+page_content=' runs along our 1000 K isotherm for  compression over 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQf-QvH/content/2301.04825v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQf-QvH/content/2301.04825v1.pdf'}
+page_content=' It is well-known that dynamic com-  pression experiment lead to a temperature rise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQf-QvH/content/2301.04825v1.pdf'}
+page_content=' Contrary  to the claim that the temperature effect is negligible by  Monteseguro et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQf-QvH/content/2301.04825v1.pdf'}
+page_content=', [10] we believe the temperatures in-  crease (not measured) along the shock compression P −V  curve is significant from our predicted EoS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQf-QvH/content/2301.04825v1.pdf'}
+page_content='  Thermal pressure measures the pressure change upon  temperature increase at constant volume, Pth(V, T) =  P(V, T) − P(V, T0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQf-QvH/content/2301.04825v1.pdf'}
+page_content=' The thermal pressure is quite linear  in T given that αKT (α and KT are the thermal expan-  sivity and the bulk modulus) is constant in the classical  regime (above the Debye temperature), expressed as  Pth(V, T) =  � T  T0  dT  �∂P  ∂T  �  V  =  � T  T0  dT αKT .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQf-QvH/content/2301.04825v1.pdf'}
+page_content='  (3)  An oversimplified linear equation (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQf-QvH/content/2301.04825v1.pdf'}
+page_content='4) could be  given to the thermal pressure Pth(T) = λT, with λ =  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQf-QvH/content/2301.04825v1.pdf'}
+page_content='0056 GPa/K for the equilibrium volume.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQf-QvH/content/2301.04825v1.pdf'}
+page_content='  One could  also see the volume dependence is weak from the bottom  right panel of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQf-QvH/content/2301.04825v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQf-QvH/content/2301.04825v1.pdf'}
+page_content='  The equilibrium bulk modulus B0 (or inverse com-  pressibility at room temperature and zero pressure) is  an important parameter in the EoS formula, such as the  Vinet EoS [35, 36].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQf-QvH/content/2301.04825v1.pdf'}
+page_content='  The fitted bulk modulus is com-  pared against earlier studies (see Table.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQf-QvH/content/2301.04825v1.pdf'}
+page_content=' III).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQf-QvH/content/2301.04825v1.pdf'}
+page_content=' We note  that Cerenius and Dubrovinsky [6] obtained similar bulk  modulus, 354 GPa versus 306 GPa, by fitting the second  order BM EoS with constraint B′  0 = 4, or third-order BM  EoS without constraint both using experimental equilib-  rium volume.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQf-QvH/content/2301.04825v1.pdf'}
+page_content=' Park et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQf-QvH/content/2301.04825v1.pdf'}
+page_content=' obtained the bulk modulus of  399 GPa and 344 GPa for the LDA and GGA functional  in DFT, respectively [37].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQf-QvH/content/2301.04825v1.pdf'}
+page_content='  We note that B0 from our  fit is close to the accepted value of about 365 GPa and  evidently smaller than Cynn’s value 383 GPa [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQf-QvH/content/2301.04825v1.pdf'}
+page_content=' The  parameter B′  0 from our model is 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQf-QvH/content/2301.04825v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQf-QvH/content/2301.04825v1.pdf'}
+page_content='  5  TABLE II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQf-QvH/content/2301.04825v1.pdf'}
+page_content=' Coefficient matrix A in atomic units for the choice of F0, the Debye model and the classical model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQf-QvH/content/2301.04825v1.pdf'}
+page_content=' The root mean  squared errors (RMS) in the fitting for the pressure P (in GPa) and the internal energy U (in mRy) are listed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQf-QvH/content/2301.04825v1.pdf'}
+page_content='  Aij  P RMS (GPa)  U RMS (mRy)  Debye model  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQf-QvH/content/2301.04825v1.pdf'}
+page_content='904  62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQf-QvH/content/2301.04825v1.pdf'}
+page_content='84  88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQf-QvH/content/2301.04825v1.pdf'}
+page_content='81  8175  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQf-QvH/content/2301.04825v1.pdf'}
+page_content='008698  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQf-QvH/content/2301.04825v1.pdf'}
+page_content='1175  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQf-QvH/content/2301.04825v1.pdf'}
+page_content='5574  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQf-QvH/content/2301.04825v1.pdf'}
+page_content='681 × 10−09 −3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQf-QvH/content/2301.04825v1.pdf'}
+page_content='752 × 10−07 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQf-QvH/content/2301.04825v1.pdf'}
+page_content='321 × 10−06 −3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQf-QvH/content/2301.04825v1.pdf'}
+page_content='439 × 10−05  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQf-QvH/content/2301.04825v1.pdf'}
+page_content='5380  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQf-QvH/content/2301.04825v1.pdf'}
+page_content='706  Classical model  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQf-QvH/content/2301.04825v1.pdf'}
+page_content='902  62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQf-QvH/content/2301.04825v1.pdf'}
+page_content='83  88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQf-QvH/content/2301.04825v1.pdf'}
+page_content='68  8176  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQf-QvH/content/2301.04825v1.pdf'}
+page_content='008676  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQf-QvH/content/2301.04825v1.pdf'}
+page_content='1171  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQf-QvH/content/2301.04825v1.pdf'}
+page_content='5549  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQf-QvH/content/2301.04825v1.pdf'}
+page_content='567 × 10−09 −3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQf-QvH/content/2301.04825v1.pdf'}
+page_content='667 × 10−07 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQf-QvH/content/2301.04825v1.pdf'}
+page_content='166 × 10−06 −3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQf-QvH/content/2301.04825v1.pdf'}
+page_content='347 × 10−05  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQf-QvH/content/2301.04825v1.pdf'}
+page_content='5378  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQf-QvH/content/2301.04825v1.pdf'}
+page_content='661  TABLE III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQf-QvH/content/2301.04825v1.pdf'}
+page_content=' Experimental equilibrium volume V0 (˚A3 per atom), bulk modulus B0 (GPa), and B′  0 at room temperature are  compared against reported theoretical results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQf-QvH/content/2301.04825v1.pdf'}
+page_content=' Data and method are briefly summarized, and the original references are given.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQf-QvH/content/2301.04825v1.pdf'}
+page_content='  Method description  V0  B0  B′  0  References  (˚A3/at)  (GPa)  Exp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQf-QvH/content/2301.04825v1.pdf'}
+page_content=' data fitted to 3rd-order BM EoS  14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQf-QvH/content/2301.04825v1.pdf'}
+page_content='120  339  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQf-QvH/content/2301.04825v1.pdf'}
+page_content='3  Monteseguro et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQf-QvH/content/2301.04825v1.pdf'}
+page_content=' [10]  Exp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQf-QvH/content/2301.04825v1.pdf'}
+page_content=' data fitted to 3rd-order BM EoS  14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQf-QvH/content/2301.04825v1.pdf'}
+page_content='145  383  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQf-QvH/content/2301.04825v1.pdf'}
+page_content='1  Cynn et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQf-QvH/content/2301.04825v1.pdf'}
+page_content=' [7]  Exp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQf-QvH/content/2301.04825v1.pdf'}
+page_content=' data fitted to  Cerenius and Dubrovinsky[6]  2nd-order BM EoS, with B′  0 = 4  14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQf-QvH/content/2301.04825v1.pdf'}
+page_content='173 (exp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQf-QvH/content/2301.04825v1.pdf'}
+page_content=' value)  354  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQf-QvH/content/2301.04825v1.pdf'}
+page_content='0  3rd-order BM EoS, without constraint  14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQf-QvH/content/2301.04825v1.pdf'}
+page_content='173 (exp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQf-QvH/content/2301.04825v1.pdf'}
+page_content=' value)  306  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQf-QvH/content/2301.04825v1.pdf'}
+page_content='8  DFT data fitted to BM EoS  Park et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQf-QvH/content/2301.04825v1.pdf'}
+page_content=' [37], Table 1 and 2  PAW LDA  13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQf-QvH/content/2301.04825v1.pdf'}
+page_content='925  399  PAW GGA  14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQf-QvH/content/2301.04825v1.pdf'}
+page_content='524  344  FPMD data fitted to 3rd-order BM EoS  14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQf-QvH/content/2301.04825v1.pdf'}
+page_content='150  366  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQf-QvH/content/2301.04825v1.pdf'}
+page_content='0  Burakovsky et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQf-QvH/content/2301.04825v1.pdf'}
+page_content=' [9]  FPMD data fitted to our EoS  14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQf-QvH/content/2301.04825v1.pdf'}
+page_content='559  361  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQf-QvH/content/2301.04825v1.pdf'}
+page_content='3  This work  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQf-QvH/content/2301.04825v1.pdf'}
+page_content='  Shock compression  High pressure high temperature conditions are gener-  ated by laser heating [38] or resistive heating [39] in a  DAC or by laser or gas gun [40] driven shock compres-  sion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQf-QvH/content/2301.04825v1.pdf'}
+page_content=' Strong shocks obey the Rankine-Hugoniot,  U − U0 + 1  2(P + P0)(V − V0) = 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQf-QvH/content/2301.04825v1.pdf'}
+page_content='  (4)  Since the analytical expression for U, P as a function of  V, T is known, for each volume V , we solve Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQf-QvH/content/2301.04825v1.pdf'}
+page_content=' (4) by  searching its root T given the experimental value V0, T0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQf-QvH/content/2301.04825v1.pdf'}
+page_content='  We compared our predicted principle Hugoniot with  available shock experimental data from several facilities  [41, 43] (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQf-QvH/content/2301.04825v1.pdf'}
+page_content='  5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQf-QvH/content/2301.04825v1.pdf'}
+page_content='  Our theoretical principle Hugoniot  agrees well with that from earlier data of Al’tshuler and  LANL March, as well as more recent data of STAR Hugo-  niot, for P < 200 GPa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQf-QvH/content/2301.04825v1.pdf'}
+page_content=' Above 200 GPa, our predicted  pressure is higher than that of LANL and STAR but  lower than Al’tshuler’s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQf-QvH/content/2301.04825v1.pdf'}
+page_content=' Our theoretical Hugoniot below  3000 K (shock temperature) are fairly reliable which cor-  respond to pressure less than 200 GPa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQf-QvH/content/2301.04825v1.pdf'}
+page_content=' The shock tem-  perature, calculated as the solution to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQf-QvH/content/2301.04825v1.pdf'}
+page_content=' (4), is shown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQf-QvH/content/2301.04825v1.pdf'}
+page_content='  C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQf-QvH/content/2301.04825v1.pdf'}
+page_content='  Equation of state parameters  Thermal EoS parameters such as thermal expansivity  α, isothermal compressibility βT , Gr¨uneisen parameter  γ, and the heat capacity CV and CP are obtained by  differentiation and algebraic manipulation of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQf-QvH/content/2301.04825v1.pdf'}
+page_content=' (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQf-QvH/content/2301.04825v1.pdf'}
+page_content=' We  now discuss some of these parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQf-QvH/content/2301.04825v1.pdf'}
+page_content=' As expected from  the thermal pressure, αKT is weakly dependent on the  volume and temperature (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQf-QvH/content/2301.04825v1.pdf'}
+page_content=' 6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQf-QvH/content/2301.04825v1.pdf'}
+page_content=' The Gr¨uneisen  parameter  γ = V  �∂P  ∂U  �  V  = V αKT  CV  (5)  is another important parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQf-QvH/content/2301.04825v1.pdf'}
+page_content=' It is used in the Mie-  Gr¨uneisen EoS, where γ is assumed independent of tem-  perature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQf-QvH/content/2301.04825v1.pdf'}
+page_content=' The span of γ as a function of temperature  reduces when the pressure increases (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQf-QvH/content/2301.04825v1.pdf'}
+page_content=' 7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQf-QvH/content/2301.04825v1.pdf'}
+page_content=' For  high compressions, it is indeed fairly temperature inde-  pendent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQf-QvH/content/2301.04825v1.pdf'}
+page_content='  The  volumetric  thermal  expansivity  α  =  −1/V (∂V/∂T)P for isotropic materials is three times the  linear thermal expansivity coefficient αL, α = 3αL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQf-QvH/content/2301.04825v1.pdf'}
+page_content=' α in  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQf-QvH/content/2301.04825v1.pdf'}
+page_content=' 8 is essentially temperature independent but rather  volume sensitive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQf-QvH/content/2301.04825v1.pdf'}
+page_content=' Our theoretical prediction is below the  reported experimental value [44], but it is noted that  around room temperature, our theory prediction gives  the right thermal expansion coefficient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQf-QvH/content/2301.04825v1.pdf'}
+page_content=' The expansivity  has downsized by a factor of 4 when the pressure goes  to 300 GPa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQf-QvH/content/2301.04825v1.pdf'}
+page_content='  The heat capacity at high pressures are  almost linear above 500 K, see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQf-QvH/content/2301.04825v1.pdf'}
+page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQf-QvH/content/2301.04825v1.pdf'}
+page_content=' The higher the  temperature, the slope of CP is smaller.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQf-QvH/content/2301.04825v1.pdf'}
+page_content=' At 0 GPa, the  predicted value 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQf-QvH/content/2301.04825v1.pdf'}
+page_content='68 J K−1 mol−1is fairly accurate and  only 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQf-QvH/content/2301.04825v1.pdf'}
+page_content='3% larger than the experimental heat capacity  25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQf-QvH/content/2301.04825v1.pdf'}
+page_content='10 J K−1 mol−1[45], given that we used the formula,  CP = CV (1 + Tαγ) (see Appendix A) with errors in  6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQf-QvH/content/2301.04825v1.pdf'}
+page_content='75  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQf-QvH/content/2301.04825v1.pdf'}
+page_content='80  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQf-QvH/content/2301.04825v1.pdf'}
+page_content='85  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQf-QvH/content/2301.04825v1.pdf'}
+page_content='90  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQf-QvH/content/2301.04825v1.pdf'}
+page_content='95  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQf-QvH/content/2301.04825v1.pdf'}
+page_content='00  V/V0  0  25  50  75  100  125  150  175  200  Pressure (GPa)  This work from FPMD  300 K  1000 K  2000 K  3000 K  T=0K DFT  T=0K Exp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQf-QvH/content/2301.04825v1.pdf'}
+page_content='  T=1000K DFT  T=1000K Exp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQf-QvH/content/2301.04825v1.pdf'}
+page_content='  T=3000K DFT  T=3000K Exp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQf-QvH/content/2301.04825v1.pdf'}
+page_content="  Khishchenko  Yusenko  Monteseguro  Cerenius  Akella  Al'tshuler  FIG." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQf-QvH/content/2301.04825v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQf-QvH/content/2301.04825v1.pdf'}
+page_content='  Theoretical and experimental EoS’s of fcc iridium  are compared.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQf-QvH/content/2301.04825v1.pdf'}
+page_content=' The DAC experimental data for Yusenko [30]  Monteseguro[10], Cerenius[6], and Akella[5] were compared  at 300 K, where the BM EoS’s were available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQf-QvH/content/2301.04825v1.pdf'}
+page_content=' Experimental  and theoretical data from Anzellini [14] are included (pink,  green, and yellow lines).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQf-QvH/content/2301.04825v1.pdf'}
+page_content='  The shockwave data (red cross)  of Al’tshuler were taken from Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQf-QvH/content/2301.04825v1.pdf'}
+page_content=' [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQf-QvH/content/2301.04825v1.pdf'}
+page_content=' Room temperature  isotherm of Khishchenko shock experiments [34] agrees very  well against our FPMD results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQf-QvH/content/2301.04825v1.pdf'}
+page_content='  0  500  1000  1500  2000  2500  3000  3500  4000  Temperature (K)  0  5  10  15  20  25  Thermal Pressure (GPa)  volume/bohr3  70  80  90  100  110  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQf-QvH/content/2301.04825v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQf-QvH/content/2301.04825v1.pdf'}
+page_content='  The thermal pressure of fcc iridium is roughly linear  in T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQf-QvH/content/2301.04825v1.pdf'}
+page_content='  CV , α, and γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQf-QvH/content/2301.04825v1.pdf'}
+page_content='  The RMSD is a critical quantity for the analysis  of the phonon vibrations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQf-QvH/content/2301.04825v1.pdf'}
+page_content='  Moseley et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQf-QvH/content/2301.04825v1.pdf'}
+page_content='  [46] pre-  sented temperature-dependent inelastic neutron scatter-  ing (INS) experiments as well as quasi-harmonic density  functional theory calculations to study the thermody-  namic properties of Ir.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQf-QvH/content/2301.04825v1.pdf'}
+page_content='  Our FPMD RMSD at 300 K  20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQf-QvH/content/2301.04825v1.pdf'}
+page_content='0  22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQf-QvH/content/2301.04825v1.pdf'}
+page_content='5  25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQf-QvH/content/2301.04825v1.pdf'}
+page_content='0  27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQf-QvH/content/2301.04825v1.pdf'}
+page_content='5  30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQf-QvH/content/2301.04825v1.pdf'}
+page_content='0  32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQf-QvH/content/2301.04825v1.pdf'}
+page_content='5  35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQf-QvH/content/2301.04825v1.pdf'}
+page_content='0  37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQf-QvH/content/2301.04825v1.pdf'}
+page_content='5  40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQf-QvH/content/2301.04825v1.pdf'}
+page_content="0  Density (g/cm3)  0  100  200  300  400  500  600  700  800  Pressure (GPa)  This work, FPMD  STAR Hugoniot  LANL Hugoniot (Marsh)  Al'tshuler  0  1  2  3  4  5  Temperature (kK)  Shock temperature  FIG." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQf-QvH/content/2301.04825v1.pdf'}
+page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQf-QvH/content/2301.04825v1.pdf'}
+page_content='  The principle Hugoniot curve from our theoretical  EoS (solid black) of fcc Ir is compared against the shockwave  experimental data (symbols).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQf-QvH/content/2301.04825v1.pdf'}
+page_content=' Round green is for the STAR  Hugoniot [41], square blue for the LANL Hugoniot [42], and  diamond orange for Al’tshuler [31, 32], respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQf-QvH/content/2301.04825v1.pdf'}
+page_content='  Com-  puted temperature from the Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQf-QvH/content/2301.04825v1.pdf'}
+page_content=' (4) is shown in red.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQf-QvH/content/2301.04825v1.pdf'}
+page_content='  The  dashed extension is beyond the simulation domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQf-QvH/content/2301.04825v1.pdf'}
+page_content='  0  500  1000  1500  2000  2500  3000  3500  4000  T (K)  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQf-QvH/content/2301.04825v1.pdf'}
+page_content='5  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQf-QvH/content/2301.04825v1.pdf'}
+page_content='0  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQf-QvH/content/2301.04825v1.pdf'}
+page_content='5  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQf-QvH/content/2301.04825v1.pdf'}
+page_content='0  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQf-QvH/content/2301.04825v1.pdf'}
+page_content='5  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQf-QvH/content/2301.04825v1.pdf'}
+page_content='0  αKT (10−3GPa/K)  0 GPa  50 GPa  100 GPa  200 GPa  300 GPa  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQf-QvH/content/2301.04825v1.pdf'}
+page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQf-QvH/content/2301.04825v1.pdf'}
+page_content='  Parameter αKT as a function of temperature for  various pressures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQf-QvH/content/2301.04825v1.pdf'}
+page_content='  0  500  1000  1500  2000  2500  3000  3500  4000  Temperature (K)  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQf-QvH/content/2301.04825v1.pdf'}
+page_content='0  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQf-QvH/content/2301.04825v1.pdf'}
+page_content='2  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQf-QvH/content/2301.04825v1.pdf'}
+page_content='4  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQf-QvH/content/2301.04825v1.pdf'}
+page_content='6  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQf-QvH/content/2301.04825v1.pdf'}
+page_content='8  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQf-QvH/content/2301.04825v1.pdf'}
+page_content='0  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQf-QvH/content/2301.04825v1.pdf'}
+page_content='2  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQf-QvH/content/2301.04825v1.pdf'}
+page_content='4  Gr ̈uneisen ̈arameter  0 G̈a  50 G̈a  100 G̈a  200 G̈a  300 G̈a  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQf-QvH/content/2301.04825v1.pdf'}
+page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQf-QvH/content/2301.04825v1.pdf'}
+page_content='  Gr¨uneisen parameter γ as a function of temperature  for various pressures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQf-QvH/content/2301.04825v1.pdf'}
+page_content='  7  0  50  100  150  200  250  300  Pressure (GPa)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQf-QvH/content/2301.04825v1.pdf'}
+page_content='50  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQf-QvH/content/2301.04825v1.pdf'}
+page_content='75  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQf-QvH/content/2301.04825v1.pdf'}
+page_content='00  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQf-QvH/content/2301.04825v1.pdf'}
+page_content='25  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQf-QvH/content/2301.04825v1.pdf'}
+page_content='50  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQf-QvH/content/2301.04825v1.pdf'}
+page_content='75  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQf-QvH/content/2301.04825v1.pdf'}
+page_content='00  α (10−5K−1)  300 K  1000 K  2000 K  3000 K  0  1000  2000  3000  Temperature (K)  0  1  2  3  4  5  α (10−5K−1)  0 GPa  50 GPa  100 GPa  200 GPa  300 GPa  Exp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQf-QvH/content/2301.04825v1.pdf'}
+page_content='  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQf-QvH/content/2301.04825v1.pdf'}
+page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQf-QvH/content/2301.04825v1.pdf'}
+page_content='  The volumetric thermal expansivity α of fcc iridium  as a function of pressure at various temperatures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQf-QvH/content/2301.04825v1.pdf'}
+page_content='  Halvor-  son and Wimber measured the linear thermal expansion as  αt = a0 + a1t + a2t2 + a3t3 with a0 = 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQf-QvH/content/2301.04825v1.pdf'}
+page_content='167 × 10−6, a1 =  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQf-QvH/content/2301.04825v1.pdf'}
+page_content='038×10−9, a2 = −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQf-QvH/content/2301.04825v1.pdf'}
+page_content='8448×10−12, a3 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQf-QvH/content/2301.04825v1.pdf'}
+page_content='5852×10−15, for t  expressed in ◦C [44] at ambient pressure (see inset solid line),  where one can show α = 3(L0/Lt)αt for isotropic materials  with reference length L0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQf-QvH/content/2301.04825v1.pdf'}
+page_content='  agrees particularly well with their experimental findings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQf-QvH/content/2301.04825v1.pdf'}
+page_content='  Their reported ⟨u2⟩ at higher temperatures (T=673 K  and 823 K, see Table I of Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQf-QvH/content/2301.04825v1.pdf'}
+page_content=' [46]) however, are higher  than our FPMD predictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQf-QvH/content/2301.04825v1.pdf'}
+page_content='  This is reasonable since  our NVT ensembles at these temperatures lead to higher  pressure and confined vibrations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQf-QvH/content/2301.04825v1.pdf'}
+page_content='  It is worth to note  that their phonon density of states (PDOS) integrates  to 1, and is not fitted well at higher energies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQf-QvH/content/2301.04825v1.pdf'}
+page_content=' Further  investigation of PDOS with a FPMD simulation to com-  pare against the experiments may give insight to the an-  harmonic effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQf-QvH/content/2301.04825v1.pdf'}
+page_content=' Debye temperatures of isochores using  Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQf-QvH/content/2301.04825v1.pdf'}
+page_content=' (2) exhibit weak temperature dependence(see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQf-QvH/content/2301.04825v1.pdf'}
+page_content='  11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQf-QvH/content/2301.04825v1.pdf'}
+page_content='  IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQf-QvH/content/2301.04825v1.pdf'}
+page_content='  CONCLUSIONS  We have performed a series of FPMD simulations for  the fcc Ir at conditions up to 3000 K and 540 GPa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQf-QvH/content/2301.04825v1.pdf'}
+page_content=' By  using a simplified model for the free-energy as a func-  tion of temperature and volume and the statistical av-  erage quantities internal energy and pressure (U, P), the  thermal EoS is obtained by globally fitting to the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQf-QvH/content/2301.04825v1.pdf'}
+page_content='  We have compared previous experimental EoS’s and pro-  vided the thermal EoS up to 3000 K, and 540 GPa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQf-QvH/content/2301.04825v1.pdf'}
+page_content=' The  P − V − T curve is reasonably agreeing with the fitted  BM EoS at low compression but differs at high com-  pression.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQf-QvH/content/2301.04825v1.pdf'}
+page_content=' Our first-principles EoS accords with the most  recent shockwave experiment work by Khishchenko We  find that αKT and the thermal pressure is quite constant  from its dependence in temperature and which turns out  to be true for a wide range of materials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQf-QvH/content/2301.04825v1.pdf'}
+page_content=' We have shown  some representative derived thermal parameters against  available experiments and found agreements and discrep-  0  500  1000  1500  2000  2500  3000  Temperature (K)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQf-QvH/content/2301.04825v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQf-QvH/content/2301.04825v1.pdf'}
+page_content='5  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQf-QvH/content/2301.04825v1.pdf'}
+page_content='0  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQf-QvH/content/2301.04825v1.pdf'}
+page_content='5  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQf-QvH/content/2301.04825v1.pdf'}
+page_content='0  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQf-QvH/content/2301.04825v1.pdf'}
+page_content='5  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQf-QvH/content/2301.04825v1.pdf'}
+page_content='0  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQf-QvH/content/2301.04825v1.pdf'}
+page_content='5  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQf-QvH/content/2301.04825v1.pdf'}
+page_content='0  CP (kB)  0 GPa  50 GPa  100 GPa  200 GPa  300 GPa  Exp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQf-QvH/content/2301.04825v1.pdf'}
+page_content=' value at 0 GPa  Classical model at 0 GPa  0  50  100  150  200  250  300  P (GPa)  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQf-QvH/content/2301.04825v1.pdf'}
+page_content='8  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQf-QvH/content/2301.04825v1.pdf'}
+page_content='0  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQf-QvH/content/2301.04825v1.pdf'}
+page_content='2  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQf-QvH/content/2301.04825v1.pdf'}
+page_content='4  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQf-QvH/content/2301.04825v1.pdf'}
+page_content='6  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQf-QvH/content/2301.04825v1.pdf'}
+page_content='8  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQf-QvH/content/2301.04825v1.pdf'}
+page_content='0  CP (kB)  300 K  1000 K  2000 K  3000 K  0  500  1000  1500  2000  2500  3000  Temperature (K)  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQf-QvH/content/2301.04825v1.pdf'}
+page_content='5  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQf-QvH/content/2301.04825v1.pdf'}
+page_content='0  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQf-QvH/content/2301.04825v1.pdf'}
+page_content='5  CV (kB)  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQf-QvH/content/2301.04825v1.pdf'}
+page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQf-QvH/content/2301.04825v1.pdf'}
+page_content='  Constant pressure heat capacity CP as a function of  temperature (top panel) and pressure (bottom panel).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQf-QvH/content/2301.04825v1.pdf'}
+page_content=' Exper-  imental value at ambient condition is 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQf-QvH/content/2301.04825v1.pdf'}
+page_content='10 J K−1 mol−1[45].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQf-QvH/content/2301.04825v1.pdf'}
+page_content='  The classical model (blue dash-dotted) for 0 GPa starts to  deviate for T below 500 K, and approaches the classical limit  as T → 0 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQf-QvH/content/2301.04825v1.pdf'}
+page_content=' Inset shows the constant volume heat capacity  CV .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQf-QvH/content/2301.04825v1.pdf'}
+page_content=' Heat capacity largely reduces when pressure goes up.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQf-QvH/content/2301.04825v1.pdf'}
+page_content='  0  500  1000  1500  2000  2500  3000  3500  4000  T (K)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQf-QvH/content/2301.04825v1.pdf'}
+page_content='05  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQf-QvH/content/2301.04825v1.pdf'}
+page_content='10  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQf-QvH/content/2301.04825v1.pdf'}
+page_content='15  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQf-QvH/content/2301.04825v1.pdf'}
+page_content='20  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQf-QvH/content/2301.04825v1.pdf'}
+page_content='25  RMSD (Å)  a/a0  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQf-QvH/content/2301.04825v1.pdf'}
+page_content='00  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQf-QvH/content/2301.04825v1.pdf'}
+page_content='96  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQf-QvH/content/2301.04825v1.pdf'}
+page_content='92  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQf-QvH/content/2301.04825v1.pdf'}
+page_content='90  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQf-QvH/content/2301.04825v1.pdf'}
+page_content='88  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQf-QvH/content/2301.04825v1.pdf'}
+page_content='87  Moseley 2020, from DPS  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQf-QvH/content/2301.04825v1.pdf'}
+page_content=' 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQf-QvH/content/2301.04825v1.pdf'}
+page_content='  RMSD as a function of temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQf-QvH/content/2301.04825v1.pdf'}
+page_content=' Experimental  data was obtained by Moseley et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQf-QvH/content/2301.04825v1.pdf'}
+page_content=' [46] for constant pressure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQf-QvH/content/2301.04825v1.pdf'}
+page_content='  ancies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQf-QvH/content/2301.04825v1.pdf'}
+page_content=' Further work might resolve these discrepancies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQf-QvH/content/2301.04825v1.pdf'}
+page_content='  V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQf-QvH/content/2301.04825v1.pdf'}
+page_content='  ACKNOWLEDGMENTS  The work was done under the auspices of the US Na-  tional Science Foundation CSEDI grant EAR-1901813 to  8  0  500  1000  1500  2000  2500  3000  3500  4000  T (K)  300  400  500  600  700  ΘD (K)  a/a0  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQf-QvH/content/2301.04825v1.pdf'}
+page_content='00  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQf-QvH/content/2301.04825v1.pdf'}
+page_content='96  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQf-QvH/content/2301.04825v1.pdf'}
+page_content='92  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQf-QvH/content/2301.04825v1.pdf'}
+page_content='90  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQf-QvH/content/2301.04825v1.pdf'}
+page_content='88  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQf-QvH/content/2301.04825v1.pdf'}
+page_content='87  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQf-QvH/content/2301.04825v1.pdf'}
+page_content=' 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQf-QvH/content/2301.04825v1.pdf'}
+page_content='  Debye temperature ΘD as a function of tempera-  ture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQf-QvH/content/2301.04825v1.pdf'}
+page_content=' Zero point motion is not included to obtain the Debye  temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQf-QvH/content/2301.04825v1.pdf'}
+page_content=' In the high temperature, classical region, our  RMSD are classical from classical FPMD (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQf-QvH/content/2301.04825v1.pdf'}
+page_content=' 10) but when  an effective classical Debye temperature is derived to model  the RMSD, there is a large change with decreasing tempera-  ture into to quantum regime in the Debye model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQf-QvH/content/2301.04825v1.pdf'}
+page_content=' A 4th order  polynomial fit is done only for higher temperature due to the  classical treatment to the ions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQf-QvH/content/2301.04825v1.pdf'}
+page_content='  R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQf-QvH/content/2301.04825v1.pdf'}
+page_content='E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQf-QvH/content/2301.04825v1.pdf'}
+page_content='C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQf-QvH/content/2301.04825v1.pdf'}
+page_content=' and the National Natural Science Foundation of  China (Grant No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQf-QvH/content/2301.04825v1.pdf'}
+page_content='  12104230) to K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQf-QvH/content/2301.04825v1.pdf'}
+page_content='L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQf-QvH/content/2301.04825v1.pdf'}
+page_content=' ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQf-QvH/content/2301.04825v1.pdf'}
+page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQf-QvH/content/2301.04825v1.pdf'}
+page_content='E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQf-QvH/content/2301.04825v1.pdf'}
+page_content='C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQf-QvH/content/2301.04825v1.pdf'}
+page_content=' is sup-  ported by the Carnegie Institution for Science and grate-  fully acknowledges the Gauss Centre for Supercomputing  e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQf-QvH/content/2301.04825v1.pdf'}
+page_content='V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQf-QvH/content/2301.04825v1.pdf'}
+page_content=' for funding this project by providing computing time  on the GCS Supercomputer Supermuc-NG at Leibniz Su-  percomputing Centre.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQf-QvH/content/2301.04825v1.pdf'}
+page_content=' All the FPMD calculations were  performed on Supermuc-NG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQf-QvH/content/2301.04825v1.pdf'}
+page_content='  Appendix A: Thermodynamic relations  Once the Helmholtz free energy F = F(V,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQf-QvH/content/2301.04825v1.pdf'}
+page_content=' T) is known  as a function of volume (V ) and temperature (T),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQf-QvH/content/2301.04825v1.pdf'}
+page_content=' the fol-  lowing thermodynamical quantities can be obtained from  it [47]:  P = −  �∂F  ∂V  �  T  ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQf-QvH/content/2301.04825v1.pdf'}
+page_content='  (A1)  S = −  �∂F  ∂T  �  V  ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQf-QvH/content/2301.04825v1.pdf'}
+page_content='  (A2)  KT = β−1  T  = −V  �∂P  ∂V  �  T  = V  �∂2F  ∂V 2  �  T  ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQf-QvH/content/2301.04825v1.pdf'}
+page_content='  (A3)  CV = T  �∂S  ∂T  �  V  = −T  �∂2F  ∂T 2  �  V  ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQf-QvH/content/2301.04825v1.pdf'}
+page_content='  (A4)  αKT = −  � ∂2F  ∂T∂V  �  ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQf-QvH/content/2301.04825v1.pdf'}
+page_content='  (A5)  γ = V  �∂P  ∂U  �  V  = V αKT  CV  ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQf-QvH/content/2301.04825v1.pdf'}
+page_content='  (A6)  CP  CV  = KS  KT  = 1 + Tαγ ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQf-QvH/content/2301.04825v1.pdf'}
+page_content='  (A7)  U = F + TS ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQf-QvH/content/2301.04825v1.pdf'}
+page_content='  (A8)  where pressure,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQf-QvH/content/2301.04825v1.pdf'}
+page_content=' entropy,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQf-QvH/content/2301.04825v1.pdf'}
+page_content=' isothermal compressibility (its  inverse is the bulk modulus KT ),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQf-QvH/content/2301.04825v1.pdf'}
+page_content=' constant volume  molar heat capacity,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQf-QvH/content/2301.04825v1.pdf'}
+page_content=' volumetric expansion coefficient,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQf-QvH/content/2301.04825v1.pdf'}
+page_content='  Gr¨uneisen parameter,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQf-QvH/content/2301.04825v1.pdf'}
+page_content=' internal energy are denoted by  P,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQf-QvH/content/2301.04825v1.pdf'}
+page_content=' S,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQf-QvH/content/2301.04825v1.pdf'}
+page_content=' βT ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQf-QvH/content/2301.04825v1.pdf'}
+page_content=' CV ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQf-QvH/content/2301.04825v1.pdf'}
+page_content=' α,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQf-QvH/content/2301.04825v1.pdf'}
+page_content=' γ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQf-QvH/content/2301.04825v1.pdf'}
+page_content=' U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQf-QvH/content/2301.04825v1.pdf'}
+page_content=' CP is the constant pressure molar  heat capacity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQf-QvH/content/2301.04825v1.pdf'}
+page_content=' The parameter B′  0 (or K′  0 = ∂K  ∂P  ��  P =0) can  thus be computed using above relations:  K′ = ∂K  ∂P  =  �∂K  ∂T  �  V  �∂T  ∂P  �  V  +  �∂K  ∂V  �  T  �∂V  ∂P  �  T  =  �∂K  ∂T  �  V  1  � ∂P  ∂T  �  V  +  �∂K  ∂V  �  T  1  � ∂P  ∂V  �  T  =  �∂K  ∂T  �  V  1  � ∂P  ∂T  �  V  −  �∂K  ∂V  �  T  V  KT  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQf-QvH/content/2301.04825v1.pdf'}
+page_content='  (A9)  Appendix B: Debye Model  The Helmholtz free energy F of a vibrating lattice at  volume V and temperature T, can be approximated as  F(V, T) = E(V ) + Fvib(V, T) + Fel(V, T) ,  (B1)  where Fvib is the vibrating energy of the lattice and Fel  is the thermal electronic free energy which is typically  negligible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQf-QvH/content/2301.04825v1.pdf'}
+page_content='  Moruzzi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQf-QvH/content/2301.04825v1.pdf'}
+page_content='  [28] proposed an empirical  Debye model with  Fvib = kBT  �  −D3(x) + 3 ln(1 − e−x)  �  + 9  8kBΘD , (B2)  with Debye temperature ΘD and dimensionless param-  eter x =  ΘD  T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQf-QvH/content/2301.04825v1.pdf'}
+page_content='  The last term is the zero-point energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQf-QvH/content/2301.04825v1.pdf'}
+page_content='  9  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQf-QvH/content/2301.04825v1.pdf'}
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+page_content='Residuals  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQf-QvH/content/2301.04825v1.pdf'}
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+page_content='Normal Q−Q  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQf-QvH/content/2301.04825v1.pdf'}
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+page_content="6  −6  −4  −2  0  2  4  6  Leverage  Standardized residuals  G  G G  G G G G  G  G  G  G  G  G  G  G  G GG  G  GG  GGG  GGG  GGG  GG  G  G  GG  G  G  G  G  G  G  G  G  G  G  G  G  G  G  G  G  G  G  G  G  G  G  G  G  G  G  G  G  G  G  G  G  G  G  G  G  G  GG  G  G  G  G  G  G  G  G  G  Cook's distance  1  0." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQf-QvH/content/2301.04825v1.pdf'}
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+page_content='5  1  Residuals vs Leverage  18  1611  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQf-QvH/content/2301.04825v1.pdf'}
+page_content=' 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQf-QvH/content/2301.04825v1.pdf'}
+page_content='  The fit summary plot from the lm function in R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQf-QvH/content/2301.04825v1.pdf'}
+page_content='  D3(x) is the third order Debye function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQf-QvH/content/2301.04825v1.pdf'}
+page_content=' The nth order  Debye function is defined as  Dn(x) =  � x  0  tn  et − 1dt, x ≥ 0 ,  (B3)  where n, a non-negative integer, is the order of the Debye  function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQf-QvH/content/2301.04825v1.pdf'}
+page_content=' The vibrational entropy is  Svib = 4kBD(x) − 3kB ln(1 − e−x) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQf-QvH/content/2301.04825v1.pdf'}
+page_content='  (B4)  Neglecting the zero-point motion, the vibrational internal  energy Uvib thus can be obtained  Uvib = Fvib + TSvib = 3kBTD(x) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQf-QvH/content/2301.04825v1.pdf'}
+page_content='  (B5)  [1] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQf-QvH/content/2301.04825v1.pdf'}
+page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQf-QvH/content/2301.04825v1.pdf'}
+page_content=' Arblaster, Platinum Metals Review 54, 93 (2010).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQf-QvH/content/2301.04825v1.pdf'}
+page_content='  [2] P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQf-QvH/content/2301.04825v1.pdf'}
+page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQf-QvH/content/2301.04825v1.pdf'}
+page_content=' Bridgman, Proceedings of the American Academy  of Arts and Sciences 71, 387 (1937).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQf-QvH/content/2301.04825v1.pdf'}
+page_content='  [3] P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQf-QvH/content/2301.04825v1.pdf'}
+page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQf-QvH/content/2301.04825v1.pdf'}
+page_content=' Bridgman, Proceedings of the American Academy  of Arts and Sciences 81, 165 (1952).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQf-QvH/content/2301.04825v1.pdf'}
+page_content='  [4] B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQf-QvH/content/2301.04825v1.pdf'}
+page_content=' Schock and Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQf-QvH/content/2301.04825v1.pdf'}
+page_content=' Johnson, Fiz .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQf-QvH/content/2301.04825v1.pdf'}
+page_content=' Metal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQf-QvH/content/2301.04825v1.pdf'}
+page_content=' Metalloved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQf-QvH/content/2301.04825v1.pdf'}
+page_content=' 31,  1101 (1971).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQf-QvH/content/2301.04825v1.pdf'}
+page_content='  [5] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQf-QvH/content/2301.04825v1.pdf'}
+page_content=' Akella, Journal of Physics and Chemistry of Solids 43,  941 (1982).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQf-QvH/content/2301.04825v1.pdf'}
+page_content='  [6] Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQf-QvH/content/2301.04825v1.pdf'}
+page_content=' Cerenius and L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQf-QvH/content/2301.04825v1.pdf'}
+page_content=' Dubrovinsky, Journal of Alloys and  Compounds 306, 26 (2000).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQf-QvH/content/2301.04825v1.pdf'}
+page_content='  [7] H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQf-QvH/content/2301.04825v1.pdf'}
+page_content=' Cynn, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQf-QvH/content/2301.04825v1.pdf'}
+page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQf-QvH/content/2301.04825v1.pdf'}
+page_content=' Klepeis, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQf-QvH/content/2301.04825v1.pdf'}
+page_content='-S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQf-QvH/content/2301.04825v1.pdf'}
+page_content=' Yoo,  and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQf-QvH/content/2301.04825v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQf-QvH/content/2301.04825v1.pdf'}
+page_content=' Young,  Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQf-QvH/content/2301.04825v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQf-QvH/content/2301.04825v1.pdf'}
+page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQf-QvH/content/2301.04825v1.pdf'}
+page_content=' 88, 135701 (2002).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQf-QvH/content/2301.04825v1.pdf'}
+page_content='  [8] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQf-QvH/content/2301.04825v1.pdf'}
+page_content=' Baroni, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQf-QvH/content/2301.04825v1.pdf'}
+page_content=' de Gironcoli, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQf-QvH/content/2301.04825v1.pdf'}
+page_content=' Dal Corso, and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQf-QvH/content/2301.04825v1.pdf'}
+page_content=' Gian-  nozzi, Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQf-QvH/content/2301.04825v1.pdf'}
+page_content=' Mod.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQf-QvH/content/2301.04825v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQf-QvH/content/2301.04825v1.pdf'}
+page_content=' 73, 515 (2001).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQf-QvH/content/2301.04825v1.pdf'}
+page_content='  [9] L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQf-QvH/content/2301.04825v1.pdf'}
+page_content=' Burakovsky, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQf-QvH/content/2301.04825v1.pdf'}
+page_content=' Burakovsky, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQf-QvH/content/2301.04825v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQf-QvH/content/2301.04825v1.pdf'}
+page_content=' Cawkwell, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQf-QvH/content/2301.04825v1.pdf'}
+page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQf-QvH/content/2301.04825v1.pdf'}
+page_content='  Preston, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQf-QvH/content/2301.04825v1.pdf'}
+page_content=' Errandonea, and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQf-QvH/content/2301.04825v1.pdf'}
+page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQf-QvH/content/2301.04825v1.pdf'}
+page_content=' Simak, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQf-QvH/content/2301.04825v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQf-QvH/content/2301.04825v1.pdf'}
+page_content=' B  94, 094112 (2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQf-QvH/content/2301.04825v1.pdf'}
+page_content='  [10] V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQf-QvH/content/2301.04825v1.pdf'}
+page_content=' Monteseguro, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQf-QvH/content/2301.04825v1.pdf'}
+page_content=' Sans, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQf-QvH/content/2301.04825v1.pdf'}
+page_content=' Cuartero, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQf-QvH/content/2301.04825v1.pdf'}
+page_content=' Cova, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQf-QvH/content/2301.04825v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQf-QvH/content/2301.04825v1.pdf'}
+page_content='  Abrikosov, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQf-QvH/content/2301.04825v1.pdf'}
+page_content=' Olovsson, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQf-QvH/content/2301.04825v1.pdf'}
+page_content=' Popescu, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQf-QvH/content/2301.04825v1.pdf'}
+page_content=' Pascarelli,  G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQf-QvH/content/2301.04825v1.pdf'}
+page_content=' Garbarino, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQf-QvH/content/2301.04825v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQf-QvH/content/2301.04825v1.pdf'}
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+Majorana Spin Current Generation by Dynamic Strain
+Yuki Yamazaki,1 Takumi Funato,2, 3 and Ai Yamakage1
+1Department of Physics, Nagoya University, Nagoya 464-8602, Japan
+2Center for Spintronics Research Network, Keio University, Yokohama 223-8522, Japan
+3Kavli Institute for Theoretical Sciences, University of Chinese Academy of Sciences, Beijing, 100190, China
+(Dated: January 11, 2023)
+Majorana fermions that emerge on the surface of topological superconductors are charge neutral but can have
+higher-rank electric multipoles by allowing for account time-reversal and crystalline symmetries. Applying
+the general classification of these multipoles, we show that the spin current of Majorana fermions is driven
+by spatially nonuniform dynamic strains on the (001) surface of superconducting antiperovskite Sr3SnO. We
+also find that the frequency dependence of the Majorana spin current reflects the energy dispersion of Majorana
+fermions. Our results suggest that the spin current can be a probe for Majorana fermions.
+Majorana fermions (MFs) are charge-neutral relativistic
+particles moving in 3D space. In addition, two types of MFs
+emerge in topological superconductors (TSCs) as gapless An-
+dreev bound states [1–7]. One type is a spatially localized
+zero-dimensional MF, which appears at the ends of nanowires
+[8–14] or in the cores of the vortices of TSCs [15–18]. MFs
+have been the subject of considerable research because of
+their potential application to fault-tolerant topological quan-
+tum computation with non-Abelian statistics [19, 20].
+Here, one can ask the following question: what are the
+physical phenomena unique to spatially extended 1D or 2D
+MFs? A typical example is a half-integer thermal quantum
+Hall effect on the surface of a TSC [21–23]. “Half-Integer” is
+a peculiarity originating from the fact that MFs are heat car-
+riers. The half-integer thermal quantum Hall effect is unique
+to MFs that are spread in 2D space and uses the thermal gra-
+dient as the “driving force”. Recently, the optical response of
+Majorana chiral edge modes has also been discussed [24, 25].
+The frequency dependence of real-part optical conductivity is
+proportional to ω2, where it can be distinguished from trivial
+superconductors or insulators and Dirac chiral edge modes.
+Then, the driving force of MFs is an electromagnetic wave.
+In superconductors, we cannot use an electric field to sur-
+vey superconducting properties since charge conservation is
+broken. The responses to acoustic waves have been studied
+extensively through measurements of ultrasonic attenuation
+[26, 27] to investigate superconducting symmetry or measure-
+ments of the temperature dependence of a superconducting
+gap [28]. Therefore, we can expect to be able to use dynamic
+strains for the driving force of MFs on the surfaces of 3D
+TSCs.
+When MFs have time-reversal symmetry, they do not ap-
+pear alone but form Kramers pairs, which are called MKPs.
+Due to the time-reversal symmetry, a single MKP is stable
+against an electrical external field such as acoustic waves with
+lattice strains. Previously, we derived the general effective
+theory for electromagnetic properties of “double” MKPs on a
+surface of a 3D TSC [29–31]. Double MKPs can have var-
+ious electrical multipole degrees of freedom that are quali-
+tatively different from those of ordinary fermions, and it has
+been shown that they can respond to static strains [31]. Double
+MKPs are always protected by crystalline symmetries in addi-
+E
+FIG. 1. This schematic shows that the Majorana spin currents flow
+on the surface of the 3D TCSC driven by a dynamic strain. The sur-
+face has four Majorana fermions that form double Majorana Kramers
+pairs (MKPs). Double MKPs have electric quadrupoles coupled to a
+dynamic strain, and the spin current of Majorana electric quadrupoles
+is generated by a dynamic strain.
+tion to time-reversal symmetry; thus, TSCs are specifically re-
+ferred to as topological crystalline superconductors (TCSCs).
+In this letter, we clarify the transport phenomena of double
+MKPs driven by spatially nonuniform dynamic strains on the
+surface of 3D TCSCs. We consider the surface of the antiper-
+ovskite TCSC Sr3SnO [32], which has double MKPs with two
+electrical multipoles on its (001) surface [31, 33]. We start
+with a 4 × 4 Dirac Hamiltonian with respect to the (001) sur-
+face point group (PG) symmetry. Then, we reveal the spin
+current of double MKPs, referred to as the “Majorana spin
+current”, generated by spatially nonuniform dynamic strains
+on the surface (see Fig. 1). When there is no gap in the dis-
+persion of double MKPs, we find that the Majorana spin cur-
+rent is caused by an intrinsic effect and does not depend on
+relaxation times. Alternatively, for the gapped case induced
+by surface magnetization, the Majorana spin current has two
+frequency regions that reflect the energy dispersion of double
+MKPs. Our study suggests that the measurement of the Ma-
+jorana spin current can be a new probe for Majorana fermion
+arXiv:2301.03937v1  [cond-mat.supr-con]  10 Jan 2023
+
+22
+TABLE I. Irreducible decomposition of the matrices, momentum, strain, and spin current in the p4m model for the A1u superconducting
+pair potential in the bulk. The matrix is transformed by g as η → D†
+gηDg. (ηi, ηj) denotes (ηi, ηj) → (−ηj, ηi) by transformation of C4z
+and (ηi, ηj) → (−ηi, ηj) by transformation of σ(xz). Electric/magnetic and PHS denote time-reversal even/odd and particle-hole symmetry,
+respectively. uij(x) = ∂iu j(x) with u the displacement field. jα
+i denotes spin σα/2 current along the ith direction.
+IR electric w/ PHS electric w/o PHS magnetic w/ PHS magnetic w/o PHS momentum strain
+spin current
+A1
+s0τ0, s0τ3
+k2
+x + k2
+y
+uxx + uyy
+jx
+y − jy
+x
+A2
+s3τ0, s3τ3
+uxy − uyx
+jx
+x + jy
+y
+B1 s3τ1
+s3τ2
+k2
+x − k2
+y
+uxx − uyy
+jx
+y + jy
+x
+B2 s0τ2
+s0τ1
+kxky
+uxy + uyx
+jx
+x − jy
+y
+E
+(s1τ1, s2τ1)
+(η5, η6)
+(η1, η2), (η3, η4)
+(kx, ky)
+(uxz, uyz)
+(jz
+y, − jz
+x)
+observation.
+Before we consider the main topic, we discuss the electro-
+magnetic properties of MKPs on the surface of a TSC. When
+there is only one MF on the surface, the single MF is strongly
+protected by particle-hole symmetry (charge-neutrality con-
+straint), and it is extremely stable against any external fields.
+If a TSC has time-reversal symmetry, then the two MFs form
+an MKP. The single MKP is protected by time-reversal sym-
+metry, and hence, it generally responds only to magnetic ex-
+ternal fields and remains stable to electrical external fields. On
+the other hand, when crystalline symmetries are taken into ac-
+count, there can exist double MKPs, and they can respond to
+an electrical perturbation that breaks the crystalline symmetry.
+The coupling of N MKPs to a spatially uniform external field
+can be represented by Ref. [31]
+ˆHsurf,ex = − ˆOF, ˆO = 1
+2
+�
+d2x
+�
+ss′
+ˆψs(x)(AF)ss′ ˆψs′(x),
+(1)
+where ˆψs(x) (s = 1, ..., 2N) are Majorana field operators and
+satisfy ˆψ†
+s(x) = ˆψs(x). AF is conjugate to F and should be
+an antisymmetric Hermite matrix since Majorana field oper-
+ators obey { ˆψs(x), ˆψs′(x′)} = δss′δ2(x − x′). When AF is the
+2 × 2 matrix, with only one MKP, there exists only AF = σy,
+which satisfies ΘsurfAFΘ−1
+surf = −AF for the time-reversal oper-
+ator Θsurf = (−iσyK). K is a complex conjugation, and hence,
+the MKP couples only to magnetic external fields. On the
+other hand, when AF is the 4 × 4 matrix, with double MKPs,
+AF, which satisfies ΘsurfAFΘ−1
+surf = AF, can be formed, and the
+double MKPs can also be coupled to the electrical external
+field by Eq. (1). Double MKPs have various electric multi-
+poles depending on each crystalline symmetry of the surface,
+i.e., wall-paper groups (WGs), and superconducting symme-
+tries. In a previous study [31], we revealed the coupling be-
+tween the electric multipoles and the spatially uniform static
+strain for each WGs.
+We consider the antiperovskite superconductor Sr3SnO as
+a concrete model.
+Antiperovskite A3BX, A = Ca, Sr, La,
+B = Pb, Sn, X = C, N, O with the space group symmetry
+Pm¯3m (No.221) [34, 35], has multiple J = 3/2 bands and is
+a candidate material for topological crystalline insulators due
+to the band inversion of two orbitals [36–38]. In particular,
+Sr3SnO has the potential to become a TSC with hole doping
+at temperatures below 5 K [32]. Interestingly, it has been sug-
+gested that double MKPs can appear in the (001) surface with
+p4m WG symmetry when the superconductor symmetry is in
+the A1u representation in bulk with Oh PG symmetry [33]. In
+the following, we consider the double MKPs on the (001) sur-
+face in the A1u state of Sr3SnO and derive the Majorana spin
+current generated by spatially nonuniform dynamic strains.
+To analyze the transport properties of double MKPs, we
+utilize the low-energy Dirac model for the (001) surface
+of Sr3SnO with p4m WG symmetry, which equals C4v PG
+symmetry.
+The surface symmetry operations are given by
+D{C4z|0} =
+−1
+√
+2(s0τ3 − is3τ3) and D{σ(xz)|0} =
+i√
+2(s2τ3 − s1τ3)
+with Dg being a representation matrix of g ∈ C4v, where siτ j’s
+are the product of Pauli matrices acting on the spin, orbital,
+and sublattice degrees of freedom. Then, we obtain the total
+symmetric Hamiltonian [30, 31]. (see Table I and Sec. I of
+Supplemental Material [39]):
+ˆH = 1
+2
+�
+k
+ˆψ†
+kH(k) ˆψk,
+H(k) = [v1(η1ky − η2kx) + v2(η3ky − η4kx)],
+(2)
+where ˆψ†
+k = ( ˆψ†
+1k, ˆψ†
+2k, ˆψ†
+3k, ˆψ†
+4k) and ˆψk =
+t( ˆψ1k, ..., ˆψ4k)
+are the Majorana creation/annihilation operators, which sat-
+isfy ˆψ†
+1k = ˆψ2−k, ˆψ†
+3k = ˆψ4−k and the subscripts 1, ..., 4 de-
+note four MFs.
+ηi’s are given by η1
+= (1/
+√
+2)(s1τ0 +
+s2τ0), η2
+=
+(−1/
+√
+2)(s1τ0 − s2τ0), η3
+=
+(1/
+√
+2)(s1τ3 +
+s2τ3), η4 = (−1/
+√
+2)(s1τ3 − s2τ3). H(k) satisfies DgH(k)D†
+g =
+H(gk) where a momentum k is transformed to gk under
+the action of g.
+The spin of MFs is represented by σ =
+(σx, σy, σz) ≡ (η5, η6, −s3τ0) where η5 = (1/
+√
+2)(s1τ2 −
+s2τ2), η6
+= (−1/
+√
+2)(s1τ2 + s2τ2), which are coupled to
+the magnetization M as M · σ.
+The time-reversal Θ,
+particle-hole C, and chiral Γ symmetries are defined by Θ =
+s2τ3K,C = s1τ0K, Γ = s3τ3 where they satisfy ΘH(k)Θ−1 =
+H(−k), CH(k)C−1 = −H(−k), ΓH(k)Γ−1 = −H(k). In our
+model, which is the C4v PG symmetric model for the A1u
+bulk’s superconducting pair potential with Oh PG symmetry,
+the double MKPs have specific electromagnetic multipoles
+with particle-hole symmetry. Table I shows that the double
+MKPs have magnetic multipoles that are represented by σ and
+electric multipoles that are represented by s3τ1 with B1 and
+s0τ2 with B2 representations. Therefore, as follows, we can
+
+3
+define the spin currents jx
+y + jy
+x and jx
+x− jy
+y generated by dynam-
+ical strains uxx − uyy and uxy + uyx, where uij(x) = ∂iuj(x) and
+uj(x) ∝ ei(q·x−ωt) denote the displacement field, respectively
+(Fig. 1). For double MKPs, we can construct the spin current
+operator only from surface effective theory, i.e., the spin cur-
+rent is defined by jα
+i ≡ 1
+2{σα, ∂H(k)/∂ki} (i = x, y), which sat-
+isfies particle-hole symmetry C jα
+i C−1 = − jα
+i . Then, we obtain
+ˆjx
+y(q) + ˆjy
+x(q) = 1
+2
+�
+k ˆψ†
+k− q
+2 [−v2s3τ1] ˆψk+ q
+2 and ˆjx
+x(q) − ˆjy
+y(q) =
+1
+2
+�
+k ˆψ†
+k− q
+2 [v1s0τ2] ˆψk+ q
+2 .
+Here, we define the operators, which are represented
+by
+ˆO(1)(q)
+≡
+1
+2
+�
+k ˆψ†
+k− q
+2 (ρ1s3τ1) ˆψk+ q
+2
+and
+ˆO(2)(q)
+≡
+1
+2
+�
+k ˆψ†
+k− q
+2 (ρ2s0τ2) ˆψk+ q
+2 . Since ˆO(1) and uxx − uyy, ˆO(2) and
+uxy +uyx share the same representation, the couplings between
+double MKPs and the dynamical strains are represented by
+ˆHsurf,ex = −
+�
+ˆO(1)(q)[uxx(q, ω) − uyy(q, ω)]
++ ˆO(2)(q)[uxy(q, ω) + uyx(q, ω)]
+�
+.
+(3)
+Then, we calculate the linear response of the spin currents
+ˆjx
+y + ˆjy
+x and ˆjx
+x − ˆjy
+y to the dynamic strains given by Eq. (3):
+⟨ˆjx
+y + ˆjy
+x⟩(q, ω) ≡ K1(q, ω)[uxx(q, ω) − uyy(q, ω)]
++ K2(q, ω)[uxy(q, ω) + uyx(q, ω)],
+(4)
+⟨ˆjx
+x − ˆjy
+y⟩(q, ω) ≡ K′
+1(q, ω)[uxx(q, ω) − uyy(q, ω)]
++ K′
+2(q, ω)[uxy(q, ω) + uyx(q, ω)],
+(5)
+where the response function Ki(q, ω) is given by Ref. [40]
+Ki(q, ω) ≡ i
+� ∞
+0
+dtei(ω+iδ)t ⟨[ˆjx
+y(q, t) + ˆjy
+x(q, t), ˆO(i)(−q, 0)]⟩,
+(6)
+where ˆA(t) = ei ˆHt ˆAe−i ˆHt and δ → +0. K′
+i (q, ω) is also de-
+fined by replacing ˆjx
+y + ˆjy
+x with ˆjx
+x − ˆjy
+y in Eq.
+(6).
+Note
+that the chemical potential µ for MFs is equal to 0 due to
+particle-hole symmetry in superconducting states. Addition-
+ally, ⟨· · · ⟩ = tr[e− ˆH/T · · · ]/tr[e− ˆH/T]. If there are no applied
+external fields, K1(q, ω) and K′
+2(q, ω) then only take finite val-
+ues since the spin currents and dynamic strains share the same
+representation of C4v. Here, we assume that the wavenumber
+q and the frequency ω are smaller than the mean free path l and
+relaxation time τ of the MFs. These conditions are represented
+by q ≪ l−1, ω ≪ τ−1 = 2γ. We expand the response function
+for ω and consider the nonequilibrium part: Ki(ω) − Ki(0).
+(the details are shown in Sec. II of the Supplemental Material
+[39]):
+K1(ω) − K1(0) ≃ iω
+8π2
+ρ1
+v1
+ln
+�(v1 − v2)2
+(v1 + v2)2
+�
+.
+(7)
+One can see that K1(ω)− K1(0) is independent of the damping
+parameter γ. Note that K′
+2(ω) − K′
+2(0) =
+v1ρ2
+−v2ρ1 (K1(ω) − K1(0))
+FIG. 2. The energy dispersion of (left) Eq. (2) and (right) Eq. (8).
+The energy gap opened by the magnetization M.
+because of chiral symmetry. The details are shown in Sec. II
+of the Supplemental Material [39].
+Next, we consider the case in which the dispersion of dou-
+ble MKPs is gapped. This situation can be realized, for ex-
+ample, by attaching ferromagnets on the surface. Then, the
+effective Hamiltonian is given by
+˜H(k) = H(k) + ˜Mzσz.
+(8)
+The second term in Eq. (8) is the Zeeman term reflected
+by coupling between the spin moment of double MKPs and
+the magnetization of ferromagnets, where we define Mz ≡
+− ˜Mz.
+Then, the energy dispersion is given by Eτ=±,± =
+±
+�
+M2z + k2(v1 + τv2)2 in Fig. 2. The applied magnetization
+opens an energy gap in the dispersion of double MKPs and
+lowers the symmetry from C4v to C4. Therefore, K2(ω) can
+take a finite value since jx
+y + jy
+x and uxy + uyx share the same
+irreducible representation of C4. The relationship between Bz
+and ω, is important; hence, we calculate the response function
+by using the Lehmann representation when the dispersion of
+double MKPs is gapped:
+K1(ω) = 1
+4
+�
+n∈occ,m∈unocc
+�⟨n| − v2s3τ1|m⟩⟨m|ρ1s3τ1|n⟩
+En − Em + ω + iδ
++
+�⟨n| − v2s3τ1|m⟩⟨m|ρ1s3τ1|n⟩
+En − Em − ω + iδ
+�∗ �
+,
+(9)
+K2(ω) = 1
+4
+�
+n∈occ,m∈unocc
+�⟨n| − v2s3τ1|m⟩⟨m|ρ2s0τ2|n⟩
+En − Em + ω + iδ
++
+�⟨n| − v2s3τ1|m⟩⟨m|ρ2s0τ2|n⟩
+En − Em − ω + iδ
+�∗ �
+.
+(10)
+where δ → +0, |n⟩ and |m⟩ denote occupied states and unoc-
+cupied states of Eq. (8), respectively. As a result, we obtain
+
+Er,± = ±VM2+ k2(V1 +TV2)2
+E
+E
+M
+2|Mz
+E一,-
+E.
+E.4
+TABLE II. Electromagnetic degrees of freedom of the double Majorana Kramers pairs (MKPs). They emerge on the surface with wallpaper-
+group (WG) symmetry, when the bulk pair potential belongs to irrep ∆. Irreps of double MKPs, magnetic operators, electric operators of
+MKPs, which couple to magnetization Mz perpendicular to the surface and strain [uij(x) = ∂iu j(x), where u(x) denotes the displacement field],
+are also shown. The last column denotes that the strain can couple to the double MKPs and the generated spin current by the strain. We adapt
+the definition for WG and irreps by Bilbao Crystallographic Server [41]. ¯Γi denotes the ith double-valued irrep of the little group on the Γ
+point. 2¯Γi = ¯Γi ⊕ ¯Γi. The result for P3m1 are the same as that for P31m.
+WG
+∆
+MKPs
+Magnetic
+Electric
+Magnetization Mz
+Strain (Spin current)
+uxx − uyy ( jx
+y + jy
+x)
+uxy + uyx (jx
+x − jy
+y)
+p4m
+A2
+¯Γ6 ⊕ ¯Γ7
+2A2 + E
+B1 + B2
+gapped
+⃝
+⃝
+p31m
+A1
+2(¯Γ4 ⊕ ¯Γ5)
+4A1
+2A2
+gapless
+×, only coupled to uxy − uyx (jx
+x + jy
+y)
+p4g
+B1
+¯M6 ⊕ ¯M7
+A2 + B1 + E
+A2 + B2
+gapped
+×, coupled to uxy − uyx ( jx
+x + jy
+y)
+⃝
+the nonequilibrium part: Ki(ω) − Ki(0).
+K1(ω) − K1(0)
+=
+�����������������
+A ω2
+Mz
+(ω ≪ 2|Mz|),
+B
+������− ω
+2 ln
+�����
+1+ 2kcv1
+ω
+1− 2kcv1
+ω
+����� +
+2Mzv2
+1
+v2
+1−v2
+2 −
+Mzv1ln
+��� v1−v2
+v1+v2
+���
+2v2
+������ + i Bπv1
+2v2 ω
+(2|Mz| ≪ ω ≪ x(kc)),
+(11)
+K2(ω) − K2(0)
+=
+�����������
+iCω
+(ω ≪ 2|Mz|),
+DπMz + iDMzln
+������
+1+ 2kcv1
+ω
+1− 2kcv1
+ω
+������
+(2|Mz| ≪ ω ≪ x(kc)),
+(12)
+A =
+2v1v2(3v2
+2 − v2
+1) − ((v2
+1 + v2
+2)2 − 4v4
+2)ln
+��� v1−v2
+v1+v2
+���
+32v3
+1v3
+2
+, B = v2ρ1
+8πv2
+1
+,
+C = v2ρ2
+4π
+−2v1v2 + v2ln
+����
+v1+v2
+v1−v2
+����
+8v2
+1v2
+2
+, D = −v2ρ2
+8π
+v2
+v3
+1 − v1v2
+2
+, (13)
+where x(kc) =
+�
+M2z + k2c(v1 + v2)2 +
+�
+M2z + k2c(v1 − v2)2 and
+kc is a cutoff value (see Sec. II of the Supplemental Material
+for detailed results [39]). Eqs. (11) and (12) show that K1(ω)
+and K2(ω) have two frequency regions with different behav-
+iors. The numerical results of Eqs. (9) and (10) are shown
+in Fig. 3, and they reproduce the approximate analytical solu-
+tions obtained in each region well. Both K1(ω) − K1(0) and
+K2(ω) − K2(0) reflect the energy dispersion of the MFs. In
+particular, there are abrupt changes in the values of (a), (b),
+and (c) in Fig. 3, which are the points at ω = 2|Mz|, just when
+the frequency is equal to the band gap. On the other hand,
+(d) is the diagram corresponding to the result of the principal
+value integration in Eq. (10), and there is no peak structure
+even at ω = 2|Mz| where the density of states becomes finite.
+This is because the chemical potential of MFs is exactly zero,
+and K2(ω) − K2(0) is the quantity that can take a finite value
+when the dispersion of MFs exhibits an energy gap with ap-
+plied magnetization. Hence, the contribution from the Fermi
+sea is considered to be dominant.
+In this letter, we show that the Majorana spin current
+generated by spatially nonuniform dynamic strain flows on
+the (001) surface of the 3D time-reversal-invariance TCSC
+Sr3SnO. For the gapless case, the spin current does not show
+FIG. 3. The numerical results of Eqs. (9) and (10). (a) and (b)
+show the amplitudes of the real and imaginary parts of K1(ω)−K1(0),
+respectively. (c) and (d) show the real and imaginary parts of K2(ω)−
+K2(0), respectively. Red, blue and green lines denote the cases of
+|Mz| = 0.3, |Mz| = 0.2 and |Mz| = 0.1, respectively. The values of the
+three black dotted lines drawn on the horizontal axis in (a), (b), (c),
+and (d) denote the size of the energy gap 2|Mz|, and from left to right
+are 0.2, 0.4, and 0.6. The parameters are ρ1 = 1.0, ρ2 = 1.0, v1 =
+0.3, v2 = 0.2, kc = 10.
+damping dependence due to the linear dispersion of the dou-
+ble MKPs and the fact that their chemical potential is exactly
+zero. In contrast, for the gapped case, the spin currents re-
+flect the energy dispersion of the MFs, and they have a unique
+frequency dependence. We emphasize again that the electro-
+magnetic multipoles formed by double MKPs determine the
+properties of the spin current responses to dynamic strains.
+To see this, we show the results of the previous study [31] in
+Table II (See the Supplemental material of Ref. [31] for the
+complete version). This table shows the electromagnetic de-
+grees of freedom of double MKPs for several WGs, irreps of
+the pair potential, and double MKPs. The case of p4m corre-
+sponds to the present model. In the case of p4m and p4g, the
+applied magnetization Mz induces the energy gap of double
+MKPs; however, in the case of p31m, the double MKPs still
+remain gapless. In addition, p4m and p4g are the same gapped
+case, but in the case of p4g, the double MKPs do not couple to
+
+(b)
+a
+Re[Ki(w) - Ki(O))
+Im[Ki() - Ki(O)
+0.3
+0.08
+Mz = 0.3
+M= 0.2
+<Mz = 0.1
+0.3
+0.2
+0.1
+c(kc)
+0
+0
+3
+3
+3
+3
+(d)
+(c)
+Re[K2(w) - K2(O))
+Im[K2(w) - K2(O))
+0
+0.1
+0.1
+0.2
+0.2
+0.3
+0.3
+-0.09
+-0.05
+0
+0
+3
+3
+3
+35
+the strain uxx − uyy but couple to uxy − uyx owing to the protec-
+tion of glide-plane symmetry. We have specifically shown the
+application of the general effective theory for the electromag-
+netic properties of surface double MKPs derived from system
+symmetries, such as crystalline and superconducting symme-
+tries, and the new dynamics of MFs characterized by these
+symmetries. We believe that our work accelerates the study of
+the coupling of given external fields with MFs and the trans-
+port phenomena arising therefrom.
+As a detection method for AC spin currents, spin wave res-
+onance can be observed by injecting spin currents into ferro-
+magnetic metals [42, 43] and the rectifying effect of magne-
+tostriction [44]. The dynamic strains uxx(r, t) and uxy(r, t) can
+be realized by a Rayleigh wave and a Love wave propagating
+in the x-direction, respectively. In relation to the experiment,
+parameters such as ρ1, ρ2, v1, v2 can be obtained by projecting
+bulk operators onto the surface. This study will be published
+in a future paper.
+The authors gratefully acknowledge M. Matsuo, Y. Nozaki,
+J. J. Nakane, Y. Imai, T. Yamaguchi, T. Matsushita, R.
+Kikuchi, and H. Kohno for valuable discussions.
+This
+work was supported by JSPS KAKENHI for Grants (No.
+JP20K03835) and the Sumitomo Foundation (190228). Y.Y.
+is supported by Grant-in-Aid for JSPS Fellows Grant No.
+22J14452.
+This work was financially supported by JST
+SPRING, Grant Number JPMJSP2125.
+The author Y.Y.
+would like to take this opportunity to thank the “Interdisci-
+plinary Frontier Next-Generation Researcher Program of the
+Tokai Higher Education and Research System.”
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+
diff --git a/vtE2T4oBgHgl3EQfggcu/content/tmp_files/load_file.txt b/vtE2T4oBgHgl3EQfggcu/content/tmp_files/load_file.txt
new file mode 100644
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+page_content=' Japan  3Kavli Institute for Theoretical Sciences,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE2T4oBgHgl3EQfggcu/content/2301.03937v1.pdf'}
+page_content=' University of Chinese Academy of Sciences,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE2T4oBgHgl3EQfggcu/content/2301.03937v1.pdf'}
+page_content=' Beijing,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE2T4oBgHgl3EQfggcu/content/2301.03937v1.pdf'}
+page_content=' 100190,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE2T4oBgHgl3EQfggcu/content/2301.03937v1.pdf'}
+page_content=' China  (Dated: January 11,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE2T4oBgHgl3EQfggcu/content/2301.03937v1.pdf'}
+page_content=' 2023)  Majorana fermions that emerge on the surface of topological superconductors are charge neutral but can have  higher-rank electric multipoles by allowing for account time-reversal and crystalline symmetries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE2T4oBgHgl3EQfggcu/content/2301.03937v1.pdf'}
+page_content=' Applying  the general classification of these multipoles, we show that the spin current of Majorana fermions is driven  by spatially nonuniform dynamic strains on the (001) surface of superconducting antiperovskite Sr3SnO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE2T4oBgHgl3EQfggcu/content/2301.03937v1.pdf'}
+page_content=' We  also find that the frequency dependence of the Majorana spin current reflects the energy dispersion of Majorana  fermions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE2T4oBgHgl3EQfggcu/content/2301.03937v1.pdf'}
+page_content=' Our results suggest that the spin current can be a probe for Majorana fermions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE2T4oBgHgl3EQfggcu/content/2301.03937v1.pdf'}
+page_content='  Majorana fermions (MFs) are charge-neutral relativistic  particles moving in 3D space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE2T4oBgHgl3EQfggcu/content/2301.03937v1.pdf'}
+page_content=' In addition, two types of MFs  emerge in topological superconductors (TSCs) as gapless An-  dreev bound states [1–7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE2T4oBgHgl3EQfggcu/content/2301.03937v1.pdf'}
+page_content=' One type is a spatially localized  zero-dimensional MF, which appears at the ends of nanowires  [8–14] or in the cores of the vortices of TSCs [15–18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE2T4oBgHgl3EQfggcu/content/2301.03937v1.pdf'}
+page_content=' MFs  have been the subject of considerable research because of  their potential application to fault-tolerant topological quan-  tum computation with non-Abelian statistics [19, 20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE2T4oBgHgl3EQfggcu/content/2301.03937v1.pdf'}
+page_content='  Here, one can ask the following question: what are the  physical phenomena unique to spatially extended 1D or 2D  MFs?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE2T4oBgHgl3EQfggcu/content/2301.03937v1.pdf'}
+page_content=' A typical example is a half-integer thermal quantum  Hall effect on the surface of a TSC [21–23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE2T4oBgHgl3EQfggcu/content/2301.03937v1.pdf'}
+page_content=' “Half-Integer” is  a peculiarity originating from the fact that MFs are heat car-  riers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE2T4oBgHgl3EQfggcu/content/2301.03937v1.pdf'}
+page_content=' The half-integer thermal quantum Hall effect is unique  to MFs that are spread in 2D space and uses the thermal gra-  dient as the “driving force”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE2T4oBgHgl3EQfggcu/content/2301.03937v1.pdf'}
+page_content=' Recently, the optical response of  Majorana chiral edge modes has also been discussed [24, 25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE2T4oBgHgl3EQfggcu/content/2301.03937v1.pdf'}
+page_content='  The frequency dependence of real-part optical conductivity is  proportional to ω2, where it can be distinguished from trivial  superconductors or insulators and Dirac chiral edge modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE2T4oBgHgl3EQfggcu/content/2301.03937v1.pdf'}
+page_content='  Then, the driving force of MFs is an electromagnetic wave.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE2T4oBgHgl3EQfggcu/content/2301.03937v1.pdf'}
+page_content='  In superconductors, we cannot use an electric field to sur-  vey superconducting properties since charge conservation is  broken.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE2T4oBgHgl3EQfggcu/content/2301.03937v1.pdf'}
+page_content=' The responses to acoustic waves have been studied  extensively through measurements of ultrasonic attenuation  [26, 27] to investigate superconducting symmetry or measure-  ments of the temperature dependence of a superconducting  gap [28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE2T4oBgHgl3EQfggcu/content/2301.03937v1.pdf'}
+page_content=' Therefore, we can expect to be able to use dynamic  strains for the driving force of MFs on the surfaces of 3D  TSCs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE2T4oBgHgl3EQfggcu/content/2301.03937v1.pdf'}
+page_content='  When MFs have time-reversal symmetry, they do not ap-  pear alone but form Kramers pairs, which are called MKPs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE2T4oBgHgl3EQfggcu/content/2301.03937v1.pdf'}
+page_content='  Due to the time-reversal symmetry, a single MKP is stable  against an electrical external field such as acoustic waves with  lattice strains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE2T4oBgHgl3EQfggcu/content/2301.03937v1.pdf'}
+page_content=' Previously, we derived the general effective  theory for electromagnetic properties of “double” MKPs on a  surface of a 3D TSC [29–31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE2T4oBgHgl3EQfggcu/content/2301.03937v1.pdf'}
+page_content=' Double MKPs can have var-  ious electrical multipole degrees of freedom that are quali-  tatively different from those of ordinary fermions, and it has  been shown that they can respond to static strains [31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE2T4oBgHgl3EQfggcu/content/2301.03937v1.pdf'}
+page_content=' Double  MKPs are always protected by crystalline symmetries in addi-  E  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE2T4oBgHgl3EQfggcu/content/2301.03937v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE2T4oBgHgl3EQfggcu/content/2301.03937v1.pdf'}
+page_content=' This schematic shows that the Majorana spin currents flow  on the surface of the 3D TCSC driven by a dynamic strain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE2T4oBgHgl3EQfggcu/content/2301.03937v1.pdf'}
+page_content=' The sur-  face has four Majorana fermions that form double Majorana Kramers  pairs (MKPs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE2T4oBgHgl3EQfggcu/content/2301.03937v1.pdf'}
+page_content=' Double MKPs have electric quadrupoles coupled to a  dynamic strain, and the spin current of Majorana electric quadrupoles  is generated by a dynamic strain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE2T4oBgHgl3EQfggcu/content/2301.03937v1.pdf'}
+page_content='  tion to time-reversal symmetry;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE2T4oBgHgl3EQfggcu/content/2301.03937v1.pdf'}
+page_content=' thus, TSCs are specifically re-  ferred to as topological crystalline superconductors (TCSCs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE2T4oBgHgl3EQfggcu/content/2301.03937v1.pdf'}
+page_content='  In this letter, we clarify the transport phenomena of double  MKPs driven by spatially nonuniform dynamic strains on the  surface of 3D TCSCs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE2T4oBgHgl3EQfggcu/content/2301.03937v1.pdf'}
+page_content=' We consider the surface of the antiper-  ovskite TCSC Sr3SnO [32], which has double MKPs with two  electrical multipoles on its (001) surface [31, 33].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE2T4oBgHgl3EQfggcu/content/2301.03937v1.pdf'}
+page_content=' We start  with a 4 × 4 Dirac Hamiltonian with respect to the (001) sur-  face point group (PG) symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE2T4oBgHgl3EQfggcu/content/2301.03937v1.pdf'}
+page_content=' Then, we reveal the spin  current of double MKPs, referred to as the “Majorana spin  current”, generated by spatially nonuniform dynamic strains  on the surface (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE2T4oBgHgl3EQfggcu/content/2301.03937v1.pdf'}
+page_content=' 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE2T4oBgHgl3EQfggcu/content/2301.03937v1.pdf'}
+page_content=' When there is no gap in the dis-  persion of double MKPs, we find that the Majorana spin cur-  rent is caused by an intrinsic effect and does not depend on  relaxation times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE2T4oBgHgl3EQfggcu/content/2301.03937v1.pdf'}
+page_content=' Alternatively, for the gapped case induced  by surface magnetization, the Majorana spin current has two  frequency regions that reflect the energy dispersion of double  MKPs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE2T4oBgHgl3EQfggcu/content/2301.03937v1.pdf'}
+page_content=' Our study suggests that the measurement of the Ma-  jorana spin current can be a new probe for Majorana fermion  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE2T4oBgHgl3EQfggcu/content/2301.03937v1.pdf'}
+page_content='03937v1  [cond-mat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE2T4oBgHgl3EQfggcu/content/2301.03937v1.pdf'}
+page_content='supr-con]  10 Jan 2023  22  TABLE I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE2T4oBgHgl3EQfggcu/content/2301.03937v1.pdf'}
+page_content=' Irreducible decomposition of the matrices, momentum, strain, and spin current in the p4m model for the A1u superconducting  pair potential in the bulk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE2T4oBgHgl3EQfggcu/content/2301.03937v1.pdf'}
+page_content=' The matrix is transformed by g as η → D†  gηDg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE2T4oBgHgl3EQfggcu/content/2301.03937v1.pdf'}
+page_content=' (ηi, ηj) denotes (ηi, ηj) → (−ηj, ηi) by transformation of C4z  and (ηi, ηj) → (−ηi, ηj) by transformation of σ(xz).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE2T4oBgHgl3EQfggcu/content/2301.03937v1.pdf'}
+page_content=' Electric/magnetic and PHS denote time-reversal even/odd and particle-hole symmetry,  respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE2T4oBgHgl3EQfggcu/content/2301.03937v1.pdf'}
+page_content=' uij(x) = ∂iu j(x) with u the displacement field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE2T4oBgHgl3EQfggcu/content/2301.03937v1.pdf'}
+page_content=' jα  i denotes spin σα/2 current along the ith direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE2T4oBgHgl3EQfggcu/content/2301.03937v1.pdf'}
+page_content='  IR electric w/ PHS electric w/o PHS magnetic w/ PHS magnetic w/o PHS momentum strain  spin current  A1  s0τ0, s0τ3  k2  x + k2  y  uxx + uyy  jx  y − jy  x  A2  s3τ0, s3τ3  uxy − uyx  jx  x + jy  y  B1 s3τ1  s3τ2  k2  x − k2  y  uxx − uyy  jx  y + jy  x  B2 s0τ2  s0τ1  kxky  uxy + uyx  jx  x − jy  y  E  (s1τ1, s2τ1)  (η5, η6)  (η1, η2), (η3, η4)  (kx, ky)  (uxz, uyz)  (jz  y, − jz  x)  observation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE2T4oBgHgl3EQfggcu/content/2301.03937v1.pdf'}
+page_content='  Before we consider the main topic, we discuss the electro-  magnetic properties of MKPs on the surface of a TSC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE2T4oBgHgl3EQfggcu/content/2301.03937v1.pdf'}
+page_content=' When  there is only one MF on the surface, the single MF is strongly  protected by particle-hole symmetry (charge-neutrality con-  straint), and it is extremely stable against any external fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE2T4oBgHgl3EQfggcu/content/2301.03937v1.pdf'}
+page_content='  If a TSC has time-reversal symmetry, then the two MFs form  an MKP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE2T4oBgHgl3EQfggcu/content/2301.03937v1.pdf'}
+page_content=' The single MKP is protected by time-reversal sym-  metry, and hence, it generally responds only to magnetic ex-  ternal fields and remains stable to electrical external fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE2T4oBgHgl3EQfggcu/content/2301.03937v1.pdf'}
+page_content=' On  the other hand, when crystalline symmetries are taken into ac-  count, there can exist double MKPs, and they can respond to  an electrical perturbation that breaks the crystalline symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE2T4oBgHgl3EQfggcu/content/2301.03937v1.pdf'}
+page_content='  The coupling of N MKPs to a spatially uniform external field  can be represented by Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE2T4oBgHgl3EQfggcu/content/2301.03937v1.pdf'}
+page_content=' [31]  ˆHsurf,ex = − ˆOF, ˆO = 1  2  �  d2x  �  ss′  ˆψs(x)(AF)ss′ ˆψs′(x),  (1)  where ˆψs(x) (s = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE2T4oBgHgl3EQfggcu/content/2301.03937v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE2T4oBgHgl3EQfggcu/content/2301.03937v1.pdf'}
+page_content=', 2N) are Majorana field operators and  satisfy ˆψ†  s(x) = ˆψs(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE2T4oBgHgl3EQfggcu/content/2301.03937v1.pdf'}
+page_content=' AF is conjugate to F and should be  an antisymmetric Hermite matrix since Majorana field oper-  ators obey { ˆψs(x), ˆψs′(x′)} = δss′δ2(x − x′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE2T4oBgHgl3EQfggcu/content/2301.03937v1.pdf'}
+page_content=' When AF is the  2 × 2 matrix, with only one MKP, there exists only AF = σy,  which satisfies ΘsurfAFΘ−1  surf = −AF for the time-reversal oper-  ator Θsurf = (−iσyK).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE2T4oBgHgl3EQfggcu/content/2301.03937v1.pdf'}
+page_content=' K is a complex conjugation, and hence,  the MKP couples only to magnetic external fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE2T4oBgHgl3EQfggcu/content/2301.03937v1.pdf'}
+page_content=' On the  other hand, when AF is the 4 × 4 matrix, with double MKPs,  AF, which satisfies ΘsurfAFΘ−1  surf = AF, can be formed, and the  double MKPs can also be coupled to the electrical external  field by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE2T4oBgHgl3EQfggcu/content/2301.03937v1.pdf'}
+page_content=' (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE2T4oBgHgl3EQfggcu/content/2301.03937v1.pdf'}
+page_content=' Double MKPs have various electric multi-  poles depending on each crystalline symmetry of the surface,  i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE2T4oBgHgl3EQfggcu/content/2301.03937v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE2T4oBgHgl3EQfggcu/content/2301.03937v1.pdf'}
+page_content=', wall-paper groups (WGs), and superconducting symme-  tries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE2T4oBgHgl3EQfggcu/content/2301.03937v1.pdf'}
+page_content=' In a previous study [31], we revealed the coupling be-  tween the electric multipoles and the spatially uniform static  strain for each WGs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE2T4oBgHgl3EQfggcu/content/2301.03937v1.pdf'}
+page_content='  We consider the antiperovskite superconductor Sr3SnO as  a concrete model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE2T4oBgHgl3EQfggcu/content/2301.03937v1.pdf'}
+page_content='  Antiperovskite A3BX, A = Ca, Sr, La,  B = Pb, Sn, X = C, N, O with the space group symmetry  Pm¯3m (No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE2T4oBgHgl3EQfggcu/content/2301.03937v1.pdf'}
+page_content='221) [34, 35], has multiple J = 3/2 bands and is  a candidate material for topological crystalline insulators due  to the band inversion of two orbitals [36–38].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE2T4oBgHgl3EQfggcu/content/2301.03937v1.pdf'}
+page_content=' In particular,  Sr3SnO has the potential to become a TSC with hole doping  at temperatures below 5 K [32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE2T4oBgHgl3EQfggcu/content/2301.03937v1.pdf'}
+page_content=' Interestingly, it has been sug-  gested that double MKPs can appear in the (001) surface with  p4m WG symmetry when the superconductor symmetry is in  the A1u representation in bulk with Oh PG symmetry [33].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE2T4oBgHgl3EQfggcu/content/2301.03937v1.pdf'}
+page_content=' In  the following, we consider the double MKPs on the (001) sur-  face in the A1u state of Sr3SnO and derive the Majorana spin  current generated by spatially nonuniform dynamic strains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE2T4oBgHgl3EQfggcu/content/2301.03937v1.pdf'}
+page_content='  To analyze the transport properties of double MKPs, we  utilize the low-energy Dirac model for the (001) surface  of Sr3SnO with p4m WG symmetry, which equals C4v PG  symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE2T4oBgHgl3EQfggcu/content/2301.03937v1.pdf'}
+page_content='  The surface symmetry operations are given by  D{C4z|0} =  −1  √  2(s0τ3 − is3τ3) and D{σ(xz)|0} =  i√  2(s2τ3 − s1τ3)  with Dg being a representation matrix of g ∈ C4v, where siτ j’s  are the product of Pauli matrices acting on the spin, orbital,  and sublattice degrees of freedom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE2T4oBgHgl3EQfggcu/content/2301.03937v1.pdf'}
+page_content=' Then, we obtain the total  symmetric Hamiltonian [30, 31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE2T4oBgHgl3EQfggcu/content/2301.03937v1.pdf'}
+page_content=' (see Table I and Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE2T4oBgHgl3EQfggcu/content/2301.03937v1.pdf'}
+page_content=' I of  Supplemental Material [39]):  ˆH = 1  2  �  k  ˆψ†  kH(k) ˆψk,  H(k) = [v1(η1ky − η2kx) + v2(η3ky − η4kx)],  (2)  where ˆψ†  k = ( ˆψ†  1k, ˆψ†  2k, ˆψ†  3k, ˆψ†  4k) and ˆψk =  t( ˆψ1k, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE2T4oBgHgl3EQfggcu/content/2301.03937v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE2T4oBgHgl3EQfggcu/content/2301.03937v1.pdf'}
+page_content=', ˆψ4k)  are the Majorana creation/annihilation operators, which sat-  isfy ˆψ†  1k = ˆψ2−k, ˆψ†  3k = ˆψ4−k and the subscripts 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE2T4oBgHgl3EQfggcu/content/2301.03937v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE2T4oBgHgl3EQfggcu/content/2301.03937v1.pdf'}
+page_content=', 4 de-  note four MFs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE2T4oBgHgl3EQfggcu/content/2301.03937v1.pdf'}
+page_content='  ηi’s are given by η1  = (1/  √  2)(s1τ0 +  s2τ0), η2  =  (−1/  √  2)(s1τ0 − s2τ0), η3  =  (1/  √  2)(s1τ3 +  s2τ3), η4 = (−1/  √  2)(s1τ3 − s2τ3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE2T4oBgHgl3EQfggcu/content/2301.03937v1.pdf'}
+page_content=' H(k) satisfies DgH(k)D†  g =  H(gk) where a momentum k is transformed to gk under  the action of g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE2T4oBgHgl3EQfggcu/content/2301.03937v1.pdf'}
+page_content='  The spin of MFs is represented by σ =  (σx, σy, σz) ≡ (η5, η6, −s3τ0) where η5 = (1/  √  2)(s1τ2 −  s2τ2), η6  = (−1/  √  2)(s1τ2 + s2τ2), which are coupled to  the magnetization M as M · σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE2T4oBgHgl3EQfggcu/content/2301.03937v1.pdf'}
+page_content='  The time-reversal Θ,  particle-hole C, and chiral Γ symmetries are defined by Θ =  s2τ3K,C = s1τ0K, Γ = s3τ3 where they satisfy ΘH(k)Θ−1 =  H(−k), CH(k)C−1 = −H(−k), ΓH(k)Γ−1 = −H(k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE2T4oBgHgl3EQfggcu/content/2301.03937v1.pdf'}
+page_content=' In our  model, which is the C4v PG symmetric model for the A1u  bulk’s superconducting pair potential with Oh PG symmetry,  the double MKPs have specific electromagnetic multipoles  with particle-hole symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE2T4oBgHgl3EQfggcu/content/2301.03937v1.pdf'}
+page_content=' Table I shows that the double  MKPs have magnetic multipoles that are represented by σ and  electric multipoles that are represented by s3τ1 with B1 and  s0τ2 with B2 representations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE2T4oBgHgl3EQfggcu/content/2301.03937v1.pdf'}
+page_content=' Therefore, as follows, we can  3  define the spin currents jx  y + jy  x and jx  x− jy  y generated by dynam-  ical strains uxx − uyy and uxy + uyx, where uij(x) = ∂iuj(x) and  uj(x) ∝ ei(q·x−ωt) denote the displacement field, respectively  (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE2T4oBgHgl3EQfggcu/content/2301.03937v1.pdf'}
+page_content=' 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE2T4oBgHgl3EQfggcu/content/2301.03937v1.pdf'}
+page_content=' For double MKPs, we can construct the spin current  operator only from surface effective theory, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE2T4oBgHgl3EQfggcu/content/2301.03937v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE2T4oBgHgl3EQfggcu/content/2301.03937v1.pdf'}
+page_content=', the spin cur-  rent is defined by jα  i ≡ 1  2{σα, ∂H(k)/∂ki} (i = x, y), which sat-  isfies particle-hole symmetry C jα  i C−1 = − jα  i .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE2T4oBgHgl3EQfggcu/content/2301.03937v1.pdf'}
+page_content=' Then, we obtain  ˆjx  y(q) + ˆjy  x(q) = 1  2  �  k ˆψ†  k− q  2 [−v2s3τ1] ˆψk+ q  2 and ˆjx  x(q) − ˆjy  y(q) =  1  2  �  k ˆψ†  k− q  2 [v1s0τ2] ˆψk+ q  2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE2T4oBgHgl3EQfggcu/content/2301.03937v1.pdf'}
+page_content='  Here, we define the operators, which are represented  by  ˆO(1)(q)  ≡  1  2  �  k ˆψ†  k− q  2 (ρ1s3τ1) ˆψk+ q  2  and  ˆO(2)(q)  ≡  1  2  �  k ˆψ†  k− q  2 (ρ2s0τ2) ˆψk+ q  2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE2T4oBgHgl3EQfggcu/content/2301.03937v1.pdf'}
+page_content=' Since ˆO(1) and uxx − uyy, ˆO(2) and  uxy +uyx share the same representation, the couplings between  double MKPs and the dynamical strains are represented by  ˆHsurf,ex = −  �  ˆO(1)(q)[uxx(q, ω) − uyy(q, ω)]  + ˆO(2)(q)[uxy(q, ω) + uyx(q, ω)]  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE2T4oBgHgl3EQfggcu/content/2301.03937v1.pdf'}
+page_content='  (3)  Then, we calculate the linear response of the spin currents  ˆjx  y + ˆjy  x and ˆjx  x − ˆjy  y to the dynamic strains given by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE2T4oBgHgl3EQfggcu/content/2301.03937v1.pdf'}
+page_content=' (3):  ⟨ˆjx  y + ˆjy  x⟩(q, ω) ≡ K1(q, ω)[uxx(q, ω) − uyy(q, ω)]  + K2(q, ω)[uxy(q, ω) + uyx(q, ω)],  (4)  ⟨ˆjx  x − ˆjy  y⟩(q, ω) ≡ K′  1(q, ω)[uxx(q, ω) − uyy(q, ω)]  + K′  2(q, ω)[uxy(q, ω) + uyx(q, ω)],  (5)  where the response function Ki(q, ω) is given by Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE2T4oBgHgl3EQfggcu/content/2301.03937v1.pdf'}
+page_content=' [40]  Ki(q, ω) ≡ i  � ∞  0  dtei(ω+iδ)t ⟨[ˆjx  y(q, t) + ˆjy  x(q, t), ˆO(i)(−q, 0)]⟩,  (6)  where ˆA(t) = ei ˆHt ˆAe−i ˆHt and δ → +0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE2T4oBgHgl3EQfggcu/content/2301.03937v1.pdf'}
+page_content=' K′  i (q, ω) is also de-  fined by replacing ˆjx  y + ˆjy  x with ˆjx  x − ˆjy  y in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE2T4oBgHgl3EQfggcu/content/2301.03937v1.pdf'}
+page_content='  (6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE2T4oBgHgl3EQfggcu/content/2301.03937v1.pdf'}
+page_content='  Note  that the chemical potential µ for MFs is equal to 0 due to  particle-hole symmetry in superconducting states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE2T4oBgHgl3EQfggcu/content/2301.03937v1.pdf'}
+page_content=' Addition-  ally, ⟨· · · ⟩ = tr[e− ˆH/T · · · ]/tr[e− ˆH/T].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE2T4oBgHgl3EQfggcu/content/2301.03937v1.pdf'}
+page_content=' If there are no applied  external fields, K1(q, ω) and K′  2(q, ω) then only take finite val-  ues since the spin currents and dynamic strains share the same  representation of C4v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE2T4oBgHgl3EQfggcu/content/2301.03937v1.pdf'}
+page_content=' Here, we assume that the wavenumber  q and the frequency ω are smaller than the mean free path l and  relaxation time τ of the MFs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE2T4oBgHgl3EQfggcu/content/2301.03937v1.pdf'}
+page_content=' These conditions are represented  by q ≪ l−1, ω ≪ τ−1 = 2γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE2T4oBgHgl3EQfggcu/content/2301.03937v1.pdf'}
+page_content=' We expand the response function  for ω and consider the nonequilibrium part: Ki(ω) − Ki(0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE2T4oBgHgl3EQfggcu/content/2301.03937v1.pdf'}
+page_content='  (the details are shown in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE2T4oBgHgl3EQfggcu/content/2301.03937v1.pdf'}
+page_content=' II of the Supplemental Material  [39]):  K1(ω) − K1(0) ≃ iω  8π2  ρ1  v1  ln  �(v1 − v2)2  (v1 + v2)2  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE2T4oBgHgl3EQfggcu/content/2301.03937v1.pdf'}
+page_content='  (7)  One can see that K1(ω)− K1(0) is independent of the damping  parameter γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE2T4oBgHgl3EQfggcu/content/2301.03937v1.pdf'}
+page_content=' Note that K′  2(ω) − K′  2(0) =  v1ρ2  −v2ρ1 (K1(ω) − K1(0))  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE2T4oBgHgl3EQfggcu/content/2301.03937v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE2T4oBgHgl3EQfggcu/content/2301.03937v1.pdf'}
+page_content=' The energy dispersion of (left) Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE2T4oBgHgl3EQfggcu/content/2301.03937v1.pdf'}
+page_content=' (2) and (right) Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE2T4oBgHgl3EQfggcu/content/2301.03937v1.pdf'}
+page_content=' (8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE2T4oBgHgl3EQfggcu/content/2301.03937v1.pdf'}
+page_content='  The energy gap opened by the magnetization M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE2T4oBgHgl3EQfggcu/content/2301.03937v1.pdf'}
+page_content='  because of chiral symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE2T4oBgHgl3EQfggcu/content/2301.03937v1.pdf'}
+page_content=' The details are shown in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE2T4oBgHgl3EQfggcu/content/2301.03937v1.pdf'}
+page_content=' II  of the Supplemental Material [39].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE2T4oBgHgl3EQfggcu/content/2301.03937v1.pdf'}
+page_content='  Next, we consider the case in which the dispersion of dou-  ble MKPs is gapped.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE2T4oBgHgl3EQfggcu/content/2301.03937v1.pdf'}
+page_content=' This situation can be realized, for ex-  ample, by attaching ferromagnets on the surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE2T4oBgHgl3EQfggcu/content/2301.03937v1.pdf'}
+page_content=' Then, the  effective Hamiltonian is given by  ˜H(k) = H(k) + ˜Mzσz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE2T4oBgHgl3EQfggcu/content/2301.03937v1.pdf'}
+page_content='  (8)  The second term in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE2T4oBgHgl3EQfggcu/content/2301.03937v1.pdf'}
+page_content=' (8) is the Zeeman term reflected  by coupling between the spin moment of double MKPs and  the magnetization of ferromagnets, where we define Mz ≡  − ˜Mz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE2T4oBgHgl3EQfggcu/content/2301.03937v1.pdf'}
+page_content='  Then, the energy dispersion is given by Eτ=±,± =  ±  �  M2z + k2(v1 + τv2)2 in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE2T4oBgHgl3EQfggcu/content/2301.03937v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE2T4oBgHgl3EQfggcu/content/2301.03937v1.pdf'}
+page_content=' The applied magnetization  opens an energy gap in the dispersion of double MKPs and  lowers the symmetry from C4v to C4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE2T4oBgHgl3EQfggcu/content/2301.03937v1.pdf'}
+page_content=' Therefore, K2(ω) can  take a finite value since jx  y + jy  x and uxy + uyx share the same  irreducible representation of C4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE2T4oBgHgl3EQfggcu/content/2301.03937v1.pdf'}
+page_content=' The relationship between Bz  and ω, is important;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE2T4oBgHgl3EQfggcu/content/2301.03937v1.pdf'}
+page_content=' hence, we calculate the response function  by using the Lehmann representation when the dispersion of  double MKPs is gapped:  K1(ω) = 1  4  �  n∈occ,m∈unocc  �⟨n| − v2s3τ1|m⟩⟨m|ρ1s3τ1|n⟩  En − Em + ω + iδ  +  �⟨n| − v2s3τ1|m⟩⟨m|ρ1s3τ1|n⟩  En − Em − ω + iδ  �∗ �  ,  (9)  K2(ω) = 1  4  �  n∈occ,m∈unocc  �⟨n| − v2s3τ1|m⟩⟨m|ρ2s0τ2|n⟩  En − Em + ω + iδ  +  �⟨n| − v2s3τ1|m⟩⟨m|ρ2s0τ2|n⟩  En − Em − ω + iδ  �∗ �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE2T4oBgHgl3EQfggcu/content/2301.03937v1.pdf'}
+page_content='  (10)  where δ → +0, |n⟩ and |m⟩ denote occupied states and unoc-  cupied states of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE2T4oBgHgl3EQfggcu/content/2301.03937v1.pdf'}
+page_content=' (8), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE2T4oBgHgl3EQfggcu/content/2301.03937v1.pdf'}
+page_content=' As a result, we obtain  Er,± = ±VM2+ k2(V1 +TV2)2  E  E  M  2|Mz  E一,-  E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE2T4oBgHgl3EQfggcu/content/2301.03937v1.pdf'}
+page_content='  E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE2T4oBgHgl3EQfggcu/content/2301.03937v1.pdf'}
+page_content='4  TABLE II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE2T4oBgHgl3EQfggcu/content/2301.03937v1.pdf'}
+page_content=' Electromagnetic degrees of freedom of the double Majorana Kramers pairs (MKPs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE2T4oBgHgl3EQfggcu/content/2301.03937v1.pdf'}
+page_content=' They emerge on the surface with wallpaper-  group (WG) symmetry, when the bulk pair potential belongs to irrep ∆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE2T4oBgHgl3EQfggcu/content/2301.03937v1.pdf'}
+page_content=' Irreps of double MKPs, magnetic operators, electric operators of  MKPs, which couple to magnetization Mz perpendicular to the surface and strain [uij(x) = ∂iu j(x), where u(x) denotes the displacement field],  are also shown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE2T4oBgHgl3EQfggcu/content/2301.03937v1.pdf'}
+page_content=' The last column denotes that the strain can couple to the double MKPs and the generated spin current by the strain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE2T4oBgHgl3EQfggcu/content/2301.03937v1.pdf'}
+page_content=' We adapt  the definition for WG and irreps by Bilbao Crystallographic Server [41].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE2T4oBgHgl3EQfggcu/content/2301.03937v1.pdf'}
+page_content=' ¯Γi denotes the ith double-valued irrep of the little group on the Γ  point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE2T4oBgHgl3EQfggcu/content/2301.03937v1.pdf'}
+page_content=' 2¯Γi = ¯Γi ⊕ ¯Γi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE2T4oBgHgl3EQfggcu/content/2301.03937v1.pdf'}
+page_content=' The result for P3m1 are the same as that for P31m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE2T4oBgHgl3EQfggcu/content/2301.03937v1.pdf'}
+page_content='  WG  ∆  MKPs  Magnetic  Electric  Magnetization Mz  Strain (Spin current)  uxx − uyy ( jx  y + jy  x)  uxy + uyx (jx  x − jy  y)  p4m  A2  ¯Γ6 ⊕ ¯Γ7  2A2 + E  B1 + B2  gapped  ⃝  ⃝  p31m  A1  2(¯Γ4 ⊕ ¯Γ5)  4A1  2A2  gapless  ×, only coupled to uxy − uyx (jx  x + jy  y)  p4g  B1  ¯M6 ⊕ ¯M7  A2 + B1 + E  A2 + B2  gapped  ×, coupled to uxy − uyx ( jx  x + jy  y)  ⃝  the nonequilibrium part: Ki(ω) − Ki(0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE2T4oBgHgl3EQfggcu/content/2301.03937v1.pdf'}
+page_content='  K1(ω) − K1(0)  =  �����������������  A ω2  Mz  (ω ≪ 2|Mz|),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE2T4oBgHgl3EQfggcu/content/2301.03937v1.pdf'}
+page_content='  B  ������− ω  2 ln  �����  1+ 2kcv1  ω  1− 2kcv1  ω  ����� +  2Mzv2  1  v2  1−v2  2 −  Mzv1ln  ��� v1−v2  v1+v2  ���  2v2  ������ + i Bπv1  2v2 ω  (2|Mz| ≪ ω ≪ x(kc)),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE2T4oBgHgl3EQfggcu/content/2301.03937v1.pdf'}
+page_content='  (11)  K2(ω) − K2(0)  =  �����������  iCω  (ω ≪ 2|Mz|),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE2T4oBgHgl3EQfggcu/content/2301.03937v1.pdf'}
+page_content='  DπMz + iDMzln  ������  1+ 2kcv1  ω  1− 2kcv1  ω  ������  (2|Mz| ≪ ω ≪ x(kc)),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE2T4oBgHgl3EQfggcu/content/2301.03937v1.pdf'}
+page_content='  (12)  A =  2v1v2(3v2  2 − v2  1) − ((v2  1 + v2  2)2 − 4v4  2)ln  ��� v1−v2  v1+v2  ���  32v3  1v3  2  ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE2T4oBgHgl3EQfggcu/content/2301.03937v1.pdf'}
+page_content=' B = v2ρ1  8πv2  1  ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE2T4oBgHgl3EQfggcu/content/2301.03937v1.pdf'}
+page_content='  C = v2ρ2  4π  −2v1v2 + v2ln  ����  v1+v2  v1−v2  ����  8v2  1v2  2  ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE2T4oBgHgl3EQfggcu/content/2301.03937v1.pdf'}
+page_content=' D = −v2ρ2  8π  v2  v3  1 − v1v2  2  ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE2T4oBgHgl3EQfggcu/content/2301.03937v1.pdf'}
+page_content=' (13)  where x(kc) =  �  M2z + k2c(v1 + v2)2 +  �  M2z + k2c(v1 − v2)2 and  kc is a cutoff value (see Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE2T4oBgHgl3EQfggcu/content/2301.03937v1.pdf'}
+page_content=' II of the Supplemental Material  for detailed results [39]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE2T4oBgHgl3EQfggcu/content/2301.03937v1.pdf'}
+page_content=' Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE2T4oBgHgl3EQfggcu/content/2301.03937v1.pdf'}
+page_content=' (11) and (12) show that K1(ω)  and K2(ω) have two frequency regions with different behav-  iors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE2T4oBgHgl3EQfggcu/content/2301.03937v1.pdf'}
+page_content=' The numerical results of Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE2T4oBgHgl3EQfggcu/content/2301.03937v1.pdf'}
+page_content=' (9) and (10) are shown  in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE2T4oBgHgl3EQfggcu/content/2301.03937v1.pdf'}
+page_content=' 3, and they reproduce the approximate analytical solu-  tions obtained in each region well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE2T4oBgHgl3EQfggcu/content/2301.03937v1.pdf'}
+page_content=' Both K1(ω) − K1(0) and  K2(ω) − K2(0) reflect the energy dispersion of the MFs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE2T4oBgHgl3EQfggcu/content/2301.03937v1.pdf'}
+page_content=' In  particular, there are abrupt changes in the values of (a), (b),  and (c) in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE2T4oBgHgl3EQfggcu/content/2301.03937v1.pdf'}
+page_content=' 3, which are the points at ω = 2|Mz|, just when  the frequency is equal to the band gap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE2T4oBgHgl3EQfggcu/content/2301.03937v1.pdf'}
+page_content=' On the other hand,  (d) is the diagram corresponding to the result of the principal  value integration in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE2T4oBgHgl3EQfggcu/content/2301.03937v1.pdf'}
+page_content=' (10), and there is no peak structure  even at ω = 2|Mz| where the density of states becomes finite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE2T4oBgHgl3EQfggcu/content/2301.03937v1.pdf'}
+page_content='  This is because the chemical potential of MFs is exactly zero,  and K2(ω) − K2(0) is the quantity that can take a finite value  when the dispersion of MFs exhibits an energy gap with ap-  plied magnetization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE2T4oBgHgl3EQfggcu/content/2301.03937v1.pdf'}
+page_content=' Hence, the contribution from the Fermi  sea is considered to be dominant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE2T4oBgHgl3EQfggcu/content/2301.03937v1.pdf'}
+page_content='  In this letter, we show that the Majorana spin current  generated by spatially nonuniform dynamic strain flows on  the (001) surface of the 3D time-reversal-invariance TCSC  Sr3SnO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE2T4oBgHgl3EQfggcu/content/2301.03937v1.pdf'}
+page_content=' For the gapless case, the spin current does not show  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE2T4oBgHgl3EQfggcu/content/2301.03937v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE2T4oBgHgl3EQfggcu/content/2301.03937v1.pdf'}
+page_content=' The numerical results of Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE2T4oBgHgl3EQfggcu/content/2301.03937v1.pdf'}
+page_content=' (9) and (10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE2T4oBgHgl3EQfggcu/content/2301.03937v1.pdf'}
+page_content=' (a) and (b)  show the amplitudes of the real and imaginary parts of K1(ω)−K1(0),  respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE2T4oBgHgl3EQfggcu/content/2301.03937v1.pdf'}
+page_content=' (c) and (d) show the real and imaginary parts of K2(ω)−  K2(0), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE2T4oBgHgl3EQfggcu/content/2301.03937v1.pdf'}
+page_content=' Red, blue and green lines denote the cases of  |Mz| = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE2T4oBgHgl3EQfggcu/content/2301.03937v1.pdf'}
+page_content='3, |Mz| = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE2T4oBgHgl3EQfggcu/content/2301.03937v1.pdf'}
+page_content='2 and |Mz| = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE2T4oBgHgl3EQfggcu/content/2301.03937v1.pdf'}
+page_content='1, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE2T4oBgHgl3EQfggcu/content/2301.03937v1.pdf'}
+page_content=' The values of the  three black dotted lines drawn on the horizontal axis in (a), (b), (c),  and (d) denote the size of the energy gap 2|Mz|, and from left to right  are 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE2T4oBgHgl3EQfggcu/content/2301.03937v1.pdf'}
+page_content='2, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE2T4oBgHgl3EQfggcu/content/2301.03937v1.pdf'}
+page_content='4, and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE2T4oBgHgl3EQfggcu/content/2301.03937v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE2T4oBgHgl3EQfggcu/content/2301.03937v1.pdf'}
+page_content=' The parameters are ρ1 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE2T4oBgHgl3EQfggcu/content/2301.03937v1.pdf'}
+page_content='0, ρ2 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE2T4oBgHgl3EQfggcu/content/2301.03937v1.pdf'}
+page_content='0, v1 =  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE2T4oBgHgl3EQfggcu/content/2301.03937v1.pdf'}
+page_content='3, v2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE2T4oBgHgl3EQfggcu/content/2301.03937v1.pdf'}
+page_content='2, kc = 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE2T4oBgHgl3EQfggcu/content/2301.03937v1.pdf'}
+page_content='  damping dependence due to the linear dispersion of the dou-  ble MKPs and the fact that their chemical potential is exactly  zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE2T4oBgHgl3EQfggcu/content/2301.03937v1.pdf'}
+page_content=' In contrast, for the gapped case, the spin currents re-  flect the energy dispersion of the MFs, and they have a unique  frequency dependence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE2T4oBgHgl3EQfggcu/content/2301.03937v1.pdf'}
+page_content=' We emphasize again that the electro-  magnetic multipoles formed by double MKPs determine the  properties of the spin current responses to dynamic strains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE2T4oBgHgl3EQfggcu/content/2301.03937v1.pdf'}
+page_content='  To see this, we show the results of the previous study [31] in  Table II (See the Supplemental material of Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE2T4oBgHgl3EQfggcu/content/2301.03937v1.pdf'}
+page_content=' [31] for the  complete version).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE2T4oBgHgl3EQfggcu/content/2301.03937v1.pdf'}
+page_content=' This table shows the electromagnetic de-  grees of freedom of double MKPs for several WGs, irreps of  the pair potential, and double MKPs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE2T4oBgHgl3EQfggcu/content/2301.03937v1.pdf'}
+page_content=' The case of p4m corre-  sponds to the present model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE2T4oBgHgl3EQfggcu/content/2301.03937v1.pdf'}
+page_content=' In the case of p4m and p4g, the  applied magnetization Mz induces the energy gap of double  MKPs;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE2T4oBgHgl3EQfggcu/content/2301.03937v1.pdf'}
+page_content=' however, in the case of p31m, the double MKPs still  remain gapless.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE2T4oBgHgl3EQfggcu/content/2301.03937v1.pdf'}
+page_content=' In addition, p4m and p4g are the same gapped  case, but in the case of p4g, the double MKPs do not couple to  (b)  a  Re[Ki(w) - Ki(O))  Im[Ki() - Ki(O)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE2T4oBgHgl3EQfggcu/content/2301.03937v1.pdf'}
+page_content='3  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE2T4oBgHgl3EQfggcu/content/2301.03937v1.pdf'}
+page_content='08  Mz = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE2T4oBgHgl3EQfggcu/content/2301.03937v1.pdf'}
+page_content='3  M= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE2T4oBgHgl3EQfggcu/content/2301.03937v1.pdf'}
+page_content='2  <Mz = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE2T4oBgHgl3EQfggcu/content/2301.03937v1.pdf'}
+page_content='1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE2T4oBgHgl3EQfggcu/content/2301.03937v1.pdf'}
+page_content='3  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE2T4oBgHgl3EQfggcu/content/2301.03937v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE2T4oBgHgl3EQfggcu/content/2301.03937v1.pdf'}
+page_content='1  c(kc)  0  0  3  3  3  3  (d)  (c)  Re[K2(w) - K2(O))  Im[K2(w) - K2(O))  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE2T4oBgHgl3EQfggcu/content/2301.03937v1.pdf'}
+page_content='1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE2T4oBgHgl3EQfggcu/content/2301.03937v1.pdf'}
+page_content='1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE2T4oBgHgl3EQfggcu/content/2301.03937v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE2T4oBgHgl3EQfggcu/content/2301.03937v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE2T4oBgHgl3EQfggcu/content/2301.03937v1.pdf'}
+page_content='3  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE2T4oBgHgl3EQfggcu/content/2301.03937v1.pdf'}
+page_content='3  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE2T4oBgHgl3EQfggcu/content/2301.03937v1.pdf'}
+page_content='09  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE2T4oBgHgl3EQfggcu/content/2301.03937v1.pdf'}
+page_content='05  0  0  3  3  3  35  the strain uxx − uyy but couple to uxy − uyx owing to the protec-  tion of glide-plane symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE2T4oBgHgl3EQfggcu/content/2301.03937v1.pdf'}
+page_content=' We have specifically shown the  application of the general effective theory for the electromag-  netic properties of surface double MKPs derived from system  symmetries, such as crystalline and superconducting symme-  tries, and the new dynamics of MFs characterized by these  symmetries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE2T4oBgHgl3EQfggcu/content/2301.03937v1.pdf'}
+page_content=' We believe that our work accelerates the study of  the coupling of given external fields with MFs and the trans-  port phenomena arising therefrom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE2T4oBgHgl3EQfggcu/content/2301.03937v1.pdf'}
+page_content='  As a detection method for AC spin currents, spin wave res-  onance can be observed by injecting spin currents into ferro-  magnetic metals [42, 43] and the rectifying effect of magne-  tostriction [44].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE2T4oBgHgl3EQfggcu/content/2301.03937v1.pdf'}
+page_content=' The dynamic strains uxx(r, t) and uxy(r, t) can  be realized by a Rayleigh wave and a Love wave propagating  in the x-direction, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE2T4oBgHgl3EQfggcu/content/2301.03937v1.pdf'}
+page_content=' In relation to the experiment,  parameters such as ρ1, ρ2, v1, v2 can be obtained by projecting  bulk operators onto the surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE2T4oBgHgl3EQfggcu/content/2301.03937v1.pdf'}
+page_content=' This study will be published  in a future paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE2T4oBgHgl3EQfggcu/content/2301.03937v1.pdf'}
+page_content='  The authors gratefully acknowledge M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE2T4oBgHgl3EQfggcu/content/2301.03937v1.pdf'}
+page_content=' Matsuo, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE2T4oBgHgl3EQfggcu/content/2301.03937v1.pdf'}
+page_content=' Nozaki,  J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE2T4oBgHgl3EQfggcu/content/2301.03937v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE2T4oBgHgl3EQfggcu/content/2301.03937v1.pdf'}
+page_content=' Nakane, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE2T4oBgHgl3EQfggcu/content/2301.03937v1.pdf'}
+page_content=' Imai, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE2T4oBgHgl3EQfggcu/content/2301.03937v1.pdf'}
+page_content=' Yamaguchi, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE2T4oBgHgl3EQfggcu/content/2301.03937v1.pdf'}
+page_content=' Matsushita, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE2T4oBgHgl3EQfggcu/content/2301.03937v1.pdf'}
+page_content='  Kikuchi, and H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE2T4oBgHgl3EQfggcu/content/2301.03937v1.pdf'}
+page_content=' Kohno for valuable discussions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE2T4oBgHgl3EQfggcu/content/2301.03937v1.pdf'}
+page_content='  This  work was supported by JSPS KAKENHI for Grants (No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE2T4oBgHgl3EQfggcu/content/2301.03937v1.pdf'}
+page_content='  JP20K03835) and the Sumitomo Foundation (190228).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE2T4oBgHgl3EQfggcu/content/2301.03937v1.pdf'}
+page_content=' Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE2T4oBgHgl3EQfggcu/content/2301.03937v1.pdf'}
+page_content='Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE2T4oBgHgl3EQfggcu/content/2301.03937v1.pdf'}
+page_content='  is supported by Grant-in-Aid for JSPS Fellows Grant No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE2T4oBgHgl3EQfggcu/content/2301.03937v1.pdf'}
+page_content='  22J14452.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE2T4oBgHgl3EQfggcu/content/2301.03937v1.pdf'}
+page_content='  This work was financially supported by JST  SPRING, Grant Number JPMJSP2125.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE2T4oBgHgl3EQfggcu/content/2301.03937v1.pdf'}
+page_content='  The author Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE2T4oBgHgl3EQfggcu/content/2301.03937v1.pdf'}
+page_content='Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE2T4oBgHgl3EQfggcu/content/2301.03937v1.pdf'}
+page_content='  would like to take this opportunity to thank the “Interdisci-  plinary Frontier Next-Generation Researcher Program of the  Tokai Higher Education and Research System.”  [1] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtE2T4oBgHgl3EQfggcu/content/2301.03937v1.pdf'}
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+Prepared for submission to JHEP
+SUp8q-QGR: Emergence of Gravity in an Infinitely
+Divisible Quantum Universe
+Houri Ziaeepoura,b
+aInstitut UTINAM, CNRS UMR 6213, Observatoire de Besan¸con, Universit´e de Franche Compt´e,
+41 bis ave. de l’Observatoire, BP 1615, 25010 Besan¸con, France
+bMullard Space Science Laboratory, University College London, Holmbury St. Mary, GU5 6NT,
+Dorking, UK
+E-mail: houriziaeepour@gmail.com
+Abstract:
+SUp8q-QGR is a foundationally quantum approach to gravity. It assumes
+that Hilbert spaces of the Universe as a whole and its subsystems represent the symme-
+try group SUp8q. The Universe is divided to infinite number of subsystems based on an
+arbitrary finite rank symmetry group G, which arises due to quantum fluctuations and
+clustering of states. After selection of two arbitrary subsystems as clock and reference
+observer, subsystems acquire a relative dynamics, and gravity emerges as a SUp8q Yang-
+Mills quantum field theory, defined on the (3+1)-dimensional parameter space of Hilbert
+spaces of the subsystems. At classical limit, this observer is perceived as the spacetime.
+Consequently,SUp8q-QGR explains both the dimension of and signature of spacetime. As
+a Yang-Mills model SUp8q-QGRis renormalizable and despite prediction of a spin-1 field
+for gravity at quantum level, at low energies it is perceived as classical Einstein model. The
+aim of the present work is to make the foundation of sqgr more mathematically rigorous
+and fill the gaps in the construction of the model reported in earlier works. In particular,
+we show that the global SUp8q symmetry manifests itself through the entanglement of ev-
+ery subsystem with the rest of the Universe. Moreover, we demonstrate irrelevance of the
+geometry of parameter space, which can be gauged out by a SUp8q gauge transformation
+up to an irrelevant constant. Therefore, SUp8q-QGR deviates from gauge-gravity duality
+models, because the perceived classical spacetime is neither quantized, nor considered to
+be non-commutative. On the other hand, using quantum uncertainty relations, we demon-
+strate that the classical spacetime and its perceived geometry present the average path of
+the ensemble of quantum states of subsystems in their parameter space. We also briefly
+discuss SUp8q-QGR specific models for dark energy.
+arXiv:2301.02813v1  [gr-qc]  7 Jan 2023
+
+Contents
+1
+Introduction
+1
+2
+A quantum Universe with infinite independent observables
+4
+2.1
+Rationale for SUp8q-QGR
+4
+2.1.1
+Symmetry as a foundational concept in quantum mechanics
+5
+2.1.2
+Universe as a composite quantum system
+6
+2.1.3
+Infinite Universe
+6
+2.1.4
+Nature of spacetime
+7
+2.2
+Axioms and construction of SUp8q-QGR
+7
+2.2.1
+Symmetry and Hilbert space
+8
+2.2.2
+Quantization
+10
+2.2.3
+Interpretation and comparison with Virasoro algebra
+12
+3
+Division to subsystems
+12
+3.1
+Infinitely divisible Universe
+13
+3.1.1
+Emergence of internal symmetries and locality in SUp8q-QGR
+13
+3.2
+Symmetries and entanglement of subsystems
+16
+3.2.1
+Global entanglement
+17
+3.2.2
+Consequences of the global entanglement
+18
+3.2.3
+Observability of entanglement
+18
+4
+Parameter space and algebra of subsystems
+19
+4.1
+Emergence of an area/size scale as a measurable
+19
+4.2
+Time
+21
+4.2.1
+General properties of a quantum clock and time parameter
+21
+4.2.2
+Quantum clock and its associated time parameter
+22
+4.2.3
+Hamiltonian associated to a clock
+23
+4.3
+Full parameter space of subsystems
+23
+4.3.1
+The sup8q algebras of subsystems and their commutation relations
+24
+– i –
+
+4.3.2
+About quantum uncertainty, nonlocality, and topological effects
+25
+4.4
+Parameter transformation
+26
+4.4.1
+Poincar´e symmetry
+27
+4.4.2
+Curvature of the parameter space
+27
+5
+Classical curved space and its relationship with quantum states of sub-
+systems
+29
+5.1
+Speed limit from quantum uncertainties
+29
+5.2
+Effective path in the parameter space
+30
+5.2.1
+Path in the full parameter space
+32
+5.2.2
+Proper (equal-time) distance and causality
+33
+5.3
+Interpretation of affine separation and its curved geometry
+34
+5.3.1
+Relation with coherence and distinguishability
+35
+6
+Evolution
+36
+6.1
+The whole Universe
+37
+6.1.1
+Formal description of dynamics
+37
+6.1.2
+A Yang-Mills whole Universe
+38
+6.1.3
+A topological Universe
+40
+6.2
+Evolution of subsystems
+41
+6.2.1
+Possibility of gravitational P, C, and CP violation
+43
+6.2.2
+Is there a conflict between a SUp8q Yang-Mills gauge theory as quan-
+tum gravity and present observations ?
+44
+6.3
+Classical limit
+45
+6.4
+Cosmological constant
+46
+6.4.1
+Static SUp8q field
+47
+6.4.2
+Θ´vacuum
+47
+6.4.3
+Manifestation of global entanglement
+49
+7
+Outlines and future perspectives
+49
+A Properties of SUp8q tensor products
+51
+A.1 About definition of symmetries of a quantum system
+52
+– ii –
+
+B About quantization
+52
+B.1
+Dual of generators in spherical harmonic representation of sup8q
+53
+B.2
+Dual of continuous parameters in spherical harmonic representation of sup8q 54
+C Criteria for definition of quantum subsystems
+54
+C.1 Symmetries and Hilbert space of subsystems
+55
+C.2 Conditions for division induced entanglement of subsystems
+55
+D Relation between reparameterization and diffeomorphism
+56
+E Coordinate independent definition of curvatures
+56
+F Expansion of density matrix and its variation
+57
+G About holographic principle and black holes
+58
+G.1 Comparison with gravity as entanglement and ADS/CFT
+59
+1
+Introduction
+So far most Quantum GRavity (QGR) candidates have been based on the classical models
+quantized in one way or another. Moreover, geometry of spacetime, sometimes extended to
+matter sector, and its quantization have had a prominent role in the construction of these
+theories. For instance, Kaluza-Klein model [1, 2] and its successors: string theory [3, 4],
+Loop Quantum Gravity (LQG) [5, 6] and its extension to Group Field Theory (GFT) [7, 8],
+and related models such as random matrices [10, 169] and tensor product models [11, 12]
+are all geometry inclined and attempt to describe gravity by quantizing D-dimensional
+spaces, where D can be 2, 3, or higher, specially in models in which matter is also treated
+geometrically.
+Of course, the reason behind this trend is the close relation of classical
+gravity with the geometry of spacetime in the Einstein general relativity.
+In geometrical approaches to QGR a classical model do exists for ℏ Ñ 0, because their
+departure point is essentially a classical model. However, for various reasons a Universe
+in which ℏ Ñ 0 cannot correspond to ours, even as an asymptotic or approximate descrip-
+tion. The reason is the innumerable experiments which have shown that matter behaves
+quantum mechanically, even at macroscopic scales. On the other hand, inconsistency of a
+classical gravity with a quantum matter is well documented [13–16]1. Thus, to construct
+1Nonetheless, skepticism still persists [17, 18].
+– 1 –
+
+a quantum model of gravity and spacetime, it seems more logical to treat the Universe as
+an intrinsically quantum system from the beginning, rather than starting with a classical
+theory and subsequently quantize it.
+Another point, which is often overlooked by the traditional approaches to quantum gravity
+and quantum cosmology, is the fact that quantum systems need an environment for de-
+coherence [19] and formation of a system, here the Universe, which seems decohered and
+random at intermediate scales. We remind that at microscopic/UV scales the Universe is
+fully quantum mechanical. Furthermore, due to the expansion of the Universe superhorizon
+scales, which are at present causally disconnected, were previously interacting and might
+have kept their quantum coherence even now [20, 21]. Decoherence of these quantum modes
+was necessarily due to an environment [22, 23], presumably, provided by the rest of the
+Universe. In other words, in a quantum model of gravity and spacetime it is imperative to
+consider the Universe as an ensemble of interacting, either coherently or causally, quantum
+subsystems [24–26]. In addition to an environment, frame-independent definition of quan-
+tum dynamics needs a quantum reference (observer) [27–29] and a quantum clock [30, 31].
+These requirements highlight the crucial role of quantum subsystems in a fundamentally
+quantum approach to the Universe and to a universal interaction such as gravity between
+its its parts. Of course, a Quantum Field Theory (QFT) by definition includes infinite
+number of subsystems - particles. But in standard formulation of QFTs the presence of
+an observer (a reference) and a quantum clock is implicit. Moreover, their quantum cor-
+relation and effects, for instance on the algebra of observables [32], decoherence through a
+change of reference frame [28], and dependence of entanglement and superposition on the
+choice of quantum reference frame [29] are not explicit.
+Quantum approaches to QGR are relatively recent additions to the jungle of QGR candi-
+dates. In absence of a hint from classical gravity, quantum field theory, or experiments,
+this class of models usually have to use new priors as their departure point. Concepts
+and conjectures used for this purpose include: the concept of locality and geometrization
+of quantum mechanics, considered to be crucial for describing black holes and their puz-
+zles [33, 34]; holography principle, which conjectures that maximum possible information
+capacity of a quantum system is proportional to the area of its null (lightlike) boundary
+rather than its volume [35–37]; and finally, the conjecture of duality between quantum
+gravity in asymptotically Anti-de-Sitter (AdS) bulk and a Conformal Field Theory (CFT)
+on the boundary. For a brief review of these proposals and their comparison with SUp8q-
+QGR see [38].
+In [39], we proposed SUp8q Quantum Gravity or in short SUp8q-QGR as a model for a
+quantum Universe. It is based on a few well motivated axioms. In particular, it assumes
+that the Universe has infinite number of independent and mutually commuting observ-
+ables. We described some of the properties of such a quantum Universe as a single system.
+Notably, we showed that it is static and topological. We also demonstrated that it becomes
+dynamics after its division to subsystems. Specifically, the definition of an operational time
+as a parameter describing variation of subsystems states with respect to that of an arbitrary
+– 2 –
+
+subsystem called a clock - `a la Page & Wootter [30] or similar procedures [31] - induces a
+dynamics. Moreover, in [38] we showed that gravity emerges as a universal quantum force
+having the form of a SUp8q Yang-Mills gauge theory defined on the (3+1)-dimensional
+parameter space of representation of SUp8q symmetry by subsystems. Therefore, in con-
+trast to other QGR models, SUp8q-QGR explains the origin of observed dimensionality
+of the classical spacetime. It is also shown that at low energies, that is when the quantum
+details of SUp8q Yang-Mills is not detectable, the action of the gravity sector of the model
+looks like that of the Einstein-Hilbert gravity.
+A discriminating aspect of SUp8q-QGR is the strict classical spacetime. In fact, there
+is no background spacetime in the model. What is called the spacetime is the parameter
+space characterizing representation of SUp8q symmetry by the Hilbert space of subsys-
+tems. For this reason quantization of spacetime is meaningless - in the same manner that
+quantization of eigen values in quantum mechanics is meaningless. In addition, we showed
+that the classical curved spacetime presents the average path of quantum subsystems in
+this parameter space.
+Regarding the above interpretation of spacetime, it is useful to remind that Einstein’s
+opinion was that without the graviton field gµνpxq, that is in absence of a gravity source,
+the spacetime itself has no physical meaning, see e.g. [40] (and references therein) for a
+concise review of the history of physical interpretation of spacetime. Einstein motivation
+for his conclusion was the failure of all experiments to detect a medium - the ether - as
+the empty space. However, general relativity does not provide any explanation for why
+and how a non-existent entity emerges in presence of matter. Therefore, the interpretation
+of coordinates of what we call spacetime as provides the missing explanation in general
+relativity. Indeed, as Einstein suggested, in SUp8q-QGR spacetime does not have any
+physical meaning without Yang-Mills field and matter. This is analogous to charge or spin,
+which cannot exist without matter and interaction mediators.
+To further advance preliminary results reported in [39] and [38], in the present work we will
+investigate in more details structure and properties of states of subsystems, parameter space
+of the representations of their SUp8q symmetry, and Lagrangian of SUp8q-QGR model.
+We will discuss in more details fundamentally quantum nature of the model and will extend
+the discussion about the parameter space and its geometry. In particular, we will prove
+that its geometry is irrelevant and can be transferred to that a flat space by a SUp8q gauge
+transformation. This property is crucial for the model, otherwise the number of degrees of
+freedom of gravity sector would be doubled and one would have to introduce new rules, for
+example Einstein equation, for fixing the geometry of parameter space. Another character-
+istic of SUp8q-QGR, which was not discussed in the previous works, is the entanglement
+of subsystems of the Universe, induced by the assumption of SUp8q symmetry at both
+local and Universe-wide scales. Additionally, we will show that symmetries at the two
+scales are not independent, but intertwined. The importance of entanglement for QGR
+is recently explored in various approaches [41–43], specially those inspired by quantum
+information[29, 32]. In SUp8q-QGR the omnipresent entanglement can be considered as
+– 3 –
+
+quantum analogous of gravitational waves memory [44–47]. Emergence of local symmetries
+and invariance (or not) of subsystems under discrete symmetries of the parameter space
+will be discussed more thoroughly. An important subject which is not discussed in the
+previous works on SUp8q-QGR is the possibility of having a constant term in the action,
+in other words a cosmological constant. We will introduce this topic here and will propose
+a summary of potential processes specific to SUp8q-QGR, which may induce a constant
+term in the effective action at classical limit.
+In Sec. 2 we review the rationale for proposing an intrinsically quantum model for grav-
+ity, axioms of SUp8q-QGR, and properties of its symmetry, algebra, Hilbert space, and
+quantization. Algebraic arguments for the emergence of local orthogonal symmetries and
+division of the Universe to subsystems are studied in Sec. 3. We also discuss the crucial
+existence of a permanent entanglement between every subsystem and the rest of the Uni-
+verse and its consequences in this section. Algebra and parameter space of the subsystems
+are discussed in Sec. 4. For this purpose we introduce an abstract quantum clock, as-
+sociate a parameter to the variation of its state, and call this parameter time. Then, we
+explain the full parameter space of subsystems and its symmetries. We also discuss the
+geometry of the parameter space. In Sec. 5 Mandelstam-Tamm uncertainty relation is
+used to introduce a parameter, which we identify as the affine parameter of the perceived
+classical spacetime. It is related to the quantum state of subsystems. Moreover, we proved
+that the signature of classical metric must be negative. These topics were discussed in [39].
+Here we provide more physical insight to its interpretation in conjunction with the global
+entanglement, coherence and causality, Finally, the dynamics and evolution of the Universe
+in SUp8q-QGR is discussed in Sec. 6.1. We also discuss classical limit of the gravitation
+sector and propose a few processes which may introduce a cosmological constant in the
+effective Lagrangian. Outlines and perspective for future research, notably application of
+SUp8q-QGR to phenomena in which quantum gravity may be important are given in Sec.
+7. Appendices A to G provide additional information and detail calculation for the subjects
+discussed in the main text.
+2
+A quantum Universe with infinite independent observables
+In this sections we first review rationale a purely quantum approach to the Universe and
+to quantum gravity. Then, we review axioms of SUp8q-QGR and construction of algebra
+and Hilbert space of the Universe based on them
+2.1
+Rationale for SUp8q-QGR
+Standard quantization methods fail to provide an inclusive formulation including not only
+a consistent and problem-free quantum gravity, but also matter and its non-gravitational
+interactions. This conundrum motivate a radically different approach to quantum gravity.
+Symmetry is one of the mathematically and physically fundamental concepts, which may
+reveal the way forward.
+– 4 –
+
+2.1.1
+Symmetry as a foundational concept in quantum mechanics
+The ideas behind SUp8q-QGR originates from the observation that Hilbert spaces of quan-
+tum systems represent their symmetries. It has led to a reformulation of the axioms of
+quantum mechanics, such that they include and emphasize the central and foundational
+role of symmetries in the mathematical description of quantum systems [26, 48, 49]. A
+short review of this rephrasing of the postulates, which from now on we call Symmetry
+Description of Quantum Mechanics (SDQM), can be found in appendices of [39].
+A consequence of this description is that the vector space of states of a quantum system
+represents its symmetries. Moreover, when the state space is parameterized with respect
+to unbounded continuous variables, such as the spacetime, its L2 integrability - its Hilbert
+space property - and the Born rule can be obtained from the axioms, rather than being
+postulated, as is the case in the standard description of quantum mechanics.
+In this
+description, the quantum coherence is interpreted as a symmetry presenting degeneracies
+of a quantum system’s state with respect to its environment, that is the rest of the Universe.
+Unitary transformations U|UPBrHs, where H is the Hilbert space of the system, and BrHs
+is the space of its (bounded) linear operators, modify the system’s coherence symmetry2
+- presumably by acting on the system or its environment. Of course, a unitary transfor-
+mation of the state can be mathematically considered as a change of the arbitrary Hilbert
+space basis. However, in a physical context such operation corresponds to changes in the
+perception of the physical world. It is why the coherence symmetry is relative not absolute.
+For example, neglecting gravity, a boost transformation of coordinates of a particle does
+not make any physical difference if the particle does not interact with something else. The
+result of a complete breaking of coherence symmetry is a projective measurement [19]. For
+long time the importance of the state coherence symmetry as a resource was overlooked
+in quantum mechanics and quantum information literature. Nonetheless, recently quan-
+tification of this symmetry and its applications as a resource have become a subject of
+interest [50, 51].
+The introduction of symmetry as a foundational concept in the axioms of quantum me-
+chanics is crucial for the Quantum First approaches [52] to the QGR, and more generally
+for modeling a quantum system without using a classical model as the departure point. In-
+deed, according to the axioms of quantum mechanics `a la Dirac [53] and von Neumann [54],
+the Hilbert space is an abstract Banach space, and the postulates do not specify how it
+should be defined or related to physical properties of the system. Nonetheless, in practice
+the Hilbert space is always chosen such that it represent symmetries of the corresponding
+classical Hamiltonian, or Lagrangian in the case of relativistic systems. Therefore, our
+2In quantum information literature what we call coherence symmetry is confusingly called asymmetry,
+emphasizing on the off-diagonal components of density matrix when a system is not completely incoherent.
+By contrast, the first expression emphasizes on the fact that superposition and interference arise due to
+quantum degeneracies, and therefore present a sort of symmetry. For this reason, in [26] the expression
+state symmetry was used. This symmetry is specific to quantum systems and additional to symmetries of
+their Lagrangians, which exist in both classical and quantum systems. Nonetheless, they are related.
+– 5 –
+
+theoretical and observational knowledge about quantum systems justify the description of
+quantum postulates based on symmetries. Appendix A.1 clarifies some issues regarding
+how symmetry of quantum systems are defined.
+2.1.2
+Universe as a composite quantum system
+According to a corollary of SDQM [26] the division of a quantum Universe to subsystems
+is crucial for its consistency. It is important to remind that many concepts in quantum
+mechanics are based on the ability to define and distinguish subsystems. Examples in-
+clude: Entanglement; Definition of a quantum clock and thereby a quantum mechanically
+consistent dynamics; System-environment discrimination and decoherence; Locality of in-
+teractions and quantum POVM measurements, etc. In the SDQM entanglement can be
+interpreted as partial breaking of state (coherence) symmetry of a composite quantum
+system, which makes entangled subsystems indivisible. This means that the Hilbert space
+of the entangled subsystems becomes an irreducible subspace of the tensor product of the
+Hilbert spaces of individual subsystems, which is not a proper subspace of one or the other.
+The concept of subsystems is specially crucial for constructing a quantum Universe3 with
+a universal interaction similar to gravity. In particular, subsystems are necessary for a
+consistent definition and implementation of the concept of locality, which is considered by
+some QGR proposals [33, 34, 41, 42, 56] to be crucial for accommodating black holes and
+explaining the paradox of information loss [57, 58].
+2.1.3
+Infinite Universe
+It is well known that a full description of quantum phenomena needs Quantum Field Theory
+(QFT), which has infinite number of degrees of freedom - in other words, infinite number
+of subsystems/particles. Even the vacuum state of a QFT can be described as a superposi-
+tion with negligibly small amplitude of all possible states [59], including Glauber coherent
+states [60] and condensates of particles. Therefore, we should consider the Universe as
+being composed of infinite number of subsystems. The Hilbert space of such a system is
+necessarily infinite dimensional, and should have infinite number of independent - mutually
+commuting - observables.
+The requirement of infinite number of mutually commuting observables is a crucial differ-
+ence between a quantum system/Universe composed of infinitely many subsystems and a
+system having an observable with a continuous spectrum, such as position, momentum or
+displacement. In both cases the Hilbert space H and space of linear operators BrHs applied
+to it are infinite dimensional. However, in the case of an observable with continuous spec-
+trum, aside the hermitian operator associated to it, other operators in BrHs are considered
+3Considering categorical description (see e.g. [55]) of quantum mechanics, the Universe is the category
+of categories.
+It incorporates all physical objects - including itself - and physical operations (functors)
+applied to them. The purpose of adjective physical here is to emphasize that these entities are not abstract
+mathematical concepts, but have detectable existence.
+– 6 –
+
+to be related to the same observable by a basis/frame transformation. On the other hand,
+tensor product of infinite number of subsystems with infinite dimensional Hilbert spaces
+is again infinite dimensional. Consequently, the discrimination between subsystems and
+their states is blurred. Thus, this property may provide a universal interaction between
+subsystems by mixing their states with those of others.
+2.1.4
+Nature of spacetime
+Observables with continuous spectra are usually related to the spacetime. In particular,
+in QFTs states of the Hilbert space and particles (subsystems) in the Fock space are
+labeled/indexed by the spacetime or its Fourier conjugate - the momentum (but not both).
+This raises the question whether the spacetime by itself, that is without matter, has any
+physical meaning, or rather what we call matter/radiation and spacetime are inseparable
+properties of the same physical reality.
+General relativity does not clarify the nature of spacetime and its difference with matter.
+As we discussed in the Introduction, Einstein considered spacetime as a quantity that
+parameterizes a spin-2 field gµν and matter fields. But, this interpretation does not not
+explain why the geometry of this parameter space, which according to Einstein by itself
+does not have any physical existence, is determined by a physical field. In any case, modern
+approaches to QGR, specially string theory, treat spacetime and matter similarly: Both
+are considered to be fields on the 2D world-sheet of strings. Loop Quantum Gravity (LQG)
+also treats space as an standalone physical entity quantized by triangulation or treated as
+a spin foam. See [38] for an extended discussion of these models and their comparison with
+SUp8q-QGR.
+Irrespective of interpretations and approaches, the undeniable fact is that without mat-
+ter (including a cosmological constant) and with trivial boundary conditions, solution of
+the Einstein equation is a constant metric, which can be also zero - corresponding to no
+spacetime at all. In addition, the choice of a null metric rather than a flat space with
+undetermined volume seems preferable, because there is no experimental evidence for the
+existence of an empty flat Minkowski spacetime at any scale.
+Therefore, we conclude
+that spacetime and matter are inseparable. Indeed, under some assumptions, in particu-
+lar holography principle [35–37], Einstein equation can be obtained and interpreted as an
+equation of state [61].
+2.2
+Axioms and construction of SUp8q-QGR
+With the above conceptual background we take a purely quantum mechanical approach to
+construction of the Universe and its content. For an extended discussion see [39].
+We begin by designing a quantum Universe base on three well motivated assumptions:
+I. Quantum mechanics is valid at all scales and applies to every entity, including the
+Universe as a whole;
+– 7 –
+
+II. Any quantum system is described by its symmetries and its Hilbert space represents
+them;
+III. The Universe has an infinite number of independent degrees of freedom, which are
+associated to mutually commuting observables.
+The last assumption means that the Hilbert space of the Universe HU is infinite dimensional
+and linear operators acting on it represent the group SUp8q. We call the vector space of
+these operators BrHUs.
+The sup8q algebra of BrHUs is defined as the following4:
+rˆLa, ˆLbs “
+iℏ
+cMP
+fc
+ab ˆLc “ iLP fc
+ab ˆLc,
+LP ”
+ℏ
+cMP
+(2.1)
+where operators ˆLα P BrHUs are generators of sup8q and anti-symmetric fc
+ab ” fabc are
+structure coefficients of the algebra. They are proportional to 3j symbols defined for three
+pairs of indices pl, mq, pl1, m1q, pl2, m2q [62]. Notice that in (2.1), operators ˆLα are nor-
+malized such that the r.h.s. explicitly depend on the Planck constant ℏ and Planck mass
+MP ”
+a
+ℏc{GN or equivalently Planck length LP ” ℏ{cMP . If ℏ Ñ 0, or MP Ñ 8,
+the length scale LP Ñ 0. In this case the algebra becomes Abelian and homomorphic to
+ÂNÑ8 Up1q. This is in agreement with symmetry of configuration space of a classical
+system with infinite number of independent observables. Thus, the quantumness of the
+Universe according to this model needs both ℏ ‰ 0 and MP ă 8. Of course at this stage
+there is no need for introducing a mass (or length) scale and the above conclusion may
+seem artificial. We show later how a dimensionful scale arises in the model and becomes
+relevant for the above algebra.
+2.2.1
+Symmetry and Hilbert space
+It is proved that:
+BrHUs – SUp8q – area preserving diffeomorphism DiffpS2q
+(2.2)
+of 2D compact Riemann surfaces S2 (should not be confused with 2D-sphere Sp2q) [62–66].
+The symbol – means homomorphic. Although all representations of SUp8q are locally
+homomorphic, they have different global topology and present globally non-equivalent rep-
+resentations of the group. Thus, class of equivalent representations of SUp8q is isomorphic
+to the class of topologically equivalent S2. Consequently, diffeomorphism of compact sur-
+faces with different genus are homomorphic to non-equivalent representation of SUp8q. As
+a shorthand we call them diffeo-surfaces. In the context of SUp8q-QGR model they cor-
+respond to globally different universes. Moreover, the topology of SUp8q representations
+would be important for nonlocal phenomena, for example when entanglement or black hole
+singularity [41] make part of the parameter space inaccessible, See Sec. 4.1 for more details.
+4In this work, all vector spaces and algebras are defined on complex numbers field C, unless explicitly
+mentioned otherwise.
+– 8 –
+
+Due to (2.2), members of sup8q algebra are parameterized by two angular coordinates
+pθ, φq of the diffeo-surface, and generators ˆLa of the algebra can be expanded with respect
+to spherical harmonic functions [62, 64]:
+ˆLlmpθ, φq “ iℏ
+ˆ BYlm
+B cos θ
+B
+Bφ ´ BYlm
+Bφ
+B
+B cos θ
+˙
+“ iℏ
+b
+|gp2q|ϵµνpBµYlmqBν,
+µ, ν P tθ, φu,
+l ě 0, ´ l ď m ď l
+(2.3)
+ˆLlmYl1m1 “ ´iℏtYlm, Yl1m1u “ ´iℏfl”m”
+lm,l1m1Yl”m”
+(2.4)
+tf, gu ”
+Bf
+B cos θ
+Bg
+Bφ ´ Bf
+Bφ
+Bg
+B cos θ
+,
+@ f, g
+(2.5)
+For the sake of simplicity from now on we will absorb the numerical factor
+ℏ
+cMP in (2.1)
+in the normalization of operators, unless when its presence is important for the discussion.
+Operation t , u in (2.5) is the Poisson algebra with respect to local coordinates of the
+diffeo-surface. The function |gp2q is the determinant of 2D metric of the diffeo-surface.
+Equalities in (2.4) show that spherical harmonic functions have the same algebra as sup8q
+under Poisson brackets operation. This property will be useful for understanding various
+properties of SUp8q-QGR.
+Cartan decomposition of SUp8q allows to write ˆLlm as a tensor product of Pauli matri-
+ces [62, 67]:
+SUp8q –
+8
+â
+SUp2q
+(2.6)
+ˆLlm “ R
+ÿ
+iα“1, 2, 3,
+α“1,¨¨¨ ,l
+apmq
+i1,¨¨¨ilσi1 ¨ ¨ ¨ σil,
+l ě 0, ´ l ď m ď l
+(2.7)
+where σiα’s are Pauli matrices [62] and R is a normalization constant. Coefficients apmq
+are determined from expansion of spherical harmonic functions with respect to spherical
+description of Cartesian coordinates [62]. We have omitted tensor product symbols be-
+tween the factors, because one can also consider σ’s as N Ñ 8 representation of Pauli
+matrices [62].
+In this case, their products would be the ordinary matrix product.
+As
+the fundamental representation of SUp2q corresponds to l “ 1{2, this relation shows that
+decomposition (2.7) with half-integer pl, mq is also a representation of SUp8q.
+The importance of SUp2q – SOp3q symmetry in QGR models is discussed in [38]. The
+obvious reason is the fact that it is the symmetry of the Euclidean Rp3q space - the classical
+physical space. However, in contrast to other QGR proposals, the decomposition (2.6) does
+not have any special physical interpretation, except that SUp2q is the smallest non-Abelian
+member of SU groups. In Appendix A we show that SUp8q can be expressed as tensor
+product of any SUpKq, K ă 8.
+It is possible - at least formally - to define generators only with respect to continuous pθ, φq
+– 9 –
+
+variables by summing over indices pl, mq (See Appendix D of [39] for more details):
+rˆLpθ1, φ1q, ˆLpθ2, φ2qs “
+ż
+dΩ3 fppθ1, φ1q, pθ2, φ2q; pθ3, φ3qq ˆLpθ3, φ3q
+(2.8)
+fppθ, φq, pθ1, φ1q; pθ”, φ”qq ”
+ÿ
+lm,l1m1,l”m”
+Y ˚
+lmpθ, φqY ˚
+l1m1pθ1, φ1qYl”m”pθ”, φ2qfl”m”
+lm,l1m1 (2.9)
+ˆLpθ, φq ”
+ÿ
+l,m
+Y ˚
+lm ˆLlm,
+dΩ ” dφ dpcos θq
+b
+|gp2q|
+(2.10)
+Thus, although the number of generators of SUp8q group ˆLlm or equivalently Ylm that
+generates diffeomorphism of diffeo-surfaces at the vicinity of a point pθ, φq is countable, the
+total number of generators is innumerable. This is because spherical harmonic functions
+Ylmpθ, φq at different points are linearly independent.
+This is an interesting property,
+because it shows that the Lagrangian for the whole Universe (6.1) or its subsystems (6.11)
+is inherently nonlocal, and algebra and parameter space of SUp8q-QGR interpreted as
+spacetime after division of the Universe to subsystems are continuous. It is in contrast
+to QGR models with symplectic geometry, which never become truly continuous. Despite
+explicit continuity of operators in (2.8) description of algebra, Cartan decomposition of
+SUp8q to SUp2q and Poisson bracket representations of SUp8q are more suitable for
+practical applications. Eigen vectors of ˆLlm and ˆLpθ, φq are discussed in Appendix E.1
+of [39].
+From (2.3) we conclude that vectors of HU are differentiable complex valued functions of
+pθ, φq. In particular, θ and φ can be themselves considered as vectors of the Hilbert space
+HU, They present an orthogonal basis and any state of the Hilbert space can be decomposed
+to orthogonal spherical harmonic functions Ylm. The relation between structure coefficients
+of sup8q algebra and Poisson brackets of Ylm in (2.4) guarantees the consistency of the
+whole structure. Moreover, from (2.9) we conclude that structure coefficient of the alge-
+bra (2.8) are generated by 3-point Green functions. Therefore, not only SUp8q-QGR is
+inherently nonlocal but also non-Gaussian.
+From now on the pair pl, mq of indices is assumed to satisfy the constraint (2.3), even if
+it is not explicitly mentioned. Moreover, if there is no risk of confusion, we simplify the
+notation by using e.g. a ” pl, mq as index of generators, i.e. ˆLa ” ˆLlm.
+2.2.2
+Quantization
+In absence of a background spacetime the canonical quantization does not apply to SUp8q-
+QGR. Nonetheless, non-commuting algebra of SUp8q can replace canonical quantization,
+see Appendix B for more details. Specifically, when the SUp8q algebra is written with
+respect to dimensionful operators proportional to ℏ, the Lie algebra commutation relation
+(2.1) plays the role of a quantization relation. On the other hand, as the algebra (2.1) is
+infinite dimensional, there should be also a commutation relation similar to the canonical
+quantization relations, that is Heisenberg uncertainty relation, between generators of sup8q
+– 10 –
+
+ˆLa and their duals ˆJb5:
+rˆLa, ˆJbs “ ´iℏδab1,
+ˆJa P BrH˚
+Us
+(2.11)
+Dual operators ˆJb act on the dual Hilbert space H˚
+U and belong to the space of (bounded)
+linear operators BrH˚
+Us – BrHUs. In particular, considering representation (2.3) for the
+generators of sup8q, it is clear that ˆJb must be complex valued functions of the parameters
+pθ, φq. We remind that vectors of the Hilbert space HU are also complex valued functions
+of pθ, φq. Therefore, their multiplication with a complex function Fpθ, φq can be described
+as an operator ˆF ” Fpθ, φq1 P BrHUs. On the other hand, spherical harmonic functions
+Ylm, which according to (2.4) have the same algebra as ˆLlm, are orthogonal functions and
+can be considered as generators of sup8q for the dual Hilbert space H˚
+U, as well.
+In Appendix B.1 we present a differential equation for ˆJlm and calculate its solutions. In
+Appendix B.2 we obtain the expression for dual of parameters pθ, φq, that is iℏB{Bpcos θq
+and iℏB{Bφ, respectively, with respect to generators ˆLlm of representation (2.3). The results
+show that they are degenerate, in the sense that they can be calculated using any ˆLlm.
+Consequently, they carry much less information about the quantum Universe constructed
+here than ˆLlm’s. In fact giving this degeneracy, it is possible to consider iℏB{Bpcos θq and
+iℏB{Bφ as being a superposition of ˆLlm operators, see Sec. 6 for details. Such interpretation
+can be used to test SUp8q-QGR, for instance in experiments sensitive to ˆLlm with lowest
+pl, mq.
+We leave details of this test proposal to a dedicated work on the experimental
+signatures of this model.
+Considering the differential form of the dual of θ and φ and the fact that HU – H˚
+U,
+operators ˆJa’s are functions of pθ, φq. Thus, they have a representations similar to (2.3)
+with respect to two parameters ppθ, pφq, which are analogous to momentum components.
+The existence of such expansions is a direct consequence of the Stone-von Neumann the-
+orem, which states that two sets of operators satisfying standard commutation relations
+are related by a unitary transformation [68]. More precisely, these operators must satisfy
+exponentiated Weyl quantization relation [69] (see e.g. [70] for a recent review) defined as:
+eixi ˆ
+Xieipi ˆPi “ eiℏx.peipi ˆPieixi ˆ
+Xi
+(2.12)
+where x and p are C-number vectors in d-dimensional space and 0 ď i ď d ´ 1, and ˆXi
+and ˆPi are the corresponding dual operators. Using Baker–Campbell–Hausdorff formula:
+e ˆ
+Xe ˆY “ e ˆZ,
+ˆZ “ ˆX ` ˆY ` 1{2r ˆX, ˆY s ` 1{12r ˆX, r ˆX, ˆY ss ` ¨ ¨ ¨ , it is straightforward to
+show that canonical commutation relation leads to (2.12). Nonetheless, this reformulation
+of the standard commutation relation is for ensuring that the domain of the unbounded
+operators ˆXi and ˆPi cover each other [68]. In SUp8q-QGR the number of independent
+operators d Ñ 8. If we use representation (2.3) of the generators of sup8q, which depend
+on numerable indices pl, mq, the exponentiated form of (2.11) becomes:
+eiLlmpθ,φqˆLlmeiJlmpθ,φq ˆJlm “ eiℏLlmJlmpθ,φqeiJlmpθ,φq ˆJlmeiLlmpθ,φqˆLlm.
+(2.13)
+5We remind that dual space V ˚ of a vector space V is the space of bijective maps v˚ : V Ñ F,
+where F is the field on which the vector space V is defined. A map v˚ P V ˚ defines a scalar product
+xv˚, vy “ v˚pvq, @ v P V .
+– 11 –
+
+In contrast to Fourier transform of functions or Hilbert space vectors defined on Rd back-
+ground spacetime, non-Abelian nature of sup8q algebra means that operators ˆLa cannot
+be considered as analogous to position vector operators ⃗ˆX in Rd, d Ñ 8 space.
+These results complete the demonstration that in SUp8q-QGR the algebra (2.1) is equiv-
+alent to the canonical quantization of unbounded operators, specially those operators that
+at low energies are interpreted as the classical spacetime. This property of SUp8q-QGR is
+distinguishable from other QGR models based on non-Abelian algebras, such as those with
+non-commutative geometry, and can be used to test and discriminate this model.
+2.2.3
+Interpretation and comparison with Virasoro algebra
+Finally, it is useful to compare the algebra (2.1) with Virasoro algebra which arises in string
+theories.
+Virasoro algebra without central extension is homomorphic to (2.1), i.e.
+the
+algebra sup8q [64, 66, 71, 72]. Moreover, similar to (2.1), Virasoro algebra is a connected
+subalgebra of area preserving diffeomorphism of torus Tp2q [71, 72]. However, it is well
+known that for its application to quantized string models it must be extended by a central
+charge, such that quantum anomalies can be canceled out. In the case of SUp8q-QGR such
+extension is not necessary, because quantization and construction of a Hilbert space are
+performed algebraically, there is no background spacetime, and no classical limit anomaly
+to be removed.
+In fact, because the diffeomorphism generated by the Poisson algebra
+(2.4) must preserve the area, it cannot preserve distances and angles. Consequently, both
+conformal and Weyl (dilaton) symmetries are broken, and the central charge which is the
+conformal symmetry charge is necessarily zero.
+SUp8q-QGR is compared with both perturbative and non-perturbative approaches to
+string theory in [38]. Here we briefly remind their main differences. Although in both
+models a 2D space seems to have an essential role, its origin and physical interpretation is
+very different. The worldsheet in string theory is a real (1+1)D spacetime, which its de-
+formation is interpreted as presenting quantum gravity of extended physical and quantum
+object - strings - embedded in a multi-dimensional quantum space. In the perturbative
+approach, coordinates of this extended space are perceived locally as fields living on the
+(1+1)D worldsheet and their dynamics is ruled by a sigma model. At least 4 of these
+quantum fields should be real and non-compactified to realize a classical spacetime far
+from Planck scale. By contrast, in SUp8q-QGR what we called diffeo-surface is a vir-
+tual surface, which its diffeomorphism algebra is homomorphic to SUp8q.
+We use its
+coordinates to parameterize an abstract Hilbert space defined on the complex numbers C.
+Therefore, SUp8q-QGR and string are profoundly different.
+3
+Division to subsystems
+According to a corollary of quantum mechanics axioms with symmetry as a foundational
+concept, an indivisible Universe is trivial [26, 39]. The necessity of distinguishable parts in a
+– 12 –
+
+Universe seems natural, because even in the whole Universe model described in the previous
+section, we recognize at least two categories of objects: states, which are vectors of the
+Hilbert HU, and linear operators, which act on them and constitute the space BrHUs –
+SUp8q.
+We should remind that in the context of a physical model, both states and
+operators applied to them are not mere mathematical objects, but have physical existence.
+Therefore, division of the Universe to entities/subsystems is inevitable. The issue, however,
+is how to discriminate these entities from each others. Specifically, consider properties of
+SUp8q group reviewed in Appendix A. Homomorphism of one and tensor product of many
+copies of SUp8q according to equation (A.3) means that they cannot be distinguished from
+each others, and an additional structure is necessary for this purpose6.
+In this section we show that such structure can arise due to quantum fluctuations - in other
+words random self SUp8q interaction. In the following subsections we describe how the
+emergence of finite rank symmetries divide the Universe to subsystems. We also discuss
+how the global SUp8q symmetry sticks together subsystems and entangle them.
+3.1
+Infinitely divisible Universe
+Necessary conditions for dividing a quantum system to subsystems are described in [25].
+They are reviewed in Appendix C. In the case of SUp8q-QGR, properties of SUp8q sym-
+metry of the Hilbert space HU and possibility of its decomposition to any finite rank SU
+group (A.5) imply that the Universe can be divided to infinite number of subsystems, each
+having its own symmetry group Gi, where index i is used to discriminate subsystems.
+The complementarity condition (c) of subsystem definition imposes the following constraint
+on the symmetries represented by subsystems:
+â
+i
+Gi – SUp8q
+(3.1)
+Division of the Hilbert space HU to those of subsystems Hi representing Gi’s is discussed in
+Appendix C.1. Considering the properties of SUp8q tensor product by itself and by other
+groups (A.7 - A.11), subsystems symmetries Gi’s can also contain infinite rank factors.
+3.1.1
+Emergence of internal symmetries and locality in SUp8q-QGR
+In [39] we explicitly demonstrated that a symmetry invariant Lagrangian-like functional
+for the whole SUp8q-QGR Universe can be written (we discuss this functional in details
+in Sec.
+6).
+After application of variational principle to this functional, one obtains a
+trivial vacuum, which is nonetheless unstable.
+This instability provides the necessary
+conditions for division of the Universe to subsystems, as described in Appendix C. To
+demonstrate the emergence of clustering and decomposition of the Hilbert space HU and
+BrHUs to approximately orthogonal subspaces, we use an operational approach rather than
+dynamical, because at this stage the model is still static.
+6On the other hand, as a quantum system with SUp8q symmetry is homomorphic to many copy of
+itself, it would be meaningful to association a probability to states of the Universe.
+– 13 –
+
+A state |ψUy P HU can be expanded with respect to spherical functions:
+|ψUy “
+ż
+dΩ ψUpθ, φq |θ, φy
+“
+ż
+dΩ
+ÿ
+lě0,
+´lďmďl
+ψUpl, m, θ, φq |θ, φy
+(3.2)
+ψUpθ, φq ”
+ÿ
+lě0,
+´lďmďl
+ψUpl, m, θ, φq ”
+ÿ
+lě0,
+´lďmďl
+ψlm
+U Ylmpθ, φq
+(3.3)
+The state |ψUy is pure because by construction no degree of freedom is traced out. There-
+fore, the density matrix of the Universe as a whole can be written as ˆρU “ |ψUyxψU|. Using
+the last equality in (3.3), we define another basis for HU:
+|ψUy “
+ż
+dΩ
+ÿ
+lě0,
+´lďmďl
+ψlm
+U |Ylmpθ, φqy,
+|Ylmpθ, φqy ” Ylmpθ, φq|θ, φy
+(3.4)
+The interest of this basis is that amplitudes ψlm
+U
+do not depend on the continuous pa-
+rameters pθ, φq. In this basis a state with constant amplitude ψlm
+U
+“ f “ cte., @ pl, mq is
+completely coherent. Such state is unique and we call it |ψccy. Unless a strict fine-tuning
+- preparation - is applied to the system, the uniqueness of |ψccy means that the proba-
+bility of its random occurrence is much smaller than partially incoherent states, which in
+general are less homogeneous, in the sense that amplitudes of their components ψlm
+U
+differ
+from each others. Thus, these states have an intrinsic clustering. Specifically, consider the
+application of ˆLl1m1 on ψpcq. Using (2.4) we find:
+ˆLl1m1pθ, φq|ψccy “ f
+ÿ
+lě0,
+´lďmďl
+ˆLl1m1pθ, φqYlmpθ, φq|θ, φy
+“ ´iℏf
+ÿ
+lě0,´lďmďl
+l1ě0,´l1ďm1ďl1
+fl1m1
+l1m1,lmYl1m1pθ, φq|θ, φy
+“ ´iℏf
+ÿ
+lě0,´lďmďl
+l1ě0,´l1ďm1ďl1
+fl1m1
+l1m1,lm|Yl1m1pθ, φqy ” |gl1m1pθ, φqy
+(3.5)
+where in the second line we have used equation (2.4). Structure coefficients fl1m1
+l1m1,lm are
+proportional to 3j symbols and depend on the indices pl, mq, pl1, m1q, pl1
+1, m1
+1q.
+Thus, in
+general the new state |gl1m1pθ, φqy is not any more a completely coherent state and some
+of the components in |Yl1m1y basis have larger (or smaller) amplitudes than others. Conse-
+quently, the density matrix ˆρU becomes more structured and approximately decomposed
+to blocks. Moreover, this grouping increases with successive application of ˆLa’s. Inversely,
+modification of |ψUy affects the distribution of ˆLa’s, because ˆρ P BrHUs and can be ex-
+panded with respect to ˆLa’s. Thus, its modification alters the probability of interaction
+and further changes. These processes are analogous to absorption of gauge bosons, which
+are generators of symmetry algebra, by matter fields - states - and re-emission of both in
+– 14 –
+
+a Compton-like interaction. In addition, non-commutative algebra of ˆLa P sup8q implies
+that they can also source themselves, in the same way as in non-Abelian Yang-Mills models
+such as QCD.
+A priori if the probability of operation/interaction with ˆLl1m1 is the same for all pl1, m1q, the
+equilibrium can be reestablished. However, due to the randomness of operations and infinite
+number of colors of ˆLa’s, the process of clustering, that is the formation of blocks inside
+ˆρU, is quasi-irreversible and probability of returning to the completely coherent state ψpcq
+is negligibly small. Clustered subspaces of ˆρU are by definition (approximately) orthogonal
+to each others and (approximately) satisfy criteria (a-c) for division of a quantum system
+to subsystems.
+Thus, after numerous action of ˆLa’s, the density ˆρU is approximately restricted to a blocked
+subspace of BrHUs:
+BrHUs Ą
+à
+i
+BrHis W ˆρU
+(3.6)
+where the symbol W means approximately belong to.
+Sub-Hilbert spaces Hi can be ap-
+proximately considered as disconnected subsystems, that is Hi X Hj « H, i ‰ j.
+In
+other words, operators BrHis Ď BrHUs, which dominantly affect Hi Ă HU, approximately
+commute with BrHjs Ď BrHUs, which dominantly affect Hj Ď HU, j ‰ i. Thus, they
+approximately satisfy the conditions (a) for dividing a quantum system to subsystems, and
+for all practical purpose the Hilbert space to which the state of the Universe belong can
+be approximated by a tensor product:
+HU ⇝
+à
+i
+Hi ⇝
+â
+i
+Hi
+(3.7)
+BrHUs ⇝
+à
+i
+BrHis ⇝
+â
+i
+BrHis
+(3.8)
+where, the symbol ⇝ means approximately lead to. It is crucial to emphasize that decom-
+position of the density matrix of the Universe ˆρU is always an approximation, otherwise
+each block can be considered as a separate universe. Moreover, one has to assume that as
+the pointer/basis states in one subspace Hi has little coherent mixing (superposition) with
+other subspaces, it acquires an independent physical reality.
+Giving the continuous nature of parameters pθ, φq of the Hilbert space HU and BrHus,
+subspaces Hi and operators BrHis must be also local in this parameter space. This property
+fulfills the second condition - criterion (b) in Appendix C - for division to subsystems.
+Moreover, the complementarity condition (c) is trivially satisfied, because any operator
+ˆO P BrHUs is either ˆO P biBrHis Ď BrHUs or belongs to a new set of operators tA1u such
+that biBrHisKtA1u and biBrHis b tA1u – BrHUs and tA1u Ď BrH1s, H1 Ď HU, BrH1s Ď
+BrHUs.
+Each set of operators tAiu Ď BrHis represents a symmetry group Gi with a rank at most
+equal to the number of linearly independent members of tAiu set. This number is nec-
+essarily finite, otherwise tAiu – BrHUs – SUp8q. In this case clustering in ˆρU was not
+enough to break the Universe approximately to subsystems. Such case bring back the state
+– 15 –
+
+of the Universe to the initial state considered for this analysis. The argument about sub-
+systems and locality arising from a global symmetry can be also inverted. Assume many
+(infinite) number of indistinguishable quantum systems, each representing a finite rank
+symmetry group G. Their Hilbert space represents G ˆ G ˆ . . . – SUp8q. Thus, SUp8q
+symmetry in SUp8q-QGR can be considered to be due to a bath of large (infinite) number
+of indistinguishable (abstract) decoherence-free quantum subsystems [73] with finite rank
+symmetries.
+The first decomposition in (3.8) is similar to expansion of a vector space with respect
+to a basis. At this stage of clustering the Universe as a whole is similar to a strongly
+coupled quantum system, where despite presence of some discernible features grouped in
+subspaces Hi, they cannot be isolated and studied separately. Consequently, unitarity,
+Born rule, and global Up1q phase symmetry of quantum states apply only to the ensemble
+of Hi. However, when these subspaces become sufficiently isolated, such that HU can be
+approximately considered as their tensor product, unitarity, Born rule, and Up1q phase
+symmetry can be approximately applied to each Hi individually.
+3.2
+Symmetries and entanglement of subsystems
+Due to the assumed global SUp8q symmetry, satisfaction of conditions (a-c) for division
+to subsystems and decomposition (3.8) of HU to tensor product of Hi’s are only approxi-
+mately true. Indeed, conservation of SUp8q means that subspaces Hi are not projected to
+themselves under application of ˆLa P BrHUs with non-zero components outside the BrHis
+blocks. Therefore, the division of the Universe to subsystems is an approximation and
+no strictly isolated subsystem exists. In the language of quantum information, maximum
+quantum complexities of the Universe and its subsystems is infinite, because preparation of
+their states needs infinite number of Q-bits and infinite number of binary-gate operations.
+Therefore, to make these properties explicit for each subsystem, we add a SUp8q factor
+to the full symmetry G1
+i of subsystem i with finite rank local symmetry Gi represented by
+its block in the density matrix of the Universe is:
+G1
+i “ Gi ˆ SUp8q
+(3.9)
+Considering the properties of tensor products containg SUp8q in (A.6 - A.10), such decom-
+position remains consistent with the global SUp8q symmetry. Without loss of generality
+we can consider the same local/internal symmetry G for all subsystems. Specifically, G can
+be chosen to be a tensor product - Cartan decomposition - containing all Gi’s. Then, some
+subsystems may represent some of the factor groups trivially. Indeed, symmetry group of
+the Standard Model of elementary particles has such a product structure. Moreover, it
+includes the two lowest rank Lie groups which cannot be further decomposed to unitary
+groups.
+– 16 –
+
+3.2.1
+Global entanglement
+The global SUp8q symmetry has a crucial role in keeping together the Universe as a
+quantum system. We discuss this property as the following proposal:
+Proposition 1: In SUp8q-QGR every subsystem is entangled to the rest of the Universe:
+Proof: This is a consequence of the preservation of the global SUp8q. To demonstrate
+this claim, consider a division of the Universe to two subsystems representing symmetry
+groups G1 and G2. In Appendix C.2 we show that a division to subsystems increases the
+dimension of the Hilbert space of the composite system with respect to before division.
+In the case of SUp8q-QGR, if we consider strictly infinite dimensional Hilbert spaces for
+the whole Universe and for the two subsystems 1 and 2, relation (3.9) and arithmetic of
+infinite numbers imply that without needing additional conditions, there is a homomor-
+phism between states of the Hilbert space HU and H1 ˆH2. Nonetheless, let’s assume that
+their dimensions N, N1, N2 are finite, but very large and approximately infinite. In this
+case, as described in more details in Appendix C.2, in general N ď N1, ˆN2. On the other
+hand, as we mentioned earlier and discussed in more details in Sec. 6, the Universe as a
+whole is static and its Lagrangian-like symmetry invariant functional (6.3) is topological.
+Therefore, its quantum properties, including its Hilbert space should not depend on the
+dynamics of its components/subsystems, which are subject to relative variations. The only
+way to establish equality between number of states of the whole Universe and ensemble of
+its subsystems is a permanent entanglement between subsystems, which decreases number
+of physically allowed states to N.
+Giving the fact that subsystems have infinite number of degrees of freedom related to their
+SUp8q symmetry, one can in principle find many untangled degrees of freedom. Therefore,
+in general projective observation of limited number of them in one subsystem would not be
+enough to project other subsystems to a given state. This is a crucial condition for having
+a dynamical Universe.
+Another way of demonstrating proposition 1 is through mutual information. Consider the
+mutual information IpA : Bq ” SpˆρAq ` SpˆρBq ´ SpˆρABq between a quantum subsystem A
+and the rest of the Universe B. The function S ě 0 presents von-Neumann entropy of each
+system in isolation. If A and B were independent from each other SpˆρABq “ SpˆρAq`SpˆρBq
+and IpA : Bq “ 0. However, due to the gauge symmetry the state of the whole Universe
+ˆρAB – 1 and SpˆρABq “ 0, and IpA : Bq ě 0. Thus, there is no isolated subsystem in
+the SUp8q-QGR Universe and subsystems are entangled. Any variation in the subsystems
+leads to a global adjustment to restore their entanglement. This is the quantum analogous
+of gravitational memory in the classical spacetime [44, 45].
+We leave quantification of
+this entanglement for future works, because it may need investigation of cosmology in the
+framework of SUp8q-QGR. Nonetheless, on the dimensional ground, we expect that it
+should be proportional to ℏ2Λ{M2
+P , where Λ is the cosmological constant. See also Sec.
+6.4 for more details.
+The always entanglement of subsystems shown here is not a surprise. Irrespective of the
+– 17 –
+
+model constructed here and in a general setup of quantum systems, There are abundant
+evidence that they are quantum mechanically correlated. For instance: Change of a quan-
+tum reference frame may lead to decoherence and/or superselection [28]; Limitations on
+the transfer of information about the reference frame are equivalent to quantum deco-
+herence [27, 29]; Entanglement and superposition of states become perspective depen-
+dent [29, 74].
+These properties of a quantum Universe and its subsystems imply that
+characteristics, such as the amount of entanglement between subsystems may depend on
+which quantum subsystem is selected as reference. The emergence of reference-dependent
+perspectives is equivalent to gauge fixing in QFT and general relativity [74, 75]. The com-
+mon aspect of these properties is the reduction of quantum coherence. In this respect, the
+global entanglement in SUp8q-QGR, which reduces quantum coherence, is of the same
+category as the above properties concluded in the context of quantum information theory.
+3.2.2
+Consequences of the global entanglement
+The ever-existing quantum correlation/entanglement between every subsystem and the rest
+of the Universe reinforces decomposition (3.9), because if there is a subsystem with only
+finite dimensional Hilbert space, its entanglement with the infinite degrees of freedom of
+the rest of the Universe makes its effective degrees of freedom, and thereby symmetry of
+its Hilbert space infinite dimensional.
+As G symmetry emerges from clustering of states, it is expected that interaction/mixing
+between subsystems through G symmetry be much stronger than via the global SUp8q.
+This property is consistent with the observed universality of gravitational interaction and its
+weakness in comparison with internal interactions of subsystems/particles. It also provides
+an explanation for the weak gravity conjecture [76].
+From (A.11) it is clear that in what concerns mathematical definition and action, there is
+no essential difference between finite rank internal symmetries of subsystems and SUp8q
+- gravity - symmetry. Nonetheless, the resemblance of gravitational interaction to a gauge
+field in SUp8q-QGR is very different from what is called gauge-gravity duality models in
+the literature [77–81], Notably, in SUp8q-QGR the gauge symmetry of quantum gravity is
+not directly related to the classical Lorentz invariance, which corresponds to the freedom
+of the choice of parameterization of the Hilbert space, and difficulties which arise in gauge-
+gravity proposals [82–84] are irrelevant for SUp8q-QGR.
+We will describe other consequences of the global entanglement of subsystems through
+SUp8q symmetry along with other properties of SUp8q-QGR in the following sections.
+3.2.3
+Observability of entanglement
+Finally, an important question is whether the global entanglement of subsystems is observ-
+able. The answer is in principle yes, but in practice no ! For such observation we need
+to consider at least 3 subsystems: the target subsystem, a quantum clock as explained in
+Sec. 4.2, and the rest of the Universe as observer/reference. Assume that the observer
+– 18 –
+
+performs a POVM on the target such that its state is projected to a pointer state defined
+in (4.6). According to the proposition 1 this operation projects the state of the rest of
+the Universe, that is the observer and the clock to a pointer state. How can the observer
+verify this expectation ? The only way is the measurement of the expected pointer state
+of the observer-clock system. However, this is impossible. In fact, it is even impossible
+to verify that the target subsystem is indeed in a projected state after the first operation.
+Such verification needs repetition of the same operation. However, a repetition implicitly
+means passage of time as a consequence of the change of clock state. But, if the clock
+state varies, the state of observer-clock subsystem changes, which in turn changes the state
+of the target subsystem according to the proposition 1. Neglecting this verification, the
+only way to verify the entanglement is either the target subsystem or the observer-clock
+themselves should perform a self-state verification test. However, none of these operations
+can be performed in isolation and without modifying some of the degrees of freedom of
+these subsystems, and thereby changing their state.
+4
+Parameter space and algebra of subsystems
+The division of the Universe to subsystems makes it possible to define a reference quantum
+observer and a quantum clock.
+In this section we study in some details the extended
+Hilbert space of subsystems and parameters that characterize its states. We also describe
+the relation between these parameters and what we perceive as the classical spacetime.
+4.1
+Emergence of an area/size scale as a measurable
+The area of diffeo-surface M of a SUp8q representation is irrelevant when only one rep-
+resentation is considered. In fact, area preserving diffeo-surfaces represent SUp8q ˆ R –
+SUp8q ˆ Up1q “ Up8q, where R is a size scale, which can be chosen to be the area of
+diffeo-surface or its square-root - a size scale. The Abelian R – Up1q group corresponds to
+the scaling symmetry Λ of diffeo-surfaces. A diffeomorphism of a compact surface can be
+decomposed to an area preserving diffeomorphism and a global scaling, that is:
+DiffpMq – ADiffpMq ˆ ΛpMq
+(4.1)
+In the context of quantum mechanics and representation of SUp8q by the Hilbert space of
+the Universe, the scaling symmetry can be also identified with the irrelevant global Up1q
+phase. On the other hand, when the Universe is divided to subsystems, each representing
+SUp8q, as we explained in Sec.
+3.1.1, the Up1q phase symmetry of the Hilbert space
+becomes local and decouples from the global scaling symmetry. Indeed, the latter breaks,
+because representations and algebras of distinguishable subsystems can be compared with
+each others through a map (a morphism). Nonetheless, the Hilbert space Hi of a subsystem
+considered in isolation is still invariant under its own global Up1q phase.
+Comparability of diffeo-surface areas of subsystems with each others does not mean that
+their areas or sizes are considered as fixed. Rather, their values become a new relative
+– 19 –
+
+Diffeo-Surface
+of the Universe
+Subsystem
+Ref. Observer
+clock
+Subsystem
+Divided Universe
+U(1) ≈ R
+r’
+r
+rc
+δt
+x  
+r
+x  
+r’
+x  rc
+δρ
+δρ’
+δρRef
+Figure 1.
+Schematic description of SUp8q-QGR. Left: 2D surfaces present diffeo-surfaces of
+the SUp8q of the Universe and its subsystems.
+Although subsystems share the Hilbert space
+associated to SUp8q symmetry, they have their own local representation of the symmetry. This
+is schematically shown by different shape of 2D surfaces. Displaced surfaces depicts the evolution
+δρ of the state of subsystems with respect to time variation δt that quantifies the change of the
+clock’s quantum state. Arrows depict homomorphism between SUp8q symmetry of subsystems
+and its associated universal gravitational interaction. Notice that infinite number of projections
+between sup8q algebras of subsystems is possible. Therefore, subsystems can be in general in a
+superposition state with respect to scaling parameter r. Different colors of subsystems depicts their
+different representation of internal symmetry G. Right: Diffeo-surface of the Universe and its
+subsystems. Parameter r P R – Up1q can be considered as a label for subsystems.
+measurable, in the same way as the distance in classical and quantum physics without
+gravity becomes a measurable when a reference - another object - is considered. Here we
+call this reference subsystem the quantum reference observer. Therefore, in addition to the
+diffeo-surface parameters pθ, φq states - vectors of the Hilbert space - of subsystems Hi’s and
+space of bounded operators applied to them BrHis’s depend on a dimensionful parameter
+r.
+But, the way we define r is arbitrary.
+For instance, the compact diffeo-surfaces of
+subsystems can be embedded in a Rp3q, because D2 ˆ R – Rp3q, where D2 is any Riemann
+2D space. Assuming parameters of the reference subsystem at an arbitrary point of Rp3q,
+the parameter r can be identified with radial coordinate in a spherical coordinate system
+having its origin at the point associated to the reference subsystem. Fig. 1 schematically
+depicts these properties.
+– 20 –
+
+4.2
+Time
+Introduction of relative time in quantum mechanics and quantum gravity is extensively
+studied in the literature [30, 85, 86], specially as a mean for solving the problem of Hamil-
+tonian constraint and the absence of time in canonical quantization of gravity. But effi-
+ciency of this approach is criticized in [87]. Equivalence of various definitions of a relative
+and dynamical time is demonstrated in [31]. Many of complexities of quantum time and
+reference frame definitions discussed in the literature originate from attempts to quantizing
+spacetime, but keeping diffeomorphism invariance. In SUp8q-QGR spacetime is a parame-
+ter space and is not quantized. Therefore, concerns about the validity of a relative time are
+irrelevant. Nonetheless, criticism of [87] will be answered in a report under preparation.
+Once the Universe is divided to subsystems, it becomes possible to define dynamics as
+relative variation of their states. In analogy to the division of the Universe to subsystem,
+our approach to the definition of a quantum clock and time parameter would be opera-
+tional. Quantum entanglement between an imperfect - not completely isolated - clock and
+systems and their effects on the definition of reference frame in a general framework is
+demonstrated in [88]. In SUp8q-QGR similar effects arise due to the omnipresent entan-
+glement of subsystems proved in the proposition 1. Consequently, successive applications
+of the same operator to a selected subsystem in general does not lead to the same outcome,
+and quantum contextuality [89, 90] becomes globally manifest. This leads to the emergence
+of an order - a time arrow - between events. In particular, relative variation of states and
+observables of subsystems can be quantified with respect to those of a single subsystem,
+called a quantum clock.
+In the following subsections we explain in more details properties of the clock and emergence
+of a dynamics, which is in spirit similar to the Page & Wootters proposal [30]. There is
+however no need for explicit conditional wave functions, because in contrast to the canonical
+quantization of QGR, there is no need for (3+1)D decomposition of formulation7.
+4.2.1
+General properties of a quantum clock and time parameter
+Without considering specifics of a quantum clock and how the change of its state is quan-
+tified, following general conclusions can be made:
+Initial state: The initial state of the clock, that is the outcome of the first operation on
+it, initializes relative dynamics of subsystems, but its value is irrelevant for further
+measurements. Only its variation in the next measurement is relevant.
+Arrow of time: Selection of any subsystem as clock induces an ordering in operations
+on subsystems. This is a direct consequence of the proposition 1.
+7When a quantum system is constructed from quantization of a classical model, there are alternative
+ways to define dynamics, for instance by using relational observables `a la Dirac [91], or what is called
+Heisenberg symmetry reduction in the phase space. Equivalence of these methods is demonstrated in [31].
+As SUp8q-QGR is not obtained from a classical model, the conditional approach of Page & Wootter is
+more suitable.
+– 21 –
+
+Monotonous time: Quantification of the state change, that is the observable proxy used
+for this purpose is arbitrary. But, the value of outcome must allow ordering. For
+instance, a vector-valued observable is not suitable, unless its module is used as time
+parameter.
+One real parameter: The above condition means that without loss of generality quan-
+tification parameter called time t can be chosen to be a real number: t P R.
+POVM time: Quantification of clock state variation must be a single-valued uniquely
+calculable function from information about the clock state, but its measurement does
+not need to be projective.
+Time clicks for everyone: Entanglement of the clock - as a subsystem - with the rest
+of the Universe means that after initialization of the clock, every other subsystem,
+and thereby the whole Universe, evolves along with the clock. This is the origin of
+the arrow of time.
+A more extended and mathematically rigorous description of necessary conditions for a
+quantum clock can be found in [31]. Nonetheless, simple properties described here are
+sufficient for our purpose of defining a relative dynamics, because we do not need to relate
+it to a classical model.
+4.2.2
+Quantum clock and its associated time parameter
+Overlooking the complexity of realistic quantum clocks and issues such as their finite size,
+which induces uncertainties [92] and decoherence [93], a quantum clock and a time param-
+eter satisfying properties described in Sec 4.2.1 can be defined by an operator ˆLc P BrHCs,
+which its application on the clock subsystem leads to the variation of its state ˆρc. If the
+clock is tomographically complete, its state and thereby the variation ˆρc as a consequence
+of ˆLc application can be determined. Then, the time parameter or more precisely its vari-
+ation δt can be defined as a monotonic scalar function of measurements. For example,
+time variation may be defined as being proportional to the amplitude of state variation
+δt 9 |δψc| ” |ψ1
+c ´ ψc| or proportional to the fidelity function:
+δt 9 Fpˆρc, ˆρ1
+cq ”
+ˆ
+tr
+ba
+ˆρc ˆρ1c
+a
+ˆρc
+˙2
+(4.2)
+If the clock is not tomographically complete, expectation value or outcome of POVM
+measurement of an observables and its variation can be used as δt. Alternatively, the time
+parameter t can be defined by parameterizing the average trajectory of the clock state ˆρC
+in its Hilbert space. The trajectory may be measured using an observable of the ˆOc. This
+approach is closer to classical definition of time than other examples described here. The
+time variation can be defined as δt 9 |tr pˆρ1
+c ˆOcq ´ tr pˆρc ˆOcq|. Obviously, variation range
+of the clock state ˆρc or the observer ˆOc is usually limited. Nonetheless, book-keeping of
+variations extends the range of t to infinity [94]. We emphasize that due to the proposition
+– 22 –
+
+1, there is no isolated clock and decomposition of the clock observable ˆOc « ˆOc ˆ 1 where
+1 presents the operation on the rest of the Universe is always an approximation.
+4.2.3
+Hamiltonian associated to a clock
+For each clock and time Parameterization t and each subsystem s, an operator ˆHs P BrHss
+called Hamiltonian can be defined such that:
+dˆρs
+dt “ ´ i
+ℏr ˆHs, ˆρss
+(4.3)
+where ˆρs is the density matrix of the subsystem s.
+Moreover, giving the fact that in
+SUp8q-QGR the whole Universe is static [39]:
+ˆÿ
+tsu
+ˆHs
+˙
+|ψUy “
+ÿ
+tsu
+ˆHs|ψsy “ 0.
+(4.4)
+where the ensemble tsu of subsystems includes also the clock and the reference subsystem
+defined in Sec. 4.1 and |ψUy “ Â
+tsu |ψsy is the state of the whole Universe. The Hamil-
+tonian ˆHs applies non-trivially only on |ψsy. Implicit in our notation is the equivalence of
+all states |ψUy up to a SUp8q symmetry transformation. Moreover, it is meaningless to
+interpret the sum over Hamiltonian operators in the l.h.s. of (4.4) as Hamiltonian of the
+Universe, because by definition and construction the Universe is static. A notable differ-
+ence between the definition of Hamiltonian here and in quantum models originated from a
+classical formulation is the fact that it is the Hamiltonian which associated to a time pa-
+rameter, not the other way around. If the time parameter is changed, Hamiltonian should
+be changed such that (4.3) remain valid. This definition decouples the Hamiltonian from
+definition of conjugate variables, which themselves is the variation of dynamical degrees of
+freedom with time.
+We emphasize that the constraint (4.4) is a consequence of the entanglement of every
+subsystem with the rest of the Universe demonstrated. In addition, the construction of
+(4.4) explicitly shows that, in contrast to ADM [95] in (3+1) dimension or other canonical
+quantization of gravity, this property is not related to diffeomorphism invariance of the
+parameter space and dynamics.
+4.3
+Full parameter space of subsystems
+Based on the arguments in Sec. 4.2.2 and other evidence discussed in the previous sections,
+we conclude that the 4 continuous quantities pt, r, θ, φq that parameterize the Hilbert space
+representing SUp8q symmetry of subsystems and their relative variations is related to
+what is perceived as spacetime in the classical limit of gravity. We give further precision
+about exact nature of this relation and what we mean by classical limit in the following
+sections. This conclusion discriminates SUp8q-QGR from QGR proposals, which try to
+– 23 –
+
+ad hocly quantize an unobservable background geometry8. This feature also implies that
+in the framework of SUp8q-QGR model, quantization of spacetime is meaningless.
+The shared parameter space pt, r, θ, φq of SUp8q symmetry of subsystems have several
+homomorphic descriptions:
+S2 ˆ R1 ˆ R1 – S2 ˆ Up1q ˆ Up1q – Rp1`3q
+(4.5)
+The middle homomorphism in (4.5) is useful for description of sup8q algebras of subsys-
+tems. Notice that in (4.5) we used Rp1`3q rather than R4. We justify this point in Sec. 5.
+In both cases the 4D parameter space is simply connected, even if S2 alone may have a
+non-trivial topology. Therefore, wave functions of subsystems can be in principle extended
+to classically singular points of the parameter space such as inside a black hole.
+In addition to SUp8q, states of subsystems represent the internal group G and depend on
+parameters that characterize its representations. we collectively call them α. Giving the
+fact that the symmetry group of subsystems is the tensor product SUp8q ˆ G, representa-
+tions of the two factors are independent and orthogonal. Therefore, state of a subsystems
+|ψsy can be decomposed to:
+|ψsy “
+ÿ
+t,r,θ,φ,α
+ψspt, r, θ, φ, αq |t, r, θ, φy b |αy
+(4.6)
+where |t, r, θ, φy b |αy is an eigen basis for the Hilbert space Hs, but its SUp8q and G
+parameters are not separable. Parameters pr, θ, φq characterize diffeo-surface of subsystems.
+Although in principle time parameter t is only related to the clock, entanglement of the
+latter with the rest of the Universe and constraint (4.4) indirectly relates t to the other
+parameters and the 4 parameters are effectively inseparable. The collective parameters α
+which characterize representations of the finite rank symmetry G are usually discrete. In
+this work we do not further discuss G symmetry and its representations by subsystems.
+The particle physics motivated by SUp8q-QGR is an important subject and needs its
+dedicated investigation, which we leave to future works.
+4.3.1
+The sup8q algebras of subsystems and their commutation relations
+Because of quantum entanglement of subsystems and coherence with respect to eigen states
+of the 4-dimensional parameter space txu - the spacetime - every pair of the parameters
+can be used to characterize SUp8q symmetry and its algebra as defined in (2.3-2.4) and
+(2.8-2.10). Although ranges of r and t are usually chosen to be the non-compact R, the
+homomorphism R – Up1q means that by identifying points at infinity, i.e. xi Ñ ˘8 i “
+0, ¨ ¨ ¨ , 3 the parameter space can be made compact, such that the surface passing through
+every pair be a 2D compact Riemann surface and its ADiff represent SUp8q [65].
+In any case, giving the emergence of locality in many-body systems due to the limited
+speed of quantum information transfer that we will discuss in Sec. 5, and the growth of
+8This trend of some QGR is models is considered by some authors as re-emergence of the infamous
+concept of ether [40, 96], which relativity wanted to disregard.
+– 24 –
+
+inhomogeneities [97], there would be negligible impact on the physical process due to the
+compactification (or not) of the parameter space. In particular, the definition of SUp8q
+generators ˆLlm in (2.3) would be still usable, because spherical harmonics are special cases
+of generalized hypergeometric functions [98]. Specifically, in (B.2) the associated Legendre
+polynomial Plm is related to the Gauss hypergeometric functions 2F1:
+Plmpηq “ p´1qm Γpl ` m ` 1qp1 ´ η2qm{2
+2mΓpl ´ m ` 1qm!
+2F1pm ´ l, m ` l ` 1; m ` 1; p1 ´ ηq{2q
+(4.7)
+with l`m ě 0 and Repηq ě ´1. Therefore, spherical harmonics and their Poisson algebra,
+which are homomorphic to sup8q, can be analytically extended outside the range of angular
+parameters pθ, φq, except for singular points at ˘1 and 8.
+Expansion and commutation relations of one of these parameters (coordinates), which we
+call η, and its dual iℏB{Bη with respect to ˆLlm generators are described in Appendix B.2.
+They apply to all parameters of SUp8q symmetry of subsystems, irrespective of how they
+are defined9. Thus, one can construct a sup8q algebra from any pair of the 4 parameters
+of the Hilbert space of subsystems:
+rˆLl,mpη, ζq, ˆLl1,m1pη, ζqs “ fl”m”
+lm,l1m1 ˆLl2,m2pη, ζq
+(4.8)
+under the condition that an appropriate associated Legendre polynomial is used in (2.3).
+Crucially, giving the similarity of commutation relations to the standard quantum mechan-
+ics, the usual uncertainty relations of quantum mechanics between parameters and their
+duals [99, 100] can be applied to the time parameter t, irrespective of how a quantum clock
+is defined. Moreover, it would be more suitable to use an expansion similar to (3.3), in
+which wave coefficients depend only on discrete parameters pl, mq rather than e.g. (3.2).
+4.3.2
+About quantum uncertainty, nonlocality, and topological effects
+Obtaining usual uncertainty relations in a model which does not follow the standard quan-
+tization procedure is remarkable. For instance, if generators ˆLlm depended on the second
+order derivatives of parameters, rather than the first order, there were no analytical ex-
+pression for iℏB{Bη as a function of ˆLlm generators. Indeed, it has been shown [101] that
+in any model with indeterminism and nonlocality, these properties and their quantification
+are related. On the other hand, nonlocality and contextuality of quantum mechanics are
+not maximal [102], in the sense that the CHSH inequalities [103] do not have the maxi-
+mum possible deviations from prediction of a local and causal model. Therefore, a priori
+a model which does not explicitly follow the standard procedure for quantization may
+have commutation relations and nonlocalities different from what is predicted by quantum
+mechanics.
+Nonlocality from indeterminism reflects itself in entanglement. But, it does not explain
+global/topological quantum effects, such as Aharanov-Bohm effect [104] and quantum Hall
+9The limiting points such as |η| Ñ 8 may need special treatment, in particular if diffeo-surfaces have
+non-trivial topology. These cases are analogous to topological material in condensed matter.
+– 25 –
+
+phenomenon, see e.g. [105] for a review. To explain these processes one has to virtually
+remove some points of the space to make it non-simply connected and topologically non-
+trivial. However, if the spacetime is a physical entity, removing or ignoring part of it to
+explain some processes is difficult to justify, because they can be detected by other obser-
+vations. For instance, in the case of Aharanov-Bohm setup, the path of a neutral particle
+can pass through the singular magnetic flux without being disrupted. By contrast, if the
+spacetime is interpreted as a parameter space, absence of some point in some phenomena
+is not controversial. In Sec. 6 we show that a symmetry invariant functional similar to
+Yang-Mills Lagrangian can be defined for SUp8q-QGR. Therefore, topological quantum
+phenomena observed in non-gravitational many-body systems may be relevant for quantum
+gravitational processes as well.
+4.4
+Parameter transformation
+As discussed earlier, parameters x ” pθ, φ, r, tq arise from different properties of SUp8q-
+QGR. Namely, pθ, φq parameterize representations of SUp8q symmetry; dimensionful pa-
+rameter r emerges from comparison of area of diffeo-surfaces; and t quantifies variation of
+the state of a chosen quantum clock. Nonetheless, these parameters are inseparable and
+define a R4, or as we will demonstrate later, a Rp3`1q space. The main reason is the fact
+the 4 components of x are interchangeable. As we discussed earlier, any pair of these pa-
+rameters can by used to represent SUp8q symmetry. Therefore, division into subsystems
+is not rigid and may change with selection of another clock or reference subsystem, such
+that they respect necessary the conditions defined in Appendix C. Specifically, due to the
+coherence symmetry and quantum superposition and indistinguishability of subsystems
+with same representations of symmetries, the set of orthogonal operators tAiu based on
+which subsystems are defined, depend on the basis for x, that is the choice of reference
+observer, clock, and coordinates of diffeo-surfaces of subsystems. Nonetheless, the Hilbert
+space Â
+s Hs of all subsystems should not depend on the choice of a basis or redefinition
+of parameters.
+The full basis of the Hilbert space Â
+s Hs is t|t, r, θ, φ, αy, @ ´ 8 ď t ď `8, 0 ď r ď
+`8, 0 ď θ ď π, 0 ď φ ď 2π, tαuu. However, in this section our concentration is on the
+SUp8q symmetry of subsystems and its representations. For this reason, we only consider
+states which their α dependence is traced out. We call the space of remaining parameters
+Ξ. A possible effect of this action is further entanglement of quantum state of subsystems
+and inseparability of remaining the parameters.
+A basis transformation of the Hilbert space of subsystems pÂ
+s Hsqα ” trαpÂ
+s Hsq (in-
+cluding clock and reference) can be written as:
+|t1, r1, θ1, φ1y “
+ÿ
+t,r,θ,φ
+ˆU|t, r, θ, φy,
+(4.9)
+ˆU P Brp
+â
+s
+Hsqαs – SUp8qN|NÑ8 – SUp8q
+(4.10)
+– 26 –
+
+The last homomorphism in (4.10) means that up to an irrelevant scaling of states, the basis
+transformation (4.9) is equivalent to application of a member of SUp8q.
+As we discussed in Sec. 2.2.1, states of Â
+s Hs (and pÂ
+s Hsqαq are complex functions on
+the parameter space. Thus, the basis transformation (4.9) of the Hilbert space can be
+equally interpreted as a diffeomorphism of the parameter space:
+pt, r, θ, φq Ñ pt1pxq, r1pxq, θ1pxq, φ1pxqq
+(4.11)
+Notice that quantum coherence, entanglement, and indistinguishability of classes of sub-
+systems - those which share the same symmetries and their representations - are crucial
+for inseparability of parameters and equivalence of basis independence of the Hilbert space
+with reparameterization and diffeomorphism invariance of the parameter space Ξ. In par-
+ticular, in absence of quantum coherence, time and distance part of the states could be
+separated from angular parameters of SUp8q symmetry.
+4.4.1
+Poincar´e symmetry
+In addition to diffeomorphism of the Rp3`1q parameter space, states of subsystems must be
+in non-trivial representations of its global Lorentz and translation transformation, which
+together constitute the Poincar´e group. A reminding of the relationship between diffeo-
+morphism and reparameterization can be found in Appendix D. Observations show that
+both the Standard Model [106] and gravity [107] are invariant under Lorentz group, which
+is isomorphic to SOp3, 1q. However, in contrast to SUp8q and more generally SUpNq@N,
+which are simply connected, SOp3, 1q is doubly connected under discrete parity P, conjuga-
+tion C, and time reversal T transformations. Considering the assumed SUp8q – SLp2, Rq
+symmetry of subsystems, it is clear that quantum superposition of subsystems must play
+a role in the preservation C, P, T symmetries. For instance, parity conserving fermions
+of SM are in p1{2, 0q ‘ p0, 1{2q representation of SOp3, 1q. Thus, each of the components
+representing a connected part of SOp3, 1q can be embedded in one SUp8q. However, only
+the superposition of disconnected sectors preserves C, P, T. This property is used by lep-
+togenesis models to explain matter-antimatter asymmetry using the observed parity and
+CP violation in the leptonic sector of the Standard Model (SM), see e,g [108] for a review.
+Although in the SM the quantum numberB ´ L, where B and L are baryonic and leptonic
+number, respectively, is conserved, at present we do not know whether these symmetries
+are conserved in the dark sector. In the framework of SUp8q-QGR, the fact that the sym-
+metry of the whole Universe as a quantum system is SUp8q, which is simply connected,
+means that the dark sector should conserve CPT. If the visible and dark sectors interact
+non-gravitationally, CPT may break - but not necessarily - in each sector. Nonetheless, it
+must be conserved be their ensemble.
+4.4.2
+Curvature of the parameter space
+Invariance of physics under a transformation similar to (4.11) means that in general the 4D
+parameter space Ξ is curved and has a nontrivial metric gµν. Reparameterization (4.11)
+– 27 –
+
+makes 4 of the 10 components of gµν arbitrary. However, the remaining 6 parameters are
+undetermined. In classical Einstein gravity these components are determined by the distri-
+bution of matter, more precisely by 6 independent components of the energy-momentum
+tensor Tµν, through Einstein equations. This implies that for a given Tµν - more precisely
+its trace T which is frame independent - diffeomorphism of the spacetime preserves its
+scalar curvature10. It is therefore legitimate to ask how the metric gµν is determined in
+SUp8q-QGR and whether a curvature preserving condition exists. Here we show that the
+curvature of parameter space Ξ is irrelevant for the physics.
+Proposition 2: The curvature of SUp8q parameter space of subsystems can be made
+trivial by a SUp8q gauge transformation, under which the Universe and its subsystems are
+invariant.
+Proof: Consider the compact version of Ξ defined in (4.5). At any point x P Ξ consider the
+orthogonal basis ei P TΞx, i “ 0, ¨ ¨ ¨ , 3, where TΞx is the tangent space of Ξ at x. Ricci
+curvature tensor and scalar curvature can be determined using sectional curvatures of 2D
+subspace of TΞx expanded by pairs pei, ejq, i ‰ j and vis-versa, see Appendix E for coordi-
+nate independent definition and relation between Ricci, Riemann, and sectional curvatures.
+Consider the 2D surface Σij Ă Ξ generated by parallel transport of pei, ejq, i ‰ j. Such
+surfaces are compact. As Ξ is the parameter space of SUp8q symmetry of subsystems,
+Σij, i, j P t0, ¨ ¨ ¨ , 3u, i ‰ j can be considered as diffeo-surfaces of subsystems and their
+ADiff deformation would be equivalent to application of SUp8q symmetry group, which
+does not change observables of the corresponding subsystems.
+Crucially, these ADiff
+do not need to preserve sectional curvatures at x. Thus, Ricci curvature tensor is not in
+general preserved, and this does not have any impact on quantum observables. A general
+diffeomorphism can be decomposed as in (4.1). Because size of diffeo-surfaces is irrele-
+vant for homomorphism of their ADiff with representations of SUp8q, scaling of Σij’s
+and thereby Ξ is similarly irrelevant for quantum observables of subsystems. Therefore,
+curvature of the parameter space Ξ is irrelevant for the dynamics of the Universe and its
+quantum subsystems. In particular, in the non-compact version of parameters or when
+areas of Σij are scaled to infinity, Ξ can be considered as a flat space.
+Proposition 2 has crucial contribution in: construction of dynamics; what we may call
+classical limit of SUp8q-QGR despite the fact that it is fundamentally quantum; its rela-
+tionship with Einstein gravity; and physical interpretation of what we perceive as curvature
+of spacetime in presence of matter and its fluctuations observed as gravitational waves. For
+instance, an immediate conclusion from this proposition is that the parameter space Ξ of
+the SUp8q symmetry of subsystems cannot be by itself the classical spacetime. Nonethe-
+less, in Sec. 5 we show that they are conceptually related. From now on we indicate the
+arbitrary metric of the 4D parameter space Ξ as ηµν (or its 2-dimensional analogue for 2D
+10In general relativity, for a given distribution of matter, e.g. vacuum, there is an equivalence class of
+metrics which lead to the same curvature. They are related by 4 constraints that present reparameterization
+(diffeomorphism) of the spacetime. The 6 d.o.f of gµν determines whether the scalar curvature changes by
+this transformation. The remaining 4 d.o.f. are associated to redefinition of coordinates without changing
+observables [109].
+– 28 –
+
+parameter space of each diffeo-surface) to distinguish it from effective classical metric gµν,
+which we obtain in the next section. Finally, Proposition 2, which demonstrates embedding
+of the geometric connection in the SUp8q gauge connection provides a natural explanation
+for gravity/gauge duality conjecture [79–81, 110–112].
+5
+Classical curved space and its relationship with quantum states of
+subsystems
+In the previous section we showed that the geometry of the parameter space Ξ of SUp8q
+symmetry representations by subsystems is arbitrary and can be gauged to be flat. This
+conclusion raises an enigma, if we want to interpret the parameter space as the spacetime.
+Thus, one might argue that SUp8q-QGR is already ruled out. For instance, deformation
+of x-ray spectral lines emitted from accretion disk of black holes [113], mirage image of
+black holes [114], and many other observed phenomena are evidence of curved spacetime
+in presence of strong gravity. In fact, using geometrical interpretation of spacetime, in
+classical physics we can measure coordinate-independent relative distance between subsys-
+tems and indirectly observe the curvature of spacetime. Nonetheless, in this section we
+relate these observations to quantum state of subsystems of the Universe and interpret the
+observered classical spacetime as effective (averaged) measure of the quantum state. For
+this purpose we begin with a brief review of speed limit in quantum mechanics, which its
+classical analogue was the reason behind development of special and general relativity, and
+eventually the interpretation of curved spacetime as gravity.
+5.1
+Speed limit from quantum uncertainties
+Introduction of non-Euclidean geometries in physics began with Hermann Minkowski, who
+unified the description of Maxwell’s electromagnetic equations using what is now called
+Minkowski metric. After observation of the constant light speed and discovery of special
+relativity by Einstein, works by Henry Poincar´e and Minkowski showed that Minkowski
+geometry can also explain Lorentz transformations.
+In general relativity the indefinite
+geometry of Minkowski space is generalized to a coordinate-dependent indefinite metric.
+Following this historical path, it seems logical to use the emergence of a finite speed of
+state change - information transfer - in quantum mechanics to investigate the relationship
+between classical spacetime and quantum state of subsystems of the Universe in SUp8q-
+QGR.
+The origin of limited speed of information transfer in quantum mechanics is the Heisenberg
+uncertainty relation between time and energy:
+∆t∆E ě ℏ{2
+(5.1)
+where ∆ means standard deviation of measurement outcomes of time t and energy E,
+respectively. In addition, the following relation applies to standard deviations of any two
+– 29 –
+
+statistical variables S1 and S2:
+∆S1∆S2 ě 1
+2pS1S2 ´ S2S1q
+(5.2)
+where the bar in the r.h.s. means statistical average. It is clear that the r.h.s. of (5.2)
+is zero unless S1 and S2 are operationally correlated, that is measurement of one of the
+quantities affects subsequent measurement of the other.
+Using (5.1-5.2), in 1945 Mandelstam and Tamm obtained a lower limit for the time neces-
+sary to transform a quantum state ˆρ1 to another perfectly distinguishable state ˆρ2 [100].
+Several other speed bounds have been later discovered [115, 116], see e.g. [117] for a con-
+cise review and references. Some of these bounds are equivalent to Mandelstam-Tamm
+limit [118] and are experimentally reachable for a set of initial states and final states [119].
+The Mandelstam-Tamm minimum time for change of a quantum state ˆρ1 to state ˆρ2 such
+that the two states are completely distinguishable, that is xˆρ1ˆρ2y “ 0, can be written
+as [51, 120–122]:
+∆t ě ℏ
+?
+2
+cos´1 Apˆρ1, ˆρ2q
+b
+Qpˆρ1, ˆHq
+,
+(5.3)
+Apˆρ1, ˆρ2q ” trp
+a
+ˆρ1
+a
+ˆρ2
+:q,
+Qpˆρ, ˆHq ” ´1
+2trpr
+a
+ˆρ, ˆHs2q
+(5.4)
+where ˆH is the system’s Hamiltonian and the geodesic distance in BrbHss used in the
+definition of Apˆρ1, ˆρ2q is the Winger-Yanase skew information metric [115]. For pure states
+this distance measure is equal to the fidelity Fpˆρ1, ˆρ2q defined in (4.2).
+The duration lower limit (5.3) gives a physical meaning to the time variable that appears
+in the Heisenberg uncertainty relation as standard deviation of time necessary for changing
+the state of a quantum system [123]. However, due to the non-uniqueness of the square-
+root of a matrix11, equation (5.3) can be applied without ambiguity only to pure states.
+For mixed states obtaining a speed limit is more complicated and several approaches and
+suggestions can be found in the literature [51, 117, 122]. In what concerns application of
+(5.3) to SUp8q-QGR in the next subsection, the density matrix ˆρ of the Universe includes
+all the subsystems. Therefore, no state is traced out and ˆρ can be in principle considered as
+pure. However, as we are only interested in the SUp8q sector of subsystems, we assumed
+that the contribution of internal symmetries to the density matrix is traced out. For this
+reason, ˆρ can be mixed. Nonetheless, for the sake of simplicity, we overlook complexities
+related to mixed states and use Mandelstam-Tamm inequality as a formal description.
+5.2
+Effective path in the parameter space
+To investigate the relationship between quantum state and apparent geometry of classical
+spacetime in the framework of SUp8q-QGR, we consider ˆρ1 to be the state of subsystems
+11It is straightforward to find examples of matrices ϱ with the same ϱ2 which satisfies properties of a
+mixed/incoherent density matrix.
+– 30 –
+
+of the Universe after selection of a reference observer and a clock. Due to the tracing of
+internal symmetries and global entanglement, the density matrix ˆρ1 comprises states of all
+subsystems, including clock and reference.
+The density operator ˆρ2 is assumed to be the infinitesimal variation of ˆρ1 for dt variation
+of the time parameter. Thus, ˆρ2 “ ˆρ1 ` δˆρ1. We assume that the clock is chosen such
+that in (5.3) the minimum time variation is achieved and (5.3) becomes an equality. This
+is possible for any state, at least asymptotically [118, 119].
+After Taylor expansion of
+functionals A and Q defined in (5.4), and insertion of the results in (5.3), we find:
+2
+ℏ2 Qp ˆH, ˆρ1qdt2 “ ´ 1
+ℏ2 trr
+a
+ˆρ1, ˆHs2dt2 “ trp
+a
+δˆρ1
+a
+δˆρ1
+:q ” ds2
+WY
+(5.5)
+Separation dsWY between states ˆρ1 and ˆρ1 ` δˆρ1 according to the Wigner-Yanase skew
+information [115, 116] is independent of the basis chosen for the Hilbert space and how
+this basis is parameterized. In the parameter space, tracing operation in the last equality
+guarantees the independence of equalities from parameterization. If the dynamics is uni-
+tary, that is if all the operations on the system are POVM, the evolution of state ˆρ1has
+a well-defined path in the Hilbert space. However, operators and functionals in the first
+equality of (5.5) are functions of the parameters pt, r, θ, φq. The dependence of equalities
+in (5.5) only on the time variation dt makes the parameterization corresponding to the
+minimum time variation for achieving dˆρ analogous to a classical reference frame at rest
+with respect to the chosen reference frame:
+Λds2
+WY ” ds2 “ gµνdxµdxν,
+x “ pt, r, θ, φq,
+g00 “ 2Λ
+ℏ2 Qp ˆH, ˆρ1q, gµν “ 0, µ||ν ‰ 0.
+(5.6)
+Notice that in (5.6) we have used Cartesian coordinates for the parameter space. The
+role of the dimensionful constant factor Λ is to ensure correct dimensionality of quantities,
+because the Fubini-Study metric is dimensionless. Giving the fact that Hilbert space of a
+quantum system is projective, one can always multiply sWY by a constant without affecting
+physical interpretations.
+A coordinate transformation x ” pr, θ, φ, tq Ñ x1 ” pr1, θ1, φ1, t1q does not change states ˆρ1
+and ˆρ1`δˆρ1 and thereby the last equality in (5.5). However, in general the variation time dt1
+with respect to the new clock parameter t1 would not be minimal and Mandelstam-Tamm
+bound takes its general form:
+2Λ
+ℏ2 Q1p ˆH1, ˆρ1qdt12 ” g1
+00dt12 ě Λ trp
+a
+δˆρ1
+a
+δˆρ1
+:q “ ds2
+(5.7)
+Moreover, the metric gµνpxq Ñ g1
+µνpx1q takes a more general form:
+ds2 “ g1
+µνdxµdxν “ g1
+00dt12 ˘ g1
+iidx1idx1i
+(5.8)
+where we have assumed a parameter transformation such that g1
+00 ą 0 and g1
+0i “ g1
+i0 “ 0.
+In addition, assuming isotropy of the spatial parameters xi, i “ 1, 2, 3 (We justify this
+assumption after discussing the physical meaning of (5.8)), g1
+ij’s should have the same sign.
+– 31 –
+
+For this reason we have assumed g1
+ij ą 0, @i, j and factorized their sign. The operator ˆH1
+in (5.7) is the Hamiltonian for the new clock parameter t1. By comparing (5.8) with (5.7)
+we find that if ds2 ě 0 (or equivalently ds2
+WY ě 0), the constraint (5.7) is satisfied only if
+the sign of spatial part in (5.8), and thereby the signature of the metric g1
+µν is negative.
+Appendix F describes expansion of state ˆρ and its variation with respect to ˆLlm generators
+of SUp8q.
+The above results were first reported in [39].
+In one hand, they demonstrate that the
+origin of the limited speed of light/signals and Lorentzian metric of the classical spacetime
+is the quantum nature of the Universe. On the other hand, although Mandelstam-Tamm
+inequality can be interpreted as the reason for speed limit, it is not enough for relating
+quantum uncertainty to the geometry of classical spacetime. The SUp8q-QGRfills the gap
+by interpreting the spacetime as parameter space of a quantum Universe.
+5.2.1
+Path in the full parameter space
+In the above formulation we assumed that the contribution of internal symmetries in the
+density matrices are traced out. Without such simplification, we should add the contribu-
+tion of collective parameter α defined in (4.6) to (5.8):
+ds12
+G ” Λ trp
+a
+δˆρG
+a
+δˆρG
+:q “ gµνpx, tαuqdxµdxν `
+ÿ
+i,jPt1,¨¨¨ ,Nu
+gijpx1, tαuqdαidαj (5.9)
+ds2 “ Λ trαp
+a
+δˆρG
+a
+δˆρG
+:q ” gµνpxqdxµdxν
+(5.10)
+where for the sake of simplicity of notation we have kept the same symbol for the effec-
+tive metric before and after tracing on G symmetry. The index G is used wherever the
+contribution of G symmetry is not traced. To be more explicit, the set tαu is assumed
+to include N parameters presented collectively as α. They characterize the representation
+the internal symmetry G by quantum states of subsystems. In (5.9) they are considered
+to be continuous. In this way, the metric (5.9) has the form of a p1 ` 3q ` N-dimensional
+Riemann geometry and ds2
+G is the affine parameter of the average path of ˆρG state in this
+extended parameter space. On the other hand, as SUp8q and G are by assumption orthog-
+onal and independent gµi “ 0 for any parameterization or reparameterization of the two
+sectors. Moreover, as G is assumed to have a finite rank, its representations have usually
+discrete parameters. Therefore, gij ‰ 0 only for discrete values of tαu. Fig. 2 schematically
+presents geometry of parameter space according to (5.9) and (5.10) and their relation12.
+In analogy with exchange of virtual particles with negative amplitude of 4-momentum
+q2 ă 0 in t-channel, states with ds2 ă 0 can be called virtual. Indeed, for pure states
+12One might claim that the relation between classical affine parameter and quantum state of Universe’s
+subsystems is not new, because in semi-classical general relativity geometric part of the Einstein equation
+is related to expectation value of energy-momentum tensor. However, the latter is a classical quantity and
+it is well known that its quantum counterpart is not well defined and has - at least naively - an infinite zero
+mode [124, 125]. By contrast, density matrix is an intrinsically quantum and well defined quantity.
+– 32 –
+
+Figure 2. A schematic presentation of SUp8q-QGR parameter space. The lowest (darker) surface
+depicts the parameter space of SUp8q symmetry - the spacetime - of subsystems. The path inside
+this surface is the average - classical - path of the quantum states of subsystems in the parameter
+space.
+Orthogonal to this surface depicts the collective parameter α of the internal symmetry
+G. Surfaces at discrete values of α remind that as a finite-rank Lie group, representations of G
+are in general characterized by discrete quantities - quantum numbers. The background presents
+parameter space of representations of SUp8q ˆ G as a continuum. The path crossing surfaces is
+the average path of subsystems in the full parameter space.
+ds2 ě 0 always. Therefore, a necessary condition for ds2 ă 0 is that the state ˆρ and its
+presumed evolution ˆρ ` dˆρ should not be pure or reachable.
+Despite the crucial role of (5.9) and (5.10) for understanding the nature of classical space-
+time and its relation to quantum realm and content of the Universe, they have little use for
+understanding evolution of subsystems, because they do not indicate what governs vari-
+ation of the quantum state ˆρ. This is similar to general relativity. Although a suitable
+form for the metric may be chosen based on symmetries of the matter distribution - for
+instance a spherically symmetric metric for a non-rotating spherically symmetric distribu-
+tion of matter - one needs a dynamical equation such as Einstein equation (in classical
+case), to relate spacetime geometry to evolution of matter. In Sec. 6 we describe dynamics
+of SUp8q-QGR.
+5.2.2
+Proper (equal-time) distance and causality
+The introduction of a size parameter r in the Hilbert space of subsystems does not specify
+how it is determined. Indeed, despite the fact that its absolute value can be redefined and
+is not a measurable, geometrical distance on 3D hypersurfaces of constant time, that is
+for a given state of quantum clock, is a measurable quantity. According to the definition
+of affine separation (5.5) and its expansion as the metric (5.8), the infinitesimal proper
+– 33 –
+
+da
+ds
+G
+dt
+ds
+dxdistance dl12 between two states ˆρ1 and ˆρ2 “ ˆρ1 ` δˆρ1 is:
+dl2
+12 ” Λ2trp
+a
+δˆρ1
+a
+δˆρ1
+:q
+ˇˇˇˇ
+dt“0
+(5.11)
+The parameter space distance between any two states is:
+l12 “ min
+ż
+γ12
+dl12,
+@γ12
+(5.12)
+where γ12 is a path in the Hilbert space connecting states ˆρ1 and ˆρ2. By construction the
+r.h.s. of (5.11) and (5.12) are independent of the Hilbert space basis and reparameteriza-
+tion.
+Locality and causality:
+In dynamical models (quantum or classical) with a background
+spacetime locality is usually defined as decreasing of interactions and correlation with dis-
+tance.
+In particular, the limited speed of light and information transfer according to
+Mandelstam-Tamm inequality (5.3) automatically limit the spatial extent of causal inter-
+actions. More generally, locality can be defined as tensor product decomposition/structure
+(TPS) similar to (3.8) such that interactions are bounded to factors [126]. Such definition
+is dynamical and depends on the resolution available to observers. Moreover, in SUp8q-
+QGRas all subsystems interact through gravity - SUp8q symmetry - locality based on
+TPS is an approximation. Using the relation between metric of classical spacetime and
+variation of quantum state of subsystems, in analogy with general relativity, we can define
+a lightcone for each state ˆρ in the Hilbert space. States ˆρ ` δˆρ are called causally related
+to ˆρ if ds2 ě 0. Two states ˆρ1 and ˆρ2 are causally related if their lightcones intersect. This
+definition is global and independent of clock, observers, and resolution of their experiments.
+5.3
+Interpretation of affine separation and its curved geometry
+We notice that in (5.7) and thereby in (5.8) the only quantity which is independent of
+the bases chosen for the Hilbert space and its parameterization is the affine separation ds.
+Although the unitary evolution of state ˆρ1 with respect to the clock follows a well defined
+path in the Hilbert space, it has no clear orbit in the parameter space, because in general
+ˆρ1 is a superposition of all eigen states |xy - in the old quantum mechanic language its path
+is not sharp. Therefore, the metric in (5.8) is not a genuine physical entity and must be
+considered as an effective representation - an order parameter representing the quantum
+state ˆρ1. Even g1
+00 in the r.h.s. of (5.7) cannot be considered as a measurable quantity,
+because its value can be redefined by changing time parameter t1. When an observer cannot
+explicitly detect SUp8q-symmetry related quantum (QGR) effects, their influence on the
+subsystems is indirectly perceived and interpreted as gravity convoyed by the curvature of
+the parameter space - the spacetime.
+Our demonstration that this effective metric has always a negative signature justifies our
+claim in previous sections that parameter space of the SUp8q-symmetry Hilbert space of
+– 34 –
+
+subsystems has R3`1 geometry. In special and general relativity, the signature of met-
+ric is dictated by the observation of the constant speed of light in classical vacuum. In
+SUp8q-QGR this is a direct consequence of quantum speed limit. Moreover, considering
+proposition (1) and the dependence of g1
+00 in (5.8) (or equivalently g00 in (5.6)) on the def-
+inition of a quantum clock, a redefinition of time can be viewed as changing entanglement
+between clock subsystem and the rest of the Universe. Despite these observations, one
+may wonder whether considering a specific signature for the parameter space is necessary.
+Indeed, giving the fact that Hilbert spaces of subsystems are defined on C, it cannot be
+ruled out that in some circumstance (applications) one may need to extend the parameter
+space, notably the time parameter, to complex plane. An example of such cases in QFT
+without gravity is a thermal model, which can be treated as a zero temperature QFT with
+complex time. There are also examples of field theory models in which a field plays the
+role of time parameter. It is shown that in such models the signature of the metric can
+change [127]. Therefore, not only the curvature but also the signature of the parameter
+space Ξ is not a physical observable. Nonetheless, for being consistent with the average
+geometry, which is an observable, it is convenient to use a Lorentzian metric for x space.
+The close relation between uncertainty relation, signature of the metric of the parameter
+space (spacetime), and its Lorentz symmetry in the framework of SUp8q-QGR shows that
+modifying Heisenberg uncertainty relation, as proposed in some QGR models [101, 102],
+may affect Lorentz invariance or equivalently the constant of speed of light and graviton,
+which are stringently constrained [107, 128, 129].
+Recalling Einstein’s opinion about the nature of spacetime described in the Introduction, he
+thought that only the affine separation has a proper physical meaning [40] (and references
+therein). However, if the spacetime is not a physical entity - otherwise there had to be the
+infamous ether - the association of the metric tensor to a geometry seems meaningless. The
+modern view of metric/gravity as a spin-2 quantum particle apparently solves this puzzle.
+But, it does not explain why this field determines the geometry of the spacetime, which
+according to Einstein does not have a physical reality of its own. SUp8q-QGR provides
+straightforward explanation for these puzzles.
+Finally, a short interpretation of holographic principle applied to black holes and ADS/CFT
+is discussed in Appendix G.
+5.3.1
+Relation with coherence and distinguishability
+The Wigner-Yanase skew information satisfies conditions defined in [50] for quantifying the
+amount of coherence of a quantum state and can be used as a measure of coherence [130].
+Therefore, the speed limit of information transfer is related to quantum coherence.
+In
+particular, for the completely incoherent state ˆρCI ” ř
+i ρi|iyxi|, with ř
+i ρi “ 1,
+ˆH|iy “
+λi|iy, QpˆρCI, ˆHq “ 0 and equation (5.3) leads to ∆t Ñ 8, meaning that the state of the
+system is stable and does not change, unless it is non-unitarily modified by application of
+an operator ˆH1, such that r ˆH, ˆH1s ‰ 0. In SUp8q-QGR, this corresponds to a state of the
+Universe where in the energy eigen basis, subsystems are decohered to a simple statistical
+– 35 –
+
+distribution with no quantum superposition. This is very far from the state of the Universe
+we are observing. Therefore, our Universe cannot be completely decohered and it should
+be eventually possible to detect the signature of quantumness in cosmological observations.
+See e.g. [21, 131, 132] for attempts to quantify such signal.
+As a measure of coherence must be monotonic [50] and there is only one completely coherent
+state ˆρCC ” NCC
+ř
+i,j |iyxj| where NCC is a normalization constant, the angle between all
+coherent states of a Hilbert space is zero and the constraint (5.3) on their evolution time
+becomes ∆t ě 0. This is a trivial constraint. Therefore, ˆρCC has fastest variation to a less
+coherent state. In SUp8q-QGR an initial completely coherent state may be - at least in
+part - responsible for the inflationary expansion of the early Universe. We leave exploration
+of such possibility for future works.
+If ˆρ1 and ˆρ2 are orthogonal states, that is trpˆρ1ˆρ2q “ 0, and both states are pure, we find
+∆t ě ℏπ{2?2Q „ ℏ{∆E, where ∆E is the variance of energy for state ˆρ1. Considering the
+statistical meaning of variance, larger ∆E means that in the energy basis, the two states
+are less distinguishable. In particular, if ∆E Ñ 8, ∆t Ñ 0. This means that despite the
+orthogonality of ˆρ1 and ˆρ2, they are not distinguishable and speed of variation from one to
+the other approaches that of completely coherent states.
+In conclusion, quantum speed limit, coherence, and distinguishability are related properties
+of quantum systems and processes. In the framework of SUp8q-QGR, the relationship
+between the average metric and quantum speed limit can be used to interpret what is
+classically called spacelike events, as unitarily unreachable states. Indeed, considering the
+ambiguity of (5.3) for mixed states described in the footnote 11, one can find pair of density
+matrices with cos´1 Apˆρ1, ˆρ2q ą π{2 of the Universe or even complex value for Apˆρ1, ˆρ2q.
+Specifically, such cases can be found when one of the matrix densities is pure and the other
+one is incoherent. In future works we will use these properties to explore physics of black
+holes in SUp8q-QGR, where horizon crossing changes the sign of ds2 of the average metric.
+6
+Evolution
+So far we constructed and studied the Hilbert space of SUp8q-QGR as a model for the
+Universe and its subsystems without considering the dynamics. In this section along with
+the review of dynamics of SUp8q-QGRpresented in [39] and with more details in [38], we
+clarify some issues which were not discussed in the previous works.
+Experience from QFT has shown that we can use variational principle at all perturbative
+order to describe dynamics of many-body quantum systems. However, in quantum me-
+chanics and QFT Lagrangians (or Hamiltonians) of models are usually constructed based
+on their classical limit model and are subsequently quantized, for instance by using path
+integral formalism or replacing canonical variables by their associated quantum operators.
+On the other hand, SUp8q-QGR is constructed as a quantum system, and in principle it
+does not have a classical analogue. In fact as we discussed in Sec. 2.2, for ℏ Ñ 0 the alge-
+bra (2.1) becomes Abelian and the model becomes trivial. Therefore, the construction of a
+– 36 –
+
+Lagrangian for SUp8q-QGRmust be based on the fact that Hilbert spaces of the Universe
+HU and its subsystems Hs represent SUp8q symmetry. For the whole Universe the evident
+candidate for a Lagrangian is a SUp8q invariant functional, describing all possible excur-
+sion of its states in the Hilbert space. For subsystems the functional must be in addition
+invariant under the finite rank G symmetry.
+6.1
+The whole Universe
+In SUp8q-QGR the Universe as a whole is by construction static.
+In fact, according
+to symmetry description of quantum mechanics axioms [26], any single closed quantum
+system without structure is static and trivial. More generally, we know that in quantum
+mechanics and quantum gravity evolution must be considered as relative concept [30] to
+avoid inconsistencies. Thus, any isolated quantum system, such as the whole Universe
+without anything outside it, is necessarily static.
+Nonetheless, a static Lagrangian - a
+general functional of parameters invariant under symmetries of the system - can be defined
+and variational principle can be applied to it. Specifically, minimization of this functional
+determines equilibrium state of the system. In the case of the whole Universe as a closed
+system in SUp8q-QGR, representations of SUp8q symmetry of the form (2.3) depend
+on continuous parameters13, and usual variational method straightforwardly applies to
+the Lagrangian. Fluctuations of parameters around their equilibrium values, obtained by
+variational method, can be interpreted as quantum fluctuations of the ground state. In the
+operational view employed in Sec. 3, these fluctuations correspond to random application
+of the members of BrHUs – SUp8q. In the next subsections we make these points more
+quantitative.
+6.1.1
+Formal description of dynamics
+In absence of a background spacetime in SUp8q-QGR the Lagrangian functional must be
+defined on the Hilbert space and must be invariant under SUp8q. In particular, for any
+SUpNq, @N symmetry group, trace of multiple hermitian generators trpT a1T a2 ¨ ¨ ¨ q are
+constant and invariant under group conjugate operation, that is g´1g1g “ g:g1g, @g, g1 P
+SUpNq. Moreover, these traces depend only on the normalization of generator trpT aT bq ”
+1{2δab, anti-symmetric structure coefficients fabc defined in (2.1), and symmetric anti-
+commutation coefficients dabc ” 2 trptT a, T buT cq. Therefore, these traces can be used to
+construct a symmetry invariant functional. Moreover, in analogy with QFT Lagrangians, it
+is enough to consider the lowest order traces of generators. Higher orders can be considered
+as higher order quantum corrections and obtained from a path integral generator functional.
+Under these considerations, the lowest order SUp8q-invariant functional can be written
+13The group SUp8q has also representations depending only on discrete parameters [64, 65]. However,
+they are dense in Rp⊭q. This is the reason for their homomorphism with Virasoro algebra without central
+charge discussed in Sec. 2.2.3.
+– 37 –
+
+as:
+LU “
+ż
+d2Ω
+„1
+2
+ÿ
+a, b
+trpL˚
+apθ, φqLbpθ, φqˆLa ˆLbq ` 1
+2
+ÿ
+a
+ˆ
+Lapθ, φqtrpˆLaˆρUq ` C.C
+˙ȷ
+(6.1)
+d2Ω ”
+b
+|ηp2q|dpcos θqdφ
+(6.2)
+where a ” pl, mq and C-number amplitudes La determine the contribution of SUp8q
+generators ˆLa to the Lagrangian. The dimensionless SUp8q coupling constant is included
+in the amplitudes. The density matrix ˆρU describes the quantum state of Universe as a
+whole. The scalar ηp2q is determinant of the metric of 2D diffeo-surface ηp2q
+µν , µ, ν P θ, φ.
+By definition the whole Universe is in a pure state, because there is nothing outside and
+entangle to it, which could have been possibly traced out. Therefore, its density matrix
+can be written as ˆρU “ |ΨUyxΨU|. We should also remind that the integration over angular
+coordinates of diffeo-surface is part of the tracing operation, because generators (2.3) of the
+SUp8q symmetry are defined at each point of the diffeo-surface. This reflects the fact that
+diffeomorphism of diffeo-surface in the neighborhood of each point is independent from
+deformation around other points up to continuity condition.
+To be consistent with Lagrangians used in QFT we assume that LU P R. Using represen-
+tation (2.3) for ˆLlm it is straightforward to see that ˆL:
+lm “ p´1qpm`1q ˆLl,´m, see Appendix
+B.2. This means that (2.3) is an anti-hermitian representation. For this reason there is
+no trpˆL:
+a ˆLbq term in (6.1) and phases of amplitudes La in the first term of (6.1) are irrel-
+evant. In addition, in the second term ˆρU is hermitian. Therefore, complex conjugate of
+trpˆLaˆρUq is not independent from itself, and Lapθ, φq must be real, otherwise LU becomes
+complex. This conclusion is not limited to the representation (2.3) and applies to any
+(anti-)hermitian representations of SUp8q. In [39] it is explicitly shown that, as expected,
+applying variational principle with respect to amplitudes La and components of ˆρU leads
+to a vacuum state as the equilibrium solution. This result is consistent with a corollary
+of the axioms of quantum mechanics reformulated to present symmetry as a foundational
+concept [26]. It states that a quantum Universe as a whole, that is when its subspaces are
+ignored, is trivial. A short summary of this approach to quantum mechanics can be found
+in appendices of [39]).
+6.1.2
+A Yang-Mills whole Universe
+The action LU is a formal description. In particular, it does not clarify how the ampli-
+tudes La’s change under global Lorentz and Poincar´e transformations of coordinates of
+the diffeo-surface pθ, φq and application of SUp8q group, which up to a global scaling of
+the area is homomorphic to local reparameterization of diffeo-surface. The Lagrangian LU
+must remain invariant under these transformations. To this end and for getting a better
+insight into the analytical form of the amplitudes, we analyze the integrand of (6.1) for its
+transformation under aforementioned symmetries.
+– 38 –
+
+First of all the surface element d2Ω is invariant under reparameterization of angular coor-
+dinates. Thus, each term in the integrand of (6.1) must be reparameterization invariant.
+Moreover, amplitudes La must be invariant under translation θ Ñ θ ` θ0, φ Ñ φ ` φ0
+for arbitrary constant shift of the coordinates origin by θ0 and φ0. This condition would
+be satisfied if La’s is a differential operator with respect to coordinates pθ, φq. In addi-
+tion, the operator must be invariant both under reparameterization of the diffeo-surface
+and SUp8q transformation. The only analytical expressions with these properties are 2-
+forms in coordinates of the diffeo-surface, which must be also singlet of SUp8q. Thus,
+La ˆLa ” F µν
+a ˆLa, µ, ν P cos θ, φ and the Lagrangian functional (6.1) takes the form of a 2D
+Yang-Mills Lagrangian for SUp8q symmetry:
+LU “
+ż
+d2Ω
+„ 1
+2 trpF µνFµνq ` 1
+2trp
+
+DˆρUq
+ȷ
+,
+µ, ν P θ, φ
+(6.3)
+Fµν ” F a
+µν ˆLa ” rDµ, Dνs,
+Dµ “ pBµ ´ Γµq1 `
+ÿ
+a
+iλAa
+µ ˆLa,
+a ” pl, mq
+(6.4)
+F a
+µνF µν
+a
+“ L˚
+aLa,
+@a.
+(6.5)
+where Dµ is 2D covariant and gauge preserving derivative with an appropriate 2D con-
+nection Γµ [133, 134] (see e.g [134] for a recent review) and λ is a dimensionless coupling
+constant. It can be included in Aa
+µ, but here we include it explicitly to keep the formulation
+similar to the standard Yang-Mills theory . The symbol 1 presents unit operator of the
+sup8q algebra. In analogy with QFT the coordinate dependent vector Aa
+µ can be called
+the SUp8q gauge field. Nonetheless, once again we emphasize that in contrast to a QFT
+on an spacetime, pθ, φq parameterize representation of SUp8q symmetry.
+In the last term of integrand in (6.3), the exact expression of the differential operator 
+
+D
+depends on the representation of 2D Euclidean group by the state |ψUy. For a scalar-type
+state 
+
+D “ ÐÝ
+DµÝÑ
+D
+µ “ ÐÝB µÝÑB
+µ. For a 2D spinor state it can be written using zweibein vectors:
+
+
+D “ iσ0σizµ
+i
+ÐÑ
+D µ
+,
+Dµ “ pBµ ` Γµq1 `
+ÿ
+a
+iλAa
+µ ˆLa,
+(6.6)
+Γµ ” 1
+8ωµijrσi, σjs ” 1
+4ωµijσij
+,
+rωµsi
+j “ zi
+ν
+ˆ
+Bµpzν
+j q ` zσ
+j Γν
+µσ
+˙
+,
+(6.7)
+σij ” i
+2rσi, σjs
+,
+ηp2q
+ij zi
+µzj
+ν “ ηp2q
+µν
+,
+ηp2q
+µν zµ
+i zν
+j “ ηp2q
+ij .
+(6.8)
+where σi i “ t1, 2u are two of the N Ñ 8-dimensional representation of Pauli matrices,
+and σ0 is the third Pauli matrix. The tensors zµ
+i ’s are zweibeins (analogous to vierbein
+(tetrads) for 2D geometry) [134, 135]. 2-dimensional metrics ηp2q
+µν and ηp2q
+ij are arbitrary and
+flat metrics of the parameter space - diffeo-surface - of the Hilbert space HU, respectively.
+Despite the presence of a connection term in (6.4), it does not affect F µν. In addition, as
+a function defined on the 2D diffeo-surface Γµ can be decomposed to spherical harmonic
+functions, which are in turn generators of SUp8q algebra, and without loss of generality
+can be included in Aa
+µ term in (6.4). This is an explicit demonstration of the Proposal
+2 applied to the Universe as a whole. Consequently, the choice of a special geometry for
+– 39 –
+
+the diffeo-surface of the Universe is equivalent to fixing SUp8q gauge and has no effect
+on measurable quantities, namely, F a
+µν - up to an unobservable rescaling of the area of
+the diffeo-surface. We should also remind that in 2D spaces the 2-form F a
+µν has only one
+independent nonzero component. Therefore, the number of degrees of freedom in the two
+sides of (6.5) is the same.
+6.1.3
+A topological Universe
+Independence of the Lagrangian (6.3) from geometry of diffeo-surface M and its parame-
+terization means that the SUp8q Yang-Mills model of the whole Universe is topological.
+Topology of 2D compact Riemann surfaces are classified by the Euler characteristics:
+ż
+d2Ω Rp2q “ 4π χpMq,
+χpMq “ 2 ´ GpMq
+(6.9)
+where Rp2q is the 2D scalar curvature of the diffeo-surface. The Euler characteristic χpMq “
+2´GpMq depends only on the surface’s genus G. The topological nature of the Lagrangian
+(6.3), in particular the pure gauge term implies that without loss of generality its integrand
+must be proportional to Rp2q:
+trpF µνFµνq 9 Rp2q
+(6.10)
+This relation can be also considered as equality, because the area of the diffeo-surface is
+arbitrary and is not an observable. Thus, (6.10) establishes a relation between geometry of
+the parameter space of SUp8q representation and amplitude of the SUp8q gauge field, and
+at lowest quantum order, variational principle relates the gauge field to the quantum state
+of the Universe ˆρU. Consequently, there is no arbitrary or unrelated quantity in the model.
+Moreover, (6.10) explicitly demonstrates the relation between the Yang-Mills Lagrangian
+(6.3) and what we perceive as gravity through curvature of the classical spacetime. Indeed,
+the number of degrees of freedom of pure Yang-Mills fields in dimension D is DpD ´
+1q{2, which is the same as the number of independent components of the Ricci curvature
+tensor Rµν after taking into account D coordinate reparameterization constraints - or
+equivalently 4 energy-momentum conservation constraints in classical Einstein theory of
+gravity. Interestingly, this relation between degrees of freedom of the two models exists
+in any dimension.
+In any case, none of these quantities are dynamic and due to the
+SUp8q symmetry, all the states are physically equivalent. Nonetheless, the Lagrangian
+(6.3) distinguishes between non-equivalent representations of SUp8q.
+In a geometrical view, if local details of the Universe are not distinguishable, only its global -
+topological - properties may characterize its states. In SUp8q-QGR the only characterizing
+global property is the topology of the diffeo-surface, which determines non-homomorphic
+representations of the SUp8q symmetry [65, 66] of the Universe. In other words, the only
+possible difference between whole universes is the representation of SUp8q by their Hilbert
+spaces. Thus, we can consider the genus of the diffeo-surface as the only initial condition for
+the Universe in SUp8q-QGR. Alternatively, one may assume that a quantum universe may
+be in a superposition of independent representations of SUp8q, that is in super position
+– 40 –
+
+of Lagrangians of type (6.3) with different values for the integral in (6.9). We give a short
+assessment of consequences of coherent mixing of global typologies in Sec. 6.4 and leave
+its detail investigation to a future work.
+6.2
+Evolution of subsystems
+As we discussed in sections 4 and 5, when the Universe is divided to subsystems and a
+reference subsystem and a quantum clock are chosen, every pair of their 4 continuous
+parameter space can be considered as parameters of a SUp8q symmetry under which
+physical properties of subsystems must be invariant. The presence of a clock means that
+in contrast to the whole Universe, the ensemble of subsystems can be indeed treated as a
+dynamical quantum system and in analogy with QFT a Lagrangian can be associated to
+it. For this purpose, we proceed in the same manner as we did for the whole Universe and
+consider the lowest order functional invariant under SUp8q ˆ G as the Lagrangian. The
+formal description of this functional has the following form:
+LUs “
+1
+4πL4
+P
+ż
+d4x
+b
+|ηp4q|
+„ 1
+4
+ˆ
+ÿ
+l,m,l1,m1
+trpL˚
+lmpxqLl1m1pxqˆLlm ˆLl1m1q `
+ÿ
+a,b
+trpT ˚
+a pxqTbpxq ˆTa ˆTbq
+˙
+`
+ÿ
+l,m,a
+trpLlmpxqTapxqˆLlm b ˆTaq `
+ÿ
+lm
+Llm trpˆLlm b 1G ˆρspxqq ` 1
+2
+ÿ
+a
+Ta trp1SUp8q b ˆTa ˆρspxqq
+ȷ
+.
+(6.11)
+where T a’s are generators of the finite rank internal symmetry G of subsystems, and LP ”
+a
+ℏGN{c2 is the Planck length. The metric ηµν is an arbitrary metric for the 4-dimensional
+parameter space presented in Cartesian form and ηp4q is its determinant. Amplitudes Llm
+and Ta are normalized to be dimensionless. The first two terms present self-interaction of
+SUp8q and G symmetries, respectively. If the generators of symmetry groups are chosen
+to be traceless, the third term in (6.11) would be zero. The forth and fifth terms present
+interaction of subsystems due to SUp8q and G symmetries, respectively. Notice that we
+have considered an arbitrary metric for the parameter space, despite Proposition 2, which
+allows the choice of a Minkowski metric. We explain this choice later in this section.
+Following the same line of arguments given for (6.1) about the invariance of parameter
+space under coordinates transformation i.e.
+Poincar´e symmetry, invariance under both
+SUp8q and G transformations, and demonstration that the resulting action should have
+the structure of a Yang-Mills model, we conclude the following final form for the Lagrangian
+– 41 –
+
+LUs:
+LUs “
+ż
+d4x
+a
+|η|
+„
+1
+16πL2
+P
+trpF µνFµνq ` 1
+4trpGµνGµνq ` 1
+2
+ÿ
+s
+trp
+
+Dˆρsq
+ȷ
+,
+µ, ν P t0, 1, 2, 3uη
+(6.12)
+Fµν ” F lm
+µν ˆLlm ” rDµ, Dνs,
+Dµ “ pBµ ´ Γµq1 ´
+ÿ
+lm
+iλAlm
+µ ˆLlm,
+L2
+P F lm
+µν F µν
+lm “ L˚
+lmLlm.
+(6.13)
+Gµν ” Ga
+µν ˆT a ” rD1
+µ, D1
+νs,
+D1
+µ “ pBµ ´ Γµq1 ´
+ÿ
+a
+iλGBa
+µ ˆT a,
+L4
+P Ga
+µνGµν
+a
+“ T ˚
+a T a. (6.14)
+As explained earlier, 
+
+D depends on the representation of Lorentz symmetry by subsystems.
+For instance, for Dirac spinors it is:
+
+
+D ” γ0γieµ
+i
+ÐÑ
+D µ
+,
+Dµ “ pBµ ´ Γµq1 ´
+ÿ
+lm
+iλsAlm
+µ ˆLlm ´
+ÿ
+a
+iλGBa
+µ ˆT a,
+(6.15)
+Γµ ” 1
+8Ωµijrγi, γjs ” 1
+4ΩµijΣij
+,
+rΩµsi
+j “ ei
+νpBµpeν
+j q ` eσ
+j Γν
+µσq
+(6.16)
+Σij ” i
+2rγi, γjs
+,
+ηijei
+µej
+ν “ ηµν
+,
+ηµνeµ
+i eν
+j “ ηij
+,
+i, j P t0, 1, 2, 3uη. (6.17)
+where γi are Dirac matrices, and other quantities in (6.16-6.17) are analogous to those of
+(6.6 - 6.8), but in the 4-dimensional parameter space of subsystems.
+As expected there is no cross term between pure gauge field terms of SUp8q and G, because
+by construction they are orthogonal and the full symmetry of subsystems is their tensor
+product. On the other hand, without any cross term, interaction of gauge fields is simply
+similar to QFT in curved spacetime. However, according to proposition 2 the metric ηµν
+of the parameter space is arbitrary. Therefore, the question arises how inSUp8q-QGRdo
+G gauge fields - including those of the Standard Model - interact with gravity ?
+To answer this question, we remind that even in Einstein gravity, the difference between
+flat and curved space, arises only when derivatives of measurable fields are involved [133].
+In particular, the definition of strength of Yang-Mills fields given in (6.13) shows that
+nonzero connection of a curved space does not affect it, and becomes relevant only in the
+field equation. Therefore, at Lagrangian level, only the metric convoys the curvature of
+spacetime (here the parameter space) to gauge fields. However, the metric can be locally
+considered to be flat. Therefore, in this respect there is difference between SUp8q-QGRand
+Einstein gravity. Moreover, in various ways an interaction between Fµν and Gµν arises in
+SUp8q-QGR.
+First of all, although trpˆLlm ˆTaq “ 0, higher order cross terms are not null. The lowest
+gauge invariant term not included in LUs is trpF µνFµνqtrpGµνGµνq, which corresponds to a
+4-vertex. It does not generate mass neither for G nor for SUp8q, because it depends on the
+field strength rather than gauge fields Aµ
+lm and Bµ
+lm
+14. Even without explicitly introducing
+14Aµ
+lm and Bµ
+a present the decomposition of the wave function of SUp8q and G gauge bosons with respect
+to generators. It is also why they appear in pair in the Lagrangian, exactly similar to ˆρ when it is pure, or
+when it is written in its canonical purification form ˆρ “ ?ˆρ?ˆρ.
+– 42 –
+
+such higher order interaction terms, due to the interaction of density matrix of subsystems
+with both SUp8q and G bosons, they will arise in the path integral constructed from the
+Lagrangian (6.12), and thereby in the effective action. In addition, in the field equation
+obtained from the Lagrangian (6.12) both ˆρs and Gµν contribute to the source term for
+the gravity field strength Fµν.
+We should also remind that even in classical physics both Einstein gravity and Yang-
+Mills action can be derived in an arbitrary background geometry from the assumption of
+consistent self-interactions [136]. Another evidence for the interaction of gravity and gauge
+bosons in general relativity is the fact that scalar curvature of a universe containing only
+radiation - gauge bosons - vanishes everywhere, because the trace of energy-momentum
+tensor is zero. This means that sectional curvatures must be null or cancel each others
+everywhere, see (E.5).
+Thus, the spacetime manifold of such universe is Ricci-flat and
+behaves like classical vacuum, despite the presence of energy.
+Finally, giving the relation between the Pontryagin topological class, average metric gµν
+defined in (5.10), and variation of the density matrix of subsystems ˆρs, it would be useful
+to use gµν rather than an arbitrary metric for calculation of physical observables such
+as Green’s functions.
+Of course, in this case all the constituents of the model become
+intertwined and self-interactions make evolution equations highly nonlinear. Nonetheless,
+in analogy with background field approaches in QFT, such as Kadanoff-Beim method which
+includes background fields, considering gµν as a background field would lead to a better
+approximation. In any case, the choice of the metric ηµν corresponds to fixing the SUp8q
+gauge and does not have any impact on the exact calculations.
+6.2.1
+Possibility of gravitational P, C, and CP violation
+Lagrangian of the whole Universe conserves parity and CP. This observation makes sense,
+because the parameter space of the SUp8q symmetry is not directional. This implies that
+even after the division of the Universe to subsystems, parity and CP must be globally
+preserved. Nonetheless, a priori subsystems may locally break these discrete symmetries.
+There are various reasons for such possibility.
+Yang-Mills theories have topologically non-trivial vacuum structure. Because this subject
+is also relevant for the crucial issue of cosmological constant, we discuss it in some details
+in Sec. 6.4. Here we only remind that in the Standard Model, the possibility of non-
+trivial vacuum in both electroweak and QCD interactions leads to what is called strong
+CP problem, see e.g. [137, 138, 157] for review and more details. This problem is closely
+related to the issue of axial Up1q anomaly. In both cases the problem can be summarized
+in so called a Θ-trem in the action:
+LΘ ” Θg2
+32π2
+ż
+d4x ?η ˜FµνFµν
+,
+˜Fµν ” 1
+2ϵµνρσFµν
+(6.18)
+where F is gauge field strength and g is its coupling.
+The integrand in this action is
+a total derivative and the integral is a topological invariant - the Pontryagin topological
+– 43 –
+
+class.
+However, the dual field ˜Fµν has a parity opposite to Fµν.
+Therefore, when it
+is added to the Yang-Mills action, the sum does not preserve the party.
+Moreover, as
+conjugation symmetry C is preserved by both actions, CP symmetry is also broken by their
+sum. Although parity symmetry is violated in electroweak interaction, there is stringent
+constraint on the parameter Θ for the QCD.
+In SUp8q-QGR the quantum mediator of gravity is a Yang-Mills gauge field. Therefore,
+complexities related to topological structure of vacua, axial Up1q anomaly which arises
+when fermions can be considered massless, and violation of P and CP due to a Θ-term
+similar to (6.18) may arise in the model.
+In particular, as the energy scale in which
+quantum gravity effects become important - presumably the Planck mass MP - is much
+larger than the mass of all SM particles, they can be considered as massless and axial Up1q
+anomaly would be more serious. Nonetheless, there are various solutions for this issue. For
+instance, at Planck scale strong non-perturbative gravitational interactions can increase
+the effective mass of particles and prevent the anomaly. On the other hand, in the strong
+gravity regime of early Universe these anomalies might have seeded baryons and leptons
+asymmetries through sphaleron-type phenomena, see e.g. [140, 141] for reviews, or at least
+contributed to them. We leave investigation of such effects to future works.
+In what concerns parity violation in gravity sector, at present there is no observation
+or constraint even for classical gravity. Nonetheless, future generations of gravitational
+wave detectors should be able to test it [142–144]. Assuming no P or CP is observed in
+gravitational sector, a possible solution for violation of discrete symmetries by Θ´term of
+SUp8q-QGR sector is introducing a pseudo-scalar gravitational axion field φg, analogous
+to a Peccei-Quinn Up1q proposed to solve the QCD strong CP problem. It replaces the
+constant Θ in an action similar to (6.18). A gravitational axion must be in close relation
+with the gravity sector.
+However, it does not seem that in SUp8q-QGR there is any
+fundamental constituent as candidate for φg.
+The only natural Up1q symmetry in the
+model, which emerges after the division of the Universe to subsystem, namely the relative
+size of the diffeo-surfaces of subsystems, is used to introduce a size (or distance) parameter
+in the model. On the other hand, in Sec. 6.4 we argue that several phenomena may act as
+an effective axion at low energies.
+In summary, many of observed and/or predicted complexities of non-Abelian Yang-Mills
+will arise in SUp8q-QGR and must be thoroughly investigated.
+6.2.2
+Is there a conflict between a SUp8q Yang-Mills gauge theory as quantum
+gravity and present observations ?
+The naive answer to this question is yes.
+In Einstein theory gravity is a spin-2 field
+and various observations, both in weak field and strong field regimes have confirmed this
+property. By contrast, irrespective of the gauge symmetry group, Yang-Mills theories are
+based on spin-1 fields. However, at present none of the many observations and tests of
+the gravity as an interaction attain the quantum gravity regime. For the time being, all
+the direct or indirect tests of gravity are based on the detection of the variation of average
+– 44 –
+
+metric (5.10) and its direct and indirect effects on matter and radiation. For instance, a
+gravitational wave temporarily changes the metric, which then detected as a change in the
+phase of a laser beam. Therefore, our knowledge of gravity properties applies only to the
+classical gravity. In the next section we show that the classical limit of SUp8q-QGR is
+exactly Einstein gravity. Thus, SUp8q-QGR is consistent with with classical gravity.
+The historical parallel of this situation is the nuclear strong force, which for a few decades
+was assumed to have mesons as intermediate particles. It was only after the discovery of
+internal structure of hadrons that the Yang-Mills Quantum Chromo-Dynamics (QCD) was
+accepted as the fundamental force, both between elementary quark and gluon fields and
+their clustering as baryons and mesons. The essential reason for this confusion was the
+fact that accelerators and cosmic ray detectors were only able to investigate low energy
+hadronized processes, rather than their fundamental partons - quarks and gluons - con-
+stituents. At present tests of quantum gravity are not even at a stage analogous to pions
+for the strong interaction.
+6.3
+Classical limit
+In quantum mechanics and QFT the classical limit of a model is defined as when ℏ Ñ
+0.
+However, as we discussed in Sec.
+2.2, in SUp8q-QGR such assumption makes the
+algebra (2.1) Abelian and the symmetry group Â8
+1 Up1q, unless at the same time MP Ñ
+8, meaning no gravity. Consequently, in absence of a background spacetime the whole
+structure of SUp8q-QGR will collapse and the model becomes meaningless. Therefore,
+rather than considering ℏ Ñ 0 limit, we consider the situation where gravity is observable,
+but its detail quantum structure, i.e. its Yang-Mills nature is not. Specifically, for an
+observer who cannot detect and discriminate the infinite number of pl, mq colors of the
+SUp8q symmetry and their corresponding quantum fields F µν
+lm , the only way to become
+aware of their presence is through the affine parameter ds in (5.10) - as Einstein suggested
+- and the effective path of subsystems in their SUp8q symmetry parameter space - the
+spacetime - perceived through an effective metric gµν. In this case, the observer sees the
+first term of the Lagrangian (6.12) as one block - just a scalar function of parameters (the
+spacetime).
+For the case of 2D parameter space of the whole Universe, we already demonstrated the
+relation between the SUp8q Yang-Mills action and 2D scalar curvature. In [38] we used
+the relation between 4D scalar curvature and sectional curvatures, along with the fact that
+surfaces generated by pairs of parameter space coordinates can be considered as diffeo-
+surfaces representing SUp8q of subsystems, to relate SUp8q Yang-Mills term in (6.12) to
+4D scalar curvature. More generally, it is demonstrated [145] that for (pseudo)-Riemannian
+spaces with dimension D ě 3, there is always a metric for every scalar function such that
+the scalar curvature obtained from that metric is equal to the function. Therefore, the first
+term of (6.12) can be always considered to be equal to a scalar curvature:
+ż
+d4x
+a
+|η| trpF µνFµνq
+classical
+GGGGGGGGGGGGA
+limit
+9
+ż
+d4x
+a
+|g|Rp4q
+(6.19)
+– 45 –
+
+Similarly, the contribution of Alm
+µ
+in covariant derivatives will be added to the parameter
+space (spacetime) connection, and the resulting Lagrangian will become similar to a G
+symmetry Yang-Mills model in a curved spacetime with Einstein-Hilbert gravity action:
+LUs Ñ Lcl.gr “
+ż
+d4x
+a
+|g|
+„
+κRp4q ` 1
+4trpGµνGµνq ` 1
+2
+ÿ
+s
+trp
+
+Dˆρsq
+ȷ
+.
+(6.20)
+where the dimensionful constant κ9M2
+P is necessary for making the Lagrangian function
+dimensionless. It is also clear that other possible scalars obtained from curvature tensor, e.g.
+RµνRµν are of higher quantum order and at least ℏ2{M2
+P times smaller than Rp4q ” gµνRµν.
+It is remarkable that we began the construction of SUp8q-QGR from abstract assumptions
+about symmetry and quantum mechanics, but rediscovered exactly Einstein gravity at
+low energies - when gravity seems classical. For instance, if we add a term proportional
+to F µν ˜Fµν to (6.12), in general it cannot be written as a function of Rp4q for the same
+metric as for F µνFµν. Therefore, it adds a completely independent pseudo-scalar function,
+which break parity unless its amplitude is controlled with yet another pseudo-scalar - a
+gravitational axion - to be consistent with the lack of parity violation in gravity sector, so
+far. Another possibility is a bimetric gravity [146–148], where the dual term is considered as
+scalar curvature of another metric. The second metric is conjectured to become significant
+at high energies. Such models are related to massive graviton models [149] and lead to the
+deviation of the speed of gravitational waves propagation from the speed of light [150]. At
+present such deviation is not consistent with present observations [107].
+6.4
+Cosmological constant
+The Lagrangian LUs does not include an explicit constant term, which can be interpreted
+as a cosmological constant. In one hand, as the nature of the observed dark energy is still
+unknown, the need for a cosmological constant in SUp8q-QGR Lagrangian is not certain.
+On the other hand, although a cosmological constant is still consistent with cosmological
+observations, the growing evidence for what is called the Hubble tension and other anomalies
+of the standard ΛCDM, see [151] for a review, may point to more elaborate models [152]
+for dark energy. Evidently, many models are suggested in the literature as alternative to
+a cosmological constant, and they are relevant in SUp8q-QGR - except those proposing
+modification of classical Einstein gravity.
+Nonetheless, in the following subsections we
+briefly discuss several processes specific to SUp8q-QGR, which potentially can lead to an
+effective cosmological constant. We leave their detail assessment to future works.
+Any candidate process should answer following questions before it can be considered to
+play a role in the observed dark energy:
+a. How does its amplitude vary with the expansion of the Universe ?
+b. Can a remnant with a roughly constant energy density consistent with the observed
+dark energy survive ?
+– 46 –
+
+c. Can it solve the coincidence problem ? In other words, can it justify dominance of
+dark energy well after galaxy formation without fine-tuning ?
+For each of the following candidate phenomena for dark energy in the framework of SUp8q-
+QGR we describe very briefly and qualitatively to which extent, if any, the above questions
+may be answered. We leave quantitative assessments to future works.
+6.4.1
+Static SUp8q field
+Although the Lagrangians (6.3) and (6.12) do not explicitly include a constant term, in
+analogy with static electromagnetic, the field strength F µν can include a constant part.
+Indeed, magnetic field is observed at cosmological scales, including in Interacluster Medium
+(ICM), but not in the CMB. Nonetheless, we know that cosmological magnetic field is
+generated by dynamo in the core of stars and by shocks.
+These processes need large
+overdensities and absence of a primordial magnetic field seems natural. By contrast, gravity
+is generated by all fields and in all circumstances. Moreover, due to the non-commutative
+nature of SUp8q, Yang-Mills graviton source themselves, which is not the case for photons.
+Condensation of gravitons and its similarity to parton swarm in high energy QCD are also
+suggested in the framework of QGR [153, 154] and black hole physics [155].
+In analogy with the QCD glueball, the Universe may be considered as a gravity-ocean
+in which other fields are immersed.
+In Einstein gravity gravitons self-interact, but its
+quantized version is nonrenormalizable.
+By contrast, as a quantum Yang-Mills theory
+SUp8q-QGR is renormalizable. This means that its self-interaction is controlled and does
+not lead to instability. It is certain that in a classical view of the spacetime, gravitons are
+everywhere, and considering their huge number, in a quantum view they form a condensate.
+In fact, in dark energy models which propose what they call vacuum of QCD or other Yang-
+Mills fields as the origin of dark energy, it is the condensation of gauge bosons (and/or
+fermions) that plays the role of dark energy. Therefore, it is rather IR modes that are able
+to preserve their quantum coherence. In this case, the energy density of the condensate is
+expected to be roughly proportional to OpV ´4q rather than usual QGR (UV) estimation
+of OpM4
+P q. Thus, the answer to the question (a) is yes. The answers to questions (b) and
+(c) are more subtle and need quantification.
+6.4.2
+Θ´vacuum
+The cosmological constant term in the classical Einstein equation has been famously in-
+terpreted as the vacuum energy [124, 156].
+Non-Abelian Yang-Mills models can have
+non-trivial vacuums, corresponding to a vanishing field strength F µν “ 0, but nonzero
+gauge field Aµ ” Aµ
+a ˆLa (for SUp8q symmetry). We remind that Aµ “ 0 is not gauge
+invariant: Aµ Ñ UAµU ´1 ` i{λpDµUqU ´1. Field configurations which can be written as
+i{λpDµUqU ´1 are called instantons and for large gauge transformation have in general non-
+trivial topologies. Topologically non-equivalent field configurations are classified by their
+winding number, which is equal to (6.18) with Θ “ 1. In fact, as we mentioned before, the
+– 47 –
+
+integrand of (6.18) is a total derivative DµKµ, where Bardeen current Kµ is:
+Kµ “
+1
+16π2 ϵµνρσtrpAνFρσ ´ AµAρAσ
+(6.21)
+The value of (6.18) with Θ “ 1 depends only on the configuration of the field Aµ on
+the boundary. This means that LΘ has the same value as a vacuum, even if F µν ‰ 0
+except for |x| Ñ 8. Different classes of field configurations represent homotopy group π3
+of SUp2q, the global rotation symmetry which remains even after fixing the gauge [157].
+As π3pSUp2q “ Z, the value of LΘ“1 P Z and a general vacuum state |Θy can be written
+as:
+|Θy “
+ÿ
+n
+e´inθg|ny
+(6.22)
+where |ny are instanton configurations with winding number n. If in the early Universe
+where QGR processes were important Yang-Mills graviton field had a Θ-vacuum configu-
+ration of the form (6.22), smooth decrease of its strength Fρσ could not change the config-
+uration of zero modes. Indeed the idea of a cosmological instanton generated by inflaton
+field is explored e.g. in [158]. But to associate cosmological constant to QGR instanton,
+significant deviation, even at classical level is necessary, because in 4D spacetime Einstein
+gravity is not topological, see e.g. [159] for an example of such models.
+In the framework of Schwinger-Dyson Closed Time Path-integral (CTP), the 1-Particle
+Irreducible (1PI) path integral of subsystems/fields can be formally written as:
+ZrJa; ˆρs ” eiWrJas “
+ż
+DΦaDΦb exp
+„
+iSpΦaq ` i
+ż
+d4x?´gJapxqΦapxq
+ȷ
+xΦa|ˆρp|Φby (6.23)
+where Φ generically presents all the fields and parameter space (spacetime) and G symmetry
+indices are implicit. Indices a, b P t`, ´u indicate fields on two opposite time branches.
+They are contracted by the diagonal tensor cab ” diagp1, ´1q and satisfy the condition
+xΦ`pxqy “ xΦ´pxqy. The last factor in (6.23) presents the projection of density matrix
+on the fields eigen vectors at initial time t0. Usually t0 Ñ ´8 is used. As in CTP every
+local observable is determined by integration of the generating functional (6.23) over the
+interval p´8, `8q and back, the past boundary condition affect observables at all time.
+In general, the result has the form of an other functional, which without loss of generality
+can be written as:
+xΦa|ˆϱ|Φby “ exppiFrΦa, Φbsδpt ´ t0qq
+(6.24)
+and the functional iFrΦa, Φbs can be added to the action iSpΦaq. Therefore, the effect of
+a non-perturbative configuration on the t0 Ñ ´8 boundary, such a instanton state (6.18)
+appear as a constant term in (6.23), and affect Green’s function.
+Concentrating on the SUp8q gauge fields in SUp8q-QGR, its instanton configuration only
+affects zero modes, that is IR variation of the fields. Thus, in principle it should be sepa-
+rated from other modes and treated as a constant phase. However, according to Proposition
+1 other modes, both gravitational and G symmetry related, remain entangled to the zero
+modes. Phenomenologically, this means that the effect of SUp8q instanton - Θ´vacuum
+– 48 –
+
+- on the subsystems can be parameterized by an amplitude φgpxq, which its effective con-
+tribution to the action (6.23) would be similar to an axion, but without a kinematic and
+mass terms of a true field. It is perceivable that at |x| ! 8, the effect of instanton settles
+to a roughly constant and behaves similar to a cosmological constant. In contrast to UV
+explanation of cosmological constant, very roughly |φg|2 „ OppMP H0q2q is expected. In
+QCD an analogous phenomenological model for the Θ-vacuum called topological liquid of
+instanton and anti-instanton offers a compelling non-perturbative description of the model
+in strong coupling regime, see e.g. [160] for a review.
+Finally, in what concerns criteria a, b, and c, the effective amplitude φg should somehow
+depend on the matter, because it is its dominant source. Satisfaction or not of the other
+two criteria needs a qualitative investigation.
+6.4.3
+Manifestation of global entanglement
+In Sec.
+6.4.2 the proposition 1 about entanglement of subsystems was crucial for our
+arguments in favour of a SUp8q topological vacuum as the origin or contributor to the
+observed dark energy. Even in the case of graviton condensation proposed in Sec. 6.4.1,
+this proposal is indirectly involved, because in absence of a global entanglement between
+subsystems, expansion of the Universe may reduce the amplitude of the quantum conden-
+sate to a negligibly small value, see e.g. simulations of a scalar field condensation in an
+expanding Universe in [161]. The question that arise is whether the global entanglement
+alone can explain the roughly constant density of dark energy.
+The content of the Universe is in large extent classical. Quantum systems and effects such
+as coherence and entanglement persist locally for a short time, before being decohered
+through interaction with the infinite environment of other subsystems/particles. In general
+decoherence increases entropy and releases energy.
+On the other hand, preservation of
+global entanglement of subsystems requires creation and/or transfer of entanglement, which
+in turn needs energy [162] (and references therein). This is presumably provided in part by
+the energy released by decoherence. This process should arrive to an equilibrium, otherwise,
+either the minimum global entanglement will be lost, which is in contradiction with the
+proposition 1, or the entanglement grows and reduces entropy, which is inconsistent with
+the second law of thermodynamics. We leave quantitative assessment of these processes,
+their consequences, and verification of criteria a-c, to future works.
+7
+Outlines and future perspectives
+In this work we investigated more thoroughly foundation and principle properties of the
+quantum gravity model dubbed SUp8q-QGR. After demonstrating that the model is truly
+quantum, we showed that what is perceived as spacetime can be interpreted as a parameter
+space inseparable and associated to the quantum subsystems of the Universe. We related
+the observed curved metric of the classical spacetime to the quantum state of the Universe
+– 49 –
+
+and demonstrated that it presents the average path of quantum subsystems in their param-
+eter space. We introduced a relative quantum dynamics and showed that its gravitational
+sector has the structure of a SUp8q Yang-Mills quantum field theory. Therefore, according
+to SUp8q-QGR the quantum mediator of gravity is a vector field like other fundamental
+interactions. Nonetheless, we demonstrated that in the classical limit - defined as when the
+fine structure of purely gravitational part of the Lagrangian and its quantum effects are not
+discernible for the observer - the resulting model is exactly Einstein gravity. We also briefly
+discussed potential explanations for the observed dark energy specific to SUp8q-QGR.
+Future works should investigate in detail these proposals as well as application of the
+SUp8q-QGR to other puzzling problems, in which quantum gravity may play an essential
+role, such as: microscopic structure of black holes and apparent loss of information behind
+their horizon; singularities, both cosmological and in black holes; the role of quantum
+gravity in early universe and inflation; observables and discriminating signals of these
+phenomena; and methods for testing SUp8q-QGR in laboratory. Regarding this last topic,
+it is important to insist that geometrical test of gravity would not be able to find the
+signature of a vector gravity. Only the detection of intrinsically quantum processes might
+be able to distinguish between the exchange of spin-1 and spin-2 force mediator.
+Regarding the construction of the model, the approach used here and in the previous works
+is in a QFT perspective. It would be interesting - and probably necessary - to study the
+model in the framework of quantum information theory, specially in what concerns reference
+frame of different subsystems, their comparison, and their perspective-dependence. This
+topic is specially crucial for studying the microphysics of black holes.
+A subject that we did not address in any detail here is the rise of the internal symmetry G.
+Although the potential for clustering and symmetry breaking is ubiquitous, it is not clear
+why and how the nature prefers some symmetries to others. SM symmetries are smallest
+non-Abelian symmetries and cannot be decomposed to yet smaller non-Abelian groups. Is
+this property related to the decomposition of SUp8q, i.e. the gravity force ? In principle
+this hypothesis can be investigated in the high energy colliders. Specifically, we should
+search for emergence of symmetries similar to what is observed in condensed matters in
+the distribution of partons and leptons at very high energies.
+Acknowledgment
+The author acknowledges the support of the Institut Henri Poincar´e (UAR 839 CNRS-
+Sorbonne Universit´e), and LabEx CARMIN (ANR-10-LABX-59-01).
+– 50 –
+
+Appendices
+A
+Properties of SUp8q tensor products
+Using decomposition of SUpNq group:
+SUpNq Ě SUpN ´ Kq ˆ SUpKq
+(A.1)
+and the fact that its r.h.s is a normal subgroup of SUpNq, it is straightforward to see that
+the repetition of (A.1) with K “ 2 leads to:
+SUpNq
+ˇˇˇˇ
+NÑ8
+–
+8
+â
+SUp2q
+(A.2)
+This is a direct demonstration of (2.6) and independent of the decomposition of associated
+generators in (2.7). Using (A.2) and the fact that n ˆ 8 “ 8 we conclude that tensor
+products of SUp8q are homomorphic to itself:
+pSUp8qqn – SUp8q
+(A.3)
+It is then straightforward to conclude that:
+SUp8q Ą
+M
+â
+SUpKq,
+M ă 8,
+@ K ă 8,
+(A.4)
+SUp8q –
+8
+â
+SUpKq,
+@ K ă 8,
+(A.5)
+where (A.4) can be concluded from repetition of (A.1) and the assumption that SUpN ´ MKq|N,MÑ8
+is represented trivially. As each SUpKq factor in turn has a Cartan decomposition,
+SUp8q –
+8
+â
+i“1
+Gi
+(A.6)
+where Gi is a finite rank Lie group. Consider one of Gi factors in (A.6). We call it G. Then,
+the symmetry of the whole Universe15 can be written as SUp8q – GˆG8, where G8 is an
+infinite rank group. For any G we can find finite N ď N1 such that SUpNq Ď G Ď SUpN1q.
+Using (A.1) and self-similarity of SUp8q [66], we conclude the following relations:
+SUpN1q ˆ G8 Ě rG ˆ G8 – SUp8qs Ě SUpNq ˆ G8
+(A.7)
+SUpN1q ˆ G8 Ě SUpN1 ´ Nq ˆ pSUpNq ˆ G8q – SUpN1 ´ Nq ˆ SUp8q – SUp8q
+(A.8)
+Thus, we conclude that G8 – SUp8q and G b SUp8q – SUp8q. If n subsystems with G
+symmetry are distinguished from the rest of the Universe, HU can be decomposed to:
+Gn ˆ SUp8q – Gn ˆ SUp8qn – pG ˆ SUp8qqn
+(A.9)
+For n Ñ 8:
+pG ˆ SUp8qqnÑ8 – SUp8q
+(A.10)
+BrHUs “
+â
+i
+BrHis –
+â
+i
+ˆ
+BrHGis b BrHSUp8qs
+˙
+–
+ˆâ
+i
+BrHGis
+˙
+b BrHSUp8qsq
+(A.11)
+15When there is no risk of confusion we identify BrHs with the symmetry group it represents.
+– 51 –
+
+A.1
+About definition of symmetries of a quantum system
+We remind that symmetries of a system are defined in two ways.
+If the dynamics is
+known and a Lagrangian is associated to it and its symmetries that is transformations
+preserving the Lagrangian, are called symmetries of the system. Alternatively, one may
+first associate a symmetry to a system, for instance, based on the observations of some
+properties, and construct a Hilbert space or a Lagrangian that preserve those symmetries.
+When the system is considered as a whole, the symmetries associated to the systems in
+these approaches are the same. However, when the system is composite, one has to be
+careful about the definition of what is called system’s symmetries. In particular, for many-
+body systems, there are local symmetries, presumably those belonging to each subsystem.
+The effective Hamiltonian or Lagrangian usually includes either strictly local - 1-point
+- interactions, or at most few-point interactions - mostly 2-point correlations. For this
+reason, symmetries preserving Lagrangian or Hamiltonian would be those of individual
+subsystems. In addition, Lagrangian or Hamiltonian may preserve some global (subsystem
+independent) symmetries.
+For example, Hamiltonian of Heisenberg spin chain/network model is invariant under the
+local SUp2q transformation of spin. In addition, if a periodic boundary condition is defined,
+it would be invariant under translation i.e. i Ñ i ` L, where L is the periodic length in
+units of lattice length. On the other hand, the Hilbert space in general represents the
+tensor product of the symmetry of subsystems. In the example of Heisenberg model, the
+symmetry of the total Hilbert space is SUp2qN, where N is the number of sites of the lattice.
+Nonetheless, quantum entanglement (correlation) may significantly reduce the symmetry
+of actual Hilbert space - the subspace of states with nonzero probability of occurrence -
+through a process known as many-body localization [97].
+B
+About quantization
+A system that satisfies axioms of quantum mechanics `a la Dirac and Von Neumann (or
+their variants) can be called quantum and does not need quantization. Indeed, it is well
+known that canonical commutation relations between conjugate operators is meaningful
+only if at least one of them is unbounded and has a continuous spectrum (see e.g. the
+chapter on uncertainty relation in [68]). The most direct evidence of this property is the
+fact that two finite dimensional hermitian operators ˆA and ˆB cannot satisfy the canonical
+quantization relation r ˆA, ˆBs “ ´iℏ1, because the trace of the l.h.s. would be null, whereas
+that of the r.h.s would be a finite nonzero value. Therefore, canonical quantization is useful
+and applicable only for quantization of classical theories, which usually include unbounded
+quantities such as spacetime or fields. Nonetheless, the Planck constant ℏ must be somehow
+involved in the construction of any quantum model, such that when ℏ Ñ 0 the model behave
+classically, that is conjugate observables, like coordinates and momentum commute with
+each other.
+– 52 –
+
+A typical example of quantum models without continuous degrees of freedom and usual
+uncertainty relation is quantum Heisenberg model of interacting spins on a lattice, which
+is widely used in condensed matters physics. The Hamiltonian of this model is ˆHH ”
+´J ř
+tiup⃗Si⃗Si`1q, where J is a coupling constant and summation is over all sites of the
+lattice.
+The Hilbert space at each site represents SUp2q symmetry and its generators
+ˆSi, i “ 1, 2, 3 can be normalized such that r ˆSi, ˆSjs “ ´iℏϵijk ˆSk. See also Appendix
+A.1 for more details about this model. It is evident that there is no pair of operators in
+this model that satisfies canonical quantization relation. Nonetheless, innumerable studies
+show that particles interacting according Heisenberg model behave as predicted by axioms
+of quantum mechanics. In particular, as expected, the 3 components of spins or angular
+momentum vectors become simultaneously measurable when ℏ Ñ 0.
+The above example shows that non-commutativity of the algebra of observables do in-
+deed replace the canonical quantization. Notably, this property is used in construction
+of several QGR models, such as: non-commutative geometry QGR [163–166], some string
+theories [167], and matrix models [168–171]. However, non-commutativity spacetime is
+stirringly constrained [172–174].
+B.1
+Dual of generators in spherical harmonic representation of sup8q
+As we discussed in Sec. 2.2.2, ˆJlm in (2.11) is a complex function. By using representation
+(2.3) for the generators of the sup8q algebra (2.1) and applying two sides of (2.11) to an
+arbitrary vector ψ P HU, which is also a complex function, we find:
+BYlm
+Bη
+B ˆJlm
+Bζ
+´ BYlm
+Bζ
+B ˆJlm
+Bη
+“ ´1.
+(B.1)
+where here η ” cos θ and ζ ” φ. Solutions of the first-order partial differential equation
+(B.1) determines dual operators (functions) ˆJlm.
+To solve equation (B.1) we first remind the definition of spherical harmonic functions Ylm:
+Ylmpη, ζq ” p´1qm
+d
+p2l ` 1q
+4π
+pl ´ mq!
+pl ` mq!Plmpηqeimζ
+(B.2)
+where Plmpηq is the associated Legendre polynomial for any l and |m| ď l. Equation (B.1)
+can be solved by factorizing the contribution of variables pη, ζq. The general solution for
+ˆJlm is:
+ˆJlmpη, ζ, η0, ζ0, ˆJlmpη0, ζ0qq “ η ´ η0 ` im ˆJlmpη0, ζ0qYlmpη0, ζ0q
+imYlmpη, ζq
+(B.3)
+where η0, ζ0 and Jlmpη0, ζ0q are integration constants. Choosing η0 “ ζ0 “ ˆJlmpη0, ζ0q “ 0,
+ˆJlmpη, ζq becomes:
+ˆJlmpη, ζq “
+η
+imYlmpη, ζq
+(B.4)
+– 53 –
+
+B.2
+Dual of continuous parameters in spherical harmonic representation of
+sup8q
+We use the representation (2.3) of the generators ˆLlmpη, ζq of sup8q, and properties of
+spherical harmonic functions Ylm and associated Legendre polynomials Plmpηq, to relate
+ˆLlm to the canonical quantization conjugates of parameters pη, ζq.
+For m “ 0, Pl0pηq ” Plpηq is the Legendre polynomial and generators ˆLl0 can be expended
+as:
+ˆLl0 “ iℏBYl0
+Bη
+B
+Bζ “ iℏ
+c
+p2l ` 1q
+4π
+BPl
+Bη
+B
+Bζ “ iℏ
+c
+p2l ` 1q
+4π
+l ` 1
+1 ´ η2
+ˆ
+ηPlpηq ´ Pl`1pηq
+˙ B
+Bζ
+“
+ˆ l ` 1
+1 ´ η2
+˙ˆ
+ηYl0pηq ´
+c
+2l ` 1
+2l ` 3Ypl`1q0pηq
+˙ˆ
+iℏ B
+Bζ
+˙
+(B.5)
+Thus:
+iℏ B
+Bζ “
+ˆ1 ´ η2
+l ` 1
+˙ˆ
+ηYl0pηq ´
+c
+2l ` 1
+2l ` 3Ypl`1q0pηq
+˙´1
+ˆLl0
+(B.6)
+Similarly,
+p´1qmeimζ ˆLl,´m ´ e´imζ ˆLlm “ ´2mℏ e´imζYlm
+B
+Bη
+(B.7)
+Using properties of spherical harmonic functions, namely Yl,´m “ p´1qmY ˚
+lm, it is straight-
+forward to show that ˆLl,´m “ p´1qm`1 ˆL:
+lm, and equation (B.7) can be written as:
+ieimζ ˆL:
+lm ` ie´imζ ˆLlm “ 2m e´imζYlm
+ˆ
+iℏ B
+Bη
+˙
+(B.8)
+Finally:
+iℏ B
+Bη “ ip2m e´imζYlmq´1peimζ ˆL:
+lm ` e´imζ ˆLlmq
+(B.9)
+It is possible to invert (B.5) and (B.8), and express the quantum conjugate of ζ and η,
+that is iℏ B{Bζ and iℏ B{Bη with respect to operators ˆLl0 and ˆLlm, respectively. However,
+their expression would be highly degenerate, because (B.5) and (B.8) are valid for all
+l ě 0 and |m| ď l. This observation demonstrates that quantization through the algebra
+(2.1) is much richer than the canonical quantization of the finite dimensional parameter
+space - the spacetime. Nonetheless, as we explained in Sec. 2.2.2, the parameters and
+their canonical conjugate can be considered as representation of operators corresponding
+to these observables.
+C
+Criteria for definition of quantum subsystems
+General conditions for the division of a quantum system with Hilbert space H to proper
+quantum subsystem is studied in details in [25]. They can be summarized as the followings:
+a. There must exist sets of operators tAiu Ă BrHs such that @ ˆa P tAiu and @ ˆb P tAju,
+and i ‰ j, rˆa,ˆbs “ 0;
+– 54 –
+
+b. Operators in each set tAiu must be local;
+c. tAiu’s must be complementary, meaning bitAiu – End pBrHsq.
+In [25] the locality condition (b) refers to locality of operations regarding the background
+spacetime. In SUp8q-QGR there is no background spacetime. Nonetheless, we can extend
+the concept of locality to the diffeo-surface of parameters, and thereby to the Hilbert space
+BrHUs.
+C.1
+Symmetries and Hilbert space of subsystems
+The division of BrHs to subspaces tAiu as subsystem makes sense only if the commuting
+subspaces can be distinguished from each others. This means that BrHs symmetry must
+be broken to a direct product Â
+i Gi, leading to BrHs Ě Â
+i gi where gi is the algebra
+associated to group Gi and H Ě Â
+i Hi, where the Hilbert space Hi represents Gi. Group
+factors Gi’s do not need to be all different.
+The ensemble of subsystems with similar
+symmetries and representations present multiplicity of their type/family. In addition, the
+order of factors in the tensor product is not important. Thus, from now on tensor products
+of subsystems are assumed to be permutation symmetrized.
+We remind that a N-dimensional Hilbert space is invariant under UpNq “ SUpNq ˆ Up1q
+transformation. The Up1q factor corresponds to a global phase transformation of states.
+However, the symmetry of the quantum system can be a subgroup of UpNq. For instance,
+Hilbert space generated by eigen vectors of position operator ⃗ˆX in non-relativistic quantum
+mechanics is infinite dimensional and invariant under SUp8q ˆ Up1q. But the symmetry
+of ⃗ˆX and a quantum system (a particle) presented by its position is homomorphic to
+Up1q ˆ Up1q ˆ Up1q ˆ SOp3q – Rp3q ˆ SOp3q Ă Up8q.
+In addition, dimension of the
+Hilbert space is determined by representation of the symmetry rather than the rank of the
+symmetry group itself. Nonetheless, in all cases the symmetry group G Ď UpNq.
+C.2
+Conditions for division induced entanglement of subsystems
+If the dimension of subspace tAiu is NAi, the minimum dimension of BrHis must be Ni
+such that N2
+i ´ 1 ě NAi. The complementary condition (c) for division of a quantum
+system to subsystems dictates that ř
+i Ni “ N, where N is the dimension of Hilbert
+space of the system. Assuming the division to two subsystem, (A.1) leads to G Ě G1 ˆ
+G2, where G1 and G2 are symmetry groups represented by Hilbert spaces of the two
+subspaces. Iteration of this relation on each subsystem demonstrates that G Ě Â
+i Gi.
+Therefore, in general, division of a quantum system reduces the symmetry. Nonetheless,
+the dimension of tensor product of Hilbert spaces of subsystem N12 ” N1 ˆ N2 ě N, for
+N1, N2 ą 1, N ě 2 and N, N1, N2 ă 8. Considering the fact that entropy, specially von-
+Neumann entropy, is roughly proportional to the number of states, increasing dimension of
+the Hilbert space after division to subsystems implies increasing quantum randomness. On
+the other hand, quantum entanglement, which may similarly breaks symmetries, reduces
+– 55 –
+
+the entropy/randomness by correlating subsystems and reducing the number of states (see
+e.g. [175]). These opposite effects of symmetry breaking in quantum systems have crucial
+roles in determining their characteristics and behaviour.
+Relation between symmetry and number of states of subsystems can be quantified using
+quantum complexity and its application as a resource [176, 177]. If states of subsystems
+1 and 2 can be prepared using n1 and n2 qubits, respectively, their maximum complexity
+Cmax in isolation is Cmax “ 2n1 ` 2n2. By contrast, if they are considered together, their
+complexity is much larger: Cmax “ 2n1`n2 " 2n1 `2n2. In the second case the SUpn1 `n2q
+symmetry represented by the Hilbert space of the ensemble of qubits of the two sub-
+systems includes SUpn1q ˆ SUpn2q symmetry of isolated subsystems, according to (A.1).
+Thus, maximum quantum complexity of a quantum system reflects breaking of states sym-
+metry (coherence) of its subsystems by their entanglement. This is not a surprise, because
+SUpNq Ě bN{2SUp2q and SUp2q is the symmetry represented by a qubit.
+D
+Relation between reparameterization and diffeomorphism
+Diffeomorphism is a continuous and invertible map of a manifold M Ñ M. Parameteriza-
+tion is a continuous map M Ñ N, where in general M ‰ N. Thus, diffeomorphism can
+be considered as a special case of parameterization when N “ M. For this reason diffeo-
+morphism and reparameterization are usually considered as the same operation. Moreover,
+diffeomorphism of N induces a diffeomorphism transformation in M, when they are related
+under a reparameterization map.
+In SUp8q-QGR, for the whole Universe M “ HU and N “ D2 – Rp2q. A basis for HU
+consists of vectors |xy, @x ” pη, ζq P Rp2q. Reparameterization corresponds to a diffeomor-
+phism in Rp2q such that: x Ñ x1pxq @x P Rp2q. An infinitesimal SUp8q transformation of
+the basis |xy is |x1pϵqy “ p1 ` ϵ ř
+pl,mq ˆLl,mpxqq|xy. A general SUp8q transformation U “
+expp
+ş
+d2Ω ř
+pl,mq ϵlm ˆLl,mpxqq for any set of constants ϵlm and p|ψy P HUq Ñ |ψ1y “ U|ψy.
+Considering the differential representation of ˆLl,mpxq in (2.3), it is clear that reparameter-
+ization, that is a diffeomorphism in D2 is equivalent to a SUp8q transformation of HU.
+After division of the Universe to subsystems N – Rp4q and reparameterization corresponds
+to diffeomorphism in Rp4q and leads to M “ Hs Ñ Hs and x Ñ x1pxq @x P Rp4q. Similar
+to the case of the whole Universe, this transformation can be described as application of
+SUp8q parameterized by only a pair of parameters. This is possible because of property
+(A.3) of SUp8q and superposition of quantum states.
+E
+Coordinate independent definition of curvatures
+For any Riemannian or pseudo-Riemannian manifold pM, gq of dimension d ě 2 equipped
+with a Levi-Civita connection ∇ the (1,3)-Riemann curvature tensor at point p P M is
+defined as:
+RppX, Y qZ “ p∇X∇Y ´ ∇Y ∇X ´ ∇rX,Y sqZ
+(E.1)
+– 56 –
+
+Vector fields X, Y, Z P TMp and TMp is the tangent space at p. When X, Y, Z are
+chosen to be Bi ” B{Bxi basis of the tangent space for coordinates xi, i “ 0, ¨ ¨ ¨ d ´ 1, one
+recovers the usual coordinate dependent definition of the Riemann curvature tensor (we
+drop p because it corresponds to the point with coordinates xi):
+RpBi, BjqBk “ Rl
+kij,
+(E.2)
+Rijkl ” RpBi, Bj, Bk, Blq ”
+B
+RpBi, BjqBk, Bl
+F
+“ gmlRm
+kij
+(E.3)
+Using the notation defined in (E.3) for (0,4)-Riemann curvature tensor, sectional curvature
+KpΠq “ KpX, Y q with respect to a 2D plane Π Ă TMp containing two vectors X, Y P TMp
+at p P M is defined as:
+KpΠq ” KpX, Y q “
+RppX, Y, X, Y q
+xX, XyxY, Y y ´ xX, Y y2
+(E.4)
+Notice that KpΠq is independent of the choice of X and Y and depends only on the plane
+passing through them.
+It can be shown that xRppX, Y qZ, Wqy can be expanded with
+respect to sectional curvatures [178]. Using relations between different curvature tensors
+of a Riemannian manifold, the scalar curvature at point p P M is defined as [179]:
+Rppq “
+ÿ
+i‰j
+Rppei, ej, ei, ejq “
+ÿ
+i‰j
+Kppei, ejq
+(E.5)
+where ei, i “ 0, ¨ ¨ ¨ , d ´ 1 is an orthonormal basis of TMp. From (E.5) we conclude that
+there is only one sectional curvature at each point of a 2D surface and it is equal to its
+scalar curvature.
+F
+Expansion of density matrix and its variation
+Following the discussion in Sec. 4.3.1 about representation of SUp8q symmetry, pairs of
+parameters pη, ζq P Ξ, the space of linear operators BrbsHss can be generated by operators
+ˆLlmpη, ζq, which satisfy the algebra (4.8).
+With this generalization, SUp8q generators
+ˆLlmpη, ζq defined in (2.3) with η “ cos θ, ζ “ φ have the structure of a 2-form. To make
+this property explicit, we may write ˆLlmpη, ζq ” ˆLµν
+lm,
+Here we consider only SUp8q symmetry and do not explicitly consider G symmetry repre-
+sentation by the Hilbert space of Hs. The state of subsystems after tracing out G symmetry
+is ˆρ P BrbsHss can be expanded as the following:
+ˆρ “
+ż
+dx4a
+|η|
+ÿ
+l,m
+ρlmpxqϵµν ˆLµν
+lm
+(F.1)
+ż
+dx4a
+|η|
+ÿ
+l,m
+ρlmpxq “ 1,
+ρlmpxq ě 0
+(F.2)
+The range of pl, mq are defined in (2.3). The constant 2-form ϵµν in the integrand of (F.1) is
+necessary to make it a scalar of the parameter space Ξ. The expression ρlmpxqϵµν ” ρlm
+µν is
+– 57 –
+
+a 2-form field on the parameter space Ξ. Although it has the same geometrical structure as
+a gauge field strength, its physical interpretation is very different. Specifically, components
+ρlm
+µν are positive probability densities and must satisfy the constraint (F.2). By contrast,
+the gauge field strength presents the number of quanta of the gauge boson.
+Variation of the density matrix δˆρ P BrbsHss can be expanded in the same manner:
+δˆρ “
+ż
+dx4a
+|η|
+ÿ
+l,m
+δρlm
+µνpxqˆLµν
+lm
+(F.3)
+This expansion can be applied to (5.10) to determine affine separation as a function of
+quantum state variation. In principle, one can write a similar expansion with respect to
+ˆLµν
+lm operators for ?δˆρ:
+a
+δˆρ ”
+ż
+dx4a
+|η|
+ÿ
+l,m
+δρ
+1lm
+µν pxqˆLµν
+lm
+(F.4)
+Using δˆρ “ ?δˆρ?δˆρ, expansions (F.3) and (F.4) and description anti-commutation of
+generators ˆLµν
+lm, see Sec.
+6.1.1 and [62], It is straightforward to see that δρ
+1lm
+µν
+modes
+depend on the mixture of all δρlm
+µν modes, but remain local with respect to parameters x -
+the spacetime.
+G
+About holographic principle and black holes
+Holography principle [35–37] can be summarized as proportionality of the maximum num-
+ber of quantum states inside a null (light-like) boundary to its area rather than its vol-
+ume [35–37, 185]. In the framework of SUp8q-QGR and its interpretation of spacetime
+as a parameter space, a quantum system to which holographic principle applies can be
+considered as an approximately isolated, which its large/infinite number of degrees of free-
+dom (subsystems) are strongly interacting.
+Black holes as defined in classical Einstein
+gravity are examples of such systems. Hilbert spaces of such systems approximately repre-
+sent SUp8q and their spacetime boundary can be identified with the diffeo-surface of the
+SUp8q representation. The dependence of the Bekenstein entropy of black holes on the
+boundary area. Of course, as discussed in Sec. 3, area of diffeo-surface of truly isolated sys-
+tem representing SUp8q is irrelevant. Thus, dependence of the Bekenstein entropy is the
+evidence that a black hole is not a fully isolated quantum system. Indeed, it is observable
+from outside of its boundary.
+In the framework of SUp8q-QGR, Bekenstein-Hawking entropy[180] can be considered as a
+relative entropy of a highly but not completely isolated subsystem of the Universe. And the
+relation between SUp8q and diffeomorphism of the horizon is consistent with the proposal
+that the Bekenstein-Hawking entropy of black holes is a manifestation of the N¨other charge
+of the spacetime diffeomorphism[181].
+– 58 –
+
+G.1
+Comparison with gravity as entanglement and ADS/CFT
+It is useful to compare the emergence of an area/distance scale in SUp8q-QGR with the
+relationship between distance and quantum entanglement proposed in [41]. Motivated by
+the holography principle and ADS/CFT conjecture (see e.g. [182] for a review) [41] estab-
+lishes an analogy between reduction of the contact surface by stretching of two connected
+regions of a flexible object with constant volume, and reduction of quantum entanglement,
+measured by the mutual quantum information capacity. Moreover, the distance between
+two gravitating sources is interpreted as a measure of their quantum entanglement: longer
+distance, smaller their entanglement.
+By contrast, in SUp8q-QGR, it is rather the quantum entanglement of subsystems which
+relates parameters of their quantum states of subsystems to each others. Relative variation
+of states induces a wandering in the parameter space - the spacetime - that characterizes
+the quantum states. When quantum effects are not distinguishable, the average path of
+subsystems in this classical parameter space is perceived and interpreted as being due to
+a curved geometry generated by their gravitational interaction..
+In [38] we showed that in the SUp8q-QGR framework subsystems are in general Type
+III quantum systems [183, 184]. Thus, no boundary can be imposed on the continuous
+parameters, in particular on the value of distance parameter r. Moreover, subsystems are
+in general in a superposition similar to (4.6). Therefore, concepts such as horizon of a black
+hole, or more generally null surfaces to which holography principle applies [61, 185] lose
+their strict meaning. In fact, it is well known that even in general relativity horizon position
+depends on the parameterization of the metric, and thereby is not a physical boundary.
+Only the singularity is physical and persistent. Additionally, (5.9) and (5.10) show that
+when quantum effects of gravity are taken into account, black holes are not simple objects
+that everything about them can be parameterized by their mass, charge, and angular
+momentum.
+As δˆρ in (5.10) includes infinite number of independent components, see
+(F.3), many quantum states may generate the same effective (classical) metric gµν, despite
+being quantum mechanically very different. Moreover, in principle, differences of block
+hole states should include both the contribution of matter G and SUp8q symmetries, and
+self-interaction of gravity. In turn, it is expected that they affect black hole evaporation.
+Evidently, the classical formulation of black holes in general relativity is not sensitive to
+these quantum details. Thus, in principle these additional aspects should be able to solve
+the problem of apparent information loss. Evidently, this claim needs detailed investigation
+and verification.
+In SUp8q-QGR black hole horizon RBH should be considered as the boundary of the
+inaccessible parameter space for the external observer, because it is where the sign of ds2
+changes. Larger RBH means larger the volume of parameter space of indistinguishable
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+the exchanged quanta/particle is virtual, that is when its 4-momentum q has q2 ă 0. In
+this case, even at tree order it would not be possible to know from which vertex a scattered
+particle is coming.
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diff --git a/xNE0T4oBgHgl3EQf-QLn/content/tmp_files/load_file.txt b/xNE0T4oBgHgl3EQf-QLn/content/tmp_files/load_file.txt
new file mode 100644
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@@ -0,0 +1,2977 @@
+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf,len=2976
+page_content='Prepared for submission to JHEP  SUp8q-QGR: Emergence of Gravity in an Infinitely  Divisible Quantum Universe  Houri Ziaeepoura,b  aInstitut UTINAM, CNRS UMR 6213, Observatoire de Besan¸con, Universit´e de Franche Compt´e,  41 bis ave.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' de l’Observatoire, BP 1615, 25010 Besan¸con, France  bMullard Space Science Laboratory, University College London, Holmbury St.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' Mary, GU5 6NT,  Dorking, UK  E-mail: houriziaeepour@gmail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='com  Abstract:  SUp8q-QGR is a foundationally quantum approach to gravity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' It assumes  that Hilbert spaces of the Universe as a whole and its subsystems represent the symme-  try group SUp8q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' The Universe is divided to infinite number of subsystems based on an  arbitrary finite rank symmetry group G, which arises due to quantum fluctuations and  clustering of states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' After selection of two arbitrary subsystems as clock and reference  observer, subsystems acquire a relative dynamics, and gravity emerges as a SUp8q Yang-  Mills quantum field theory, defined on the (3+1)-dimensional parameter space of Hilbert  spaces of the subsystems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' At classical limit, this observer is perceived as the spacetime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='  Consequently,SUp8q-QGR explains both the dimension of and signature of spacetime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' As  a Yang-Mills model SUp8q-QGRis renormalizable and despite prediction of a spin-1 field  for gravity at quantum level, at low energies it is perceived as classical Einstein model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' The  aim of the present work is to make the foundation of sqgr more mathematically rigorous  and fill the gaps in the construction of the model reported in earlier works.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' In particular,  we show that the global SUp8q symmetry manifests itself through the entanglement of ev-  ery subsystem with the rest of the Universe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' Moreover, we demonstrate irrelevance of the  geometry of parameter space, which can be gauged out by a SUp8q gauge transformation  up to an irrelevant constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' Therefore, SUp8q-QGR deviates from gauge-gravity duality  models, because the perceived classical spacetime is neither quantized, nor considered to  be non-commutative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' On the other hand, using quantum uncertainty relations, we demon-  strate that the classical spacetime and its perceived geometry present the average path of  the ensemble of quantum states of subsystems in their parameter space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' We also briefly  discuss SUp8q-QGR specific models for dark energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='02813v1  [gr-qc]  7 Jan 2023  Contents  1  Introduction  1  2  A quantum Universe with infinite independent observables  4  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='1  Rationale for SUp8q-QGR  4  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='1  Symmetry as a foundational concept in quantum mechanics  5  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='2  Universe as a composite quantum system  6  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='3  Infinite Universe  6  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='4  Nature of spacetime  7  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='2  Axioms and construction of SUp8q-QGR  7  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='1  Symmetry and Hilbert space  8  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='2  Quantization  10  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='3  Interpretation and comparison with Virasoro algebra  12  3  Division to subsystems  12  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='1  Infinitely divisible Universe  13  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='1  Emergence of internal symmetries and locality in SUp8q-QGR  13  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='2  Symmetries and entanglement of subsystems  16  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='1  Global entanglement  17  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='2  Consequences of the global entanglement  18  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='3  Observability of entanglement  18  4  Parameter space and algebra of subsystems  19  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='1  Emergence of an area/size scale as a measurable  19  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='2  Time  21  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='1  General properties of a quantum clock and time parameter  21  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='2  Quantum clock and its associated time parameter  22  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='3  Hamiltonian associated to a clock  23  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='3  Full parameter space of subsystems  23  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='1  The sup8q algebras of subsystems and their commutation relations  24  – i –  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='2  About quantum uncertainty, nonlocality, and topological effects  25  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='4  Parameter transformation  26  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='1  Poincar´e symmetry  27  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='2  Curvature of the parameter space  27  5  Classical curved space and its relationship with quantum states of sub-  systems  29  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='1  Speed limit from quantum uncertainties  29  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='2  Effective path in the parameter space  30  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='1  Path in the full parameter space  32  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='2  Proper (equal-time) distance and causality  33  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='3  Interpretation of affine separation and its curved geometry  34  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='1  Relation with coherence and distinguishability  35  6  Evolution  36  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='1  The whole Universe  37  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='1  Formal description of dynamics  37  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='2  A Yang-Mills whole Universe  38  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='3  A topological Universe  40  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='2  Evolution of subsystems  41  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='1  Possibility of gravitational P, C, and CP violation  43  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='2  Is there a conflict between a SUp8q Yang-Mills gauge theory as quan-  tum gravity and present observations ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='  44  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='3  Classical limit  45  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='4  Cosmological constant  46  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='1  Static SUp8q field  47  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='2  Θ´vacuum  47  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='3  Manifestation of global entanglement  49  7  Outlines and future perspectives  49  A Properties of SUp8q tensor products  51  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='1 About definition of symmetries of a quantum system  52  – ii –  B About quantization  52  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='1  Dual of generators in spherical harmonic representation of sup8q  53  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='2  Dual of continuous parameters in spherical harmonic representation of sup8q 54  C Criteria for definition of quantum subsystems  54  C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='1 Symmetries and Hilbert space of subsystems  55  C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='2 Conditions for division induced entanglement of subsystems  55  D Relation between reparameterization and diffeomorphism  56  E Coordinate independent definition of curvatures  56  F Expansion of density matrix and its variation  57  G About holographic principle and black holes  58  G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='1 Comparison with gravity as entanglement and ADS/CFT  59  1  Introduction  So far most Quantum GRavity (QGR) candidates have been based on the classical models  quantized in one way or another.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' Moreover, geometry of spacetime, sometimes extended to  matter sector, and its quantization have had a prominent role in the construction of these  theories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' For instance, Kaluza-Klein model [1, 2] and its successors: string theory [3, 4],  Loop Quantum Gravity (LQG) [5, 6] and its extension to Group Field Theory (GFT) [7, 8],  and related models such as random matrices [10, 169] and tensor product models [11, 12]  are all geometry inclined and attempt to describe gravity by quantizing D-dimensional  spaces, where D can be 2, 3, or higher, specially in models in which matter is also treated  geometrically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='  Of course, the reason behind this trend is the close relation of classical  gravity with the geometry of spacetime in the Einstein general relativity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='  In geometrical approaches to QGR a classical model do exists for ℏ Ñ 0, because their  departure point is essentially a classical model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' However, for various reasons a Universe  in which ℏ Ñ 0 cannot correspond to ours, even as an asymptotic or approximate descrip-  tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' The reason is the innumerable experiments which have shown that matter behaves  quantum mechanically, even at macroscopic scales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' On the other hand, inconsistency of a  classical gravity with a quantum matter is well documented [13–16]1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' Thus, to construct  1Nonetheless, skepticism still persists [17, 18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='  – 1 –  a quantum model of gravity and spacetime, it seems more logical to treat the Universe as  an intrinsically quantum system from the beginning, rather than starting with a classical  theory and subsequently quantize it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='  Another point, which is often overlooked by the traditional approaches to quantum gravity  and quantum cosmology, is the fact that quantum systems need an environment for de-  coherence [19] and formation of a system, here the Universe, which seems decohered and  random at intermediate scales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' We remind that at microscopic/UV scales the Universe is  fully quantum mechanical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' Furthermore, due to the expansion of the Universe superhorizon  scales, which are at present causally disconnected, were previously interacting and might  have kept their quantum coherence even now [20, 21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' Decoherence of these quantum modes  was necessarily due to an environment [22, 23], presumably, provided by the rest of the  Universe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' In other words, in a quantum model of gravity and spacetime it is imperative to  consider the Universe as an ensemble of interacting, either coherently or causally, quantum  subsystems [24–26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' In addition to an environment, frame-independent definition of quan-  tum dynamics needs a quantum reference (observer) [27–29] and a quantum clock [30, 31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='  These requirements highlight the crucial role of quantum subsystems in a fundamentally  quantum approach to the Universe and to a universal interaction such as gravity between  its its parts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' Of course, a Quantum Field Theory (QFT) by definition includes infinite  number of subsystems - particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' But in standard formulation of QFTs the presence of  an observer (a reference) and a quantum clock is implicit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' Moreover, their quantum cor-  relation and effects, for instance on the algebra of observables [32], decoherence through a  change of reference frame [28], and dependence of entanglement and superposition on the  choice of quantum reference frame [29] are not explicit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='  Quantum approaches to QGR are relatively recent additions to the jungle of QGR candi-  dates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' In absence of a hint from classical gravity, quantum field theory, or experiments,  this class of models usually have to use new priors as their departure point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' Concepts  and conjectures used for this purpose include: the concept of locality and geometrization  of quantum mechanics, considered to be crucial for describing black holes and their puz-  zles [33, 34];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' holography principle, which conjectures that maximum possible information  capacity of a quantum system is proportional to the area of its null (lightlike) boundary  rather than its volume [35–37];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' and finally, the conjecture of duality between quantum  gravity in asymptotically Anti-de-Sitter (AdS) bulk and a Conformal Field Theory (CFT)  on the boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' For a brief review of these proposals and their comparison with SUp8q-  QGR see [38].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='  In [39], we proposed SUp8q Quantum Gravity or in short SUp8q-QGR as a model for a  quantum Universe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' It is based on a few well motivated axioms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' In particular, it assumes  that the Universe has infinite number of independent and mutually commuting observ-  ables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' We described some of the properties of such a quantum Universe as a single system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='  Notably, we showed that it is static and topological.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' We also demonstrated that it becomes  dynamics after its division to subsystems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' Specifically, the definition of an operational time  as a parameter describing variation of subsystems states with respect to that of an arbitrary  – 2 –  subsystem called a clock - `a la Page & Wootter [30] or similar procedures [31] - induces a  dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' Moreover, in [38] we showed that gravity emerges as a universal quantum force  having the form of a SUp8q Yang-Mills gauge theory defined on the (3+1)-dimensional  parameter space of representation of SUp8q symmetry by subsystems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' Therefore, in con-  trast to other QGR models, SUp8q-QGR explains the origin of observed dimensionality  of the classical spacetime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' It is also shown that at low energies, that is when the quantum  details of SUp8q Yang-Mills is not detectable, the action of the gravity sector of the model  looks like that of the Einstein-Hilbert gravity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='  A discriminating aspect of SUp8q-QGR is the strict classical spacetime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' In fact, there  is no background spacetime in the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' What is called the spacetime is the parameter  space characterizing representation of SUp8q symmetry by the Hilbert space of subsys-  tems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' For this reason quantization of spacetime is meaningless - in the same manner that  quantization of eigen values in quantum mechanics is meaningless.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' In addition, we showed  that the classical curved spacetime presents the average path of quantum subsystems in  this parameter space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='  Regarding the above interpretation of spacetime, it is useful to remind that Einstein’s  opinion was that without the graviton field gµνpxq, that is in absence of a gravity source,  the spacetime itself has no physical meaning, see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' [40] (and references therein) for a  concise review of the history of physical interpretation of spacetime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' Einstein motivation  for his conclusion was the failure of all experiments to detect a medium - the ether - as  the empty space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' However, general relativity does not provide any explanation for why  and how a non-existent entity emerges in presence of matter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' Therefore, the interpretation  of coordinates of what we call spacetime as provides the missing explanation in general  relativity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' Indeed, as Einstein suggested, in SUp8q-QGR spacetime does not have any  physical meaning without Yang-Mills field and matter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' This is analogous to charge or spin,  which cannot exist without matter and interaction mediators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='  To further advance preliminary results reported in [39] and [38], in the present work we will  investigate in more details structure and properties of states of subsystems, parameter space  of the representations of their SUp8q symmetry, and Lagrangian of SUp8q-QGR model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='  We will discuss in more details fundamentally quantum nature of the model and will extend  the discussion about the parameter space and its geometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' In particular, we will prove  that its geometry is irrelevant and can be transferred to that a flat space by a SUp8q gauge  transformation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' This property is crucial for the model, otherwise the number of degrees of  freedom of gravity sector would be doubled and one would have to introduce new rules, for  example Einstein equation, for fixing the geometry of parameter space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' Another character-  istic of SUp8q-QGR, which was not discussed in the previous works, is the entanglement  of subsystems of the Universe, induced by the assumption of SUp8q symmetry at both  local and Universe-wide scales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' Additionally, we will show that symmetries at the two  scales are not independent, but intertwined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' The importance of entanglement for QGR  is recently explored in various approaches [41–43], specially those inspired by quantum  information[29, 32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' In SUp8q-QGR the omnipresent entanglement can be considered as  – 3 –  quantum analogous of gravitational waves memory [44–47].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' Emergence of local symmetries  and invariance (or not) of subsystems under discrete symmetries of the parameter space  will be discussed more thoroughly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' An important subject which is not discussed in the  previous works on SUp8q-QGR is the possibility of having a constant term in the action,  in other words a cosmological constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' We will introduce this topic here and will propose  a summary of potential processes specific to SUp8q-QGR, which may induce a constant  term in the effective action at classical limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='  In Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' 2 we review the rationale for proposing an intrinsically quantum model for grav-  ity, axioms of SUp8q-QGR, and properties of its symmetry, algebra, Hilbert space, and  quantization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' Algebraic arguments for the emergence of local orthogonal symmetries and  division of the Universe to subsystems are studied in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' We also discuss the crucial  existence of a permanent entanglement between every subsystem and the rest of the Uni-  verse and its consequences in this section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' Algebra and parameter space of the subsystems  are discussed in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' For this purpose we introduce an abstract quantum clock, as-  sociate a parameter to the variation of its state, and call this parameter time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' Then, we  explain the full parameter space of subsystems and its symmetries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' We also discuss the  geometry of the parameter space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' In Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' 5 Mandelstam-Tamm uncertainty relation is  used to introduce a parameter, which we identify as the affine parameter of the perceived  classical spacetime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' It is related to the quantum state of subsystems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' Moreover, we proved  that the signature of classical metric must be negative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' These topics were discussed in [39].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='  Here we provide more physical insight to its interpretation in conjunction with the global  entanglement, coherence and causality, Finally, the dynamics and evolution of the Universe  in SUp8q-QGR is discussed in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' We also discuss classical limit of the gravitation  sector and propose a few processes which may introduce a cosmological constant in the  effective Lagrangian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' Outlines and perspective for future research, notably application of  SUp8q-QGR to phenomena in which quantum gravity may be important are given in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' Appendices A to G provide additional information and detail calculation for the subjects  discussed in the main text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='  2  A quantum Universe with infinite independent observables  In this sections we first review rationale a purely quantum approach to the Universe and  to quantum gravity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' Then, we review axioms of SUp8q-QGR and construction of algebra  and Hilbert space of the Universe based on them  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='1  Rationale for SUp8q-QGR  Standard quantization methods fail to provide an inclusive formulation including not only  a consistent and problem-free quantum gravity, but also matter and its non-gravitational  interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' This conundrum motivate a radically different approach to quantum gravity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='  Symmetry is one of the mathematically and physically fundamental concepts, which may  reveal the way forward.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='  – 4 –  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='1  Symmetry as a foundational concept in quantum mechanics  The ideas behind SUp8q-QGR originates from the observation that Hilbert spaces of quan-  tum systems represent their symmetries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' It has led to a reformulation of the axioms of  quantum mechanics, such that they include and emphasize the central and foundational  role of symmetries in the mathematical description of quantum systems [26, 48, 49].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' A  short review of this rephrasing of the postulates, which from now on we call Symmetry  Description of Quantum Mechanics (SDQM), can be found in appendices of [39].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='  A consequence of this description is that the vector space of states of a quantum system  represents its symmetries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' Moreover, when the state space is parameterized with respect  to unbounded continuous variables, such as the spacetime, its L2 integrability - its Hilbert  space property - and the Born rule can be obtained from the axioms, rather than being  postulated, as is the case in the standard description of quantum mechanics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='  In this  description, the quantum coherence is interpreted as a symmetry presenting degeneracies  of a quantum system’s state with respect to its environment, that is the rest of the Universe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='  Unitary transformations U|UPBrHs, where H is the Hilbert space of the system, and BrHs  is the space of its (bounded) linear operators, modify the system’s coherence symmetry2  presumably by acting on the system or its environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' Of course, a unitary transfor-  mation of the state can be mathematically considered as a change of the arbitrary Hilbert  space basis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' However, in a physical context such operation corresponds to changes in the  perception of the physical world.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' It is why the coherence symmetry is relative not absolute.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='  For example, neglecting gravity, a boost transformation of coordinates of a particle does  not make any physical difference if the particle does not interact with something else.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' The  result of a complete breaking of coherence symmetry is a projective measurement [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' For  long time the importance of the state coherence symmetry as a resource was overlooked  in quantum mechanics and quantum information literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' Nonetheless, recently quan-  tification of this symmetry and its applications as a resource have become a subject of  interest [50, 51].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='  The introduction of symmetry as a foundational concept in the axioms of quantum me-  chanics is crucial for the Quantum First approaches [52] to the QGR, and more generally  for modeling a quantum system without using a classical model as the departure point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' In-  deed, according to the axioms of quantum mechanics `a la Dirac [53] and von Neumann [54],  the Hilbert space is an abstract Banach space, and the postulates do not specify how it  should be defined or related to physical properties of the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' Nonetheless, in practice  the Hilbert space is always chosen such that it represent symmetries of the corresponding  classical Hamiltonian, or Lagrangian in the case of relativistic systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' Therefore, our  2In quantum information literature what we call coherence symmetry is confusingly called asymmetry,  emphasizing on the off-diagonal components of density matrix when a system is not completely incoherent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='  By contrast, the first expression emphasizes on the fact that superposition and interference arise due to  quantum degeneracies, and therefore present a sort of symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' For this reason, in [26] the expression  state symmetry was used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' This symmetry is specific to quantum systems and additional to symmetries of  their Lagrangians, which exist in both classical and quantum systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' Nonetheless, they are related.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='  – 5 –  theoretical and observational knowledge about quantum systems justify the description of  quantum postulates based on symmetries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='1 clarifies some issues regarding  how symmetry of quantum systems are defined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='2  Universe as a composite quantum system  According to a corollary of SDQM [26] the division of a quantum Universe to subsystems  is crucial for its consistency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' It is important to remind that many concepts in quantum  mechanics are based on the ability to define and distinguish subsystems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' Examples in-  clude: Entanglement;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' Definition of a quantum clock and thereby a quantum mechanically  consistent dynamics;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' System-environment discrimination and decoherence;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' Locality of in-  teractions and quantum POVM measurements, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' In the SDQM entanglement can be  interpreted as partial breaking of state (coherence) symmetry of a composite quantum  system, which makes entangled subsystems indivisible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' This means that the Hilbert space  of the entangled subsystems becomes an irreducible subspace of the tensor product of the  Hilbert spaces of individual subsystems, which is not a proper subspace of one or the other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='  The concept of subsystems is specially crucial for constructing a quantum Universe3 with  a universal interaction similar to gravity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' In particular, subsystems are necessary for a  consistent definition and implementation of the concept of locality, which is considered by  some QGR proposals [33, 34, 41, 42, 56] to be crucial for accommodating black holes and  explaining the paradox of information loss [57, 58].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='3  Infinite Universe  It is well known that a full description of quantum phenomena needs Quantum Field Theory  (QFT), which has infinite number of degrees of freedom - in other words, infinite number  of subsystems/particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' Even the vacuum state of a QFT can be described as a superposi-  tion with negligibly small amplitude of all possible states [59], including Glauber coherent  states [60] and condensates of particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' Therefore, we should consider the Universe as  being composed of infinite number of subsystems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' The Hilbert space of such a system is  necessarily infinite dimensional, and should have infinite number of independent - mutually  commuting - observables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='  The requirement of infinite number of mutually commuting observables is a crucial differ-  ence between a quantum system/Universe composed of infinitely many subsystems and a  system having an observable with a continuous spectrum, such as position, momentum or  displacement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' In both cases the Hilbert space H and space of linear operators BrHs applied  to it are infinite dimensional.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' However, in the case of an observable with continuous spec-  trum, aside the hermitian operator associated to it, other operators in BrHs are considered  3Considering categorical description (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' [55]) of quantum mechanics, the Universe is the category  of categories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='  It incorporates all physical objects - including itself - and physical operations (functors)  applied to them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' The purpose of adjective physical here is to emphasize that these entities are not abstract  mathematical concepts, but have detectable existence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='  – 6 –  to be related to the same observable by a basis/frame transformation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' On the other hand,  tensor product of infinite number of subsystems with infinite dimensional Hilbert spaces  is again infinite dimensional.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' Consequently, the discrimination between subsystems and  their states is blurred.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' Thus, this property may provide a universal interaction between  subsystems by mixing their states with those of others.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='4  Nature of spacetime  Observables with continuous spectra are usually related to the spacetime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' In particular,  in QFTs states of the Hilbert space and particles (subsystems) in the Fock space are  labeled/indexed by the spacetime or its Fourier conjugate - the momentum (but not both).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='  This raises the question whether the spacetime by itself, that is without matter, has any  physical meaning, or rather what we call matter/radiation and spacetime are inseparable  properties of the same physical reality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='  General relativity does not clarify the nature of spacetime and its difference with matter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='  As we discussed in the Introduction, Einstein considered spacetime as a quantity that  parameterizes a spin-2 field gµν and matter fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' But, this interpretation does not not  explain why the geometry of this parameter space, which according to Einstein by itself  does not have any physical existence, is determined by a physical field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' In any case, modern  approaches to QGR, specially string theory, treat spacetime and matter similarly: Both  are considered to be fields on the 2D world-sheet of strings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' Loop Quantum Gravity (LQG)  also treats space as an standalone physical entity quantized by triangulation or treated as  a spin foam.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' See [38] for an extended discussion of these models and their comparison with  SUp8q-QGR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='  Irrespective of interpretations and approaches, the undeniable fact is that without mat-  ter (including a cosmological constant) and with trivial boundary conditions, solution of  the Einstein equation is a constant metric, which can be also zero - corresponding to no  spacetime at all.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' In addition, the choice of a null metric rather than a flat space with  undetermined volume seems preferable, because there is no experimental evidence for the  existence of an empty flat Minkowski spacetime at any scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='  Therefore, we conclude  that spacetime and matter are inseparable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' Indeed, under some assumptions, in particu-  lar holography principle [35–37], Einstein equation can be obtained and interpreted as an  equation of state [61].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='2  Axioms and construction of SUp8q-QGR  With the above conceptual background we take a purely quantum mechanical approach to  construction of the Universe and its content.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' For an extended discussion see [39].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='  We begin by designing a quantum Universe base on three well motivated assumptions:  I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' Quantum mechanics is valid at all scales and applies to every entity, including the  Universe as a whole;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='  – 7 –  II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' Any quantum system is described by its symmetries and its Hilbert space represents  them;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='  III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' The Universe has an infinite number of independent degrees of freedom, which are  associated to mutually commuting observables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='  The last assumption means that the Hilbert space of the Universe HU is infinite dimensional  and linear operators acting on it represent the group SUp8q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' We call the vector space of  these operators BrHUs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='  The sup8q algebra of BrHUs is defined as the following4:  rˆLa, ˆLbs “  iℏ  cMP  fc  ab ˆLc “ iLP fc  ab ˆLc,  LP ”  ℏ  cMP  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='1)  where operators ˆLα P BrHUs are generators of sup8q and anti-symmetric fc  ab ” fabc are  structure coefficients of the algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' They are proportional to 3j symbols defined for three  pairs of indices pl, mq, pl1, m1q, pl2, m2q [62].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' Notice that in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='1), operators ˆLα are nor-  malized such that the r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' explicitly depend on the Planck constant ℏ and Planck mass  MP ”  a  ℏc{GN or equivalently Planck length LP ” ℏ{cMP .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' If ℏ Ñ 0, or MP Ñ 8,  the length scale LP Ñ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' In this case the algebra becomes Abelian and homomorphic to  ÂNÑ8 Up1q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' This is in agreement with symmetry of configuration space of a classical  system with infinite number of independent observables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' Thus, the quantumness of the  Universe according to this model needs both ℏ ‰ 0 and MP ă 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' Of course at this stage  there is no need for introducing a mass (or length) scale and the above conclusion may  seem artificial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' We show later how a dimensionful scale arises in the model and becomes  relevant for the above algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='1  Symmetry and Hilbert space  It is proved that:  BrHUs – SUp8q – area preserving diffeomorphism DiffpS2q  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='2)  of 2D compact Riemann surfaces S2 (should not be confused with 2D-sphere Sp2q) [62–66].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='  The symbol – means homomorphic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' Although all representations of SUp8q are locally  homomorphic, they have different global topology and present globally non-equivalent rep-  resentations of the group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' Thus, class of equivalent representations of SUp8q is isomorphic  to the class of topologically equivalent S2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' Consequently, diffeomorphism of compact sur-  faces with different genus are homomorphic to non-equivalent representation of SUp8q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' As  a shorthand we call them diffeo-surfaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' In the context of SUp8q-QGR model they cor-  respond to globally different universes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' Moreover, the topology of SUp8q representations  would be important for nonlocal phenomena, for example when entanglement or black hole  singularity [41] make part of the parameter space inaccessible, See Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='1 for more details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='  4In this work, all vector spaces and algebras are defined on complex numbers field C, unless explicitly  mentioned otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='  – 8 –  Due to (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='2), members of sup8q algebra are parameterized by two angular coordinates  pθ, φq of the diffeo-surface, and generators ˆLa of the algebra can be expanded with respect  to spherical harmonic functions [62, 64]:  ˆLlmpθ, φq “ iℏ  ˆ BYlm  B cos θ  B  Bφ ´ BYlm  Bφ  B  B cos θ  ˙  “ iℏ  b  |gp2q|ϵµνpBµYlmqBν,  µ, ν P tθ, φu,  l ě 0, ´ l ď m ď l  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='3)  ˆLlmYl1m1 “ ´iℏtYlm, Yl1m1u “ ´iℏfl”m”  lm,l1m1Yl”m”  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='4)  tf, gu ”  Bf  B cos θ  Bg  Bφ ´ Bf  Bφ  Bg  B cos θ  ,  @ f, g  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='5)  For the sake of simplicity from now on we will absorb the numerical factor  ℏ  cMP in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='1)  in the normalization of operators, unless when its presence is important for the discussion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='  Operation t , u in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='5) is the Poisson algebra with respect to local coordinates of the  diffeo-surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' The function |gp2q is the determinant of 2D metric of the diffeo-surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='  Equalities in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='4) show that spherical harmonic functions have the same algebra as sup8q  under Poisson brackets operation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' This property will be useful for understanding various  properties of SUp8q-QGR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='  Cartan decomposition of SUp8q allows to write ˆLlm as a tensor product of Pauli matri-  ces [62, 67]:  SUp8q –  8  â  SUp2q  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='6)  ˆLlm “ R  ÿ  iα“1, 2, 3,  α“1,¨¨¨ ,l  apmq  i1,¨¨¨ilσi1 ¨ ¨ ¨ σil,  l ě 0, ´ l ď m ď l  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='7)  where σiα’s are Pauli matrices [62] and R is a normalization constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' Coefficients apmq  are determined from expansion of spherical harmonic functions with respect to spherical  description of Cartesian coordinates [62].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' We have omitted tensor product symbols be-  tween the factors, because one can also consider σ’s as N Ñ 8 representation of Pauli  matrices [62].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='  In this case, their products would be the ordinary matrix product.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='  As  the fundamental representation of SUp2q corresponds to l “ 1{2, this relation shows that  decomposition (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='7) with half-integer pl, mq is also a representation of SUp8q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='  The importance of SUp2q – SOp3q symmetry in QGR models is discussed in [38].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' The  obvious reason is the fact that it is the symmetry of the Euclidean Rp3q space - the classical  physical space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' However, in contrast to other QGR proposals, the decomposition (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='6) does  not have any special physical interpretation, except that SUp2q is the smallest non-Abelian  member of SU groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' In Appendix A we show that SUp8q can be expressed as tensor  product of any SUpKq, K ă 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='  It is possible - at least formally - to define generators only with respect to continuous pθ, φq  – 9 –  variables by summing over indices pl, mq (See Appendix D of [39] for more details):  rˆLpθ1, φ1q, ˆLpθ2, φ2qs “  ż  dΩ3 fppθ1, φ1q, pθ2, φ2q;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' pθ3, φ3qq ˆLpθ3, φ3q  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='8)  fppθ, φq, pθ1, φ1q;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' pθ”, φ”qq ”  ÿ  lm,l1m1,l”m”  Y ˚  lmpθ, φqY ˚  l1m1pθ1, φ1qYl”m”pθ”, φ2qfl”m”  lm,l1m1 (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='9)  ˆLpθ, φq ”  ÿ  l,m  Y ˚  lm ˆLlm,  dΩ ” dφ dpcos θq  b  |gp2q|  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='10)  Thus, although the number of generators of SUp8q group ˆLlm or equivalently Ylm that  generates diffeomorphism of diffeo-surfaces at the vicinity of a point pθ, φq is countable, the  total number of generators is innumerable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' This is because spherical harmonic functions  Ylmpθ, φq at different points are linearly independent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='  This is an interesting property,  because it shows that the Lagrangian for the whole Universe (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='1) or its subsystems (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='11)  is inherently nonlocal, and algebra and parameter space of SUp8q-QGR interpreted as  spacetime after division of the Universe to subsystems are continuous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' It is in contrast  to QGR models with symplectic geometry, which never become truly continuous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' Despite  explicit continuity of operators in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='8) description of algebra, Cartan decomposition of  SUp8q to SUp2q and Poisson bracket representations of SUp8q are more suitable for  practical applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' Eigen vectors of ˆLlm and ˆLpθ, φq are discussed in Appendix E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='1  of [39].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='  From (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='3) we conclude that vectors of HU are differentiable complex valued functions of  pθ, φq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' In particular, θ and φ can be themselves considered as vectors of the Hilbert space  HU, They present an orthogonal basis and any state of the Hilbert space can be decomposed  to orthogonal spherical harmonic functions Ylm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' The relation between structure coefficients  of sup8q algebra and Poisson brackets of Ylm in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='4) guarantees the consistency of the  whole structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' Moreover, from (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='9) we conclude that structure coefficient of the alge-  bra (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='8) are generated by 3-point Green functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' Therefore, not only SUp8q-QGR is  inherently nonlocal but also non-Gaussian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='  From now on the pair pl, mq of indices is assumed to satisfy the constraint (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='3), even if  it is not explicitly mentioned.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' Moreover, if there is no risk of confusion, we simplify the  notation by using e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' a ” pl, mq as index of generators, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' ˆLa ” ˆLlm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='2  Quantization  In absence of a background spacetime the canonical quantization does not apply to SUp8q-  QGR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' Nonetheless, non-commuting algebra of SUp8q can replace canonical quantization,  see Appendix B for more details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' Specifically, when the SUp8q algebra is written with  respect to dimensionful operators proportional to ℏ, the Lie algebra commutation relation  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='1) plays the role of a quantization relation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' On the other hand, as the algebra (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='1) is  infinite dimensional, there should be also a commutation relation similar to the canonical  quantization relations, that is Heisenberg uncertainty relation, between generators of sup8q  – 10 –  ˆLa and their duals ˆJb5:  rˆLa, ˆJbs “ ´iℏδab1,  ˆJa P BrH˚  Us  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='11)  Dual operators ˆJb act on the dual Hilbert space H˚  U and belong to the space of (bounded)  linear operators BrH˚  Us – BrHUs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' In particular, considering representation (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='3) for the  generators of sup8q, it is clear that ˆJb must be complex valued functions of the parameters  pθ, φq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' We remind that vectors of the Hilbert space HU are also complex valued functions  of pθ, φq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' Therefore, their multiplication with a complex function Fpθ, φq can be described  as an operator ˆF ” Fpθ, φq1 P BrHUs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' On the other hand, spherical harmonic functions  Ylm, which according to (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='4) have the same algebra as ˆLlm, are orthogonal functions and  can be considered as generators of sup8q for the dual Hilbert space H˚  U, as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='  In Appendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='1 we present a differential equation for ˆJlm and calculate its solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' In  Appendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='2 we obtain the expression for dual of parameters pθ, φq, that is iℏB{Bpcos θq  and iℏB{Bφ, respectively, with respect to generators ˆLlm of representation (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' The results  show that they are degenerate, in the sense that they can be calculated using any ˆLlm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='  Consequently, they carry much less information about the quantum Universe constructed  here than ˆLlm’s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' In fact giving this degeneracy, it is possible to consider iℏB{Bpcos θq and  iℏB{Bφ as being a superposition of ˆLlm operators, see Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' 6 for details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' Such interpretation  can be used to test SUp8q-QGR, for instance in experiments sensitive to ˆLlm with lowest  pl, mq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='  We leave details of this test proposal to a dedicated work on the experimental  signatures of this model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='  Considering the differential form of the dual of θ and φ and the fact that HU – H˚  U,  operators ˆJa’s are functions of pθ, φq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' Thus, they have a representations similar to (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='3)  with respect to two parameters ppθ, pφq, which are analogous to momentum components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='  The existence of such expansions is a direct consequence of the Stone-von Neumann the-  orem, which states that two sets of operators satisfying standard commutation relations  are related by a unitary transformation [68].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' More precisely, these operators must satisfy  exponentiated Weyl quantization relation [69] (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' [70] for a recent review) defined as:  eixi ˆ  Xieipi ˆPi “ eiℏx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='peipi ˆPieixi ˆ  Xi  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='12)  where x and p are C-number vectors in d-dimensional space and 0 ď i ď d ´ 1, and ˆXi  and ˆPi are the corresponding dual operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' Using Baker–Campbell–Hausdorff formula:  e ˆ  Xe ˆY “ e ˆZ,  ˆZ “ ˆX ` ˆY ` 1{2r ˆX, ˆY s ` 1{12r ˆX, r ˆX, ˆY ss ` ¨ ¨ ¨ , it is straightforward to  show that canonical commutation relation leads to (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' Nonetheless, this reformulation  of the standard commutation relation is for ensuring that the domain of the unbounded  operators ˆXi and ˆPi cover each other [68].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' In SUp8q-QGR the number of independent  operators d Ñ 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' If we use representation (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='3) of the generators of sup8q, which depend  on numerable indices pl, mq, the exponentiated form of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='11) becomes:  eiLlmpθ,φqˆLlmeiJlmpθ,φq ˆJlm “ eiℏLlmJlmpθ,φqeiJlmpθ,φq ˆJlmeiLlmpθ,φqˆLlm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='13)  5We remind that dual space V ˚ of a vector space V is the space of bijective maps v˚ : V Ñ F,  where F is the field on which the vector space V is defined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' A map v˚ P V ˚ defines a scalar product  xv˚, vy “ v˚pvq, @ v P V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='  – 11 –  In contrast to Fourier transform of functions or Hilbert space vectors defined on Rd back-  ground spacetime, non-Abelian nature of sup8q algebra means that operators ˆLa cannot  be considered as analogous to position vector operators ⃗ˆX in Rd, d Ñ 8 space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='  These results complete the demonstration that in SUp8q-QGR the algebra (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='1) is equiv-  alent to the canonical quantization of unbounded operators, specially those operators that  at low energies are interpreted as the classical spacetime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' This property of SUp8q-QGR is  distinguishable from other QGR models based on non-Abelian algebras, such as those with  non-commutative geometry, and can be used to test and discriminate this model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='3  Interpretation and comparison with Virasoro algebra  Finally, it is useful to compare the algebra (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='1) with Virasoro algebra which arises in string  theories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='  Virasoro algebra without central extension is homomorphic to (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='1), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='  the  algebra sup8q [64, 66, 71, 72].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' Moreover, similar to (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='1), Virasoro algebra is a connected  subalgebra of area preserving diffeomorphism of torus Tp2q [71, 72].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' However, it is well  known that for its application to quantized string models it must be extended by a central  charge, such that quantum anomalies can be canceled out.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' In the case of SUp8q-QGR such  extension is not necessary, because quantization and construction of a Hilbert space are  performed algebraically, there is no background spacetime, and no classical limit anomaly  to be removed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='  In fact, because the diffeomorphism generated by the Poisson algebra  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='4) must preserve the area, it cannot preserve distances and angles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' Consequently, both  conformal and Weyl (dilaton) symmetries are broken, and the central charge which is the  conformal symmetry charge is necessarily zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='  SUp8q-QGR is compared with both perturbative and non-perturbative approaches to  string theory in [38].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' Here we briefly remind their main differences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' Although in both  models a 2D space seems to have an essential role, its origin and physical interpretation is  very different.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' The worldsheet in string theory is a real (1+1)D spacetime, which its de-  formation is interpreted as presenting quantum gravity of extended physical and quantum  object - strings - embedded in a multi-dimensional quantum space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' In the perturbative  approach, coordinates of this extended space are perceived locally as fields living on the  (1+1)D worldsheet and their dynamics is ruled by a sigma model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' At least 4 of these  quantum fields should be real and non-compactified to realize a classical spacetime far  from Planck scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' By contrast, in SUp8q-QGR what we called diffeo-surface is a vir-  tual surface, which its diffeomorphism algebra is homomorphic to SUp8q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='  We use its  coordinates to parameterize an abstract Hilbert space defined on the complex numbers C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='  Therefore, SUp8q-QGR and string are profoundly different.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='  3  Division to subsystems  According to a corollary of quantum mechanics axioms with symmetry as a foundational  concept, an indivisible Universe is trivial [26, 39].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' The necessity of distinguishable parts in a  – 12 –  Universe seems natural, because even in the whole Universe model described in the previous  section, we recognize at least two categories of objects: states, which are vectors of the  Hilbert HU, and linear operators, which act on them and constitute the space BrHUs –  SUp8q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='  We should remind that in the context of a physical model, both states and  operators applied to them are not mere mathematical objects, but have physical existence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='  Therefore, division of the Universe to entities/subsystems is inevitable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' The issue, however,  is how to discriminate these entities from each others.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' Specifically, consider properties of  SUp8q group reviewed in Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' Homomorphism of one and tensor product of many  copies of SUp8q according to equation (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='3) means that they cannot be distinguished from  each others, and an additional structure is necessary for this purpose6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='  In this section we show that such structure can arise due to quantum fluctuations - in other  words random self SUp8q interaction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' In the following subsections we describe how the  emergence of finite rank symmetries divide the Universe to subsystems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' We also discuss  how the global SUp8q symmetry sticks together subsystems and entangle them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='1  Infinitely divisible Universe  Necessary conditions for dividing a quantum system to subsystems are described in [25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='  They are reviewed in Appendix C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' In the case of SUp8q-QGR, properties of SUp8q sym-  metry of the Hilbert space HU and possibility of its decomposition to any finite rank SU  group (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='5) imply that the Universe can be divided to infinite number of subsystems, each  having its own symmetry group Gi, where index i is used to discriminate subsystems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='  The complementarity condition (c) of subsystem definition imposes the following constraint  on the symmetries represented by subsystems:  â  i  Gi – SUp8q  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='1)  Division of the Hilbert space HU to those of subsystems Hi representing Gi’s is discussed in  Appendix C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' Considering the properties of SUp8q tensor product by itself and by other  groups (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='7 - A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='11), subsystems symmetries Gi’s can also contain infinite rank factors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='1  Emergence of internal symmetries and locality in SUp8q-QGR  In [39] we explicitly demonstrated that a symmetry invariant Lagrangian-like functional  for the whole SUp8q-QGR Universe can be written (we discuss this functional in details  in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='  6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='  After application of variational principle to this functional, one obtains a  trivial vacuum, which is nonetheless unstable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='  This instability provides the necessary  conditions for division of the Universe to subsystems, as described in Appendix C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' To  demonstrate the emergence of clustering and decomposition of the Hilbert space HU and  BrHUs to approximately orthogonal subspaces, we use an operational approach rather than  dynamical, because at this stage the model is still static.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='  6On the other hand, as a quantum system with SUp8q symmetry is homomorphic to many copy of  itself, it would be meaningful to association a probability to states of the Universe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='  – 13 –  A state |ψUy P HU can be expanded with respect to spherical functions:  |ψUy “  ż  dΩ ψUpθ, φq |θ, φy  “  ż  dΩ  ÿ  lě0,  ´lďmďl  ψUpl, m, θ, φq |θ, φy  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='2)  ψUpθ, φq ”  ÿ  lě0,  ´lďmďl  ψUpl, m, θ, φq ”  ÿ  lě0,  ´lďmďl  ψlm  U Ylmpθ, φq  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='3)  The state |ψUy is pure because by construction no degree of freedom is traced out.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' There-  fore, the density matrix of the Universe as a whole can be written as ˆρU “ |ψUyxψU|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' Using  the last equality in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='3), we define another basis for HU:  |ψUy “  ż  dΩ  ÿ  lě0,  ´lďmďl  ψlm  U |Ylmpθ, φqy,  |Ylmpθ, φqy ” Ylmpθ, φq|θ, φy  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='4)  The interest of this basis is that amplitudes ψlm  U  do not depend on the continuous pa-  rameters pθ, φq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' In this basis a state with constant amplitude ψlm  U  “ f “ cte.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=', @ pl, mq is  completely coherent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' Such state is unique and we call it |ψccy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' Unless a strict fine-tuning  preparation - is applied to the system, the uniqueness of |ψccy means that the proba-  bility of its random occurrence is much smaller than partially incoherent states, which in  general are less homogeneous, in the sense that amplitudes of their components ψlm  U  differ  from each others.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' Thus, these states have an intrinsic clustering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' Specifically, consider the  application of ˆLl1m1 on ψpcq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' Using (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='4) we find:  ˆLl1m1pθ, φq|ψccy “ f  ÿ  lě0,  ´lďmďl  ˆLl1m1pθ, φqYlmpθ, φq|θ, φy  “ ´iℏf  ÿ  lě0,´lďmďl  l1ě0,´l1ďm1ďl1  fl1m1  l1m1,lmYl1m1pθ, φq|θ, φy  “ ´iℏf  ÿ  lě0,´lďmďl  l1ě0,´l1ďm1ďl1  fl1m1  l1m1,lm|Yl1m1pθ, φqy ” |gl1m1pθ, φqy  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='5)  where in the second line we have used equation (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' Structure coefficients fl1m1  l1m1,lm are  proportional to 3j symbols and depend on the indices pl, mq, pl1, m1q, pl1  1, m1  1q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='  Thus, in  general the new state |gl1m1pθ, φqy is not any more a completely coherent state and some  of the components in |Yl1m1y basis have larger (or smaller) amplitudes than others.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' Conse-  quently, the density matrix ˆρU becomes more structured and approximately decomposed  to blocks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' Moreover, this grouping increases with successive application of ˆLa’s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' Inversely,  modification of |ψUy affects the distribution of ˆLa’s, because ˆρ P BrHUs and can be ex-  panded with respect to ˆLa’s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' Thus, its modification alters the probability of interaction  and further changes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' These processes are analogous to absorption of gauge bosons, which  are generators of symmetry algebra, by matter fields - states - and re-emission of both in  – 14 –  a Compton-like interaction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' In addition, non-commutative algebra of ˆLa P sup8q implies  that they can also source themselves, in the same way as in non-Abelian Yang-Mills models  such as QCD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='  A priori if the probability of operation/interaction with ˆLl1m1 is the same for all pl1, m1q, the  equilibrium can be reestablished.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' However, due to the randomness of operations and infinite  number of colors of ˆLa’s, the process of clustering, that is the formation of blocks inside  ˆρU, is quasi-irreversible and probability of returning to the completely coherent state ψpcq  is negligibly small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' Clustered subspaces of ˆρU are by definition (approximately) orthogonal  to each others and (approximately) satisfy criteria (a-c) for division of a quantum system  to subsystems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='  Thus, after numerous action of ˆLa’s, the density ˆρU is approximately restricted to a blocked  subspace of BrHUs:  BrHUs Ą  à  i  BrHis W ˆρU  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='6)  where the symbol W means approximately belong to.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='  Sub-Hilbert spaces Hi can be ap-  proximately considered as disconnected subsystems, that is Hi X Hj « H, i ‰ j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='  In  other words, operators BrHis Ď BrHUs, which dominantly affect Hi Ă HU, approximately  commute with BrHjs Ď BrHUs, which dominantly affect Hj Ď HU, j ‰ i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' Thus, they  approximately satisfy the conditions (a) for dividing a quantum system to subsystems, and  for all practical purpose the Hilbert space to which the state of the Universe belong can  be approximated by a tensor product:  HU ⇝  à  i  Hi ⇝  â  i  Hi  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='7)  BrHUs ⇝  à  i  BrHis ⇝  â  i  BrHis  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='8)  where, the symbol ⇝ means approximately lead to.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' It is crucial to emphasize that decom-  position of the density matrix of the Universe ˆρU is always an approximation, otherwise  each block can be considered as a separate universe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' Moreover, one has to assume that as  the pointer/basis states in one subspace Hi has little coherent mixing (superposition) with  other subspaces, it acquires an independent physical reality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='  Giving the continuous nature of parameters pθ, φq of the Hilbert space HU and BrHus,  subspaces Hi and operators BrHis must be also local in this parameter space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' This property  fulfills the second condition - criterion (b) in Appendix C - for division to subsystems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='  Moreover, the complementarity condition (c) is trivially satisfied, because any operator  ˆO P BrHUs is either ˆO P biBrHis Ď BrHUs or belongs to a new set of operators tA1u such  that biBrHisKtA1u and biBrHis b tA1u – BrHUs and tA1u Ď BrH1s, H1 Ď HU, BrH1s Ď  BrHUs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='  Each set of operators tAiu Ď BrHis represents a symmetry group Gi with a rank at most  equal to the number of linearly independent members of tAiu set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' This number is nec-  essarily finite, otherwise tAiu – BrHUs – SUp8q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' In this case clustering in ˆρU was not  enough to break the Universe approximately to subsystems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' Such case bring back the state  – 15 –  of the Universe to the initial state considered for this analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' The argument about sub-  systems and locality arising from a global symmetry can be also inverted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' Assume many  (infinite) number of indistinguishable quantum systems, each representing a finite rank  symmetry group G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' Their Hilbert space represents G ˆ G ˆ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' – SUp8q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' Thus, SUp8q  symmetry in SUp8q-QGR can be considered to be due to a bath of large (infinite) number  of indistinguishable (abstract) decoherence-free quantum subsystems [73] with finite rank  symmetries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='  The first decomposition in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='8) is similar to expansion of a vector space with respect  to a basis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' At this stage of clustering the Universe as a whole is similar to a strongly  coupled quantum system, where despite presence of some discernible features grouped in  subspaces Hi, they cannot be isolated and studied separately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' Consequently, unitarity,  Born rule, and global Up1q phase symmetry of quantum states apply only to the ensemble  of Hi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' However, when these subspaces become sufficiently isolated, such that HU can be  approximately considered as their tensor product, unitarity, Born rule, and Up1q phase  symmetry can be approximately applied to each Hi individually.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='2  Symmetries and entanglement of subsystems  Due to the assumed global SUp8q symmetry, satisfaction of conditions (a-c) for division  to subsystems and decomposition (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='8) of HU to tensor product of Hi’s are only approxi-  mately true.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' Indeed, conservation of SUp8q means that subspaces Hi are not projected to  themselves under application of ˆLa P BrHUs with non-zero components outside the BrHis  blocks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' Therefore, the division of the Universe to subsystems is an approximation and  no strictly isolated subsystem exists.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' In the language of quantum information, maximum  quantum complexities of the Universe and its subsystems is infinite, because preparation of  their states needs infinite number of Q-bits and infinite number of binary-gate operations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='  Therefore, to make these properties explicit for each subsystem, we add a SUp8q factor  to the full symmetry G1  i of subsystem i with finite rank local symmetry Gi represented by  its block in the density matrix of the Universe is:  G1  i “ Gi ˆ SUp8q  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='9)  Considering the properties of tensor products containg SUp8q in (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='6 - A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='10), such decom-  position remains consistent with the global SUp8q symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' Without loss of generality  we can consider the same local/internal symmetry G for all subsystems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' Specifically, G can  be chosen to be a tensor product - Cartan decomposition - containing all Gi’s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' Then, some  subsystems may represent some of the factor groups trivially.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' Indeed, symmetry group of  the Standard Model of elementary particles has such a product structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' Moreover, it  includes the two lowest rank Lie groups which cannot be further decomposed to unitary  groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='  – 16 –  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='1  Global entanglement  The global SUp8q symmetry has a crucial role in keeping together the Universe as a  quantum system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' We discuss this property as the following proposal:  Proposition 1: In SUp8q-QGR every subsystem is entangled to the rest of the Universe:  Proof: This is a consequence of the preservation of the global SUp8q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' To demonstrate  this claim, consider a division of the Universe to two subsystems representing symmetry  groups G1 and G2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' In Appendix C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='2 we show that a division to subsystems increases the  dimension of the Hilbert space of the composite system with respect to before division.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='  In the case of SUp8q-QGR, if we consider strictly infinite dimensional Hilbert spaces for  the whole Universe and for the two subsystems 1 and 2, relation (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='9) and arithmetic of  infinite numbers imply that without needing additional conditions, there is a homomor-  phism between states of the Hilbert space HU and H1 ˆH2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' Nonetheless, let’s assume that  their dimensions N, N1, N2 are finite, but very large and approximately infinite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' In this  case, as described in more details in Appendix C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='2, in general N ď N1, ˆN2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' On the other  hand, as we mentioned earlier and discussed in more details in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' 6, the Universe as a  whole is static and its Lagrangian-like symmetry invariant functional (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='3) is topological.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='  Therefore, its quantum properties, including its Hilbert space should not depend on the  dynamics of its components/subsystems, which are subject to relative variations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' The only  way to establish equality between number of states of the whole Universe and ensemble of  its subsystems is a permanent entanglement between subsystems, which decreases number  of physically allowed states to N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='  Giving the fact that subsystems have infinite number of degrees of freedom related to their  SUp8q symmetry, one can in principle find many untangled degrees of freedom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' Therefore,  in general projective observation of limited number of them in one subsystem would not be  enough to project other subsystems to a given state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' This is a crucial condition for having  a dynamical Universe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='  Another way of demonstrating proposition 1 is through mutual information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' Consider the  mutual information IpA : Bq ” SpˆρAq ` SpˆρBq ´ SpˆρABq between a quantum subsystem A  and the rest of the Universe B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' The function S ě 0 presents von-Neumann entropy of each  system in isolation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' If A and B were independent from each other SpˆρABq “ SpˆρAq`SpˆρBq  and IpA : Bq “ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' However, due to the gauge symmetry the state of the whole Universe  ˆρAB – 1 and SpˆρABq “ 0, and IpA : Bq ě 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' Thus, there is no isolated subsystem in  the SUp8q-QGR Universe and subsystems are entangled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' Any variation in the subsystems  leads to a global adjustment to restore their entanglement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' This is the quantum analogous  of gravitational memory in the classical spacetime [44, 45].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='  We leave quantification of  this entanglement for future works, because it may need investigation of cosmology in the  framework of SUp8q-QGR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' Nonetheless, on the dimensional ground, we expect that it  should be proportional to ℏ2Λ{M2  P , where Λ is the cosmological constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' See also Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='4 for more details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='  The always entanglement of subsystems shown here is not a surprise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' Irrespective of the  – 17 –  model constructed here and in a general setup of quantum systems, There are abundant  evidence that they are quantum mechanically correlated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' For instance: Change of a quan-  tum reference frame may lead to decoherence and/or superselection [28];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' Limitations on  the transfer of information about the reference frame are equivalent to quantum deco-  herence [27, 29];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' Entanglement and superposition of states become perspective depen-  dent [29, 74].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='  These properties of a quantum Universe and its subsystems imply that  characteristics, such as the amount of entanglement between subsystems may depend on  which quantum subsystem is selected as reference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' The emergence of reference-dependent  perspectives is equivalent to gauge fixing in QFT and general relativity [74, 75].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' The com-  mon aspect of these properties is the reduction of quantum coherence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' In this respect, the  global entanglement in SUp8q-QGR, which reduces quantum coherence, is of the same  category as the above properties concluded in the context of quantum information theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='2  Consequences of the global entanglement  The ever-existing quantum correlation/entanglement between every subsystem and the rest  of the Universe reinforces decomposition (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='9), because if there is a subsystem with only  finite dimensional Hilbert space, its entanglement with the infinite degrees of freedom of  the rest of the Universe makes its effective degrees of freedom, and thereby symmetry of  its Hilbert space infinite dimensional.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='  As G symmetry emerges from clustering of states, it is expected that interaction/mixing  between subsystems through G symmetry be much stronger than via the global SUp8q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='  This property is consistent with the observed universality of gravitational interaction and its  weakness in comparison with internal interactions of subsystems/particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' It also provides  an explanation for the weak gravity conjecture [76].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='  From (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='11) it is clear that in what concerns mathematical definition and action, there is  no essential difference between finite rank internal symmetries of subsystems and SUp8q  gravity - symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' Nonetheless, the resemblance of gravitational interaction to a gauge  field in SUp8q-QGR is very different from what is called gauge-gravity duality models in  the literature [77–81], Notably, in SUp8q-QGR the gauge symmetry of quantum gravity is  not directly related to the classical Lorentz invariance, which corresponds to the freedom  of the choice of parameterization of the Hilbert space, and difficulties which arise in gauge-  gravity proposals [82–84] are irrelevant for SUp8q-QGR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='  We will describe other consequences of the global entanglement of subsystems through  SUp8q symmetry along with other properties of SUp8q-QGR in the following sections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='3  Observability of entanglement  Finally, an important question is whether the global entanglement of subsystems is observ-  able.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' The answer is in principle yes, but in practice no !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' For such observation we need  to consider at least 3 subsystems: the target subsystem, a quantum clock as explained in  Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='2, and the rest of the Universe as observer/reference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' Assume that the observer  – 18 –  performs a POVM on the target such that its state is projected to a pointer state defined  in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' According to the proposition 1 this operation projects the state of the rest of  the Universe, that is the observer and the clock to a pointer state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' How can the observer  verify this expectation ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' The only way is the measurement of the expected pointer state  of the observer-clock system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' However, this is impossible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' In fact, it is even impossible  to verify that the target subsystem is indeed in a projected state after the first operation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='  Such verification needs repetition of the same operation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' However, a repetition implicitly  means passage of time as a consequence of the change of clock state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' But, if the clock  state varies, the state of observer-clock subsystem changes, which in turn changes the state  of the target subsystem according to the proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' Neglecting this verification, the  only way to verify the entanglement is either the target subsystem or the observer-clock  themselves should perform a self-state verification test.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' However, none of these operations  can be performed in isolation and without modifying some of the degrees of freedom of  these subsystems, and thereby changing their state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='  4  Parameter space and algebra of subsystems  The division of the Universe to subsystems makes it possible to define a reference quantum  observer and a quantum clock.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='  In this section we study in some details the extended  Hilbert space of subsystems and parameters that characterize its states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' We also describe  the relation between these parameters and what we perceive as the classical spacetime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='1  Emergence of an area/size scale as a measurable  The area of diffeo-surface M of a SUp8q representation is irrelevant when only one rep-  resentation is considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' In fact, area preserving diffeo-surfaces represent SUp8q ˆ R –  SUp8q ˆ Up1q “ Up8q, where R is a size scale, which can be chosen to be the area of  diffeo-surface or its square-root - a size scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' The Abelian R – Up1q group corresponds to  the scaling symmetry Λ of diffeo-surfaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' A diffeomorphism of a compact surface can be  decomposed to an area preserving diffeomorphism and a global scaling, that is:  DiffpMq – ADiffpMq ˆ ΛpMq  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='1)  In the context of quantum mechanics and representation of SUp8q by the Hilbert space of  the Universe, the scaling symmetry can be also identified with the irrelevant global Up1q  phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' On the other hand, when the Universe is divided to subsystems, each representing  SUp8q, as we explained in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='1, the Up1q phase symmetry of the Hilbert space  becomes local and decouples from the global scaling symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' Indeed, the latter breaks,  because representations and algebras of distinguishable subsystems can be compared with  each others through a map (a morphism).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' Nonetheless, the Hilbert space Hi of a subsystem  considered in isolation is still invariant under its own global Up1q phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='  Comparability of diffeo-surface areas of subsystems with each others does not mean that  their areas or sizes are considered as fixed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' Rather, their values become a new relative  – 19 –  Diffeo-Surface  of the Universe  Subsystem  Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' Observer  clock  Subsystem  Divided Universe  U(1) ≈ R  r’  r  rc  δt  x  r  x  r’  x  rc  δρ  δρ’  δρRef  Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='  Schematic description of SUp8q-QGR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' Left: 2D surfaces present diffeo-surfaces of  the SUp8q of the Universe and its subsystems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='  Although subsystems share the Hilbert space  associated to SUp8q symmetry, they have their own local representation of the symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' This  is schematically shown by different shape of 2D surfaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' Displaced surfaces depicts the evolution  δρ of the state of subsystems with respect to time variation δt that quantifies the change of the  clock’s quantum state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' Arrows depict homomorphism between SUp8q symmetry of subsystems  and its associated universal gravitational interaction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' Notice that infinite number of projections  between sup8q algebras of subsystems is possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' Therefore, subsystems can be in general in a  superposition state with respect to scaling parameter r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' Different colors of subsystems depicts their  different representation of internal symmetry G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' Right: Diffeo-surface of the Universe and its  subsystems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' Parameter r P R – Up1q can be considered as a label for subsystems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='  measurable, in the same way as the distance in classical and quantum physics without  gravity becomes a measurable when a reference - another object - is considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' Here we  call this reference subsystem the quantum reference observer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' Therefore, in addition to the  diffeo-surface parameters pθ, φq states - vectors of the Hilbert space - of subsystems Hi’s and  space of bounded operators applied to them BrHis’s depend on a dimensionful parameter  r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='  But, the way we define r is arbitrary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='  For instance, the compact diffeo-surfaces of  subsystems can be embedded in a Rp3q, because D2 ˆ R – Rp3q, where D2 is any Riemann  2D space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' Assuming parameters of the reference subsystem at an arbitrary point of Rp3q,  the parameter r can be identified with radial coordinate in a spherical coordinate system  having its origin at the point associated to the reference subsystem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' 1 schematically  depicts these properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='  – 20 –  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='2  Time  Introduction of relative time in quantum mechanics and quantum gravity is extensively  studied in the literature [30, 85, 86], specially as a mean for solving the problem of Hamil-  tonian constraint and the absence of time in canonical quantization of gravity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' But effi-  ciency of this approach is criticized in [87].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' Equivalence of various definitions of a relative  and dynamical time is demonstrated in [31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' Many of complexities of quantum time and  reference frame definitions discussed in the literature originate from attempts to quantizing  spacetime, but keeping diffeomorphism invariance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' In SUp8q-QGR spacetime is a parame-  ter space and is not quantized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' Therefore, concerns about the validity of a relative time are  irrelevant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' Nonetheless, criticism of [87] will be answered in a report under preparation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='  Once the Universe is divided to subsystems, it becomes possible to define dynamics as  relative variation of their states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' In analogy to the division of the Universe to subsystem,  our approach to the definition of a quantum clock and time parameter would be opera-  tional.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' Quantum entanglement between an imperfect - not completely isolated - clock and  systems and their effects on the definition of reference frame in a general framework is  demonstrated in [88].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' In SUp8q-QGR similar effects arise due to the omnipresent entan-  glement of subsystems proved in the proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' Consequently, successive applications  of the same operator to a selected subsystem in general does not lead to the same outcome,  and quantum contextuality [89, 90] becomes globally manifest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' This leads to the emergence  of an order - a time arrow - between events.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' In particular, relative variation of states and  observables of subsystems can be quantified with respect to those of a single subsystem,  called a quantum clock.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='  In the following subsections we explain in more details properties of the clock and emergence  of a dynamics, which is in spirit similar to the Page & Wootters proposal [30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' There is  however no need for explicit conditional wave functions, because in contrast to the canonical  quantization of QGR, there is no need for (3+1)D decomposition of formulation7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='1  General properties of a quantum clock and time parameter  Without considering specifics of a quantum clock and how the change of its state is quan-  tified, following general conclusions can be made:  Initial state: The initial state of the clock, that is the outcome of the first operation on  it, initializes relative dynamics of subsystems, but its value is irrelevant for further  measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' Only its variation in the next measurement is relevant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='  Arrow of time: Selection of any subsystem as clock induces an ordering in operations  on subsystems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' This is a direct consequence of the proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='  7When a quantum system is constructed from quantization of a classical model, there are alternative  ways to define dynamics, for instance by using relational observables `a la Dirac [91], or what is called  Heisenberg symmetry reduction in the phase space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' Equivalence of these methods is demonstrated in [31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='  As SUp8q-QGR is not obtained from a classical model, the conditional approach of Page & Wootter is  more suitable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='  – 21 –  Monotonous time: Quantification of the state change, that is the observable proxy used  for this purpose is arbitrary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' But, the value of outcome must allow ordering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' For  instance, a vector-valued observable is not suitable, unless its module is used as time  parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='  One real parameter: The above condition means that without loss of generality quan-  tification parameter called time t can be chosen to be a real number: t P R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='  POVM time: Quantification of clock state variation must be a single-valued uniquely  calculable function from information about the clock state, but its measurement does  not need to be projective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='  Time clicks for everyone: Entanglement of the clock - as a subsystem - with the rest  of the Universe means that after initialization of the clock, every other subsystem,  and thereby the whole Universe, evolves along with the clock.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' This is the origin of  the arrow of time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='  A more extended and mathematically rigorous description of necessary conditions for a  quantum clock can be found in [31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' Nonetheless, simple properties described here are  sufficient for our purpose of defining a relative dynamics, because we do not need to relate  it to a classical model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='2  Quantum clock and its associated time parameter  Overlooking the complexity of realistic quantum clocks and issues such as their finite size,  which induces uncertainties [92] and decoherence [93], a quantum clock and a time param-  eter satisfying properties described in Sec 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='1 can be defined by an operator ˆLc P BrHCs,  which its application on the clock subsystem leads to the variation of its state ˆρc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' If the  clock is tomographically complete, its state and thereby the variation ˆρc as a consequence  of ˆLc application can be determined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' Then, the time parameter or more precisely its vari-  ation δt can be defined as a monotonic scalar function of measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' For example,  time variation may be defined as being proportional to the amplitude of state variation  δt 9 |δψc| ” |ψ1  c ´ ψc| or proportional to the fidelity function:  δt 9 Fpˆρc, ˆρ1  cq ”  ˆ  tr  ba  ˆρc ˆρ1c  a  ˆρc  ˙2  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='2)  If the clock is not tomographically complete, expectation value or outcome of POVM  measurement of an observables and its variation can be used as δt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' Alternatively, the time  parameter t can be defined by parameterizing the average trajectory of the clock state ˆρC  in its Hilbert space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' The trajectory may be measured using an observable of the ˆOc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' This  approach is closer to classical definition of time than other examples described here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' The  time variation can be defined as δt 9 |tr pˆρ1  c ˆOcq ´ tr pˆρc ˆOcq|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' Obviously, variation range  of the clock state ˆρc or the observer ˆOc is usually limited.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' Nonetheless, book-keeping of  variations extends the range of t to infinity [94].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' We emphasize that due to the proposition  – 22 –  1, there is no isolated clock and decomposition of the clock observable ˆOc « ˆOc ˆ 1 where  1 presents the operation on the rest of the Universe is always an approximation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='3  Hamiltonian associated to a clock  For each clock and time Parameterization t and each subsystem s, an operator ˆHs P BrHss  called Hamiltonian can be defined such that:  dˆρs  dt “ ´ i  ℏr ˆHs, ˆρss  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='3)  where ˆρs is the density matrix of the subsystem s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='  Moreover, giving the fact that in  SUp8q-QGR the whole Universe is static [39]:  ˆÿ  tsu  ˆHs  ˙  |ψUy “  ÿ  tsu  ˆHs|ψsy “ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='4)  where the ensemble tsu of subsystems includes also the clock and the reference subsystem  defined in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='1 and |ψUy “ Â  tsu |ψsy is the state of the whole Universe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' The Hamil-  tonian ˆHs applies non-trivially only on |ψsy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' Implicit in our notation is the equivalence of  all states |ψUy up to a SUp8q symmetry transformation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' Moreover, it is meaningless to  interpret the sum over Hamiltonian operators in the l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' of (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='4) as Hamiltonian of the  Universe, because by definition and construction the Universe is static.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' A notable differ-  ence between the definition of Hamiltonian here and in quantum models originated from a  classical formulation is the fact that it is the Hamiltonian which associated to a time pa-  rameter, not the other way around.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' If the time parameter is changed, Hamiltonian should  be changed such that (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='3) remain valid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' This definition decouples the Hamiltonian from  definition of conjugate variables, which themselves is the variation of dynamical degrees of  freedom with time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='  We emphasize that the constraint (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='4) is a consequence of the entanglement of every  subsystem with the rest of the Universe demonstrated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' In addition, the construction of  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='4) explicitly shows that, in contrast to ADM [95] in (3+1) dimension or other canonical  quantization of gravity, this property is not related to diffeomorphism invariance of the  parameter space and dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='3  Full parameter space of subsystems  Based on the arguments in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='2 and other evidence discussed in the previous sections,  we conclude that the 4 continuous quantities pt, r, θ, φq that parameterize the Hilbert space  representing SUp8q symmetry of subsystems and their relative variations is related to  what is perceived as spacetime in the classical limit of gravity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' We give further precision  about exact nature of this relation and what we mean by classical limit in the following  sections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' This conclusion discriminates SUp8q-QGR from QGR proposals, which try to  – 23 –  ad hocly quantize an unobservable background geometry8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' This feature also implies that  in the framework of SUp8q-QGR model, quantization of spacetime is meaningless.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='  The shared parameter space pt, r, θ, φq of SUp8q symmetry of subsystems have several  homomorphic descriptions:  S2 ˆ R1 ˆ R1 – S2 ˆ Up1q ˆ Up1q – Rp1`3q  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='5)  The middle homomorphism in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='5) is useful for description of sup8q algebras of subsys-  tems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' Notice that in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='5) we used Rp1`3q rather than R4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' We justify this point in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='  In both cases the 4D parameter space is simply connected, even if S2 alone may have a  non-trivial topology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' Therefore, wave functions of subsystems can be in principle extended  to classically singular points of the parameter space such as inside a black hole.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='  In addition to SUp8q, states of subsystems represent the internal group G and depend on  parameters that characterize its representations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' we collectively call them α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' Giving the  fact that the symmetry group of subsystems is the tensor product SUp8q ˆ G, representa-  tions of the two factors are independent and orthogonal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' Therefore, state of a subsystems  |ψsy can be decomposed to:  |ψsy “  ÿ  t,r,θ,φ,α  ψspt, r, θ, φ, αq |t, r, θ, φy b |αy  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='6)  where |t, r, θ, φy b |αy is an eigen basis for the Hilbert space Hs, but its SUp8q and G  parameters are not separable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' Parameters pr, θ, φq characterize diffeo-surface of subsystems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='  Although in principle time parameter t is only related to the clock, entanglement of the  latter with the rest of the Universe and constraint (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='4) indirectly relates t to the other  parameters and the 4 parameters are effectively inseparable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' The collective parameters α  which characterize representations of the finite rank symmetry G are usually discrete.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' In  this work we do not further discuss G symmetry and its representations by subsystems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='  The particle physics motivated by SUp8q-QGR is an important subject and needs its  dedicated investigation, which we leave to future works.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='1  The sup8q algebras of subsystems and their commutation relations  Because of quantum entanglement of subsystems and coherence with respect to eigen states  of the 4-dimensional parameter space txu - the spacetime - every pair of the parameters  can be used to characterize SUp8q symmetry and its algebra as defined in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='3-2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='4) and  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='8-2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' Although ranges of r and t are usually chosen to be the non-compact R, the  homomorphism R – Up1q means that by identifying points at infinity, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' xi Ñ ˘8 i “  0, ¨ ¨ ¨ , 3 the parameter space can be made compact, such that the surface passing through  every pair be a 2D compact Riemann surface and its ADiff represent SUp8q [65].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='  In any case, giving the emergence of locality in many-body systems due to the limited  speed of quantum information transfer that we will discuss in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' 5, and the growth of  8This trend of some QGR is models is considered by some authors as re-emergence of the infamous  concept of ether [40, 96], which relativity wanted to disregard.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='  – 24 –  inhomogeneities [97], there would be negligible impact on the physical process due to the  compactification (or not) of the parameter space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' In particular, the definition of SUp8q  generators ˆLlm in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='3) would be still usable, because spherical harmonics are special cases  of generalized hypergeometric functions [98].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' Specifically, in (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='2) the associated Legendre  polynomial Plm is related to the Gauss hypergeometric functions 2F1:  Plmpηq “ p´1qm Γpl ` m ` 1qp1 ´ η2qm{2  2mΓpl ´ m ` 1qm!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='  2F1pm ´ l, m ` l ` 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' m ` 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' p1 ´ ηq{2q  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='7)  with l`m ě 0 and Repηq ě ´1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' Therefore, spherical harmonics and their Poisson algebra,  which are homomorphic to sup8q, can be analytically extended outside the range of angular  parameters pθ, φq, except for singular points at ˘1 and 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='  Expansion and commutation relations of one of these parameters (coordinates), which we  call η, and its dual iℏB{Bη with respect to ˆLlm generators are described in Appendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='  They apply to all parameters of SUp8q symmetry of subsystems, irrespective of how they  are defined9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' Thus, one can construct a sup8q algebra from any pair of the 4 parameters  of the Hilbert space of subsystems:  rˆLl,mpη, ζq, ˆLl1,m1pη, ζqs “ fl”m”  lm,l1m1 ˆLl2,m2pη, ζq  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='8)  under the condition that an appropriate associated Legendre polynomial is used in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='  Crucially, giving the similarity of commutation relations to the standard quantum mechan-  ics, the usual uncertainty relations of quantum mechanics between parameters and their  duals [99, 100] can be applied to the time parameter t, irrespective of how a quantum clock  is defined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' Moreover, it would be more suitable to use an expansion similar to (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='3), in  which wave coefficients depend only on discrete parameters pl, mq rather than e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='2  About quantum uncertainty, nonlocality, and topological effects  Obtaining usual uncertainty relations in a model which does not follow the standard quan-  tization procedure is remarkable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' For instance, if generators ˆLlm depended on the second  order derivatives of parameters, rather than the first order, there were no analytical ex-  pression for iℏB{Bη as a function of ˆLlm generators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' Indeed, it has been shown [101] that  in any model with indeterminism and nonlocality, these properties and their quantification  are related.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' On the other hand, nonlocality and contextuality of quantum mechanics are  not maximal [102], in the sense that the CHSH inequalities [103] do not have the maxi-  mum possible deviations from prediction of a local and causal model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' Therefore, a priori  a model which does not explicitly follow the standard procedure for quantization may  have commutation relations and nonlocalities different from what is predicted by quantum  mechanics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='  Nonlocality from indeterminism reflects itself in entanglement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' But, it does not explain  global/topological quantum effects, such as Aharanov-Bohm effect [104] and quantum Hall  9The limiting points such as |η| Ñ 8 may need special treatment, in particular if diffeo-surfaces have  non-trivial topology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' These cases are analogous to topological material in condensed matter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='  – 25 –  phenomenon, see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' [105] for a review.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' To explain these processes one has to virtually  remove some points of the space to make it non-simply connected and topologically non-  trivial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' However, if the spacetime is a physical entity, removing or ignoring part of it to  explain some processes is difficult to justify, because they can be detected by other obser-  vations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' For instance, in the case of Aharanov-Bohm setup, the path of a neutral particle  can pass through the singular magnetic flux without being disrupted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' By contrast, if the  spacetime is interpreted as a parameter space, absence of some point in some phenomena  is not controversial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' In Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' 6 we show that a symmetry invariant functional similar to  Yang-Mills Lagrangian can be defined for SUp8q-QGR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' Therefore, topological quantum  phenomena observed in non-gravitational many-body systems may be relevant for quantum  gravitational processes as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='4  Parameter transformation  As discussed earlier, parameters x ” pθ, φ, r, tq arise from different properties of SUp8q-  QGR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' Namely, pθ, φq parameterize representations of SUp8q symmetry;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' dimensionful pa-  rameter r emerges from comparison of area of diffeo-surfaces;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' and t quantifies variation of  the state of a chosen quantum clock.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' Nonetheless, these parameters are inseparable and  define a R4, or as we will demonstrate later, a Rp3`1q space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' The main reason is the fact  the 4 components of x are interchangeable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' As we discussed earlier, any pair of these pa-  rameters can by used to represent SUp8q symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' Therefore, division into subsystems  is not rigid and may change with selection of another clock or reference subsystem, such  that they respect necessary the conditions defined in Appendix C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' Specifically, due to the  coherence symmetry and quantum superposition and indistinguishability of subsystems  with same representations of symmetries, the set of orthogonal operators tAiu based on  which subsystems are defined, depend on the basis for x, that is the choice of reference  observer, clock, and coordinates of diffeo-surfaces of subsystems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' Nonetheless, the Hilbert  space   s Hs of all subsystems should not depend on the choice of a basis or redefinition  of parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='  The full basis of the Hilbert space   s Hs is t|t, r, θ, φ, αy, @ ´ 8 ď t ď `8, 0 ď r ď  `8, 0 ď θ ď π, 0 ď φ ď 2π, tαuu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' However, in this section our concentration is on the  SUp8q symmetry of subsystems and its representations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' For this reason, we only consider  states which their α dependence is traced out.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' We call the space of remaining parameters  Ξ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' A possible effect of this action is further entanglement of quantum state of subsystems  and inseparability of remaining the parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='  A basis transformation of the Hilbert space of subsystems p  s Hsqα ” trαp  s Hsq (in-  cluding clock and reference) can be written as:  |t1, r1, θ1, φ1y “  ÿ  t,r,θ,φ  ˆU|t, r, θ, φy,  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='9)  ˆU P Brp  â  s  Hsqαs – SUp8qN|NÑ8 – SUp8q  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='10)  – 26 –  The last homomorphism in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='10) means that up to an irrelevant scaling of states, the basis  transformation (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='9) is equivalent to application of a member of SUp8q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='  As we discussed in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='1, states of   s Hs (and p  s Hsqαq are complex functions on  the parameter space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' Thus, the basis transformation (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='9) of the Hilbert space can be  equally interpreted as a diffeomorphism of the parameter space:  pt, r, θ, φq Ñ pt1pxq, r1pxq, θ1pxq, φ1pxqq  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='11)  Notice that quantum coherence, entanglement, and indistinguishability of classes of sub-  systems - those which share the same symmetries and their representations - are crucial  for inseparability of parameters and equivalence of basis independence of the Hilbert space  with reparameterization and diffeomorphism invariance of the parameter space Ξ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' In par-  ticular, in absence of quantum coherence, time and distance part of the states could be  separated from angular parameters of SUp8q symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='1  Poincar´e symmetry  In addition to diffeomorphism of the Rp3`1q parameter space, states of subsystems must be  in non-trivial representations of its global Lorentz and translation transformation, which  together constitute the Poincar´e group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' A reminding of the relationship between diffeo-  morphism and reparameterization can be found in Appendix D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' Observations show that  both the Standard Model [106] and gravity [107] are invariant under Lorentz group, which  is isomorphic to SOp3, 1q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' However, in contrast to SUp8q and more generally SUpNq@N,  which are simply connected, SOp3, 1q is doubly connected under discrete parity P, conjuga-  tion C, and time reversal T transformations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' Considering the assumed SUp8q – SLp2, Rq  symmetry of subsystems, it is clear that quantum superposition of subsystems must play  a role in the preservation C, P, T symmetries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' For instance, parity conserving fermions  of SM are in p1{2, 0q ‘ p0, 1{2q representation of SOp3, 1q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' Thus, each of the components  representing a connected part of SOp3, 1q can be embedded in one SUp8q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' However, only  the superposition of disconnected sectors preserves C, P, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' This property is used by lep-  togenesis models to explain matter-antimatter asymmetry using the observed parity and  CP violation in the leptonic sector of the Standard Model (SM), see e,g [108] for a review.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='  Although in the SM the quantum numberB ´ L, where B and L are baryonic and leptonic  number, respectively, is conserved, at present we do not know whether these symmetries  are conserved in the dark sector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' In the framework of SUp8q-QGR, the fact that the sym-  metry of the whole Universe as a quantum system is SUp8q, which is simply connected,  means that the dark sector should conserve CPT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' If the visible and dark sectors interact  non-gravitationally, CPT may break - but not necessarily - in each sector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' Nonetheless, it  must be conserved be their ensemble.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='2  Curvature of the parameter space  Invariance of physics under a transformation similar to (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='11) means that in general the 4D  parameter space Ξ is curved and has a nontrivial metric gµν.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' Reparameterization (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='11)  – 27 –  makes 4 of the 10 components of gµν arbitrary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' However, the remaining 6 parameters are  undetermined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' In classical Einstein gravity these components are determined by the distri-  bution of matter, more precisely by 6 independent components of the energy-momentum  tensor Tµν, through Einstein equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' This implies that for a given Tµν - more precisely  its trace T which is frame independent - diffeomorphism of the spacetime preserves its  scalar curvature10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' It is therefore legitimate to ask how the metric gµν is determined in  SUp8q-QGR and whether a curvature preserving condition exists.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' Here we show that the  curvature of parameter space Ξ is irrelevant for the physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='  Proposition 2: The curvature of SUp8q parameter space of subsystems can be made  trivial by a SUp8q gauge transformation, under which the Universe and its subsystems are  invariant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='  Proof: Consider the compact version of Ξ defined in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' At any point x P Ξ consider the  orthogonal basis ei P TΞx, i “ 0, ¨ ¨ ¨ , 3, where TΞx is the tangent space of Ξ at x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' Ricci  curvature tensor and scalar curvature can be determined using sectional curvatures of 2D  subspace of TΞx expanded by pairs pei, ejq, i ‰ j and vis-versa, see Appendix E for coordi-  nate independent definition and relation between Ricci, Riemann, and sectional curvatures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='  Consider the 2D surface Σij Ă Ξ generated by parallel transport of pei, ejq, i ‰ j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' Such  surfaces are compact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' As Ξ is the parameter space of SUp8q symmetry of subsystems,  Σij, i, j P t0, ¨ ¨ ¨ , 3u, i ‰ j can be considered as diffeo-surfaces of subsystems and their  ADiff deformation would be equivalent to application of SUp8q symmetry group, which  does not change observables of the corresponding subsystems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='  Crucially, these ADiff  do not need to preserve sectional curvatures at x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' Thus, Ricci curvature tensor is not in  general preserved, and this does not have any impact on quantum observables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' A general  diffeomorphism can be decomposed as in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' Because size of diffeo-surfaces is irrele-  vant for homomorphism of their ADiff with representations of SUp8q, scaling of Σij’s  and thereby Ξ is similarly irrelevant for quantum observables of subsystems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' Therefore,  curvature of the parameter space Ξ is irrelevant for the dynamics of the Universe and its  quantum subsystems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' In particular, in the non-compact version of parameters or when  areas of Σij are scaled to infinity, Ξ can be considered as a flat space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='  Proposition 2 has crucial contribution in: construction of dynamics;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' what we may call  classical limit of SUp8q-QGR despite the fact that it is fundamentally quantum;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' its rela-  tionship with Einstein gravity;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' and physical interpretation of what we perceive as curvature  of spacetime in presence of matter and its fluctuations observed as gravitational waves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' For  instance, an immediate conclusion from this proposition is that the parameter space Ξ of  the SUp8q symmetry of subsystems cannot be by itself the classical spacetime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' Nonethe-  less, in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' 5 we show that they are conceptually related.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' From now on we indicate the  arbitrary metric of the 4D parameter space Ξ as ηµν (or its 2-dimensional analogue for 2D  10In general relativity, for a given distribution of matter, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' vacuum, there is an equivalence class of  metrics which lead to the same curvature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' They are related by 4 constraints that present reparameterization  (diffeomorphism) of the spacetime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' The 6 d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='o.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='f of gµν determines whether the scalar curvature changes by  this transformation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' The remaining 4 d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='o.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' are associated to redefinition of coordinates without changing  observables [109].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='  – 28 –  parameter space of each diffeo-surface) to distinguish it from effective classical metric gµν,  which we obtain in the next section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' Finally, Proposition 2, which demonstrates embedding  of the geometric connection in the SUp8q gauge connection provides a natural explanation  for gravity/gauge duality conjecture [79–81, 110–112].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='  5  Classical curved space and its relationship with quantum states of  subsystems  In the previous section we showed that the geometry of the parameter space Ξ of SUp8q  symmetry representations by subsystems is arbitrary and can be gauged to be flat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' This  conclusion raises an enigma, if we want to interpret the parameter space as the spacetime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='  Thus, one might argue that SUp8q-QGR is already ruled out.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' For instance, deformation  of x-ray spectral lines emitted from accretion disk of black holes [113], mirage image of  black holes [114], and many other observed phenomena are evidence of curved spacetime  in presence of strong gravity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' In fact, using geometrical interpretation of spacetime, in  classical physics we can measure coordinate-independent relative distance between subsys-  tems and indirectly observe the curvature of spacetime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' Nonetheless, in this section we  relate these observations to quantum state of subsystems of the Universe and interpret the  observered classical spacetime as effective (averaged) measure of the quantum state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' For  this purpose we begin with a brief review of speed limit in quantum mechanics, which its  classical analogue was the reason behind development of special and general relativity, and  eventually the interpretation of curved spacetime as gravity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='1  Speed limit from quantum uncertainties  Introduction of non-Euclidean geometries in physics began with Hermann Minkowski, who  unified the description of Maxwell’s electromagnetic equations using what is now called  Minkowski metric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' After observation of the constant light speed and discovery of special  relativity by Einstein, works by Henry Poincar´e and Minkowski showed that Minkowski  geometry can also explain Lorentz transformations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='  In general relativity the indefinite  geometry of Minkowski space is generalized to a coordinate-dependent indefinite metric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='  Following this historical path, it seems logical to use the emergence of a finite speed of  state change - information transfer - in quantum mechanics to investigate the relationship  between classical spacetime and quantum state of subsystems of the Universe in SUp8q-  QGR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='  The origin of limited speed of information transfer in quantum mechanics is the Heisenberg  uncertainty relation between time and energy:  ∆t∆E ě ℏ{2  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='1)  where ∆ means standard deviation of measurement outcomes of time t and energy E,  respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' In addition, the following relation applies to standard deviations of any two  – 29 –  statistical variables S1 and S2:  ∆S1∆S2 ě 1  2pS1S2 ´ S2S1q  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='2)  where the bar in the r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' means statistical average.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' It is clear that the r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' of (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='2)  is zero unless S1 and S2 are operationally correlated, that is measurement of one of the  quantities affects subsequent measurement of the other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='  Using (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='1-5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='2), in 1945 Mandelstam and Tamm obtained a lower limit for the time neces-  sary to transform a quantum state ˆρ1 to another perfectly distinguishable state ˆρ2 [100].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='  Several other speed bounds have been later discovered [115, 116], see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' [117] for a con-  cise review and references.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' Some of these bounds are equivalent to Mandelstam-Tamm  limit [118] and are experimentally reachable for a set of initial states and final states [119].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='  The Mandelstam-Tamm minimum time for change of a quantum state ˆρ1 to state ˆρ2 such  that the two states are completely distinguishable, that is xˆρ1ˆρ2y “ 0, can be written  as [51, 120–122]:  ∆t ě ℏ  ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='  2  cos´1 Apˆρ1, ˆρ2q  b  Qpˆρ1, ˆHq  ,  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='3)  Apˆρ1, ˆρ2q ” trp  a  ˆρ1  a  ˆρ2  :q,  Qpˆρ, ˆHq ” ´1  2trpr  a  ˆρ, ˆHs2q  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='4)  where ˆH is the system’s Hamiltonian and the geodesic distance in BrbHss used in the  definition of Apˆρ1, ˆρ2q is the Winger-Yanase skew information metric [115].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' For pure states  this distance measure is equal to the fidelity Fpˆρ1, ˆρ2q defined in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='  The duration lower limit (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='3) gives a physical meaning to the time variable that appears  in the Heisenberg uncertainty relation as standard deviation of time necessary for changing  the state of a quantum system [123].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' However, due to the non-uniqueness of the square-  root of a matrix11, equation (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='3) can be applied without ambiguity only to pure states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='  For mixed states obtaining a speed limit is more complicated and several approaches and  suggestions can be found in the literature [51, 117, 122].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' In what concerns application of  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='3) to SUp8q-QGR in the next subsection, the density matrix ˆρ of the Universe includes  all the subsystems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' Therefore, no state is traced out and ˆρ can be in principle considered as  pure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' However, as we are only interested in the SUp8q sector of subsystems, we assumed  that the contribution of internal symmetries to the density matrix is traced out.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' For this  reason, ˆρ can be mixed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' Nonetheless, for the sake of simplicity, we overlook complexities  related to mixed states and use Mandelstam-Tamm inequality as a formal description.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='2  Effective path in the parameter space  To investigate the relationship between quantum state and apparent geometry of classical  spacetime in the framework of SUp8q-QGR, we consider ˆρ1 to be the state of subsystems  11It is straightforward to find examples of matrices ϱ with the same ϱ2 which satisfies properties of a  mixed/incoherent density matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='  – 30 –  of the Universe after selection of a reference observer and a clock.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' Due to the tracing of  internal symmetries and global entanglement, the density matrix ˆρ1 comprises states of all  subsystems, including clock and reference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='  The density operator ˆρ2 is assumed to be the infinitesimal variation of ˆρ1 for dt variation  of the time parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' Thus, ˆρ2 “ ˆρ1 ` δˆρ1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' We assume that the clock is chosen such  that in (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='3) the minimum time variation is achieved and (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='3) becomes an equality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' This  is possible for any state, at least asymptotically [118, 119].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='  After Taylor expansion of  functionals A and Q defined in (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='4), and insertion of the results in (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='3), we find:  2  ℏ2 Qp ˆH, ˆρ1qdt2 “ ´ 1  ℏ2 trr  a  ˆρ1, ˆHs2dt2 “ trp  a  δˆρ1  a  δˆρ1  :q ” ds2  WY  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='5)  Separation dsWY between states ˆρ1 and ˆρ1 ` δˆρ1 according to the Wigner-Yanase skew  information [115, 116] is independent of the basis chosen for the Hilbert space and how  this basis is parameterized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' In the parameter space, tracing operation in the last equality  guarantees the independence of equalities from parameterization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' If the dynamics is uni-  tary, that is if all the operations on the system are POVM, the evolution of state ˆρ1has  a well-defined path in the Hilbert space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' However, operators and functionals in the first  equality of (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='5) are functions of the parameters pt, r, θ, φq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' The dependence of equalities  in (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='5) only on the time variation dt makes the parameterization corresponding to the  minimum time variation for achieving dˆρ analogous to a classical reference frame at rest  with respect to the chosen reference frame:  Λds2  WY ” ds2 “ gµνdxµdxν,  x “ pt, r, θ, φq,  g00 “ 2Λ  ℏ2 Qp ˆH, ˆρ1q, gµν “ 0, µ||ν ‰ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='6)  Notice that in (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='6) we have used Cartesian coordinates for the parameter space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' The  role of the dimensionful constant factor Λ is to ensure correct dimensionality of quantities,  because the Fubini-Study metric is dimensionless.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' Giving the fact that Hilbert space of a  quantum system is projective, one can always multiply sWY by a constant without affecting  physical interpretations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='  A coordinate transformation x ” pr, θ, φ, tq Ñ x1 ” pr1, θ1, φ1, t1q does not change states ˆρ1  and ˆρ1`δˆρ1 and thereby the last equality in (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' However, in general the variation time dt1  with respect to the new clock parameter t1 would not be minimal and Mandelstam-Tamm  bound takes its general form:  2Λ  ℏ2 Q1p ˆH1, ˆρ1qdt12 ” g1  00dt12 ě Λ trp  a  δˆρ1  a  δˆρ1  :q “ ds2  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='7)  Moreover, the metric gµνpxq Ñ g1  µνpx1q takes a more general form:  ds2 “ g1  µνdxµdxν “ g1  00dt12 ˘ g1  iidx1idx1i  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='8)  where we have assumed a parameter transformation such that g1  00 ą 0 and g1  0i “ g1  i0 “ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='  In addition, assuming isotropy of the spatial parameters xi, i “ 1, 2, 3 (We justify this  assumption after discussing the physical meaning of (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='8)), g1  ij’s should have the same sign.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='  – 31 –  For this reason we have assumed g1  ij ą 0, @i, j and factorized their sign.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' The operator ˆH1  in (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='7) is the Hamiltonian for the new clock parameter t1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' By comparing (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='8) with (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='7)  we find that if ds2 ě 0 (or equivalently ds2  WY ě 0), the constraint (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='7) is satisfied only if  the sign of spatial part in (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='8), and thereby the signature of the metric g1  µν is negative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='  Appendix F describes expansion of state ˆρ and its variation with respect to ˆLlm generators  of SUp8q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='  The above results were first reported in [39].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='  In one hand, they demonstrate that the  origin of the limited speed of light/signals and Lorentzian metric of the classical spacetime  is the quantum nature of the Universe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' On the other hand, although Mandelstam-Tamm  inequality can be interpreted as the reason for speed limit, it is not enough for relating  quantum uncertainty to the geometry of classical spacetime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' The SUp8q-QGRfills the gap  by interpreting the spacetime as parameter space of a quantum Universe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='1  Path in the full parameter space  In the above formulation we assumed that the contribution of internal symmetries in the  density matrices are traced out.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' Without such simplification, we should add the contribu-  tion of collective parameter α defined in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='6) to (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='8):  ds12  G ” Λ trp  a  δˆρG  a  δˆρG  :q “ gµνpx, tαuqdxµdxν `  ÿ  i,jPt1,¨¨¨ ,Nu  gijpx1, tαuqdαidαj (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='9)  ds2 “ Λ trαp  a  δˆρG  a  δˆρG  :q ” gµνpxqdxµdxν  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='10)  where for the sake of simplicity of notation we have kept the same symbol for the effec-  tive metric before and after tracing on G symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' The index G is used wherever the  contribution of G symmetry is not traced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' To be more explicit, the set tαu is assumed  to include N parameters presented collectively as α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' They characterize the representation  the internal symmetry G by quantum states of subsystems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' In (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='9) they are considered  to be continuous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' In this way, the metric (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='9) has the form of a p1 ` 3q ` N-dimensional  Riemann geometry and ds2  G is the affine parameter of the average path of ˆρG state in this  extended parameter space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' On the other hand, as SUp8q and G are by assumption orthog-  onal and independent gµi “ 0 for any parameterization or reparameterization of the two  sectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' Moreover, as G is assumed to have a finite rank, its representations have usually  discrete parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' Therefore, gij ‰ 0 only for discrete values of tαu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' 2 schematically  presents geometry of parameter space according to (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='9) and (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='10) and their relation12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='  In analogy with exchange of virtual particles with negative amplitude of 4-momentum  q2 ă 0 in t-channel, states with ds2 ă 0 can be called virtual.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' Indeed, for pure states  12One might claim that the relation between classical affine parameter and quantum state of Universe’s  subsystems is not new, because in semi-classical general relativity geometric part of the Einstein equation  is related to expectation value of energy-momentum tensor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' However, the latter is a classical quantity and  it is well known that its quantum counterpart is not well defined and has - at least naively - an infinite zero  mode [124, 125].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' By contrast, density matrix is an intrinsically quantum and well defined quantity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='  – 32 –  Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' A schematic presentation of SUp8q-QGR parameter space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' The lowest (darker) surface  depicts the parameter space of SUp8q symmetry - the spacetime - of subsystems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' The path inside  this surface is the average - classical - path of the quantum states of subsystems in the parameter  space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='  Orthogonal to this surface depicts the collective parameter α of the internal symmetry  G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' Surfaces at discrete values of α remind that as a finite-rank Lie group, representations of G  are in general characterized by discrete quantities - quantum numbers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' The background presents  parameter space of representations of SUp8q ˆ G as a continuum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' The path crossing surfaces is  the average path of subsystems in the full parameter space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='  ds2 ě 0 always.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' Therefore, a necessary condition for ds2 ă 0 is that the state ˆρ and its  presumed evolution ˆρ ` dˆρ should not be pure or reachable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='  Despite the crucial role of (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='9) and (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='10) for understanding the nature of classical space-  time and its relation to quantum realm and content of the Universe, they have little use for  understanding evolution of subsystems, because they do not indicate what governs vari-  ation of the quantum state ˆρ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' This is similar to general relativity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' Although a suitable  form for the metric may be chosen based on symmetries of the matter distribution - for  instance a spherically symmetric metric for a non-rotating spherically symmetric distribu-  tion of matter - one needs a dynamical equation such as Einstein equation (in classical  case), to relate spacetime geometry to evolution of matter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' In Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' 6 we describe dynamics  of SUp8q-QGR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='2  Proper (equal-time) distance and causality  The introduction of a size parameter r in the Hilbert space of subsystems does not specify  how it is determined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' Indeed, despite the fact that its absolute value can be redefined and  is not a measurable, geometrical distance on 3D hypersurfaces of constant time, that is  for a given state of quantum clock, is a measurable quantity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' According to the definition  of affine separation (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='5) and its expansion as the metric (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='8), the infinitesimal proper  – 33 –  da  ds  G  dt  ds  dxdistance dl12 between two states ˆρ1 and ˆρ2 “ ˆρ1 ` δˆρ1 is:  dl2  12 ” Λ2trp  a  δˆρ1  a  δˆρ1  :q  ˇˇˇˇ  dt“0  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='11)  The parameter space distance between any two states is:  l12 “ min  ż  γ12  dl12,  @γ12  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='12)  where γ12 is a path in the Hilbert space connecting states ˆρ1 and ˆρ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' By construction the  r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' of (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='11) and (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='12) are independent of the Hilbert space basis and reparameteriza-  tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='  Locality and causality:  In dynamical models (quantum or classical) with a background  spacetime locality is usually defined as decreasing of interactions and correlation with dis-  tance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='  In particular, the limited speed of light and information transfer according to  Mandelstam-Tamm inequality (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='3) automatically limit the spatial extent of causal inter-  actions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' More generally, locality can be defined as tensor product decomposition/structure  (TPS) similar to (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='8) such that interactions are bounded to factors [126].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' Such definition  is dynamical and depends on the resolution available to observers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' Moreover, in SUp8q-  QGRas all subsystems interact through gravity - SUp8q symmetry - locality based on  TPS is an approximation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' Using the relation between metric of classical spacetime and  variation of quantum state of subsystems, in analogy with general relativity, we can define  a lightcone for each state ˆρ in the Hilbert space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' States ˆρ ` δˆρ are called causally related  to ˆρ if ds2 ě 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' Two states ˆρ1 and ˆρ2 are causally related if their lightcones intersect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' This  definition is global and independent of clock, observers, and resolution of their experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='3  Interpretation of affine separation and its curved geometry  We notice that in (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='7) and thereby in (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='8) the only quantity which is independent of  the bases chosen for the Hilbert space and its parameterization is the affine separation ds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='  Although the unitary evolution of state ˆρ1 with respect to the clock follows a well defined  path in the Hilbert space, it has no clear orbit in the parameter space, because in general  ˆρ1 is a superposition of all eigen states |xy - in the old quantum mechanic language its path  is not sharp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' Therefore, the metric in (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='8) is not a genuine physical entity and must be  considered as an effective representation - an order parameter representing the quantum  state ˆρ1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' Even g1  00 in the r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' of (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='7) cannot be considered as a measurable quantity,  because its value can be redefined by changing time parameter t1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' When an observer cannot  explicitly detect SUp8q-symmetry related quantum (QGR) effects, their influence on the  subsystems is indirectly perceived and interpreted as gravity convoyed by the curvature of  the parameter space - the spacetime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='  Our demonstration that this effective metric has always a negative signature justifies our  claim in previous sections that parameter space of the SUp8q-symmetry Hilbert space of  – 34 –  subsystems has R3`1 geometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' In special and general relativity, the signature of met-  ric is dictated by the observation of the constant speed of light in classical vacuum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' In  SUp8q-QGR this is a direct consequence of quantum speed limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' Moreover, considering  proposition (1) and the dependence of g1  00 in (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='8) (or equivalently g00 in (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='6)) on the def-  inition of a quantum clock, a redefinition of time can be viewed as changing entanglement  between clock subsystem and the rest of the Universe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' Despite these observations, one  may wonder whether considering a specific signature for the parameter space is necessary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='  Indeed, giving the fact that Hilbert spaces of subsystems are defined on C, it cannot be  ruled out that in some circumstance (applications) one may need to extend the parameter  space, notably the time parameter, to complex plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' An example of such cases in QFT  without gravity is a thermal model, which can be treated as a zero temperature QFT with  complex time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' There are also examples of field theory models in which a field plays the  role of time parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' It is shown that in such models the signature of the metric can  change [127].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' Therefore, not only the curvature but also the signature of the parameter  space Ξ is not a physical observable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' Nonetheless, for being consistent with the average  geometry, which is an observable, it is convenient to use a Lorentzian metric for x space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='  The close relation between uncertainty relation, signature of the metric of the parameter  space (spacetime), and its Lorentz symmetry in the framework of SUp8q-QGR shows that  modifying Heisenberg uncertainty relation, as proposed in some QGR models [101, 102],  may affect Lorentz invariance or equivalently the constant of speed of light and graviton,  which are stringently constrained [107, 128, 129].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='  Recalling Einstein’s opinion about the nature of spacetime described in the Introduction, he  thought that only the affine separation has a proper physical meaning [40] (and references  therein).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' However, if the spacetime is not a physical entity - otherwise there had to be the  infamous ether - the association of the metric tensor to a geometry seems meaningless.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' The  modern view of metric/gravity as a spin-2 quantum particle apparently solves this puzzle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='  But, it does not explain why this field determines the geometry of the spacetime, which  according to Einstein does not have a physical reality of its own.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' SUp8q-QGR provides  straightforward explanation for these puzzles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='  Finally, a short interpretation of holographic principle applied to black holes and ADS/CFT  is discussed in Appendix G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='1  Relation with coherence and distinguishability  The Wigner-Yanase skew information satisfies conditions defined in [50] for quantifying the  amount of coherence of a quantum state and can be used as a measure of coherence [130].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='  Therefore, the speed limit of information transfer is related to quantum coherence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='  In  particular, for the completely incoherent state ˆρCI ” ř  i ρi|iyxi|, with ř  i ρi “ 1,  ˆH|iy “  λi|iy, QpˆρCI, ˆHq “ 0 and equation (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='3) leads to ∆t Ñ 8, meaning that the state of the  system is stable and does not change, unless it is non-unitarily modified by application of  an operator ˆH1, such that r ˆH, ˆH1s ‰ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' In SUp8q-QGR, this corresponds to a state of the  Universe where in the energy eigen basis, subsystems are decohered to a simple statistical  – 35 –  distribution with no quantum superposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' This is very far from the state of the Universe  we are observing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' Therefore, our Universe cannot be completely decohered and it should  be eventually possible to detect the signature of quantumness in cosmological observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='  See e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' [21, 131, 132] for attempts to quantify such signal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='  As a measure of coherence must be monotonic [50] and there is only one completely coherent  state ˆρCC ” NCC  ř  i,j |iyxj| where NCC is a normalization constant, the angle between all  coherent states of a Hilbert space is zero and the constraint (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='3) on their evolution time  becomes ∆t ě 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' This is a trivial constraint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' Therefore, ˆρCC has fastest variation to a less  coherent state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' In SUp8q-QGR an initial completely coherent state may be - at least in  part - responsible for the inflationary expansion of the early Universe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' We leave exploration  of such possibility for future works.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='  If ˆρ1 and ˆρ2 are orthogonal states, that is trpˆρ1ˆρ2q “ 0, and both states are pure, we find  ∆t ě ℏπ{2?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='2Q „ ℏ{∆E, where ∆E is the variance of energy for state ˆρ1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' Considering the  statistical meaning of variance, larger ∆E means that in the energy basis, the two states  are less distinguishable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' In particular, if ∆E Ñ 8, ∆t Ñ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' This means that despite the  orthogonality of ˆρ1 and ˆρ2, they are not distinguishable and speed of variation from one to  the other approaches that of completely coherent states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='  In conclusion, quantum speed limit, coherence, and distinguishability are related properties  of quantum systems and processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' In the framework of SUp8q-QGR, the relationship  between the average metric and quantum speed limit can be used to interpret what is  classically called spacelike events, as unitarily unreachable states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' Indeed, considering the  ambiguity of (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='3) for mixed states described in the footnote 11, one can find pair of density  matrices with cos´1 Apˆρ1, ˆρ2q ą π{2 of the Universe or even complex value for Apˆρ1, ˆρ2q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='  Specifically, such cases can be found when one of the matrix densities is pure and the other  one is incoherent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' In future works we will use these properties to explore physics of black  holes in SUp8q-QGR, where horizon crossing changes the sign of ds2 of the average metric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='  6  Evolution  So far we constructed and studied the Hilbert space of SUp8q-QGR as a model for the  Universe and its subsystems without considering the dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' In this section along with  the review of dynamics of SUp8q-QGRpresented in [39] and with more details in [38], we  clarify some issues which were not discussed in the previous works.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='  Experience from QFT has shown that we can use variational principle at all perturbative  order to describe dynamics of many-body quantum systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' However, in quantum me-  chanics and QFT Lagrangians (or Hamiltonians) of models are usually constructed based  on their classical limit model and are subsequently quantized, for instance by using path  integral formalism or replacing canonical variables by their associated quantum operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='  On the other hand, SUp8q-QGR is constructed as a quantum system, and in principle it  does not have a classical analogue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' In fact as we discussed in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='2, for ℏ Ñ 0 the alge-  bra (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='1) becomes Abelian and the model becomes trivial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' Therefore, the construction of a  – 36 –  Lagrangian for SUp8q-QGRmust be based on the fact that Hilbert spaces of the Universe  HU and its subsystems Hs represent SUp8q symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' For the whole Universe the evident  candidate for a Lagrangian is a SUp8q invariant functional, describing all possible excur-  sion of its states in the Hilbert space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' For subsystems the functional must be in addition  invariant under the finite rank G symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='1  The whole Universe  In SUp8q-QGR the Universe as a whole is by construction static.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='  In fact, according  to symmetry description of quantum mechanics axioms [26], any single closed quantum  system without structure is static and trivial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' More generally, we know that in quantum  mechanics and quantum gravity evolution must be considered as relative concept [30] to  avoid inconsistencies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' Thus, any isolated quantum system, such as the whole Universe  without anything outside it, is necessarily static.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='  Nonetheless, a static Lagrangian - a  general functional of parameters invariant under symmetries of the system - can be defined  and variational principle can be applied to it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' Specifically, minimization of this functional  determines equilibrium state of the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' In the case of the whole Universe as a closed  system in SUp8q-QGR, representations of SUp8q symmetry of the form (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='3) depend  on continuous parameters13, and usual variational method straightforwardly applies to  the Lagrangian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' Fluctuations of parameters around their equilibrium values, obtained by  variational method, can be interpreted as quantum fluctuations of the ground state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' In the  operational view employed in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' 3, these fluctuations correspond to random application  of the members of BrHUs – SUp8q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' In the next subsections we make these points more  quantitative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='1  Formal description of dynamics  In absence of a background spacetime in SUp8q-QGR the Lagrangian functional must be  defined on the Hilbert space and must be invariant under SUp8q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' In particular, for any  SUpNq, @N symmetry group, trace of multiple hermitian generators trpT a1T a2 ¨ ¨ ¨ q are  constant and invariant under group conjugate operation, that is g´1g1g “ g:g1g, @g, g1 P  SUpNq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' Moreover, these traces depend only on the normalization of generator trpT aT bq ”  1{2δab, anti-symmetric structure coefficients fabc defined in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='1), and symmetric anti-  commutation coefficients dabc ” 2 trptT a, T buT cq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' Therefore, these traces can be used to  construct a symmetry invariant functional.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' Moreover, in analogy with QFT Lagrangians, it  is enough to consider the lowest order traces of generators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' Higher orders can be considered  as higher order quantum corrections and obtained from a path integral generator functional.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='  Under these considerations, the lowest order SUp8q-invariant functional can be written  13The group SUp8q has also representations depending only on discrete parameters [64, 65].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' However,  they are dense in Rp⊭q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' This is the reason for their homomorphism with Virasoro algebra without central  charge discussed in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='  – 37 –  as:  LU “  ż  d2Ω  „1  2  ÿ  a, b  trpL˚  apθ, φqLbpθ, φqˆLa ˆLbq ` 1  2  ÿ  a  ˆ  Lapθ, φqtrpˆLaˆρUq ` C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='C  ˙ȷ  (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='1)  d2Ω ”  b  |ηp2q|dpcos θqdφ  (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='2)  where a ” pl, mq and C-number amplitudes La determine the contribution of SUp8q  generators ˆLa to the Lagrangian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' The dimensionless SUp8q coupling constant is included  in the amplitudes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' The density matrix ˆρU describes the quantum state of Universe as a  whole.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' The scalar ηp2q is determinant of the metric of 2D diffeo-surface ηp2q  µν , µ, ν P θ, φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='  By definition the whole Universe is in a pure state, because there is nothing outside and  entangle to it, which could have been possibly traced out.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' Therefore, its density matrix  can be written as ˆρU “ |ΨUyxΨU|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' We should also remind that the integration over angular  coordinates of diffeo-surface is part of the tracing operation, because generators (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='3) of the  SUp8q symmetry are defined at each point of the diffeo-surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' This reflects the fact that  diffeomorphism of diffeo-surface in the neighborhood of each point is independent from  deformation around other points up to continuity condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='  To be consistent with Lagrangians used in QFT we assume that LU P R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' Using represen-  tation (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='3) for ˆLlm it is straightforward to see that ˆL:  lm “ p´1qpm`1q ˆLl,´m, see Appendix  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' This means that (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='3) is an anti-hermitian representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' For this reason there is  no trpˆL:  a ˆLbq term in (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='1) and phases of amplitudes La in the first term of (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='1) are irrel-  evant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' In addition, in the second term ˆρU is hermitian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' Therefore, complex conjugate of  trpˆLaˆρUq is not independent from itself, and Lapθ, φq must be real, otherwise LU becomes  complex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' This conclusion is not limited to the representation (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='3) and applies to any  (anti-)hermitian representations of SUp8q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' In [39] it is explicitly shown that, as expected,  applying variational principle with respect to amplitudes La and components of ˆρU leads  to a vacuum state as the equilibrium solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' This result is consistent with a corollary  of the axioms of quantum mechanics reformulated to present symmetry as a foundational  concept [26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' It states that a quantum Universe as a whole, that is when its subspaces are  ignored, is trivial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' A short summary of this approach to quantum mechanics can be found  in appendices of [39]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='2  A Yang-Mills whole Universe  The action LU is a formal description.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' In particular, it does not clarify how the ampli-  tudes La’s change under global Lorentz and Poincar´e transformations of coordinates of  the diffeo-surface pθ, φq and application of SUp8q group, which up to a global scaling of  the area is homomorphic to local reparameterization of diffeo-surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' The Lagrangian LU  must remain invariant under these transformations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' To this end and for getting a better  insight into the analytical form of the amplitudes, we analyze the integrand of (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='1) for its  transformation under aforementioned symmetries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='  – 38 –  First of all the surface element d2Ω is invariant under reparameterization of angular coor-  dinates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' Thus, each term in the integrand of (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='1) must be reparameterization invariant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='  Moreover, amplitudes La must be invariant under translation θ Ñ θ ` θ0, φ Ñ φ ` φ0  for arbitrary constant shift of the coordinates origin by θ0 and φ0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' This condition would  be satisfied if La’s is a differential operator with respect to coordinates pθ, φq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' In addi-  tion, the operator must be invariant both under reparameterization of the diffeo-surface  and SUp8q transformation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' The only analytical expressions with these properties are 2-  forms in coordinates of the diffeo-surface, which must be also singlet of SUp8q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' Thus,  La ˆLa ” F µν  a ˆLa, µ, ν P cos θ, φ and the Lagrangian functional (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='1) takes the form of a 2D  Yang-Mills Lagrangian for SUp8q symmetry:  LU “  ż  d2Ω  „ 1  2 trpF µνFµνq ` 1  2trp\x1a  \x1a  DˆρUq  ȷ  ,  µ, ν P θ, φ  (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='3)  Fµν ” F a  µν ˆLa ” rDµ, Dνs,  Dµ “ pBµ ´ Γµq1 `  ÿ  a  iλAa  µ ˆLa,  a ” pl, mq  (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='4)  F a  µνF µν  a  “ L˚  aLa,  @a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='  (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='5)  where Dµ is 2D covariant and gauge preserving derivative with an appropriate 2D con-  nection Γµ [133, 134] (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='g [134] for a recent review) and λ is a dimensionless coupling  constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' It can be included in Aa  µ, but here we include it explicitly to keep the formulation  similar to the standard Yang-Mills theory .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' The symbol 1 presents unit operator of the  sup8q algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' In analogy with QFT the coordinate dependent vector Aa  µ can be called  the SUp8q gauge field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' Nonetheless, once again we emphasize that in contrast to a QFT  on an spacetime, pθ, φq parameterize representation of SUp8q symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='  In the last term of integrand in (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='3), the exact expression of the differential operator \x1a  \x1a  D  depends on the representation of 2D Euclidean group by the state |ψUy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' For a scalar-type  state \x1a  \x1a  D “ ÐÝ  DµÝÑ  D  µ “ ÐÝB µÝÑB  µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' For a 2D spinor state it can be written using zweibein vectors:  \x1a  \x1a  D “ iσ0σizµ  i  ÐÑ  D µ  ,  Dµ “ pBµ ` Γµq1 `  ÿ  a  iλAa  µ ˆLa,  (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='6)  Γµ ” 1  8ωµijrσi, σjs ” 1  4ωµijσij  ,  rωµsi  j “ zi  ν  ˆ  Bµpzν  j q ` zσ  j Γν  µσ  ˙  ,  (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='7)  σij ” i  2rσi, σjs  ,  ηp2q  ij zi  µzj  ν “ ηp2q  µν  ,  ηp2q  µν zµ  i zν  j “ ηp2q  ij .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='  (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='8)  where σi i “ t1, 2u are two of the N Ñ 8-dimensional representation of Pauli matrices,  and σ0 is the third Pauli matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' The tensors zµ  i ’s are zweibeins (analogous to vierbein  (tetrads) for 2D geometry) [134, 135].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' 2-dimensional metrics ηp2q  µν and ηp2q  ij are arbitrary and  flat metrics of the parameter space - diffeo-surface - of the Hilbert space HU, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='  Despite the presence of a connection term in (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='4), it does not affect F µν.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' In addition, as  a function defined on the 2D diffeo-surface Γµ can be decomposed to spherical harmonic  functions, which are in turn generators of SUp8q algebra, and without loss of generality  can be included in Aa  µ term in (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' This is an explicit demonstration of the Proposal  2 applied to the Universe as a whole.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' Consequently, the choice of a special geometry for  – 39 –  the diffeo-surface of the Universe is equivalent to fixing SUp8q gauge and has no effect  on measurable quantities, namely, F a  µν - up to an unobservable rescaling of the area of  the diffeo-surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' We should also remind that in 2D spaces the 2-form F a  µν has only one  independent nonzero component.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' Therefore, the number of degrees of freedom in the two  sides of (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='5) is the same.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='3  A topological Universe  Independence of the Lagrangian (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='3) from geometry of diffeo-surface M and its parame-  terization means that the SUp8q Yang-Mills model of the whole Universe is topological.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='  Topology of 2D compact Riemann surfaces are classified by the Euler characteristics:  ż  d2Ω Rp2q “ 4π χpMq,  χpMq “ 2 ´ GpMq  (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='9)  where Rp2q is the 2D scalar curvature of the diffeo-surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' The Euler characteristic χpMq “  2´GpMq depends only on the surface’s genus G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' The topological nature of the Lagrangian  (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='3), in particular the pure gauge term implies that without loss of generality its integrand  must be proportional to Rp2q:  trpF µνFµνq 9 Rp2q  (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='10)  This relation can be also considered as equality, because the area of the diffeo-surface is  arbitrary and is not an observable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' Thus, (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='10) establishes a relation between geometry of  the parameter space of SUp8q representation and amplitude of the SUp8q gauge field, and  at lowest quantum order, variational principle relates the gauge field to the quantum state  of the Universe ˆρU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' Consequently, there is no arbitrary or unrelated quantity in the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='  Moreover, (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='10) explicitly demonstrates the relation between the Yang-Mills Lagrangian  (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='3) and what we perceive as gravity through curvature of the classical spacetime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' Indeed,  the number of degrees of freedom of pure Yang-Mills fields in dimension D is DpD ´  1q{2, which is the same as the number of independent components of the Ricci curvature  tensor Rµν after taking into account D coordinate reparameterization constraints - or  equivalently 4 energy-momentum conservation constraints in classical Einstein theory of  gravity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' Interestingly, this relation between degrees of freedom of the two models exists  in any dimension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='  In any case, none of these quantities are dynamic and due to the  SUp8q symmetry, all the states are physically equivalent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' Nonetheless, the Lagrangian  (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='3) distinguishes between non-equivalent representations of SUp8q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='  In a geometrical view, if local details of the Universe are not distinguishable, only its global -  topological - properties may characterize its states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' In SUp8q-QGR the only characterizing  global property is the topology of the diffeo-surface, which determines non-homomorphic  representations of the SUp8q symmetry [65, 66] of the Universe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' In other words, the only  possible difference between whole universes is the representation of SUp8q by their Hilbert  spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' Thus, we can consider the genus of the diffeo-surface as the only initial condition for  the Universe in SUp8q-QGR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' Alternatively, one may assume that a quantum universe may  be in a superposition of independent representations of SUp8q, that is in super position  – 40 –  of Lagrangians of type (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='3) with different values for the integral in (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' We give a short  assessment of consequences of coherent mixing of global typologies in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='4 and leave  its detail investigation to a future work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='2  Evolution of subsystems  As we discussed in sections 4 and 5, when the Universe is divided to subsystems and a  reference subsystem and a quantum clock are chosen, every pair of their 4 continuous  parameter space can be considered as parameters of a SUp8q symmetry under which  physical properties of subsystems must be invariant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' The presence of a clock means that  in contrast to the whole Universe, the ensemble of subsystems can be indeed treated as a  dynamical quantum system and in analogy with QFT a Lagrangian can be associated to  it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' For this purpose, we proceed in the same manner as we did for the whole Universe and  consider the lowest order functional invariant under SUp8q ˆ G as the Lagrangian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' The  formal description of this functional has the following form:  LUs “  1  4πL4  P  ż  d4x  b  |ηp4q|  „ 1  4  ˆ  ÿ  l,m,l1,m1  trpL˚  lmpxqLl1m1pxqˆLlm ˆLl1m1q `  ÿ  a,b  trpT ˚  a pxqTbpxq ˆTa ˆTbq  ˙  `  ÿ  l,m,a  trpLlmpxqTapxqˆLlm b ˆTaq `  ÿ  lm  Llm trpˆLlm b 1G ˆρspxqq ` 1  2  ÿ  a  Ta trp1SUp8q b ˆTa ˆρspxqq  ȷ  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='  (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='11)  where T a’s are generators of the finite rank internal symmetry G of subsystems, and LP ”  a  ℏGN{c2 is the Planck length.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' The metric ηµν is an arbitrary metric for the 4-dimensional  parameter space presented in Cartesian form and ηp4q is its determinant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' Amplitudes Llm  and Ta are normalized to be dimensionless.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' The first two terms present self-interaction of  SUp8q and G symmetries, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' If the generators of symmetry groups are chosen  to be traceless, the third term in (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='11) would be zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' The forth and fifth terms present  interaction of subsystems due to SUp8q and G symmetries, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' Notice that we  have considered an arbitrary metric for the parameter space, despite Proposition 2, which  allows the choice of a Minkowski metric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' We explain this choice later in this section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='  Following the same line of arguments given for (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='1) about the invariance of parameter  space under coordinates transformation i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='  Poincar´e symmetry, invariance under both  SUp8q and G transformations, and demonstration that the resulting action should have  the structure of a Yang-Mills model, we conclude the following final form for the Lagrangian  – 41 –  LUs:  LUs “  ż  d4x  a  |η|  „  1  16πL2  P  trpF µνFµνq ` 1  4trpGµνGµνq ` 1  2  ÿ  s  trp\x1a  \x1a  Dˆρsq  ȷ  ,  µ, ν P t0, 1, 2, 3uη  (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='12)  Fµν ” F lm  µν ˆLlm ” rDµ, Dνs,  Dµ “ pBµ ´ Γµq1 ´  ÿ  lm  iλAlm  µ ˆLlm,  L2  P F lm  µν F µν  lm “ L˚  lmLlm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='  (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='13)  Gµν ” Ga  µν ˆT a ” rD1  µ, D1  νs,  D1  µ “ pBµ ´ Γµq1 ´  ÿ  a  iλGBa  µ ˆT a,  L4  P Ga  µνGµν  a  “ T ˚  a T a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='14)  As explained earlier, \x1a  \x1a  D depends on the representation of Lorentz symmetry by subsystems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='  For instance, for Dirac spinors it is:  \x1a  \x1a  D ” γ0γieµ  i  ÐÑ  D µ  ,  Dµ “ pBµ ´ Γµq1 ´  ÿ  lm  iλsAlm  µ ˆLlm ´  ÿ  a  iλGBa  µ ˆT a,  (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='15)  Γµ ” 1  8Ωµijrγi, γjs ” 1  4ΩµijΣij  ,  rΩµsi  j “ ei  νpBµpeν  j q ` eσ  j Γν  µσq  (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='16)  Σij ” i  2rγi, γjs  ,  ηijei  µej  ν “ ηµν  ,  ηµνeµ  i eν  j “ ηij  ,  i, j P t0, 1, 2, 3uη.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='17)  where γi are Dirac matrices, and other quantities in (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='16-6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='17) are analogous to those of  (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='6 - 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='8), but in the 4-dimensional parameter space of subsystems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='  As expected there is no cross term between pure gauge field terms of SUp8q and G, because  by construction they are orthogonal and the full symmetry of subsystems is their tensor  product.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' On the other hand, without any cross term, interaction of gauge fields is simply  similar to QFT in curved spacetime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' However, according to proposition 2 the metric ηµν  of the parameter space is arbitrary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' Therefore, the question arises how inSUp8q-QGRdo  G gauge fields - including those of the Standard Model - interact with gravity ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='  To answer this question, we remind that even in Einstein gravity, the difference between  flat and curved space, arises only when derivatives of measurable fields are involved [133].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='  In particular, the definition of strength of Yang-Mills fields given in (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='13) shows that  nonzero connection of a curved space does not affect it, and becomes relevant only in the  field equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' Therefore, at Lagrangian level, only the metric convoys the curvature of  spacetime (here the parameter space) to gauge fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' However, the metric can be locally  considered to be flat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' Therefore, in this respect there is difference between SUp8q-QGRand  Einstein gravity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' Moreover, in various ways an interaction between Fµν and Gµν arises in  SUp8q-QGR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='  First of all, although trpˆLlm ˆTaq “ 0, higher order cross terms are not null.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' The lowest  gauge invariant term not included in LUs is trpF µνFµνqtrpGµνGµνq, which corresponds to a  4-vertex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' It does not generate mass neither for G nor for SUp8q, because it depends on the  field strength rather than gauge fields Aµ  lm and Bµ  lm  14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' Even without explicitly introducing  14Aµ  lm and Bµ  a present the decomposition of the wave function of SUp8q and G gauge bosons with respect  to generators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' It is also why they appear in pair in the Lagrangian, exactly similar to ˆρ when it is pure, or  when it is written in its canonical purification form ˆρ “ ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='ˆρ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='ˆρ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='  – 42 –  such higher order interaction terms, due to the interaction of density matrix of subsystems  with both SUp8q and G bosons, they will arise in the path integral constructed from the  Lagrangian (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='12), and thereby in the effective action.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' In addition, in the field equation  obtained from the Lagrangian (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='12) both ˆρs and Gµν contribute to the source term for  the gravity field strength Fµν.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='  We should also remind that even in classical physics both Einstein gravity and Yang-  Mills action can be derived in an arbitrary background geometry from the assumption of  consistent self-interactions [136].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' Another evidence for the interaction of gravity and gauge  bosons in general relativity is the fact that scalar curvature of a universe containing only  radiation - gauge bosons - vanishes everywhere, because the trace of energy-momentum  tensor is zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' This means that sectional curvatures must be null or cancel each others  everywhere, see (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='  Thus, the spacetime manifold of such universe is Ricci-flat and  behaves like classical vacuum, despite the presence of energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='  Finally, giving the relation between the Pontryagin topological class, average metric gµν  defined in (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='10), and variation of the density matrix of subsystems ˆρs, it would be useful  to use gµν rather than an arbitrary metric for calculation of physical observables such  as Green’s functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='  Of course, in this case all the constituents of the model become  intertwined and self-interactions make evolution equations highly nonlinear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' Nonetheless,  in analogy with background field approaches in QFT, such as Kadanoff-Beim method which  includes background fields, considering gµν as a background field would lead to a better  approximation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' In any case, the choice of the metric ηµν corresponds to fixing the SUp8q  gauge and does not have any impact on the exact calculations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='1  Possibility of gravitational P, C, and CP violation  Lagrangian of the whole Universe conserves parity and CP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' This observation makes sense,  because the parameter space of the SUp8q symmetry is not directional.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' This implies that  even after the division of the Universe to subsystems, parity and CP must be globally  preserved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' Nonetheless, a priori subsystems may locally break these discrete symmetries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='  There are various reasons for such possibility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='  Yang-Mills theories have topologically non-trivial vacuum structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' Because this subject  is also relevant for the crucial issue of cosmological constant, we discuss it in some details  in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' Here we only remind that in the Standard Model, the possibility of non-  trivial vacuum in both electroweak and QCD interactions leads to what is called strong  CP problem, see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' [137, 138, 157] for review and more details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' This problem is closely  related to the issue of axial Up1q anomaly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' In both cases the problem can be summarized  in so called a Θ-trem in the action:  LΘ ” Θg2  32π2  ż  d4x ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='η ˜FµνFµν  ,  ˜Fµν ” 1  2ϵµνρσFµν  (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='18)  where F is gauge field strength and g is its coupling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='  The integrand in this action is  a total derivative and the integral is a topological invariant - the Pontryagin topological  – 43 –  class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='  However, the dual field ˜Fµν has a parity opposite to Fµν.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='  Therefore, when it  is added to the Yang-Mills action, the sum does not preserve the party.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='  Moreover, as  conjugation symmetry C is preserved by both actions, CP symmetry is also broken by their  sum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' Although parity symmetry is violated in electroweak interaction, there is stringent  constraint on the parameter Θ for the QCD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='  In SUp8q-QGR the quantum mediator of gravity is a Yang-Mills gauge field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' Therefore,  complexities related to topological structure of vacua, axial Up1q anomaly which arises  when fermions can be considered massless, and violation of P and CP due to a Θ-term  similar to (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='18) may arise in the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='  In particular, as the energy scale in which  quantum gravity effects become important - presumably the Planck mass MP - is much  larger than the mass of all SM particles, they can be considered as massless and axial Up1q  anomaly would be more serious.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' Nonetheless, there are various solutions for this issue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' For  instance, at Planck scale strong non-perturbative gravitational interactions can increase  the effective mass of particles and prevent the anomaly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' On the other hand, in the strong  gravity regime of early Universe these anomalies might have seeded baryons and leptons  asymmetries through sphaleron-type phenomena, see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' [140, 141] for reviews, or at least  contributed to them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' We leave investigation of such effects to future works.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='  In what concerns parity violation in gravity sector, at present there is no observation  or constraint even for classical gravity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' Nonetheless, future generations of gravitational  wave detectors should be able to test it [142–144].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' Assuming no P or CP is observed in  gravitational sector, a possible solution for violation of discrete symmetries by Θ´term of  SUp8q-QGR sector is introducing a pseudo-scalar gravitational axion field φg, analogous  to a Peccei-Quinn Up1q proposed to solve the QCD strong CP problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' It replaces the  constant Θ in an action similar to (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='18).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' A gravitational axion must be in close relation  with the gravity sector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='  However, it does not seem that in SUp8q-QGR there is any  fundamental constituent as candidate for φg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='  The only natural Up1q symmetry in the  model, which emerges after the division of the Universe to subsystem, namely the relative  size of the diffeo-surfaces of subsystems, is used to introduce a size (or distance) parameter  in the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' On the other hand, in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='4 we argue that several phenomena may act as  an effective axion at low energies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='  In summary, many of observed and/or predicted complexities of non-Abelian Yang-Mills  will arise in SUp8q-QGR and must be thoroughly investigated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='2  Is there a conflict between a SUp8q Yang-Mills gauge theory as quantum  gravity and present observations ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='  The naive answer to this question is yes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='  In Einstein theory gravity is a spin-2 field  and various observations, both in weak field and strong field regimes have confirmed this  property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' By contrast, irrespective of the gauge symmetry group, Yang-Mills theories are  based on spin-1 fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' However, at present none of the many observations and tests of  the gravity as an interaction attain the quantum gravity regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' For the time being, all  the direct or indirect tests of gravity are based on the detection of the variation of average  – 44 –  metric (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='10) and its direct and indirect effects on matter and radiation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' For instance, a  gravitational wave temporarily changes the metric, which then detected as a change in the  phase of a laser beam.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' Therefore, our knowledge of gravity properties applies only to the  classical gravity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' In the next section we show that the classical limit of SUp8q-QGR is  exactly Einstein gravity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' Thus, SUp8q-QGR is consistent with with classical gravity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='  The historical parallel of this situation is the nuclear strong force, which for a few decades  was assumed to have mesons as intermediate particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' It was only after the discovery of  internal structure of hadrons that the Yang-Mills Quantum Chromo-Dynamics (QCD) was  accepted as the fundamental force, both between elementary quark and gluon fields and  their clustering as baryons and mesons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' The essential reason for this confusion was the  fact that accelerators and cosmic ray detectors were only able to investigate low energy  hadronized processes, rather than their fundamental partons - quarks and gluons - con-  stituents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' At present tests of quantum gravity are not even at a stage analogous to pions  for the strong interaction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='3  Classical limit  In quantum mechanics and QFT the classical limit of a model is defined as when ℏ Ñ  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='  However, as we discussed in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='2, in SUp8q-QGR such assumption makes the  algebra (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='1) Abelian and the symmetry group Â8  1 Up1q, unless at the same time MP Ñ  8, meaning no gravity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' Consequently, in absence of a background spacetime the whole  structure of SUp8q-QGR will collapse and the model becomes meaningless.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' Therefore,  rather than considering ℏ Ñ 0 limit, we consider the situation where gravity is observable,  but its detail quantum structure, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' its Yang-Mills nature is not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' Specifically, for an  observer who cannot detect and discriminate the infinite number of pl, mq colors of the  SUp8q symmetry and their corresponding quantum fields F µν  lm , the only way to become  aware of their presence is through the affine parameter ds in (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='10) - as Einstein suggested  and the effective path of subsystems in their SUp8q symmetry parameter space - the  spacetime - perceived through an effective metric gµν.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' In this case, the observer sees the  first term of the Lagrangian (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='12) as one block - just a scalar function of parameters (the  spacetime).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='  For the case of 2D parameter space of the whole Universe, we already demonstrated the  relation between the SUp8q Yang-Mills action and 2D scalar curvature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' In [38] we used  the relation between 4D scalar curvature and sectional curvatures, along with the fact that  surfaces generated by pairs of parameter space coordinates can be considered as diffeo-  surfaces representing SUp8q of subsystems, to relate SUp8q Yang-Mills term in (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='12) to  4D scalar curvature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' More generally, it is demonstrated [145] that for (pseudo)-Riemannian  spaces with dimension D ě 3, there is always a metric for every scalar function such that  the scalar curvature obtained from that metric is equal to the function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' Therefore, the first  term of (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='12) can be always considered to be equal to a scalar curvature:  ż  d4x  a  |η| trpF µνFµνq  classical  GGGGGGGGGGGGA  limit  9  ż  d4x  a  |g|Rp4q  (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='19)  – 45 –  Similarly, the contribution of Alm  µ  in covariant derivatives will be added to the parameter  space (spacetime) connection, and the resulting Lagrangian will become similar to a G  symmetry Yang-Mills model in a curved spacetime with Einstein-Hilbert gravity action:  LUs Ñ Lcl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='gr “  ż  d4x  a  |g|  „  κRp4q ` 1  4trpGµνGµνq ` 1  2  ÿ  s  trp\x1a  \x1a  Dˆρsq  ȷ  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='  (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='20)  where the dimensionful constant κ9M2  P is necessary for making the Lagrangian function  dimensionless.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' It is also clear that other possible scalars obtained from curvature tensor, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='  RµνRµν are of higher quantum order and at least ℏ2{M2  P times smaller than Rp4q ” gµνRµν.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='  It is remarkable that we began the construction of SUp8q-QGR from abstract assumptions  about symmetry and quantum mechanics, but rediscovered exactly Einstein gravity at  low energies - when gravity seems classical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' For instance, if we add a term proportional  to F µν ˜Fµν to (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='12), in general it cannot be written as a function of Rp4q for the same  metric as for F µνFµν.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' Therefore, it adds a completely independent pseudo-scalar function,  which break parity unless its amplitude is controlled with yet another pseudo-scalar - a  gravitational axion - to be consistent with the lack of parity violation in gravity sector, so  far.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' Another possibility is a bimetric gravity [146–148], where the dual term is considered as  scalar curvature of another metric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' The second metric is conjectured to become significant  at high energies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' Such models are related to massive graviton models [149] and lead to the  deviation of the speed of gravitational waves propagation from the speed of light [150].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' At  present such deviation is not consistent with present observations [107].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='4  Cosmological constant  The Lagrangian LUs does not include an explicit constant term, which can be interpreted  as a cosmological constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' In one hand, as the nature of the observed dark energy is still  unknown, the need for a cosmological constant in SUp8q-QGR Lagrangian is not certain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='  On the other hand, although a cosmological constant is still consistent with cosmological  observations, the growing evidence for what is called the Hubble tension and other anomalies  of the standard ΛCDM, see [151] for a review, may point to more elaborate models [152]  for dark energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' Evidently, many models are suggested in the literature as alternative to  a cosmological constant, and they are relevant in SUp8q-QGR - except those proposing  modification of classical Einstein gravity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='  Nonetheless, in the following subsections we  briefly discuss several processes specific to SUp8q-QGR, which potentially can lead to an  effective cosmological constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' We leave their detail assessment to future works.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='  Any candidate process should answer following questions before it can be considered to  play a role in the observed dark energy:  a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' How does its amplitude vary with the expansion of the Universe ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='  b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' Can a remnant with a roughly constant energy density consistent with the observed  dark energy survive ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='  – 46 –  c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' Can it solve the coincidence problem ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' In other words, can it justify dominance of  dark energy well after galaxy formation without fine-tuning ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='  For each of the following candidate phenomena for dark energy in the framework of SUp8q-  QGR we describe very briefly and qualitatively to which extent, if any, the above questions  may be answered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' We leave quantitative assessments to future works.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='1  Static SUp8q field  Although the Lagrangians (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='3) and (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='12) do not explicitly include a constant term, in  analogy with static electromagnetic, the field strength F µν can include a constant part.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='  Indeed, magnetic field is observed at cosmological scales, including in Interacluster Medium  (ICM), but not in the CMB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' Nonetheless, we know that cosmological magnetic field is  generated by dynamo in the core of stars and by shocks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='  These processes need large  overdensities and absence of a primordial magnetic field seems natural.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' By contrast, gravity  is generated by all fields and in all circumstances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' Moreover, due to the non-commutative  nature of SUp8q, Yang-Mills graviton source themselves, which is not the case for photons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='  Condensation of gravitons and its similarity to parton swarm in high energy QCD are also  suggested in the framework of QGR [153, 154] and black hole physics [155].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='  In analogy with the QCD glueball, the Universe may be considered as a gravity-ocean  in which other fields are immersed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='  In Einstein gravity gravitons self-interact, but its  quantized version is nonrenormalizable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='  By contrast, as a quantum Yang-Mills theory  SUp8q-QGR is renormalizable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' This means that its self-interaction is controlled and does  not lead to instability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' It is certain that in a classical view of the spacetime, gravitons are  everywhere, and considering their huge number, in a quantum view they form a condensate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='  In fact, in dark energy models which propose what they call vacuum of QCD or other Yang-  Mills fields as the origin of dark energy, it is the condensation of gauge bosons (and/or  fermions) that plays the role of dark energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' Therefore, it is rather IR modes that are able  to preserve their quantum coherence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' In this case, the energy density of the condensate is  expected to be roughly proportional to OpV ´4q rather than usual QGR (UV) estimation  of OpM4  P q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' Thus, the answer to the question (a) is yes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' The answers to questions (b) and  (c) are more subtle and need quantification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='2  Θ´vacuum  The cosmological constant term in the classical Einstein equation has been famously in-  terpreted as the vacuum energy [124, 156].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='  Non-Abelian Yang-Mills models can have  non-trivial vacuums, corresponding to a vanishing field strength F µν “ 0, but nonzero  gauge field Aµ ” Aµ  a ˆLa (for SUp8q symmetry).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' We remind that Aµ “ 0 is not gauge  invariant: Aµ Ñ UAµU ´1 ` i{λpDµUqU ´1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' Field configurations which can be written as  i{λpDµUqU ´1 are called instantons and for large gauge transformation have in general non-  trivial topologies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' Topologically non-equivalent field configurations are classified by their  winding number, which is equal to (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='18) with Θ “ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' In fact, as we mentioned before, the  – 47 –  integrand of (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='18) is a total derivative DµKµ, where Bardeen current Kµ is:  Kµ “  1  16π2 ϵµνρσtrpAνFρσ ´ AµAρAσ  (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='21)  The value of (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='18) with Θ “ 1 depends only on the configuration of the field Aµ on  the boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' This means that LΘ has the same value as a vacuum, even if F µν ‰ 0  except for |x| Ñ 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' Different classes of field configurations represent homotopy group π3  of SUp2q, the global rotation symmetry which remains even after fixing the gauge [157].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='  As π3pSUp2q “ Z, the value of LΘ“1 P Z and a general vacuum state |Θy can be written  as:  |Θy “  ÿ  n  e´inθg|ny  (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='22)  where |ny are instanton configurations with winding number n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' If in the early Universe  where QGR processes were important Yang-Mills graviton field had a Θ-vacuum configu-  ration of the form (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='22), smooth decrease of its strength Fρσ could not change the config-  uration of zero modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' Indeed the idea of a cosmological instanton generated by inflaton  field is explored e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' in [158].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' But to associate cosmological constant to QGR instanton,  significant deviation, even at classical level is necessary, because in 4D spacetime Einstein  gravity is not topological, see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' [159] for an example of such models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='  In the framework of Schwinger-Dyson Closed Time Path-integral (CTP), the 1-Particle  Irreducible (1PI) path integral of subsystems/fields can be formally written as:  ZrJa;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' ˆρs ” eiWrJas “  ż  DΦaDΦb exp  „  iSpΦaq ` i  ż  d4x?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='´gJapxqΦapxq  ȷ  xΦa|ˆρp|Φby (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='23)  where Φ generically presents all the fields and parameter space (spacetime) and G symmetry  indices are implicit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' Indices a, b P t`, ´u indicate fields on two opposite time branches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='  They are contracted by the diagonal tensor cab ” diagp1, ´1q and satisfy the condition  xΦ`pxqy “ xΦ´pxqy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' The last factor in (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='23) presents the projection of density matrix  on the fields eigen vectors at initial time t0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' Usually t0 Ñ ´8 is used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' As in CTP every  local observable is determined by integration of the generating functional (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='23) over the  interval p´8, `8q and back, the past boundary condition affect observables at all time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='  In general, the result has the form of an other functional, which without loss of generality  can be written as:  xΦa|ˆϱ|Φby “ exppiFrΦa, Φbsδpt ´ t0qq  (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='24)  and the functional iFrΦa, Φbs can be added to the action iSpΦaq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' Therefore, the effect of  a non-perturbative configuration on the t0 Ñ ´8 boundary, such a instanton state (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='18)  appear as a constant term in (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='23), and affect Green’s function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='  Concentrating on the SUp8q gauge fields in SUp8q-QGR, its instanton configuration only  affects zero modes, that is IR variation of the fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' Thus, in principle it should be sepa-  rated from other modes and treated as a constant phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' However, according to Proposition  1 other modes, both gravitational and G symmetry related, remain entangled to the zero  modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' Phenomenologically, this means that the effect of SUp8q instanton - Θ´vacuum  – 48 –  on the subsystems can be parameterized by an amplitude φgpxq, which its effective con-  tribution to the action (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='23) would be similar to an axion, but without a kinematic and  mass terms of a true field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' It is perceivable that at |x| !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' 8, the effect of instanton settles  to a roughly constant and behaves similar to a cosmological constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' In contrast to UV  explanation of cosmological constant, very roughly |φg|2 „ OppMP H0q2q is expected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' In  QCD an analogous phenomenological model for the Θ-vacuum called topological liquid of  instanton and anti-instanton offers a compelling non-perturbative description of the model  in strong coupling regime, see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' [160] for a review.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='  Finally, in what concerns criteria a, b, and c, the effective amplitude φg should somehow  depend on the matter, because it is its dominant source.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' Satisfaction or not of the other  two criteria needs a qualitative investigation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='3  Manifestation of global entanglement  In Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='2 the proposition 1 about entanglement of subsystems was crucial for our  arguments in favour of a SUp8q topological vacuum as the origin or contributor to the  observed dark energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' Even in the case of graviton condensation proposed in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='1,  this proposal is indirectly involved, because in absence of a global entanglement between  subsystems, expansion of the Universe may reduce the amplitude of the quantum conden-  sate to a negligibly small value, see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' simulations of a scalar field condensation in an  expanding Universe in [161].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' The question that arise is whether the global entanglement  alone can explain the roughly constant density of dark energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='  The content of the Universe is in large extent classical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' Quantum systems and effects such  as coherence and entanglement persist locally for a short time, before being decohered  through interaction with the infinite environment of other subsystems/particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' In general  decoherence increases entropy and releases energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='  On the other hand, preservation of  global entanglement of subsystems requires creation and/or transfer of entanglement, which  in turn needs energy [162] (and references therein).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' This is presumably provided in part by  the energy released by decoherence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' This process should arrive to an equilibrium, otherwise,  either the minimum global entanglement will be lost, which is in contradiction with the  proposition 1, or the entanglement grows and reduces entropy, which is inconsistent with  the second law of thermodynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' We leave quantitative assessment of these processes,  their consequences, and verification of criteria a-c, to future works.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='  7  Outlines and future perspectives  In this work we investigated more thoroughly foundation and principle properties of the  quantum gravity model dubbed SUp8q-QGR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' After demonstrating that the model is truly  quantum, we showed that what is perceived as spacetime can be interpreted as a parameter  space inseparable and associated to the quantum subsystems of the Universe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' We related  the observed curved metric of the classical spacetime to the quantum state of the Universe  – 49 –  and demonstrated that it presents the average path of quantum subsystems in their param-  eter space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' We introduced a relative quantum dynamics and showed that its gravitational  sector has the structure of a SUp8q Yang-Mills quantum field theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' Therefore, according  to SUp8q-QGR the quantum mediator of gravity is a vector field like other fundamental  interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' Nonetheless, we demonstrated that in the classical limit - defined as when the  fine structure of purely gravitational part of the Lagrangian and its quantum effects are not  discernible for the observer - the resulting model is exactly Einstein gravity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' We also briefly  discussed potential explanations for the observed dark energy specific to SUp8q-QGR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='  Future works should investigate in detail these proposals as well as application of the  SUp8q-QGR to other puzzling problems, in which quantum gravity may play an essential  role, such as: microscopic structure of black holes and apparent loss of information behind  their horizon;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' singularities, both cosmological and in black holes;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' the role of quantum  gravity in early universe and inflation;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' observables and discriminating signals of these  phenomena;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' and methods for testing SUp8q-QGR in laboratory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' Regarding this last topic,  it is important to insist that geometrical test of gravity would not be able to find the  signature of a vector gravity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' Only the detection of intrinsically quantum processes might  be able to distinguish between the exchange of spin-1 and spin-2 force mediator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='  Regarding the construction of the model, the approach used here and in the previous works  is in a QFT perspective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' It would be interesting - and probably necessary - to study the  model in the framework of quantum information theory, specially in what concerns reference  frame of different subsystems, their comparison, and their perspective-dependence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' This  topic is specially crucial for studying the microphysics of black holes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='  A subject that we did not address in any detail here is the rise of the internal symmetry G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='  Although the potential for clustering and symmetry breaking is ubiquitous, it is not clear  why and how the nature prefers some symmetries to others.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' SM symmetries are smallest  non-Abelian symmetries and cannot be decomposed to yet smaller non-Abelian groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' Is  this property related to the decomposition of SUp8q, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' the gravity force ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' In principle  this hypothesis can be investigated in the high energy colliders.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' Specifically, we should  search for emergence of symmetries similar to what is observed in condensed matters in  the distribution of partons and leptons at very high energies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='  Acknowledgment  The author acknowledges the support of the Institut Henri Poincar´e (UAR 839 CNRS-  Sorbonne Universit´e), and LabEx CARMIN (ANR-10-LABX-59-01).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='  – 50 –  Appendices  A  Properties of SUp8q tensor products  Using decomposition of SUpNq group:  SUpNq Ě SUpN ´ Kq ˆ SUpKq  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='1)  and the fact that its r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='s is a normal subgroup of SUpNq, it is straightforward to see that  the repetition of (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='1) with K “ 2 leads to:  SUpNq  ˇˇˇˇ  NÑ8  –  8  â  SUp2q  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='2)  This is a direct demonstration of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='6) and independent of the decomposition of associated  generators in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' Using (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='2) and the fact that n ˆ 8 “ 8 we conclude that tensor  products of SUp8q are homomorphic to itself:  pSUp8qqn – SUp8q  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='3)  It is then straightforward to conclude that:  SUp8q Ą  M  â  SUpKq,  M ă 8,  @ K ă 8,  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='4)  SUp8q –  8  â  SUpKq,  @ K ă 8,  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='5)  where (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='4) can be concluded from repetition of (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='1) and the assumption that SUpN ´ MKq|N,MÑ8  is represented trivially.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' As each SUpKq factor in turn has a Cartan decomposition,  SUp8q –  8  â  i“1  Gi  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='6)  where Gi is a finite rank Lie group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' Consider one of Gi factors in (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' We call it G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' Then,  the symmetry of the whole Universe15 can be written as SUp8q – GˆG8, where G8 is an  infinite rank group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' For any G we can find finite N ď N1 such that SUpNq Ď G Ď SUpN1q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='  Using (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='1) and self-similarity of SUp8q [66], we conclude the following relations:  SUpN1q ˆ G8 Ě rG ˆ G8 – SUp8qs Ě SUpNq ˆ G8  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='7)  SUpN1q ˆ G8 Ě SUpN1 ´ Nq ˆ pSUpNq ˆ G8q – SUpN1 ´ Nq ˆ SUp8q – SUp8q  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='8)  Thus, we conclude that G8 – SUp8q and G b SUp8q – SUp8q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' If n subsystems with G  symmetry are distinguished from the rest of the Universe, HU can be decomposed to:  Gn ˆ SUp8q – Gn ˆ SUp8qn – pG ˆ SUp8qqn  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='9)  For n Ñ 8:  pG ˆ SUp8qqnÑ8 – SUp8q  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='10)  BrHUs “  â  i  BrHis –  â  i  ˆ  BrHGis b BrHSUp8qs  ˙  –  ˆâ  i  BrHGis  ˙  b BrHSUp8qsq  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='11)  15When there is no risk of confusion we identify BrHs with the symmetry group it represents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='  – 51 –  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='1  About definition of symmetries of a quantum system  We remind that symmetries of a system are defined in two ways.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='  If the dynamics is  known and a Lagrangian is associated to it and its symmetries that is transformations  preserving the Lagrangian, are called symmetries of the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' Alternatively, one may  first associate a symmetry to a system, for instance, based on the observations of some  properties, and construct a Hilbert space or a Lagrangian that preserve those symmetries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='  When the system is considered as a whole, the symmetries associated to the systems in  these approaches are the same.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' However, when the system is composite, one has to be  careful about the definition of what is called system’s symmetries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' In particular, for many-  body systems, there are local symmetries, presumably those belonging to each subsystem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='  The effective Hamiltonian or Lagrangian usually includes either strictly local - 1-point  interactions, or at most few-point interactions - mostly 2-point correlations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' For this  reason, symmetries preserving Lagrangian or Hamiltonian would be those of individual  subsystems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' In addition, Lagrangian or Hamiltonian may preserve some global (subsystem  independent) symmetries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='  For example, Hamiltonian of Heisenberg spin chain/network model is invariant under the  local SUp2q transformation of spin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' In addition, if a periodic boundary condition is defined,  it would be invariant under translation i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' i Ñ i ` L, where L is the periodic length in  units of lattice length.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' On the other hand, the Hilbert space in general represents the  tensor product of the symmetry of subsystems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' In the example of Heisenberg model, the  symmetry of the total Hilbert space is SUp2qN, where N is the number of sites of the lattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='  Nonetheless, quantum entanglement (correlation) may significantly reduce the symmetry  of actual Hilbert space - the subspace of states with nonzero probability of occurrence -  through a process known as many-body localization [97].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='  B  About quantization  A system that satisfies axioms of quantum mechanics `a la Dirac and Von Neumann (or  their variants) can be called quantum and does not need quantization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' Indeed, it is well  known that canonical commutation relations between conjugate operators is meaningful  only if at least one of them is unbounded and has a continuous spectrum (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' the  chapter on uncertainty relation in [68]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' The most direct evidence of this property is the  fact that two finite dimensional hermitian operators ˆA and ˆB cannot satisfy the canonical  quantization relation r ˆA, ˆBs “ ´iℏ1, because the trace of the l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' would be null, whereas  that of the r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='s would be a finite nonzero value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' Therefore, canonical quantization is useful  and applicable only for quantization of classical theories, which usually include unbounded  quantities such as spacetime or fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' Nonetheless, the Planck constant ℏ must be somehow  involved in the construction of any quantum model, such that when ℏ Ñ 0 the model behave  classically, that is conjugate observables, like coordinates and momentum commute with  each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='  – 52 –  A typical example of quantum models without continuous degrees of freedom and usual  uncertainty relation is quantum Heisenberg model of interacting spins on a lattice, which  is widely used in condensed matters physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' The Hamiltonian of this model is ˆHH ”  ´J ř  tiup⃗Si⃗Si`1q, where J is a coupling constant and summation is over all sites of the  lattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='  The Hilbert space at each site represents SUp2q symmetry and its generators  ˆSi, i “ 1, 2, 3 can be normalized such that r ˆSi, ˆSjs “ ´iℏϵijk ˆSk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' See also Appendix  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='1 for more details about this model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' It is evident that there is no pair of operators in  this model that satisfies canonical quantization relation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' Nonetheless, innumerable studies  show that particles interacting according Heisenberg model behave as predicted by axioms  of quantum mechanics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' In particular, as expected, the 3 components of spins or angular  momentum vectors become simultaneously measurable when ℏ Ñ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='  The above example shows that non-commutativity of the algebra of observables do in-  deed replace the canonical quantization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' Notably, this property is used in construction  of several QGR models, such as: non-commutative geometry QGR [163–166], some string  theories [167], and matrix models [168–171].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' However, non-commutativity spacetime is  stirringly constrained [172–174].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='1  Dual of generators in spherical harmonic representation of sup8q  As we discussed in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='2, ˆJlm in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='11) is a complex function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' By using representation  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='3) for the generators of the sup8q algebra (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='1) and applying two sides of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='11) to an  arbitrary vector ψ P HU, which is also a complex function, we find:  BYlm  Bη  B ˆJlm  Bζ  ´ BYlm  Bζ  B ˆJlm  Bη  “ ´1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='  (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='1)  where here η ” cos θ and ζ ” φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' Solutions of the first-order partial differential equation  (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='1) determines dual operators (functions) ˆJlm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='  To solve equation (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='1) we first remind the definition of spherical harmonic functions Ylm:  Ylmpη, ζq ” p´1qm  d  p2l ` 1q  4π  pl ´ mq!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='  pl ` mq!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='Plmpηqeimζ  (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='2)  where Plmpηq is the associated Legendre polynomial for any l and |m| ď l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' Equation (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='1)  can be solved by factorizing the contribution of variables pη, ζq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' The general solution for  ˆJlm is:  ˆJlmpη, ζ, η0, ζ0, ˆJlmpη0, ζ0qq “ η ´ η0 ` im ˆJlmpη0, ζ0qYlmpη0, ζ0q  imYlmpη, ζq  (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='3)  where η0, ζ0 and Jlmpη0, ζ0q are integration constants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' Choosing η0 “ ζ0 “ ˆJlmpη0, ζ0q “ 0,  ˆJlmpη, ζq becomes:  ˆJlmpη, ζq “  η  imYlmpη, ζq  (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='4)  – 53 –  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='2  Dual of continuous parameters in spherical harmonic representation of  sup8q  We use the representation (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='3) of the generators ˆLlmpη, ζq of sup8q, and properties of  spherical harmonic functions Ylm and associated Legendre polynomials Plmpηq, to relate  ˆLlm to the canonical quantization conjugates of parameters pη, ζq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='  For m “ 0, Pl0pηq ” Plpηq is the Legendre polynomial and generators ˆLl0 can be expended  as:  ˆLl0 “ iℏBYl0  Bη  B  Bζ “ iℏ  c  p2l ` 1q  4π  BPl  Bη  B  Bζ “ iℏ  c  p2l ` 1q  4π  l ` 1  1 ´ η2  ˆ  ηPlpηq ´ Pl`1pηq  ˙ B  Bζ  “  ˆ l ` 1  1 ´ η2  ˙ˆ  ηYl0pηq ´  c  2l ` 1  2l ` 3Ypl`1q0pηq  ˙ˆ  iℏ B  Bζ  ˙  (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='5)  Thus:  iℏ B  Bζ “  ˆ1 ´ η2  l ` 1  ˙ˆ  ηYl0pηq ´  c  2l ` 1  2l ` 3Ypl`1q0pηq  ˙´1  ˆLl0  (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='6)  Similarly,  p´1qmeimζ ˆLl,´m ´ e´imζ ˆLlm “ ´2mℏ e´imζYlm  B  Bη  (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='7)  Using properties of spherical harmonic functions, namely Yl,´m “ p´1qmY ˚  lm, it is straight-  forward to show that ˆLl,´m “ p´1qm`1 ˆL:  lm, and equation (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='7) can be written as:  ieimζ ˆL:  lm ` ie´imζ ˆLlm “ 2m e´imζYlm  ˆ  iℏ B  Bη  ˙  (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='8)  Finally:  iℏ B  Bη “ ip2m e´imζYlmq´1peimζ ˆL:  lm ` e´imζ ˆLlmq  (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='9)  It is possible to invert (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='5) and (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='8), and express the quantum conjugate of ζ and η,  that is iℏ B{Bζ and iℏ B{Bη with respect to operators ˆLl0 and ˆLlm, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' However,  their expression would be highly degenerate, because (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='5) and (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='8) are valid for all  l ě 0 and |m| ď l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' This observation demonstrates that quantization through the algebra  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='1) is much richer than the canonical quantization of the finite dimensional parameter  space - the spacetime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' Nonetheless, as we explained in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='2, the parameters and  their canonical conjugate can be considered as representation of operators corresponding  to these observables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='  C  Criteria for definition of quantum subsystems  General conditions for the division of a quantum system with Hilbert space H to proper  quantum subsystem is studied in details in [25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' They can be summarized as the followings:  a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' There must exist sets of operators tAiu Ă BrHs such that @ ˆa P tAiu and @ ˆb P tAju,  and i ‰ j, rˆa,ˆbs “ 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='  – 54 –  b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' Operators in each set tAiu must be local;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='  c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' tAiu’s must be complementary, meaning bitAiu – End pBrHsq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='  In [25] the locality condition (b) refers to locality of operations regarding the background  spacetime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' In SUp8q-QGR there is no background spacetime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' Nonetheless, we can extend  the concept of locality to the diffeo-surface of parameters, and thereby to the Hilbert space  BrHUs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='  C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='1  Symmetries and Hilbert space of subsystems  The division of BrHs to subspaces tAiu as subsystem makes sense only if the commuting  subspaces can be distinguished from each others.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' This means that BrHs symmetry must  be broken to a direct product   i Gi, leading to BrHs Ě Â  i gi where gi is the algebra  associated to group Gi and H Ě Â  i Hi, where the Hilbert space Hi represents Gi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' Group  factors Gi’s do not need to be all different.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='  The ensemble of subsystems with similar  symmetries and representations present multiplicity of their type/family.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' In addition, the  order of factors in the tensor product is not important.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' Thus, from now on tensor products  of subsystems are assumed to be permutation symmetrized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='  We remind that a N-dimensional Hilbert space is invariant under UpNq “ SUpNq ˆ Up1q  transformation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' The Up1q factor corresponds to a global phase transformation of states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='  However, the symmetry of the quantum system can be a subgroup of UpNq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' For instance,  Hilbert space generated by eigen vectors of position operator ⃗ˆX in non-relativistic quantum  mechanics is infinite dimensional and invariant under SUp8q ˆ Up1q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' But the symmetry  of ⃗ˆX and a quantum system (a particle) presented by its position is homomorphic to  Up1q ˆ Up1q ˆ Up1q ˆ SOp3q – Rp3q ˆ SOp3q Ă Up8q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='  In addition, dimension of the  Hilbert space is determined by representation of the symmetry rather than the rank of the  symmetry group itself.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' Nonetheless, in all cases the symmetry group G Ď UpNq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='  C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='2  Conditions for division induced entanglement of subsystems  If the dimension of subspace tAiu is NAi, the minimum dimension of BrHis must be Ni  such that N2  i ´ 1 ě NAi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' The complementary condition (c) for division of a quantum  system to subsystems dictates that ř  i Ni “ N, where N is the dimension of Hilbert  space of the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' Assuming the division to two subsystem, (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='1) leads to G Ě G1 ˆ  G2, where G1 and G2 are symmetry groups represented by Hilbert spaces of the two  subspaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' Iteration of this relation on each subsystem demonstrates that G Ě Â  i Gi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='  Therefore, in general, division of a quantum system reduces the symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' Nonetheless,  the dimension of tensor product of Hilbert spaces of subsystem N12 ” N1 ˆ N2 ě N, for  N1, N2 ą 1, N ě 2 and N, N1, N2 ă 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' Considering the fact that entropy, specially von-  Neumann entropy, is roughly proportional to the number of states, increasing dimension of  the Hilbert space after division to subsystems implies increasing quantum randomness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' On  the other hand, quantum entanglement, which may similarly breaks symmetries, reduces  – 55 –  the entropy/randomness by correlating subsystems and reducing the number of states (see  e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' [175]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' These opposite effects of symmetry breaking in quantum systems have crucial  roles in determining their characteristics and behaviour.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='  Relation between symmetry and number of states of subsystems can be quantified using  quantum complexity and its application as a resource [176, 177].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' If states of subsystems  1 and 2 can be prepared using n1 and n2 qubits, respectively, their maximum complexity  Cmax in isolation is Cmax “ 2n1 ` 2n2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' By contrast, if they are considered together, their  complexity is much larger: Cmax “ 2n1`n2 " 2n1 `2n2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' In the second case the SUpn1 `n2q  symmetry represented by the Hilbert space of the ensemble of qubits of the two sub-  systems includes SUpn1q ˆ SUpn2q symmetry of isolated subsystems, according to (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='  Thus, maximum quantum complexity of a quantum system reflects breaking of states sym-  metry (coherence) of its subsystems by their entanglement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' This is not a surprise, because  SUpNq Ě bN{2SUp2q and SUp2q is the symmetry represented by a qubit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='  D  Relation between reparameterization and diffeomorphism  Diffeomorphism is a continuous and invertible map of a manifold M Ñ M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' Parameteriza-  tion is a continuous map M Ñ N, where in general M ‰ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' Thus, diffeomorphism can  be considered as a special case of parameterization when N “ M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' For this reason diffeo-  morphism and reparameterization are usually considered as the same operation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' Moreover,  diffeomorphism of N induces a diffeomorphism transformation in M, when they are related  under a reparameterization map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='  In SUp8q-QGR, for the whole Universe M “ HU and N “ D2 – Rp2q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' A basis for HU  consists of vectors |xy, @x ” pη, ζq P Rp2q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' Reparameterization corresponds to a diffeomor-  phism in Rp2q such that: x Ñ x1pxq @x P Rp2q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' An infinitesimal SUp8q transformation of  the basis |xy is |x1pϵqy “ p1 ` ϵ ř  pl,mq ˆLl,mpxqq|xy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' A general SUp8q transformation U “  expp  ş  d2Ω ř  pl,mq ϵlm ˆLl,mpxqq for any set of constants ϵlm and p|ψy P HUq Ñ |ψ1y “ U|ψy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='  Considering the differential representation of ˆLl,mpxq in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='3), it is clear that reparameter-  ization, that is a diffeomorphism in D2 is equivalent to a SUp8q transformation of HU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='  After division of the Universe to subsystems N – Rp4q and reparameterization corresponds  to diffeomorphism in Rp4q and leads to M “ Hs Ñ Hs and x Ñ x1pxq @x P Rp4q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' Similar  to the case of the whole Universe, this transformation can be described as application of  SUp8q parameterized by only a pair of parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' This is possible because of property  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='3) of SUp8q and superposition of quantum states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='  E  Coordinate independent definition of curvatures  For any Riemannian or pseudo-Riemannian manifold pM, gq of dimension d ě 2 equipped  with a Levi-Civita connection ∇ the (1,3)-Riemann curvature tensor at point p P M is  defined as:  RppX, Y qZ “ p∇X∇Y ´ ∇Y ∇X ´ ∇rX,Y sqZ  (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='1)  – 56 –  Vector fields X, Y, Z P TMp and TMp is the tangent space at p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' When X, Y, Z are  chosen to be Bi ” B{Bxi basis of the tangent space for coordinates xi, i “ 0, ¨ ¨ ¨ d ´ 1, one  recovers the usual coordinate dependent definition of the Riemann curvature tensor (we  drop p because it corresponds to the point with coordinates xi):  RpBi, BjqBk “ Rl  kij,  (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='2)  Rijkl ” RpBi, Bj, Bk, Blq ”  B  RpBi, BjqBk, Bl  F  “ gmlRm  kij  (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='3)  Using the notation defined in (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='3) for (0,4)-Riemann curvature tensor, sectional curvature  KpΠq “ KpX, Y q with respect to a 2D plane Π Ă TMp containing two vectors X, Y P TMp  at p P M is defined as:  KpΠq ” KpX, Y q “  RppX, Y, X, Y q  xX, XyxY, Y y ´ xX, Y y2  (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='4)  Notice that KpΠq is independent of the choice of X and Y and depends only on the plane  passing through them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='  It can be shown that xRppX, Y qZ, Wqy can be expanded with  respect to sectional curvatures [178].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' Using relations between different curvature tensors  of a Riemannian manifold, the scalar curvature at point p P M is defined as [179]:  Rppq “  ÿ  i‰j  Rppei, ej, ei, ejq “  ÿ  i‰j  Kppei, ejq  (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='5)  where ei, i “ 0, ¨ ¨ ¨ , d ´ 1 is an orthonormal basis of TMp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' From (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='5) we conclude that  there is only one sectional curvature at each point of a 2D surface and it is equal to its  scalar curvature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='  F  Expansion of density matrix and its variation  Following the discussion in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='1 about representation of SUp8q symmetry, pairs of  parameters pη, ζq P Ξ, the space of linear operators BrbsHss can be generated by operators  ˆLlmpη, ζq, which satisfy the algebra (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='  With this generalization, SUp8q generators  ˆLlmpη, ζq defined in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='3) with η “ cos θ, ζ “ φ have the structure of a 2-form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' To make  this property explicit, we may write ˆLlmpη, ζq ” ˆLµν  lm,  Here we consider only SUp8q symmetry and do not explicitly consider G symmetry repre-  sentation by the Hilbert space of Hs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' The state of subsystems after tracing out G symmetry  is ˆρ P BrbsHss can be expanded as the following:  ˆρ “  ż  dx4a  |η|  ÿ  l,m  ρlmpxqϵµν ˆLµν  lm  (F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='1)  ż  dx4a  |η|  ÿ  l,m  ρlmpxq “ 1,  ρlmpxq ě 0  (F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='2)  The range of pl, mq are defined in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' The constant 2-form ϵµν in the integrand of (F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='1) is  necessary to make it a scalar of the parameter space Ξ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' The expression ρlmpxqϵµν ” ρlm  µν is  – 57 –  a 2-form field on the parameter space Ξ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' Although it has the same geometrical structure as  a gauge field strength, its physical interpretation is very different.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' Specifically, components  ρlm  µν are positive probability densities and must satisfy the constraint (F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' By contrast,  the gauge field strength presents the number of quanta of the gauge boson.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='  Variation of the density matrix δˆρ P BrbsHss can be expanded in the same manner:  δˆρ “  ż  dx4a  |η|  ÿ  l,m  δρlm  µνpxqˆLµν  lm  (F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='3)  This expansion can be applied to (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='10) to determine affine separation as a function of  quantum state variation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' In principle, one can write a similar expansion with respect to  ˆLµν  lm operators for ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='δˆρ:  a  δˆρ ”  ż  dx4a  |η|  ÿ  l,m  δρ  1lm  µν pxqˆLµν  lm  (F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='4)  Using δˆρ “ ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='δˆρ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='δˆρ, expansions (F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='3) and (F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='4) and description anti-commutation of  generators ˆLµν  lm, see Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='1 and [62], It is straightforward to see that δρ  1lm  µν  modes  depend on the mixture of all δρlm  µν modes, but remain local with respect to parameters x -  the spacetime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='  G  About holographic principle and black holes  Holography principle [35–37] can be summarized as proportionality of the maximum num-  ber of quantum states inside a null (light-like) boundary to its area rather than its vol-  ume [35–37, 185].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' In the framework of SUp8q-QGR and its interpretation of spacetime  as a parameter space, a quantum system to which holographic principle applies can be  considered as an approximately isolated, which its large/infinite number of degrees of free-  dom (subsystems) are strongly interacting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='  Black holes as defined in classical Einstein  gravity are examples of such systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' Hilbert spaces of such systems approximately repre-  sent SUp8q and their spacetime boundary can be identified with the diffeo-surface of the  SUp8q representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' The dependence of the Bekenstein entropy of black holes on the  boundary area.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' Of course, as discussed in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' 3, area of diffeo-surface of truly isolated sys-  tem representing SUp8q is irrelevant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' Thus, dependence of the Bekenstein entropy is the  evidence that a black hole is not a fully isolated quantum system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' Indeed, it is observable  from outside of its boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='  In the framework of SUp8q-QGR, Bekenstein-Hawking entropy[180] can be considered as a  relative entropy of a highly but not completely isolated subsystem of the Universe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' And the  relation between SUp8q and diffeomorphism of the horizon is consistent with the proposal  that the Bekenstein-Hawking entropy of black holes is a manifestation of the N¨other charge  of the spacetime diffeomorphism[181].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='  – 58 –  G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='1  Comparison with gravity as entanglement and ADS/CFT  It is useful to compare the emergence of an area/distance scale in SUp8q-QGR with the  relationship between distance and quantum entanglement proposed in [41].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' Motivated by  the holography principle and ADS/CFT conjecture (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' [182] for a review) [41] estab-  lishes an analogy between reduction of the contact surface by stretching of two connected  regions of a flexible object with constant volume, and reduction of quantum entanglement,  measured by the mutual quantum information capacity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' Moreover, the distance between  two gravitating sources is interpreted as a measure of their quantum entanglement: longer  distance, smaller their entanglement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='  By contrast, in SUp8q-QGR, it is rather the quantum entanglement of subsystems which  relates parameters of their quantum states of subsystems to each others.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' Relative variation  of states induces a wandering in the parameter space - the spacetime - that characterizes  the quantum states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' When quantum effects are not distinguishable, the average path of  subsystems in this classical parameter space is perceived and interpreted as being due to  a curved geometry generated by their gravitational interaction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='.  In [38] we showed that in the SUp8q-QGR framework subsystems are in general Type  III quantum systems [183, 184].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' Thus, no boundary can be imposed on the continuous  parameters, in particular on the value of distance parameter r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' Moreover, subsystems are  in general in a superposition similar to (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' Therefore, concepts such as horizon of a black  hole, or more generally null surfaces to which holography principle applies [61, 185] lose  their strict meaning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' In fact, it is well known that even in general relativity horizon position  depends on the parameterization of the metric, and thereby is not a physical boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='  Only the singularity is physical and persistent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' Additionally, (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='9) and (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='10) show that  when quantum effects of gravity are taken into account, black holes are not simple objects  that everything about them can be parameterized by their mass, charge, and angular  momentum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='  As δˆρ in (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='10) includes infinite number of independent components, see  (F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='3), many quantum states may generate the same effective (classical) metric gµν, despite  being quantum mechanically very different.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' Moreover, in principle, differences of block  hole states should include both the contribution of matter G and SUp8q symmetries, and  self-interaction of gravity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' In turn, it is expected that they affect black hole evaporation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='  Evidently, the classical formulation of black holes in general relativity is not sensitive to  these quantum details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' Thus, in principle these additional aspects should be able to solve  the problem of apparent information loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' Evidently, this claim needs detailed investigation  and verification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='  In SUp8q-QGR black hole horizon RBH should be considered as the boundary of the  inaccessible parameter space for the external observer, because it is where the sign of ds2  changes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' Larger RBH means larger the volume of parameter space of indistinguishable  (entangled) quantum states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' This is analogous to scattering of elementary particles when  the exchanged quanta/particle is virtual, that is when its 4-momentum q has q2 ă 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' In  this case, even at tree order it would not be possible to know from which vertex a scattered  particle is coming.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='  – 59 –  References  [1] Th.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' Kaluza, Zum Unit¨atsproblem der Physik, Sitz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' Preuss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' Akad.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' Wiss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' K1  (1921) 966.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='  [2] O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' Klein, Quantentheori und f¨unfdimensionale Relativist¨atstheori, Zeits Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' 37 (1926) 895.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='  [3] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content=' Green, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
+page_content='H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQf-QLn/content/2301.02813v1.pdf'}
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+Restricted Orthogonal Gradient Projection for Continual Learning
+Zeyuan Yang 1 Zonghan Yang 1 Peng Li 2 Yang Liu 1 2 3 4 5 6
+Abstract
+Continual learning aims to avoid catastrophic for-
+getting and effectively leverage learned experi-
+ences to master new knowledge. Existing gradient
+projection approaches impose hard constraints on
+the optimization space for new tasks to minimize
+interference, which simultaneously hinders for-
+ward knowledge transfer. To address this issue,
+recent methods reuse frozen parameters with a
+growing network, resulting in high computational
+costs. Thus, it remains a challenge whether we
+can improve forward knowledge transfer for gra-
+dient projection approaches using a fixed network
+architecture. In this work, we propose the Re-
+stricted Orthogonal Gradient prOjection (ROGO)
+framework. The basic idea is to adopt a restricted
+orthogonal constraint allowing parameters opti-
+mized in the direction oblique to the whole frozen
+space to facilitate forward knowledge transfer
+while consolidating previous knowledge. Our
+framework requires neither data buffers nor extra
+parameters. Extensive experiments have demon-
+strated the superiority of our framework over sev-
+eral strong baselines. We also provide theoretical
+guarantees for our relaxing strategy.
+1. Introduction
+A critical capability for intelligent systems is to continu-
+ally learn given a sequence of tasks (Thrun & Mitchell,
+1995; McCloskey & Cohen, 1989). Unlike human beings,
+vanilla neural networks straightforwardly update parame-
+ters regarding current data distribution when learning new
+tasks, suffering from catastrophic forgetting (McCloskey
+& Cohen, 1989; Ratcliff, 1990; Kirkpatrick et al., 2017).
+1Department of Computer Science and Technology, Insti-
+tute of AI, Tsinghua University 2Institute for AI Industry Re-
+search, Tsinghua University 3Beijing National Research Cen-
+ter for Information Science and Technology 4Beijing Academy
+of Artificial Intelligence 5International Innovation Center of Ts-
+inghua University 6Quan Cheng Laboratory. Correspondence
+to:
+Peng Li <lipeng@air.tsinghua.edu.cn>, Yang Liu <li-
+uyang2011@tsinghua.edu.cn>.
+Work in progress.
+Continual learning without forgetting has thus gained in-
+creasing attention in recent years (Kurle et al., 2019; Ehret
+et al., 2020; Ramesh & Chaudhari, 2021; Liu & Liu, 2022;
+Teng et al., 2022). Moreover, an ideal continual learner
+is expected to not only avoid catastrophic forgetting but
+also facilitate forward knowledge transfer (Lopez-Paz &
+Ranzato, 2017), i.e., leveraging past learning experiences
+to master new knowledge efficiently and effectively (Parisi
+et al., 2019; Finn et al., 2019). Without forward knowledge
+transfer, approaches may have limited performance even
+with less forgetting (Kemker et al., 2018).
+To mitigate forgetting and facilitate forward knowledge
+transfer, replay-based methods (Lopez-Paz & Ranzato,
+2017; Shin et al., 2017; Choi et al., 2021) stores some
+old samples in the memory, and expansion-based meth-
+ods (Rusu et al., 2016; Yoon et al., 2017; 2019) expand
+the model structure to accommodate incoming knowl-
+edge. However, these methods require either extra memory
+buffers (Parisi et al., 2019) or a growing network architec-
+ture as new tasks continually arrive (Kong et al., 2022),
+which are always computationally expansive (De Lange
+et al., 2021). Thus, promoting performance within a fixed
+network capacity remains challenging.
+Regularization-
+based methods (Kirkpatrick et al., 2017; De Lange et al.,
+2021; Aljundi et al., 2018) penalize the transformation of
+parameters regarding the corresponding plasticity via regu-
+larization terms. Instead of constraining individual neurons
+explicitly, recent gradient projection methods (Zeng et al.,
+2019; Saha et al., 2021; Wang et al., 2021) further con-
+strain the directions of gradient update, obtaining superior
+performance. However, in spite of effectively mitigating
+forgetting, the limited optimization space also hinders the
+capability of learning new tasks (Zeng et al., 2019), resulting
+in insufficient forward knowledge transfer.
+Particularly, we illustrate the optimizing process of tradi-
+tional gradient projection methods by the black dashed line
+in Figure 1. As shown in Figure 1, the strict orthogonal
+constraint may exclude the global optimum (the red star)
+from the optimization space. In other words, constraining
+the directions of gradient update fails on the plasticity in
+the stability-plasticity dilemma (French, 1997). TRGP (Lin
+et al., 2022) tackles this problem by allocating new parame-
+ters within the selected subspace of old tasks as trust regions,
+suffering a similar computational cost burden as expansion-
+arXiv:2301.12131v1  [cs.LG]  28 Jan 2023
+
+Restricted Orthogonal Gradient Projection for Continual Learning
+Task 1
+Task 2
+SGD
+ROGO
+(Ours)
+Orthogonal
+Gradient Projection
+(Previous Work)
+Frozen
+Figure 1: Illustration of our ROGO framework. The lines
+with arrow denote the gradients on task 2 (pale blue area)
+after learning task 1 (pale yellow area). We indicated the
+global optimum by the red star (⋆). The black straight
+line denotes the frozen space, to which gradient projection
+methods constrain the gradients to be orthogonal.
+based methods (Wang et al., 2021). Therefore, promoting
+forward knowledge transfer within a fixed network capacity
+remains a key challenge for gradient projection methods.
+To address this challenge, we propose our Restricted Orthog-
+onal Gradient prOjection (ROGO) framework to facilitate
+forward knowledge transfer. Instead of optimizing in an
+orthogonal direction, our method adopts a restricted orthog-
+onal constraint. As illustrated by the red line in Figure 1,
+our framework allows the parameters to be updated in the di-
+rection oblique to the original frozen space, which explores
+a larger optimization space and thus obtains better perfor-
+mance on new tasks. Specifically, we design a simple yet
+effective strategy to find the critical subspace, namely the
+relaxing space, within the frozen space. The parameters are
+jointly optimized in the relaxing and original optimization
+spaces to facilitate forward knowledge transfer. A compli-
+mentary regularization term is introduced to consolidate
+previous knowledge as well. Extensive experiments on var-
+ious continual learning benchmarks demonstrate that our
+ROGO framework promotes forward knowledge transfer
+and achieves better classification performance compared
+with related state-of-the-art approaches. Moreover, our
+framework can also be extended as an expansion-based
+method by storing the parameters of the selected relaxing
+space, universally surpassing TRGP (Lin et al., 2022) and
+other expansion-based approaches. We also provide theoret-
+ical proof to guarantee the efficiency of our strategy.
+2. Related Work
+2.1. Gradient Projection Methods
+Gradient projection methods constrain the gradients to be
+orthogonal to the frozen space constructed by previous
+tasks to overcome forgetting. OWM (Zeng et al., 2019)
+first proposed to modify the gradients upon projector matri-
+ces. OGD (Farajtabar et al., 2020) further keeps the gradi-
+ents orthogonal to the space spanned by previous gradients,
+whereas GPM (Saha et al., 2021) computes the frozen space
+based on old data. NCL (Kao et al., 2021) combines the idea
+of gradient projection and Bayesian weight regularization
+to mitigate catastrophic forgetting. In spite of minimizing
+backward interference, these approaches suffer poor for-
+ward knowledge transfer and lack plasticity (Kong et al.,
+2022). TRGP (Lin et al., 2022) expands the model with
+trust regions to achieve better performance on new tasks by
+introducing additional scale parameters. On the contrary,
+we focus on facilitating forward knowledge transfer within
+a fixed capacity network by optimizing the parameters in
+the direction oblique to the frozen space.
+2.2. Regularization-based Methods
+Regularization-based methods introduce regularization
+terms to the objective function to penalize the modifica-
+tion of parameters, requiring neither data buffers nor extra
+parameters as gradient projection methods. EWC (Kirk-
+patrick et al., 2017) first proposes to constrain the change
+based on approximated importance weights. HAT (Serra
+et al., 2018) learns task-based hard attention to identify im-
+portant parameters. Other methods, also called parameter-
+isolation methods, defy forgetting via freezing the gradient
+updates of particular parameters (De Lange et al., 2021).
+PackNet (Mallya & Lazebnik, 2018) iteratively prunes and
+allocates parameters subset to incoming tasks, whereas Ku-
+mar et al. (2021) facilitate inter-task transfer with the Indian
+Buffet Process (Griffiths & Ghahramani, 2011). Instead of
+restricting individual parameters update, the main idea of
+our approach is constraining the direction of gradients.
+2.3. Other Methods
+Replay-based methods (Lopez-Paz & Ranzato, 2017;
+Chaudhry et al., 2018) maintain a complementary mem-
+ory for old data, which are replayed during learning new
+tasks. Recent approaches (Chenshen et al., 2018; Cong et al.,
+2020) deploy auxiliary deep generative models to synthesize
+pseudo data. However, including extra data into the current
+task introduces excessive training time (De Lange et al.,
+2021). Expansion-based methods (Yoon et al., 2017; 2019;
+Douillard et al., 2022) dynamically allocate new parameters
+or modules to learn new tasks. While these methods face
+capacity explosion inevitably after learning a long sequence
+of tasks. In contrast, our method maintains a fixed network
+architecture and requires no previous data.
+3. Restricted Orthogonal Gradient Projection
+3.1. Preliminaries
+In a continual learning setting, we consider T tasks arriving
+as a sequence. When learning the current task, the datasets
+
+Restricted Orthogonal Gradient Projection for Continual Learning
+of old tasks are inaccessible. We use an L-layer neural
+network with fixed capacity, and parameters defined as W =
+{W l}L
+l=1, where W l denotes the parameters in the l-th layer.
+To mitigate catastrophic forgetting, gradient projection meth-
+ods construct the representative space, namely the frozen
+space, based on previous tasks and optimize the parame-
+ters orthogonal to the frozen space. Specifically, for each
+task t, Saha et al. (2021) obtain the layer-wise represen-
+tation space Rl
+t by compressing the representation matrix
+Hl
+t = [hl
+t,1, ..., hl
+t,Nt], where Nt denotes the number of
+samples in task t and hl
+t,j denotes the intermediate repre-
+sentation of the j-th input. The constructed representation
+spaces of previous tasks are then concatenated to be the
+frozen gradient space {U l
+t}L
+l=1 for future training.
+Ut = {U l
+t}L
+l=1 = {Rl
+1 ∪ · · · ∪ Rl
+t}L
+l=1
+(1)
+During training task t, gradients gl
+t are layer-wise con-
+strained to be orthogonal to U l
+t−1. Particularly, assuming
+Bl
+t−1 = [ul
+t−1,1, ..., ul
+t−1,N] as the total N orthogonal basis
+of U l
+t−1, GPM (Saha et al., 2021) modifies the gradients as:
+gl
+t = gl
+t − ProjU l
+t−1(gl
+t)
+= gl
+t − Bl
+t−1(Bl
+t−1)T gl
+t
+(2)
+In spite of alleviating forgetting, the strict orthogonal con-
+straint on parameters hinders the forward knowledge transfer
+by the limited optimization space, thus compromising the
+performance of new tasks. TRGP (Lin et al., 2022) tackles
+this problem by selecting old tasks relevant to the current
+task and expanding corresponding frozen spaces as the trust
+regions. The scaled weight projection is further introduced
+for memory-efficient updating and storing the parameters
+by scaling the basis. Considering that task i is selected as
+the trust region, the scaled weight projection is:
+ProjSl
+i
+U l
+i(gl
+t) = Bl
+iSl
+i(Bl
+i)
+T gl
+t
+(3)
+where Sl
+i denotes the scale matrix. For each task t, TRGP
+selects several different tasks as the trust regions U l
+i. During
+the training phase, gradient gl
+t is modified as:
+gl
+t = gl
+t − ProjU l
+t−1(gl
+t) +
+�
+i
+ProjSl
+i
+U l
+i(gl
+t)
+(4)
+The parameters in the trust regions are retrained and the
+learned scale matrices are stored in the memory. During the
+inference phase, TRGP retrieves the model training on each
+task by replacing the parameters with the scale matrices.
+The parameters W l
+t,I used for inference on task t are:
+W l
+t,I = W l −
+�
+i
+ProjU l
+i(W l) +
+�
+i
+ProjSl
+i
+U l
+i(W l)
+(5)
+However, as tasks come, storing scaling matrices introduces
+increasing extra parameters. Our experiments demonstrate
+that TRGP requires around 5,000% amount of the parame-
+ters regarding the initial network after 20 tasks on MiniIma-
+geNet, see Figure 3-(b). Therefore, we propose to facilitate
+forward knowledge transfer within a fixed network capacity
+by a restricted orthogonal constraint, allowing parameters
+updated in the direction oblique to the whole frozen space.
+3.2. Overview
+In this section, we introduce our Restricted Orthogonal
+Gradient prOjection (ROGO) framework. To better charac-
+terize the relationship between the gradient direction and
+the frozen space, we first define the angle between a given
+space and a vector in Definition 3.1.
+Definition 3.1. (Angle between vector and space) We de-
+note the angle between two inputs as Θ(·) and the inner
+product between two vectors as ⟨·, ·⟩. The angle between
+a vector v ∈ Rn and a space U n×c ⊂ Rn is defined as the
+minimum angle between the given vector v and any unit
+vector u ∈ U:
+Θ(v, U) = arccos min
+u∈U
+⟨v, u⟩
+∥v∥
+(6)
+Instead of 90◦ in traditional gradient projection methods,
+we aim to relax the angle between the frozen space and the
+modified gradient to be an acute angle. However, directly
+rotating a vector in a high-dimensional space is too flexible
+to control. Hence, we notice the Proposition 3.2.
+Proposition 3.2. Given the full space R and a random
+space U ⊂ R, for any vector v ∈ U ⊥, where U ⊥ = R\U
+denotes the orthogonal complement of U, it can be modified
+to be oblique to the given space U in any angle, by scaling
+and combining a vector v′ within a selected subspace U ′ ⊆
+U.
+According to Proposition 3.2, we can manipulate the gradi-
+ent direction to be oblique to the entire frozen space U l
+t−1
+in any target angle by modifying the parameters within an
+appropriate subspace of U l
+t−1. Therefore, for each task t,
+instead of directly rotating the gradient, we introduce the
+layer-wise relaxing space V l
+t ⊆ U l
+t−1. During the training
+phase, we modify the gradients as:
+gl
+t = gl
+t − ProjU l
+t−1\V l
+t (gl
+t)
+(7)
+By jointly optimizing the parameters within the selected
+relaxing space V l
+t and the original optimization space, the
+strict orthogonal constraint is relaxed. Hence, with a null
+relaxing space, namely dim(V l
+t ) = 0, our framework works
+exactly the same as GPM, under the orthogonal constraint.
+Therefore, our framework can be viewed as a generalization
+of GPM by regulating the relaxing space V l
+t . The relaxing
+space selection thus becomes the key procedure.
+In Section 3.3, we introduce our searching strategy for the
+relaxing space. Moreover, given the relaxing space V l
+t , we
+
+Restricted Orthogonal Gradient Projection for Continual Learning
+involve complementary regularization loss to better consol-
+idate previous knowledge. The detailed procedure is pro-
+vided in Section 3.4. For better illustration, we provide the
+procedure of our ROGO framework in Algorithm 1. Besides,
+we propose ROGO-Exp in Section 3.5, involving additional
+parameters as expansion-based methods, to further validate
+the flexibility and effectiveness of our framework.
+Algorithm 1 Restricted orthogonal gradient projection
+1: Initiate the frozen spaces U0 = {U l
+0}L
+l=1 as ∅s and
+optimize W1 for task 1.
+2: Compute frozen space U1 with representation matrices.
+3: for t ∈ 2, ..., T do
+4:
+Initiate the relaxing space {V l
+t }L
+l=1 as ∅s.
+5:
+repeat
+6:
+Fine-tune for predefined et epochs.
+7:
+Determine additional relaxing space {V l,new
+t
+}L
+l=1
+by the searching strategy in Section 3.3.
+8:
+V l
+t ← V l
+t ∪ V l,new
+t
+.
+9:
+until V l,new
+t
+is ∅.
+10:
+Optimize {W l
+t}L
+l=1 within the space orthogonal to
+U l
+t−1\V l
+t with the regularization terms in Section 3.4.
+11:
+Determine the representation space {Rl
+t}L
+l=1.
+12:
+Update the frozen space by U l
+t = U l
+t−1 ∪ Rl
+t.
+13: end for
+3.3. Searching the Relaxing Space
+To determine the relaxing space V l
+t , we first construct the
+representation space Rl
+g,t by the top-k principal eigenvec-
+tors of the representative matrix [gl
+t,1, ..., gl
+t,Nt], where gl
+t,j
+denotes the gradient of the j-th input. The estimated impor-
+tance is further characterized by the angle from Rl
+g,t. Here
+we define that a vector d is relaxable when:
+Θ(d, Rl
+g,t) ≤ γl
+t
+(8)
+where γl
+t is a predefined threshold. For task t, we aim to
+find the relaxing space V l
+t ⊆ U l
+t−1 spanned all by relaxable
+vectors in U l
+t−1, namely:
+�
+�
+�
+�
+�
+max
+u∈V l
+t
+Θ(u, Rl
+g,t) ≤ γl
+t
+min
+v∈U l,c
+t−1
+Θ(v, Rl
+g,t) > γl
+t
+(9)
+where U l,c
+t−1 = U l
+t−1\V l
+t denotes the complemented sub-
+space of V l
+t with respect to U l
+t−1. Criterion (9) guarantees
+that all vectors in V l
+t are relaxable and there remains none
+relaxable vector in U l,c
+t−1. However, it is hard to construct
+the appropriate V l
+t directly from the high-dimensional U l
+t−1.
+Therefore, we propose a simple yet efficient strategy to
+find V l
+t . Initiating V l
+t as ∅, we iteratively select the clos-
+est vector to Rl
+g,t within U l,c
+t−1 by arg minΘ(d, Rl
+g,t) and
+then append it into V l
+t as the basis if it is relaxable. This
+procedure is repeated until no relaxable vector is left, thus
+obtaining the target V l
+t consisting of all relaxable vectors
+within U l
+t−1. The pseudo-code of our searching strategy is
+provided in Algorithm 2 in Appendix D.
+Theorem 3.3. Denote S = {u|Θ(u, Rl
+g,t) ≤ γl
+t and u ∈
+U l
+t−1} as the whole solution set, the obtained relaxing space
+V l
+t takes up the maximum space in S.
+To further substantiate the effectiveness of our searching
+strategy, here we introduce Theorem 3.3. According to
+Theorem 3.3, our strategy guarantees to find the maximum
+space within the whole solution set satisfying criterion (9).
+The detailed proof is provided in Appendix A.2.
+Theorem 3.4. Denote kr as the dimension of the represen-
+tation space Rl
+g,t and kv as the dimension of the relaxing
+space V l
+t , kv ≤ kr, regardless of the frozen space U l
+t−1.
+Moreover, we provide theoretical analysis on the upper
+bound of the dimension of V l
+t , which is also the number of
+iterations, to investigate the efficiency of our strategy. Here
+we introduce Theorem 3.4, which guarantees that the dimen-
+sion of V l
+t is no more than of the representation space Rl
+g,t.
+In other words, the number of iterations is bounded. Hence,
+as Rl
+g,t is constructed by the top-k principal eigenvectors,
+the dimension of V l
+t is further regulated by hyperparameters,
+of which the ablation study is provided in Section 5. We
+also include the detailed proof in Appendix A.3.
+3.4. Constrained Update in the Relaxing Space
+Given the selected relaxing space V l
+t , the optimization space
+is enlarged, thus facilitating forward knowledge transfer.
+In the meanwhile, we also want to consolidate previous
+knowledge stored within V l
+t . One direct way is to fine-tune
+the parameters with additional regularization terms such as
+EWC (Kirkpatrick et al., 2017). However, regularization
+terms are designed for explicit parameters, which are not
+applicable to implicit subspace in our framework. Therefore,
+we borrow the scaled weight projection (Lin et al., 2022) to
+modify explicit parameters instead. With the scale matrix
+Sl
+t, the gradient gl
+t is modified as:
+gl
+t = gl
+t − ProjU l
+t−1(gl
+t) + ProjSl
+t
+V l
+t (gl
+t)
+(10)
+Notice that we update the parameters within Vl
+t with Sl
+t
+after training each task, instead of storing the parameters
+for inference as TRGP. Parameters within Vl
+t are then re-
+strictedly fine-tuned by adding regularization terms on Sl
+t of
+instead of directly on parameters. Specifically, the objective
+function of task t is:
+Lt = L(Wt, D(t)) +
+L
+�
+l=1
+βl∥Sl
+t − 1(Sl
+t)∥2
+2
+(11)
+where 1(·) denotes the identity matrix with the same shape
+as the input matrix and βl is the weight of the regulariza-
+
+Restricted Orthogonal Gradient Projection for Continual Learning
+tion term for layer l. During backpropagation, gradients
+within the relaxing space V l
+t are regulated by Sl
+t. To further
+validate our strategy, we also equip TRGP with the above
+regularization terms for better comparison in Section 4.3.
+Generally, in our framework, we adopt our searching strat-
+egy to determine the relaxing space and modify those pa-
+rameters with constraints on the scaling matrices.
+However, during training, the direction of gradients shifts
+sharply and frequently due to the steep learning scope of
+deep neural networks. Diverse subspaces would be selected
+in different training phases. Therefore, we iteratively ex-
+amine whether there remains relaxable vectors within the
+remaining frozen space U l,c
+t−1 after limited epochs and then
+search for additional relaxing space V l,new
+t
+. If additional
+relaxing space V l,new
+t
+is involved, we expand the scaling
+matrix with identity matrices to accommodate the increased
+space. On the contrary, TRGP maintains a fixed-size scaling
+matrix throughout training. As the U l
+t−1 is fixed for each
+task, the number of iterations of our strategy is naturally
+limited. In the implementation, we further constrain the
+maximum number of iterations for efficiency.
+In general, by iteratively searching for the relaxing space
+and modifying those gradients with additional regularization
+loss, our restricted orthogonal method optimizes the param-
+eters in the direction oblique to the whole frozen space, thus
+facilitating forward knowledge transfer while consolidating
+previous knowledge within a fixed network capacity.
+3.5. Extensions
+To further validate our framework, we propose ROGO-Exp,
+a modified version of our proposed ROGO, directly stor-
+ing the parameters within the relaxing space. Similar to
+TRGP, for each task t, we retrieve the corresponding relax-
+ing space {V l
+t }L
+l=1 and the scale matrices {Sl
+t}L
+l=1 during
+the inference phase. The modified parameters W l
+t,I used for
+inference on task t are:
+W l
+t,I = W l − ProjV l
+t (W l) + ProjSl
+t
+V l
+t (W l)
+(12)
+where W l denotes the parameters of layer l of current net-
+work. By replacing the parameters in the relaxing space
+with the parameters optimized in task t, the model achieves
+better performance, at the price of expansive computational
+cost in long task sequences. Further experimental results
+validate the efficiency of our ROGO-Exp against state-of-
+the-art expansion-based methods.
+4. Experiments
+4.1. Experimental Setup
+Datasets: Following Saha et al. (2021), we evaluate our
+framework on CIFAR-100 Split (Krizhevsky & Hinton,
+2009), MiniImageNet (Vinyals et al., 2016), Permuted
+MNIST (PMNIST) (Kirkpatrick et al., 2017). Moreover, we
+introduce the Mixture benchmark, first proposed by (Serra
+et al., 2018), consisting of eight datasets. In this work, we
+conduct experiments on Mixture with the seven tasks as a
+sequence except for TrafficSigns1. Details and statistics of
+the datasets can be found in Appendix B.1. Moreover, we
+include the details of network architectures in Appendix B.2.
+Baselines:
+We compare ROGO with competitive and
+well-established methods within a fixed network capac-
+ity. For regularization-based methods, we compare against
+EWC (Kirkpatrick et al., 2017) and HAT (Serra et al., 2018).
+For gradient projection methods, we consider OGD (Fara-
+jtabar et al., 2020) and GPM (Saha et al., 2021). For replay-
+based methods, we adopt ER Res (Chaudhry et al., 2019)
+and A-GEM (Chaudhry et al., 2018). The memory buffer
+size for PMNIST, CIFAR-100 Split, MiniImageNet, and
+Mixture are 1,000, 2,000, 500 and 3,000, respectively. More-
+over, we consider TRGP (Lin et al., 2022) as a competitive
+approach under an expansion setting. Implementation de-
+tails are listed in Appendix B.3.
+Metrics:
+Denote Ai,j as the test accuracy of task j after learning task
+i and bi as the test accuracy of task i at random initialization.
+We employ the following three evaluation metrics.
+• Average Accuracy (ACC) (Mirzadeh et al., 2020): the
+average test accuracy evaluated after learning all tasks,
+defined as 1
+T
+�T
+i=1 AT,i.
+• Backward Transfer (BWT) (Lopez-Paz & Ranzato,
+2017): the average accuracy decrease after learning
+all tasks, defined as
+1
+T −1
+�T −1
+i=1 (AT,i − Ai,i).
+• Forward Transfer (Ωnew) (Kemker et al., 2018):
+the average accuracy of new tasks, defined as
+1
+T −1
+�T
+i=2(Ai,i − bi). As bi stays still across different
+approaches, we consider
+1
+T −1
+�T
+i=2 Ai,i.
+Besides, FWT (Lopez-Paz & Ranzato, 2017) reflects the
+influence of the observed tasks on new tasks in a zero-shot
+manner, defined as
+1
+T −1
+�T
+i=2(Ai−1,i − bi). In this paper,
+we mainly focus on Ωnew, while FWT results are also pro-
+vided. Detailed definitions are provided in Appendix B.4.
+4.2. Main Results
+We show the comparative results on four benchmarks in
+Table 1. The accuracy results of major baselines are adopted
+from GPM (Saha et al., 2021) and we implement all the
+baselines to get the forward knowledge transfer and for-
+getting results. Implementation details are provided in Ap-
+pendix B.3. We run each experiment five times and report
+1We fail to obtain TrafficSigns, as all the links provided in (Stal-
+lkamp et al., 2011; Serra et al., 2018; Saha et al., 2021) are expired.
+
+Restricted Orthogonal Gradient Projection for Continual Learning
+Table 1: Comparison of average final accuracy ACC and forward knowledge transfer Ωnew. ⋆ denotes under the non-
+incremental setting and † denotes requiring an extra data buffer. For each task, we mark the best and the second best
+performance in bold and underline respectively. All results reported are averaged over 5 runs.
+Method
+CIFAR-100 Split
+MiniImageNet
+PMNIST
+Mixture
+ACC (%)
+Ωnew (%)
+ACC (%)
+Ωnew (%)
+ACC (%)
+Ωnew (%)
+ACC (%)
+Ωnew (%)
+Multitask⋆
+79.58 ± 0.54
+-
+69.46 ± 0.62
+-
+96.70 ± 0.02
+-
+81.29 ± 0.23
+-
+A-GEM†
+63.98 ± 1.22
+77.48 ± 0.40
+57.24 ± 0.72
+67.55 ± 1.20
+83.56 ± 0.16
+97.42 ± 0.03
+59.86 ± 1.01
+85.87 ± 0.42
+ER Res†
+71.73 ± 0.63
+77.13 ± 0.18
+58.94 ± 0.85
+69.48 ± 0.42
+87.24 ± 0.53
+97.37 ± 0.05
+75.07 ± 0.55
+86.35 ± 0.22
+EWC
+68.80 ± 0.88
+69.87 ± 1.09
+52.01 ± 2.53
+63.45 ± 2.80
+89.97 ± 0.57
+93.13 ± 0.70
+69.62 ± 2.69
+74.41 ± 1.00
+HAT
+72.06 ± 0.50
+71.50 ± 0.62
+59.78 ± 0.57
+62.63 ± 0.63
+-
+-
+77.54 ± 0.18
+79.26 ± 0.15
+OGD
+70.96 ± 0.32
+73.86 ± 0.57
+59.83 ± 0.62
+62.02 ± 1.24
+82.56 ± 0.66
+97.40 ± 0.06
+74.38 ± 0.27
+81.97 ± 0.53
+GPM
+72.48 ± 0.40
+71.53 ± 0.54
+60.41 ± 0.61
+61.63 ± 0.60
+93.91 ± 0.16
+96.56 ± 0.04
+77.49 ± 0.68
+82.13 ± 0.37
+ROGO
+74.04 ± 0.35
+75.22 ± 0.36
+63.66 ± 1.24
+63.81 ± 0.39
+94.20 ± 0.11
+96.26 ± 0.10
+77.91 ± 0.45
+82.21 ± 0.42
+(a)
+(b)
+(c)
+Test Accuracy (%) 
+Task ID
+Test Accuracy (%) 
+Task ID
+�
+Test Accuracy (%) 
+Test Accuracy (%) 
+Task ID
+(d)
+Figure 2: Results on CIFAR-100 Split setting: (a) average accuracy after learning each task; (b) accuracy evolution of a
+randomly selected task; (c) accuracy tested on task i after learning task i; (d) average accuracy of different ϵ. The optimum ϵ
+value of GPM is annotated by a red circle “o”, and the shaded area indicates the standard deviation.
+the mean results and the standard deviation. Other results
+including the forgetting are provided in Appendix C.2.
+According to Table 1, ROGO dominates EWC across all
+benchmarks and generally outperforms HAT. Although HAT
+obtains comparable accuracy on Mixture, ROGO gains 2.7%
+better Ωnew on average. For gradient projection methods,
+compared with GPM, ROGO achieves over 1% and 3%
+higher ACC on CIFAR-100 Split and MiniImageNet respec-
+tively while improving the forward knowledge transfer. We
+observe that OGD+ achieves the better Ωnew on several
+datasets. Whereas, it fails on the accuracy with significant
+forgetting. Also, we notice that A-GEM and ER Res gain
+better Ωnew, in spite of the inferior accuracy compared with
+GPM and especially ROGO. We assume that this is because
+replay-based methods explore the full optimization space
+with extra data, while both regularization-based and gradi-
+ent projection methods constrain the optimization space to
+mitigate forgetting. Other results including BWT and FWT
+are provided in Appendix C.3. In general, ROGO obtains
+the best accuracy and improves forward knowledge transfer
+without extra data buffers across all datasets.
+Moreover, we compare the average accuracy after learn-
+ing each task on CIFAR-100 Split with GPM, as GPM
+achieves the highest accuracy among selected baselines. As
+shown in Figure 2-(a), ROGO increasingly gains superior
+performance by a larger margin, as the optimization space
+of GPM is gradually constrained by accumulated frozen
+spaces. We provide detailed results on other benchmarks in
+Appendix C.1. For better illustration, we also exhibit the ac-
+curacy evolution of specific tasks during sequential training
+in Figure 2-(b). Here we choose the second task on CIFAR-
+100 Split. According to the results, ROGO universally out-
+performs GPM over the task sequence. Results of three ran-
+domly selected tasks are provided in Appendix C.4 respec-
+tively. In addition, we observe the accuracy on new tasks
+in Figure 2-(c). Similar to the average accuracy, ROGO
+achieves increasingly better Ωnew by relaxing the strict or-
+thogonal constraint. Results on other benchmarks included
+in Appendix C.3 further substantiate this phenomenon.
+To understand our relaxing strategy better, we further con-
+duct experiments on different thresholds ϵ, which regulate
+the dimension of the frozen space. Saha et al. (2021) argue
+that ϵ mediates the stability-plasticity dilemma by control-
+ling the frozen space and thus is critical for GPM. However,
+ROGO enables the frozen space to be adaptively relaxed
+regarding the current task. Therefore, ϵ plays a much less
+important role in ROGO. We present the performance of
+different ϵ on CIFAR-100 Split in Figure 2-(d). As shown
+in Figure 2-(d), the performance of GPM drops significantly
+when ϵ ≥ 0.97, the optimal value reported in GPM, while
+ROGO consistently performs well even with ϵ = 0.98. Gen-
+
+Restricted Orthogonal Gradient Projection for Continual Learning
+Table 2: Comparison of average accuracy and forward knowledge transfer with TRGP under an expansion setting. The
+percentages indicate the ratios of the rank of the relaxing space with respect to the frozen space. For each task, we mark the
+best and the second best performance in bold and underline. Detailed results are provided in Appendix C.7.
+Methods
+TRGP
+ROGO-Exp
+50%
+80%
+T%
+ACC (%)
+Ωnew (%)
+ACC (%)
+Ωnew (%)
+ACC (%)
+Ωnew (%)
+ACC (%)
+Ωnew (%)
+CIFAR
+74.46 ± 0.22
+75.01 ± 0.22
+74.90 ± 0.37
+75.68 ± 0.43
+74.97 ± 0.30
+75.46 ± 0.17
+75.34 ± 0.61
+75.86 ± 0.42
+PMNIST
+96.34 ± 0.11
+97.07 ± 0.14
+96.44 ± 0.20
+97.07 ± 0.10
+96.76 ± 0.15
+97.20 ± 0.08
+97.01 ± 0.08
+97.26 ± 0.05
+MiniImageNet
+61.78 ± 1.94
+63.08 ± 1.40
+63.46 ± 0.85
+62.76 ± 0.64
+62.57 ± 1.32
+62.32 ± 1.19
+62.78 ± 1.00
+62.42 ± 1.28
+Mixture
+83.54 ± 1.15
+84.88 ± 0.95
+82.45 ± 0.49
+84.13 ± 0.25
+83.33 ± 0.31
+84.60 ± 0.20
+83.62 ± 0.26
+84.44 ± 0.42
+Table 3: Comparison of average accuracy and forward knowledge transfer with TRGP within a fixed network capacity. We
+modify TRGP as TRGP-Reg with similar regularization terms. β indicates the regularization weight. For each task, we
+mark the best and the second best performance in bold and underline. Detailed results are provided in Appendix C.7.
+Methods
+TRGP-Reg
+ROGO
+β = 1
+β = 5
+β = 50
+ACC (%)
+Ωnew (%)
+ACC (%)
+Ωnew (%)
+ACC (%)
+Ωnew (%)
+ACC (%)
+Ωnew (%)
+CIFAR
+72.01 ± 0.30
+73.17 ± 0.24
+71.96 ± 0.40
+72.64 ± 0.67
+72.49 ± 0.09
+72.67 ± 0.26
+74.04 ± 0.35
+75.22 ± 0.36
+PMNIST
+76.57 ± 2.69
+94.95 ± 0.23
+75.50 ± 3.96
+95.47 ± 0.30
+76.56 ± 3.83
+95.88 ± 0.19
+94.20 ± 0.11
+96.26 ± 0.10
+MiniImageNet
+55.81 ± 2.43
+59.14 ± 1.00
+58.81 ± 2.24
+62.05 ± 0.74
+22.69 ± 0.39
+20.39 ± 0.36
+63.66 ± 1.24
+63.81 ± 0.39
+Mixture
+73.31 ± 1.05
+83.64 ± 0.38
+74.71 ± 0.85
+83.27 ± 0.24
+17.36 ± 0.86
+7.27 ± 0.98
+77.91 ± 0.45
+82.21 ± 0.42
+erally, ROGO is more robust on the threshold ϵ.
+In brief, our approach universally outperforms selected base-
+lines without extra data buffers. Achieving better average
+accuracy, ROGO also improves forward knowledge trans-
+fer by exploring a larger optimization space than GPM. To
+validate the efficiency of our relaxing strategy, we further
+compare ROGO-Exp with well-established and competitive
+expansion-based methods in the next section.
+4.3. Comparison with Expansion-based Methods
+The above experiments exhibit the superiority of our ap-
+proach when maintaining a fixed network capacity. How-
+ever, under the scenario with no limit on the network capac-
+ity, expansion-based methods achieve great performance by
+allocating new neurons or modules. Therefore, to further
+validate our strategy, we compare ROGO and ROGO-Exp
+with relative expansion-based methods.
+In this section, we adopt TRGP (Lin et al., 2022), which
+expands the optimization space by retraining parameters
+within the selected trust regions, achieving superior perfor-
+mance. In the inference phase, TRGP reuses the parameters
+in corresponding trust regions memorized after learning this
+task. In contrast, GPM and our ROGO only store the repre-
+sentation of the frozen space. Therefore, although indeed
+a stable network capacity is allocated for each task, the en-
+tire memory size of TRGP grows continually. As shown in
+Figure 3-(b), after learning the last task on MiniImageNet,
+TRGP requires around 5,000% extra parameters with re-
+spect to the network capacity. Results on other benchmarks
+provided in Appendix C.6 further substantiate that TRGP
+introduces a significant number of extra parameters. Thus,
+we categorize TRGP as an expansion-based method here.
+In this setting, the main difference between ROGO-Exp and
+TRGP is the strategy of deciding which part of the frozen
+space to reuse. We conduct experiments on all four bench-
+marks and report the results in Table 2. The percentages
+indicate the ratios of the rank of the relaxing space with
+respect to the corresponding frozen space. We evaluate two
+constant ratios and further use the ratios in TRGP, denoted
+as T%. As TRGP selects the top 2 tasks as the trust regions,
+T% is larger than 80% at most times. According to Table 2,
+IRGP-Exp universally outperforms TRGP, even by relaxing
+only 50% of the frozen space. Moreover, we observe that
+ROGO gains superior performance over both TRGP and
+ROGO-Exp on MiniImageNet. We assume that the reason
+is the positive backward knowledge transfer. Training the
+network on new tasks also benefits the previous task. In
+General, our approach achieves better performance with a
+comparable size of the relaxing space, which substantiates
+the efficiency of our searching strategy.
+We further modify TRGP as TRGP-Reg with similar reg-
+ularization terms on the scale matrices as our ROGO to
+compare the relaxing strategies. We report the results on
+four benchmarks with three representative regularization
+weights β on TRGP-Reg in Table 3. As shown in Table 3,
+ROGO significantly outperforms TRGP-Reg, especially on
+PMNIST, gaining around 20% ACC improvement. Gener-
+ally, ROGO achieves better or comparable Ωnew than TRGP
+under or without the constraint of a fixed network capacity.
+
+Restricted Orthogonal Gradient Projection for Continual Learning
+Table 4: Ablation study of ζ = cos γ and β on CIFAR100-Split, where γ is the threshold for the relaxing strategy and β is
+the regularization weight. ζconv and ζfc denote the threshold for convolutional and fully connected layers, respectively.
+(a) Ablation study of ζ. For each ζ combination, we report the relaxing ratio of each layer.
+ζcong
+ζfc
+Conv1
+Conv2
+Conv3
+Fc1
+Fc2
+ACC
+BWT
+0.20
+0.20
+38.09%
+34.63%
+51.11%
+80.75%
+60.56%
+71.97%
+-2.97%
+0.50
+0.50
+30.26%
+28.75%
+40.53%
+34.78%
+11.26%
+72.58%
+-2.41%
+0.80
+0.80
+28.51%
+18.71%
+25.68%
+5.94%
+0.23%
+73.34%
+-2.04%
+0.90
+0.90
+27.87%
+14.16%
+18.65%
+1.08%
+0.07%
+73.68%
+-1.76%
+0.95
+0.90
+26.31%
+10.51%
+13.63%
+0.97%
+0.04%
+74.04%
+-1.43%
+0.95
+0.95
+24.12%
+10.52%
+13.83%
+0.08%
+0.00%
+73.87%
+-1.53%
+0.99
+0.99
+18.66%
+5.51%
+9.51%
+0.00%
+0.00%
+73.70%
+-1.34%
+(b) Ablation study of β.
+β
+ACC
+BWT
+0.0
+72.43%
+-3.03%
+0.1
+72.94%
+-2.57%
+0.5
+73.68%
+-1.88%
+1.0
+74.04%
+-1.34%
+5.0
+73.77%
+-1.42%
+10.0
+73.42%
+-0.87%
+50.0
+72.65%
+-0.83%
+Ratio (%) 
+(a)
+(b)
+Ratio (%)
+Task ID
+Task ID
+Figure 3: (a) Relaxing ratios of the last layer on CIFAR-
+100 Split, MiniImageNet, and PMNIST. (b) Ratios of the
+amount of extra parameters concerning the amount of the
+parameters of the network architecture on MiniImageNet.
+Detailed results are included in Appendix C.7.
+5. Analysis and Discussion
+To gain a deeper insight into ROGO, we investigate the
+trend of scales of the space relaxed by our strategy. With the
+theoretical upper bound of the rank of the relaxed subspace
+provided in Theorem 3.4, we further inspect the ratios of
+the relaxing spaces concerning corresponding frozen spaces
+in practice. Results of the last layer on three different set-
+tings are provided in Figure 3-(a). As shown in Figure 3-(a),
+relaxing ratios generally maintain a stable trend, fluctuating
+smoothly within a small range over sequential tasks on both
+benchmarks. As different tasks explore different optimiza-
+tion directions, ideal relaxing spaces vary across tasks, in
+accordance with the fluctuation of our results. To further
+investigate our framework, we provide the ablation study of
+related hyper-parameters on CIFAR-100 Split in Table 4.
+Ablation study of the relaxing ratios. ROGO mediates the
+stability-plasticity dilemma by controlling the dimension
+and flexibility of the relaxing space by γ in Equation (9).
+For better illustration, here we represent γ by ζ = cos γ.
+We first conduct an ablation study of ζ, reporting both the
+performance and the relaxing ratios in Table 4(a), where
+ζconv denotes the hyper-parameter for convolutional layers
+and ζfc denotes the hyper-parameter for fully connected
+layers. As shown in Table 4(a), increasing ζ gradually
+constrains the scale of the relaxing space, thus alleviating the
+forgetting. On the other hand, a small ζ guarantees sufficient
+forward knowledge transfer, while fails on consolidating
+previous knowledge. Generally, our method persists in a
+superb performance with ζ ≥ 0.80.
+Ablation study of the regularization weights. Moreover,
+we observe the performance of ROGO with various regular-
+ization weights β in Table 4(b). In this work, we adopt a
+consistent β for the entire network. According to Table 4(b),
+Similarly, we observe less forgetting on larger β, which con-
+strains the update of parameters within the relaxing space
+more strictly. However, strict constraints also lead to limited
+performance on new tasks as discussed in Section 4. In a
+nutshell, ζ and β operate together in ROGO to overcome
+catastrophic forgetting and enhance forward transfer.
+Training time. The dimension of the frozen spaces keeps
+growing as the tasks accumulate, leading to expanding
+range for searching relaxing spaces. Therefore, the compu-
+tation complexity and time consumption are supposed to
+increase gradually. We further reported the time consump-
+tion comparison on CIFAR-100 Split and MiniImageNet in
+Appendix C.5. According to Table 14, ROGO takes around
+50% more time than GPM, similar to TRGP. In general, the
+practical efficiency of our approach is acceptable.
+6. Conclusion
+In this paper, we proposed ROGO, a novel continual learn-
+ing framework that facilitates forward knowledge transfer
+in gradient projection methods within a fixed network ca-
+pacity. ROGO imposes a restricted orthogonal constraint
+allowing parameters to be updated in the direction oblique
+to the whole frozen space, which thus explores a larger op-
+timization space. Extensive experiments demonstrate that
+our ROGO framework surpasses related state-of-the-art ap-
+proaches on diverse benchmarks. Moreover, we proposed
+ROGO-Exp that allows network expansion with the relax-
+ing space, which achieves better performance than related
+expansion-based methods. We also provided theoretical
+analysis validating the efficiency of our algorithm.
+
+Restricted Orthogonal Gradient Projection for Continual Learning
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+Restricted Orthogonal Gradient Projection for Continual Learning
+A. Proof
+In this section, we sequentially provide the proof of Theorem 3.3 and 3.4. For better comprehension, we first introduce
+complementary Lemma A.1. For simplification, all vectors here are assumed to be unit vectors, namely ∥v∥ = 1.
+A.1. Proof of Lemma A.1
+Lemma A.1. Denote the relaxed space as V = span{v1, v2, ..., vn}, where vn is the last base included in V . Given
+representation space U, ∀v ∈ V , we have Θ(v, U) ≤ Θ(vn, U).
+Depicting the angle by projection form, Lemma A.1 can be expressed as:
+∀v ∈ V, ∥ProjU(v)∥ ≥ ∥ProjU(vn)∥
+(13)
+Denote B = [u1, ..., um] as the representative matrix of the representation space U = span{u1, ..., um}, where ui is the
+i-th normalized orthogonal base of U. Lemma A.1 can be further expressed as:
+∀v ∈ V, vT BBT v ≥ vT
+n BBT vn
+(14)
+As vis are the basis sequentially appended by our searching strategy, for any i ≤ j, we have:
+vT
+i BBT vi ≥ vT
+j BBT vj
+(15)
+Consider a special case where the current relaxed subspace has only one base, denoted by V = span{v1}, namely n = 1.
+Obviously, Lemma A.1 is true in this case. Thus, we consider the general case that n ≥ 2.
+We provide proof by inductive reasoning. Note that as we assume all vectors to be unit vectors, we have �
+i ∥wi∥2
+2 = 1.
+First, we consider a special case V = span{v1, v2}, namely n = 2. Assume there exists v = w1v1 + w2v2 that
+vT BBT v < vT
+2 BBT v2, with w2
+1 + w2
+2 = 1, we have:
+w2
+1vT
+2 BBT v2 > w2
+1vT
+1 BBT v1 + 2w1w2vT
+1 BBT v2
+(16)
+Construct v′ = w2v1 − w1v2, we have:
+v′T BBT v′ = w2
+2vT
+1 BBT v1 + w2
+1vT
+2 BBT v2 − 2w1w2vT
+1 BBT v2
+> w2
+2vT
+1 BBT v1 + w2
+1vT
+2 BBT v2 + w2
+1vT
+1 BBT v1 − w2
+1vT
+2 BBT v2
+= vT
+1 BBT v1
+(17)
+which contradicts Θ(v1, U) ≤ Θ(v, U) for ∀v ∈ span{v1, v2}.
+Then we consider the general case V = span{v1, ..., vt}. For ∀v ∈ V , we have Θ(v, U) ≤ Θ(vt, U). After vt+1 is
+included, we assume that there exists v ∈ V that Θ(v, U) > Θ(vt+1, U). Then we can find the maximum s satisfying
+that there exists v = �s
+i=1 wivi + wt+1vt+1 that Θ(v, U) > Θ(vt+1, U) and for ∀v′ = �s−1
+i=1 wivi + wt+1vt+1, we have
+Θ(v′, U) ≤ Θ(vt+1, U). When s = 1, it is similar to the special case, so the proof is omitted. Thus, we consider the case
+where s ≥ 2. For simplification, we express v as v = c0v0 + c1vs + c2vt+1 with v0 = �s−1
+i=1 aivi, where cis and ais are
+coefficients. We have:
+(c0v0 + c1vs + c2vt+1)T BBT (c0v0 + c1vs + c2vt+1) < vT
+t+1BBT vt+1
+(18)
+As Θ(w1v0 + w2vs, U) ≤ Θ(vs, U) ≤ Θ(vt+1, U), we have:
+(w1v0 + w2vs)T BBT (w1v0 + w2vs) ≥ vT
+s BBT vs
+(19)
+
+Restricted Orthogonal Gradient Projection for Continual Learning
+which is:
+w2
+1vT
+0 BBT v0 + 2w1w2vT
+0 BBT vs ≥ w2
+1vT
+s BBT vs ≥ w2
+1vT
+t BBT vt
+(20)
+Similarly we have:
+w2
+1vT
+0 BBT v0 + 2w1w2vT
+0 BBT vt ≥ w2
+1vT
+t BBT vt
+(21)
+Then we can express Equation (18) as:
+vT
+t+1BBT vt+1 > (c2
+0 + c2
+2)vT
+t BBT vt + c2
+1vT
+s BBT vs + 2c1c2vT
+s BBT vt
+(22)
+As ∥v∥ = ∥vi∥ = 1, c2
+0 + c2
+1 + c2
+2 = 1. Then we have:
+−2c1c2vT
+s BBT vt > c2
+1vT
+s BBT vs − c2
+1vT
+t+1BBT vt+1
+(23)
+Construct v′ = c2vs−c1vt+1
+√
+c2
+1+c2
+2
+, we have:
+v′T BBT v′ =
+1
+c2
+1 + c2
+2
+(c2
+2vT
+s BBT vs + c2
+1vT
+t+1BBT vt+1 − 2c1c2vT
+s BBT vt)
+>
+1
+c2
+1 + c2
+2
+(c2
+2vT
+s BBT vs + c2
+1vT
+t+1BBT vt+1 + c2
+1vT
+s BBT vs − c2
+1vT
+t+1BBT vt+1)
+= vT
+s BBT vs
+(24)
+which contradicts Θ(vs, U) ≤ Θ(v, U) for ∀v ∈ span{vs, vs+1, ..., vt, vt+1}.
+Thus, for ∀v ∈ V = span{v1, ..., vn}, we have Θ(v, U) ≤ Θ(vn, U).
+A.2. Proof of Theorem 3.3
+Here we provide proof of Theorem 3.4. Denote the whole solution set as S = {u|Θ(u, U) ≤ γ and u ∈ U f} where U is
+the representation space, U f is the frozen space and γ is the threshold. Theorem 3.3 can be expressed as that all subspace
+V ′ ⊆ S satisfies dim(V ′) ≤ dim(V ), where V is the relaxing space obtained by our searching strategy.
+For the sake of contradiction, we assume there exists V ′ ⊆ S that dim(V ′) > dim(V ). Denote V c
+f as the orthogonal
+complement of V with respect to the frozen space U f, as the relaxing space is a subspace of the frozen space. According to
+Lemmoa A.1, for ∀v ∈ V c
+f , we have Θ(v, U) > γ. We also have:
+dim(V ′ ∩ V c
+f ) = dim(V ′) + dim(V c
+f ) − dim(V ′ + V c
+f ) > 0
+(25)
+Thus, there exists v′ ∈ V ′ that v′ ∈ U c
+f, namely there exists v′ ∈ S that Θ(v′, U) > γ, which is contradict. Therefore, the
+relaxing space obtained by our searching strategy takes up the maximum subspace of the whole solution set.
+A.3. Proof of Theorem 3.4
+Here we provide proof of Theorem 3.4.
+Denote the relaxing space and the representation space as V and U =
+span{u1, ..., ukr} respectively. With Lemma A.1, we have ∀v ∈ V , Θ(v, U) ≤ Θ(vkv, U) <
+π
+2 , where kv denotes
+dim(V ), the dimension of V . In other words,
+∀v ∈ V, v ̸⊥ U
+(26)
+
+Restricted Orthogonal Gradient Projection for Continual Learning
+Similar to Theorem 3.4, assume the dimension of V is larger than kr, namely dim(V ) > dim(U) = kr. Denote U c as the
+orthogonal complement of U with respect to the whole space R. Obviously, dim(U c) = n − kr, where n is the dimension
+of R. Then we have:
+dim(V ∩ U c) = dim(V ) + dim(U c) − dim(V + U c)
+> kr + (n − kr) − n = 0
+(27)
+Thus, there exists v′ ∈ V such that v′ ∈ U c too. As U c is the orthogonal complement, ∀u ∈ U c, u ⊥ U. Then we have
+v′ ⊥ U, which contradicts Equation (26). Therefore, the assumption is aborted. We have that kv ≤ kr. The upper bound of
+the dimension of V is kr, namely the dimension of the representation space U.
+B. Experimental Setup
+B.1. Datasets
+Here we introduce the datasets we use for evaluation. 1) CIFAR-100 Split Saha et al. (2021) constructed CIFAR-100
+Split, by splitting CIFAR100 (Krizhevsky & Hinton, 2009) into 10 tasks where each task has 10 classes. 2) MiniImageNet
+Following Saha et al. (2021), we split MiniImageNet (Vinyals et al., 2016) into 20 sequential tasks with 5 classes each.
+3) Permuted MNIST (PMNIST) PMNIST (Kirkpatrick et al., 2017) is a variant of MNIST (LeCun et al., 1998) where
+each task has a different permutation of inputting images, consists of 10 sequential tasks with 10 classes each. 4) Mixture
+Serra et al. (2018) first proposed Mixture consisting of 8 datasets, including CIFAR-10 (Krizhevsky & Hinton, 2009),
+MNIST (LeCun et al., 1998), CIFAR-100 (Krizhevsky & Hinton, 2009), SVHN (Netzer et al., 2011), FashionMNIST (Xiao
+et al., 2017), TrafficSigns (Stallkamp et al., 2011), FaceScrub (Ng & Winkler, 2014), and NotMNIST (Bulatov, 2011), from
+which Ebrahimi et al. (2020) further constructed 5-Datasets. Here we follow the original harder benchmark. Particularly, we
+consider all tasks as a sequence except TrafficSigns (Stallkamp et al., 2011), which we failed to access. 5) CIFAR-100 Sup
+In addition, following Yoon et al. (2019), we adopt CIFAR-100 Sup consisting of 20 superclasses as sequential tasks. Here
+we adopt the five different predefined orders proposed by Yoon et al. (2019) and report the main results in Appendix C.7.
+Among all evaluated datasets, PMNIST is a benchmark under the domain-incremental scenario, while other four datasets are
+under the task-incremental scenario.
+Moreover, we provide the statistics of selected datasets in Table 5 and Table 6. For the Mixture benchmark, the images of
+MNIST, FashionMNIST, and notMNIST are replicated across all RGB channels following Serra et al. (2018).
+Table 5: Statistics of CIFAR-100 Split, MiniImageNet, and PMNIST.
+CIFAR-100 Split
+CIFAR-100 Sup
+MiniImageNet
+PMNIST
+Image Size
+32 × 32
+32 × 32
+84 × 84
+28 × 28
+Channels
+3
+3
+3
+1
+Classes
+100
+100
+100
+10
+Tasks
+10
+20
+20
+10
+Classes/task
+10
+5
+5
+10
+Training Samples/task
+4,750
+2,375
+2,375
+54,000
+Validation Samples/task
+250
+125
+125
+6,000
+Testing Samples/task
+1,000
+500
+500
+10,000
+B.2. Model Details
+MLP architecture: We adopt a 3-layer model including two hidden layers with 100 neurons each for the PMNIST setting,
+the same as Lopez-Paz & Ranzato (2017). ReLU is used as the activate function here and for all other architectures. Also,
+we use softmax with cross entropy loss on all settings.
+AlexNet architecture: For CIFAR-100 Split setting, we adopt the same network as Serra et al. (2018) with batch normal-
+ization, including two fully connected layers and three convolutional layers. The convolutional layers have 4 × 4, 3 × 3, and
+2 × 2 kernel sizes with 64, 128, and 256 filters respectively. After each convolutional layer, we add batch normalization and
+
+Restricted Orthogonal Gradient Projection for Continual Learning
+Table 6: Statistics of Mixture benchmark.
+Dataset
+Classes
+# Taining
+# Validation
+# Testing
+CIFAR-10 (Krizhevsky & Hinton, 2009)
+10
+47,500
+2,500
+10,000
+MNIST (LeCun et al., 1998)
+10
+57,000
+3,000
+10,000
+CIFAR-100 (Krizhevsky & Hinton, 2009)
+100
+47,500
+2,500
+10,000
+SVHN (Netzer et al., 2011)
+10
+69,595
+3,662
+26,032
+FashionMNIST (Xiao et al., 2017)
+10
+57,000
+3,000
+10,000
+FaceScrub (Ng & Winkler, 2014)
+100
+19,570
+1,030
+2,289
+NotMNIST (Bulatov, 2011)
+10
+16,011
+842
+1,873
+2 × 2 max-pooling. Each fully connected layer has 2048 units. For the first two layers, we use the dropout of 0.2, and for
+the rest layers, we use the dropout of 0.5.
+Modified LeNet-5 architecture: For the CIFAR-100 Sup setting, a modified LeNet-5 architecture consisting of two
+convolutional layers and two fully connected layers is adopted, similar to Saha et al. (2021). Max-pooling of 3 × 2 is used
+after each convolutional layer. The last two layers have 800 and 500 units respectively.
+Reduced ResNet-18 architecture: We adopt the same reduced ResNet-18 architecture as Saha et al. (2021) for the
+MiniImageNet and Mixture settings, using 2 × 2 average-pooling before the classifier layer instead of the 4 × 4 average-
+pooling used by Lopez-Paz & Ranzato (2017). Moreover, we present the dimension of the representation space of each layer
+of our architectures in Table 7.
+Table 7: Dimension of the representation space of each layer.
+Network
+Depth
+Dimension of the representation space
+MLP
+3 layers
+784; 100; 100
+AlexNet
+5 layers
+48; 576; 512; 1,024; 2,048
+LeNet-5
+4 layers
+75; 500; 3,200; 800
+ResNet-18
+17 layers and 3 short-
+cut connections
+27; 180; 180; 180; 180; 180; 360; 20; 360; 360; 360; 720;
+40; 720; 720; 720; 1,440; 80; 1,440; 1,440
+B.3. Implementation Details
+We use the official implementation of GPM (Saha et al., 2021), HAT (Serra et al., 2018), and TRGP (Lin et al., 2022). We
+implement A-GEM and ER Res with the official implementation by Chaudhry et al. (2018) and implement EWC with the
+implementation by Serra et al. (2018). For OGD (Farajtabar et al., 2020), we use the implementation by Bennani et al.
+(2020). Following Saha et al. (2021) and Lin et al. (2022), we run all experiments five times on an established seed without
+fixing the cuda settings for a fair comparison. Particularly, we use five random seeds on PMNIST where there is no diversity
+on a single seed. For CIFAR-100 Sup, we use five different orders provided by Yoon et al. (2019). Following Saha et al.
+(2021), we report the experimental results of replay-base methods A-GEM and ER Res on the Mixture dataset with the
+same buffer size as GPM and ROGO, which is 8.98M in terms of the number of parameters for Resnet18 architecture.
+On CIFAR-100 Split, MiniImageNet, and PMNIST, we follow the hyper-parameters utilized by Saha et al. (2021) and Lin
+et al. (2022), including learning rate, batch size, and the threshold ϵ. On Mixture, as we adopt the same network architecture
+Saha et al. (2021) use on their 5-Dataset setting, we follow the provided learning rate and batch size as well.
+As discussed in Section 5, the threshold ζ = cos γ controls the criterion of the relaxing space. For CIFAR100-Split and
+PMNIST, we use ζ = 0.95 for convolutional layers and ζ = 0.9 for fully connected layers. For MiniImageNet and Mixture,
+we use the same ζ for all layers, 0.95 and 0.9 respectively. The regularization weight β is set as 5 for the ResNet18
+architecture and 1 for others. Moreover, we limit the max iteration times to 2 for CIFAR100-Split and MiniImageNet for
+efficiency. Particularly, we run all the experiments on a single NVIDIA GeForce RTX 2080 Ti GPU.
+
+Restricted Orthogonal Gradient Projection for Continual Learning
+B.4. Metrics
+Here we present the detailed definitions of the metrics evaluating the forward knowledge transfer. Ωnew (Kemker et al.,
+2018). Denote bi as the test accuracy of task i at random initialization, FWT, first proposed by Lopez-Paz & Ranzato (2017),
+is defined as FWT =
+1
+T −1
+�T
+i=2(Ai−1,i − bi), evaluating the zero-shot performance of the initialization with respect to the
+observed tasks. While Ωnew, first proposed by Kemker et al. (2018), is defined as Ωnew =
+1
+T −1
+�T
+i=2(Ai,i − bi), reflecting
+the test accuracy on new tasks based on the learnt knowledge. As bi stays still across different approaches, we consider
+Ωnew =
+1
+T −1
+�T
+i=2 Ai,i for simplicity. For this simplified Ωnew, we have: Ωnew =
+T
+T −1ACC − BWT −
+1
+T −1A1,1, with
+the ACC and BWT defined in Section 4.1.
+C. Experimental Results
+C.1. Final Accuracy
+We provide the test accuracy after learning each task on other benchmarks here. As discussed in Section 4, our ROGO
+universally outperforms GPM over the task sequence on all benchmarks.
+(a)
+(b)
+(c)
+Test Accuracy (%) 
+Task ID
+Test Accuracy (%) 
+Task ID
+Task ID
+Test Accuracy (%) 
+Test Accuracy (%) 
+Task ID
+(d)
+Figure 4: Average accuracy after learning each task on (a) CIFAR100-Split, (b) MiniImageNet, (c) PMNIST, and (d)
+Mixture.
+C.2. Backward Knowledge Trasnfer
+As mentioned in Section 4.1, final accuracy (ACC), forgetting (BWT) and forward knowledge transfer (Ωnew) are jointly
+considered to evaluate a continual learner. According to Table 8, ROGO achieves the best forgetting on MiniImageNet and
+PMNIST datasets. Despite our relaxing strategy focusing on facilitating the forward knowledge transfer, ROGO gains better
+final accuracy with comparable forgetting over all benchmarks. Generally speaking, ROGO achieves superior performance
+than previous baselines with a fixed network capacity.
+Table 8: Comparison of average accuracy and forgetting tested after learning all tasks. Multitask is under the non-incremental
+setting. For each task, we mark the best and the second best performance in bold and underline respectively. All results
+reported are averaged over 5 runs.
+Method
+CIFAR-100 Split
+MiniImageNet
+PMNIST
+Mixture
+ACC (%)
+BWT (%)
+ACC (%)
+BWT (%)
+ACC (%)
+BWT (%)
+ACC (%)
+BWT (%)
+Multitask
+79.58 ± 0.54
+-
+69.46 ± 0.62
+-
+96.70 ± 0.02
+-
+81.29 ± 0.23
+-
+A-GEM
+63.98 ± 1.22
+-16.30 ± 1.19
+57.24 ± 0.72
+-10.95 ± 1.29
+83.56 ± 0.16
+-14.57 ± 2.33
+59.86 ± 1.01
+-29.37 ± 1.11
+ER Res
+71.73 ± 0.63
+-5.50 ± 0.76
+58.94 ± 0.85
+-8.84 ± 0.56
+87.24 ± 0.53
+-10.24 ± 1.58
+75.07 ± 0.55
+-11.81 ± 0.82
+EWC
+68.80 ± 0.88
+-2.31 ± 0.76
+52.01 ± 2.53
+-12.14 ± 2.28
+89.97 ± 0.57
+-3.58 ± 1.22
+69.62 ± 2.69
+-6.00 ± 4.09
+HAT
+72.06 ± 0.50
+-0.12 ± 0.42
+59.78 ± 0.57
+-3.31 ± 0.05
+-
+-
+77.54 ± 0.18
+-0.55 ± 0.28
+GPM
+72.48 ± 0.40
+-0.72 ± 0.27
+60.41 ± 0.61
+-1.54 ± 0.34
+93.91 ± 0.16
+-3.30 ± 0.09
+77.49 ± 0.68
+-4.70 ± 0.50
+ROGO
+74.04 ± 0.35
+-1.43 ± 0.43
+63.66 ± 1.24
+-0.09 ± 0.94
+94.20 ± 0.11
+-2.44 ± 0.13
+77.91 ± 0.45
+-4.55 ± 0.36
+
+Restricted Orthogonal Gradient Projection for Continual Learning
+C.3. Forward Knowledge Transfer
+We provide detailed forward knowledge transfer performance on all four benchmarks here. First, we present the results of
+Ωnew and the detailed accuracy of each task after learning it in Table 9 to 12. According to Table 11 and Table 12, ROGO
+achieves a similar forward knowledge transfer compared with GPM. For other benchmarks, ROGO improves Ωnew by 2.7%
+and 1.8% on CIFAR-100 Split and MiniImageNet respectively, as shown in Table 9 and Table 10.
+Table 9: The accuracy tested on task i after learning task i and Ωnew on CIFAR-100 Split.
+Method
+1
+2
+3
+4
+5
+6
+7
+8
+9
+Avg (Ωnew)
+GPM
+67.7
+72.5
+70.1
+73.6
+71.8
+70.3
+71.0
+71.8
+73.7
+71.5
+ROGO
+71.7
+74.8
+73.6
+76.9
+75.8
+74.7
+74.0
+75.9
+79.3
+75.2
+Table 10: The accuracy tested on task i after learning task i and Ωnew on MiniImageNet.
+Method
+1
+2
+3
+4
+5
+6
+7
+8
+9
+10
+11
+12
+13
+14
+15
+16
+17
+18
+19
+Avg (Ωnew)
+GPM
+59.0
+57.1
+65.6
+59.6
+76.2
+57.2
+66.5
+72.8
+81.5
+42.0
+59.6
+62.1
+62.3
+58.9
+57.2
+59.0
+50.7
+65.7
+57.3
+61.6
+ROGO
+57.1
+55.1
+66.2
+61.3
+79.8
+61.4
+66.6
+73.5
+83.6
+42.8
+65.1
+63.4
+67.1
+64.6
+61.6
+61.6
+50.0
+68.2
+60.8
+63.7
+Table 11: The accuracy tested on task i after learning task i and Ωnew on PMNIST.
+Method
+1
+2
+3
+4
+5
+6
+7
+8
+9
+Avg (Ωnew)
+GPM
+97.5
+97.4
+97.0
+96.8
+96.5
+96.2
+96.2
+95.8
+95.1
+96.5
+ROGO
+97.4
+97.2
+97.0
+96.5
+96.3
+96.1
+95.7
+94.9
+94.8
+96.2
+Table 12: The accuracy tested on task i after learning task i and Ωnew on Mixture.
+Method
+1
+2
+3
+4
+5
+6
+Avg (Ωnew)
+GPM
+99.0
+43.7
+87.3
+99.1
+69.9
+93.4
+82.1
+ROGO
+99.1
+42.9
+87.6
+99.1
+70.6
+93.5
+82.2
+Moreover, we provide the result of FWT (using the definition in (Lopez-Paz & Ranzato, 2017)) on all benchmarks in
+Table 13. According to Table 13, although our method facilitates the forward knowledge transfer reflected by Ωnew, ROGO
+achieves better FWT than GPM on all three task-incremental benchmarks, namely better zero-shot performance.
+Table 13: Comparison of forward knowledge transfer between GPM and ROGO, evaluated by FWT.
+Methods
+CIFAR-100 Split
+MiniImageNet
+PMNIST
+Mixture
+GPM
+-0.65 ± 0.18
+-0.26 ± 0.88
++0.66 ± 1.17
+-0.52 ± 1.49
+ROGO
++0.20 ± 0.36
++0.30 ± 0.29
+-0.63 ± 0.97
++0.13 ± 0.77
+C.4. Accuracy Evolution
+Here we present the accuracy tested on three randomly selected tasks immediately after learning them. We select the 2nd,
+4th, and 6th tasks for all benchmarks. Generally, ROGO outperforms GPM on selected tasks over the sequence. We further
+notice that the improvement is more significant on later tasks owing to a larger relaxing space, as discussed in Section 5.
+
+Restricted Orthogonal Gradient Projection for Continual Learning
+(a)
+(b)
+(c)
+Test Accuracy (%) 
+Task ID
+Test Accuracy (%) 
+Task ID
+Task ID
+Test Accuracy (%) 
+Test Accuracy (%) 
+Task ID
+(d)
+Figure 5: Accuracy evolution of the 2nd task on (a) CIFAR-100 Split, (b) MiniImageNet, (c) PMNIST, and (d) Mixture.
+(a)
+(b)
+(c)
+Test Accuracy (%) 
+Task ID
+Test Accuracy (%) 
+Task ID
+Task ID
+Test Accuracy (%) 
+Test Accuracy (%) 
+Task ID
+(d)
+Figure 6: Accuracy evolution of the 4th task on (a) CIFAR-100 Split, (b) MiniImageNet, (c) PMNIST, and (d) Mixture.
+(a)
+(b)
+(c)
+Test Accuracy (%) 
+Task ID
+Test Accuracy (%) 
+Task ID
+Task ID
+Test Accuracy (%) 
+Test Accuracy (%) 
+Task ID
+(d)
+Figure 7: Accuracy evolution of the 6th task on (a) CIFAR-100 Split, (b) MiniImageNet, (c) PMNIST, and (d) Mixture.
+C.5. Time Consumption
+We report the time consumption of ROGO on two benchmarks compared with relative baselines. TRGP and ROGO are both
+evaluated on a single NVIDIA GeForce RTX 2080 Ti GPU and we report the results according to (Lin et al., 2022). As
+discussed in Section 5, ROGO takes acceptable extra time compared with GPM on both datasets. For both dataset, ROGO
+tasks similar time as TRGP, which is similar to EWC and much less than A-GEM and OWM.
+Table 14: Time comparison evaluated on two benchmarks. We use the results reported in (Lin et al., 2022) and the time is
+normalized with respect to GPM.
+Datasets
+Methods
+OWM
+EWC
+HAT
+A-GEM
+ER Res
+GPM
+TRGP
+ROGO
+CIFAR-100
+2.41
+1.76
+1.62
+3.48
+1.49
+1.00
+1.65
+1.62
+MiniImageNet
+-
+1.22
+0.91
+1.79
+0.82
+1.00
+1.34
+1.43
+C.6. Memory Usage
+We provide a comparison between TRGP and ROGO on the ratio of the amount of extra parameters concerning the amount
+of the parameters of the initial network architecture. According to Figure 8, TRGP requires at least 200% of the number of
+
+Restricted Orthogonal Gradient Projection for Continual Learning
+extra parameters after learning all tasks on all four benchmarks, while ROGO only stores the representation of the frozen
+space, which can further be released in the inference phase.
+(a)
+(b)
+(c)
+Ratio (%) 
+Task ID
+Ratio (%) 
+Task ID
+Task ID
+Ratio (%) 
+Ratio (%) 
+Task ID
+(d)
+Figure 8: Ratio of the amount of extra parameters concerning the amount of the parameters of the initial network architecture
+on (a) CIFAR100-Split, (b) MiniImageNet, (c) PMNIST, and (d) Mixture.
+C.7. Other Results
+Here we provide detailed results including the forgetting performance compared with TRGP in Table 15 and 16. Similar to
+the discussion in Section 4.3, our ROGO framework obtains superior accuracy performance with comparable forgetting
+under or without the constraint of a fixed network capacity.
+Table 15: Comparison of average accuracy and forgetting with TRGP under an expansion setting. The percentages indicate
+the ratios of the rank of the relaxing space with respect to the frozen space. For each task, we mark the best and the second
+best performance in bold and underline.
+Methods
+TRGP
+ROGO-Exp
+50%
+80%
+T%
+ACC (%)
+BWT (%)
+ACC (%)
+BWT (%)
+ACC (%)
+BWT (%)
+ACC (%)
+BWT (%)
+CIFAR
+74.46 ± 0.22
+-0.42 ± 0.20
+74.90 ± 0.37
+-0.99 ± 0.27
+74.97 ± 0.30
+-0.68 ± 0.39
+75.34 ± 0.61
+-0.63 ± 0.57
+PMNIST
+96.34 ± 0.11
+-0.58 ± 0.10
+96.44 ± 0.20
+-0.75 ± 0.13
+96.76 ± 0.15
+-0.46 ± 0.08
+97.01 ± 0.08
+-0.31 ± 0.05
+MiniImageNet
+61.78 ± 1.94
+-1.01 ± 0.58
+63.46 ± 0.85
+0.50 ± 0.39
+62.57 ± 1.32
+0.42 ± 0.57
+62.78 ± 1.00
+0.45 ± 0.35
+Mixture
+83.54 ± 1.15
+-0.80 ± 1.10
+82.45 ± 0.49
+-0.74 ± 0.16
+83.33 ± 0.31
+-0.98 ± 0.23
+83.62 ± 0.26
+-0.33 ± 0.14
+Table 16: Comparison of average accuracy and forgetting with TRGP within a fixed network capacity. We modify TRGP as
+TRGP-Reg with similar regularization terms. β indicates the regularization weight. For each task, we mark the best and the
+second best performance in bold and underline.
+Methods
+TRGP-Reg
+ROGO
+β = 1
+β = 5
+β = 50
+ACC (%)
+Ωnew (%)
+ACC (%)
+Ωnew (%)
+ACC (%)
+Ωnew (%)
+ACC (%)
+Ωnew (%)
+CIFAR
+72.01 ± 0.30
+-1.63 ± 0.20
+71.96 ± 0.40
+-1.09 ± 0.37
+72.49 ± 0.09
+-0.69 ± 0.24
+74.04 ± 0.35
+-1.43 ± 0.43
+PMNIST
+76.57 ± 2.69
+-20.69 ± 2.94
+75.50 ± 3.96
+-22.41 ± 4.24
+76.56 ± 3.83
+-21.63 ± 4.13
+94.20 ± 0.11
+-0.09 ± 0.94
+MiniImageNet
+55.81 ± 2.43
+-3.69 ± 2.19
+58.81 ± 2.24
+-2.85 ± 1.54
+22.69 ± 0.39
+0.01 ± 0.07
+63.66 ± 1.24
+-2.44 ± 0.13
+Mixture
+73.31 ± 1.05
+-11.05 ± 1.43
+74.71 ± 0.85
+-9.33 ± 1.05
+17.36 ± 0.86
+-0.01 ± 0.03
+77.91 ± 0.45
+-4.55 ± 0.36
+Moreover, we illustrate the test accuracy over the task sequence in Figure 9. As shown in Figure 9-(a) and Figure 9-(b),
+our ROGO-Exp dominates TRGP on both benchmarks relaxing either 80% or T% of the frozen space under an expansion
+setting. We further compare ROGO with TRGP modified with the same regularization terms. As shown in Figure 9-(c) and
+Figure 9-(d), the performance of TRGP drops significantly constrained in a fixed network capacity on both benchmarks.
+Following GPM (Saha et al., 2021), we conduct experiments on the CIFAR-100 Sup dataset as well. The implementation
+details are included in Appendix B.3 as well. Here we report the accuracy performance in Table 17. Note that the selected
+baselines for this setting all require extra parameters other than GPM and we directly use the results of the baselines
+reported in GPM and TRGP. In accordance with the results on the other four benchmarks discussed in Section 4.2, ROGO
+
+Restricted Orthogonal Gradient Projection for Continual Learning
+(a)
+(b)
+(c)
+Test Accuracy (%) 
+Task ID
+Test Accuracy (%) 
+Task ID
+Task ID
+Test Accuracy (%) 
+Test Accuracy (%) 
+Task ID
+(d)
+Figure 9: Test accuracy after learning each task under an expansion setting on (a) CIFAR-100 Split and (b) PMNIST, and
+within a fixed network capacity on (c) CIFAR-100 Split and (d) PMNIST.
+outperforms GPM and other related baselines by a significant margin. Particularly, we notice ROGO directly obtains superior
+performance than TRGP within a fixed network capacity under this setting.
+Table 17: Results of ACC (%) on CIFAR-100 Sup setting. STL is under a non-incremental setting. All baselines require
+extra network capacity except GPM. We also report the standard deviation result of ROGO for comparison.
+Metric
+Methods
+STL*
+PNN
+DEN
+RCL
+APD
+GPM
+TRGP
+ROGO
+ACC (%)
+61.00
+50.76
+51.10
+51.99
+56.81
+57.72
+58.25
+58.74 ± 0.60
+D. Algorithm
+We present the pseudo-code of our searching strategy here.
+Algorithm 2 Relaxing Space Searching
+Input: gradient {gl
+t}L
+l=1, frozen space {U l
+t−1}L
+l=1 and thresholds {ϵl
+th, γl
+t}L
+l=1
+Output: relaxing space {V l
+t }L
+l=1
+1: for l ∈ 1, ..., L do
+2:
+Construct the significant representation space Rl
+g,t from gradients gl
+t with top-k eigenvectors.
+3:
+V l
+t ← ∅
+4:
+repeat
+5:
+d ← arg min
+d∈U l,c
+t−1
+Θ(d, Rl
+g,t)
+6:
+if Θ(d, Rl
+g,t) ≤ γl
+t then
+7:
+V l
+t ← V l
+t ∪ d
+8:
+U l,c
+t−1 ← U l
+t−1\V l
+t
+9:
+end if
+10:
+until Θ(d, Rl
+g,t) > γl
+t
+11: end for
+
diff --git a/z9FLT4oBgHgl3EQfoS-L/content/tmp_files/load_file.txt b/z9FLT4oBgHgl3EQfoS-L/content/tmp_files/load_file.txt
new file mode 100644
index 0000000000000000000000000000000000000000..7333c8f8acd48a35bdb3af3daa7fb562c56f2597
--- /dev/null
+++ b/z9FLT4oBgHgl3EQfoS-L/content/tmp_files/load_file.txt
@@ -0,0 +1,1803 @@
+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf,len=1802
+page_content='Restricted Orthogonal Gradient Projection for Continual Learning  Zeyuan Yang 1 Zonghan Yang 1 Peng Li 2 Yang Liu 1 2 3 4 5 6  Abstract  Continual learning aims to avoid catastrophic for-  getting and effectively leverage learned experi-  ences to master new knowledge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content=' Existing gradient  projection approaches impose hard constraints on  the optimization space for new tasks to minimize  interference, which simultaneously hinders for-  ward knowledge transfer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content=' To address this issue,  recent methods reuse frozen parameters with a  growing network, resulting in high computational  costs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content=' Thus, it remains a challenge whether we  can improve forward knowledge transfer for gra-  dient projection approaches using a fixed network  architecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content=' In this work, we propose the Re-  stricted Orthogonal Gradient prOjection (ROGO)  framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content=' The basic idea is to adopt a restricted  orthogonal constraint allowing parameters opti-  mized in the direction oblique to the whole frozen  space to facilitate forward knowledge transfer  while consolidating previous knowledge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content=' Our  framework requires neither data buffers nor extra  parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content=' Extensive experiments have demon-  strated the superiority of our framework over sev-  eral strong baselines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content=' We also provide theoretical  guarantees for our relaxing strategy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content=' Introduction  A critical capability for intelligent systems is to continu-  ally learn given a sequence of tasks (Thrun & Mitchell,  1995;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content=' McCloskey & Cohen, 1989).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content=' Unlike human beings,  vanilla neural networks straightforwardly update parame-  ters regarding current data distribution when learning new  tasks, suffering from catastrophic forgetting (McCloskey  & Cohen, 1989;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content=' Ratcliff, 1990;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content=' Kirkpatrick et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content=', 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='  1Department of Computer Science and Technology, Insti-  tute of AI, Tsinghua University 2Institute for AI Industry Re-  search, Tsinghua University 3Beijing National Research Cen-  ter for Information Science and Technology 4Beijing Academy  of Artificial Intelligence 5International Innovation Center of Ts-  inghua University 6Quan Cheng Laboratory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content=' Correspondence  to:  Peng Li <lipeng@air.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='tsinghua.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='cn>, Yang Liu <li-  uyang2011@tsinghua.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='cn>.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='  Work in progress.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='  Continual learning without forgetting has thus gained in-  creasing attention in recent years (Kurle et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content=', 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content=' Ehret  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content=', 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content=' Ramesh & Chaudhari, 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content=' Liu & Liu, 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='  Teng et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content=', 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content=' Moreover, an ideal continual learner  is expected to not only avoid catastrophic forgetting but  also facilitate forward knowledge transfer (Lopez-Paz &  Ranzato, 2017), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content=', leveraging past learning experiences  to master new knowledge efficiently and effectively (Parisi  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content=', 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content=' Finn et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content=', 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content=' Without forward knowledge  transfer, approaches may have limited performance even  with less forgetting (Kemker et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content=', 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='  To mitigate forgetting and facilitate forward knowledge  transfer, replay-based methods (Lopez-Paz & Ranzato,  2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content=' Shin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content=', 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content=' Choi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content=', 2021) stores some  old samples in the memory, and expansion-based meth-  ods (Rusu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content=', 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content=' Yoon et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content=', 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content=' 2019) expand  the model structure to accommodate incoming knowl-  edge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content=' However, these methods require either extra memory  buffers (Parisi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content=', 2019) or a growing network architec-  ture as new tasks continually arrive (Kong et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content=', 2022),  which are always computationally expansive (De Lange  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content=', 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content=' Thus, promoting performance within a fixed  network capacity remains challenging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='  Regularization-  based methods (Kirkpatrick et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content=', 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content=' De Lange et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content=',  2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content=' Aljundi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content=', 2018) penalize the transformation of  parameters regarding the corresponding plasticity via regu-  larization terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content=' Instead of constraining individual neurons  explicitly, recent gradient projection methods (Zeng et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content=',  2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content=' Saha et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content=', 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content=' Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content=', 2021) further con-  strain the directions of gradient update, obtaining superior  performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content=' However, in spite of effectively mitigating  forgetting, the limited optimization space also hinders the  capability of learning new tasks (Zeng et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content=', 2019), resulting  in insufficient forward knowledge transfer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='  Particularly, we illustrate the optimizing process of tradi-  tional gradient projection methods by the black dashed line  in Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content=' As shown in Figure 1, the strict orthogonal  constraint may exclude the global optimum (the red star)  from the optimization space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content=' In other words, constraining  the directions of gradient update fails on the plasticity in  the stability-plasticity dilemma (French, 1997).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content=' TRGP (Lin  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content=', 2022) tackles this problem by allocating new parame-  ters within the selected subspace of old tasks as trust regions,  suffering a similar computational cost burden as expansion-  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='12131v1  [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='LG]  28 Jan 2023  Restricted Orthogonal Gradient Projection for Continual Learning  Task 1  Task 2  SGD  ROGO  (Ours)  Orthogonal  Gradient Projection  (Previous Work)  Frozen  Figure 1: Illustration of our ROGO framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content=' The lines  with arrow denote the gradients on task 2 (pale blue area)  after learning task 1 (pale yellow area).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content=' We indicated the  global optimum by the red star (⋆).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content=' The black straight  line denotes the frozen space, to which gradient projection  methods constrain the gradients to be orthogonal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='  based methods (Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content=', 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content=' Therefore, promoting  forward knowledge transfer within a fixed network capacity  remains a key challenge for gradient projection methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='  To address this challenge, we propose our Restricted Orthog-  onal Gradient prOjection (ROGO) framework to facilitate  forward knowledge transfer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content=' Instead of optimizing in an  orthogonal direction, our method adopts a restricted orthog-  onal constraint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content=' As illustrated by the red line in Figure 1,  our framework allows the parameters to be updated in the di-  rection oblique to the original frozen space, which explores  a larger optimization space and thus obtains better perfor-  mance on new tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content=' Specifically, we design a simple yet  effective strategy to find the critical subspace, namely the  relaxing space, within the frozen space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content=' The parameters are  jointly optimized in the relaxing and original optimization  spaces to facilitate forward knowledge transfer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content=' A compli-  mentary regularization term is introduced to consolidate  previous knowledge as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content=' Extensive experiments on var-  ious continual learning benchmarks demonstrate that our  ROGO framework promotes forward knowledge transfer  and achieves better classification performance compared  with related state-of-the-art approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content=' Moreover, our  framework can also be extended as an expansion-based  method by storing the parameters of the selected relaxing  space, universally surpassing TRGP (Lin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content=', 2022) and  other expansion-based approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content=' We also provide theoret-  ical proof to guarantee the efficiency of our strategy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content=' Related Work  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content=' Gradient Projection Methods  Gradient projection methods constrain the gradients to be  orthogonal to the frozen space constructed by previous  tasks to overcome forgetting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content=' OWM (Zeng et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content=', 2019)  first proposed to modify the gradients upon projector matri-  ces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content=' OGD (Farajtabar et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content=', 2020) further keeps the gradi-  ents orthogonal to the space spanned by previous gradients,  whereas GPM (Saha et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content=', 2021) computes the frozen space  based on old data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content=' NCL (Kao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content=', 2021) combines the idea  of gradient projection and Bayesian weight regularization  to mitigate catastrophic forgetting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content=' In spite of minimizing  backward interference, these approaches suffer poor for-  ward knowledge transfer and lack plasticity (Kong et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content=',  2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content=' TRGP (Lin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content=', 2022) expands the model with  trust regions to achieve better performance on new tasks by  introducing additional scale parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content=' On the contrary,  we focus on facilitating forward knowledge transfer within  a fixed capacity network by optimizing the parameters in  the direction oblique to the frozen space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content=' Regularization-based Methods  Regularization-based methods introduce regularization  terms to the objective function to penalize the modifica-  tion of parameters, requiring neither data buffers nor extra  parameters as gradient projection methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content=' EWC (Kirk-  patrick et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content=', 2017) first proposes to constrain the change  based on approximated importance weights.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content=' HAT (Serra  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content=', 2018) learns task-based hard attention to identify im-  portant parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content=' Other methods, also called parameter-  isolation methods, defy forgetting via freezing the gradient  updates of particular parameters (De Lange et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content=', 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='  PackNet (Mallya & Lazebnik, 2018) iteratively prunes and  allocates parameters subset to incoming tasks, whereas Ku-  mar et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content=' (2021) facilitate inter-task transfer with the Indian  Buffet Process (Griffiths & Ghahramani, 2011).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content=' Instead of  restricting individual parameters update, the main idea of  our approach is constraining the direction of gradients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content=' Other Methods  Replay-based methods (Lopez-Paz & Ranzato, 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='  Chaudhry et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content=', 2018) maintain a complementary mem-  ory for old data, which are replayed during learning new  tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content=' Recent approaches (Chenshen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content=', 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content=' Cong et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content=',  2020) deploy auxiliary deep generative models to synthesize  pseudo data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content=' However, including extra data into the current  task introduces excessive training time (De Lange et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content=',  2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content=' Expansion-based methods (Yoon et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content=', 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content=' 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='  Douillard et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content=', 2022) dynamically allocate new parameters  or modules to learn new tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content=' While these methods face  capacity explosion inevitably after learning a long sequence  of tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content=' In contrast, our method maintains a fixed network  architecture and requires no previous data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content=' Restricted Orthogonal Gradient Projection  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content=' Preliminaries  In a continual learning setting, we consider T tasks arriving  as a sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content=' When learning the current task, the datasets  Restricted Orthogonal Gradient Projection for Continual Learning  of old tasks are inaccessible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content=' We use an L-layer neural  network with fixed capacity, and parameters defined as W =  {W l}L  l=1, where W l denotes the parameters in the l-th layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='  To mitigate catastrophic forgetting, gradient projection meth-  ods construct the representative space, namely the frozen  space, based on previous tasks and optimize the parame-  ters orthogonal to the frozen space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content=' Specifically, for each  task t, Saha et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content=' (2021) obtain the layer-wise represen-  tation space Rl  t by compressing the representation matrix  Hl  t = [hl  t,1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content=', hl  t,Nt], where Nt denotes the number of  samples in task t and hl  t,j denotes the intermediate repre-  sentation of the j-th input.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content=' The constructed representation  spaces of previous tasks are then concatenated to be the  frozen gradient space {U l  t}L  l=1 for future training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='  Ut = {U l  t}L  l=1 = {Rl  1 ∪ · · · ∪ Rl  t}L  l=1  (1)  During training task t, gradients gl  t are layer-wise con-  strained to be orthogonal to U l  t−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content=' Particularly, assuming  Bl  t−1 = [ul  t−1,1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content=', ul  t−1,N] as the total N orthogonal basis  of U l  t−1, GPM (Saha et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content=', 2021) modifies the gradients as:  gl  t = gl  t − ProjU l  t−1(gl  t)  = gl  t − Bl  t−1(Bl  t−1)T gl  t  (2)  In spite of alleviating forgetting, the strict orthogonal con-  straint on parameters hinders the forward knowledge transfer  by the limited optimization space, thus compromising the  performance of new tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content=' TRGP (Lin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content=', 2022) tackles  this problem by selecting old tasks relevant to the current  task and expanding corresponding frozen spaces as the trust  regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content=' The scaled weight projection is further introduced  for memory-efficient updating and storing the parameters  by scaling the basis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content=' Considering that task i is selected as  the trust region, the scaled weight projection is:  ProjSl  i  U l  i(gl  t) = Bl  iSl  i(Bl  i)  T gl  t  (3)  where Sl  i denotes the scale matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content=' For each task t, TRGP  selects several different tasks as the trust regions U l  i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content=' During  the training phase, gradient gl  t is modified as:  gl  t = gl  t − ProjU l  t−1(gl  t) +  �  i  ProjSl  i  U l  i(gl  t)  (4)  The parameters in the trust regions are retrained and the  learned scale matrices are stored in the memory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content=' During the  inference phase, TRGP retrieves the model training on each  task by replacing the parameters with the scale matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='  The parameters W l  t,I used for inference on task t are:  W l  t,I = W l −  �  i  ProjU l  i(W l) +  �  i  ProjSl  i  U l  i(W l)  (5)  However, as tasks come, storing scaling matrices introduces  increasing extra parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content=' Our experiments demonstrate  that TRGP requires around 5,000% amount of the parame-  ters regarding the initial network after 20 tasks on MiniIma-  geNet, see Figure 3-(b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content=' Therefore, we propose to facilitate  forward knowledge transfer within a fixed network capacity  by a restricted orthogonal constraint, allowing parameters  updated in the direction oblique to the whole frozen space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content=' Overview  In this section, we introduce our Restricted Orthogonal  Gradient prOjection (ROGO) framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content=' To better charac-  terize the relationship between the gradient direction and  the frozen space, we first define the angle between a given  space and a vector in Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='  Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content=' (Angle between vector and space) We de-  note the angle between two inputs as Θ(·) and the inner  product between two vectors as ⟨·, ·⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content=' The angle between  a vector v ∈ Rn and a space U n×c ⊂ Rn is defined as the  minimum angle between the given vector v and any unit  vector u ∈ U:  Θ(v, U) = arccos min  u∈U  ⟨v, u⟩  ∥v∥  (6)  Instead of 90◦ in traditional gradient projection methods,  we aim to relax the angle between the frozen space and the  modified gradient to be an acute angle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content=' However, directly  rotating a vector in a high-dimensional space is too flexible  to control.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content=' Hence, we notice the Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='  Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content=' Given the full space R and a random  space U ⊂ R, for any vector v ∈ U ⊥, where U ⊥ = R\\U  denotes the orthogonal complement of U, it can be modified  to be oblique to the given space U in any angle, by scaling  and combining a vector v′ within a selected subspace U ′ ⊆  U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='  According to Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='2, we can manipulate the gradi-  ent direction to be oblique to the entire frozen space U l  t−1  in any target angle by modifying the parameters within an  appropriate subspace of U l  t−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content=' Therefore, for each task t,  instead of directly rotating the gradient, we introduce the  layer-wise relaxing space V l  t ⊆ U l  t−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content=' During the training  phase, we modify the gradients as:  gl  t = gl  t − ProjU l  t−1\\V l  t (gl  t)  (7)  By jointly optimizing the parameters within the selected  relaxing space V l  t and the original optimization space, the  strict orthogonal constraint is relaxed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content=' Hence, with a null  relaxing space, namely dim(V l  t ) = 0, our framework works  exactly the same as GPM, under the orthogonal constraint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='  Therefore, our framework can be viewed as a generalization  of GPM by regulating the relaxing space V l  t .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content=' The relaxing  space selection thus becomes the key procedure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='  In Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='3, we introduce our searching strategy for the  relaxing space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content=' Moreover, given the relaxing space V l  t , we  Restricted Orthogonal Gradient Projection for Continual Learning  involve complementary regularization loss to better consol-  idate previous knowledge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content=' The detailed procedure is pro-  vided in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content=' For better illustration, we provide the  procedure of our ROGO framework in Algorithm 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content=' Besides,  we propose ROGO-Exp in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='5, involving additional  parameters as expansion-based methods, to further validate  the flexibility and effectiveness of our framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='  Algorithm 1 Restricted orthogonal gradient projection  1: Initiate the frozen spaces U0 = {U l  0}L  l=1 as ∅s and  optimize W1 for task 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='  2: Compute frozen space U1 with representation matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='  3: for t ∈ 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content=', T do  4:  Initiate the relaxing space {V l  t }L  l=1 as ∅s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='  5:  repeat  6:  Fine-tune for predefined et epochs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='  7:  Determine additional relaxing space {V l,new  t  }L  l=1  by the searching strategy in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='  8:  V l  t ← V l  t ∪ V l,new  t  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='  9:  until V l,new  t  is ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='  10:  Optimize {W l  t}L  l=1 within the space orthogonal to  U l  t−1\\V l  t with the regularization terms in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='  11:  Determine the representation space {Rl  t}L  l=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='  12:  Update the frozen space by U l  t = U l  t−1 ∪ Rl  t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='  13: end for  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content=' Searching the Relaxing Space  To determine the relaxing space V l  t , we first construct the  representation space Rl  g,t by the top-k principal eigenvec-  tors of the representative matrix [gl  t,1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content=', gl  t,Nt], where gl  t,j  denotes the gradient of the j-th input.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content=' The estimated impor-  tance is further characterized by the angle from Rl  g,t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content=' Here  we define that a vector d is relaxable when:  Θ(d, Rl  g,t) ≤ γl  t  (8)  where γl  t is a predefined threshold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content=' For task t, we aim to  find the relaxing space V l  t ⊆ U l  t−1 spanned all by relaxable  vectors in U l  t−1, namely:  �  �  �  �  �  max  u∈V l  t  Θ(u, Rl  g,t) ≤ γl  t  min  v∈U l,c  t−1  Θ(v, Rl  g,t) > γl  t  (9)  where U l,c  t−1 = U l  t−1\\V l  t denotes the complemented sub-  space of V l  t with respect to U l  t−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content=' Criterion (9) guarantees  that all vectors in V l  t are relaxable and there remains none  relaxable vector in U l,c  t−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content=' However, it is hard to construct  the appropriate V l  t directly from the high-dimensional U l  t−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='  Therefore, we propose a simple yet efficient strategy to  find V l  t .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content=' Initiating V l  t as ∅, we iteratively select the clos-  est vector to Rl  g,t within U l,c  t−1 by arg minΘ(d, Rl  g,t) and  then append it into V l  t as the basis if it is relaxable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content=' This  procedure is repeated until no relaxable vector is left, thus  obtaining the target V l  t consisting of all relaxable vectors  within U l  t−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content=' The pseudo-code of our searching strategy is  provided in Algorithm 2 in Appendix D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='  Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content=' Denote S = {u|Θ(u, Rl  g,t) ≤ γl  t and u ∈  U l  t−1} as the whole solution set, the obtained relaxing space  V l  t takes up the maximum space in S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='  To further substantiate the effectiveness of our searching  strategy, here we introduce Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content=' According to  Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='3, our strategy guarantees to find the maximum  space within the whole solution set satisfying criterion (9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='  The detailed proof is provided in Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='  Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content=' Denote kr as the dimension of the represen-  tation space Rl  g,t and kv as the dimension of the relaxing  space V l  t , kv ≤ kr, regardless of the frozen space U l  t−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='  Moreover, we provide theoretical analysis on the upper  bound of the dimension of V l  t , which is also the number of  iterations, to investigate the efficiency of our strategy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content=' Here  we introduce Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='4, which guarantees that the dimen-  sion of V l  t is no more than of the representation space Rl  g,t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='  In other words, the number of iterations is bounded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content=' Hence,  as Rl  g,t is constructed by the top-k principal eigenvectors,  the dimension of V l  t is further regulated by hyperparameters,  of which the ablation study is provided in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content=' We  also include the detailed proof in Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content=' Constrained Update in the Relaxing Space  Given the selected relaxing space V l  t , the optimization space  is enlarged, thus facilitating forward knowledge transfer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='  In the meanwhile, we also want to consolidate previous  knowledge stored within V l  t .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content=' One direct way is to fine-tune  the parameters with additional regularization terms such as  EWC (Kirkpatrick et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content=', 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content=' However, regularization  terms are designed for explicit parameters, which are not  applicable to implicit subspace in our framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content=' Therefore,  we borrow the scaled weight projection (Lin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content=', 2022) to  modify explicit parameters instead.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content=' With the scale matrix  Sl  t, the gradient gl  t is modified as:  gl  t = gl  t − ProjU l  t−1(gl  t) + ProjSl  t  V l  t (gl  t)  (10)  Notice that we update the parameters within Vl  t with Sl  t  after training each task, instead of storing the parameters  for inference as TRGP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content=' Parameters within Vl  t are then re-  strictedly fine-tuned by adding regularization terms on Sl  t of  instead of directly on parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content=' Specifically, the objective  function of task t is:  Lt = L(Wt, D(t)) +  L  �  l=1  βl∥Sl  t − 1(Sl  t)∥2  2  (11)  where 1(·) denotes the identity matrix with the same shape  as the input matrix and βl is the weight of the regulariza-  Restricted Orthogonal Gradient Projection for Continual Learning  tion term for layer l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content=' During backpropagation, gradients  within the relaxing space V l  t are regulated by Sl  t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content=' To further  validate our strategy, we also equip TRGP with the above  regularization terms for better comparison in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='  Generally, in our framework, we adopt our searching strat-  egy to determine the relaxing space and modify those pa-  rameters with constraints on the scaling matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='  However, during training, the direction of gradients shifts  sharply and frequently due to the steep learning scope of  deep neural networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content=' Diverse subspaces would be selected  in different training phases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content=' Therefore, we iteratively ex-  amine whether there remains relaxable vectors within the  remaining frozen space U l,c  t−1 after limited epochs and then  search for additional relaxing space V l,new  t  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content=' If additional  relaxing space V l,new  t  is involved, we expand the scaling  matrix with identity matrices to accommodate the increased  space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content=' On the contrary, TRGP maintains a fixed-size scaling  matrix throughout training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content=' As the U l  t−1 is fixed for each  task, the number of iterations of our strategy is naturally  limited.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content=' In the implementation, we further constrain the  maximum number of iterations for efficiency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='  In general, by iteratively searching for the relaxing space  and modifying those gradients with additional regularization  loss, our restricted orthogonal method optimizes the param-  eters in the direction oblique to the whole frozen space, thus  facilitating forward knowledge transfer while consolidating  previous knowledge within a fixed network capacity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content=' Extensions  To further validate our framework, we propose ROGO-Exp,  a modified version of our proposed ROGO, directly stor-  ing the parameters within the relaxing space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content=' Similar to  TRGP, for each task t, we retrieve the corresponding relax-  ing space {V l  t }L  l=1 and the scale matrices {Sl  t}L  l=1 during  the inference phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content=' The modified parameters W l  t,I used for  inference on task t are:  W l  t,I = W l − ProjV l  t (W l) + ProjSl  t  V l  t (W l)  (12)  where W l denotes the parameters of layer l of current net-  work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content=' By replacing the parameters in the relaxing space  with the parameters optimized in task t, the model achieves  better performance, at the price of expansive computational  cost in long task sequences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content=' Further experimental results  validate the efficiency of our ROGO-Exp against state-of-  the-art expansion-based methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content=' Experiments  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content=' Experimental Setup  Datasets: Following Saha et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content=' (2021), we evaluate our  framework on CIFAR-100 Split (Krizhevsky & Hinton,  2009), MiniImageNet (Vinyals et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content=', 2016), Permuted  MNIST (PMNIST) (Kirkpatrick et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content=', 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content=' Moreover, we  introduce the Mixture benchmark, first proposed by (Serra  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content=', 2018), consisting of eight datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content=' In this work, we  conduct experiments on Mixture with the seven tasks as a  sequence except for TrafficSigns1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content=' Details and statistics of  the datasets can be found in Appendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content=' Moreover, we  include the details of network architectures in Appendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='  Baselines:  We compare ROGO with competitive and  well-established methods within a fixed network capac-  ity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content=' For regularization-based methods, we compare against  EWC (Kirkpatrick et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content=', 2017) and HAT (Serra et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content=', 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='  For gradient projection methods, we consider OGD (Fara-  jtabar et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content=', 2020) and GPM (Saha et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content=', 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content=' For replay-  based methods, we adopt ER Res (Chaudhry et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content=', 2019)  and A-GEM (Chaudhry et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content=', 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content=' The memory buffer  size for PMNIST, CIFAR-100 Split, MiniImageNet, and  Mixture are 1,000, 2,000, 500 and 3,000, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content=' More-  over, we consider TRGP (Lin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content=', 2022) as a competitive  approach under an expansion setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content=' Implementation de-  tails are listed in Appendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='  Metrics:  Denote Ai,j as the test accuracy of task j after learning task  i and bi as the test accuracy of task i at random initialization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='  We employ the following three evaluation metrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='  Average Accuracy (ACC) (Mirzadeh et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content=', 2020): the  average test accuracy evaluated after learning all tasks,  defined as 1  T  �T  i=1 AT,i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='  Backward Transfer (BWT) (Lopez-Paz & Ranzato,  2017): the average accuracy decrease after learning  all tasks, defined as  1  T −1  �T −1  i=1 (AT,i − Ai,i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='  Forward Transfer (Ωnew) (Kemker et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content=', 2018):  the average accuracy of new tasks, defined as  1  T −1  �T  i=2(Ai,i − bi).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content=' As bi stays still across different  approaches, we consider  1  T −1  �T  i=2 Ai,i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='  Besides, FWT (Lopez-Paz & Ranzato, 2017) reflects the  influence of the observed tasks on new tasks in a zero-shot  manner, defined as  1  T −1  �T  i=2(Ai−1,i − bi).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content=' In this paper,  we mainly focus on Ωnew, while FWT results are also pro-  vided.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content=' Detailed definitions are provided in Appendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content=' Main Results  We show the comparative results on four benchmarks in  Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content=' The accuracy results of major baselines are adopted  from GPM (Saha et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content=', 2021) and we implement all the  baselines to get the forward knowledge transfer and for-  getting results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content=' Implementation details are provided in Ap-  pendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content=' We run each experiment five times and report  1We fail to obtain TrafficSigns, as all the links provided in (Stal-  lkamp et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content=', 2011;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content=' Serra et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content=', 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content=' Saha et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content=', 2021) are expired.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='  Restricted Orthogonal Gradient Projection for Continual Learning  Table 1: Comparison of average final accuracy ACC and forward knowledge transfer Ωnew.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content=' ⋆ denotes under the non-  incremental setting and † denotes requiring an extra data buffer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content=' For each task, we mark the best and the second best  performance in bold and underline respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content=' All results reported are averaged over 5 runs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='  Method  CIFAR-100 Split  MiniImageNet  PMNIST  Mixture  ACC (%)  Ωnew (%)  ACC (%)  Ωnew (%)  ACC (%)  Ωnew (%)  ACC (%)  Ωnew (%)  Multitask⋆  79.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='58 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='54 69.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='46 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='62 96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='70 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='02 81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='29 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='23 A-GEM†  63.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='98 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='22  77.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='48 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='40  57.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='24 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='72  67.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='55 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='20  83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='56 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='16  97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='42 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='03  59.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='86 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='01  85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='87 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='42  ER Res†  71.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='73 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='63  77.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='13 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='18  58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='94 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='85  69.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='48 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='42  87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='24 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='53  97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='37 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='05  75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='07 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='55  86.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='35 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='22  EWC  68.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='80 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='88  69.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='87 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='09  52.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='01 ± 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='53  63.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='45 ± 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='80  89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='97 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='57  93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='13 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='70  69.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='62 ± 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='69  74.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='41 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='00  HAT  72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='06 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='50  71.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='50 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='62  59.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='78 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='57  62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='63 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='63 77.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='54 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='18  79.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='26 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='15  OGD  70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='96 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='32  73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='86 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='57  59.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='83 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='62  62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='02 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='24  82.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='56 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='66  97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='40 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='06  74.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='38 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='27  81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='97 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='53  GPM  72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='48 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='40  71.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='53 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='54  60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='41 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='61  61.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='63 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='60  93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='91 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='16  96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='56 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='04  77.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='49 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='68  82.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='13 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='37  ROGO  74.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='04 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='35  75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='22 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='36  63.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='66 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='24  63.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='81 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='39  94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='20 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='11  96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='26 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='10  77.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='91 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='45  82.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='21 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='42  (a)  (b)  (c)  Test Accuracy (%)  Task ID  Test Accuracy (%)  Task ID  �  Test Accuracy (%)  Test Accuracy (%)  Task ID  (d)  Figure 2: Results on CIFAR-100 Split setting: (a) average accuracy after learning each task;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content=' (b) accuracy evolution of a  randomly selected task;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content=' (c) accuracy tested on task i after learning task i;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content=' (d) average accuracy of different ϵ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content=' The optimum ϵ  value of GPM is annotated by a red circle “o”, and the shaded area indicates the standard deviation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='  the mean results and the standard deviation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content=' Other results  including the forgetting are provided in Appendix C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='  According to Table 1, ROGO dominates EWC across all  benchmarks and generally outperforms HAT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content=' Although HAT  obtains comparable accuracy on Mixture, ROGO gains 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='7%  better Ωnew on average.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content=' For gradient projection methods,  compared with GPM, ROGO achieves over 1% and 3%  higher ACC on CIFAR-100 Split and MiniImageNet respec-  tively while improving the forward knowledge transfer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content=' We  observe that OGD+ achieves the better Ωnew on several  datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content=' Whereas, it fails on the accuracy with significant  forgetting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content=' Also, we notice that A-GEM and ER Res gain  better Ωnew, in spite of the inferior accuracy compared with  GPM and especially ROGO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content=' We assume that this is because  replay-based methods explore the full optimization space  with extra data, while both regularization-based and gradi-  ent projection methods constrain the optimization space to  mitigate forgetting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content=' Other results including BWT and FWT  are provided in Appendix C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content=' In general, ROGO obtains  the best accuracy and improves forward knowledge transfer  without extra data buffers across all datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='  Moreover, we compare the average accuracy after learn-  ing each task on CIFAR-100 Split with GPM, as GPM  achieves the highest accuracy among selected baselines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content=' As  shown in Figure 2-(a), ROGO increasingly gains superior  performance by a larger margin, as the optimization space  of GPM is gradually constrained by accumulated frozen  spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content=' We provide detailed results on other benchmarks in  Appendix C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content=' For better illustration, we also exhibit the ac-  curacy evolution of specific tasks during sequential training  in Figure 2-(b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content=' Here we choose the second task on CIFAR-  100 Split.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content=' According to the results, ROGO universally out-  performs GPM over the task sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content=' Results of three ran-  domly selected tasks are provided in Appendix C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='4 respec-  tively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content=' In addition, we observe the accuracy on new tasks  in Figure 2-(c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content=' Similar to the average accuracy, ROGO  achieves increasingly better Ωnew by relaxing the strict or-  thogonal constraint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content=' Results on other benchmarks included  in Appendix C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='3 further substantiate this phenomenon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='  To understand our relaxing strategy better, we further con-  duct experiments on different thresholds ϵ, which regulate  the dimension of the frozen space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content=' Saha et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content=' (2021) argue  that ϵ mediates the stability-plasticity dilemma by control-  ling the frozen space and thus is critical for GPM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content=' However,  ROGO enables the frozen space to be adaptively relaxed  regarding the current task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content=' Therefore, ϵ plays a much less  important role in ROGO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content=' We present the performance of  different ϵ on CIFAR-100 Split in Figure 2-(d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content=' As shown  in Figure 2-(d), the performance of GPM drops significantly  when ϵ ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='97, the optimal value reported in GPM, while  ROGO consistently performs well even with ϵ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content=' Gen-  Restricted Orthogonal Gradient Projection for Continual Learning  Table 2: Comparison of average accuracy and forward knowledge transfer with TRGP under an expansion setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content=' The  percentages indicate the ratios of the rank of the relaxing space with respect to the frozen space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content=' For each task, we mark the  best and the second best performance in bold and underline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content=' Detailed results are provided in Appendix C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='  Methods  TRGP  ROGO-Exp  50%  80%  T%  ACC (%)  Ωnew (%)  ACC (%)  Ωnew (%)  ACC (%)  Ωnew (%)  ACC (%)  Ωnew (%)  CIFAR  74.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='46 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='22  75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='01 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='22  74.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='90 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='37  75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='68 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='43  74.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='97 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='30  75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='46 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='17  75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='34 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='61  75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='86 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='42  PMNIST  96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='34 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='11  97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='07 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='14  96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='44 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='20  97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='07 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='10  96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='76 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='15  97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='20 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='08  97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='01 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='08  97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='26 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='05  MiniImageNet  61.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='78 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='94  63.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='08 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='40  63.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='46 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='85  62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='76 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='64  62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='57 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='32  62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='32 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='19  62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='78 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='00  62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='42 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='28  Mixture  83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='54 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='15  84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='88 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='95  82.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='45 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='49  84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='13 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='25  83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='33 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='31  84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='60 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='20  83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='62 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='26  84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='44 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='42  Table 3: Comparison of average accuracy and forward knowledge transfer with TRGP within a fixed network capacity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content=' We  modify TRGP as TRGP-Reg with similar regularization terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content=' β indicates the regularization weight.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content=' For each task, we  mark the best and the second best performance in bold and underline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content=' Detailed results are provided in Appendix C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='  Methods  TRGP-Reg  ROGO  β = 1  β = 5  β = 50  ACC (%)  Ωnew (%)  ACC (%)  Ωnew (%)  ACC (%)  Ωnew (%)  ACC (%)  Ωnew (%)  CIFAR  72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='01 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='30  73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='17 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='24  71.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='96 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='40  72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='64 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='67  72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='49 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='09  72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='67 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='26  74.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='04 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='35  75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='22 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='36  PMNIST  76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='57 ± 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='69  94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='95 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='23  75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='50 ± 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='96  95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='47 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='30  76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='56 ± 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='83  95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='88 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='19  94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='20 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='11  96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='26 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='10  MiniImageNet  55.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='81 ± 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='43  59.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='14 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='00  58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='81 ± 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='24  62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='05 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='74  22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='69 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='39  20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='39 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='36  63.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='66 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='24  63.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='81 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='39  Mixture  73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='31 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='05  83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='64 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='38  74.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='71 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='85  83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='27 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='24  17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='36 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='86  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='27 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='98  77.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='91 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='45  82.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='21 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='42  erally, ROGO is more robust on the threshold ϵ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='  In brief, our approach universally outperforms selected base-  lines without extra data buffers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content=' Achieving better average  accuracy, ROGO also improves forward knowledge trans-  fer by exploring a larger optimization space than GPM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content=' To  validate the efficiency of our relaxing strategy, we further  compare ROGO-Exp with well-established and competitive  expansion-based methods in the next section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content=' Comparison with Expansion-based Methods  The above experiments exhibit the superiority of our ap-  proach when maintaining a fixed network capacity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content=' How-  ever, under the scenario with no limit on the network capac-  ity, expansion-based methods achieve great performance by  allocating new neurons or modules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content=' Therefore, to further  validate our strategy, we compare ROGO and ROGO-Exp  with relative expansion-based methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='  In this section, we adopt TRGP (Lin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content=', 2022), which  expands the optimization space by retraining parameters  within the selected trust regions, achieving superior perfor-  mance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content=' In the inference phase, TRGP reuses the parameters  in corresponding trust regions memorized after learning this  task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content=' In contrast, GPM and our ROGO only store the repre-  sentation of the frozen space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content=' Therefore, although indeed  a stable network capacity is allocated for each task, the en-  tire memory size of TRGP grows continually.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content=' As shown in  Figure 3-(b), after learning the last task on MiniImageNet,  TRGP requires around 5,000% extra parameters with re-  spect to the network capacity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content=' Results on other benchmarks  provided in Appendix C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='6 further substantiate that TRGP  introduces a significant number of extra parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content=' Thus,  we categorize TRGP as an expansion-based method here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='  In this setting, the main difference between ROGO-Exp and  TRGP is the strategy of deciding which part of the frozen  space to reuse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content=' We conduct experiments on all four bench-  marks and report the results in Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content=' The percentages  indicate the ratios of the rank of the relaxing space with  respect to the corresponding frozen space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content=' We evaluate two  constant ratios and further use the ratios in TRGP, denoted  as T%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content=' As TRGP selects the top 2 tasks as the trust regions,  T% is larger than 80% at most times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content=' According to Table 2,  IRGP-Exp universally outperforms TRGP, even by relaxing  only 50% of the frozen space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content=' Moreover, we observe that  ROGO gains superior performance over both TRGP and  ROGO-Exp on MiniImageNet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content=' We assume that the reason  is the positive backward knowledge transfer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content=' Training the  network on new tasks also benefits the previous task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content=' In  General, our approach achieves better performance with a  comparable size of the relaxing space, which substantiates  the efficiency of our searching strategy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='  We further modify TRGP as TRGP-Reg with similar reg-  ularization terms on the scale matrices as our ROGO to  compare the relaxing strategies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content=' We report the results on  four benchmarks with three representative regularization  weights β on TRGP-Reg in Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content=' As shown in Table 3,  ROGO significantly outperforms TRGP-Reg, especially on  PMNIST, gaining around 20% ACC improvement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content=' Gener-  ally, ROGO achieves better or comparable Ωnew than TRGP  under or without the constraint of a fixed network capacity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='  Restricted Orthogonal Gradient Projection for Continual Learning  Table 4: Ablation study of ζ = cos γ and β on CIFAR100-Split, where γ is the threshold for the relaxing strategy and β is  the regularization weight.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content=' ζconv and ζfc denote the threshold for convolutional and fully connected layers, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='  (a) Ablation study of ζ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content=' For each ζ combination, we report the relaxing ratio of each layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='  ζcong  ζfc  Conv1  Conv2  Conv3  Fc1  Fc2  ACC  BWT  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
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+page_content='87%  50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
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+page_content='83%  Ratio (%)  (a)  (b)  Ratio (%)  Task ID  Task ID  Figure 3: (a) Relaxing ratios of the last layer on CIFAR-  100 Split, MiniImageNet, and PMNIST.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content=' (b) Ratios of the  amount of extra parameters concerning the amount of the  parameters of the network architecture on MiniImageNet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='  Detailed results are included in Appendix C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content=' Analysis and Discussion  To gain a deeper insight into ROGO, we investigate the  trend of scales of the space relaxed by our strategy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content=' With the  theoretical upper bound of the rank of the relaxed subspace  provided in Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='4, we further inspect the ratios of  the relaxing spaces concerning corresponding frozen spaces  in practice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content=' Results of the last layer on three different set-  tings are provided in Figure 3-(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content=' As shown in Figure 3-(a),  relaxing ratios generally maintain a stable trend, fluctuating  smoothly within a small range over sequential tasks on both  benchmarks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content=' As different tasks explore different optimiza-  tion directions, ideal relaxing spaces vary across tasks, in  accordance with the fluctuation of our results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content=' To further  investigate our framework, we provide the ablation study of  related hyper-parameters on CIFAR-100 Split in Table 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='  Ablation study of the relaxing ratios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content=' ROGO mediates the  stability-plasticity dilemma by controlling the dimension  and flexibility of the relaxing space by γ in Equation (9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='  For better illustration, here we represent γ by ζ = cos γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='  We first conduct an ablation study of ζ, reporting both the  performance and the relaxing ratios in Table 4(a), where  ζconv denotes the hyper-parameter for convolutional layers  and ζfc denotes the hyper-parameter for fully connected  layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content=' As shown in Table 4(a), increasing ζ gradually  constrains the scale of the relaxing space, thus alleviating the  forgetting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content=' On the other hand, a small ζ guarantees sufficient  forward knowledge transfer, while fails on consolidating  previous knowledge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content=' Generally, our method persists in a  superb performance with ζ ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='  Ablation study of the regularization weights.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content=' Moreover,  we observe the performance of ROGO with various regular-  ization weights β in Table 4(b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content=' In this work, we adopt a  consistent β for the entire network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content=' According to Table 4(b),  Similarly, we observe less forgetting on larger β, which con-  strains the update of parameters within the relaxing space  more strictly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content=' However, strict constraints also lead to limited  performance on new tasks as discussed in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content=' In a  nutshell, ζ and β operate together in ROGO to overcome  catastrophic forgetting and enhance forward transfer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='  Training time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content=' The dimension of the frozen spaces keeps  growing as the tasks accumulate, leading to expanding  range for searching relaxing spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content=' Therefore, the compu-  tation complexity and time consumption are supposed to  increase gradually.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content=' We further reported the time consump-  tion comparison on CIFAR-100 Split and MiniImageNet in  Appendix C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content=' According to Table 14, ROGO takes around  50% more time than GPM, similar to TRGP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content=' In general, the  practical efficiency of our approach is acceptable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content=' Conclusion  In this paper, we proposed ROGO, a novel continual learn-  ing framework that facilitates forward knowledge transfer  in gradient projection methods within a fixed network ca-  pacity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content=' ROGO imposes a restricted orthogonal constraint  allowing parameters to be updated in the direction oblique  to the whole frozen space, which thus explores a larger op-  timization space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content=' Extensive experiments demonstrate that  our ROGO framework surpasses related state-of-the-art ap-  proaches on diverse benchmarks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content=' Moreover, we proposed  ROGO-Exp that allows network expansion with the relax-  ing space, which achieves better performance than related  expansion-based methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content=' We also provided theoretical  analysis validating the efficiency of our algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
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+page_content=', Lu, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content=', Ram, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content=',  and Horesh, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content=' Overcoming catastrophic forgetting via  direction-constrained optimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content=' In European Confer-  ence on Machine Learning and Principles and Practice  of Knowledge Discovery in Databases, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='  Thrun, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
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+page_content='  Robotics and autonomous systems, 15(1-2):25–46, 1995.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
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+page_content=' Advances in  neural information processing systems, 29, 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='  Wang, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
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+page_content=' Training networks in  null space of feature covariance for continual learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content=' In  Proceedings of the IEEE/CVF Conference on Computer  Vision and Pattern Recognition, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
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+page_content='  Xiao, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content=', Rasul, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content=', and Vollgraf, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content=' Fashion-mnist: a  novel image dataset for benchmarking machine learning  algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content=' arXiv preprint arXiv:1708.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='07747, 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='  Yoon, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content=', Yang, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content=', Lee, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content=', and Hwang, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content=' Lifelong  learning with dynamically expandable networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content=' arXiv  preprint arXiv:1708.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='01547, 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='  Yoon, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content=', Kim, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content=', Yang, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content=', and Hwang, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content=' Scalable and  order-robust continual learning with additive parameter  decomposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content=' arXiv preprint arXiv:1902.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='09432, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='  Zeng, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content=', Chen, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content=', Cui, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content=', and Yu, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content=' Continual learning of  context-dependent processing in neural networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content=' Nature  Machine Intelligence, 1(8):364–372, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='  Restricted Orthogonal Gradient Projection for Continual Learning  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content=' Proof  In this section, we sequentially provide the proof of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='3 and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content=' For better comprehension, we first introduce  complementary Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content=' For simplification, all vectors here are assumed to be unit vectors, namely ∥v∥ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content=' Proof of Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='1  Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content=' Denote the relaxed space as V = span{v1, v2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content=', vn}, where vn is the last base included in V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content=' Given  representation space U, ∀v ∈ V , we have Θ(v, U) ≤ Θ(vn, U).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='  Depicting the angle by projection form, Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='1 can be expressed as:  ∀v ∈ V, ∥ProjU(v)∥ ≥ ∥ProjU(vn)∥  (13)  Denote B = [u1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content=', um] as the representative matrix of the representation space U = span{u1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content=', um}, where ui is the  i-th normalized orthogonal base of U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content=' Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='1 can be further expressed as:  ∀v ∈ V, vT BBT v ≥ vT  n BBT vn  (14)  As vis are the basis sequentially appended by our searching strategy, for any i ≤ j, we have:  vT  i BBT vi ≥ vT  j BBT vj  (15)  Consider a special case where the current relaxed subspace has only one base, denoted by V = span{v1}, namely n = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='  Obviously, Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='1 is true in this case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content=' Thus, we consider the general case that n ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='  We provide proof by inductive reasoning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content=' Note that as we assume all vectors to be unit vectors, we have �  i ∥wi∥2  2 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='  First, we consider a special case V = span{v1, v2}, namely n = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content=' Assume there exists v = w1v1 + w2v2 that  vT BBT v < vT  2 BBT v2, with w2  1 + w2  2 = 1, we have:  w2  1vT  2 BBT v2 > w2  1vT  1 BBT v1 + 2w1w2vT  1 BBT v2  (16)  Construct v′ = w2v1 − w1v2, we have:  v′T BBT v′ = w2  2vT  1 BBT v1 + w2  1vT  2 BBT v2 − 2w1w2vT  1 BBT v2  > w2  2vT  1 BBT v1 + w2  1vT  2 BBT v2 + w2  1vT  1 BBT v1 − w2  1vT  2 BBT v2  = vT  1 BBT v1  (17)  which contradicts Θ(v1, U) ≤ Θ(v, U) for ∀v ∈ span{v1, v2}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='  Then we consider the general case V = span{v1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content=', vt}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content=' For ∀v ∈ V , we have Θ(v, U) ≤ Θ(vt, U).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content=' After vt+1 is  included, we assume that there exists v ∈ V that Θ(v, U) > Θ(vt+1, U).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content=' Then we can find the maximum s satisfying  that there exists v = �s  i=1 wivi + wt+1vt+1 that Θ(v, U) > Θ(vt+1, U) and for ∀v′ = �s−1  i=1 wivi + wt+1vt+1, we have  Θ(v′, U) ≤ Θ(vt+1, U).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content=' When s = 1, it is similar to the special case, so the proof is omitted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content=' Thus, we consider the case  where s ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content=' For simplification, we express v as v = c0v0 + c1vs + c2vt+1 with v0 = �s−1  i=1 aivi, where cis and ais are  coefficients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content=' We have:  (c0v0 + c1vs + c2vt+1)T BBT (c0v0 + c1vs + c2vt+1) < vT  t+1BBT vt+1  (18)  As Θ(w1v0 + w2vs,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content=' U) ≤ Θ(vs,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content=' U) ≤ Θ(vt+1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content=' U),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content=' we have:  (w1v0 + w2vs)T BBT (w1v0 + w2vs) ≥ vT  s BBT vs  (19)  Restricted Orthogonal Gradient Projection for Continual Learning  which is:  w2  1vT  0 BBT v0 + 2w1w2vT  0 BBT vs ≥ w2  1vT  s BBT vs ≥ w2  1vT  t BBT vt  (20)  Similarly we have:  w2  1vT  0 BBT v0 + 2w1w2vT  0 BBT vt ≥ w2  1vT  t BBT vt  (21)  Then we can express Equation (18) as:  vT  t+1BBT vt+1 > (c2  0 + c2  2)vT  t BBT vt + c2  1vT  s BBT vs + 2c1c2vT  s BBT vt  (22)  As ∥v∥ = ∥vi∥ = 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content=' c2  0 + c2  1 + c2  2 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content=' Then we have:  −2c1c2vT  s BBT vt > c2  1vT  s BBT vs − c2  1vT  t+1BBT vt+1  (23)  Construct v′ = c2vs−c1vt+1  √  c2  1+c2  2  , we have:  v′T BBT v′ =  1  c2  1 + c2  2  (c2  2vT  s BBT vs + c2  1vT  t+1BBT vt+1 − 2c1c2vT  s BBT vt)  >  1  c2  1 + c2  2  (c2  2vT  s BBT vs + c2  1vT  t+1BBT vt+1 + c2  1vT  s BBT vs − c2  1vT  t+1BBT vt+1)  = vT  s BBT vs  (24)  which contradicts Θ(vs, U) ≤ Θ(v, U) for ∀v ∈ span{vs, vs+1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content=', vt, vt+1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='  Thus, for ∀v ∈ V = span{v1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content=', vn}, we have Θ(v, U) ≤ Θ(vn, U).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content=' Proof of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='3  Here we provide proof of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content=' Denote the whole solution set as S = {u|Θ(u, U) ≤ γ and u ∈ U f} where U is  the representation space, U f is the frozen space and γ is the threshold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='3 can be expressed as that all subspace  V ′ ⊆ S satisfies dim(V ′) ≤ dim(V ), where V is the relaxing space obtained by our searching strategy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='  For the sake of contradiction, we assume there exists V ′ ⊆ S that dim(V ′) > dim(V ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content=' Denote V c  f as the orthogonal  complement of V with respect to the frozen space U f, as the relaxing space is a subspace of the frozen space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content=' According to  Lemmoa A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='1, for ∀v ∈ V c  f , we have Θ(v, U) > γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content=' We also have:  dim(V ′ ∩ V c  f ) = dim(V ′) + dim(V c  f ) − dim(V ′ + V c  f ) > 0  (25)  Thus, there exists v′ ∈ V ′ that v′ ∈ U c  f, namely there exists v′ ∈ S that Θ(v′, U) > γ, which is contradict.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content=' Therefore, the  relaxing space obtained by our searching strategy takes up the maximum subspace of the whole solution set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content=' Proof of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='4  Here we provide proof of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='  Denote the relaxing space and the representation space as V and U =  span{u1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content=', ukr} respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content=' With Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='1, we have ∀v ∈ V , Θ(v, U) ≤ Θ(vkv, U) <  π  2 , where kv denotes  dim(V ), the dimension of V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content=' In other words,  ∀v ∈ V, v ̸⊥ U  (26)  Restricted Orthogonal Gradient Projection for Continual Learning  Similar to Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='4, assume the dimension of V is larger than kr, namely dim(V ) > dim(U) = kr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content=' Denote U c as the  orthogonal complement of U with respect to the whole space R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content=' Obviously, dim(U c) = n − kr, where n is the dimension  of R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content=' Then we have:  dim(V ∩ U c) = dim(V ) + dim(U c) − dim(V + U c)  > kr + (n − kr) − n = 0  (27)  Thus, there exists v′ ∈ V such that v′ ∈ U c too.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content=' As U c is the orthogonal complement, ∀u ∈ U c, u ⊥ U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content=' Then we have  v′ ⊥ U, which contradicts Equation (26).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content=' Therefore, the assumption is aborted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content=' We have that kv ≤ kr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content=' The upper bound of  the dimension of V is kr, namely the dimension of the representation space U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content=' Experimental Setup  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content=' Datasets  Here we introduce the datasets we use for evaluation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content=' 1) CIFAR-100 Split Saha et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content=' (2021) constructed CIFAR-100  Split, by splitting CIFAR100 (Krizhevsky & Hinton, 2009) into 10 tasks where each task has 10 classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content=' 2) MiniImageNet  Following Saha et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content=' (2021), we split MiniImageNet (Vinyals et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content=', 2016) into 20 sequential tasks with 5 classes each.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='  3) Permuted MNIST (PMNIST) PMNIST (Kirkpatrick et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content=', 2017) is a variant of MNIST (LeCun et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content=', 1998) where  each task has a different permutation of inputting images, consists of 10 sequential tasks with 10 classes each.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content=' 4) Mixture  Serra et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content=' (2018) first proposed Mixture consisting of 8 datasets, including CIFAR-10 (Krizhevsky & Hinton, 2009),  MNIST (LeCun et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content=', 1998), CIFAR-100 (Krizhevsky & Hinton, 2009), SVHN (Netzer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content=', 2011), FashionMNIST (Xiao  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content=', 2017), TrafficSigns (Stallkamp et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content=', 2011), FaceScrub (Ng & Winkler, 2014), and NotMNIST (Bulatov, 2011), from  which Ebrahimi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content=' (2020) further constructed 5-Datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content=' Here we follow the original harder benchmark.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content=' Particularly, we  consider all tasks as a sequence except TrafficSigns (Stallkamp et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content=', 2011), which we failed to access.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content=' 5) CIFAR-100 Sup  In addition, following Yoon et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content=' (2019), we adopt CIFAR-100 Sup consisting of 20 superclasses as sequential tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content=' Here  we adopt the five different predefined orders proposed by Yoon et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content=' (2019) and report the main results in Appendix C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='  Among all evaluated datasets, PMNIST is a benchmark under the domain-incremental scenario, while other four datasets are  under the task-incremental scenario.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='  Moreover, we provide the statistics of selected datasets in Table 5 and Table 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content=' For the Mixture benchmark, the images of  MNIST, FashionMNIST, and notMNIST are replicated across all RGB channels following Serra et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content=' (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='  Table 5: Statistics of CIFAR-100 Split, MiniImageNet, and PMNIST.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='  CIFAR-100 Split  CIFAR-100 Sup  MiniImageNet  PMNIST  Image Size  32 × 32  32 × 32  84 × 84  28 × 28  Channels  3  3  3  1  Classes  100  100  100  10  Tasks  10  20  20  10  Classes/task  10  5  5  10  Training Samples/task  4,750  2,375  2,375  54,000  Validation Samples/task  250  125  125  6,000  Testing Samples/task  1,000  500  500  10,000  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content=' Model Details  MLP architecture: We adopt a 3-layer model including two hidden layers with 100 neurons each for the PMNIST setting,  the same as Lopez-Paz & Ranzato (2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content=' ReLU is used as the activate function here and for all other architectures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content=' Also,  we use softmax with cross entropy loss on all settings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='  AlexNet architecture: For CIFAR-100 Split setting, we adopt the same network as Serra et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content=' (2018) with batch normal-  ization, including two fully connected layers and three convolutional layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content=' The convolutional layers have 4 × 4, 3 × 3, and  2 × 2 kernel sizes with 64, 128, and 256 filters respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content=' After each convolutional layer, we add batch normalization and  Restricted Orthogonal Gradient Projection for Continual Learning  Table 6: Statistics of Mixture benchmark.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='  Dataset  Classes  # Taining  # Validation  # Testing  CIFAR-10 (Krizhevsky & Hinton, 2009)  10  47,500  2,500  10,000  MNIST (LeCun et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content=', 1998)  10  57,000  3,000  10,000  CIFAR-100 (Krizhevsky & Hinton, 2009)  100  47,500  2,500  10,000  SVHN (Netzer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content=', 2011)  10  69,595  3,662  26,032  FashionMNIST (Xiao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content=', 2017)  10  57,000  3,000  10,000  FaceScrub (Ng & Winkler, 2014)  100  19,570  1,030  2,289  NotMNIST (Bulatov, 2011)  10  16,011  842  1,873  2 × 2 max-pooling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content=' Each fully connected layer has 2048 units.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content=' For the first two layers, we use the dropout of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='2, and for  the rest layers, we use the dropout of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='  Modified LeNet-5 architecture: For the CIFAR-100 Sup setting, a modified LeNet-5 architecture consisting of two  convolutional layers and two fully connected layers is adopted, similar to Saha et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content=' (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content=' Max-pooling of 3 × 2 is used  after each convolutional layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content=' The last two layers have 800 and 500 units respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='  Reduced ResNet-18 architecture: We adopt the same reduced ResNet-18 architecture as Saha et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content=' (2021) for the  MiniImageNet and Mixture settings, using 2 × 2 average-pooling before the classifier layer instead of the 4 × 4 average-  pooling used by Lopez-Paz & Ranzato (2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content=' Moreover, we present the dimension of the representation space of each layer  of our architectures in Table 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='  Table 7: Dimension of the representation space of each layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='  Network  Depth  Dimension of the representation space  MLP  3 layers  784;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content=' 100;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content=' 100  AlexNet  5 layers  48;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content=' 576;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content=' 512;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content=' 1,024;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content=' 2,048  LeNet-5  4 layers  75;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content=' 500;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content=' 3,200;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content=' 800  ResNet-18  17 layers and 3 short-  cut connections  27;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content=' 180;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
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+page_content=' 20;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content=' 360;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
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+page_content=' 720;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='  40;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content=' 720;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content=' 720;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content=' 720;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content=' 1,440;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content=' 80;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content=' 1,440;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content=' 1,440  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content=' Implementation Details  We use the official implementation of GPM (Saha et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content=', 2021), HAT (Serra et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content=', 2018), and TRGP (Lin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content=', 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content=' We  implement A-GEM and ER Res with the official implementation by Chaudhry et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content=' (2018) and implement EWC with the  implementation by Serra et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content=' (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content=' For OGD (Farajtabar et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content=', 2020), we use the implementation by Bennani et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='  (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content=' Following Saha et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content=' (2021) and Lin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content=' (2022), we run all experiments five times on an established seed without  fixing the cuda settings for a fair comparison.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content=' Particularly, we use five random seeds on PMNIST where there is no diversity  on a single seed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content=' For CIFAR-100 Sup, we use five different orders provided by Yoon et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content=' (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content=' Following Saha et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='  (2021), we report the experimental results of replay-base methods A-GEM and ER Res on the Mixture dataset with the  same buffer size as GPM and ROGO, which is 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='98M in terms of the number of parameters for Resnet18 architecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='  On CIFAR-100 Split, MiniImageNet, and PMNIST, we follow the hyper-parameters utilized by Saha et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content=' (2021) and Lin  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content=' (2022), including learning rate, batch size, and the threshold ϵ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content=' On Mixture, as we adopt the same network architecture  Saha et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content=' (2021) use on their 5-Dataset setting, we follow the provided learning rate and batch size as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='  As discussed in Section 5, the threshold ζ = cos γ controls the criterion of the relaxing space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content=' For CIFAR100-Split and  PMNIST, we use ζ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='95 for convolutional layers and ζ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='9 for fully connected layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content=' For MiniImageNet and Mixture,  we use the same ζ for all layers, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='95 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='9 respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content=' The regularization weight β is set as 5 for the ResNet18  architecture and 1 for others.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content=' Moreover, we limit the max iteration times to 2 for CIFAR100-Split and MiniImageNet for  efficiency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content=' Particularly, we run all the experiments on a single NVIDIA GeForce RTX 2080 Ti GPU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='  Restricted Orthogonal Gradient Projection for Continual Learning  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content=' Metrics  Here we present the detailed definitions of the metrics evaluating the forward knowledge transfer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content=' Ωnew (Kemker et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content=',  2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content=' Denote bi as the test accuracy of task i at random initialization, FWT, first proposed by Lopez-Paz & Ranzato (2017),  is defined as FWT =  1  T −1  �T  i=2(Ai−1,i − bi), evaluating the zero-shot performance of the initialization with respect to the  observed tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content=' While Ωnew, first proposed by Kemker et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content=' (2018), is defined as Ωnew =  1  T −1  �T  i=2(Ai,i − bi), reflecting  the test accuracy on new tasks based on the learnt knowledge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content=' As bi stays still across different approaches, we consider  Ωnew =  1  T −1  �T  i=2 Ai,i for simplicity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content=' For this simplified Ωnew, we have: Ωnew =  T  T −1ACC − BWT −  1  T −1A1,1, with  the ACC and BWT defined in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='  C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content=' Experimental Results  C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content=' Final Accuracy  We provide the test accuracy after learning each task on other benchmarks here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content=' As discussed in Section 4, our ROGO  universally outperforms GPM over the task sequence on all benchmarks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='  (a)  (b)  (c)  Test Accuracy (%)  Task ID  Test Accuracy (%)  Task ID  Task ID  Test Accuracy (%)  Test Accuracy (%)  Task ID  (d)  Figure 4: Average accuracy after learning each task on (a) CIFAR100-Split, (b) MiniImageNet, (c) PMNIST, and (d)  Mixture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='  C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content=' Backward Knowledge Trasnfer  As mentioned in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='1, final accuracy (ACC), forgetting (BWT) and forward knowledge transfer (Ωnew) are jointly  considered to evaluate a continual learner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content=' According to Table 8, ROGO achieves the best forgetting on MiniImageNet and  PMNIST datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content=' Despite our relaxing strategy focusing on facilitating the forward knowledge transfer, ROGO gains better  final accuracy with comparable forgetting over all benchmarks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content=' Generally speaking, ROGO achieves superior performance  than previous baselines with a fixed network capacity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='  Table 8: Comparison of average accuracy and forgetting tested after learning all tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content=' Multitask is under the non-incremental  setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content=' For each task, we mark the best and the second best performance in bold and underline respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content=' All results  reported are averaged over 5 runs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='  Method  CIFAR-100 Split  MiniImageNet  PMNIST  Mixture  ACC (%)  BWT (%)  ACC (%)  BWT (%)  ACC (%)  BWT (%)  ACC (%)  BWT (%)  Multitask  79.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='58 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='54 69.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='46 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='62 96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='70 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='02 81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='29 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='23 A-GEM  63.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='98 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='22  16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='30 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='19  57.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='24 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='72  10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='95 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='29  83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='56 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='16  14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='57 ± 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='33  59.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='86 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='01  29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='37 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='11  ER Res  71.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='73 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='63  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='50 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='76  58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='94 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='85  8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='84 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='56  87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='24 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='53  10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='24 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='58  75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='07 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='55  11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='81 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='82  EWC  68.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='80 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='88  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='31 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='76  52.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='01 ± 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='53  12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='14 ± 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='28  89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='97 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='57  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='58 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='22  69.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='62 ± 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='69  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='00 ± 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='09  HAT  72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='06 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='50  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='12 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='42  59.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='78 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='57  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='31 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='05 77.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='54 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='18  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='55 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='28  GPM  72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='48 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='40  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='72 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='27  60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='41 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='61  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='54 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='34  93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='91 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='16  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='30 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='09  77.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='49 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='68  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='70 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='50  ROGO  74.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='04 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='35  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='43 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='43  63.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='66 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='24  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='09 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='94  94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='20 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='11  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='44 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='13  77.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='91 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='45  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='55 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='36  Restricted Orthogonal Gradient Projection for Continual Learning  C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content=' Forward Knowledge Transfer  We provide detailed forward knowledge transfer performance on all four benchmarks here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content=' First, we present the results of  Ωnew and the detailed accuracy of each task after learning it in Table 9 to 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content=' According to Table 11 and Table 12, ROGO  achieves a similar forward knowledge transfer compared with GPM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content=' For other benchmarks, ROGO improves Ωnew by 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='7%  and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='8% on CIFAR-100 Split and MiniImageNet respectively, as shown in Table 9 and Table 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='  Table 9: The accuracy tested on task i after learning task i and Ωnew on CIFAR-100 Split.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='  Method  1  2  3  4  5  6  7  8  9  Avg (Ωnew)  GPM  67.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='7  72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='5  70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='1  73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='6  71.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='8  70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='3  71.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='0  71.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='8  73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='7  71.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='5  ROGO  71.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='7  74.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='8  73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='6  76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='9  75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='8  74.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='7  74.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='0  75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='9  79.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='3  75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='2  Table 10: The accuracy tested on task i after learning task i and Ωnew on MiniImageNet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='  Method  1  2  3  4  5  6  7  8  9  10  11  12  13  14  15  16  17  18  19  Avg (Ωnew)  GPM  59.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='0  57.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='1  65.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='6  59.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='6  76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='2  57.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='2  66.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='5  72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='8  81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='5  42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='0  59.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='6  62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='1  62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='3  58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='9  57.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='2  59.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='0  50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='7  65.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='7  57.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='3  61.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='6  ROGO  57.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='1  55.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='1  66.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='2  61.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='3  79.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='8  61.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='4  66.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='6  73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='5  83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='6  42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='8  65.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='1  63.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='4  67.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='1  64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='6  61.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='6  61.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='6  50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='0  68.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='2  60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='8  63.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='7  Table 11: The accuracy tested on task i after learning task i and Ωnew on PMNIST.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='  Method  1  2  3  4  5  6  7  8  9  Avg (Ωnew)  GPM  97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='5  97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='4  97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='0  96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='8  96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='5  96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='2  96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='2  95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='8  95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='1  96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='5  ROGO  97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='4  97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='2  97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='0  96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='5  96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='3  96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='1  95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='7  94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='9  94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='8  96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='2  Table 12: The accuracy tested on task i after learning task i and Ωnew on Mixture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='  Method  1  2  3  4  5  6  Avg (Ωnew)  GPM  99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='0  43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='7  87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='3  99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='1  69.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='9  93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='4  82.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='1  ROGO  99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='1  42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='9  87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='6  99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='1  70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='6  93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='5  82.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='2  Moreover, we provide the result of FWT (using the definition in (Lopez-Paz & Ranzato, 2017)) on all benchmarks in  Table 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content=' According to Table 13, although our method facilitates the forward knowledge transfer reflected by Ωnew, ROGO  achieves better FWT than GPM on all three task-incremental benchmarks, namely better zero-shot performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='  Table 13: Comparison of forward knowledge transfer between GPM and ROGO, evaluated by FWT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='  Methods  CIFAR-100 Split  MiniImageNet  PMNIST  Mixture  GPM  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='65 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='18  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='26 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='88  +0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='66 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='17  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='52 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='49  ROGO  +0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='20 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='36  +0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='30 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='29  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='63 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='97  +0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='13 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='77  C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content=' Accuracy Evolution  Here we present the accuracy tested on three randomly selected tasks immediately after learning them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content=' We select the 2nd,  4th, and 6th tasks for all benchmarks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content=' Generally, ROGO outperforms GPM on selected tasks over the sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content=' We further  notice that the improvement is more significant on later tasks owing to a larger relaxing space, as discussed in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='  Restricted Orthogonal Gradient Projection for Continual Learning  (a)  (b)  (c)  Test Accuracy (%)  Task ID  Test Accuracy (%)  Task ID  Task ID  Test Accuracy (%)  Test Accuracy (%)  Task ID  (d)  Figure 5: Accuracy evolution of the 2nd task on (a) CIFAR-100 Split, (b) MiniImageNet, (c) PMNIST, and (d) Mixture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='  (a)  (b)  (c)  Test Accuracy (%)  Task ID  Test Accuracy (%)  Task ID  Task ID  Test Accuracy (%)  Test Accuracy (%)  Task ID  (d)  Figure 6: Accuracy evolution of the 4th task on (a) CIFAR-100 Split, (b) MiniImageNet, (c) PMNIST, and (d) Mixture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='  (a)  (b)  (c)  Test Accuracy (%)  Task ID  Test Accuracy (%)  Task ID  Task ID  Test Accuracy (%)  Test Accuracy (%)  Task ID  (d)  Figure 7: Accuracy evolution of the 6th task on (a) CIFAR-100 Split, (b) MiniImageNet, (c) PMNIST, and (d) Mixture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='  C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content=' Time Consumption  We report the time consumption of ROGO on two benchmarks compared with relative baselines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content=' TRGP and ROGO are both  evaluated on a single NVIDIA GeForce RTX 2080 Ti GPU and we report the results according to (Lin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content=', 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content=' As  discussed in Section 5, ROGO takes acceptable extra time compared with GPM on both datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content=' For both dataset, ROGO  tasks similar time as TRGP, which is similar to EWC and much less than A-GEM and OWM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='  Table 14: Time comparison evaluated on two benchmarks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content=' We use the results reported in (Lin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content=', 2022) and the time is  normalized with respect to GPM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='  Datasets  Methods  OWM  EWC  HAT  A-GEM  ER Res  GPM  TRGP  ROGO  CIFAR-100  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='41  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='76  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='62  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='48  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='49  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='00  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='65  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='62  MiniImageNet 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='22  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='91  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='79  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='82  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='00  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='34  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='43  C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content=' Memory Usage  We provide a comparison between TRGP and ROGO on the ratio of the amount of extra parameters concerning the amount  of the parameters of the initial network architecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content=' According to Figure 8, TRGP requires at least 200% of the number of  Restricted Orthogonal Gradient Projection for Continual Learning  extra parameters after learning all tasks on all four benchmarks, while ROGO only stores the representation of the frozen  space, which can further be released in the inference phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='  (a)  (b)  (c)  Ratio (%)  Task ID  Ratio (%)  Task ID  Task ID  Ratio (%)  Ratio (%)  Task ID  (d)  Figure 8: Ratio of the amount of extra parameters concerning the amount of the parameters of the initial network architecture  on (a) CIFAR100-Split, (b) MiniImageNet, (c) PMNIST, and (d) Mixture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='  C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content=' Other Results  Here we provide detailed results including the forgetting performance compared with TRGP in Table 15 and 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content=' Similar to  the discussion in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='3, our ROGO framework obtains superior accuracy performance with comparable forgetting  under or without the constraint of a fixed network capacity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='  Table 15: Comparison of average accuracy and forgetting with TRGP under an expansion setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content=' The percentages indicate  the ratios of the rank of the relaxing space with respect to the frozen space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content=' For each task, we mark the best and the second  best performance in bold and underline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='  Methods  TRGP  ROGO-Exp  50%  80%  T%  ACC (%)  BWT (%)  ACC (%)  BWT (%)  ACC (%)  BWT (%)  ACC (%)  BWT (%)  CIFAR  74.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='46 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='22  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='42 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='20  74.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='90 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='37  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='99 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='27  74.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='97 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='30  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='68 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='39  75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='34 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='61  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='63 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='57  PMNIST  96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='34 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='11  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='58 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='10  96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='44 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='20  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='75 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='13  96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='76 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='15  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='46 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='08  97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='01 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='08  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='31 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='05  MiniImageNet  61.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='78 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='94  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='01 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='58  63.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='46 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='85  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='50 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='39  62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='57 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='32  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='42 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='57  62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='78 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='00  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='45 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='35  Mixture  83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='54 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='15  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='80 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='10  82.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='45 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='49  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='74 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='16  83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='33 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='31  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='98 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='23  83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='62 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='26  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='33 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='14  Table 16: Comparison of average accuracy and forgetting with TRGP within a fixed network capacity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content=' We modify TRGP as  TRGP-Reg with similar regularization terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content=' β indicates the regularization weight.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content=' For each task, we mark the best and the  second best performance in bold and underline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='  Methods  TRGP-Reg  ROGO  β = 1  β = 5  β = 50  ACC (%)  Ωnew (%)  ACC (%)  Ωnew (%)  ACC (%)  Ωnew (%)  ACC (%)  Ωnew (%)  CIFAR  72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='01 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='30  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='63 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='20  71.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='96 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='40  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='09 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='37  72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='49 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='09  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='69 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='24  74.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='04 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='35  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='43 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='43  PMNIST  76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='57 ± 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='69  20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='69 ± 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='94  75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='50 ± 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='96  22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='41 ± 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='24  76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='56 ± 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='83  21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='63 ± 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='13  94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='20 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='11  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='09 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='94  MiniImageNet  55.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='81 ± 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='43  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='69 ± 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='19  58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='81 ± 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='24  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='85 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='54  22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='69 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='39  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='01 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='07  63.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='66 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='24  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='44 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='13  Mixture  73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='31 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='05  11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='05 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='43  74.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='71 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='85  9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='33 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='05  17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='36 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='86  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='01 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='03  77.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='91 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='45  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='55 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='36  Moreover, we illustrate the test accuracy over the task sequence in Figure 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content=' As shown in Figure 9-(a) and Figure 9-(b),  our ROGO-Exp dominates TRGP on both benchmarks relaxing either 80% or T% of the frozen space under an expansion  setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content=' We further compare ROGO with TRGP modified with the same regularization terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content=' As shown in Figure 9-(c) and  Figure 9-(d), the performance of TRGP drops significantly constrained in a fixed network capacity on both benchmarks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='  Following GPM (Saha et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content=', 2021), we conduct experiments on the CIFAR-100 Sup dataset as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content=' The implementation  details are included in Appendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='3 as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content=' Here we report the accuracy performance in Table 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content=' Note that the selected  baselines for this setting all require extra parameters other than GPM and we directly use the results of the baselines  reported in GPM and TRGP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content=' In accordance with the results on the other four benchmarks discussed in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='2, ROGO  Restricted Orthogonal Gradient Projection for Continual Learning  (a)  (b)  (c)  Test Accuracy (%)  Task ID  Test Accuracy (%)  Task ID  Task ID  Test Accuracy (%)  Test Accuracy (%)  Task ID  (d)  Figure 9: Test accuracy after learning each task under an expansion setting on (a) CIFAR-100 Split and (b) PMNIST, and  within a fixed network capacity on (c) CIFAR-100 Split and (d) PMNIST.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='  outperforms GPM and other related baselines by a significant margin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content=' Particularly, we notice ROGO directly obtains superior  performance than TRGP within a fixed network capacity under this setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='  Table 17: Results of ACC (%) on CIFAR-100 Sup setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content=' STL is under a non-incremental setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content=' All baselines require  extra network capacity except GPM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content=' We also report the standard deviation result of ROGO for comparison.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='  Metric  Methods  STL*  PNN  DEN  RCL  APD  GPM  TRGP  ROGO  ACC (%)  61.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='00  50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='76  51.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='10  51.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='99  56.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='81  57.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='72  58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='25  58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='74 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='60  D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content=' Algorithm  We present the pseudo-code of our searching strategy here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='  Algorithm 2 Relaxing Space Searching  Input: gradient {gl  t}L  l=1, frozen space {U l  t−1}L  l=1 and thresholds {ϵl  th, γl  t}L  l=1  Output: relaxing space {V l  t }L  l=1  1: for l ∈ 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content=', L do  2:  Construct the significant representation space Rl  g,t from gradients gl  t with top-k eigenvectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
+page_content='  3:  V l  t ← ∅  4:  repeat  5:  d ← arg min  d∈U l,c  t−1  Θ(d, Rl  g,t)  6:  if Θ(d, Rl  g,t) ≤ γl  t then  7:  V l  t ← V l  t ∪ d  8:  U l,c  t−1 ← U l  t−1\\V l  t  9:  end if  10:  until Θ(d, Rl  g,t) > γl  t  11: end for' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9FLT4oBgHgl3EQfoS-L/content/2301.12131v1.pdf'}
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+Magneto-optical chirality in a coherently coupled exciton-plasmon system
+Samarth Vadia,1, 2, 3, ∗ Johannes Scherzer,1, ∗ Kenji Watanabe,4 Takashi Taniguchi,5 and Alexander H¨ogele1, 2
+1Fakult¨at f¨ur Physik, Munich Quantum Center, and Center for NanoScience (CeNS),
+Ludwig-Maximilians-Universit¨at M¨unchen, Geschwister-Scholl-Platz 1, 80539 M¨unchen, Germany
+2Munich Center for Quantum Science and Technology (MCQST), Schellingtr. 4, 80799 M¨unchen, Germany
+3attocube Systems AG, Eglfinger Weg 2, 85540 Haar, Germany
+4Research Center for Functional Materials, National Institute for Materials Science, 1-1 Namiki, Tsukuba 305-0044, Japan
+5International Center for Materials Nanoarchitectonics,
+National Institute for Materials Science, 1-1 Namiki, Tsukuba 305-0044, Japan
+(Dated: January 10, 2023)
+Chirality is a fundamental asymmetry phenomenon, with chiral optical elements exhibiting asymmetric re-
+sponse in reflection or absorption of circularly polarized light. Recent realizations of such elements include
+nanoplasmonic systems with broken mirror symmetry and polarization-contrasting optical absorption known
+as circular dichroism. An alternative route to circular dichroism is provided by spin-valley polarized excitons
+in atomically thin semiconductors. In the presence of magnetic fields, they exhibit an imbalanced coupling to
+circularly polarized photons and thus circular dichroism. Here, we demonstrate that polarization-contrasting
+optical transitions associated with excitons in monolayer WSe2 can be transferred to proximal plasmonic nan-
+odisks by coherent coupling. The coupled exciton-plasmon system exhibits magneto-induced circular dichroism
+in a spectrally narrow window of Fano interference, which we model in a master equation framework. Our work
+motivates exciton-plasmon interfaces as building blocks of chiral metasurfaces for applications in information
+processing, non-linear optics and sensing.
+Direct band gap and reduced dielectric screening in
+semiconducting monolayer transition metal dichalcogenides
+(TMDs) [1, 2] give rise to tightly bound excitons [3] with siz-
+able light-matter interactions that facilitate efficient coupling
+to dielectric or plasmonic systems [4–6]. Capitalizing on the
+large oscillator strength of TMD excitons and the flexibility
+of combining them with plasmonic structures, recent exam-
+ples of coupled exciton-plasmon systems include realizations
+in the strong and weak coupling regimes [7–10]. While the
+former is characterized by the formation of exciton-plasmon
+polaritons, the latter is distinguished by Fano-type interfer-
+ence spectra, as discussed in early work on various exciton-
+plasmon coupled systems [11–13].
+In this framework, the
+dipolar selection rules of spin-valley polarized excitons in
+TMD monolayers [14–18] provide a route to chiral optical
+phenomena, as the valley degeneracy can be lifted by mag-
+netic field to induce spectrally imbalanced coupling to left-
+and right-handed circularly polarized photons [19–23]. This
+opto-valleytronic feature of TMD monolayer excitons has
+been utilized to demonstrate chiral effects such as directional
+coupling of light in silver nanowires on WS2 [24], spatial sep-
+aration of valley-polarized excitons by silver nanogroove ar-
+rays [25], or second-harmonic generation of circularly polar-
+ized photons in gold-WS2 metasurfaces [26].
+Here, we study magneto-optical characteristics of an
+exciton-plasmon metasurface based on a WSe2 monolayer
+and gold (Au) nanodisks. We elucidate the effect of Fano
+interference as a function of exciton-plasmon spectral reso-
+nance detuning in the weak coupling regime [10, 27–31], and
+study both experimentally and theoretically the polarization
+properties of the coherently coupled system in the presence
+of external magnetic fields.
+Remarkably, the coupled sys-
+∗ These authors contributed equally to this work
+tem exhibits magnetic circular dichroism that is distinct from
+the characteristics of the fundamental valley-polarized exci-
+ton transition in monolayer WSe2. The resulting chiral be-
+havior of the Fano-coupled metasurface manifests in the form
+of a spectral window with polarization dependent reflectivity
+in an otherwise broadband opaque medium. The observations
+are substantiated by a master equation analysis with excel-
+lent quantitative agreement with experimental findings. Our
+work provides insight into the underlying coherent interfer-
+ence phenomena and can serve as a guideline to the design
+of exciton-plasmon metasurfaces with optical chirality in the
+visible spectral range.
+The exciton-plasmon interface was fabricated by encapsu-
+lating a monolayer WSe2 in hexagonal boron nitride (hBN)
+and placing the resulting heterostructure on top of a plasmonic
+Au nanodisk array on a SiO2/Si substrate, as illustrated in
+Fig. 1a (see Methods for sample details). The sample fea-
+tures regions of encapsulated WSe2 monolayer, Au nanodisk
+arrays, as well as regions where both elements are combined
+in vertical proximity. To characterize the optical responses of
+the bare exciton and plasmon systems, and the regime of their
+coupling, we performed differential reflection spectroscopy
+at cryogenic temperatures (see Methods for experimental de-
+tails). The corresponding spectra are shown in Fig. 1c, d and
+e, where differential reflection DR = (Rsub − R)/(Rsub)
+was measured relative to the reflection Rsub of the SiO2/Si
+substrate. The coupled system can be modelled in a three-
+level system framework with relevant states and rates shown
+in Fig. 1e. The DR spectra in Fig. 1c, d and e are representa-
+tive for the excitation from the ground state |g⟩ to the exciton
+state |e⟩ with resonance and Rabi frequency ωeg and Ωeg and
+decay rate Γeg (Fig. 1c), the optical excitation to the plasmon
+state |p⟩ with resonance and Rabi frequencies ωpg and Ωpg
+and decay rate Γpg (Fig. 1d), and the simultaneous excitation
+of the interacting exciton-plasmon system with coherent cou-
+arXiv:2301.03275v1  [physics.optics]  9 Jan 2023
+
+2
+0.0
+0.5
+1.0
+0.0
+0.5
+1.0
+1.6
+1.7
+1.8
+0.0
+0.5
+1.0
+Energy [eV]
+ۧ
+ȁ𝑔
+ۧ
+ȁ𝑒
+ۧ
+ȁ𝑝
+Ω𝑒𝑔
+Ω𝑝𝑔
+Γ𝑒𝑔
+Γ𝑝𝑔
+Ωc
+a
+f
+b
+c
+d
+σ+
+σ-
+Differential reflection, DR
+ω𝑒𝑔
+ω𝑝𝑔
+K
+K'
+σ+
+σ-
+e
+CB
+VB
+FIG. 1. Fano interference in a coupled exciton-plasmon system.
+a, Schematic illustration of monolayer WSe2 encapsulated in hBN,
+placed on gold nanodisks and probed with circularly polarized light.
+b, Band structure schematics of monolayer WSe2 at two opposite
+corners of the hexagonal Brillouin zone, with σ+ (σ−) circularly
+polarized optical transitions between spin-up (spin-down) polarized
+states at the K (K’) valleys of the conduction band (CB) and valence
+band (VB). c, d, and e, Differential reflection spectra of monolayer
+WSe2, gold nanodisk, and the coupled system, respectively. f, En-
+ergy levels of the coupled system with |g⟩, |e⟩ and |p⟩ denoting the
+ground, exciton and plasmon state, respectively, and the correspond-
+ing exciton and plasmon optical transition frequencies ωeg and ωpg,
+Rabi frequencies Ωeg and Ωpg, and radiative decay rates Γeg and
+Γpg, as well as the exciton-plasmon coupling strength Ωc.
+pling constant Ωc (Fig. 1e).
+The band structure of WSe2 monolayer is shown schemat-
+ically in Fig. 1b, together with polarization-contrasting op-
+tical transitions of the fundamental exciton X0 in the K
+and K′ valleys of the hexagonal Brillouin zone. The tran-
+sitions are degenerate at zero magnetic field, resulting in a
+single Lorentzian peak in the DR spectrum of Fig. 1b at
+ℏωeg ≃ 1.723 eV. The full-width at half-maximum (FWHM)
+linewidth of the exciton transition ℏΓeg ≃ 8 meV is substan-
+tially smaller than the plasmon linewidth of ℏΓpg ≃ 180 meV
+obtained from the region of a bare Au nanodisk array in Fig-
+ure 1d with resonance energy at ℏωpg = 1.72 eV. Due to
+variations in the dielectric environment of the nanodisk ar-
+ray, the plasmon resonance energy ℏωpg varies in the range
+of 100 meV (see Supplementary Section 1 for the description
+of gold nanodisk arrays). In our studies, this variation bene-
+ficially provides position-dependent spectral energy detuning
+δ = ℏωpg − ℏωeg of the plasmon resonance energy with re-
+spect to the exciton resonance which has negligible variations
+across the sample.
+The spectrum of the coupled system in Fig. 1e is character-
+ized by a Fano interference lineshape [27, 28, 32, 33] with a
+narrow reflection dip in the broad plasmonic extinction peak.
+A closer inspection of the spectrum reveals an additional peak
+superimposed on the reflection dip, which we ascribe to the
+contribution from uncoupled excitons that are located within
+the optical spot yet sufficiently far away from plasmonic nan-
+odisks. To interpret the resulting lineshape, we inspected sys-
+tem realizations with different resonance detunings at differ-
+ent spatial positions of the interfaced array. Two representa-
+tive DR spectra for negative and positive resonance detuning
+δ are shown in Fig. 2a and b, respectively. In both spectra, the
+position of the dip remains essentially constant due to small
+variation in the exciton energy across the sample, which also
+holds for the uncoupled exciton peak inside the Fano dip. Due
+to the asymmetric character of the Fano interference, however,
+the overall spectral shape is strongly modified at different res-
+onance conditions.
+We model this intricate optical response in the framework
+of two coherently coupled oscillators using a classical light
+field interacting with exciton and plasmon dipolar excitations.
+The dipole moments of the respective optical transitions are
+given as the imaginary components of the quantum coherence
+obtained from the master equation analysis (see Supplemen-
+tary Section 2 for theoretical modeling of the extinction spec-
+trum). All main parameters of the system including the decay
+rates Γeg, Γpg and the Rabi frequencies Ωeg, Ωpg were de-
+termined from experiments on bare system components. To
+quantify the coupling strength Ωc, we plot the spectral posi-
+tion of the two maxima enclosing the dip in the Fano spec-
+tra as a function of detuning δ in Fig. 2e, with their energy
+separation reproduced by the theoretical model for a coupling
+strength ℏΩc = 28 meV, as shown by solid lines in Fig. 2e.
+With this coupling, our model yields the normalized extinc-
+tion spectra shown in Fig. 2c and d for resonance detunings
+δ = −38 and 32 meV as extracted from the spectra in Fig. 2a
+and b. All features of the optical response are reproduced with
+good agreement by the theoretical model.
+With this understanding of the Fano interference phenom-
+ena, we elucidate in the following the magneto-optical re-
+sponse of the coupled exciton-plasmon system.
+First, we
+quantify the degree of circular dichroism (CD) associated with
+the exciton valley Zeeman effect in monolayer WSe2 [20–23].
+In the presence of an out-of-plane magnetic field of 9 T, circu-
+larly polarized DR spectra of Fig. 3a and b reveal two exciton
+resonances associated with K and K′ transitions that couple
+to σ+ and σ− polarized light, respectively. The valley Zee-
+
+R3
+0.0
+0.5
+1.0
+d = -38 meV
+d = 32 meV
+1.6
+1.7
+1.8
+0.0
+0.5
+1.0
+Energy (eV)
+1.6
+1.7
+1.8
+Energy (eV)
+a
+c
+b
+e
+Ω𝑐 = 28 meV
+d
+Differential reflection, DR
+Normalized extinction
+-40
+-20
+0
+20
+40
+1.68
+1.70
+1.72
+1.74
+1.76
+Energy (eV)
+Detuning d (meV)
+FIG. 2. Exciton-plasmon Fano interference in experiment and
+theory. a and b, Normalized differential reflection spectra of the
+coupled system for −38 and 32 meV energy detuning from the
+exciton-plasmon spectral resonance condition.
+The spectra were
+recorded on two positions of the nanodisk array with different plas-
+mon energies ℏωpg (determined from Lorentzian fits) for a weakly
+varying exciton energy ℏωeg. c and d, Respective Fano model spec-
+tra with exciton-plasmon coupling strength ℏΩc = 28 meV. e, Evo-
+lution of the exciton-plasmon coupling as a function of resonance
+energy detuning δ, with red data points corresponding to plasmonic
+array regions with different ℏωpg and respective model results (black
+lines) obtained with the bare exciton energy ℏωeg = 1.723 eV (dot-
+ted line) and ℏΩc = 28 meV.
+man splitting of 2.1 meV corresponds to the exciton Land´e
+factor with an absolute value of 4 as expected from previous
+experiments [21–23] and theory [34–37].
+The polarization-contrasting response of the two valleys
+is quantified by CD, which calculates as CD = (DR− −
+DR+)/(DR− + DR+), where DR+ and DR− are the the σ+
+and σ− polarized DR spectra respectively. Figure 3c shows
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+-0.4
+-0.2
+0.0
+0.2
+0.4
+1.71
+1.73
+1.75
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+Energy (eV)
+1.71
+1.73
+1.75
+-0.4
+-0.2
+0.0
+0.2
+0.4
+Energy (eV)
+ 
+a
+c
+Circular dichroism, CD
+Differential reflection, DR
+d
+b
+9T
+-9T
+9T
+-9T
+K
+K'
+σ+
+σ-
+σ+
+σ-
+FIG. 3. Magnetic circular dichroism of the exciton transition.
+a and b, Valley-selective differential reflection spectra at 9 T for
+σ+ and σ− polarized excitation, respectively (solid lines show
+Lorentzian fits). c and d, Circular dichroism of the exciton transition
+at 9 (red) and −9 T (blue) from experiment and theory, respectively.
+the CD at 9 T as red solid line, where DR+ and DR− are the
+Lorentzian fits to the σ+ and σ− polarized DR spectra shown
+as solid lines in Fig. 3a and b, respectively. It shows a rever-
+sal in polarity around the resonance energy of the exciton at
+0 T with maximum CD of ∼ 20%. The CD obtained from
+the corresponding experiments at −9 T, shown in Fig. 3c in
+blue, is reversed in sign for the entire spectral range (devia-
+tions from the mirror-symmetry around the exciton energy at
+zero field stem from sample inhomogeneities sampled by spa-
+tial displacements in magnetic field over a range of 18 T). All
+main features of the spectra are captured by our model CD
+spectra in Fig. 3d obtained from extinction.
+Next, we study the polarization-dependent optical response
+of the interface region in the presence of a magnetic field. The
+valley Zeeman effect of the bare exciton is imprinted on the
+coupled exciton-plasmon system in an intricate way and man-
+ifests as polarization-dependent reflectance. Our theory cap-
+tures the experimentally observed features of the CD spectra
+and their evolution with the magnetic field, as evidenced by
+comparing experimental and theoretical data shown in Fig. 4a
+and b and Fig. 4c and d, respectively. The evolution of CD
+with increasing magnetic fields of 3, 6 and 9 T, recorded in the
+region of the Fano interference with −15 meV resonance de-
+tuning, is shown in Fig. 4a. The magneto-induced dichroism
+becomes increasingly pronounced with increasing magnetic
+field as for the bare WSe2 monolayer. Notably, the spectra in
+Fig. 4c show a sign reversal of the CD response for the cou-
+
+4
+ 
+-0.08
+-0.04
+0.00
+0.04
+0.08
+1.71
+1.73
+-0.08
+-0.04
+0.00
+0.04
+0.08
+Energy (eV)
+1.71
+1.73
+Energy (eV)
+a
+b
+c
+d
+Circular dichroism, CD
+9T
+-9T
+9T
+-9T
+3T
+6T
+9T
+3T
+6T
+9T
+FIG. 4.
+Magnetic circular dichroism of the coupled exciton-
+plasmon system. a and b, Experimental and theoretical circular
+dichroism spectra of the coupled exciton-plasmon system at 3, 6 and
+9 T. c and d, Experimental and theoretical circular dichroism spectra
+in magnetic fields of 9 (red) and −9 T (blue).
+pled system in comparison to the bare exciton case in Fig. 3c.
+While the monolayer features a peak in the exciton DR spec-
+trum, the coupled exciton-plasmon spectrum is characterized
+by a narrow dip around the exciton resonance as a result of
+Fano interference, leading to reversed CD response and pro-
+nounced asymmetry for finite exciton-plasmon detunings. In
+the calculated spectra of Fig. 4d, the asymmetry manifests in
+the form of an additional dip in the CD spectra, in qualitative
+agreement with the experimental spectra in Fig. 4c. The con-
+tribution of uncoupled excitons within the optical focal spot
+with a reversed sign of CD explains this observation. A max-
+imum CD of up to ∼ 7 % is achieved in the coupled exciton-
+plasmon system, indicating a significant transfer of the opto-
+valleytronic exciton features onto the coupled system. Con-
+sistently, the CD spectra exhibit a sign reversal at magnetic
+fields of ±9 T.
+Our observation of magneto-optical effects in a coherently
+coupled exciton-plasmon system and their detailed quantita-
+tive understanding provide a pathway to design ultrathin meta-
+surfaces for chiral spectral filtering, which could be also ex-
+ploited to control nonreciprocal phenomena with magnetic
+field for uni-directional flow of circularly polarized photons
+as required for information transfer in quantum networks
+[38, 39]. An obvious way to create a permanent chiral exciton-
+plasmon metasurface is to utilize layered ferromagnets that in-
+duce sizable exciton valley Zeeman splittings of several meV,
+equivalent to external magnetic fields well above 10 T [40–
+44]. Furthermore, spectrally narrow response from TMD ex-
+citons in the limit of atomically thin mirror [45–48] would
+yield spectrally sharp Fano reflectance or windows of trans-
+parency in a broad extinction response. As such, spectral re-
+gions with perfect destructive quantum interference and nega-
+tive refractive index equivalent to electromagnetically induced
+transparency could be realized for chiral slow light and infor-
+mation storage [49–51].
+Methods
+The sample was fabricated by depositing monolayer WSe2
+(HQ Graphene) embedded in high-quality hBN (NIMS) onto
+a gold nanodisk array. The array was fabricated by standard
+electron-beam lithography and gold evaporation on a Si/SiO2
+substrate. All measurements were carried out at cryogenic
+temperatures. The data in Fig. 1 and Fig. 2 were recorded in
+a helium bath cryostat at 4.2 K, whereas the magnetic field
+measurements of Fig. 3 were performed in a closed-cycle
+magneto-cryostat (attocube systems, attoDRY1000) at 3.5 K.
+White-light reflection spectroscopy was performed using a
+halogen lamp (Ocean Optics, HL-2000) or a supercontinuum
+laser (NKT, SuperK Extreme EXR-4) focused to a spot of ∼
+1 µm diameter in a home-built confocal microscope equipped
+with cryogenic nano-positioners (attocube systems, ANP100
+and ANP101 series) and micro-objectives (attocube systems,
+LT-APO/VISIR/0.82 or LT-APO/LWD/NIR/0.63).
+The re-
+flected light was spectrally dispersed by a monochromator
+(Roper Scientific, Acton SP2500 or Acton 300i) and detected
+by a CCD (Roper Scientific, Spec 10:100BR/LN or Andor,
+iDus 416).
+Acknowledgements
+The authors thank A. O. Govorov for fruitful discussions,
+and P. Altpeter and C. Obermayer for continuous support
+in the clean room.
+This research was funded by the Eu-
+ropean Research Council (ERC) under the Grant Agree-
+ment No. 772195 as well as the Deutsche Forschungsgemein-
+schaft (DFG, German Research Foundation) within the Prior-
+ity Programme SPP 2244 2DMP (Project No. 443405595)
+and the Germany’s Excellence Strategy Munich Center for
+Quantum Science and Technology (MCQST) EXC-2111-
+390814868.
+S. V. acknowledges funding from the Euro-
+pean Union’s Horizon 2020 research and innovation pro-
+gramme under the Marie Skłodowska-Curie Grant Agree-
+ment Spin-NANO No. 676108. K. W. and T. T. acknowledge
+support from JSPS KAKENHI (Grant Numbers 19H05790,
+20H00354 and 21H05233).
+Corresponding authors
+S. V.
+(samarth.vadia@physik.lmu.de),
+J, S.
+(jo-
+hannes.scherzer@physik.lmu.de),
+and
+A. H.
+(alexan-
+der.hoegele@lmu.de).
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+Magneto-optical chirality in a coherently coupled exciton-plasmon system
+Supporting Information
+Samarth Vadia,1, 2, 3, ∗ Johannes Scherzer,1, ∗ Kenji
+Watanabe,4 Takashi Taniguchi,5 and Alexander H¨ogele1, 2
+1Fakult¨at f¨ur Physik, Munich Quantum Center, and Center
+for NanoScience (CeNS), Ludwig-Maximilians-Universit¨at
+M¨unchen, Geschwister-Scholl-Platz 1, 80539 M¨unchen, Germany
+2Munich Center for Quantum Science and Technology (MCQST),
+Schellingtr. 4, 80799 M¨unchen, Germany
+3attocube Systems AG, Eglfinger Weg 2, 85540 Haar, Germany
+4Research Center for Functional Materials,
+National Institute for Materials Science,
+1-1 Namiki, Tsukuba 305-0044, Japan
+5International Center for Materials Nanoarchitectonics,
+National Institute for Materials Science,
+1-1 Namiki, Tsukuba 305-0044, Japan
+(Dated: January 10, 2023)
+∗ These authors contributed equally to this work
+1
+arXiv:2301.03275v1  [physics.optics]  9 Jan 2023
+
+SUPPORTING NOTE 1: GOLD NANODISK ARRAYS
+The metasurface discussed in this work consists of a monolayer WSe2 encapsulated in hexag-
+onal boron nitride (hBN) and placed on a gold (Au) nanodisk array. The individual metallic
+nanoparticles support plasmon resonaces localized to the size of the particle. They are charac-
+terized by electric field enhancement in their direct proximity under optical excitation, which we
+use to couple exciton and plasmon resonances. The spectral position of the plasmon resonance
+depends on the shape and size of the particles, as well as on their dielectric environment.
+The design of plasmonic nanodisk arrays respects the following basic considerations. First, the
+plasmon resonance of the Au nanodisks has to match the energy of the exciton transition in WSe2
+to enable near-resonance coherent coupling. Second, the TMD monolayer has to be placed in close
+proximity to the Au nanodisks for the near-field enhancement of the exciton-plasmon interaction.
+Third, the height of the nanodisks should be as small as possible to avoid exciton localization in
+the form of quantum dots due to the variation in lateral strain in the TMD monolayer [1, 2]. And
+fourth, finally, the number of nanodisks within the focal spot under optical excitation should be
+large to increase the effective coupling strength.
+Following these considerations, we performed finite-difference time-domain (FDTD) simula-
+tions with a commercial software (Lumerical) to find the optimal geometrical parameters for our
+system. A SiO2 coated silver mirror was used as substrate for the nanostructure both in simu-
+lations and in actual samples. A thin layer with refractive index n = 1.76 simulates the hBN
+interface at the top surface of the nanodisks. The main geometrical parameters for tuning the plas-
+mon resonance of the disks are their diameter and height. With the consideration to minimize the
+nanodisk height in mind, we fixed the height at ∼ 15 nm where evaporated gold is expected to
+form smooth surfaces. The plasmon resonance can be shifted to smaller energies by increasing the
+disk diameter, while a blue-shift accompanies diameter reduction. Figure 1a shows this behavior
+when simulating the extinction of an optical plane wave propagating through gold nanodisks with
+different diameters. To enhance the near-field effect, we used a 6 nm thin bottom hBN layer in
+our van der Waals heterostack defining the distance between the TMD monolayer and the Au nan-
+odisk array. To estimate the magnitude of near-field enhancement at the monolayer position, we
+evaluated the intensity of the electric field in FDTD simulation in the monolayer plane as plotted
+in Fig. 2a. The mean intensity value across the entire area of a nanodisk is shown by the dashed
+line in the intensity line profile in Fig. 2b and serves as an input parameter for the model of the
+2
+
+1.6
+1.8
+2.0
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+Extinction (norm.)
+Energy (eV)
+200 nm
+a = 300 nm
+c
+a
+b
+D = 118 nm
+1.6
+1.8
+2.0
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+Differential Reflection, DR
+Energy (eV)
+FIG. 1. a, Extinction spectra of two different gold nanodisk arrays on a silver/SiO2 substrate obtained from
+FDTD simulation. The reflected (R) and transmitted (T) intensities were evaluated at the both ends of the
+computational cell, and normalized to the signal of the plane wave source. The extinction is defined as
+1 − R − T. Both arrays have a lattice constant a of 300 nm and height h of 15 nm. Red and blue spectra
+correspond to diameters D of 114 and 106 nm, respectively. b, Differential reflection spectra of two gold
+nanodisk arrays fabricated via e-beam lithography on a silver/SiO2 substrate. The geometrical parameters
+correspond to the ones used for the simulation in a. c, Scanning electron micrograph of an array section of
+the sample investigated in the main text with mean nanodisk diameter D of 118 nm.
+coupled exciton-plasmon system. The plasmonic response in differential reflection varies within a
+range of 100 meV across the metasurface. The shift of the plasmon resonance is significant only
+in regions where the array was covered by hBN, which indicates that the variation of the dielectric
+environment, related to inhomogeneities in the distance between the hBN layer and the plasmonic
+structure, dominates the variation of the plasmon energy.
+Using the simulation results as input parameters, we fabricated gold nanodisk arrays using elec-
+3
+
+-100
+-50
+0
+50
+100
+0
+20
+40
+60
+80
+100
+120
+Field Intensity (a.u.)
+x (nm)
+-50
+0
+50
+-50
+-25
+0
+25
+50
+y (nm)
+x (nm)
+25
+50
+75
+100
+Field
+Intensity 
+(a.u.) 
+a
+b
+FIG. 2. a, Simulated field intensity at the cross-section area of the FDTD computational cell 6 nm above an
+Au nanodisk (at the position of the monolayer) at the plasmon resonance. The intensity is normalized to the
+intensity of the excitation plane wave source, thus effectively determining the intensity enhancement factor
+for the near-field of the plasmonic nanodisk. b, One-dimensional line cut (dashed line in a) of the intensity
+cross-section area shown in a. The dashed line at field intensity of 50 represents the approximation of the
+intensity within the overlap area Ac of the monolayer and the plasmonic near-field enhancement shown in
+a.
+tron beam lithography. Figure 1c shows a scanning electron micrograph of the Au nanodisk array
+used in the experiments of the main text. The optimized interparticle distance for a maximum den-
+sity of nanodisks within the periodic array was a = 300 nm. We found that smaller array spacings
+compromised the lift-off process for nanodisk diameters larger than 100 nm. The resolution of our
+electron beam lithography process was dependent on the e-beam voltage with a limit of ∼ 10 nm.
+This limited resolution determined the range of variations in the disk diameters in an array of
+otherwise identical fabrication parameters. The variation of individual nanodisk diameters in turn
+resulted in inhomogeneous broadening of the plasmon resonances measured in differential reflec-
+tion (see Fig.1b) as compared to the results of corresponding FDTD simulations shown in Fig. 1a.
+The experimental full-width at half-maximum (FWHM) linewidth of the plasmon resonances was
+FWHM ≃ 180 meV, and was used as the plasmon decoherence rate in the model detailed in the
+following section.
+4
+
+SUPPORTING NOTE 2: COHERENT INTERFERENCE IN THE COUPLED-OSCILLATOR
+MODEL
+To model the coupled system of excitons in WSe2 monolayer and localized surface plasmons in
+Au nanodisk arrays interacting with a light field of frequency ω, we consider a three-level system
+comprising |g⟩, |e⟩ and |p⟩ as the ground, exciton and plasmon state, respectively. The energy
+separation between the states |g⟩ and |e⟩ and the states |g⟩ and |p⟩ is ℏωeg and ℏωpg, respectively.
+The Rabi frequencies for the respective transitions are Ωeg and Ωpg. In addition, the coherent
+coupling between the states |e⟩ and |p⟩ is denoted by Ωc. The Hamiltonian of the system is written
+as:
+H = ℏ
+�
+�
+�
+�
+�
+0
+1
+2Ωegeiωt
+1
+2Ωpgeiωt
+1
+2Ωege−iωt
+ωeg
+1
+2Ωc
+1
+2Ωpge−iωt
+1
+2Ωc
+ωpg
+�
+�
+�
+�
+� .
+(1)
+To describe the coherent dynamics of the Hamiltonian including the decoherence and dissipa-
+tion channels, we use the master equation for the density matrix, given by [3]:
+˙ρ = − i
+ℏ [H, ρ] + Lρ,
+(2)
+where the density matrix ρ with matrix elements ρij describes the statistical state of the system.
+The diagonal matrix elements ρij (with i = j) are the populations of the states, with i = g, e or
+p, and the off-diagonal matrix elements ρij (with i ̸= j) are the coherences between the states i
+and j with complex conjugate ¯ρij = ρji. The second term in the equation, Lρ, accounts for the
+dissipation in the Lindblad form.
+First, we apply the rotating-wave approximation (RWA) to remove the time-dependent terms
+in the Hamiltonian of Eq. 1. Using ρii = �ρii, ρeg = �ρege−iωt, ρpg = �ρpge−iωt, and ρep = �ρep, the
+Hamiltonian in RWA modifies to:
+�H = ℏ
+�
+�
+�
+�
+�
+0
+1
+2Ωeg
+1
+2Ωpg
+1
+2Ωeg
+∆eg
+1
+2Ωc
+1
+2Ωpg
+1
+2Ωc
+∆pg
+�
+�
+�
+�
+� ,
+(3)
+where ∆eg = ωeg − ω and ∆pg = ωpg − ω are the respective detunings between the transition
+frequencies ωeg and ωpg and the laser frequency ω. Omitting the tilde superscript in the following
+for simplicity, we describe the dynamic evolution of the density matrix elements by the master
+5
+
+equation in the Lindblad form as:
+˙ρgg = −i
+�Ωeg
+2 (ρge − ρeg) + Ωpg
+2 (ρpg − ρgp)
+�
++ Γeeρee + Γppρpp
+˙ρee = −i
+�Ωeg
+2 (ρeg − ρge) + Ωc
+2 (ρpg − ρgp)
+�
+− Γeeρee
+˙ρpp = −i
+�Ωpg
+2 (ρgp − ρpg) + Ωc
+2 (ρep − ρpe)
+�
+− Γppρpp
+˙ρge = −i
+�Ωeg
+2 (ρee − ρgg) − ∆egρge + Ωpg
+2 ρpe − Ωc
+2 ρgp
+�
+− Γegρge
+2
+˙ρgp = −i
+�Ωpg
+2 (ρpp − ρgg) + Ωeg
+2 ρep − Ωc
+2 ρge − ∆pgρgp
+�
+− Γpgρgp
+2
+˙ρep = −i
+�Ωeg
+2 ρgp − Ωpg
+2 ρeg + ∆egρep + Ωc
+2 (ρpp − ρee) − ∆pgρep
+�
+.
+(4)
+Dissipation is introduced in the above master equation via population decay rates, Γee and Γpp,
+and the decoherence rates, Γeg and Γpg [4]. The population decay rate includes contributions from
+radiative and non-radiative decay channels, Γii = Γr
+ii + Γnr
+ii , respectively. The decoherence rate is
+related to the population decay rate as Γig = Γii + 2γig,deph where γig,deph is the pure dephasing
+rate.
+The Rabi frequency Ωig between the excited state |i⟩ (|i⟩ = |e⟩ or |p⟩) and the ground state |g⟩
+is given by:
+Ωig = µigE
+ℏ
+=
+�
+6πc2ΓiiP
+ℏnω3
+igA
+(5)
+where µig is the dipole moment of the transition, E is the electric field, c is the speed of light, Γii
+is the population decay rate of the state |i⟩, P is the illumination laser power, n is the refractive
+index, ωig is the transition frequency from the excited state |i⟩ to the ground state |g⟩, and A is
+the illumination area. For the transition between the exciton state |e⟩ and the ground state |g⟩,
+there are two distinct Rabi frequencies, namely the coupled Rabi frequency Ωig,c frequency for
+excitons in the near-field of plasmonic nanodisks affected by the plasmonic enhancement of the
+electric field, and the uncoupled Rabi frequency Ωig,u of excitons away from Au nanodisk but still
+excited within the optical focal spot. The Rabi frequency of coupled exciton fraction is related to
+the Rabi frequency of the uncoupled fraction as Ωeg,c = Ωeg,u
+√
+FAc/Au, where Ac is the overlap
+area between the monolayer and the nanodisk with plasmonic near-field enhancement, Au is the
+area of uncoupled excitons within the optical focal spot, and F is the enhancement factor of the
+electric field obtained from the FDTD calculation.
+6
+
+Solving the set of coupled equations for steady state where ˙ρij = 0, and using population con-
+servation in the three-level system as ρgg + ρee + ρpp = 1, we obtain analytical expressions for the
+populations and coherences in the coupled system as a function of experimental parameters. The
+optical response is obtained from the imaginary part of the coherence matrix elements [5]. In our
+system, the focal spot consists of an exciton-plasmon hybrid region showing coherent interference
+and also a region with uncoupled excitons in monolayer away from the nanodisk. Therefore, we
+obtain the extinction E in the optical response as the sum of the coherence matrix elements of two
+coupled states and the uncoupled exciton state as follows:
+E ∝ αe
+Ωeg,cΓr
+ee
+Ω2
+eg,c + Ω2
+pg
+Im(ρge) + αp
+ΩpgΓr
+pp
+Ω2
+eg,c + Ω2
+pg
+Im(ρgp) + αe
+Γr
+ee
+Ωeg,uc
+Im(ρge,u),
+(6)
+where αe and αp correspond to the maximum on-resonance scattering contrast of the exciton and
+plasmon transition with radiative decay rates Γr
+ee and Γr
+pp. The respective contrasts αi are obtained
+as [6]:
+αi = 3
+2π
+c2
+n2ω2
+igA.
+(7)
+We note here that the extinction is not quantitative as all reflecting surfaces of the sample are not
+taken into account to calculate the Rayleigh scattering field based on the procedure detailed in [6].
+However, the response spectrum is proportional to the imaginary part of the coupled coherences
+ρeg and ρpg, and the uncoupled exciton coherence ρeg,u as a function of frequency, and we obtain
+correct qualitative description of differential reflection.
+The parameters to obtain the extinction are known from the experiment or literature. For the
+exciton state |e⟩, the population decay due to radiative recombination is ℏΓee = ℏΓr
+ee ≃ 1 meV
+[7, 8]. The experimental FWHM linewidth is determined for the monolayer TMD region (Fig.
+1c) as ℏΓeg ≃ 8 meV, and we treat the broadening beyond the transform-limit of 1 meV as pure
+dephasing. For the plasmon state |p⟩, the experimental FWHM linewidth is ℏΓpg ≃ 180 meV.
+We neglect pure dephasing processes such that Γpg = Γpp [9]. The radiative contribution in the
+plasmonic state |p⟩ for the gold nanodisk is estimated to be ≃ 46 %, such that ℏΓr
+pp ≃ 82 meV [9].
+The Rabi frequencies Ωeg,u and Ωpg are calculated using an illumination laser power P = 1 nW,
+illumination area A ≃ 1 µm2, and the corresponding linewidth and resonance frequency. The Rabi
+frequency of excitons coupled to the plasmons Ωeg,c is calculated using the enhancement factor,
+F = 50, and the coupled and uncoupled areas, Ac and Au, determined from the FDTD simulation
+shown in Fig. 2. The coupling constant is estimated from the experiment to be ℏΩc ≃ 28 meV
+(see main text, Fig. 2e). A comparison between experimental differential reflectance spectra and
+7
+
+0.0
+0.5
+1.0
+d = -38 meV
+d = -15 meV
+d = -10 meV
+d = 13 meV
+d = 32 meV
+1.6
+1.7
+1.8
+0.0
+0.5
+1.0
+Energy (eV)
+1.6
+1.7
+1.8
+Energy (eV)
+1.6
+1.7
+1.8
+Energy (eV)
+1.6
+1.7
+1.8
+Energy (eV)
+1.6
+1.7
+1.8
+Energy (eV)
+Differential reflection, DR
+Normalized extinction
+a
+b
+c
+d
+e
+f
+g
+h
+i
+j
+FIG. 3. a – e, Differential reflectance spectra of the coupled system for various detunings δ = ℏωpg −
+ℏωeg. The spectra were recorded on different spatial positions with different plasmon frequencies ωpg and
+a weakly varying exciton frequency ωeg. f – j, Respective normalized extinction obtained from the Fano
+model with exciton-plasmon coupling strength ℏΩc = 28 meV.
+extinction response from the model across different detunings is presented in Fig. 3 with very good
+agreement between the experiment and the model across a wide range of energy detunings.
+To determine the magneto-induced circular dichroism (CD), we calculate the extinction for
+different polarizations of light. Light with σ+ (σ−) polarization only couples to the K (K′) valley
+of the monolayer exciton transition. These opposite valleys have different resonance energies ωeg
+in the presence of an out-of-plane magnetic field according to the valley Zeeman splitting with the
+characteristic exciton g-factor of 4 [10–16]. The optical response for σ+ and σ− polarization is
+normalized by the sum of two optical responses in the calculation of CD, leading to the cancellation
+of the terms due to various reflecting surfaces in the device. Therefore, the calculated model CD
+values shown in main text are quantitative.
+8
+
+SUPPORTING NOTE 3: MAGNETO INDUCED CHIRALITY OF THE COUPLED SYSTEM
+IN DIFFERENTIAL REFLECTION
+σ+
+σ-
+9T
+Circular dichroism, CD
+Differential reflection, DR
+0.6
+0.8
+1.0
+1.71
+1.73
+-0.08
+-0.04
+0.00
+0.04
+0.08
+Energy (eV)
+FIG. 4. a, Differential reflectance spectra of the coupled system around the Fano interference induced dip
+for different circular polarizations at a magnetic field of 9T. The transfer of the magneto induced chirality
+from the monolayer exciton to the coupled exciton-plasmon system is observed as an energy shift of the
+dip position for different polarization. The data of the two spectra for σ+ and σ− detection lead to the CD
+spectrum shown in Fig.4c in the main text and in b here, for better comparability.
+References
+[1] Palacios-Berraquero, C. et al. Large-scale quantum-emitter arrays in atomically thin semiconductors.
+Nat. Commun. 8, 1 (2017).
+[2] Branny, A., Kumar, S., Proux, R. & Gerardot, B. D. Deterministic strain-induced arrays of quantum
+emitters in a two-dimensional semiconductor. Nat. Commun. 8, 1 (2017).
+[3] Loudon, R. The Quantum Theory of Light (Oxford University Press, 2000).
+[4] Rand, S. Lectures on Light: Nonlinear and Quantum Optics using the Density Matrix (Oxford Uni-
+versity Press, 2016).
+9
+
+[5] Gerardot, B. D. et al. Optical pumping of a single hole spin in a quantum dot. Nature 451, 441 (2008).
+[6] Karrai, K. & Warburton, R. J. Optical transmission and reflection spectroscopy of single quantum
+dots. Superlattices Microst. 33, 311 (2003).
+[7] Glazov, M. M. et al. Exciton fine structure and spin decoherence in monolayers of transition metal
+dichalcogenides. Phys. Rev. B 89, 201302 (2014).
+[8] Moody, G. et al. Intrinsic homogeneous linewidth and broadening mechanisms of excitons in mono-
+layer transition metal dichalcogenides. Nat. Commun. 6, 1 (2015).
+[9] S¨onnichsen, C. et al. Drastic reduction of plasmon damping in gold nanorods. Phys. Rev. Lett. 88,
+774021 (2002).
+[10] Srivastava, A. et al. Valley Zeeman effect in elementary optical excitations of monolayer WSe2. Nat.
+Phys. 11, 141 (2015).
+[11] Wang, G. et al. Magneto-optics in transition metal diselenide monolayers. 2D Mater. 2, 034002
+(2015).
+[12] Koperski, M. et al. Orbital, spin and valley contributions to Zeeman splitting of excitonic resonances
+in MoSe2, WSe2 and WS2 monolayers. 2D Mater. 6, 015001 (2018).
+[13] Wo´zniak, T., Faria Junior, P. E., Seifert, G., Chaves, A. & Kunstmann, J. Exciton g factors of van der
+waals heterostructures from first-principles calculations. Phys. Rev. B 101, 235408 (2020).
+[14] F¨orste, J. et al. Exciton g-factors in monolayer and bilayer WSe2 from experiment and theory. Nat.
+Commun. 11, 4539 (2020).
+[15] Deilmann, T., Kr¨uger, P. & Rohlfing, M. Ab initio studies of exciton g factors: Monolayer transition
+metal dichalcogenides in magnetic fields. Phys. Rev. Lett. 124, 226402 (2020).
+[16] Xuan, F. & Quek, S. Y. Valley Zeeman effect and Landau levels in two-dimensional transition metal
+dichalcogenides. Phys. Rev. Research 2, 033256 (2020).
+10
+
diff --git a/ztE1T4oBgHgl3EQfkgSy/content/tmp_files/load_file.txt b/ztE1T4oBgHgl3EQfkgSy/content/tmp_files/load_file.txt
new file mode 100644
index 0000000000000000000000000000000000000000..e07eaa4a3822776af82f12e698cbf089df0303fb
--- /dev/null
+++ b/ztE1T4oBgHgl3EQfkgSy/content/tmp_files/load_file.txt
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+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfkgSy/content/2301.03275v1.pdf,len=933
+page_content='Magneto-optical chirality in a coherently coupled exciton-plasmon system  Samarth Vadia,1, 2, 3, ∗ Johannes Scherzer,1, ∗ Kenji Watanabe,4 Takashi Taniguchi,5 and Alexander H¨ogele1, 2  1Fakult¨at f¨ur Physik, Munich Quantum Center, and Center for NanoScience (CeNS),  Ludwig-Maximilians-Universit¨at M¨unchen, Geschwister-Scholl-Platz 1, 80539 M¨unchen, Germany  2Munich Center for Quantum Science and Technology (MCQST), Schellingtr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfkgSy/content/2301.03275v1.pdf'}
+page_content=' 4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfkgSy/content/2301.03275v1.pdf'}
+page_content=' 80799 M¨unchen,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfkgSy/content/2301.03275v1.pdf'}
+page_content=' Germany  3attocube Systems AG,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfkgSy/content/2301.03275v1.pdf'}
+page_content=' Eglfinger Weg 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfkgSy/content/2301.03275v1.pdf'}
+page_content=' 85540 Haar,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfkgSy/content/2301.03275v1.pdf'}
+page_content=' Germany  4Research Center for Functional Materials,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfkgSy/content/2301.03275v1.pdf'}
+page_content=' National Institute for Materials Science,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfkgSy/content/2301.03275v1.pdf'}
+page_content=' 1-1 Namiki,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfkgSy/content/2301.03275v1.pdf'}
+page_content=' Tsukuba 305-0044,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfkgSy/content/2301.03275v1.pdf'}
+page_content=' Japan  5International Center for Materials Nanoarchitectonics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfkgSy/content/2301.03275v1.pdf'}
+page_content='  National Institute for Materials Science,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfkgSy/content/2301.03275v1.pdf'}
+page_content=' 1-1 Namiki,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfkgSy/content/2301.03275v1.pdf'}
+page_content=' Tsukuba 305-0044,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfkgSy/content/2301.03275v1.pdf'}
+page_content=' Japan  (Dated: January 10,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfkgSy/content/2301.03275v1.pdf'}
+page_content=' 2023)  Chirality is a fundamental asymmetry phenomenon,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfkgSy/content/2301.03275v1.pdf'}
+page_content=' with chiral optical elements exhibiting asymmetric re-  sponse in reflection or absorption of circularly polarized light.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfkgSy/content/2301.03275v1.pdf'}
+page_content=' Recent realizations of such elements include  nanoplasmonic systems with broken mirror symmetry and polarization-contrasting optical absorption known  as circular dichroism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfkgSy/content/2301.03275v1.pdf'}
+page_content=' An alternative route to circular dichroism is provided by spin-valley polarized excitons  in atomically thin semiconductors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfkgSy/content/2301.03275v1.pdf'}
+page_content=' In the presence of magnetic fields, they exhibit an imbalanced coupling to  circularly polarized photons and thus circular dichroism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfkgSy/content/2301.03275v1.pdf'}
+page_content=' Here, we demonstrate that polarization-contrasting  optical transitions associated with excitons in monolayer WSe2 can be transferred to proximal plasmonic nan-  odisks by coherent coupling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfkgSy/content/2301.03275v1.pdf'}
+page_content=' The coupled exciton-plasmon system exhibits magneto-induced circular dichroism  in a spectrally narrow window of Fano interference, which we model in a master equation framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfkgSy/content/2301.03275v1.pdf'}
+page_content=' Our work  motivates exciton-plasmon interfaces as building blocks of chiral metasurfaces for applications in information  processing, non-linear optics and sensing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfkgSy/content/2301.03275v1.pdf'}
+page_content='  Direct band gap and reduced dielectric screening in  semiconducting monolayer transition metal dichalcogenides  (TMDs) [1, 2] give rise to tightly bound excitons [3] with siz-  able light-matter interactions that facilitate efficient coupling  to dielectric or plasmonic systems [4–6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfkgSy/content/2301.03275v1.pdf'}
+page_content=' Capitalizing on the  large oscillator strength of TMD excitons and the flexibility  of combining them with plasmonic structures, recent exam-  ples of coupled exciton-plasmon systems include realizations  in the strong and weak coupling regimes [7–10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfkgSy/content/2301.03275v1.pdf'}
+page_content=' While the  former is characterized by the formation of exciton-plasmon  polaritons, the latter is distinguished by Fano-type interfer-  ence spectra, as discussed in early work on various exciton-  plasmon coupled systems [11–13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfkgSy/content/2301.03275v1.pdf'}
+page_content='  In this framework, the  dipolar selection rules of spin-valley polarized excitons in  TMD monolayers [14–18] provide a route to chiral optical  phenomena, as the valley degeneracy can be lifted by mag-  netic field to induce spectrally imbalanced coupling to left-  and right-handed circularly polarized photons [19–23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfkgSy/content/2301.03275v1.pdf'}
+page_content=' This  opto-valleytronic feature of TMD monolayer excitons has  been utilized to demonstrate chiral effects such as directional  coupling of light in silver nanowires on WS2 [24], spatial sep-  aration of valley-polarized excitons by silver nanogroove ar-  rays [25], or second-harmonic generation of circularly polar-  ized photons in gold-WS2 metasurfaces [26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfkgSy/content/2301.03275v1.pdf'}
+page_content='  Here, we study magneto-optical characteristics of an  exciton-plasmon metasurface based on a WSe2 monolayer  and gold (Au) nanodisks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfkgSy/content/2301.03275v1.pdf'}
+page_content=' We elucidate the effect of Fano  interference as a function of exciton-plasmon spectral reso-  nance detuning in the weak coupling regime [10, 27–31], and  study both experimentally and theoretically the polarization  properties of the coherently coupled system in the presence  of external magnetic fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfkgSy/content/2301.03275v1.pdf'}
+page_content='  Remarkably, the coupled sys-  ∗ These authors contributed equally to this work  tem exhibits magnetic circular dichroism that is distinct from  the characteristics of the fundamental valley-polarized exci-  ton transition in monolayer WSe2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfkgSy/content/2301.03275v1.pdf'}
+page_content=' The resulting chiral be-  havior of the Fano-coupled metasurface manifests in the form  of a spectral window with polarization dependent reflectivity  in an otherwise broadband opaque medium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfkgSy/content/2301.03275v1.pdf'}
+page_content=' The observations  are substantiated by a master equation analysis with excel-  lent quantitative agreement with experimental findings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfkgSy/content/2301.03275v1.pdf'}
+page_content=' Our  work provides insight into the underlying coherent interfer-  ence phenomena and can serve as a guideline to the design  of exciton-plasmon metasurfaces with optical chirality in the  visible spectral range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfkgSy/content/2301.03275v1.pdf'}
+page_content='  The exciton-plasmon interface was fabricated by encapsu-  lating a monolayer WSe2 in hexagonal boron nitride (hBN)  and placing the resulting heterostructure on top of a plasmonic  Au nanodisk array on a SiO2/Si substrate, as illustrated in  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfkgSy/content/2301.03275v1.pdf'}
+page_content=' 1a (see Methods for sample details).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfkgSy/content/2301.03275v1.pdf'}
+page_content=' The sample fea-  tures regions of encapsulated WSe2 monolayer, Au nanodisk  arrays, as well as regions where both elements are combined  in vertical proximity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfkgSy/content/2301.03275v1.pdf'}
+page_content=' To characterize the optical responses of  the bare exciton and plasmon systems, and the regime of their  coupling, we performed differential reflection spectroscopy  at cryogenic temperatures (see Methods for experimental de-  tails).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfkgSy/content/2301.03275v1.pdf'}
+page_content=' The corresponding spectra are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfkgSy/content/2301.03275v1.pdf'}
+page_content=' 1c, d and  e, where differential reflection DR = (Rsub − R)/(Rsub)  was measured relative to the reflection Rsub of the SiO2/Si  substrate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfkgSy/content/2301.03275v1.pdf'}
+page_content=' The coupled system can be modelled in a three-  level system framework with relevant states and rates shown  in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfkgSy/content/2301.03275v1.pdf'}
+page_content=' 1e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfkgSy/content/2301.03275v1.pdf'}
+page_content=' The DR spectra in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfkgSy/content/2301.03275v1.pdf'}
+page_content=' 1c, d and e are representa-  tive for the excitation from the ground state |g⟩ to the exciton  state |e⟩ with resonance and Rabi frequency ωeg and Ωeg and  decay rate Γeg (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfkgSy/content/2301.03275v1.pdf'}
+page_content=' 1c), the optical excitation to the plasmon  state |p⟩ with resonance and Rabi frequencies ωpg and Ωpg  and decay rate Γpg (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfkgSy/content/2301.03275v1.pdf'}
+page_content=' 1d), and the simultaneous excitation  of the interacting exciton-plasmon system with coherent cou-  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfkgSy/content/2301.03275v1.pdf'}
+page_content='03275v1  [physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfkgSy/content/2301.03275v1.pdf'}
+page_content='optics]  9 Jan 2023  2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfkgSy/content/2301.03275v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfkgSy/content/2301.03275v1.pdf'}
+page_content='5  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfkgSy/content/2301.03275v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfkgSy/content/2301.03275v1.pdf'}
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+page_content='5  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfkgSy/content/2301.03275v1.pdf'}
+page_content='0  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfkgSy/content/2301.03275v1.pdf'}
+page_content='6  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfkgSy/content/2301.03275v1.pdf'}
+page_content='7  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfkgSy/content/2301.03275v1.pdf'}
+page_content='8  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfkgSy/content/2301.03275v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfkgSy/content/2301.03275v1.pdf'}
+page_content='5  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfkgSy/content/2301.03275v1.pdf'}
+page_content="0  Energy [eV]  ۧ  ȁ𝑔  ۧ  ȁ𝑒  ۧ  ȁ𝑝  Ω𝑒𝑔  Ω𝑝𝑔  Γ𝑒𝑔  Γ𝑝𝑔  Ωc  a  f  b  c  d  σ+  σ-  Differential reflection, DR  ω𝑒𝑔  ω𝑝𝑔  K  K'  σ+  σ-  e  CB  VB  FIG." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfkgSy/content/2301.03275v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfkgSy/content/2301.03275v1.pdf'}
+page_content=' Fano interference in a coupled exciton-plasmon system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfkgSy/content/2301.03275v1.pdf'}
+page_content='  a, Schematic illustration of monolayer WSe2 encapsulated in hBN,  placed on gold nanodisks and probed with circularly polarized light.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfkgSy/content/2301.03275v1.pdf'}
+page_content='  b, Band structure schematics of monolayer WSe2 at two opposite  corners of the hexagonal Brillouin zone, with σ+ (σ−) circularly  polarized optical transitions between spin-up (spin-down) polarized  states at the K (K’) valleys of the conduction band (CB) and valence  band (VB).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfkgSy/content/2301.03275v1.pdf'}
+page_content=' c, d, and e, Differential reflection spectra of monolayer  WSe2, gold nanodisk, and the coupled system, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfkgSy/content/2301.03275v1.pdf'}
+page_content=' f, En-  ergy levels of the coupled system with |g⟩, |e⟩ and |p⟩ denoting the  ground, exciton and plasmon state, respectively, and the correspond-  ing exciton and plasmon optical transition frequencies ωeg and ωpg,  Rabi frequencies Ωeg and Ωpg, and radiative decay rates Γeg and  Γpg, as well as the exciton-plasmon coupling strength Ωc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfkgSy/content/2301.03275v1.pdf'}
+page_content='  pling constant Ωc (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfkgSy/content/2301.03275v1.pdf'}
+page_content=' 1e).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfkgSy/content/2301.03275v1.pdf'}
+page_content='  The band structure of WSe2 monolayer is shown schemat-  ically in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfkgSy/content/2301.03275v1.pdf'}
+page_content=' 1b, together with polarization-contrasting op-  tical transitions of the fundamental exciton X0 in the K  and K′ valleys of the hexagonal Brillouin zone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfkgSy/content/2301.03275v1.pdf'}
+page_content=' The tran-  sitions are degenerate at zero magnetic field, resulting in a  single Lorentzian peak in the DR spectrum of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfkgSy/content/2301.03275v1.pdf'}
+page_content=' 1b at  ℏωeg ≃ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfkgSy/content/2301.03275v1.pdf'}
+page_content='723 eV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfkgSy/content/2301.03275v1.pdf'}
+page_content=' The full-width at half-maximum (FWHM)  linewidth of the exciton transition ℏΓeg ≃ 8 meV is substan-  tially smaller than the plasmon linewidth of ℏΓpg ≃ 180 meV  obtained from the region of a bare Au nanodisk array in Fig-  ure 1d with resonance energy at ℏωpg = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfkgSy/content/2301.03275v1.pdf'}
+page_content='72 eV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfkgSy/content/2301.03275v1.pdf'}
+page_content=' Due to  variations in the dielectric environment of the nanodisk ar-  ray, the plasmon resonance energy ℏωpg varies in the range  of 100 meV (see Supplementary Section 1 for the description  of gold nanodisk arrays).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfkgSy/content/2301.03275v1.pdf'}
+page_content=' In our studies, this variation bene-  ficially provides position-dependent spectral energy detuning  δ = ℏωpg − ℏωeg of the plasmon resonance energy with re-  spect to the exciton resonance which has negligible variations  across the sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfkgSy/content/2301.03275v1.pdf'}
+page_content='  The spectrum of the coupled system in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfkgSy/content/2301.03275v1.pdf'}
+page_content=' 1e is character-  ized by a Fano interference lineshape [27, 28, 32, 33] with a  narrow reflection dip in the broad plasmonic extinction peak.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfkgSy/content/2301.03275v1.pdf'}
+page_content='  A closer inspection of the spectrum reveals an additional peak  superimposed on the reflection dip, which we ascribe to the  contribution from uncoupled excitons that are located within  the optical spot yet sufficiently far away from plasmonic nan-  odisks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfkgSy/content/2301.03275v1.pdf'}
+page_content=' To interpret the resulting lineshape, we inspected sys-  tem realizations with different resonance detunings at differ-  ent spatial positions of the interfaced array.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfkgSy/content/2301.03275v1.pdf'}
+page_content=' Two representa-  tive DR spectra for negative and positive resonance detuning  δ are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfkgSy/content/2301.03275v1.pdf'}
+page_content=' 2a and b, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfkgSy/content/2301.03275v1.pdf'}
+page_content=' In both spectra, the  position of the dip remains essentially constant due to small  variation in the exciton energy across the sample, which also  holds for the uncoupled exciton peak inside the Fano dip.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfkgSy/content/2301.03275v1.pdf'}
+page_content=' Due  to the asymmetric character of the Fano interference, however,  the overall spectral shape is strongly modified at different res-  onance conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfkgSy/content/2301.03275v1.pdf'}
+page_content='  We model this intricate optical response in the framework  of two coherently coupled oscillators using a classical light  field interacting with exciton and plasmon dipolar excitations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfkgSy/content/2301.03275v1.pdf'}
+page_content='  The dipole moments of the respective optical transitions are  given as the imaginary components of the quantum coherence  obtained from the master equation analysis (see Supplemen-  tary Section 2 for theoretical modeling of the extinction spec-  trum).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfkgSy/content/2301.03275v1.pdf'}
+page_content=' All main parameters of the system including the decay  rates Γeg, Γpg and the Rabi frequencies Ωeg, Ωpg were de-  termined from experiments on bare system components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfkgSy/content/2301.03275v1.pdf'}
+page_content=' To  quantify the coupling strength Ωc, we plot the spectral posi-  tion of the two maxima enclosing the dip in the Fano spec-  tra as a function of detuning δ in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfkgSy/content/2301.03275v1.pdf'}
+page_content=' 2e, with their energy  separation reproduced by the theoretical model for a coupling  strength ℏΩc = 28 meV, as shown by solid lines in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfkgSy/content/2301.03275v1.pdf'}
+page_content=' 2e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfkgSy/content/2301.03275v1.pdf'}
+page_content='  With this coupling, our model yields the normalized extinc-  tion spectra shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfkgSy/content/2301.03275v1.pdf'}
+page_content=' 2c and d for resonance detunings  δ = −38 and 32 meV as extracted from the spectra in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfkgSy/content/2301.03275v1.pdf'}
+page_content=' 2a  and b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfkgSy/content/2301.03275v1.pdf'}
+page_content=' All features of the optical response are reproduced with  good agreement by the theoretical model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfkgSy/content/2301.03275v1.pdf'}
+page_content='  With this understanding of the Fano interference phenom-  ena, we elucidate in the following the magneto-optical re-  sponse of the coupled exciton-plasmon system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfkgSy/content/2301.03275v1.pdf'}
+page_content='  First, we  quantify the degree of circular dichroism (CD) associated with  the exciton valley Zeeman effect in monolayer WSe2 [20–23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfkgSy/content/2301.03275v1.pdf'}
+page_content='  In the presence of an out-of-plane magnetic field of 9 T, circu-  larly polarized DR spectra of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfkgSy/content/2301.03275v1.pdf'}
+page_content=' 3a and b reveal two exciton  resonances associated with K and K′ transitions that couple  to σ+ and σ− polarized light, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfkgSy/content/2301.03275v1.pdf'}
+page_content=' The valley Zee-  R3  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfkgSy/content/2301.03275v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfkgSy/content/2301.03275v1.pdf'}
+page_content='5  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfkgSy/content/2301.03275v1.pdf'}
+page_content='0  d = -38 meV  d = 32 meV  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfkgSy/content/2301.03275v1.pdf'}
+page_content='6  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfkgSy/content/2301.03275v1.pdf'}
+page_content='7  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfkgSy/content/2301.03275v1.pdf'}
+page_content='8  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfkgSy/content/2301.03275v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfkgSy/content/2301.03275v1.pdf'}
+page_content='5  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfkgSy/content/2301.03275v1.pdf'}
+page_content='0  Energy (eV)  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfkgSy/content/2301.03275v1.pdf'}
+page_content='6  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfkgSy/content/2301.03275v1.pdf'}
+page_content='7  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfkgSy/content/2301.03275v1.pdf'}
+page_content='8  Energy (eV)  a  c  b  e  Ω𝑐 = 28 meV  d  Differential reflection, DR  Normalized extinction  40  20  0  20  40  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfkgSy/content/2301.03275v1.pdf'}
+page_content='68  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfkgSy/content/2301.03275v1.pdf'}
+page_content='70  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfkgSy/content/2301.03275v1.pdf'}
+page_content='72  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfkgSy/content/2301.03275v1.pdf'}
+page_content='74  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfkgSy/content/2301.03275v1.pdf'}
+page_content='76  Energy (eV)  Detuning d (meV)  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfkgSy/content/2301.03275v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfkgSy/content/2301.03275v1.pdf'}
+page_content=' Exciton-plasmon Fano interference in experiment and  theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfkgSy/content/2301.03275v1.pdf'}
+page_content=' a and b, Normalized differential reflection spectra of the  coupled system for −38 and 32 meV energy detuning from the  exciton-plasmon spectral resonance condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfkgSy/content/2301.03275v1.pdf'}
+page_content='  The spectra were  recorded on two positions of the nanodisk array with different plas-  mon energies ℏωpg (determined from Lorentzian fits) for a weakly  varying exciton energy ℏωeg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfkgSy/content/2301.03275v1.pdf'}
+page_content=' c and d, Respective Fano model spec-  tra with exciton-plasmon coupling strength ℏΩc = 28 meV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfkgSy/content/2301.03275v1.pdf'}
+page_content=' e, Evo-  lution of the exciton-plasmon coupling as a function of resonance  energy detuning δ, with red data points corresponding to plasmonic  array regions with different ℏωpg and respective model results (black  lines) obtained with the bare exciton energy ℏωeg = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfkgSy/content/2301.03275v1.pdf'}
+page_content='723 eV (dot-  ted line) and ℏΩc = 28 meV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfkgSy/content/2301.03275v1.pdf'}
+page_content='  man splitting of 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfkgSy/content/2301.03275v1.pdf'}
+page_content='1 meV corresponds to the exciton Land´e  factor with an absolute value of 4 as expected from previous  experiments [21–23] and theory [34–37].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfkgSy/content/2301.03275v1.pdf'}
+page_content='  The polarization-contrasting response of the two valleys  is quantified by CD, which calculates as CD = (DR− −  DR+)/(DR− + DR+), where DR+ and DR− are the the σ+  and σ− polarized DR spectra respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfkgSy/content/2301.03275v1.pdf'}
+page_content=' Figure 3c shows  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfkgSy/content/2301.03275v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfkgSy/content/2301.03275v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfkgSy/content/2301.03275v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfkgSy/content/2301.03275v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfkgSy/content/2301.03275v1.pdf'}
+page_content='8  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfkgSy/content/2301.03275v1.pdf'}
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+page_content='0  Energy (eV)  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfkgSy/content/2301.03275v1.pdf'}
+page_content='71  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfkgSy/content/2301.03275v1.pdf'}
+page_content='73  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfkgSy/content/2301.03275v1.pdf'}
+page_content='75  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfkgSy/content/2301.03275v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfkgSy/content/2301.03275v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfkgSy/content/2301.03275v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfkgSy/content/2301.03275v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfkgSy/content/2301.03275v1.pdf'}
+page_content="4  Energy (eV)  a  c  Circular dichroism, CD  Differential reflection, DR  d  b  9T  9T  9T  9T  K  K'  σ+  σ-  σ+  σ-  FIG." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfkgSy/content/2301.03275v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfkgSy/content/2301.03275v1.pdf'}
+page_content=' Magnetic circular dichroism of the exciton transition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfkgSy/content/2301.03275v1.pdf'}
+page_content='  a and b, Valley-selective differential reflection spectra at 9 T for  σ+ and σ− polarized excitation, respectively (solid lines show  Lorentzian fits).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfkgSy/content/2301.03275v1.pdf'}
+page_content=' c and d, Circular dichroism of the exciton transition  at 9 (red) and −9 T (blue) from experiment and theory, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfkgSy/content/2301.03275v1.pdf'}
+page_content='  the CD at 9 T as red solid line, where DR+ and DR− are the  Lorentzian fits to the σ+ and σ− polarized DR spectra shown  as solid lines in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfkgSy/content/2301.03275v1.pdf'}
+page_content=' 3a and b, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfkgSy/content/2301.03275v1.pdf'}
+page_content=' It shows a rever-  sal in polarity around the resonance energy of the exciton at  0 T with maximum CD of ∼ 20%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfkgSy/content/2301.03275v1.pdf'}
+page_content=' The CD obtained from  the corresponding experiments at −9 T, shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfkgSy/content/2301.03275v1.pdf'}
+page_content=' 3c in  blue, is reversed in sign for the entire spectral range (devia-  tions from the mirror-symmetry around the exciton energy at  zero field stem from sample inhomogeneities sampled by spa-  tial displacements in magnetic field over a range of 18 T).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfkgSy/content/2301.03275v1.pdf'}
+page_content=' All  main features of the spectra are captured by our model CD  spectra in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfkgSy/content/2301.03275v1.pdf'}
+page_content=' 3d obtained from extinction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfkgSy/content/2301.03275v1.pdf'}
+page_content='  Next, we study the polarization-dependent optical response  of the interface region in the presence of a magnetic field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfkgSy/content/2301.03275v1.pdf'}
+page_content=' The  valley Zeeman effect of the bare exciton is imprinted on the  coupled exciton-plasmon system in an intricate way and man-  ifests as polarization-dependent reflectance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfkgSy/content/2301.03275v1.pdf'}
+page_content=' Our theory cap-  tures the experimentally observed features of the CD spectra  and their evolution with the magnetic field, as evidenced by  comparing experimental and theoretical data shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfkgSy/content/2301.03275v1.pdf'}
+page_content=' 4a  and b and Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfkgSy/content/2301.03275v1.pdf'}
+page_content=' 4c and d, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfkgSy/content/2301.03275v1.pdf'}
+page_content=' The evolution of CD  with increasing magnetic fields of 3, 6 and 9 T, recorded in the  region of the Fano interference with −15 meV resonance de-  tuning, is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfkgSy/content/2301.03275v1.pdf'}
+page_content=' 4a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfkgSy/content/2301.03275v1.pdf'}
+page_content=' The magneto-induced dichroism  becomes increasingly pronounced with increasing magnetic  field as for the bare WSe2 monolayer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfkgSy/content/2301.03275v1.pdf'}
+page_content=' Notably, the spectra in  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfkgSy/content/2301.03275v1.pdf'}
+page_content=' 4c show a sign reversal of the CD response for the cou-  4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfkgSy/content/2301.03275v1.pdf'}
+page_content='08  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfkgSy/content/2301.03275v1.pdf'}
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+page_content='04  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfkgSy/content/2301.03275v1.pdf'}
+page_content='08  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfkgSy/content/2301.03275v1.pdf'}
+page_content='71  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfkgSy/content/2301.03275v1.pdf'}
+page_content='73  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfkgSy/content/2301.03275v1.pdf'}
+page_content='08  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfkgSy/content/2301.03275v1.pdf'}
+page_content='04  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfkgSy/content/2301.03275v1.pdf'}
+page_content='00  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfkgSy/content/2301.03275v1.pdf'}
+page_content='04  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfkgSy/content/2301.03275v1.pdf'}
+page_content='08  Energy (eV)  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfkgSy/content/2301.03275v1.pdf'}
+page_content='71  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfkgSy/content/2301.03275v1.pdf'}
+page_content='73  Energy (eV)  a  b  c  d  Circular dichroism, CD  9T  9T  9T  9T  3T  6T  9T  3T  6T  9T  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfkgSy/content/2301.03275v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfkgSy/content/2301.03275v1.pdf'}
+page_content='  Magnetic circular dichroism of the coupled exciton-  plasmon system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfkgSy/content/2301.03275v1.pdf'}
+page_content=' a and b, Experimental and theoretical circular  dichroism spectra of the coupled exciton-plasmon system at 3, 6 and  9 T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfkgSy/content/2301.03275v1.pdf'}
+page_content=' c and d, Experimental and theoretical circular dichroism spectra  in magnetic fields of 9 (red) and −9 T (blue).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfkgSy/content/2301.03275v1.pdf'}
+page_content='  pled system in comparison to the bare exciton case in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfkgSy/content/2301.03275v1.pdf'}
+page_content=' 3c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfkgSy/content/2301.03275v1.pdf'}
+page_content='  While the monolayer features a peak in the exciton DR spec-  trum, the coupled exciton-plasmon spectrum is characterized  by a narrow dip around the exciton resonance as a result of  Fano interference, leading to reversed CD response and pro-  nounced asymmetry for finite exciton-plasmon detunings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfkgSy/content/2301.03275v1.pdf'}
+page_content=' In  the calculated spectra of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfkgSy/content/2301.03275v1.pdf'}
+page_content=' 4d, the asymmetry manifests in  the form of an additional dip in the CD spectra, in qualitative  agreement with the experimental spectra in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfkgSy/content/2301.03275v1.pdf'}
+page_content=' 4c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfkgSy/content/2301.03275v1.pdf'}
+page_content=' The con-  tribution of uncoupled excitons within the optical focal spot  with a reversed sign of CD explains this observation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfkgSy/content/2301.03275v1.pdf'}
+page_content=' A max-  imum CD of up to ∼ 7 % is achieved in the coupled exciton-  plasmon system, indicating a significant transfer of the opto-  valleytronic exciton features onto the coupled system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfkgSy/content/2301.03275v1.pdf'}
+page_content=' Con-  sistently, the CD spectra exhibit a sign reversal at magnetic  fields of ±9 T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfkgSy/content/2301.03275v1.pdf'}
+page_content='  Our observation of magneto-optical effects in a coherently  coupled exciton-plasmon system and their detailed quantita-  tive understanding provide a pathway to design ultrathin meta-  surfaces for chiral spectral filtering, which could be also ex-  ploited to control nonreciprocal phenomena with magnetic  field for uni-directional flow of circularly polarized photons  as required for information transfer in quantum networks  [38, 39].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfkgSy/content/2301.03275v1.pdf'}
+page_content=' An obvious way to create a permanent chiral exciton-  plasmon metasurface is to utilize layered ferromagnets that in-  duce sizable exciton valley Zeeman splittings of several meV,  equivalent to external magnetic fields well above 10 T [40–  44].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfkgSy/content/2301.03275v1.pdf'}
+page_content=' Furthermore, spectrally narrow response from TMD ex-  citons in the limit of atomically thin mirror [45–48] would  yield spectrally sharp Fano reflectance or windows of trans-  parency in a broad extinction response.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfkgSy/content/2301.03275v1.pdf'}
+page_content=' As such, spectral re-  gions with perfect destructive quantum interference and nega-  tive refractive index equivalent to electromagnetically induced  transparency could be realized for chiral slow light and infor-  mation storage [49–51].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfkgSy/content/2301.03275v1.pdf'}
+page_content='  Methods  The sample was fabricated by depositing monolayer WSe2  (HQ Graphene) embedded in high-quality hBN (NIMS) onto  a gold nanodisk array.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfkgSy/content/2301.03275v1.pdf'}
+page_content=' The array was fabricated by standard  electron-beam lithography and gold evaporation on a Si/SiO2  substrate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfkgSy/content/2301.03275v1.pdf'}
+page_content=' All measurements were carried out at cryogenic  temperatures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfkgSy/content/2301.03275v1.pdf'}
+page_content=' The data in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfkgSy/content/2301.03275v1.pdf'}
+page_content=' 1 and Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfkgSy/content/2301.03275v1.pdf'}
+page_content=' 2 were recorded in  a helium bath cryostat at 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfkgSy/content/2301.03275v1.pdf'}
+page_content='2 K, whereas the magnetic field  measurements of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfkgSy/content/2301.03275v1.pdf'}
+page_content=' 3 were performed in a closed-cycle  magneto-cryostat (attocube systems, attoDRY1000) at 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfkgSy/content/2301.03275v1.pdf'}
+page_content='5 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfkgSy/content/2301.03275v1.pdf'}
+page_content='  White-light reflection spectroscopy was performed using a  halogen lamp (Ocean Optics, HL-2000) or a supercontinuum  laser (NKT, SuperK Extreme EXR-4) focused to a spot of ∼  1 µm diameter in a home-built confocal microscope equipped  with cryogenic nano-positioners (attocube systems, ANP100  and ANP101 series) and micro-objectives (attocube systems,  LT-APO/VISIR/0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfkgSy/content/2301.03275v1.pdf'}
+page_content='82 or LT-APO/LWD/NIR/0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfkgSy/content/2301.03275v1.pdf'}
+page_content='63).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfkgSy/content/2301.03275v1.pdf'}
+page_content='  The re-  flected light was spectrally dispersed by a monochromator  (Roper Scientific, Acton SP2500 or Acton 300i) and detected  by a CCD (Roper Scientific, Spec 10:100BR/LN or Andor,  iDus 416).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfkgSy/content/2301.03275v1.pdf'}
+page_content='  Acknowledgements  The authors thank A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfkgSy/content/2301.03275v1.pdf'}
+page_content=' O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfkgSy/content/2301.03275v1.pdf'}
+page_content=' Govorov for fruitful discussions,  and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfkgSy/content/2301.03275v1.pdf'}
+page_content=' Altpeter and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfkgSy/content/2301.03275v1.pdf'}
+page_content=' Obermayer for continuous support  in the clean room.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfkgSy/content/2301.03275v1.pdf'}
+page_content='  This research was funded by the Eu-  ropean Research Council (ERC) under the Grant Agree-  ment No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfkgSy/content/2301.03275v1.pdf'}
+page_content=' 772195 as well as the Deutsche Forschungsgemein-  schaft (DFG, German Research Foundation) within the Prior-  ity Programme SPP 2244 2DMP (Project No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfkgSy/content/2301.03275v1.pdf'}
+page_content=' 443405595)  and the Germany’s Excellence Strategy Munich Center for  Quantum Science and Technology (MCQST) EXC-2111-  390814868.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfkgSy/content/2301.03275v1.pdf'}
+page_content='  S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfkgSy/content/2301.03275v1.pdf'}
+page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfkgSy/content/2301.03275v1.pdf'}
+page_content=' acknowledges funding from the Euro-  pean Union’s Horizon 2020 research and innovation pro-  gramme under the Marie Skłodowska-Curie Grant Agree-  ment Spin-NANO No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfkgSy/content/2301.03275v1.pdf'}
+page_content=' 676108.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfkgSy/content/2301.03275v1.pdf'}
+page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfkgSy/content/2301.03275v1.pdf'}
+page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfkgSy/content/2301.03275v1.pdf'}
+page_content=' and T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfkgSy/content/2301.03275v1.pdf'}
+page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfkgSy/content/2301.03275v1.pdf'}
+page_content=' acknowledge  support from JSPS KAKENHI (Grant Numbers 19H05790,  20H00354 and 21H05233).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfkgSy/content/2301.03275v1.pdf'}
+page_content='  Corresponding authors  S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfkgSy/content/2301.03275v1.pdf'}
+page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfkgSy/content/2301.03275v1.pdf'}
+page_content='  (samarth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfkgSy/content/2301.03275v1.pdf'}
+page_content='vadia@physik.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfkgSy/content/2301.03275v1.pdf'}
+page_content='lmu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfkgSy/content/2301.03275v1.pdf'}
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+page_content='scherzer@physik.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfkgSy/content/2301.03275v1.pdf'}
+page_content='lmu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfkgSy/content/2301.03275v1.pdf'}
+page_content='de),  and  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfkgSy/content/2301.03275v1.pdf'}
+page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfkgSy/content/2301.03275v1.pdf'}
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+page_content=' Nat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfkgSy/content/2301.03275v1.pdf'}
+page_content=' Commun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfkgSy/content/2301.03275v1.pdf'}
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+page_content=', Shan, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfkgSy/content/2301.03275v1.pdf'}
+page_content=' & Heinz, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfkgSy/content/2301.03275v1.pdf'}
+page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfkgSy/content/2301.03275v1.pdf'}
+page_content=' Control of val-  ley polarization in monolayer MoS2 by optical helicity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfkgSy/content/2301.03275v1.pdf'}
+page_content=' Nat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfkgSy/content/2301.03275v1.pdf'}
+page_content='  Nanotechnol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfkgSy/content/2301.03275v1.pdf'}
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+page_content=', Yao, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfkgSy/content/2301.03275v1.pdf'}
+page_content=', Xiao, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfkgSy/content/2301.03275v1.pdf'}
+page_content=' & Cui, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfkgSy/content/2301.03275v1.pdf'}
+page_content=' Valley polariza-  tion in MoS2 monolayers by optical pumping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfkgSy/content/2301.03275v1.pdf'}
+page_content=' Nat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfkgSy/content/2301.03275v1.pdf'}
+page_content=' Nanotech-  nol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfkgSy/content/2301.03275v1.pdf'}
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+page_content=' Robust optical emission polarization in MoS2  monolayers through selective valley excitation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfkgSy/content/2301.03275v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfkgSy/content/2301.03275v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfkgSy/content/2301.03275v1.pdf'}
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+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfkgSy/content/2301.03275v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfkgSy/content/2301.03275v1.pdf'}
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+page_content=' Nat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfkgSy/content/2301.03275v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfkgSy/content/2301.03275v1.pdf'}
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+page_content=' Nat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfkgSy/content/2301.03275v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfkgSy/content/2301.03275v1.pdf'}
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+page_content=' 2D Mater.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfkgSy/content/2301.03275v1.pdf'}
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+page_content=' 2D Mater.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfkgSy/content/2301.03275v1.pdf'}
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+page_content=' Nanoscale chiral valley-photon interface through  optical spin-orbit coupling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfkgSy/content/2301.03275v1.pdf'}
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+page_content=' Nat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfkgSy/content/2301.03275v1.pdf'}
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+page_content='  Realization of an electrically tunable narrow-bandwidth atomi-  cally thin mirror using monolayer mose2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfkgSy/content/2301.03275v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfkgSy/content/2301.03275v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfkgSy/content/2301.03275v1.pdf'}
+page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfkgSy/content/2301.03275v1.pdf'}
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+page_content='  [49] Boller, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfkgSy/content/2301.03275v1.pdf'}
+page_content='-J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfkgSy/content/2301.03275v1.pdf'}
+page_content=', Imamoglu, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfkgSy/content/2301.03275v1.pdf'}
+page_content=' & Harris, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfkgSy/content/2301.03275v1.pdf'}
+page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfkgSy/content/2301.03275v1.pdf'}
+page_content='  Observation of  electromagnetically induced transparency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfkgSy/content/2301.03275v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfkgSy/content/2301.03275v1.pdf'}
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+page_content='  [50] Hau, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfkgSy/content/2301.03275v1.pdf'}
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+page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfkgSy/content/2301.03275v1.pdf'}
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+page_content=' Behroozi, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfkgSy/content/2301.03275v1.pdf'}
+page_content=' Light  speed reduction to 17 meters per second in ultracold atomic  gases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfkgSy/content/2301.03275v1.pdf'}
+page_content=' Nature 397, 594 (1999).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfkgSy/content/2301.03275v1.pdf'}
+page_content='  [51] Lukin, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfkgSy/content/2301.03275v1.pdf'}
+page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfkgSy/content/2301.03275v1.pdf'}
+page_content=' & Imamoglu, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfkgSy/content/2301.03275v1.pdf'}
+page_content=' Controlling photons using elec-  tromagnetically induced transparency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfkgSy/content/2301.03275v1.pdf'}
+page_content=' Nature 413, 273 (2001).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfkgSy/content/2301.03275v1.pdf'}
+page_content='  Magneto-optical chirality in a coherently coupled exciton-plasmon system  Supporting Information  Samarth Vadia,1, 2, 3, ∗ Johannes Scherzer,1, ∗ Kenji  Watanabe,4 Takashi Taniguchi,5 and Alexander H¨ogele1, 2  1Fakult¨at f¨ur Physik, Munich Quantum Center, and Center  for NanoScience (CeNS), Ludwig-Maximilians-Universit¨at  M¨unchen, Geschwister-Scholl-Platz 1, 80539 M¨unchen, Germany  2Munich Center for Quantum Science and Technology (MCQST),  Schellingtr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfkgSy/content/2301.03275v1.pdf'}
+page_content=' 4, 80799 M¨unchen, Germany  3attocube Systems AG, Eglfinger Weg 2, 85540 Haar, Germany  4Research Center for Functional Materials,  National Institute for Materials Science,  1-1 Namiki, Tsukuba 305-0044, Japan  5International Center for Materials Nanoarchitectonics,  National Institute for Materials Science,  1-1 Namiki, Tsukuba 305-0044, Japan  (Dated: January 10, 2023)  ∗ These authors contributed equally to this work  1  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfkgSy/content/2301.03275v1.pdf'}
+page_content='03275v1  [physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfkgSy/content/2301.03275v1.pdf'}
+page_content='optics]  9 Jan 2023  SUPPORTING NOTE 1: GOLD NANODISK ARRAYS  The metasurface discussed in this work consists of a monolayer WSe2 encapsulated in hexag-  onal boron nitride (hBN) and placed on a gold (Au) nanodisk array.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfkgSy/content/2301.03275v1.pdf'}
+page_content=' The individual metallic  nanoparticles support plasmon resonaces localized to the size of the particle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfkgSy/content/2301.03275v1.pdf'}
+page_content=' They are charac-  terized by electric field enhancement in their direct proximity under optical excitation, which we  use to couple exciton and plasmon resonances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfkgSy/content/2301.03275v1.pdf'}
+page_content=' The spectral position of the plasmon resonance  depends on the shape and size of the particles, as well as on their dielectric environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfkgSy/content/2301.03275v1.pdf'}
+page_content='  The design of plasmonic nanodisk arrays respects the following basic considerations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfkgSy/content/2301.03275v1.pdf'}
+page_content=' First, the  plasmon resonance of the Au nanodisks has to match the energy of the exciton transition in WSe2  to enable near-resonance coherent coupling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfkgSy/content/2301.03275v1.pdf'}
+page_content=' Second, the TMD monolayer has to be placed in close  proximity to the Au nanodisks for the near-field enhancement of the exciton-plasmon interaction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfkgSy/content/2301.03275v1.pdf'}
+page_content='  Third, the height of the nanodisks should be as small as possible to avoid exciton localization in  the form of quantum dots due to the variation in lateral strain in the TMD monolayer [1, 2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfkgSy/content/2301.03275v1.pdf'}
+page_content=' And  fourth, finally, the number of nanodisks within the focal spot under optical excitation should be  large to increase the effective coupling strength.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfkgSy/content/2301.03275v1.pdf'}
+page_content='  Following these considerations, we performed finite-difference time-domain (FDTD) simula-  tions with a commercial software (Lumerical) to find the optimal geometrical parameters for our  system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfkgSy/content/2301.03275v1.pdf'}
+page_content=' A SiO2 coated silver mirror was used as substrate for the nanostructure both in simu-  lations and in actual samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfkgSy/content/2301.03275v1.pdf'}
+page_content=' A thin layer with refractive index n = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfkgSy/content/2301.03275v1.pdf'}
+page_content='76 simulates the hBN  interface at the top surface of the nanodisks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfkgSy/content/2301.03275v1.pdf'}
+page_content=' The main geometrical parameters for tuning the plas-  mon resonance of the disks are their diameter and height.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfkgSy/content/2301.03275v1.pdf'}
+page_content=' With the consideration to minimize the  nanodisk height in mind, we fixed the height at ∼ 15 nm where evaporated gold is expected to  form smooth surfaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfkgSy/content/2301.03275v1.pdf'}
+page_content=' The plasmon resonance can be shifted to smaller energies by increasing the  disk diameter, while a blue-shift accompanies diameter reduction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfkgSy/content/2301.03275v1.pdf'}
+page_content=' Figure 1a shows this behavior  when simulating the extinction of an optical plane wave propagating through gold nanodisks with  different diameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfkgSy/content/2301.03275v1.pdf'}
+page_content=' To enhance the near-field effect, we used a 6 nm thin bottom hBN layer in  our van der Waals heterostack defining the distance between the TMD monolayer and the Au nan-  odisk array.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfkgSy/content/2301.03275v1.pdf'}
+page_content=' To estimate the magnitude of near-field enhancement at the monolayer position, we  evaluated the intensity of the electric field in FDTD simulation in the monolayer plane as plotted  in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfkgSy/content/2301.03275v1.pdf'}
+page_content=' 2a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfkgSy/content/2301.03275v1.pdf'}
+page_content=' The mean intensity value across the entire area of a nanodisk is shown by the dashed  line in the intensity line profile in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfkgSy/content/2301.03275v1.pdf'}
+page_content=' 2b and serves as an input parameter for the model of the  2  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfkgSy/content/2301.03275v1.pdf'}
+page_content='6  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfkgSy/content/2301.03275v1.pdf'}
+page_content='8  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfkgSy/content/2301.03275v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfkgSy/content/2301.03275v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfkgSy/content/2301.03275v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfkgSy/content/2301.03275v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfkgSy/content/2301.03275v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfkgSy/content/2301.03275v1.pdf'}
+page_content='8  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfkgSy/content/2301.03275v1.pdf'}
+page_content='0  Extinction (norm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfkgSy/content/2301.03275v1.pdf'}
+page_content=')  Energy (eV)  200 nm  a = 300 nm  c  a  b  D = 118 nm  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfkgSy/content/2301.03275v1.pdf'}
+page_content='6  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfkgSy/content/2301.03275v1.pdf'}
+page_content='8  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfkgSy/content/2301.03275v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfkgSy/content/2301.03275v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfkgSy/content/2301.03275v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfkgSy/content/2301.03275v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfkgSy/content/2301.03275v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfkgSy/content/2301.03275v1.pdf'}
+page_content='8  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfkgSy/content/2301.03275v1.pdf'}
+page_content='0  Differential Reflection, DR  Energy (eV)  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfkgSy/content/2301.03275v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfkgSy/content/2301.03275v1.pdf'}
+page_content=' a, Extinction spectra of two different gold nanodisk arrays on a silver/SiO2 substrate obtained from  FDTD simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfkgSy/content/2301.03275v1.pdf'}
+page_content=' The reflected (R) and transmitted (T) intensities were evaluated at the both ends of the  computational cell, and normalized to the signal of the plane wave source.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfkgSy/content/2301.03275v1.pdf'}
+page_content=' The extinction is defined as  1 − R − T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfkgSy/content/2301.03275v1.pdf'}
+page_content=' Both arrays have a lattice constant a of 300 nm and height h of 15 nm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfkgSy/content/2301.03275v1.pdf'}
+page_content=' Red and blue spectra  correspond to diameters D of 114 and 106 nm, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfkgSy/content/2301.03275v1.pdf'}
+page_content=' b, Differential reflection spectra of two gold  nanodisk arrays fabricated via e-beam lithography on a silver/SiO2 substrate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfkgSy/content/2301.03275v1.pdf'}
+page_content=' The geometrical parameters  correspond to the ones used for the simulation in a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfkgSy/content/2301.03275v1.pdf'}
+page_content=' c, Scanning electron micrograph of an array section of  the sample investigated in the main text with mean nanodisk diameter D of 118 nm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfkgSy/content/2301.03275v1.pdf'}
+page_content='  coupled exciton-plasmon system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfkgSy/content/2301.03275v1.pdf'}
+page_content=' The plasmonic response in differential reflection varies within a  range of 100 meV across the metasurface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfkgSy/content/2301.03275v1.pdf'}
+page_content=' The shift of the plasmon resonance is significant only  in regions where the array was covered by hBN, which indicates that the variation of the dielectric  environment, related to inhomogeneities in the distance between the hBN layer and the plasmonic  structure, dominates the variation of the plasmon energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfkgSy/content/2301.03275v1.pdf'}
+page_content='  Using the simulation results as input parameters, we fabricated gold nanodisk arrays using elec-  3  100  50  0  50  100  0  20  40  60  80  100  120  Field Intensity (a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfkgSy/content/2301.03275v1.pdf'}
+page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfkgSy/content/2301.03275v1.pdf'}
+page_content=')  x (nm)  50  0  50  50  25  0  25  50  y (nm)  x (nm)  25  50  75  100  Field  Intensity  (a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfkgSy/content/2301.03275v1.pdf'}
+page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfkgSy/content/2301.03275v1.pdf'}
+page_content=')  a  b  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfkgSy/content/2301.03275v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfkgSy/content/2301.03275v1.pdf'}
+page_content=' a, Simulated field intensity at the cross-section area of the FDTD computational cell 6 nm above an  Au nanodisk (at the position of the monolayer) at the plasmon resonance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfkgSy/content/2301.03275v1.pdf'}
+page_content=' The intensity is normalized to the  intensity of the excitation plane wave source, thus effectively determining the intensity enhancement factor  for the near-field of the plasmonic nanodisk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfkgSy/content/2301.03275v1.pdf'}
+page_content=' b, One-dimensional line cut (dashed line in a) of the intensity  cross-section area shown in a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfkgSy/content/2301.03275v1.pdf'}
+page_content=' The dashed line at field intensity of 50 represents the approximation of the  intensity within the overlap area Ac of the monolayer and the plasmonic near-field enhancement shown in  a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfkgSy/content/2301.03275v1.pdf'}
+page_content='  tron beam lithography.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfkgSy/content/2301.03275v1.pdf'}
+page_content=' Figure 1c shows a scanning electron micrograph of the Au nanodisk array  used in the experiments of the main text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfkgSy/content/2301.03275v1.pdf'}
+page_content=' The optimized interparticle distance for a maximum den-  sity of nanodisks within the periodic array was a = 300 nm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfkgSy/content/2301.03275v1.pdf'}
+page_content=' We found that smaller array spacings  compromised the lift-off process for nanodisk diameters larger than 100 nm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfkgSy/content/2301.03275v1.pdf'}
+page_content=' The resolution of our  electron beam lithography process was dependent on the e-beam voltage with a limit of ∼ 10 nm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfkgSy/content/2301.03275v1.pdf'}
+page_content='  This limited resolution determined the range of variations in the disk diameters in an array of  otherwise identical fabrication parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfkgSy/content/2301.03275v1.pdf'}
+page_content=' The variation of individual nanodisk diameters in turn  resulted in inhomogeneous broadening of the plasmon resonances measured in differential reflec-  tion (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfkgSy/content/2301.03275v1.pdf'}
+page_content='1b) as compared to the results of corresponding FDTD simulations shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfkgSy/content/2301.03275v1.pdf'}
+page_content=' 1a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfkgSy/content/2301.03275v1.pdf'}
+page_content='  The experimental full-width at half-maximum (FWHM) linewidth of the plasmon resonances was  FWHM ≃ 180 meV, and was used as the plasmon decoherence rate in the model detailed in the  following section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfkgSy/content/2301.03275v1.pdf'}
+page_content='  4  SUPPORTING NOTE 2: COHERENT INTERFERENCE IN THE COUPLED-OSCILLATOR  MODEL  To model the coupled system of excitons in WSe2 monolayer and localized surface plasmons in  Au nanodisk arrays interacting with a light field of frequency ω, we consider a three-level system  comprising |g⟩, |e⟩ and |p⟩ as the ground, exciton and plasmon state, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfkgSy/content/2301.03275v1.pdf'}
+page_content=' The energy  separation between the states |g⟩ and |e⟩ and the states |g⟩ and |p⟩ is ℏωeg and ℏωpg, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfkgSy/content/2301.03275v1.pdf'}
+page_content='  The Rabi frequencies for the respective transitions are Ωeg and Ωpg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfkgSy/content/2301.03275v1.pdf'}
+page_content=' In addition, the coherent  coupling between the states |e⟩ and |p⟩ is denoted by Ωc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfkgSy/content/2301.03275v1.pdf'}
+page_content=' The Hamiltonian of the system is written  as:  H = ℏ  �  �  �  �  �  0  1  2Ωegeiωt  1  2Ωpgeiωt  1  2Ωege−iωt  ωeg  1  2Ωc  1  2Ωpge−iωt  1  2Ωc  ωpg  �  �  �  �  � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfkgSy/content/2301.03275v1.pdf'}
+page_content='  (1)  To describe the coherent dynamics of the Hamiltonian including the decoherence and dissipa-  tion channels, we use the master equation for the density matrix, given by [3]:  ˙ρ = − i  ℏ [H, ρ] + Lρ,  (2)  where the density matrix ρ with matrix elements ρij describes the statistical state of the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfkgSy/content/2301.03275v1.pdf'}
+page_content='  The diagonal matrix elements ρij (with i = j) are the populations of the states, with i = g, e or  p, and the off-diagonal matrix elements ρij (with i ̸= j) are the coherences between the states i  and j with complex conjugate ¯ρij = ρji.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfkgSy/content/2301.03275v1.pdf'}
+page_content=' The second term in the equation, Lρ, accounts for the  dissipation in the Lindblad form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfkgSy/content/2301.03275v1.pdf'}
+page_content='  First, we apply the rotating-wave approximation (RWA) to remove the time-dependent terms  in the Hamiltonian of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfkgSy/content/2301.03275v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfkgSy/content/2301.03275v1.pdf'}
+page_content=' Using ρii = �ρii, ρeg = �ρege−iωt, ρpg = �ρpge−iωt, and ρep = �ρep, the  Hamiltonian in RWA modifies to:  �H = ℏ  �  �  �  �  �  0  1  2Ωeg  1  2Ωpg  1  2Ωeg  ∆eg  1  2Ωc  1  2Ωpg  1  2Ωc  ∆pg  �  �  �  �  � ,  (3)  where ∆eg = ωeg − ω and ∆pg = ωpg − ω are the respective detunings between the transition  frequencies ωeg and ωpg and the laser frequency ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfkgSy/content/2301.03275v1.pdf'}
+page_content=' Omitting the tilde superscript in the following  for simplicity,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfkgSy/content/2301.03275v1.pdf'}
+page_content=' we describe the dynamic evolution of the density matrix elements by the master  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfkgSy/content/2301.03275v1.pdf'}
+page_content='5  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfkgSy/content/2301.03275v1.pdf'}
+page_content='equation in the Lindblad form as:  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfkgSy/content/2301.03275v1.pdf'}
+page_content='˙ρgg = −i  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfkgSy/content/2301.03275v1.pdf'}
+page_content='�Ωeg  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfkgSy/content/2301.03275v1.pdf'}
+page_content='2 (ρge − ρeg) + Ωpg  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfkgSy/content/2301.03275v1.pdf'}
+page_content='2 (ρpg − ρgp)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfkgSy/content/2301.03275v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfkgSy/content/2301.03275v1.pdf'}
+page_content='+ Γeeρee + Γppρpp  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfkgSy/content/2301.03275v1.pdf'}
+page_content='˙ρee = −i  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfkgSy/content/2301.03275v1.pdf'}
+page_content='�Ωeg  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfkgSy/content/2301.03275v1.pdf'}
+page_content='2 (ρeg − ρge) + Ωc  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfkgSy/content/2301.03275v1.pdf'}
+page_content='2 (ρpg − ρgp)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfkgSy/content/2301.03275v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfkgSy/content/2301.03275v1.pdf'}
+page_content='− Γeeρee  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfkgSy/content/2301.03275v1.pdf'}
+page_content='˙ρpp = −i  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfkgSy/content/2301.03275v1.pdf'}
+page_content='�Ωpg  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfkgSy/content/2301.03275v1.pdf'}
+page_content='2 (ρgp − ρpg) + Ωc  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfkgSy/content/2301.03275v1.pdf'}
+page_content='2 (ρep − ρpe)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfkgSy/content/2301.03275v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfkgSy/content/2301.03275v1.pdf'}
+page_content='− Γppρpp  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfkgSy/content/2301.03275v1.pdf'}
+page_content='˙ρge = −i  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfkgSy/content/2301.03275v1.pdf'}
+page_content='�Ωeg  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfkgSy/content/2301.03275v1.pdf'}
+page_content='2 (ρee − ρgg) − ∆egρge + Ωpg  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfkgSy/content/2301.03275v1.pdf'}
+page_content='2 ρpe − Ωc  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfkgSy/content/2301.03275v1.pdf'}
+page_content='2 ρgp  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfkgSy/content/2301.03275v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfkgSy/content/2301.03275v1.pdf'}
+page_content='− Γegρge  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfkgSy/content/2301.03275v1.pdf'}
+page_content='2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfkgSy/content/2301.03275v1.pdf'}
+page_content='˙ρgp = −i  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfkgSy/content/2301.03275v1.pdf'}
+page_content='�Ωpg  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfkgSy/content/2301.03275v1.pdf'}
+page_content='2 (ρpp − ρgg) + Ωeg  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfkgSy/content/2301.03275v1.pdf'}
+page_content='2 ρep − Ωc  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfkgSy/content/2301.03275v1.pdf'}
+page_content='2 ρge − ∆pgρgp  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfkgSy/content/2301.03275v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfkgSy/content/2301.03275v1.pdf'}
+page_content='− Γpgρgp  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfkgSy/content/2301.03275v1.pdf'}
+page_content='2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfkgSy/content/2301.03275v1.pdf'}
+page_content='˙ρep = −i  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfkgSy/content/2301.03275v1.pdf'}
+page_content='�Ωeg  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfkgSy/content/2301.03275v1.pdf'}
+page_content='2 ρgp − Ωpg  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfkgSy/content/2301.03275v1.pdf'}
+page_content='2 ρeg + ∆egρep + Ωc  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfkgSy/content/2301.03275v1.pdf'}
+page_content='2 (ρpp − ρee) − ∆pgρep  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfkgSy/content/2301.03275v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfkgSy/content/2301.03275v1.pdf'}
+page_content='.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfkgSy/content/2301.03275v1.pdf'}
+page_content='  (4)  Dissipation is introduced in the above master equation via population decay rates, Γee and Γpp,  and the decoherence rates, Γeg and Γpg [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfkgSy/content/2301.03275v1.pdf'}
+page_content=' The population decay rate includes contributions from  radiative and non-radiative decay channels, Γii = Γr  ii + Γnr  ii , respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfkgSy/content/2301.03275v1.pdf'}
+page_content=' The decoherence rate is  related to the population decay rate as Γig = Γii + 2γig,deph where γig,deph is the pure dephasing  rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfkgSy/content/2301.03275v1.pdf'}
+page_content='  The Rabi frequency Ωig between the excited state |i⟩ (|i⟩ = |e⟩ or |p⟩) and the ground state |g⟩  is given by:  Ωig = µigE  ℏ  =  �  6πc2ΓiiP  ℏnω3  igA  (5)  where µig is the dipole moment of the transition, E is the electric field, c is the speed of light, Γii  is the population decay rate of the state |i⟩, P is the illumination laser power, n is the refractive  index, ωig is the transition frequency from the excited state |i⟩ to the ground state |g⟩, and A is  the illumination area.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfkgSy/content/2301.03275v1.pdf'}
+page_content=' For the transition between the exciton state |e⟩ and the ground state |g⟩,  there are two distinct Rabi frequencies, namely the coupled Rabi frequency Ωig,c frequency for  excitons in the near-field of plasmonic nanodisks affected by the plasmonic enhancement of the  electric field, and the uncoupled Rabi frequency Ωig,u of excitons away from Au nanodisk but still  excited within the optical focal spot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfkgSy/content/2301.03275v1.pdf'}
+page_content=' The Rabi frequency of coupled exciton fraction is related to  the Rabi frequency of the uncoupled fraction as Ωeg,c = Ωeg,u  √  FAc/Au, where Ac is the overlap  area between the monolayer and the nanodisk with plasmonic near-field enhancement, Au is the  area of uncoupled excitons within the optical focal spot, and F is the enhancement factor of the  electric field obtained from the FDTD calculation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfkgSy/content/2301.03275v1.pdf'}
+page_content='  6  Solving the set of coupled equations for steady state where ˙ρij = 0, and using population con-  servation in the three-level system as ρgg + ρee + ρpp = 1, we obtain analytical expressions for the  populations and coherences in the coupled system as a function of experimental parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfkgSy/content/2301.03275v1.pdf'}
+page_content=' The  optical response is obtained from the imaginary part of the coherence matrix elements [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfkgSy/content/2301.03275v1.pdf'}
+page_content=' In our  system, the focal spot consists of an exciton-plasmon hybrid region showing coherent interference  and also a region with uncoupled excitons in monolayer away from the nanodisk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfkgSy/content/2301.03275v1.pdf'}
+page_content=' Therefore, we  obtain the extinction E in the optical response as the sum of the coherence matrix elements of two  coupled states and the uncoupled exciton state as follows:  E ∝ αe  Ωeg,cΓr  ee  Ω2  eg,c + Ω2  pg  Im(ρge) + αp  ΩpgΓr  pp  Ω2  eg,c + Ω2  pg  Im(ρgp) + αe  Γr  ee  Ωeg,uc  Im(ρge,u),  (6)  where αe and αp correspond to the maximum on-resonance scattering contrast of the exciton and  plasmon transition with radiative decay rates Γr  ee and Γr  pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfkgSy/content/2301.03275v1.pdf'}
+page_content=' The respective contrasts αi are obtained  as [6]:  αi = 3  2π  c2  n2ω2  igA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfkgSy/content/2301.03275v1.pdf'}
+page_content='  (7)  We note here that the extinction is not quantitative as all reflecting surfaces of the sample are not  taken into account to calculate the Rayleigh scattering field based on the procedure detailed in [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfkgSy/content/2301.03275v1.pdf'}
+page_content='  However, the response spectrum is proportional to the imaginary part of the coupled coherences  ρeg and ρpg, and the uncoupled exciton coherence ρeg,u as a function of frequency, and we obtain  correct qualitative description of differential reflection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfkgSy/content/2301.03275v1.pdf'}
+page_content='  The parameters to obtain the extinction are known from the experiment or literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfkgSy/content/2301.03275v1.pdf'}
+page_content=' For the  exciton state |e⟩, the population decay due to radiative recombination is ℏΓee = ℏΓr  ee ≃ 1 meV  [7, 8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfkgSy/content/2301.03275v1.pdf'}
+page_content=' The experimental FWHM linewidth is determined for the monolayer TMD region (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfkgSy/content/2301.03275v1.pdf'}
+page_content='  1c) as ℏΓeg ≃ 8 meV, and we treat the broadening beyond the transform-limit of 1 meV as pure  dephasing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfkgSy/content/2301.03275v1.pdf'}
+page_content=' For the plasmon state |p⟩, the experimental FWHM linewidth is ℏΓpg ≃ 180 meV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfkgSy/content/2301.03275v1.pdf'}
+page_content='  We neglect pure dephasing processes such that Γpg = Γpp [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfkgSy/content/2301.03275v1.pdf'}
+page_content=' The radiative contribution in the  plasmonic state |p⟩ for the gold nanodisk is estimated to be ≃ 46 %, such that ℏΓr  pp ≃ 82 meV [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfkgSy/content/2301.03275v1.pdf'}
+page_content='  The Rabi frequencies Ωeg,u and Ωpg are calculated using an illumination laser power P = 1 nW,  illumination area A ≃ 1 µm2, and the corresponding linewidth and resonance frequency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfkgSy/content/2301.03275v1.pdf'}
+page_content=' The Rabi  frequency of excitons coupled to the plasmons Ωeg,c is calculated using the enhancement factor,  F = 50, and the coupled and uncoupled areas, Ac and Au, determined from the FDTD simulation  shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfkgSy/content/2301.03275v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfkgSy/content/2301.03275v1.pdf'}
+page_content=' The coupling constant is estimated from the experiment to be ℏΩc ≃ 28 meV  (see main text, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfkgSy/content/2301.03275v1.pdf'}
+page_content=' 2e).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfkgSy/content/2301.03275v1.pdf'}
+page_content=' A comparison between experimental differential reflectance spectra and  7  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfkgSy/content/2301.03275v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfkgSy/content/2301.03275v1.pdf'}
+page_content='5  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfkgSy/content/2301.03275v1.pdf'}
+page_content='0  d = -38 meV  d = -15 meV  d = -10 meV  d = 13 meV  d = 32 meV  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfkgSy/content/2301.03275v1.pdf'}
+page_content='6  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfkgSy/content/2301.03275v1.pdf'}
+page_content='7  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfkgSy/content/2301.03275v1.pdf'}
+page_content='8  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfkgSy/content/2301.03275v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfkgSy/content/2301.03275v1.pdf'}
+page_content='5  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfkgSy/content/2301.03275v1.pdf'}
+page_content='0  Energy (eV)  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfkgSy/content/2301.03275v1.pdf'}
+page_content='6  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfkgSy/content/2301.03275v1.pdf'}
+page_content='7  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfkgSy/content/2301.03275v1.pdf'}
+page_content='8  Energy (eV)  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfkgSy/content/2301.03275v1.pdf'}
+page_content='6  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfkgSy/content/2301.03275v1.pdf'}
+page_content='7  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfkgSy/content/2301.03275v1.pdf'}
+page_content='8  Energy (eV)  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfkgSy/content/2301.03275v1.pdf'}
+page_content='6  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfkgSy/content/2301.03275v1.pdf'}
+page_content='7  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfkgSy/content/2301.03275v1.pdf'}
+page_content='8  Energy (eV)  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfkgSy/content/2301.03275v1.pdf'}
+page_content='6  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfkgSy/content/2301.03275v1.pdf'}
+page_content='7  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfkgSy/content/2301.03275v1.pdf'}
+page_content='8  Energy (eV)  Differential reflection, DR  Normalized extinction  a  b  c  d  e  f  g  h  i  j  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfkgSy/content/2301.03275v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfkgSy/content/2301.03275v1.pdf'}
+page_content=' a – e, Differential reflectance spectra of the coupled system for various detunings δ = ℏωpg −  ℏωeg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfkgSy/content/2301.03275v1.pdf'}
+page_content=' The spectra were recorded on different spatial positions with different plasmon frequencies ωpg and  a weakly varying exciton frequency ωeg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfkgSy/content/2301.03275v1.pdf'}
+page_content=' f – j, Respective normalized extinction obtained from the Fano  model with exciton-plasmon coupling strength ℏΩc = 28 meV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfkgSy/content/2301.03275v1.pdf'}
+page_content='  extinction response from the model across different detunings is presented in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfkgSy/content/2301.03275v1.pdf'}
+page_content=' 3 with very good  agreement between the experiment and the model across a wide range of energy detunings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfkgSy/content/2301.03275v1.pdf'}
+page_content='  To determine the magneto-induced circular dichroism (CD), we calculate the extinction for  different polarizations of light.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfkgSy/content/2301.03275v1.pdf'}
+page_content=' Light with σ+ (σ−) polarization only couples to the K (K′) valley  of the monolayer exciton transition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfkgSy/content/2301.03275v1.pdf'}
+page_content=' These opposite valleys have different resonance energies ωeg  in the presence of an out-of-plane magnetic field according to the valley Zeeman splitting with the  characteristic exciton g-factor of 4 [10–16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfkgSy/content/2301.03275v1.pdf'}
+page_content=' The optical response for σ+ and σ− polarization is  normalized by the sum of two optical responses in the calculation of CD, leading to the cancellation  of the terms due to various reflecting surfaces in the device.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfkgSy/content/2301.03275v1.pdf'}
+page_content=' Therefore, the calculated model CD  values shown in main text are quantitative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfkgSy/content/2301.03275v1.pdf'}
+page_content='  8  SUPPORTING NOTE 3: MAGNETO INDUCED CHIRALITY OF THE COUPLED SYSTEM  IN DIFFERENTIAL REFLECTION  σ+  σ-  9T  Circular dichroism, CD  Differential reflection, DR  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfkgSy/content/2301.03275v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfkgSy/content/2301.03275v1.pdf'}
+page_content='8  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfkgSy/content/2301.03275v1.pdf'}
+page_content='0  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfkgSy/content/2301.03275v1.pdf'}
+page_content='71  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfkgSy/content/2301.03275v1.pdf'}
+page_content='73  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfkgSy/content/2301.03275v1.pdf'}
+page_content='08  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfkgSy/content/2301.03275v1.pdf'}
+page_content='04  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfkgSy/content/2301.03275v1.pdf'}
+page_content='00  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfkgSy/content/2301.03275v1.pdf'}
+page_content='04  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfkgSy/content/2301.03275v1.pdf'}
+page_content='08  Energy (eV)  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfkgSy/content/2301.03275v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfkgSy/content/2301.03275v1.pdf'}
+page_content=' a, Differential reflectance spectra of the coupled system around the Fano interference induced dip  for different circular polarizations at a magnetic field of 9T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfkgSy/content/2301.03275v1.pdf'}
+page_content=' The transfer of the magneto induced chirality  from the monolayer exciton to the coupled exciton-plasmon system is observed as an energy shift of the  dip position for different polarization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfkgSy/content/2301.03275v1.pdf'}
+page_content=' The data of the two spectra for σ+ and σ− detection lead to the CD  spectrum shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfkgSy/content/2301.03275v1.pdf'}
+page_content='4c in the main text and in b here, for better comparability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfkgSy/content/2301.03275v1.pdf'}
+page_content='  References  [1] Palacios-Berraquero, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfkgSy/content/2301.03275v1.pdf'}
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+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfkgSy/content/2301.03275v1.pdf'}
+page_content=' B 101, 235408 (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfkgSy/content/2301.03275v1.pdf'}
+page_content='  [14] F¨orste, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfkgSy/content/2301.03275v1.pdf'}
+page_content=' et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfkgSy/content/2301.03275v1.pdf'}
+page_content=' Exciton g-factors in monolayer and bilayer WSe2 from experiment and theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfkgSy/content/2301.03275v1.pdf'}
+page_content=' Nat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfkgSy/content/2301.03275v1.pdf'}
+page_content='  Commun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfkgSy/content/2301.03275v1.pdf'}
+page_content=' 11, 4539 (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfkgSy/content/2301.03275v1.pdf'}
+page_content='  [15] Deilmann, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfkgSy/content/2301.03275v1.pdf'}
+page_content=', Kr¨uger, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfkgSy/content/2301.03275v1.pdf'}
+page_content=' & Rohlfing, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfkgSy/content/2301.03275v1.pdf'}
+page_content=' Ab initio studies of exciton g factors: Monolayer transition  metal dichalcogenides in magnetic fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfkgSy/content/2301.03275v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfkgSy/content/2301.03275v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfkgSy/content/2301.03275v1.pdf'}
+page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfkgSy/content/2301.03275v1.pdf'}
+page_content=' 124, 226402 (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfkgSy/content/2301.03275v1.pdf'}
+page_content='  [16] Xuan, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfkgSy/content/2301.03275v1.pdf'}
+page_content=' & Quek, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfkgSy/content/2301.03275v1.pdf'}
+page_content=' Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfkgSy/content/2301.03275v1.pdf'}
+page_content=' Valley Zeeman effect and Landau levels in two-dimensional transition metal  dichalcogenides.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfkgSy/content/2301.03275v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfkgSy/content/2301.03275v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfkgSy/content/2301.03275v1.pdf'}
+page_content=' Research 2, 033256 (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfkgSy/content/2301.03275v1.pdf'}
+page_content='  10' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfkgSy/content/2301.03275v1.pdf'}